University of Wollongong Research Online

University of Wollongong Thesis Collection University of Wollongong Thesis Collections

2011 Tuning the horn: a composer-performer's perspective on using extended just intonation Michael H. Dixon University of Wollongong

Recommended Citation Dixon, Michael H., Tuning the horn: a composer-performer's perspective on using extended just intonation, Doctor of Creative Arts thesis, Faculty of Creative Arts, University of Wollongong, 2011. http://ro.uow.edu.au/theses/3419

Research Online is the open access institutional repository for the University of Wollongong. For further information contact Manager Repository Services: [email protected].

TUNING THE HORN

A COMPOSER-PERFORMER’S PERSPECTIVE

ON USING EXTENDED JUST INTONATION

A thesis submitted in partial fulfilment of

the requirements for the award of the degree

DOCTOR OF CREATIVE ARTS

from

UNIVERSITY OF WOLLONGONG

by

MICHAEL H DIXON, MMus, MPhil

FACULTY OF CREATIVE ARTS, SCHOOL OF MUSIC,

2011 Certification

I, Michael Hugh Dixon, declare that this thesis, submitted in partial fulfillment of the requirements for the award of Doctor of Creative Arts, in the Faculty of Creative Arts, University of Wollongong, is wholly my own work unless otherwise referenced or acknowledged. The document has not been submitted for qualifications at any other academic institution.

1 March 2011

i Table of Contents

List of Tables iv List of Examples v List of Illustrations vii Abstract viii Acknowledgments ix Prologue xi

Chapter 1 1 The extended xenharmonic family of labrosones 1.2 3rd harmonic 13 1.3 5th harmonic 18 1.4 7th harmonic 20 1.5 11th harmonic 29 1.6 13th harmonic 40 1.7 17th harmonic and beyond 46 1.8 Summary 53

Chapter 2 55 Labrosone harmonics: composing in extended just intonation

2.1 7-limit just intonation: natural labrosones 57 2.2 11-limit just intonation: natural labrosones 65 2.3 13-limit just intonation: natural labrosones 82 2.4 13-limit just intonation: finger hole labrosones 86 2.5 Brass instruments in 7, 11 & 13-limit just 93 intonation composition 2.6 Standard brass: observations regarding 99 performance in non-standard tunings

ii Chapter 3 101 A new repertoire: French horn and close relatives

3.1 Extended just intonation compositions: Group A 111 3.2 Extended just intonation compositions: Group B 153 3.3 Extended just intonation compositions: Group C 179

Epilogue 203

Endnotes 211

References 215

Appendix A 165, 9-note scales A1-8

Appendix B HSIS-13 B1-8

Appendix C Notation: HEWM C1-3

Appendix D Method for horn D1-119

Folio of Compositions Folio 1-170

Track List: Tuning Cribs & Performances Folio 171-174

CDs 1, 2 & 3 after the Folio

iii List of Tables

Table 1 Classification of labrosones 6 Table 2 Musical scenario summary - Chapter Two 56 Table 3 Alphorn pitches - Chapter Two 84 Table 4 Pitch relations in 3.1.3 121 Table 5 Valve slide settings for 3.3.2 188 Table 6 Valve slide tunings for 3.3.3 196 Table 7 Bassoon pitches in 3.3.3 200 Table 8 Valve slide tunings for 3.3.3 202

iv List of Examples

Example 1 Slurring across the grain, Martin’s Ballade 23

Example 2 Harmonics in Britten’s Serenade 35 Example 3 Harmonics for I wouldn’t mind dyin’ 36 Example 4 7th harmonics for quartertone effect 37 Example 5 Multi-crook approach by Ligeti 41 Example 6 Harmonics in Mandelbaum’s Wind Quintet No.2 45 Example 7 High written G, 24th harmonic 47

Example 8 Horn harmonics 1-31 in Chemchok Heruka 52 Example 9 Pitch ratio allocation for each rozhok 88 Example 10 Trombone tuning plan for Scenario Six score 97 Example 11 Scale 1 from the set of 165 103 Example 12 Scale 2 from the set of 165 104 Example 13 Scale 3 from the set of 165 104 Example 14 Scale 165 from the set of 165 104 Example 15 Harmonic series & Derivation 1 (HSIS-13) 108 Example 16 Harmonic series & Derivation 1 (HSIS-13) 109 Example 17 Harmonic series & Derivation 1 (HSIS-13) 109 Example 18 Horn tuning in 3.1.1 113 Example 19 Keyboard tuning in 3.1.1 116 Example 20 Tuning parameters in 3.1.2 118 Example 21 Koto string arrangement in 3.1.3 122 Example 22 Horn entry - harmonics in 3.1.3 123 Example 23 Dyads used in 3.1.4 126 Example 24 Pitch groups in 3.1.5 129 Example 25 Vowel map in 3.1.6 133 Example 26 Ratio 45:28 with harmonisations in 3.1.6 133 Example 27 Chord changes in 3.1.6 134

v Example 28 Feature chord, ratios 6:7:9:11, in 3.1.7 139

Example 29 Vowel map in 3.1.7 140

Example 30 Chord progression in 3.1.7 142

Example 31 Tonic chord in 3.1.8 148

Example 32 Horn entry - 13th harmonic in 3.1.8 151

Example 33 Ratio scales in 3.2.1 155

Example 34 Off-unison, pitch beating in 3.2.1 156

Example 35 Symmetrical 9-note scales in 3.2.2 158

Example 36 Descending line of 24 ratios in 3.2.2 159

Example 37 3 sets of harmonics: 4, 5, 6, 7 in 3.2.3 163

Example 38 9-note scale with 3 tonics in 3.2.3 164

Example 39 Interval scales in 3.2.4 167

Example 40 Harmonic series relations in 3.2.4 169

Example 41 Melodic fragments (harmonics) in 3.2.4 171

Example 42 Pitch beatings in 3.2.4 171

Example 43 Multi-crook use in 3.2.5 176

Example 44 Notation, fingering & harmonics in 3.3.1 184

Example 45 Arpeggios difficult to achieve in 3.3.2 191

Example 46 Converging harmonics in 3.3.3 198

Example 47 Microtonal shift in 3.3.3 199

vi List of Illustrations

Illustration 1 Bugle 58

Illustration 2 Russian Horn Capella 66

Illustration 3 Blowing horn 71

Illustration 4 Trutruka 75

Illustration 5 Alphorn & Arkady Shilkloper 82

Illustration 6 Rozhok 88

Illustration 7 Valve slide settings for 3.1.1 114

Illustration 8 Xenophone 125

Illustration 9 Valve slide settings for 3.2.4 169

Illustration 10 Valve slide settings for 3.3.2 188

Illustration 11 Valve slide settings for 3.3.3 201

vii Abstract

There is a large repertoire for the (French) horn yet it is mostly confined to standard tuning systems. Development of new repertoire, as demonstrated in this thesis, may assist in bringing non-standard tunings out of the margins and into the mainstream. A significant compositional resource is provided by the available harmonics of the standard double horn, 13-24 on each of 16 tube lengths, along with traditional techniques of lip, right hand and valve slide adjustments. The performance of compositions in extended just intonation is discovered to be within the realm of expertise of players using standard brass instruments. A deeper understanding of the use of harmonics produced on lip- vibrated instruments or labrosones is gained by classifying them according to just intonation prime-limits. The use of harmonics is therefore linked with the long and rich tradition of just intonation and other microtonal exploration. A method specifically for training horn players in extended just intonation completes this thesis.

viii Acknowledgments

My gratitude goes to my Principal Supervisor Greg Schiemer for his guidance throughout my candidature and in particular for his expert advice regarding the preparation of this thesis. I also acknowledge the suggestions by my

Associate Supervisor Stephen Ingham in the early part of my candidature. I had many fruitful discussions concerning instrumental and vocal technique with a number of musicians. In particular: trombone players Will Farmer, Brett Favell,

Scott Kinmont, Jack Machin, Greg van der Struik; trumpet players Joshua Clark,

John Foster, Andrew Evans, Leanne Sullivan, Julie Waters; tuba player Angus

Charlton; euphonium player Matt van Emmerick; bassoon player Matt

Ockenden; oboe player Kirsten Barry; singer Wendy Dixon. I am also grateful to

Lee Bracegirdle for translating some Spanish text; to Joy Goodsell for proof- reading the main text; to Wendy Dixon for proof-reading the Appendices and to

Terumi Narushima for assistance with referencing.

For some years a microtonal, diamond shaped metallaphone was housed at

UOW. I made use of this xenophone with the kind permission of designer/ builder Christian van der Vyver. The preparation of tuning cribs that allowed for computer generated sound files was an integral part of my work. Warren

Burt provided assistance in tuning the Kurzweil K2000. The software Li’l Miss’

ix Scale Oven was an extremely useful tool and the technical support from it’s creator Jeff Scott was most beneficial to me.

A number of composers allowed me to include their works in concerts I organised: Michael Barkl, Dennis Bathory-Kitsch, Warren Burt, Hugh Dixon,

Kraig Grady, Terumi Narushima, Greg Schiemer. The performances were memorable.

None of my work for this DCA would have any meaning without the performances of my compositions. Many performers graciously gave of their time and energy to prepare and present live performances and recordings. They are listed with the commentary on each composition in the portfolio. Also my thanks to Wirripang Pty Ltd for publishing the compositions I selected.

The support and understanding of my wife, Lisa Wynne-Allen, was paramount especially in the final year of thesis preparation.

x Prologue

Over the past 15 years my work as a composer and instrumental performer has been increasingly driven by my interest in exploring and understanding the world of just intonation, an area so often taken for granted or misunderstood by many musicians. As a (French) horn player I have had first-hand encounters with the harmonic series and in particular with harmonics higher in the series than those on which the simplest just intonation or Pythagorean diatonic tuning systems are built. This gradually led me to realise that another world of intonation is possible beyond the simple just intonation tunings which are traditionally created from low-order harmonics, a world that almost disappeared with the advent of the equal tempered tuning system. It is also a world of intonation that has until now been rediscovered mainly through the building of new instruments as part of the musical exploration of composers like Partch and others who explored just tuning systems based on harmonics higher than the sixth. Yet as a horn player I was aware that my instrument already gives musicians access to the world of extended just intonation.

Accordingly my thesis is focused on the composition of a new repertoire of extended just intonation works that feature the horn. This arose out of a desire to explore the little-known world of extended just intonation and to develop a repertoire that will allow such works to become part of the historical trajectory

xi of a conventional orchestral instrument. The repertoire includes chamber works in which the horn is used in ensemble with other standard brass instruments and voice. The thesis discusses a number of tools and techniques developed to bring the new repertoire to performance. It is supported by information about the tradition of writing and performing for brass in just intonation and an exploration of the potential brass instruments have to extend just intonation in contemporary composition. The appendix also includes what I hope will be useful as an instruction manual for playing the horn in extended just intonation.

Chapter One expounds the tradition of using brass instruments and is a broad commentary on the classification of lip vibrated instruments (labrosones) by available harmonics in a global context. Instruments include those made from seashells, cane and other woods, plastic and various non-brass metals. These have a variety of forms, from natural instruments to those with modifications such as finger holes, keys, slides or valves. The natural instruments cover a wide range from conch-shells which are barely changed from their natural state to the sophisticated design and manufacture of the twice-folded trumpets used in the high art of the European renaissance and baroque. The instruments with modifications include finger hole instruments played in folk music and in renaissance art music; brass band and other similar valved instruments and the three brass most pertinent to this thesis: the horn, trumpet and trombone. All of these instruments produce one or more modes of vibration, called harmonics

xii for convenience, in accordance with the science of acoustics and harmonics. On some instruments it is possible to produce only one harmonic satisfactorily, whilst on standard modern valved brass players can reliably produce between 8 and 16 harmonics.

Chapter Two speculatively explores the musical possibilities of a range of labrosones. A few compositions, selected from the new repertoire in the accompanying portfolio, have been transcribed for natural and finger hole instruments. In some cases reproductions of the pitches in these transcriptions would require natural instruments manufactured to very specific lengths. In other cases finger hole instruments would require very precise placement of the holes. This chapter shows that standard brass instruments could be used to achieve the same pitch results with appropriate planning. Some observations are made regarding the use of standard brass in non-standard tunings.

Chapter Three is an examination of the compositions I created from the viewpoint of the specific tools and training required to fulfil the tuning requirements. Compositions within the DCA portfolio are arranged according to three ways in which I have composed for brass in just intonation. The first is represented by compositions (A) in which the performer has control over how to realise the tuning intentions and therefore the responsibility to choose a suitable technique. Composer Lou Harrison's used this approach in Main

xiii Bersama-sama (2006/1978) for horn and gamelan and James Tenney in Spectrum

2 for Wind Quintet (1995). The second is represented by compositions (B) in which the composer takes some control of the tuning, making suggestions based on a general knowledge of instrumental design and technique yet expecting the performer to do the fine-tuning. This can save some preparation time for the performer. I have used this approach in my works such as Door of

Many Mysteries and Stillness of Unceasing Activity. The third is represented by compositions (C) in which the composer takes most of the intonation control based on highly specific knowledge for setting-up the tuning of the instrument as a multi-crooked collection of tubes. Hairy, Hippy, Happy by Marc Sabat

(2010/2006) uses this approach as well as Horn by Larry Polansky (1993). A

Hundred Valleys I for trumpet uses a combination of the second and third approaches and A Hundred Valleys II for horn uses all the main tuning techniques outlined in this thesis.

Supplementary material includes full scores, recordings from live performances and computer generated practice files. There is value of having accurate sound

files as a tuning reference to assist intonation during practice and for this purpose the software Lil’ Miss Scale Oven was the tuning agent. In the scores of the accompanying folio and in many of the examples throughout the thesis prescribed intonations are presented in ways that accommodate notational

xiv ambiguities associated with quartertones; some of these are discussed in

Chapter Three (p.106) and Appendix D (p.8)

A significant appendix (D) illustrates the technical issues involved in preparing and performing such contemporary compositions through my Method for playing the horn in extended just intonation. Sets of exercises were created to develop skills for work through to the 16th harmonic though could be easily extended or reduced to fit the range of other brass instruments. They will serve as a useful guide for composers wishing to understand microtonal brass techniques pertinent to extended just intonation. It begins by exploring harmonics available from the sixteen separate tube lengths available on a standard double horn. It also provides exercises based on various harmonics in thirds and provides exercises with wider intervals.

The thesis draws on my experience as a performer for more than 30 years and a composer for 15 years. I have taken advantage of the instrument's sixteen harmonic series and its just intonation capabilities. This knowledge became the starting point for the compositional practice.

Most of the research was conducted through the usual mixture of library and online searches. Additionally a number of instructive video clips have been posted on the website You Tube. Some of the research then led to associated websites with further information as referenced. Not all persons posting a video

xv on You Tube had an associated website and some posted only the name of their

You Tube channel. All video sources have been clearly noted in the list of

References in standard citation format.

xvi Chapter 1! The extended xenharmonic family of labrosones

The French Horn has been the focus of my musicianship for the past 35 years.

As a composer, performer and teacher it has been my passport in the profession as I explore the world of microtonal music. As part of my exegesis I have written a folio of compositions which explore the microtonal resources of the

French Horn. The microtonal basis for the folio extends an approach to tuning that was defined as early as the first century AD and it is aligned with a modern tradition best represented by the work of composer, instrument-builder and theoretician Harry Partch (1974). It will be referred to throughout the thesis as extended just intonation, a term first used by Ben Johnston in his 1967 article

‘Three attacks on a problem’ (2006). I have appropriated the term xenharmonic to describe the strange sounding qualities of just tunings based on the prime numbered harmonics 7, 11 and 13; however Ivor Darreg, the microtonal composer, instrument-builder and theorist who first coined the term used it to describe qualities of music in various non-twelve equal divisions of the octave rather than extended just intonation tunings (Darreg 1982).

In order to organise this folio it has been necessary to understand the instrument in a way that shows where the French Horn fits within the pantheon of microtonal instruments. This requires new scholarship related to the

1 approach to microtonality. It also requires a new classification of wind instruments that use vibration of the lips to produce sound.

The traditional approach of organology - musical instrument classification - sheds some light on the place of a modern orchestral brass instrument like the

French Horn in a post-Partch world of microtonality. The Sachs-Hornborstal classification uses the term "trumpet" to describe all wind instruments that use vibration of the lips to produce sound (Brown, H M, Palmer, F 178).

Ethnomusicologist Dale Olsen prefers the term “lip concussion instruments” to avoid associations with a particular material together with associated techniques of instrument construction by artisans working in a foundry (Olsen

1998: 29). Such all-inclusive definitions acknowledge the commonalities that exist between instruments used by various peoples in various places and periods by identifying similarities between the way the sound is made.

Anthony Baines, an authority on brass instruments, mainly uses the term

"brass" though makes passing reference to the term labrosones as a generic name (1993: 40). For this thesis “labrosones” will be used because it provides a succinct definition.

A more inclusive way to understand microtonal capabilities of an orchestral instrument like the French Horn in a global musical context would be to classify labrosones by the overtones they produce. For this purpose I will draw on the

2 harmonic theories of Partch (1974) and classify various instruments discussed in this chapter according to tuning limit. As a system of classification it is one that may have uses for other wind instruments. However a system for the classification of all wind instruments is beyond the scope of this thesis. It is applied only to wind instruments that use vibration of the lips to produce sound.

Just intonation music is often categorised by the upper prime numbered harmonic series limit. Prime numbered harmonics above the 2nd are significant because each provides a basic interval that has a unique tuning. Even numbered harmonics simply provide octaves of any odd numbered harmonic. Number two is the first prime number and the first even number in a numerical sequence. It provides the octave above the fundamental. The 3rd harmonic provides the interval of the perfect fifth from the 2nd harmonic, which can be expressed in ratio form as 3:2. Combining two 3:2 intervals gives the ratio interval 9:8, which also can be expressed as the relation between the 9th and 8th harmonics. This is why the odd numbered harmonic, the 9th, is less significant than say the 7th, 11th or 13th. The 9th harmonic is derived from the 3rd harmonic.

A succession of 3:2 ratio intervals can be presented as a Pythagorean diatonic scale and if extended far enough will very closely approximate almost any required interval. However it is usually simpler to use higher prime numbered

3 harmonics because the basic intervals of each can be clearly perceived and often easily played.

I will refer to harmonics played on labrosones throughout this thesis whilst acknowledging that I am using the term in an imprecise way. Musicians are used to the term as it is applied to playing their instruments. Technically it is more correct to refer to the “operating mode of an instrument” as Joe Wolfe explains in his article How harmonic are harmonics? (n.d.c). It is more correct because these modes are not in perfect harmonic series relations due to factors such as imperfect consistency of bore size and how well the instrument is made regarding the relationship between the bore, leadpipe and bell. Also the different lengths of tube within each instrument may not all provide a harmonic series with the same precision. It is also worth noting that the 1st harmonic on some lip vibrated instruments is not available. This includes the trumpet and trombone. For them the series of notes actually begins with the 2nd harmonic, yet pedal tones can be successfully played. On the trombone they sound one octave lower than the 2nd harmonic, giving the allusion that the fundamental is present. This phenomenon is called heterodyning by Wolfe (n.d.a) or a cooperative regime by Myers (1997: 21) and the pitches produced called factitious notes by Baines (1993: 36). On the trumpet the pedal tones are more than an octave below the 2nd harmonic, though can be lipped up to the octave.

However they don’t achieve a high level of resonance so are not often used for

4 musical purposes, especially where evenness of tone is desired. The tuba, other conical brass and the horn do have clear, stable and usable 1st harmonics.

The manipulation of the embouchure to adjust pitches is called “lipping.” This skill is common to brass and wind players alike. For instance players of baroque and classical oboes can “lip” up to a fourth away from the optimal resonance of a some pitches.1 Lipping can be used for intonation adjustments, fine-tuning, as well as the production of “factitious notes” mentioned above.

Table 1.1 shows the new classification. In the left hand column the prime-limit harmonic defines each group of instruments. These are treated as chapter sub- headings. A player of a particular instrument may be able to produce a harmonic higher than the listed prime number yet not as high as the next prime number. For instance players of instruments in the 3rd harmonic category can usually produce the 4th harmonic yet not necessarily the 5th harmonic or if they can it is not traditionally used. The middle column has lists of selected instruments that represent the prime-limit categories. The right had column indicates the length of tube for selected instruments. In comparison, the three standard brass that feature in this Chapter Three have the following tube lengths: the B! trumpet has a length of 1.31 metres; the B! trombone and B! horn

2.7 metres and the F horn 3.77 metres. Calculations for the length of an

5 instrument are approximate due to complications in making measurements as described in The Cambridge Companion to Brass Instruments (1997: 283-4).

! Table 1. Classification of labrosones by the available harmonics with ! approximate tube lengths. Harmonics Instruments Lengths (approx.) Fundamental Conch trumpets 20 to 60cm + 2nd harmonic Kul; Mabu 60cm approx. Agwara; Waza Up to 3m; 50-180cm Russian horn bands; Ragai 10cm to 2.5m Dung-chen Up to 5m Didjeridu 1 to 1.5m Finger hole horns 25 to 80cm 3rd harmonic Kakaki 2 to 4m 11th Century Up to 1m Ancient Mediterranean 50cm to 1.5m Shofar and other animal horns 25cm to 1.2m Trumpets, Later Middle Ages (Europe) Up to 2m Cornett 60cm for treble Hora 60cm approx. 5th harmonic La-pa Up to 1.5m P!k"ea Up to 2.5m Field trumpets Up to 1.5m 7th harmonic Serpent; Ophicleide 2.5m; 2.65m Bugle 1.31 to 1.6m Standard brass 1.3 to 5.3m 11th harmonic Clarin 3m Trutruca 2-5m Standard horn (frequently) 2.7 to 5m Standard trombone (rarely) 2.7 to 4m Standard trumpet (lead in jazz ensemble) 1.3 to 2m 13th harmonic Cors de chasse 4.4m Alphorn 3.5 to 4m Baroque natural trumpet 2.12m for D trumpet Orchestral natural horn from 1750 2.3 to 5.4m Some trumpet jazz performance 1.31m 17th harmonic Natural horn (Waldhorn) (in solos) 2.3 to 4.42m and beyond Natural trumpet (in solos) 2.12m Standard horn contemporary (rare) 5m

6 It should be noted that the cornett, in the second section of Table 1.1 is a significantly different instrument to another instrument with a very similar spelling. The cornet with one t is the treble brass band instrument whilst the cornett with two ts is the finger hole labrosone that featured in much music of the Middle Ages and Renaissance in Europe.

Regarding the presentation of information throughout this chapter, italics will be used when introducing a non-standard instrument and per the usual convention for titles of books and so on.

7 1.1.! Fundamental and 2nd harmonic

There are vast numbers of labrosones that are played using one and only one harmonic. Conchs are a pertinent example. Baines states that it is the fundamental of a conch that is most likely to be sounded, due to its resonance

(1993: 42). Sometimes a second, non-harmonic note is played up to a fourth lower than the main note. This is achieved by allowing the embouchure to vibrate below the optimal resonance of the harmonic. The technique is called playing in falset by Baines and the pitch is referred to as a factitious note

(1993: 36). The same technique is used on standard brass instruments. Another technique of conch players involves placing the hand, or an object, in the bell to change the pitch and tone colour as done in traditional Maori conch playing

(Flintoff 2004: 50). Players throughout the world use this technique, some adding wooden or bamboo mouthpieces to provide lower pitches (Baines

1993: 43). Larger conchs may be sixty centimetres long due to the spiral shape inside and therefore have three or four harmonics available (Baines 1993: 42).

Examples of such extended use can be heard in Japan (1.2).

Many other instruments are traditionally played sounding one pitch only such as the wooden kul of Papua New Guinea (Sentā & Hakubutsukan 1989) and the mabu of the Solomon Islands (Baines 1993: 41), both being used to instil a

fighting spirit. Players of longer instruments may be able to produce a number

8 of pitches yet be bound by tradition to sound but one. This is the case for players of the titiru trumpets in Wayana, South America. Playing more than one pitch is considered bad taste (Olsen & Sheehy 1998: 165)!

An excellent example of ensemble use of labrosones is found in the Alur region of Uganda. Agwara can be as long as three metres though only one pitch is played on each. A pentatonic scale is created from the combined pitches, melodies arising in a hocket fashion (Rycroft 2001: 353). An excerpt from the

JVC/Smithsonian Folkways Video Anthology shows all this clearly (Yamamoto c1996). Similar types of ensembles are present in other parts of Africa. A complete ensemble of waza has ten, sometimes twelve instruments in Sudan

(Simon 2007; Simon 1998: 565). Bands using the ivory horns in Congo, also present melodies in hocket fashion (Baines 45-6), as do the ensembles in Ghana playing the ntahera (Nketia 1962: 49-50; DjeDje 1998: 464).

Over a period of sixty to seventy years from 1757 in Russia, bands of hunting horns were used to present standard western music in a similar manner to the ensembles mentioned above. The players were instructed to produce only one harmonic. Later each player learnt to provided two harmonics an octave apart.

Additionally they inserted the hand in the bell to produce pitches a semitone below each harmonic. Thus each player could provide four pitches. One band had thirty-two players, giving the arranger a pitch range of several octaves

9 (Baines, 1993: 176-7). This tradition has been reinvigorated in recent times by

Sergey Peschansky and the group Russian Horn Cappella as documented on their website (2010). A similar tradition exists in Lithuania with the ensemble members playing the wooden ragai and forming a scale by the pitches of the various sized instruments (Karaška 1984: v2 978; Goertzen 1998: 511).

The dung-chen in Tibet are made of telescoped metal sections and would seem to have a potentially large range due to their length of up to five metres. However, the very wide and flat mouthpiece inhibits this.2 In ritualistic settings two notes are played in a powerful manner, an octave apart (Clark 2005: 123). Precision with pitch does not seem to be a significant issue since dung-chen, customarily played in pairs, are often not in tune with each other (Kaufmann 1975: 16). Just as the monk’s vocalisations are part of a meditative and mantric practice wherein each individual sings his lowest note with no direct relation to others, the low note of the dung-chen does not need to have a direct relation to musical surroundings. The importance of the sound is the metaphorical relation to the

“terror deities” (Kaufmann 1975: 16). It does not matter if the 1st harmonic is not a perfect octave below the 2nd or in tune with the rest of the natural notes. Non- conical trumpets generally do not have a good 1st harmonic due to a lack of optimal resonance, which also does not matter in this context. One source on

Tibetan music notation discussed in Grove suggests that three pitches are played (Helffer 2001: v25: 449). However, Kaufmann states that two of the

10 sounds are dynamic/timbral qualities of one pitch. The lowest note is played either as a distant, mellow sound or a rough roaring sound. The other, higher pitched note is only used in instrumental pieces (Kaufmann 1975: 16). This implies the use of the 1st and 2nd harmonics.

Didjeridu players produce a continuous drone through the technique of circular breathing. The main pitch is the fundamental. In addition to a tremendous rhythmic variety provided by players a wonderful repertoire of sounds is made through the interactions of vocalisations with a drone. Generally cylindrical instruments have only the odd numbered harmonics available. The interval between the 1st and 3rd harmonics will be anywhere between a stretched octave and a narrow twelfth (Knopoff 2007; Wolfe n.d.b). The 3rd harmonic is most often played at the end of a composition. The next harmonic playable on the didjeridu is the 5th, which will not conform to harmonic series ratios either, and is rarely heard.

Finger holes have been applied to animal horns or short wooden labrosones for at least a millennium. A twenty-two centimetre long ox horn instrument from

Sweden is dated to the tenth century (Baines & Dickey 2007). Finger hole labrosones when used in folk music are often played without overblowing, requiring only the 1st harmonic. They include the ožragis of Lithuania with two to six finger holes (Karaška 1984: v2 978), the sarv of Finland (Dournon 2000:

11 133), the bans or b!s or bugīr of India (Arnold 1998: 916) whilst the aza rag of

Latvia and the rozhok3 of Russia may have added mouthpieces (Marcuse 1975:

771).

12 1.2! 3rd harmonic

Players of the kakaki of west Africa use a musical style with abrupt rhythms and pitch changes between the 2nd and 3rd harmonics. The interval between these harmonics is often wider than a perfect 5th. The kakaki are made of various metals from brass to the tin from petrol cans (Baines 1993: 76). Many instruments in Asia would seem to be played in the same way as the kakaki such as the kombu from the Kerala district of India which is C-shaped and “is treated as a percussion instrument adding to the orchestral clamour” says Reck

(2003: 181). There are other names for similar instruments throughout Asia such as karna, plural karnai (Baines 1993: 80) or karnāl (Arnold 1998: 699-700).

Baines suggests the tradition of the kakaki goes back at least to the fifteenth century and probably back to the eleventh century. Trumpets were only the straight type in the eleventh century (Tarr 1988: 50) and it is assumed that players used one or two notes, the 2nd and 3rd harmonics. At that time there was little difference between European and Moslem trumpets (Baines 1993: 79-80). although the trumpet was perhaps better established in Moslem countries, providing music “on a grandiose scale” (Farmer 1957: 434). It wasn’t until after the resurrection of the Roman tuba from the tenth to twelfth century in North

Italy that trumpets in Europe began their long development and the range increased (Baines 1993: 73).

13 Long trumpets had much the same function in the ancient world of the

Mediterranean: they were used as signalling devices and to indicate military strength. Some of the trumpets we know of are the Egyptian šnb; a silver and a bronze trumpet from Tutankamen’s tomb; the Assyrian and Hebrew ḥaṣoṣerah and the Greek salpinx (Sarkissian 1997: 13). The latter was also used for musical contests (Scott 1986: 406-7). All of these were straight trumpets between 47 and

58 centimetres long (Sarkissian 2007). The interchangeable names lituus and bucina stood for a bronze hooked instrument that came to Rome from the

Etruscans before the first century BC, later the name referred to an animal horn instrument (Meucci 1989: 88). A surviving bronze lituus in the Vatican is 1.4 metres long. Experiments with replicas show that 6 harmonics can be produced on it (McKinnon 2001: 11).4 There were two other types of Roman trumpets, the tuba and cornu, also originating from the Etruscans (Meucci 1989: 88). The cylindrical tuba was up to 1.3 metres long and sounded the attack and retreat; the bronze, curved cornu sounded the relief for sentries.

In the Middle East sometimes a name appears in more than one context and applies to distinctly different instruments. This contrasts with a prior example in which the same instrument has a number of names. Meucci and Ibsen al

Faruqi commented that “many Arabic words at various times indicated a straight trumpet” (Sarkissian 2007). Nafir is a term for trumpets also implying

14 war. Other names include anfar and būq, the latter more likely to refer to a horn from an animal (Sarkissian 2007).

Some African instruments made from the horns of animal such as the sable antelope, the kudu or the gemsbok have an available 1st harmonic as well as the

2nd and 3rd harmonics. Just as the metal or wood instruments have a stretched interval between the 2nd and 3rd harmonics, these animal horn instruments have a stretched octave between the 1st and 2nd harmonics (Baines 1993: 45). Shofar is the name for the instrument fashioned from an animal horn by Jewish groups such as the Yemenites and Ashkenazim. The Yemenites use the horn from the kudu whilst other Jewish groups use the shorter horn from a goat or ram

(Montagu 2001). Many groups play them at any time of rejoicing though the

Ashkenazim use them only for ceremonial events. In such cases the shofar player presents the four main calls using two pitches: harmonics 2 and 3.

Montagu acknowledges the interval between the two pitches on any particular horn may be as narrow as a fourth or as wide as a sixth “yet they are always regarded as the 2nd and 3rd harmonics” (Montagu 2001).

There is no evidence to suggest that players of long trumpets in Europe ventured higher than the 3rd harmonic before the eleventh century. Sometime after that the first four harmonics were used. Indications come from current experts’ understanding of the earliest written music and a clear statement to

15 that effect made by Johannes de Grocheo c1300 (Sarkissian 2007). In the late fourteenth century instrument makers developed the skill of bending and folding metal. This allowed important changes to trumpet design. Up till then players were confined to holding up straight trumpets with a useable length of around 150 centimetres (Steele-Perkins 2001: 5-7). With the new craft of metal bending the overall tube length could be greatly extended and players experimented with a larger number of available harmonics. The earliest depiction of an ‘S’ shaped trumpet is thought to be 1379 (Sarkissian 2007). The twice-folded ‘natural’ trumpet was produced early in the fifteenth century

(Smithers 1973: 35) scarcely changing through to its ‘golden age’ in the baroque period. Also of interest are the trumpets with slides which were being built no later than 1411 (Sarkissian 2010) and perhaps as early as 1379 (Polk 1989: 395).

There seems little doubt that the early Renaissance slide trumpet was the direct ancestor of the trombone and no doubt it was used as a bass instrument in ensembles with two shawms (Steele-Perkins 2001: 26; Polk 1997: 42). Late in the fourteen hundreds the more practical trombone with its U-shaped slide took over the bass role in such ensembles. The trombone was closely associated with the cornett for a couple of centuries although the trombone's role was not restricted to one consort (Herbert 1997: 76).

From the eleventh century in Europe the cornett has been played with fingering holes like a recorder yet blown as a brass instrument (Dickey 1997: 51). It

16 evolved from a basic animal horn, is conical in bore and made of wood or ivory

(Dickey 1997: 51-4). The curved, treble cornett is about 60 centimetres in length and its lowest note, an A just below middle C, is produced with all finger holes closed and can be lipped lower to a G (Baines & Dickey 2007). Virtuosi of the cornett appeared from 1450 and would have needed at least harmonics 1 to 3 to cover a range of two octaves, whilst production of the 4th harmonic would allow for almost three octaves (Dickey 1997: 54).

The final example in this category is the conch as used in Japan. There it is called the hora by the Buddhist priests who traditionally use harmonics 2, 3 and

4 as well as a falset below the 2nd (Clark 2005: 60).

17 1.3! 5th harmonic

There are less instruments in this category than in 1.1 and 1.2 because the vast variety of labrosones are played using only the 1st and 2nd harmonics or the 2nd and 3rd.

The la-pa in China also known as the tongjiao (Thrasher 1984: v2 493) has a range of harmonics 2 to 5 (Sachs 1964: 238). It is made of telescoped metal sections, like the Tibetan dung-chen (1.1), yet is shorter. The same instrument in

Taiwan is used in shadow play accompaniments. Harmonics 3 and 4 are most played, occasional the 5th harmonic is added (Baines 1993: 78n + 79).

Many labrosones are made of wood yet those previously discussed are played producing only one or two harmonics. Carved and fashioned from a manuka tree in New Zealand, the pūkāea can be as long as 2.5 metres with 5 or 6 harmonics available, the player “readily sliding between these to create a spine- chilling sound” (Flintoff 2005: 78). This sliding technique seems to be unique, as far as this author knows.

Field or military trumpets can be viewed as part of an evolutionary process in the use of brass instruments in Europe. Up to the eleventh century players went no higher than the 3rd harmonic. Between then and the thirteenth century players introduced the 4th harmonic. As higher harmonics continued to be

18 introduced players began to specialise in particular registers. In trumpet ensemble music for five or ten players of the late 1500’s, the alto e basso part contained only the 3rd, 4th and 5th harmonics. Below this the players of the basso part provided the 2nd harmonic and the vulgano part the 3rd harmonic. Above these three the sonata part (quinta or principale) provided the main melody on harmonics 4 to 8 and the clarion part (soprano) was improvised using harmonics 8 to 13 (Tarr 1997: 85). The players of field or military trumpets in the

1500s used a combined range of the lower three parts of the ensembles just referred to, producing calls with harmonics from the 2nd to the 5th (Tarr 1988:

78).

An interesting example of the use of modern instruments being played like

Field trumpets is in Ben Neill’s work Antiphony. Four trumpet parts each have notes equivalent to harmonics 3, 4 and 5 (Gann 1994: 47). The lowest trumpet part is in the key of B!, the next in C, then D, then F. Antiphony can be viewed as a polytonal work or as a combination of tones all in the same key. The 4th harmonic from each trumpet part makes up a ratio of pitches that form harmonics 8, 9, 10 and 12 of the B! series.

19 1.4! 7th harmonic

For the player of a standard brass instrument, knowledge of lower harmonics from early years comes through lip slurring practice although this is done primarily for muscle flexibility and strength building rather than tuning. Few give much attention to the odd numbered harmonics beyond 3 and 5. Some will play the 7th harmonic in arpeggios as part of a warm-up routine, yet usually consider it an out-of-tune version of a standard pitch.

Many authors of tutor books for the horn recommend practicing slurs between harmonics going as high as the 16th yet avoiding the 7th, 11th, 13th and even the

15th. They simply use standard arpeggio notes. Examples abound from Oscar

Franz’s late nineteenth century volume (Franz 1942) to the well-known mid- twentieth century volume of Philip Farkas (1956) and Barry Tuckwell’s Playing the Horn (1978). Recently Jeffrey Agrell drew attention to this and suggested slurring with all the harmonics to allow the lip muscles to find their way around adjacent harmonics before learning to skip over particular ones (2008).

Christopher Leuba, in his short 1962 work on intonation, advocates using the 7th harmonic for dominant seventh chords (2004: 13). Richard Merewether, horn player and instrument designer, argued for the use of the narrow minor third interval in doleful cadences such as the end of the Second Movement of

Beethoven’s Eroica Symphony. In his pamphlet The Horn he referred to the ratio

20 7:6, the interval between the 6th and 7th harmonics (Merewether 1978: 39). Some unpublished warmups and teaching aids, such as Hector McDonald’s warmups

(2005) and teaching aids produced by Dominic Harvey of the ANU School of

Music (1999), include the 7th harmonic. Additionally I have heard a number of professionals slurring through to the 8th harmonic with no omissions. Also any player working through the publication by the Australian Music Examinations

Board for Horn Grade Two and Three (Lawrance 1992) will have noticed that the slurring exercises are designed to use the natural harmonics including the

7th and Grade Four includes the 11th harmonic (Lawrance 1992). Interestingly, the early nineteenth century horn methods of Dauprat (1994/1824), Domnich

(1985/1807) and Duvernoy (1802) do not place any emphasis on harmonics either in writing or in exercises. They simply refer to diatonic sequences and begin with exercises that require right hand technique to provide certain notes not available otherwise, in other words to fill the gaps between harmonics 2 to

3, 4 to 5, 5 to 6 and so on. It is clear that the tradition of horn playing has been that of tempering harmonics to fit with standard music practice. A few composers have used the natural 7th harmonic such as Randall E. Faust in his works Harmonielehre (1996) and Prelude (1977). Works by composers Britten and

Ligeti are referred to in later sections (1.5, 1.6).

A similar situation is evident for other brass instruments: the 7th harmonic is included in slurring practice by some authors of tutor or exercise books. The

21 few examples I have come across of trumpet music using the 7th harmonic are studies or slurring exercises. The recent book by John Foster includes some 7th and 14th harmonics in the section titled “Studies on the harmonic series” (2010:

21). A 1938 tutor book by Earl Irons includes the 7th harmonic in ten groups of slurring exercises (1966). Richard Hodges technical studies for treble clef brass instruments include the 7th harmonics in eleven exercises (2005: 57-62). I have not come across any compositions for trumpet that ask for the 7th harmonic to be played as it naturally occurs.

Trombonists do not shy away from using the 7th harmonic in slide positions other than first position though will adjust the slide a bit higher than usual to temper it. They use the 7th harmonic for pitching ease, articulation clarity and slide movement minimisation. Higher harmonics are generally more difficult to pitch than lower ones or articulate as clearly therefore selecting the 7th harmonic rather than the 9th or 10th can help. Additionally, minimising slide movement can make many passages more manageable.

An interesting effect called “slurring against the grain” is used in a few trombone solo works. The slide movement is identical to playing a descending chromatic scale, however the lips pitch a higher note with each shift. Example 1, below, shows the slide position, harmonic and the fundamental to which the harmonic belongs from a well-known trombone work. Players will temper the

22 7th harmonics and usually place a ‘!’ next to the relevant slide position as a reminder to raise the pitch. In this example the ‘!’ signs are placed above.

Example 1. Slurring across the grain in Frank Martin’s Ballade for Trombone and Piano.

The phrase begins on the note C played in third position as the 5th harmonic of the A" series. The next note is D, played in fourth position as the 6th harmonic of the G series. This is followed by E in position five as the 7th harmonic of the F! series, then F in position six as the 8th harmonic, with the phrase ending on F! in seventh position as the 9th harmonic of the E series. The next sequence slurs through harmonics 5 to 9 and the final sequence uses harmonics 6 to 10.

An example of trombone playing that does not temper the 7th harmonic is found in Stuart Dempster’s recording Underground Overlays from the Cistern

Chapel (1995) and in other compositions in live performances he has given. His piece for the MiniMax Festival 29 July 2002 in Brisbane was titled Twenty

Tantalizing Titillating Trombones Tooting Turbine Tonics (2002). In the same concert

23 a trombone quartet performed Robert Davidson’s Tibrogargan Round (2002) also using natural 7th harmonics.

Exploration by brass players of the many intervals and chords that can be associated with the 7th harmonic is extremely rare. Higher prime numbered harmonics receive even less examination. A musician wishing to use the 7th harmonic in musical situations, not only in exercises, requires an aural acceptance that the harmonic is completely valid without altering the tuning.

Certainly pitching the 7th harmonic in new musical contexts requires additional practice.

Groups of instrumentalists that use the 7th harmonic unashamedly are the cors de chasse (trompes de chasse). Alphorn players also play the natural 7th harmonic and both groups of players will use harmonics as high as the 16th therefore these instruments will receive more attention in Section 1.6.

The bugle is a natural labrosone that has an available 7th harmonic. This conically shaped instrument is very well known in many western countries especially for its signalling and ceremonial use by the military. Most calls use harmonics 2 to 6 (Baines & Herbert 2007) though some rare older calls used harmonics 1 and 2 (Baines 1993: 27). The instrument has much less resonance higher in the harmonic series therefore the high written C, 8th harmonic, is never used. This is due to it having a larger bore than a trumpet (Bate 1966: 6-7).

24 It was bugle notation that was adopted by brass bands at their inception in the nineteenth century. The C just below the treble staff was and still is played on the 2nd harmonic (Baines 1993: 27). Orchestral trumpets and horns at that time equated this written note with the 4th harmonic, therefore their high written C was played on the 16th harmonic. Horn writing was not influenced by brass band conventions even after the almost universal adoption of the valved instrument. However, writers of trumpet music adopted the convention for the cornet (Tarr 2007). This occurred for a number of reasons. Trumpets and cornets often worked side by side in orchestras throughout much of the nineteenth century and once the valves for the trumpet became the standard setup there seemed little point in treating the notation differently. The introduction of the cornet also influenced trumpet players to change from using longer F instruments to the shorter B! and C instruments, a shift that occurred between

1850-1890 (Tarr 2007). Some musicians played all the orchestral trumpet repertoire on the cornet (Baines & Herbert 2007) though others used the keyed bugle or the keyed trumpet. The latter was used more for solo repertoire than ensembles, as was the slide trumpet in England (Dudgeon 1997: 131-9). Much orchestral music in the nineteenth century was written for a valve trumpet in F, mid-way in length between the natural trumpets and the modern trumpet or cornet. Players of this F trumpet would have needed to produce harmonics up to the 8th or 9th harmonic to cover standard repertoire.

25 As mentioned most orchestral trumpet players transferred to the short B! trumpet but now much orchestral repertoire is played on the C trumpet. For clarity and the security gained by greater pitching accuracy orchestral players began using even shorter trumpets. The D trumpet was first made in Brussels,

1861 and the piccolo G trumpet by Besson in France 1885 (Tarr 2001: 837).

Stravinsky took advantage of both in his composition Le Sacre du Printemps

(1965/1913). For orchestral playing the object has been to gain maximum clarity, cleanliness of articulation, accuracy and power. The pitch limits depend upon context but for some orchestral situations a C trumpet will be used up to the

10th harmonic for its fuller tone rather than a shorter trumpet.

The need to extend much higher than the 8th harmonic has not been a large part of trombone history. With the slide as its characteristic feature there have always been a significant number of notes available to fit in with standard western music writing. Even the very high passage in Schumann’s Symphony no.3 (n.d./1850) does not require higher than the 8th harmonic for the E! alto trombone5 though some players choose to use the tenor trombone for this work.6 This would require the 12th harmonic of the B! instrument. A few other examples of higher writing to the 12th harmonic are dealt with later. The range used in solo repertoire is not significantly different to large ensemble writing. In brass bands solo material usually goes to the cornet or euphonium therefore the

26 pitch range for the trombone in this genre is more limited than in orchestral writing.7

There is, or used to be, a certain psychological aspect to considering the 8th harmonic the upper limit of modern brass instruments other than the horn. The

8th harmonic on the open tube is the high written C above the staff for all brass instruments reading treble clef except the horn. This includes all the brass band instruments other than the bass trombone, which uses the bass clef. The notes leading up to high C from the G at the top of the staff all require 8th harmonics from the following sequence of gradually shortening tube lengths: valves 2 + 3;

1 + 2; 1; 2; 0 or slide positions 5, 4, 3, 2, 1. To go higher would seem to require a new sequence. However this is not the case as the C! can be simply played as the 9th harmonic of valve 2 and the D as the 9th harmonic of the open tube, slide positions 2 and 1 respectively. Going higher still, playing the written D! and E as 10th harmonics may result in intonation that sounds too low for many contexts. These notes could be played as 12th harmonics on longer tubes, making the tuning sharper. The disadvantage is that the 12th harmonics are more difficult to produce with as much pitching accuracy.

The vast body of western repertoire for standard brass other than the horn does not extend beyond the use of the 8th harmonic. Going higher is too demanding

27 for a usable tessitura as the most comfortable range falls between harmonics 2 to 6 and 2 to 12 for the horn.

The serpent and the ophicleide had a part to play in European music history and they both have current exponents. The serpent was derived from the tenor version of the cornett at the end of the sixteenth century (Bevan 1978: 47). It was principally used in wind ensembles for its bass register (Bevan 1997: 144). Up to

8 harmonics are playable with all tone holes covered. However with the tone holes uncovered the tone deteriorates therefore the range is usually restricted to that between the 1st and 4th harmonics (Baines 1993: 49-50). The serpent also has considerable intonation issues with many a fingering choice being used for a number of adjacent pitches (Morley-Pegge, Bate & Weston 2001). In contrast the ophicleide has keys controlling the covering of holes which makes the

“intonation much better than the serpent” as Bevan says (1978: 61). This gives it a large usable range up to the 8th harmonic.

28 1.5! 11th harmonic

There are a handful of non-standard instruments that use harmonics in the higher range, two of which are the clarin and the trutruca. Both are made of cane, the clarin is straight and the trutruca often looped.

The sound of the clarin can be accessed in a number of recordings. The main harmonics used are 6, 7, 8 and 9 with the 12th harmonic played as a short note and as the upper tonic to the 6th harmonic. This use can be heard in a recording made by Dale Olsen of two players who present an engaging melodic and rhythmic motif (Olsen & Sheehy 1979). An example uploaded to You Tube without the 12th harmonic is a melody played on harmonics 6 to 9 in which the

7th harmonic is the tonic (Music Tradicional del Peru 2008). The video titled

Musica hecha con clarin cajamarquino, cajon y flauta (2007) has musicians playing in the same pitch range. Also of interest is a video showing a musician explaining the use of the clarin in Spanish and playing examples of current and traditional use with pitches ranging from the 6th to 12th harmonics (Reportaje al

Clarin Cajamarquino Patrimonio del Peru 2008). In all of these musical examples the 11th harmonic is never heard yet there is no reason to suspect it cannot be played.

The trutruca (sometimes spelt trutruka) is played with a similar range of harmonics as the clarin. A characteristic motif is a short blast of a downward

29 glissando from harmonic 9 to 6 as heard in an album by the Chilean group Las

Araucanas (Cuecas de una Misma Tierra 2009). European military signals have made their way into traditional settings probably because many of the players spent time in the army (Olsen & Sheehy 1998: 234). Recorded in July 2009 is a piece called Nguillatun (2009) in which the trutruca player produces harmonics

5 to 10 as well as the 12th. The segments sound like bugle calls and are in harmony with other instruments. In the video titled Mapuche and Mack Yidhaky

(2006) an unnamed trutruca player uses an extended range including the 5th and 4th harmonics. Further extension of harmonics was reported by Anthony

Baines who had access to a recording of a player who produced a passage mostly on harmonics 4 to 7, then finished with a glissando from the 4th to 14th

(1993: 50).

A few examples of orchestral repertoire will suffice to show the usual limits for standard brass. Trumpet writing that has the 9th harmonic as the upper limit can be found such as Shostakovitch’s Symphony 5 (1994/1937) with its high written

D’s (sounding C) for trumpets 1 and 2 in B!. In the modern orchestral trumpet repertoire Benjamin Britten asked for a written D for the D instrument

(sounding E), the 9th harmonic, in A Midsummer Night’s Dream (1960).8 The use of higher harmonics on modern trumpets in orchestral repertoire is very rare.

However, players use the standard B! instrument in Wind Bands (Concert

30 Band) and jazz ensembles. The use of the 12th harmonic is not unusual in these contexts.9 The upper limit for jazz players is generally the 12th harmonic though specialist high range players will extend that to the 16th harmonic (Tarr 2007).10

In history the slide trumpet was known specially for its use by J. S. Bach in the baroque period. Bach used it in six cantatas partly to double the soprano line in choruses yet also as a separate musical line. It could be played chromatically and diatonically much lower in the range than the natural trumpet and horn.

The writing in BWV 162 (1887/1715) takes advantage of this possibility with a range of only one octave from the 2nd to 4th harmonics (middle C to the next C) in the opening movement and doubling the soprano line up to F (above the soprano clef) in the closing chorus (Bach 1887/1715). The writing in BWV 46

(1860/1723), BWV 67 (1868/1724) and BWV 77 (1870/1723) looks more like the writing for natural trumpet, even requiring the 18th harmonic in BWV 67, with a few pitches that require the slide.11

A few examples highlight writing for the trombone. Jean Francaix’s orchestration for chamber orchestra of Francis Poulenc’s Barbar the Elephant has a high written C (1963).12 This is played as the 9th harmonic in first position.

Shostakovich wrote the same pitch for the 1st trombone in his Symphony no 5

(1994/1937).13 This is an octave lower than the trumpet in the excerpt mentioned earlier. A famous example from Beethoven’s Symphony no 5 (n.d.a/

1808) occurs where the player of the alto instrument must produce a high F,14

31 which can be done on 1st position as the 9th harmonic or 3rd position as the 10th harmonic. Some other composers requiring high harmonics on the trombone are Richard Strauss and Alban Berg who both wrote a high E in the opera scores

Die Frau ohne Schatten (1996/1917)15 and Wozzeck (1955/1922)16 respectively.

This pitch is played in 2nd position with the 12th harmonic. Of more interest is the high D! in Berg’s Lulu (1978/1937)17 and repeated E"s in Drei Orchesterstücke

(1954/1915)18 because some trombone players produce using the 11th harmonic in a sharp 2nd position. Brahms’ Symphony no 2 (1991/1877) with a D, the tonic, can be played on 1st position as the 10th harmonic. However, if the intonation is too low for the tonic the player may use 4th position and the 12th harmonic or even a sharp 3rd position and the 11th harmonic.

The other brass instrument in the modern orchestra is the tuba. A few examples illustrate the upper range of harmonics that players use. A passage in the overture to Die Meistersinger (1976/1868) by Richard Wagner was written for a

C tuba extending to high E, the 10th harmonic.19 Hector Berlioz wrote for the B" above middle C in his Symphony Fantastique (1972/1830)20 though this was originally written for the ophicleide. The part is customarily played on an F tuba, the high note is therefore played as the 12th harmonic on first valve. Less power and more use of the higher range is expected from the tuba in concert bands in contrast to orchestras.21 Writing up to F above the bass staff is not rare,

32 with parts occasionally going to the B! above, though that pitch would probably be played on an E! instrument using the 12th harmonic.

Horn players will often need to use the 10th harmonic, even by those predominantly using the B! instrument. Written A!‘s and A’s are required pitches in music examinations from grade six and on. These notes can be played on the B! instrument as 10th harmonics with second valve and no valves respectively. The same notes will be slightly sharper when played as 12th harmonics using valves 2 and 3 for A! and valves 1 and 2 (or 3) for A. Writing for horn in the orchestra will rarely go higher than the C above the treble staff. I don’t know of any band repertoire that does so. Even the high C can be played as the 12th harmonic. As with the other standard brass a shorter instrument may be used to retain clarity and accuracy in the high range. For the horn this is a descant instrument pitched in F or E! alto. The latter’s 12th harmonic is a written

F an octave above the treble staff.22

A rare example of the use of the 11th harmonic is found in a work for brass band and euphonium solo. The composer Philip Wilby indicates that the soloist should produce natural harmonics 8 to 12 in the cadenza from Dove Descending:

Sonata in Open Tones (1999).23

33 Possibly the most well known example in Western art music that uses the natural 7th and 11th harmonics is the “Prologue” and “Epilogue” from Britten’s

Serenade op.31 for tenor, horn and strings (1944). Here the melody of the unaccompanied horn’s natural harmonics evokes an other-worldly quality. The horn player must produce harmonics 4-12 and the 14th, the octave above the 7th.

Strangely the higher note is not notated an octave higher as a B! but as an A.

The pitch is clearly meant to be the 14th harmonic as evidenced in the aural tradition stemming from the performances by Dennis Brain, for whom it was written.24 The 11th harmonic is particularly potent in the music and is the first truly non-standard pitch presented. It first occurs approached from the tonic, written C 8th harmonic. The two notes are heard as a neutral interval, neither a perfect fourth nor an augmented fourth, as if Britten was suggesting the twilight between day and night. The second occurrence is part of a stepwise melody from harmonics 7 to 11 with the goal note, the high G 12th harmonic, interrupted by the repetition of the 8th harmonic. Example 2 shows the music in

Britten’s notation with the addition of harmonic series numbers for clarity.

34 !"#$"%&'q'(')*'+&,-.&'$#'/010%2, , ˙™ !"#$%&$%' œ œ™ œ œ œ œ œ œ™ ˙ Œ &) J J J œ ))J /01,23+456 ! "# ! $ % "& ! ' (")"%*))+,- , œ œ œ œ œ œ œ J œ œ œ™ ˙ Œ j ˙™ & J 41754- J J œ ))J )+89:œ œ™ ! "" "& $ % "& ! % $ ( *$&.*$/" > > j >œ j > œ œ & œ œ bœ œ œ œ œ Œ œ œ bœ œ œ œ J œ J J ™ ,2;<2941754-J J % ( ! % $ % % ( ! % $ % "& %

"& *%0+.(" , , - w> ˙™ œ ˙™ & Œ œ œ™ œ œ™ œ J : ::J ))J J "" ! "# ! "' "# ". .",0"%#*,,- œ œ œ U & œ œ J œ *+,-J ))) w ! $ % "& '

Example 2. Harmonics of the horn in the “Prologue” from Britten’s Serenade for tenor, horn and strings.

35 The 11th harmonic of the horn could well be used for blues and Arabic music.

With some horn crooks, or valve choices, the 11th harmonic is not too high in the overall pitch range. The melody, I wouldn’t mind dyin’, but I gotta go by myself

(1993/1928) as sung by early blues musician Rev. I. B. Ware, has a neutral 4th as well as a neutral 3rd. The song transcribes well for the horn, shown below in

Example 3. In the key of F (C for horn) the neutral 4th can be played using the

11th harmonic of the open F horn, the neutral 3rd with the third valve, the 11th harmonic of the D horn.25

Example 3. Harmonics on the horn used to play I wouldn’t mind dyin’ by Rev. I. B. Ware.

One other use for the 11th harmonic on the horn is to negotiate quartertone passages. Douglas Hill makes this very suggestion in his book Extended techniques for the horn. He presents a fingering chart of almost two octaves by using the 7th, 11th and 13th harmonics (1983: 65-67). Hill also suggests tuning one side of a double horn a quartertone from the other to achieve a larger range.

The latter method is certainly more precise in tuning than using the 7th and 13th harmonics. Each 12-tone equal tempered semitone interval is 100 cents and each

36 24-tone equal tempered quartertone interval is 50 cents. The 11th harmonic when used in place of a true quartertone is only 1 cent higher whereas the 7th harmonic is 19 cents higher and the 13th harmonic is 9 cents lower. The 7th and

13th harmonics will give a flavour of non-standard pitch but are not strict quartertones.

Thea Musgrave asked for the use the 7th harmonic as a means to achieve quartertones in her Horn Concerto (1971). The score is clearly marked with arrows indicating a quartertone flat from the standard notation as shown in

Example 4. The fingerings supplied are for the 7th harmonics of both sides of the double horn. The musical effect is achieved even though the two elements do not equate exactly.

Example 4. 7th harmonics used to give the effect of quartertones.

A very precise solution to playing quartertones on the horn has been achieved and is used by Samuel Stoll as observed on video of a lecture for students at

Goldsmith College (Microtonalprojects 2010). His horn has an adapted valve

37 slide used in conjunction with an additional valve to achieve quartertones on the F and B! horns. His instrument also achieves eighth tones on the B! side.

Stoll can also be heard in a duet with trumpeter Stephen Altoft performing

Donald Bousted’s Duo-1 for quarter-tone trumpet and quarter-tone horn at UK

MicroFest3 (2009). Altoft had already developed a design with Johannes Radeke and Sigmar Fischar to convert his “existing trumpets into microtonal instruments” (Altoft 2006). Altoft’s work in 19-tone, 24-tone and 48-tone divisions can be heard on the CD 24 Microtonal Studies (2008) and extensive study materials are available on the website The Microtonal Trumpet Project

(n.d.).

Trumpet players have used the quartertone for over a century and developed a significant tradition. Instruments have existed since 1893 in Russia although the purpose of those particular ones is not now known (Davies 2007). For his opera

Makta, premiered in 1931, Czech composer Hába required two quartertone trumpets (Battan 1980: 101). A few decades later Czech composer Pavel Blatny wrote for trumpeter Jaromir Hnilicka (Ellis 1975: 17). Hnilicka’s work intrigued jazz artist Don Ellis who commissioned Holton to make a quartertone tone trumpet, which was designed by Larry Ramirez. It was produced in 1965 and

Ellis asserted the instrument allowed for better tuning in just intonation (Ellis

1975: 6). Many of his performances such as “Indian Lady” and “Turkish Bath” from the album Electric Bath (1998/1867) and “Variations for Trumpet” from

38 Don Ellis Autumn (2007/1968) feature his quartertone playing. Another microtonal modification came in 1974 with the Firebird trumpet designed by

Ramirez for Maynard Ferguson (Bevan 1994: 1220). It has a small trombone-like slide and has also been used by Rajesh Mehta to play Indian ragas (Mehta n.d.).

Arabic music has at least two trumpet playing advocates, father and son

Nassim and Ibrahim Maalouf. Instrument designer Michel Wikrikaz produced quartertone trumpets for Nassim in the 1970’s (Mortenson: 15) and one more recently for Ibrahim (Mortenson 15-16). An example of Nassim’s work can be viewed in the documentary Between Two Notes (2007). Ibrahim’s work can be accessed via his own website (2010). Other exponents of quartertone trumpets include Marco Blaauw (n.d.), Jeff Kaiser (2010), Adir Kochavi (Laar 2009),

Anders Nykvist26 and Markus Stockhausen with his quartertone flugelhorn

(2011). The term quartertone is used loosely by these players and use of the interval varies considerably from player to player, evident by the genres they play in: Jazz; Indian; Arabic and contemporary Western.

39 1.6! The 13th harmonic

György Ligeti is another composer who has called for natural harmonics on the horn. In 1953 he asked for harmonics 4 to 8 in the horn part of Movement III of his Sechs Bagatelles (1953). The 7th harmonic in the arpeggiated ostinato adds a lovely colour to the texture. Both this and Britten’s Prologue (p 34) are in the same key, F and do not modulate. A few decades later Ligeti went further and asked for the 11th and 13th harmonics in his Trio for violin, horn and piano

(1984/1982). He used a multi-crook approach within a 12-tone equal tempered framework. The horn player needs to tune to the piano yet produce non- tempered harmonics with many different lengths of tube. Each valve or valve combination is treated as a crook tuned to a particular key. It is then selected and retained for each segment. All of this is marked in the score by Ligeti. In

Example 5 the fingerings required by the horn player to execute the part are indicated below the staves. For reference the harmonic series numbers are also indicated.

Ligeti extended this type of writing in his Hamburg Konzert, premiered in 2001, for four obligato natural horn players as well as the horn soloist using the same approach as in the Trio.

40 ,,-./"012"33"45.45675.&"74"258.1.9:56%.;1&.<.=+ (b ( Bb "" 2-05-61789:;/7289:83*.< - #>œ œ- œbœ !"#$%&$%' œ bœ œ #œ™ œ nœ œ ™ bœ œ &8 œ bœ ‰ œ ™ Œ * bœ™ )* !"#$%&"#$' (-b') (b'* (b'+ ,-.)/01234 ! " # $ % $ &' ( # % $ &' && % ( ( "$ 78==18./ . bœ nœ œ j nœ. . #œ & œ œ œ #œ j ‰ ‰ #œ ‰ ‰ ‰ œ ‰ ‰ J ‰ ‰ j ‰ - * J ˙™ œ **. J J œ (b'* >. " % # " " % ! " % # $ ! #+ Eb j . #œ. j . j ‰ ‰ nœ ‰ ‰ œ. ‰ J ‰ ‰ j ‰ ‰ ‰ nœ ‰ ‰ j ‰ ‰bœ. ‰ j ‰ ‰bœ ‰ ‰ œ. ‰ & #œ. J J œ #œ. J œ. J œ . J >. !') . " % # $ ! " % # ( >" % $ ) #% 2-05-61789:;/7289:83*.< j - j ‰ ‰ bœ. ‰ ‰ j ‰ bœ ‰ ‰ œ. ‰ ‰ #œ œ™ nœ #œ & œ. J œ . J * #œ w . !'* œ # ( >" % $ &' % $ ! "

Example 5. Multi-crook approach to horn writing with harmonics to the 13th. Ligeti Trio, “Scherzo.” Horn part, bars 55-70. All Ligeti's markings are retained.

41 A rare example of a non-western labrosone going beyond the 12th harmonic is described in Baines’ history of brass. In India, 1963, a player of the baqa, a long straight instrument, produced many quasi-glissandi from the 8th to 12th harmonics, beginning the musical fragment with a grace note from the 13th to

12th harmonics (Baines 1993: 89). Upper harmonics have been mostly the domain of Western practitioners relying on extraordinarily well-designed instruments to gain consistency.

For the trumpet in Europe the first use above the 12th harmonic is detailed in

Bendinelli’s instruction book The Entire Art of Trumpet Playing of 1614

(Tarr 1975). Most music for natural trumpets prior to this time was not notated.

Players memorised and improvised their music and learnt their craft through an aural tradition. Bendinelli’s Sonata of 1584 requires the 13th harmonic as do the works In dulci jubilo of 1619 by Praetorius and L’Orfeo of 1607 by Monteverdi

(Smithers 1973: 135; Steele-Perkins 2001: 17). Also of great interest is the use of the pitch b in the middle of the treble staff, in Bendinelli’s 1584 Sonata. This has to be lipped down a semitone from the 8th harmonic and is the first written indication of the necessity for such a technique (Tarr 1975: 17).

Players of the hunting horn, cors de chasse (trompes de chasse), will use a range from the 2nd to 16th harmonics. Some repertoire can be heard on recordings such as Jagdmusik (1982). Giacomo Rossini composed a short work for hunting horns

42 in 1828 called Le Rendez-vouz de Chasse (n.d.). The players engaged to give the premier performance did not believe that the written B! existed on the hunting horn. Rossini had to demonstrate this note, the 7th harmonic, to convince the players (Tarr 2007).

The alphorn is most often associated with the Swiss people though it exists in various sizes in many parts of Europe. The instrument of 1.5 metres in length and dates back to the fourteenth century (Baines 1993: 51). Georg Rhau provided the first known printed work for alphorn in 1545, which shows motifs using a range between the 6th and 12th harmonics (Jones 2010: 42). Alphorn motifs are traditionally played in free time and improvised unaccompanied

(Jones 2010: 41). Motifs have been imitated or quoted by many composers from

Haydn and Rossini to Meyerbeer and Wagner (Jones 2010: 43-5). One use of the alphorn was to indicate the end of a storm. Beethoven’s theme in Movement 5,

“Hirtengesang” (The Shepherd’s Hymn) of Symphony no 6 (The Pastoral) clearly does this (n.d.b/1808). Johannes Brahms made one alphorn tune famous by including a tempered version of it in his Symphony no 1 (1986/1876). The actual instrument has been used very rarely in Western art music. One example is in the opera Daphne by Richard Strauss which includes three alphorn each in a separate key (Jones 45-6; Strauss 1938).27 Each instrument uses a range from the

5th to 12th harmonics. Contemporary use of the instrument can be heard from groups such as Mnozil Brass (2009). Gerhard Füssi includes the 7th and 11th

43 harmonics and is accompanied throughout by the tubist who plays arpeggiated

figures on the tonic chord. The Moscow Art Trio performed at the Sydney

Festival and Arkady Shilkloper included a very lively improvisation on the alphorn including all harmonics from the fundamental to the 16th (2009).

Recently Salvador Brotons composed for horn player Javier Bonet to perform on the horagai, alphorn, natural horn and valved horn in the work Concerto for

Horn and Orchestra: Ab Origine (2009). Balthasa Streitt of the contemporary duo

Stimmhorn not only plays the alphorn in a virtuosic manner to the 20th harmonic but performs it in a music theatre context (2010).

Some other works featuring higher horn harmonics are Joseph Perhson’s

Harmonic Etude (1987) which requires harmonics 1 to 14 on some tube lengths;

Douglass Hill’s Thoughtful Wanderings for natural horn and percussion (1990) and

Jörg Widmann’s Air (1973) with many 11th and 13th harmonics, going as high as the 20th for horn in E!. The horn part in the opening of Joel Mandelbaum’s Wind

Quintet No.2 uses harmonics 2, 6 to 14 and 16 on the F horn. This is shown below in Example 6 with the harmonic series numbers printed below each note.

44 Example 6. Horn part in Wind Quintet No.2 by Joel Mandelbaum, bars 1-10. Harmonic series numbers are indicated below each note. Horn part notated with permission from the composer.

45 1.7! The 17th harmonic and beyond

Producing harmonics beyond the 16th is considered the realm of extreme high register brass playing. These harmonics are almost never called on in modern music. However there are many examples for the trumpet and horn from the eighteenth century.

The range of harmonics for players of the horn and trumpet was identical in the baroque period. The only difference between the two was that the horn sounded one octave lower. Trumpet players already had a long history of playing art music whereas the hunting horn was a recent addition. It first appeared in opera scenes playing hunting signals from 1700 (Hiebert 1997: 104).

In many instances in the early eighteenth century the trumpet players would have performed the horn parts. It was also very rare to have trumpets and horn playing simultaneously, one example being Georg Frederich Handel’s Royal

Fireworks Music (1962/1749). The larger bore waldhorn was invented by the

Leichmanbschneider brothers in 1700 (Fitzpatrick 1970: 26). The waldhorn became the choice of the professional and would have further differentiated the tone colour of the horn from the trumpet.

Some works requiring an extreme high range for horn in the baroque period are the Capriccios by Jan Dismas Zelenka (1977) written between 1717 and 1729. The

46 composer wrote for the 24th harmonic from the first horn in his Capriccio I in D shown in Example 7.28

Example 7. High written G played as the 24th harmonic in Zelenka’s Capriccio I in D.

Capriccio IV in A goes higher in actual pitch yet only to the 18th harmonic for the horn in A.29 This harmonic was the upper limit that Johann Sebastian Bach wrote for. Examples are in various cantatas including BWV 1 (1851/1725) for horn in F30 and the same harmonic a fourth higher for horn in B! in Cantata

BWV 14 (1852b/1735).31 The first trumpet player for Bach’s works also required a range up to the 18th harmonic with one 20th harmonic appearing in Cantata no.

31 (Oldham, Campbell & Greated 2007; J S Bach 1857/1715).32 The most extreme trumpet parts were written for Johann Heinisch between 1727 and 1750.

47 Antonio Caldara, Johann Joseph Fux and Georg von Reutter the Younger wrote for Heinisch with harmonics up to the 20th and even the 24th (Tarr 2007a).

In the classical period the trumpet became relegated to a subsidiary role.

Composers found more use for the mellower sound of the horn. In orchestral writing the middle range of the horn became relied on to underpin harmonic movement. However, a number of works require the extremities of the horn range. Most extreme are the 24th harmonics in Wolfgang Amadeus Mozart’s

Horn Duets. (n.d./1786)33 for horn in E!. Other examples include Joseph

Haydn’s Symphony 51 (1951/1774) and Divertimento in Es (2004/1767) which have parts extending to the 21st harmonic for horn in E!;34 Symphony 31

(1964/1765) to the 20th harmonic for horn in D35 and Concerto per il Corno di caccia (1985/1762) to the 18th harmonic.36 There were a number of extraordinary horn players in Bohemia from the 1750s, many chamber works attesting to this.

Some wind octets in the classical period such as those written in 1770 by Josef

Mysleviček (1962) and Václav Vincenc Mašek (1977), date of compostion unknown, include 20th harmonics for the first horn part in E!. This writing style was used by Ludwig van Beethoven in his Sextet for two horns and strings

(1999/1794). The high horn player, first horn, ascends to the 18th harmonic on an E! instrument. In the same work the low horn player, second horn, has to produce notes in descending steps from the 2nd harmonic to a perfect fourth

48 lower, the low written G. Haydn, in his Symphony 51 (1951/1774) also requires the low horn player to lower the 2nd harmonic, in this work by an augmented fourth, a very advanced lipping technique. The pitch a fourth below the 2nd harmonic is a factitious note, similar to the pedal tones on the trumpet and trombone. The pitches in-between this factitious note and the 2nd harmonic have even less resonance yet some degree of clarity. Carl Maria von Weber in his Concertino for horn (c1975/18015), requires the player of a horn in E to lip down from the 2nd harmonic in steps to an augmented fourth below in the cadenza. In the final phrase of the entire work Weber wrote a high E above the treble staff, played as the 20th harmonic. Some other examples of very high writing come from Robert Schumann in his Konzerstück for vier hörner und orchestra (1968/1849). He wrote up to the 20th harmonic for the first horn in F,37 and the same harmonic for the 3rd horn in C38 in the 3rd (Rhenish) Symphony

(c1882/1850). A few decades later Richard Strauss wrote the same harmonic, also for horn in F, in the first horn part of his Symphonia Domestica (1904).39

Mostly the extreme high range has not been required from the nineteenth century onwards due to the greater use of hand-stopping on the horn and to a lesser extent on the trumpet. Also the introduction of valves allowed for lyrical, scalic and chromatic passages to be played in the most soloistic register, harmonics 6 to 12 of the F horn.

49 Rare examples of extremely high writing are found in the later part of the twentieth century. Players would most likely use a descant horn half the length as a standard F horn, thus the high C above the treble stave would be the 8th harmonic rather than the 12th of the B! horn, or the 16th harmonic of the long F horn. The use of the descant horn may not make the pitch easier to produce but will increase the likelihood of pitching it accurately.

Larry Polansky’s Horn has an extensive use of harmonics: through to the 13th on the B! horn, 17th on the F horn and 20th on the D horn (1992). Polansky shifted some of the pitches an octave lower they naturally occur because they would be too high in the horn range. For instance the B! horn series harmonics 1 to 13 are played as they naturally occur and the next four pitches are dropped by one octave: the 14th harmonic is played as the 7th harmonic; the 15th harmonic can be played as the 10th harmonic of the F horn; the 16th harmonic is played as the 8th harmonic of the B! horn and the 17th harmonic as the 9th harmonic of the A horn.

All the D harmonic series pitches can be played on the D horn, though harmonic series numbers 1 to 10 are doubled because the tube length of the instrument is twice as long as the writing demands. These ten harmonics are played as harmonics 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 then harmonics 11 to 17 are played in the octave they naturally occur. All the F series pitches, harmonics 1

50 to 17, are played as they naturally occur. The composition can be played unaccompanied or with electronic tape.

A detailed study of just intonation through to the 23rd harmonic has been by undertaken by Marc Sabat and Robin Hayward (2006). The valve slides are tuned in just intonation relations to each other to provide a large number of ratio intervals. Robin Hayward on tuba and Dianna Gaetjens on horn tested their microtonal capacity with excellent results (2005) and have also performed

Hayward’s Monophones (2006), Sabat’s Wonderful Scatter (2005), and Hairy,

Hippy, Happy (2010), the latter with Daniel Ploeger on trombone. The more recent studio recording of Hairy, Hippy, Happy has different instrumentalists

(Sabat 2010/2006). In these compositions and the Sabat/Hayward study on a

23-limit harmonic space the musicians are not expected to produce 23rd harmonics as they occur in the harmonic series. Rather, they produce the musical intervals based on the harmonic series relations.

The highest harmonics required from a brass instrument that I know of are in my work Chemchok Heruka for horn and cello (Dixon 2003). All the harmonics between the fundamental and the 29th of the long C horn are called for in the

first movement. This fundamental is the lowest pitch written for horn that I am aware of. It is one octave below the cello C string and very difficult to produce.

For the CD recording I played the note a number of times for later editing

51 (Locana 2003). Additionally, I was using a mouthpiece with a larger inner diameter than in recent years, which made it easier to play low in the range. I did not attempt the low C in a recent recital.40 Example 8 illustrates these harmonics. Dynamics are omitted here, leaving space for the harmonic series numbers to be clearly seen below each pitch.

52 for Matthew Chemchok Heruka

I Vajrasattva Michael Hugh Dixon Intently focussed q = 20; Tonic = C 2003 Use C harmonics throughout this movement œ Horn in C ? 2 œbœnœœµœ#œœ#œ B 4 œœ & œœBœbœnœœ#œ Œ mp-mf œ bœœœœµ (harmonics) Harmonics: 1œ œ œ œ 26

3 œ œ#œœ#œBœ ∑ œ ? Œ œ bœnœœ#œœbœnœ µ & œœ & œœµœœBœ p mp-mfœ œ œ œ œbœœ 20 2 œ 1

5:4 7 5:4 tightly focussed œ#œBœnœbœbœ. nœ œ. œ#œBœnœbœbœ Œ Œ ‰ œbœ µ Œ ‰ & ˙ µœ 12 11 18 19 24 25 26 27 28 29 24 22 24 25 26 27 28 29

12 opt. 8va ------3 3 3 3 mf œ œ œ. bœ œ µœ.#œ µœ ? œbœ œ œ œ œ œ œ & œ œ œ Œ œ œ & 15 12 18 19 22 23 22 1œ 5 6 7 9

16 3 3 3 3 3 3 3 3 3

& œµœœ œœœ œµœœBœœœœœœœ œ œ œœµœœœœœœœ œœœœœœœœœœœœœœœœœœ œbœ 11 œœœ œ 13 9 9 9 round tone, 18 3 hoo attack f ^ ^ ^ ? œ bœ nœ µœ œ & ≈ bœ nœ bœ j Ù & Œ -œ œ 14 15 14 3 18 19 20 22

Example 8. Horn harmonics 1-31, bars 1 to 19, of Chemchok Heruka. Harmonic series numbers are indicated below each note.

53 Summary

Labrosones throughout the world provide a rich resource for the composer of just intonation. Traditions in Africa and Russia have made good use of natural instruments that only play one harmonic. Players of the clarin and trutruca in

South America have a strong tradition of using the natural 7th harmonic and could produce the 11th. In Europe players of the alphorn and cor de chasse readily produce harmonics up to the 16th including prime numbered 13th harmonic. All these natural instruments have the potential to be used in a way that extends their traditions. Techniques such as lipping, hand adjustments in the bell and modifications such as fingerholes have extended the pitch and tuning resources of those, or similar instruments.

Artisans in Europe adapted instruments from the late fourteenth century devising ways to enable greater length, add slides, crooks, keys and valves.

Now valved brass and the trombone are prevalent throughout the world and players can apply techniques developed over millennia. Technical expertise combined with the inclusion of many natural harmonics opens up new possibilities for performing contemporary music in extended just intonation.

54 Chapter 2! Labrosone harmonics: composing in extended just ! intonation

A number of musical scenarios in three just intonation prime-limits, 7, 11 and 13 demonstrate the compositional uses of labrosones. Reference will be made to natural instruments and those modified by finger holes, slides or valves. Of most interest to this study are instruments that can provide one or more of the lesser known 7th, 11th and 13th harmonics. Players can gain direct experiential knowledge of intervals associated with these harmonics as they occur naturally on the instrument. Players can then apply this knowledge in other just intonation contexts. Modified natural instruments are those with valves or slides such as the horn, trumpet and trombone as well as finger hole labrosones.

Valves were applied to natural instruments in the early nineteenth century and used by many as a quick means of changing crooks, tube lengths (Evans 1997:

208). Standard brass instruments can still be viewed in this way, even the trombone with each slide position providing a different tube length. The trombone began its history playing simple diatonic bass lines (Polk 1997: 47) and in the Renaissance was often used in consort with cornetts (Herbert 1997:

76). Some brass players learn to play the cornett, even though the technique of using finger holes is significantly different to using valves, as a way to engage with historically based performance of Renaissance music.

55 Six musical scenarios provide the framework for this chapter. The first scenario is in 7-limit just intonation. The second, third and fourth scenarios are all in 11- limit just intonation. The fifth and sixth are in 13-limit just intonation. Scenarios

One to Five are scored for natural labrosones: bugles, Russian horns, animal horns, trutruca and alphorns respectively. Scenario Six is scored for the finger hole labrosones, rozhok. The scenarios are summarised in Table 2.

Table 2. Summary of the instruments, harmonics used and the tuning limit applied to each musical scenario.

Extended just Natural Scenario Instruments intonation limit harmonics

1 Bugles 7 2,3,4,5,6,7

2 Russian horns 11 1

3 Animal horns 11 2,3

4 Trutruca 11 6,8,9,11,23

5 Alphorns 13 1 to 18

6 Rozhok 13 1

In this chapter there are no specific scenarios for slide or valve instruments. The discussion shows how they can be used in place of any natural labrosone as each slide and valves instrument is, in essence, a collection of natural instruments within the one body. For example a standard double horn is an aggregate of several hunting horns. The chapter concludes with some observations on the use of standard brass for composition in extended just intonation.

56 2.1! 7-limit just intonation: natural labrosones

Some music in 7-limit just intonation can be played simply using available harmonics. Many of the instruments discussed in Chapter One have an accessible 7th harmonic including all those featured in 1.4 (7-limit), 1.5 (11-limit) and 1.6 (17-limit and beyond).

Scenario One

The musical scenario in this category is scored for three bugles. Any competent bugle player can produce a 7th harmonic. Trumpet players are often called on to play the bugle and for many the harmonic above, the 8th, is within their normal compass. This musical scenario requires two purpose-built bugles of specific lengths. The first bugle part is for a standard B! instrument, however the second bugle needs to be tuned a wide whole tone lower, to a low A!. This note is an

8:7 ratio below B!, the relation between the 8th and 7th harmonics. In cent values the interval between the two notes is 231 cents rather than the 200 cents for an equal-tempered whole tone. The third bugle needs to be tuned a further 8:7 ratio lower to a very low G!, notated in the score as G". The interval between bugles one and three is a 64:49 ratio, the lower note with a cent value of 462 cents from B! rather than 400 cents for an equal-tempered major third. This also means that the second note for bugle three is the ratio 80:49 therefore a different

57 B! to that for bugle one. The cent value for the bugle three B! is minus 76, or

1124, compared with the bugle one pitch that has a cent value of 0 or 1200. All pitches for bugle three are notated as three-quarter flats. All three bugles have a range from the 2nd to 7th harmonics.

!!

Illustration 1. Bugle. Source: http://en.wikipedia.org/wiki/Image:Bugle-rhside-large.jpg !!Accessed 14 December 2010. !!Attribution: User: Kalibos.

58 SCORE IN C Scenario One score Michael H Dixon 2010 Tempo I q ! "# Tempo II q ! $# Tempo III q ! %& septuplet x ! sextuplet x sextuplet x ! quintuplet x

Bugle 1 b>œ b>œ > ° 1 œ œ œ œ œ œ œ œ œ œ œ Bb &4 bœ œ bœ bœ œ f 7:4 6:4 5:4 > > Bugle 2 1 bœ bœ œ bœ bœ œ b>œ Ab &4 bœ œ œ œ bœ œ œ bœ œ œ œ f 7:4 6:4 5:4 5:4 Bugle 3 1 bœ Bb>œ œ bœ Bb>œ œ bœ Gb &4 bœBbœB œ œ bœ Bbœ B œ bœ Bbœ B œ œ f B B B ¢ 7:4 6:4 >

Tempo IV q ! '#( Tempo V q ! '&# quintuplet x ! x x ! triplet e 4 3:2 > b>œ 1 ° œ œ œ œ bœ bœ œ œ 3 & bœ bœ bœ œ 4 b˙ Œ ∑ 3:2 3:2 > 3:2 2 b>œ bœ bœ bœ & bœ œ œ bœ œ œ bœ œ 43 b˙ Œ ∑ 3:2 3:2 bœ > 3:2 3 b>œ bœ Bbœ bœ 3 & Bbœ Bbœ B œ Bbœ Bbœ B B Bbœ Bbœ Bbœ 4Bb˙ Œ ∑ ¢ 3:2 3:2 Bbœ

Verse One 10 expressively 1 ° & b˙ Œ Œ bœ™ œ Œ bœ Œ Œ ∑ Œ bœ œ ‰ b˙ Œ mp b˙ ™ b˙ J 2 & ∑ ∑ ∑ ∑ ∑ b˙ Œ bœ Œ Œ ∑ mp

3 b˙ & ∑ ∑ ∑ ∑ Œ B ∑ ∑ ∑ ¢ mp

18 1 ° œ™ œ™ ˙ & ˙ Œ bœ™ bœ™ Œ™ bœ™ Œ™ bœ™ Œ b˙ Œ ∑

2 & ∑ ∑ Œ™ bœ™ ∑ ∑ ∑ ∑ b˙ Œ

3 ∑ ∑ ∑ Œ™ Bbœ™ ∑ ∑ ∑ ∑ ¢&

59 26 1 ° & bœ™ Œ™ ∑ ∑ ∑ Œ b˙ ˙™ bœ Œ Œ ∑ 4

2 4 & Œ™ bœ™ ∑ Ó bœ ˙ Œ bœ Œ Œ ∑ Œ b˙ ∑ 4

3 ∑ Bbœ œ Œ ∑ ∑ ∑ ∑ Bb˙™ 4 ¢& Bb˙ Bbœ 4

Tempo IV q ! "#$ 34 bœ œ 1 ° œ œ œ œ œ œ œ & 4b˙ Ó b˙ œ bœ Œ Ó 43 3:2 3:2 3:2

2 & 4 Ó b˙ ∑ ∑ Œ bœ œ Œ 43

3 4 ∑ ∑ ∑ Ó Œ 3 ¢& 4 Bbœ 4

Tempo III q ! %& 38 5:4 1 ° œ & 43 ∑ bœ œ œ œ œ 4bœ œ œ œ œ Œ ∑ 43 p mp 5:4 j j 2 & 43bœ œ ‰ ‰ œ ∑ 4 ∑ ‰ ‰ ‰ bœ œ œ œ 43 p mp 5:4 3 bœ & 43 Œ ‰ bœ ‰ ∑ 4 Ó™ B bœ œ œ ‰ ‰ Ó 43 ¢ Bp mp B

Tempo II q ! '# 42 6:4 b>œ 6:4 bœ 1 ° 3 œ œ œ œ œ œ œ œ 1 & 4 Œ Œ bœ œ œ œ œ œ œ œ J ‰ 4 5:4 > > f 2 bœbœ œ & 43 Œ œ œ Œ ∑ 41 5:4 5:4 b>œ 3 3BbœBbœB œ œ Œ Œ ∑ 1 ¢& 4 4

60 Tempo I q ! "# Tempo II q ! $# Tempo III q ! %& 44 b>œ b>œ > 1 ° œ œ œ œ œ &41 bœ œ œ œ bœ œ œ bœ œ œ œ f 7:4 6:4 5:4 > > 2 bœ bœ œ bœ bœ œ b>œ &41 bœ œ œ œ bœ œ œ bœ œ œ œ f 7:4 6:4 5:4 > > 3 1 Bbœ Bbœ œ Bbœ Bbœ œ Bbœ &4Bbœ Bbœ œ œ Bbœ Bbœ œ Bbœ Bbœ œ œ ¢ f 7:4 6:4 5:4>

Tempo IV q ! '#( Tempo V q ! '&# Verse Two 47 3:2 > b>œ 1 ° œ œ œ œ bœ bœ œ œ 6 & bœ bœ bœ œ b˙ Œ ∑ ∑ 8 3:2 3:2 > 3:2 2 b>œ bœ bœ bœ & bœ œ œ bœ œ œ bœ œ b˙ Œ ∑ ∑ 68 3:2 3:2 bœ > 3:2 3 b>œ bœ Bbœ bœ b˙ 6 & Bbœ Bbœ B œ BbœBbœB B Bbœ Bbœ Bbœ Bb˙ Œ Bb˙ Œ B Œ 8 ¢ 3:2 3:2 Bbœ mf expressively

54 e = e 2 ° j j &68 ∑ ∑ ∑ ∑ ∑ Œ™ bœ œ ‰bœ Œ™ Œ™ bœ mf ™ f ™ 3 6 Bbœ™ bœ™ bœ™Bbœ™ bœ &8 bœ™ B Bbœ™ B bœ™ Œ™ bœ™ Œ™ ‰B ‰ bœ™ œ™ Œ™ ¢ B B p Bmf J B Bbœ™ œ™

62 2 ° Œ ∑ Œ™ ∑ 3 ∑ ∑ & bœ™ ™ bœ™ 4 3:2 3:2 3 bœ 3 & Œ™ bœ Bbœ B ™ Œ™ Œ™ 4 b˙™ œ œ bœ œ œ œ œ ¢ B f J bœ p Bmp B B ™ Bbœ™ œ™

68 3:2 3:2 3:2 2 ° b˙ bœ œ bœ œ œ bœ œ & œ™ Œ™ ∑ ™ J ‰ ‰ J ‰ ‰ Œ 45 mf mf 3:2 3:2 3:2 3:2 3:2 3:2 3 b˙ œ œ bœ œ Bbœ œ œ œ œ 5 & Bbœ™ bœ™ œ œ bœ œ œ œ œ B ™ B bœ œ œ œ œ 4 ¢ mf Bmp B mf B

61 Tempo III q ! "# 73 5:4 5:4 2 ° bœ œ œ œ œ & 45 Œ Œ Œ bœ ∑ 4 bœœœœœŒ Ó 45 f mp mf 5:4 5:4 3 Bbœ bœ Bbœ œ œ œ œ bœ bœ œ & 45 Œ B bœ Œ B Bb˙ bœ 4 Œ b˙ B Bbœœœ 45 ¢ f B mf B p B mp 76 Tempo IV q ! $%& Tempo II q ! '% bœ œ 1 ° & 45 ‰ Œ Ó™ 4 ∑ mf

2 bœ œ œ bœ & 45 ‰ Ó™ 4 Œ Œ œ bœ Œ mf 3:2 > > 3 5Bbœ œ œ Bbœ œ 4 BbœBbœ œ œ œ œ & 4 ‰ Œ BbœBbœ œ4BbœBbœ œ œ œ œ Œ Œ ¢ mf mp 6:4 6:4

78 > 2 ° bœ œ bœbœ œ & Œ Œ bœ œ œ œ Œ Œ bœ œ œ œ 43 3:2 3:2 6:4 > > > 3 BbœBbœ œ œ œ œ BbœBbœ œ 3 & BbœBbœ œ œ œ œ Ó Œ BbœBbœ œ œ Œ 4 ¢ 6:4 6:4 6:4

80 Tempo I q ! (% 1 ° 3 Œ Œ bœ ‰ Œ Œ bœ ‰ 4 & 4 mf J J 4 7:4 > 2 3 j > bœbœ œ 4 & 4 Œ bœ™ œ Œ Œ bœ œ œ œ Œ 4 mf 7:4 > > 3 3 bœBbœ œ bœBbœ œ 4 & 4BbœBbœB œ œ Œ Œ BbœBbœB œ œ Œ Œ 4 ¢ mf 7:4 7:4

82 b>œ 1 ° œ œ œ œ & 4 Œ bœ œ Ó f 7:4 > > 2 bœbœ œ œ œ œ & 4bœ œ œ œ Œ œ œ œ œ Œ f 7:4 7:4 > 3 bœ Bbœ œ & 4 Œ Œ Œ bœBbœB œ œ ¢ fB 7:4

62 Tempo II q ! "# 83 b>œ 1 ° œ œ & Œ bœ œ œ Ó > 6:4 > 2 bœ bœ œ œ œ œ & bœ œ œ Œ œ œ œ Œ 6:4 6:4 > 3 BbœBbœ œ & Œ Œ Œ BbœBbœ œ ¢ 6:4

Tempo I q ! $# 84 b>œ 1 ° œ œ & Œ bœ œ œ œ Ó 43 7:4 > > 2 bœ bœ œ œ œ œ & bœ œ œ œ Œ œ œ œ œ Œ 43 7:4 7:4 b>œ 3 Œ Œ Œ bœBbœB œ œ 3 ¢& BbœB œ 4 7:4 85 b>œ >œ b˙ 1 ° œ œ œ œ œ œ & 43 Œ bœ œ œ Œ 7:4 7:4 > > > 2 bœ bœ œ œ œ œ œ œ œ & 43 bœ œ œ œ œ- œ œ œ œ- œ œ œ b˙ Œ 7:4 7:4 7:4 > > 3 Bbœ œ 3 BbœBbœ œ œ œ œ œ œ Œ ∑ ¢& 4Bbœ œ œ œ 7:4 7:4 Verse Three 87 7:4 7:4 1 ° > œ œ œ œ & ‰™bœ œ Œ ∑ ‰™ bœ Œ Œ 83 J ‰ ‰ 43 f mf 7:4 7:4 > 7:4 2 j bœbœ bœ > œ œbœ bœ bœ bœ bœ Œ œ Œ bœ œ Œ Œ bœ œ œ œ B 3 ‰ ‰ 3 & f ™ 8 pJ 4

3 Bbœ & ∑ Œ Bbœ Œ Œ Bbœ œ 83 ‰Bbœ ‰ 43 ¢ ff mpJ 91 7:4 bœ œ™ 1 ° œ & 43 ‰™bœ Œ Œ Œ œ Œ Œ 83 43 ∑ f ff 7:4 7:4 7:4 7:4 2 3 j bœ œ bœ 3 œ™ 3 j & 4bœ™ Œ Œ Œ œ œ œ Œ Œ Œ œ œ ‰ Œ ‰ J 8 4bœ™ Œ f mf bœ 7:4 7:4 bœ 3 bœ Bbœ Bbœ™ & 43 Œ Bbœ Œ ‰™ B Œ Œ ‰™ ‰ J Œ Œ 83 43 ∑ ¢ f piu f

63 96 7:4 7:4 7:4 bœ 1 ° œ™ œ & ∑ ≈ ‰ ‰bœ œ ‰ ‰ ‰ Œ Œ J mp J R ff 7:4 7:4 7:4 7:4 7:4 2 r bœ > œœ œ bœ & bœ œœ Œ bœ ‰ œ œ œ œ ‰ ‰ Œ œbœ œœ J ™ ™bœ œ bœ f ff bœ mp 7:4 7:4 7:4 7:4 bœ 3 ∑ ≈ bœBbœ œ œ Bbœ bœ œœŒ œ™ œ B & Bbœ œ œB B Bbœ bœœ J ¢ mp mf B > f ff

99 b>œ >œ œ 1 ° œ œ œ œ & Œ bœ œ œ œ œ œ œ œ J ‰ f 7:4 7:4 f > > > 2 bœ bœ œ œ œ œ œ œ œ j & bœ œ œ œ œ œ œ œ œ œ œ œ œ ‰ 7:4 7:4 7:4 > 3 BbœBbœ œ & Œ Œ BbœBbœ œ œ Œ ¢ f 7:4

100 b>œ >œ œ 1 ° œ œ œ œ & Œ bœ œ œ œ œ œ œ œ J ‰ 43 7:4 7:4 > > > 2 bœ bœ œ œ œ œ œ œ œ j & bœ œ œ œ œ œ œ œ œ œ œ œ œ ‰ 43 7:4 7:4 7:4 > 3 BbœBbœ œ 3 & Œ Œ BbœBbœ œ œ Œ 4 ¢ 7:4

101 b>œ >œ >œ 1 ° œ œ œ œ œ œ & 43 bœ œ œ œ œ œ œ œ œ œ œ œ 42 b˙ 7:4 7:4 7:4 > > > 2 bœ bœ œ œ œ œ œ œ œ & 43 bœ œ œ œ œ œ œ œ œ œ œ œ 42 b˙ 7:4 7:4 7:4 > > > 3 3 bœBbœ œ œ œ œ œ œ œ 2 & 4BbœBbœB œ œ œ œ œ œ œ œ œ œ 4Bb˙ ¢ 7:4 7:4 7:4

64 2.2! 11-limit just intonation: natural labrosones

Musical scenarios in this section are written for three different instrumental consorts. Each is set in a pitch context outside the traditional use for the instruments. The number of instruments that have an accessible 11th harmonic is less than those having an accessible 7th harmonic. One of the scenarios in this category is scored for an instrument that can produce 11th harmonics.

Two scenarios demonstrate a different method of achieving 11-limit just intonation. Instruments are used that have only one or two available harmonics and each would need to be purpose-built to the length required to produce the specific pitch. The instruments are then used in the same manner as for African and Russian horn bands.

The same composition is used for each scenario as the pointillistic melody and the small pitch set make the isorhythmic organisation clear.

Scenario Two

Thirteen “Russian” horns play the nine ratio pitches with some pitches doubled at the octave. It is not necessary to see a complete score as the writing is similar throughout. The two excerpts show the extremes of the idiom, the most sparse writing in bars 1 to 9 and the most dense writing in bars 53 to 55.

65 Illustration 2. Russian Horn Capella performing in Paris Source: http://www.horncapella.com/gallery/1/199 Accessed 14 December 2010 Permission granted in an email by Sergey Peschansky, 16 December 2010.

66 SCORE IN C Scenario Two score Michael H Dixon 1 Focussed; energetic q. = 51 2010 1 ° & 98 ∑ ∑ ∑

2 & 98 ∑ ∑ ∑

3 µœ & 98 ∑ Œ™ ‰ ‰ Œ™ ∑ mp 4 & 98 ∑ ∑ ∑

5 & 98 ∑ ∑ ∑

6 & 98 Œ™ Œ™ ‰ ‰ µœ ∑ ∑ mp 7 & 98 ∑ ∑ ∑

8 & 98 Œ™ œ ‰ ‰ Œ™ ∑ ‰ ‰ œ Œ™ Œ™ mpµ µ 9 & 98 ∑ ∑ Œ™ Œ™ ‰ œ ‰ mp 10 & 98 ∑ ‰ œ ‰ Œ™ Œ™ ∑ mpµ 11 9 & 8 ∑ Œ™ Œ™ ‰ ‰ ∑ mpœ 12 9 & 8 ‰ ‰ Œ™ Œ™ ∑ ∑ mpœ 13 9 ∑ ∑ ∑ ¢& 8

67 4 5 ° œ & µ ‰ ‰ Œ™ Œ™ ∑ ∑ mp 7 & ∑ ‰ œ ‰ Œ™ Œ™ Œ™ Œ™ ‰ ‰ œ mp

8 & ∑ Œ™ Œ™ ‰ µœ ‰ ∑

9 & ∑ Œ™ œ ‰ ‰ Œ™ ∑

10 & ∑ ∑ Œ™ ‰ µœ ‰ Œ™

11 Œ™ ‰ ‰ Œ™ ∑ ∑ ¢& œ

= 7 53 ° œ µœ 15 & ‰ ‰ µ Œ™ Œ™ Œ™ Œ™ ‰ ‰ ∑ 158

6 & ∑ ∑ ‰ µœ ‰ Œ™ Œ™ 158 mp 7 & ∑ ∑ Œ™ ‰ ‰ œ Œ™ 158

8 15 & ∑ µœ ‰ ‰ Œ™ Œ™ ∑ 8

9 & Œ™ Œ™ œ ‰ ‰ ∑ ∑ 158

10 15 & ∑ ∑ Œ™ Œ™ ‰ ‰ µœ 8

11 ∑ Œ™ ‰ ‰ Œ™ ∑ 15 ¢& œ 8

68 53 œ 1 ° µ 9 &158 ‰ ‰ Œ™ Œ Œ™ Œ Œ 8

2 µœ µœ 9 &158 ‰ ‰ Œ™ ‰ J Œ™ Œ Œ 8

3 µœ µœ 9 &158 ‰ ‰ Œ™ Œ Œ™ J ‰ Œ 8

4 œ 9 &158 Œ™ Œ™ Œ Œ™ Œ J ‰ 8

5 15 œ œ 9 & 8 Œ™ Œ™ ‰ µJ Œ™ Œ µJ ‰ 8 6 15 œ ‰ Œ 9 & 8 Œ™ Œ™ Œ Œ™ µJ 8

7 9 &158 œ ‰ ‰ Œ™ Œ ‰ ‰ œ Œ Œ 8

8 15 j 9 & 8 Œ™ ‰ ‰ µœ Œ Œ™ Œ Œ 8

9 9 &158 Œ™ Œ™ Œ ‰ œ ‰ Œ Œ 8

10 15 9 & 8 Œ™ ‰ µœ ‰ Œ Œ™ Œ Œ 8 11 15 j 9 & 8 œ ‰ ‰ Œ™ ‰ œ Œ™ Œ Œ 8

12 15 9 & 8 ∑ 8

13 15 Œ™ ‰ ‰ Œ Œ™ Œ Œ 9 ¢& 8 8 œ

69 54 œ œ œ 1 ° 12 µ &98 ∑ 8 Œ™ Œ™ Œ™

2 µœ 12 µœ œ œ &98 Œ™ ‰ ‰ Œ™ 8 Œ™ Œ™ Œ™

3 µœ 12 µœ µœ œ œ &98 Œ™ ‰ ‰ Œ™ 8 Œ™ ‰ ‰ Œ™

4 œ 12 œ &98 ‰ ‰ Œ™ Œ™ 8 J ‰ ‰ Œ™ Œ™ Œ™

5 12 &98 Œ™ ‰ ‰ µœ Œ™ 8 Œ™ ‰ µœ ‰ Œ™ µœ œ œ

6 12 &98 Œ™ Œ™ µœ ‰ ‰ 8 ‰ µœ ‰ Œ™ Œ™ Œ™

7 12 &98 ‰ ‰ œ Œ™ Œ™ 8 Œ™ ‰ ‰ œ Œ™ Œ™

8 9 12 &8 ‰ µœ ‰ Œ™ Œ™ 8 Œ™ Œ™ µœ ‰ ‰ Œ™

9 12 &98 Œ™ Œ™ ‰ œ ‰ 8 ‰ ‰ œ Œ™ Œ™ œ œ œ

10 12 &98 ∑ 8 ∑

11 9 12 Œ Œ Œ &8 ∑ 8 ™ ™ ™ œ œ œ 12 9 Œ™ Œ™ ‰ ‰ 12 Œ™ Œ™ ‰ ‰ &8 œ 8 œ œ œ œ 13 9 Œ™ Œ™ ‰ ‰ 12 ∑ ¢&8 8 œ

70 Scenario Three

Most animal horns produce two notes which approximate the perfect fifth of the harmonic series. However, if there is a sufficient quantity of horns to choose from, instruments may be found that do offer true perfect fifths. Only six instruments are needed to cover all the notes in the score. The scenario assumes the instruments have a true perfect fifth between the 2nd and 3rd harmonics. It is also assumed that the fundamental cannot be produced therefore there are no low Ds in this score. A comparison with the previous scenario is served well enough with the same excerpt, the opening bars, 1 to 9 and the final three bars,

53 to 55.

!

! Illustration 3. Blowing horn. Fishergate, York, YO10 4UA UK Source: http://www.jelldragon.com/bugles.htm Permission granted in email by Rob, Jelling Dragon, York [email protected]

71 SCORE IN C Scenario Three score Michael H Dixon 1 Focussed; energetic q. = 51 2010 1 ° µœ &98 ∑ Œ™ ‰ ‰ Œ™ ∑ mp

2 &98 Œ™ Œ™ ‰ ‰ µœ ∑ ∑ mp

3 &98 Œ™ œ ‰ ‰ Œ™ ∑ ‰ ‰ œ Œ™ Œ™ mpµ µ

4 &98 ∑ ‰ œ ‰ Œ™ Œ™ ∑ mpµ

5 &98 ∑ Œ™ Œ™ ‰ ‰ ∑ mpœ

6 &98 ‰ ‰ Œ™ Œ™ ∑ Œ™ Œ™ ‰ œ ‰ ¢ mpœ mp

4 3 ° & ∑ Œ™ Œ™ ‰ µœ ‰ ∑

4 & µœ ‰ ‰ Œ™ Œ™ ∑ Œ™ ‰ œ ‰ Œ™ mp µ

5 & Œ™ ‰ ‰ Œ™ ‰ œ ‰ Œ™ Œ™ Œ™ Œ™ ‰ ‰ œ œ mp 6 ∑ Œ™ œ ‰ ‰ Œ™ ∑ ¢&

7 1 ° µœ & ∑ Œ™ Œ™ ‰ ‰ ∑ 158

2 & ∑ ∑ ‰ µœ ‰ Œ™ Œ™ 158 mp

3 15 & ∑ µœ ‰ ‰ Œ™ Œ™ ∑ 8

4 µœ 15 & ‰ ‰ Œ™ Œ™ ∑ Œ™ Œ™ ‰ ‰ µœ 8

5 15 & ∑ Œ™ ‰ œ ‰ Œ™ Œ™ ‰ ‰ œ Œ™ 8

6 Œ™ Œ™ œ ‰ ‰ ∑ ∑ 15 ¢& 8

72 53 1 ° µœ µœ 9 &158 ‰ ‰ Œ™ Œ Œ™ J ‰ Œ 8 œ œ 2 15 µ ‰ ‰ Œ ‰ µJ Œ œ 9 & 8 ™ ™ µJ Œ ‰ 8 œ 3 15µ 9 & 8 ‰ ‰ ‰ ‰ µœ Œ Œ™ Œ Œ 8

4 15 œ œ 9 & 8 Œ™ ‰ µœ ‰ ‰ µJ Œ™ Œ µJ ‰ 8 5 15 j œ Œ œ ‰ 9 & 8 œ ‰ ‰ Œ™ ‰ œ ‰ ‰ J 8 6 15 Œ™ ‰ ‰ Œ ‰ œ ‰ ‰ Œ™ 9 ¢& 8 œ 8

54 1 ° 9 µœ 12 µœ µœ œ œ &8 Œ™ ‰ ‰ Œ™ 8 Œ™ ‰ ‰ Œ™

2 9 µœ 12 µœ œ œ &8 Œ™ ‰ ‰ µœ ‰ ‰ 8 ‰ µœ ‰ Œ™ Œ™ œ œ œ 3 9 12 µ &8 ‰ µœ ‰ Œ™ Œ™ 8 Œ™ Œ™ µœ ‰ ‰

4 9 œ 12 œ µœ œ œ &8 Œ™ ‰ ‰ µ Œ™ 8 Œ™ ‰ µ ‰ Œ™

5 9 œ ‰ œ Œ Œ 12 œ ‰ ‰ ‰ ‰ œ Œ &8 ™ ™ 8 ™ œ œ œ

6 9 Œ™ Œ™ ‰ œ 12 ‰ ‰ œ Œ™ ‰ ‰ ¢&8 œ 8 œ œ œ œ

73 Scenario Four

The south American clarin and the trutruca are both usually played with a pitch range between harmonics 6 and 12. However in all recordings reviewed in

Chapter One the 11th harmonic was never played. There would be no physical reason for such an omission therefore the following score includes the 11th harmonic. Four trutruca are required, purpose-built in such lengths to form specific ratios to each other. The longest, trutruca four is pitched in C, with the next longest, trutruca three pitched a pure whole tone higher in D, ratio 9:8 to C.

Trutruca two, shorter again is a perfect fourth higher than the longest, therefore in G, the ratio 4:3 to C. The shortest instrument, trutruca one is the ratio 44:27 to the C of trutruca four. This ratio is also a pure perfect fourth from the 11th harmonic of trutruca four.

Trutruca one is notated as horn in A!. All 11th harmonics are notated as half- sharps or half-flats. The harmonics that each player needs to play are clearly shown at the far left of the score, before bar 1. The entire score is included as it could be played on natural horns or standard horns.

74 !!

!!Illustration 4. Trutruka. Source: http://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/ Accessed 14 December 2010 Attribution: User: man77 (Own work) [CC-BY-SA-3.0] (www.creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons

75 Scenario Four score

Section 1 Michael H Dixon Score in C 2010 Focussed; energetic q. = 51 Trutruca in A ° 9 œ œ half-flat &8BBœ Œ™ Œ™ ‰ ‰Bœ Œ™ ‰B ‰ Œ™ ∑ Jh9+12 mpgradually increase the intensity till the end j Trutruca in G œ &98 µœœ ∑ ∑ Œ™ Œ™ ‰ œ ‰ h6,8,9,11,12œ mp j Trutruca in D &98 œœ Œ™ œ ‰ ‰ Œ™ Œ™ Œ™ ‰ ‰ ‰ ‰ œ Œ™ Œ™ µh8,9,11,12œœ µmp gradually increase the intensityœ till the endµ Trutruca in C 9 j ‰ ‰ Œ™ Œ™ ‰ ‰ Œ™ Œ™ ∑ ¢&8 µœœ œ µœ h8,9,11,12 mp gradually increase the intensity till the end

4 G ° µœ ‰ ‰ Œ™ Œ™ ∑ ∑ & gradually increase the intensity till the end

D Œ™ ‰ ‰ Œ™ ‰ ‰ Œ™ ‰ ‰ Œ™ Œ™ ‰ ‰ & œ œ µœ œ C ∑ Œ™ ‰ ‰ Œ™ Œ™ ‰ ‰ Œ™ ¢& œ µœ

7 A half- ° Bœ flat & ∑ Œ™ Œ™ ‰ ‰ ‰ Bœ ‰ Œ™ Œ™

G & ‰ ‰ µœ Œ™ œ ‰ ‰ ∑ ∑

D ∑ ‰ ‰ ‰ ‰ Œ™ ∑ & µœ œ

C ∑ ∑ Œ™ ‰ ‰ œ ‰ ‰ ¢& µœ

Section 2 A half- ° 10 Bœ flat & Œ™ Œ™ ‰ ‰ Bœ 8 Œ™ ‰ ‰ Œ Œ Œ™ Œ™ Œ Œ

D Œ™ ‰ ‰ Œ™ 10 Œ™ Œ™ j ‰ ‰ j ‰ ‰ ‰ ‰ Œ Œ & µœ 8 œ µœ µœ œ C ‰ ‰ Œ™ Œ™ 10 ‰ ‰ Œ™ Œ Œ Œ™ Œ™ Œ j ‰ ¢& œ 8 µœ œ

76 13 G ° & µœ ‰ ‰ Œ™ Œ Œ Œ Œ Œ™ Œ™ 98 Œ™ Œ™ ‰ ‰ œ 108 j D Œ™ ‰ ‰ Œ Œ Œ œ ‰ ‰ ‰ ‰ ‰ 9 Œ™ Œ™ Œ™ 10 & œ œ µœ 8 8 C Œ™ Œ™ Œ j ‰ j ‰ Œ Œ™ Œ™ 9 Œ™ ‰ ‰ Œ™ 10 ¢& œ µœ 8 µœ 8

16 A half- ° 10 Bœ 9 flat & 8 Œ Œ Œ™ Œ™ Œ Œ Œ™ ‰ ‰ ‰Bœ ‰ Œ™ Œ Œ 8

G 10 j œ 9 & 8 œ ‰ µJ ‰ Œ™ œ ‰ ‰ Œ Œ Œ™ Œ™ Œ™ Œ™ Œ Œ 8

D 10 j j 9 & 8 Œ Œ Œ™ Œ™ µœ ‰ Œ Œ™ Œ™ Œ™ ‰ ‰ œ ‰ µœ Œ 8

C 10 Œ Œ Œ™ Œ™ Œ j ‰ ‰ ‰ Œ™ Œ™ Œ™ Œ ‰ j 9 ¢& 8 µœ œ µœ 8

Section 3 A half- ° 9 10 Bœ flat &8 Œ™ Œ™ ‰ ‰ Bœ 8 Œ™ ‰ ‰ Œ Œ Œ™ Œ™ Œ Œ

G &98 ∑ 108 Œ™ Œ™ Œ Œ Œ™ Œ™ Œ Œ

D 9 Œ™ ‰ ‰ Œ™ 10 Œ™ Œ™ j ‰ ‰ j ‰ ‰ ‰ ‰ Œ Œ &8 µœ 8 œ µœ µœ œ

C j &9 ‰ ‰ Œ™ Œ™ 10 ‰ ‰ Œ™ Œ Œ Œ™ Œ™ Œ œ ‰ ¢ 8 œ 8 µœ

22 G ° œ œ j ‰ Œ Œ 9 Œ Œ ‰ ‰ 13 & µ ‰ ‰ Œ™ Œ Œ µJ ‰ œ ™ ™ 8 ™ ™ œ 8

D Œ™ ‰ ‰ Œ Œ Œ Œ Œ™ ‰ ‰ 9 ∑ 13 & œ µœ 8 8 C ‰ ‰ Œ™ Œ j ‰ j ‰ Œ ‰ ‰ Œ™ 9 Œ™ ‰ ‰ Œ™ 13 ¢& œ œ µœ œ 8 µœ 8

77 25 A half- ° 13 œ œ 15 flat ‰ B Œ Œ™ Œ œ ‰ Œ Œ Œ Œ™ Œ ‰ B ‰ ‰ & 8 J BJ 8 G 13 j œ j œ 15 & 8 œ ‰ µJ ‰ Œ™ Œ Œ Œ Œ Œ Œ Œ œ ‰ ‰ ‰ µ 8 D 13 Œ Œ Œ™ Œ Œ j ‰ j ‰ Œ Œ j ‰ Œ Œ™ 15 & 8 œ µœ œ 8

C j &13 Œ Œ Œ™ œ ‰ Œ Œ Œ j ‰ j ‰ Œ Œ Œ™ 15 ¢ 8 œ µœ 8 27 A half- ° 15 œ 11 flat Œ œ ‰ Œ B ‰ Œ Œ Œ™ & 8 BJ J 8 G 15 Œ Œ j ‰ Œ œ ‰ Œ Œ 11 & 8 œ µJ ™ 8 D 15 j ‰ Œ Œ Œ Œ j ‰ ‰ ‰ 11 & 8 œ œ µœ 8 C &15 Œ Œ Œ Œ Œ Œ ‰ 11 ¢ 8 œ µœ 8

Section 4 A half- ° 11 Bœ 12 flat & 8 Œ Œ Œ Œ ‰ ‰Bœ Œ™ Œ™ J ‰ Œ™ ∑ 8

G 11 Œ Œ ‰ j Œ Œ ‰ ‰ Œ Œ Œ Œ œ j ‰ 12 & 8 œ ™ œ ™ ™ µJ ‰ Œ Œ™ œ 8 D 11 Œ Œ j ‰ Œ Œ™ Œ™ Œ™ Œ ‰ Œ ‰ j j ‰ Œ™ Œ 12 & 8 µœ œ µœ µœ œ 8 C 11 Œ Œ Œ™ Œ ‰ Œ™ Œ Œ™ ∑ 128 ¢& 8 œ œ œ µœ 31 A half- ° 12 Bœ 13 flat & 8 Œ Œ Œ™ Œ™ Œ Œ Œ Œ ‰ ‰ Œ™ Œ™ Œ Œ Œ Œ™ 8

G 12 œ œ j 13 & 8 µJ ‰ Œ Œ™ Œ™ Œ µJ ‰ œ ‰ Œ œ ‰ ‰ Œ™ ‰ œ ‰ Œ Œ Œ Œ™ 8 D 12 Œ j‰ ‰ ‰ Œ™ Œ Œ Œ j‰ Œ™ ‰ ‰ Œ™ Œ Œ j‰ ‰ 13 & 8 µœ œ œ µœ µœ œ œ 8 C 12 j‰ Œ Œ™ ‰ ‰ j‰ j‰ Œ Œ Œ™ Œ™ Œ™ Œ j‰ Œ Œ™ 13 ¢& 8 œ µœ œ µœ µœ 8

78 34 A half- ° 13 œ 15 œ flat ‰ B Œ Œ™ Œ œ ‰ Œ Œ Œ Œ œ ‰ Œ Œ ‰ B ‰ & 8 J BJ 8 BJ G 13 Œ œ j 15 j œ & 8 µJ ‰ Œ™ œ ‰ Œ Œ 8 Œ Œ Œ Œ Œ œ ‰ œ ‰ µ j D 13 œ ‰ Œ Œ™ Œ Œ j ‰ 15 j ‰ Œ Œ Œ j ‰ Œ Œ™ & 8 œ 8 µœ œ C &13 j ‰ Œ Œ™ Œ Œ Œ 15 Œ j ‰ j ‰ Œ Œ Œ Œ™ ¢ 8 µœ 8 œ µœ 36 A half- ° œ flat Œ œ ‰ Œ B ‰ Œ Œ Œ™ & BJ J G Œ Œ Œ Œ œ ‰ Œ Œ & µJ ™

D j j j & µœ ‰ Œ Œ Œ Œ œ ‰ µœ ‰ ‰

C j & j ‰ Œ œ ‰ Œ Œ Œ ‰ ¢ œ œ µœ

Section 5 A half- ° œ œ flat Œ™ B ‰ ‰ Œ Œ™ Œ œ ‰ Œ Œ Œ Œ B œ ‰ Œ Œ & BJ B G Œ Œ j ‰ œ j ‰ Œ j ‰ Œ j ‰ Œ Œ Œ œ & ™ ™ œ µJ ‰ ‰ œ œ œ ™ µJ ‰ D ‰ ‰ ‰ ‰ Œ Œ™ Œ Œ Œ Œ Œ Œ Œ™ j ‰ Œ & œ µœ œ C ‰ ‰ Œ™ Œ ‰ ‰ Œ Œ j ‰ j ‰ Œ j ‰ Œ™ Œ Œ ¢& œ µœ œ µœ œ 39 A half- ° Bœ 12 flat & Bœ ‰ ‰ Œ™ Œ™ Œ™ ‰ ‰ 8

G & œ ‰ ‰ µœ ‰ ‰ Œ™ Œ™ œ ‰ ‰ 128

D Œ™ ‰ ‰ ‰ Œ™ Œ™ 12 & µœ œ œ 8 C ‰ ‰ Œ™ Œ™ ‰ Œ™ 12 ¢& œ µœ œ 8

79 40 A half- ° 12 14 œ 13 flat Œ Œ Œ™ Œ™ Œ Œ™ Œ™ ‰ œB ‰ Œ Œ & 8 8 BJ J 8 G 12 œ 14 13 & 8 µJ ‰ Œ Œ™ Œ™ Œ 8 Œ™ œ ‰ œ Œ Œ Œ Œ 8 D 12 Œ j ‰ ‰ ‰ Œ™ Œ 14 Œ™ ‰ ‰ Œ Œ j ‰ Œ 13 & 8 µœ œ 8 œ µœ 8

C j j &12 œ ‰ Œ Œ™ ‰ ‰ œ ‰ 14 ‰ ‰ Œ™ Œ Œ Œ j ‰ 13 ¢ 8 µœ 8 µœ œ 8 42 A half- ° 13 œ œ 15 flat B ‰ Œ Œ™ Œ™ Œ™ ‰ B Œ Œ Œ™ œ ‰ Œ & 8 J J BJ 8 j G 13 Œ j ‰ Œ Œ ‰ ‰ ‰ œ ‰ Œ ‰ ‰ Œ Œ 15 & 8 œ ™ ™ œ œ µJ œ 8 D 13 Œ Œ ‰ ‰ Œ™ ‰ ‰ Œ Œ Œ Œ™ Œ j ‰ 15 & 8 µœ œ œ 8 C j &13 Œ Œ ‰ ‰ Ó Œ ‰ Œ Œ ‰ ‰ Œ Œ 15 ¢ 8 µœ µœ œ 8 44 A half- ° 15 œ œ flat Œ Œ ‰ œ Œ Œ™ Œ B ‰ Œ œ ‰ Œ B ‰ ‰ Œ Œ Œ & 8 BJ J BJ G 15 Œ Œ Œ Œ ‰ Œ ‰ œ j ‰ ‰ œ j ‰ Œ Œ & 8 œ œ µJ Œ Œ œ µ ‰ œ D 15 j ‰ Œ Œ Œ ‰ ‰ Œ Œ j ‰ Œ Œ Œ™ Œ j ‰ Œ & 8 µœ œ µœ µœ C 15 j & 8 Œ j ‰ ‰ Œ Œ™ Œ Œ j ‰ Œ Œ Œ™ Œ Œ ¢ œ µœ œ œ µœ Section 6 A half- ° œ œ œ flat Œ™ œB ‰ ‰ Œ œ ‰ ‰ Œ œ ‰ Œ Œ B ‰ Œ B œ ‰ Œ Œ & B B BJ J B j j j j G & Œ™ Œ™ œ ‰ µœ ‰ ‰ œ ‰ Œ Œ Œ œ ‰ Œ Œ™ œ ‰ µœ œ j D Œ™ ‰ ‰ Œ Œ™ Œ Œ Œ j ‰ Œ j ‰ Œ™ j ‰ ‰ & µœ œ µœ œ œ C ‰ Œ™ j ‰ ‰ ‰ Œ Œ j ‰ j ‰ Œ j ‰ Œ™ Œ Œ ¢& œ œ œ µœ œ µœ œ

80 48 A half- ° œ œ flat œ ‰ ‰ Œ™ Œ™ œ ‰ ‰ ‰ ‰B œ ‰ Œ Œ œ ‰ ‰ ‰ ‰B ‰ Œ & B BJ BJ BJ G ‰ ‰ µœ ‰ ‰ ‰ ‰ µœ ‰ ‰ ‰ ‰ µœ ‰ Œ µœ ‰ Œ ‰ œ ‰ ‰ ‰ j & œ œ œ J J œ œ œ D Œ™ ‰ œ ‰ Œ™ Œ™ Œ Œ j ‰ j ‰ j ‰ Œ™ Œ & œµœ µœ µœ µœ œ C & ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰ œ ‰ Œ Œ Œ ‰ ‰ ‰ ‰ ¢ œ µœ œ µœ œ œ œ œ µœ 50 A half- ° œ œ flat œ ‰ ‰ Œ™ Œ œ ‰B ‰ ‰ Œ B ‰ ‰ Œ Œ Œ Œ œ ‰ Œ & BJ BJ BJ G œ ‰ ‰ ‰ Œ Œ ‰ œ ‰ Œ œ ‰ ‰ j ‰ œ ‰ Œ Œ Œ œ ‰ & µJ œ œ µ µ œ µJ J j D Œ™ ‰ ‰ Œ Œ ‰ ‰ Œ Œ™ Œ Œ Œ j ‰ Œ ‰ & œ µœ µœ œ C j j & ‰ ‰ Œ™ Œ ‰ Œ™ ‰ j ‰ ‰ Œ ‰ Œ Œ Œ ¢ µœ œµœ œ µœ œ œ µœ 52 A half- ° œ œ œ œ œ 9 flat B ‰B ‰ Œ™ Œ Œ œ ‰ Œ B ‰ ‰ Œ™ Œ™ Œ B ‰B ‰ & J J BJ J J 8

G j j j µœ ‰ ‰µœ Œ™ ‰ ‰ œ ‰ œ ‰ œ‰ ‰ Œ™ ‰ µœ ‰ œ œ ‰µœœ‰ 9 & J J œ œ J J œ œ 8 D j‰ Œ ‰ ‰ j ‰ j ‰ Œ Œ ‰ ‰ ‰ ‰ ‰ ‰ Œ Œ Œ 9 & œ µœ œ µœ µœ µœ œµœ 8 C j‰ ‰ ‰ ‰ ‰ j ‰ j ‰ Œ j‰ ‰ ‰ ‰ ‰ ‰ Œ Œ Œ 9 ¢& µœ µœ œ œ µœ µœ µœ µœ 8 œ 54 A half- ° 9 Bœ 12 Bœ Bœ œ œ flat &8 Œ™ ‰ ‰ Bœ ‰ ‰ 8 ‰ Bœ ‰ ‰ ‰ Œ™

G œ œ &98 ‰ ‰ ‰ ‰ µœ ‰ œ ‰ 128 ‰ œ ‰ µœ ‰ Œ™ µœ œ œ

D 9 ‰ œ Œ™ Œ™ 12 Œ™ ‰ ‰ œ ‰ ‰ &8 µœ 8 µœ œµœ œ œ œ œ C &9 Œ™ ‰ ‰ ‰ ‰ j 12 Œ™ Œ™ ‰ ‰ œ œ œ ¢ 8 µœ œ 8 œ œ œ œ

81 2.3! 13-limit just intonation: natural labrosones

There are few labrosones with available 13th harmonics.

Scenario Five uses a consort of alphorns. These instruments have an available

13th harmonic though most players rarely use it, as detailed in Chapter 1, pages

43-44.

!!

! Illustration 5. Alphorn. Source: Arkady Shilkloper. ! Permission granted by Arkady Shilkloper in an email 16 December 2010.

82 Scenario Five

Alphorns have a large range of available harmonics. The 2nd to 12th harmonics are the most used in traditional contexts though competent players can produce the 1st and up to the 16th. Alphorns are also made in a variety of lengths therefore the instruments could be used in a similar way to the trutruca in

Scenario Four yet with more extensive use of harmonics. This score extends the use of alphorns beyond their normal use. Six players would be required, each with an instrument in a different length and key, listed here from shortest to longest: player 1 in G, player 2 in F!, player 3 in F", player 4 in E, player 5 in E#, player 6 in C. Players 4, 5, and 6 would need to produce harmonics up to the

18th; player 3 up to the 15th; player 2 up to the 14th and player 1 up to the 13th.

Some of the ratios in the score could be achieved by lipping the harmonics by small amounts. Player 1 would need to lip up the 8th harmonic by 7 cents at bar

20 to enable an interval of ratio 10:2 to the E# in the part for player 5. It would be best if the instrument built for player 6 was a length to produce the note C as

508 cents relative to the G of Alphorn 1. The entire phrase at bars 25 to 28 is tuned higher than the phrase at bar 12 to 15. It is easier to lip down than up.

The phrase at bars 12 to 15 could be lipped 10 cents lower than the harmonics expected from the instrument’s length.

83 Table 3 shows the tuning relations between instruments: the note name for each instrument; the ratio to the tonic G 1:1; the amount in cents each instrument’s fundamental is from G and the nearest equal tempered cent value for comparison.

Table 3. Alphorn pitch specifications for Scenario Five.

Pitch ratio Alphorn Note name from G Cents from G 12tet cents 1 G 1:1 0 (1200) 0

2 F 13:7 1072 1100

3 F! 26:15 952 1000

4 E 5:3 884 900

5 E" 45:28 821 800

169:126 508 6 C 4:3 498 500

Dynamics in the score for Scenario Five are omitted here so that the harmonic series numbers can be clearly seen below each pitch.

84 Scenario Five score Score in C Michael H Dixon 2010 Section 1 Rubato with some feeling of spaciousness; low notes never rushed. Cents: 204 702 0 841 Alphorn 1 8 ? U 6 œ ?U 7 œ œ 8 in G ° 8 œ 8 œ B 8 œ B ™ ™ 8 & œ œ™ & œ œ™ & J Harmonics: 9 3 2 1 œ™ 9 12 13 2 12 13 Alphorn 2 8 6 7 8 in F# & 8 ∑ 8 ∑ 8 ∑ 8

Cents: 952 Alphorn 3 8 6 7 ? 8 in F half-flat & 8 ∑ 8 ∑ 8 Œ™ Bœ Bœ™ & 8 Harmonics: J8 2

Alphorn 4 8 6 7 8 in E & 8 ∑ 8 ∑ 8 ∑ 8

Alphorn 5 8 6 7 8 in Eb & 8 ∑ 8 ∑ 8 ∑ 8

Alphorn 6 8 6 7 8 in C 8 ∑ 8 ∑ 8 ∑ 8 ¢&

4 702 0 1 in G ° 8 œ œ œ 18 ˙ ˙ ˙ œ œ 8 & 8 ™ ™ 8 œ Œ 8 12 8 Cents: 1072 574 258 841 8 18 j 8 2 in F# 8 ? 8 ? ? #œ Bœ 8 & Œ™ #œ & j &µœ & #œ µœ J ‰ Œ Œ Harmonics: 8 #œ2 ™ #œ6 #œ2 ™ 10 #œ2 ™ #œ6 8 10 12 14

3 in F 8 18 8 half-flat & 8Bœ Œ Œ Œ™ 8 ∑ 8 ¢ 8J

6 Cents: 884 386 70 1088 155 773 4 in E ° 8 ? ? 7 œ ? 6 #œ ? 10 & 8 j œ™ &#œ œ œ & 8 #œ œ B œ™ & 8 œ œ™ & 8 Harmonics: œ8 3 10 12 3 #œ9 10 12 14 3 12 15 #œ9 3

Cents: 884 7 6 10 6 in C ? 8 ˙ ˙ 8 ˙ œ™ 8 ˙™ 8 Harmonics:¢ 5

9 œ 10 #œ 8 B 12 4 in E ° 8 œ#œ œ œ ? 8 j ?#œ™ J #œ™ 8 œ#œ œ#œ ? & #œ#œ œ™& #œ & œ#œ#œ Œ & 9 10 12 15 18 15 12 3 9 5 7 5 8 9 10 12 15 16 12 œ2 ™

10 8 6 in C ? 8 œ™ œ œ œ™ 8 ˙ ˙ 128 ˙ œ™ œ™ Œ ¢

85 12 1 in G ° 8 6 7 8 9 & 8 8 8 8 ‰ 8 4œ œ™ œ™ œ™ œ™ ˙ œ™ ˙ œ™ 702 œ 0 498 884 267 1049 139 6 in C 8 œ 6 ? 7 œ Bbœ j 8 9 ? 8 & 8 8 œ & 8 œ ‰ 8 ¢ œ œ œ J œ™ œ µœ œ µœ œB 9 6 œ2 ™ œ™ 10 9 10 œ2 ™ 6 7 4 3 Bbœ7 9 11 9 11 12 13 lip the entire phrase down by 10 cents 702 204 16 841 702 1 in G ° 9 6 œ œ 8 ˙ œ 6 œ ? & 8 ∑ 8B ™ ™ 8 Œ 8 œ ‰ œ™ & 13 12 12 9 3 952 454 139 841 3 in F 9 ? ? 6 ? 8 œ 6 half-flat & 8 ‰ & & 8BœBœ & 8BœBœB Ó 8 ∑ Bœ2 ™ œ6 5 Bœ2 ™ 10 12 œ6 Bœ2 ™ 10 12 15 B Bœ B 590 9 6 8 6 5 in Eb & 8 ∑ 8 ∑ 8 ∑ 8 ‰ ‰ #œ Œ™ 14J

6 in C 9 6 8 6 & 8B˙ œ œ™ 8 ∑ 8 ∑ 8 ∑ ¢ 13

20 7 841 8 6 7 1 in G ° 8 8 8 Bœ ˙ 8 & Œ™ œ™ ˙™ w ˙™ ‰ lip up 8 13 normal lip setting 139 1157 952 3 in F 8 6 7 j 8 half-flat & ∑ ∑ 8 ∑ 8 ∑ 8 ‰ ‰Bœ œ œ œ 8 10 B9 Bœ8 9 10 323 7 821 1025 8 6 œ 7 œ 8 5 in Eb #œ ? 8 j ? j ? 8 bœ 8 ‰ ‰ Ó 8 & bœ œ bœ™ & œ & œ bœ&‰ bœ œ œbœ J ¢ 14 12 10 bœ2 ™ 3 bœ2 ™ 9 bœ2 ™ 9 3 8 9 10 12 14 18

25 712

2 in F# ° 8 5 œ œ 10 ˙ ˙ 7 & 8 ∑ 8 ™ 8 ™ 8 13 277 10 895 508 712

6 in C 8 ? ? 5 10 ? 7 & 8bœ œ & 8 Œ & 8 œ œ & 8 ¢ œ œ œ œ œ œ œ ™ ™ 14 12 10 œ2 ™ 8 9 œ2 ™ 8 10 9 10 4

28 841 702 0 1 in G ° 7 5 6 œ œ 10 œ œ œ ? & 8 ∑ 8 ∑ 8 ‰ B ™ 8 Œ Œ 13 12 1 574 1072 841 œ™ 7 5 6 2 in F# 8 œ™ ˙ 8 ? 8 Bœ ? 108 Bœ & #œ#œ & #œ #œ & #œ ‰ Œ™ Œ Œ 12 8 #œ2 ™ 12 14 8 #œ2 ™ 12 14

7 5 6 6 in C 8 bœ œ œ 8 ∑ 8 ∑ 108 ∑ & œ œ ¢ œ8 œ9 10 12 14 16 18

86 2.4! 13-limit just intonation: finger hole labrosones

Knowledge and experience with finger hole labrosones has increased over the last fifty years or so as some players of standard brass instruments have developed further skills in historical performance practice of European works.

Scenario Six

Scenario Six uses a consort of finger hole horns with each instrument having only the 1st harmonic available and therefore covering less than an octave in range. Each of the five rozhok composed for in this scenario would require precise placement of the finger holes to achieve the desired tuning. Although the technique of lipping is familiar to players of the rozhok, this is not recommended to achieve the tuning. Lipping reduces optimal resonance of the vibration within the instrument. Lipping also requires more aural astuteness from the performer to know how far to adjust the pitch.

In this score only instruments one and two have the same length. Example 9 shows the ratios for each part above the assigned pitches. The note C is the ratio

1:1 and all other pitches are in strict relation to it.

Each instrument has a range of less than an octave. The top note for each is played with all holes open, the lowest note with all holes closed. The entire composition is included to demonstrate the allocation of pitches throughout.

87 !!Example 9 .! Pitch ratio allocation for each rozhok.

! ! Illustration 6. Rozhok. Source: Mizynec, V 1987, Folk Instruments of the Ukraine, Bayda Books, Australia. Permission granted in an email 9 February 2011.

88 Scenario Six score Steady Michael H Dixon e = 91 2010 œ 1 ° 8 Œ ‰ J Œ Œ 13 ∑ 12 ∑ 13 &8 16 13:8 16 16 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ 2 J J J B 12 13 &8 Œ ‰ ‰ ‰ 1613 16 ∑ 16

3 12 œ™ œ 13 &8 ∑ 1613 ∑ 16 ‰ J ‰ 16

4 8 ‰ œ Œ Œ œ ‰ 13 ∑ 12 Œ j ‰ j 13 &8 J J 16 16 ™ œ œ 16 5 8 œ ‰ Œ ‰ œ Œ 13 ∑ 12 ∑ 13 ¢&8 J J 16 16 16 4 8:7 13:7 Bœ œ œ Bœ œ Bœ 1 ° 13 6 œ™ 5 œ™ 4 œ™ 5 &16 ∑ ∑ 16 µJ 16 µJ 16 µ 16 16:13 2 13 œœœœœœœœœœœœœ 6 5 4 5 &16 ∑ B 16 ∑ 16 ∑ 16 ∑ 16 39:32 3 13Bœœœœœœœœœœœœœ ∑ 6 ∑ 5 ∑ 4 ∑ 5 ¢&16 16 16 16 16 9 Bœ œ Bœ œ œ Bœ œ 1 ° 5 œ™ 6 œ™ 5 œ™ 7 5 8 4 &16 µJ 16 µJ 16 µJ 16 ∑ 16 ∑ 8 ∑ 16 ˙ œ œ œ œ œ œ œ œ 2 5 6 5 7 5 8 4 &16 ∑ 16 ∑ 16 ∑ 16 ∑ 16 ∑ 8 16

3 5 6 5 7 5 8 œ œ 4 &16 ∑ 16 ∑ 16 ∑ 16 ∑ 16 ∑ 8 Œ Œ J ‰ 16 4:3 13:12 4 œ™ œ œ™ œ œ™œ 5 J J 6 Bœœœœ œ œ 5 J J 7 Bœœœœœœœ 5 J J 8 Ó ‰ j ‰ j 4 ¢&16 16 16 16 16 8 œ œ 16 15 16:9 13:9 bœ bœ œ bœ œ bœ œ bœ bœ 2 ° 4 Bœ œ 5 Bœ 6 Bœ œ4 Bœœ œœ 5 Bœœœ 6 Bœ 9 &16 J 16 J 16 8 16 16 ‰™ 16 7:4 14:13 3 4 bœ™ Bœ 5 bœ™ Bœœ 6 bœ™ œ™ 4 Bœœœœ bœ™ œ 5 bœ™ Bœœ 6 Bœ œœ bœ™ 9 ¢&16B 16B J 16B J J 8 B 16B J 16 B J 16 21 8:5 13:10 bœ bœ bœ bœ œ 1 ° 9 Bœ œ œ œ œ 10 Bœ œ œ Bœ œ œ 7 Bœ œ 4 &16 ∑ 16 ≈ R ≈ R ≈ R ‰™ 8 Œ 4 51 20:13 5 9 ‰ ‰™ 10 Œ r≈ r≈ r≈ ‰™ 7 Œ 4 ¢&16 œ µœ œ œ œ œ œ 16 µœ œ œ œ µœ œ œ 8 œ µœ œ œ 4

89 26 8:7 œ œ œ œ œ 1 ° 4 4 6 J J 4 J 8 J 4 J œ™ 7 &4 ∑ 8 ∑ 16 ≈ ≈ 8 Œ ‰ 16 ‰™ ‰™ 8 ‰ ≈µJ 16 7:4 14:13 œ 3 4 œ œ 4 6 4 œ 8 œ 4 œ bœ Bœ 7 &4 Œ 8 ∑ 16 ∑ 8 J ‰ Œ 16 J ≈‰ ‰™ 8 ‰ JB ™ 16

4 6 4 8 4 j 7 &4 Œ œ œ œ 48 ∑ 16 ∑ 8 ∑ 16 ∑ 8 ‰ œ œ ≈‰ 16 9:8 18:13 R 5 4 Ó œ œ 4 œ œ 6 œ œ 4 œ œ 8 œ œ 4 œ ‰ Œ 7 ¢&4 8 µœœ œœ16 µœ œ 8 µœœ œœ16 µœJ œœœ 8 J 16 32 13:7 13:7 Bœ œ œ Bœ œ Bœ œ œ 1 ° 7 5 6 7 9 5 &16 Œ 16 µœ™ ∑ 16 ∑ 16 ∑ 16 ‰™ ‰™ 16 J 16:9 13:9 bœ bœ œ bœ œ 2 7 5 6 Bœ œ œ 7 Bœ œœ 9 Bœ œ 5 &16 ∑ 16 ∑ J ‰™ 16 ≈ R ≈R ≈R 16 16 ‰™ 16

3 7 ∑ 5 ∑ ‰ bœ™ 6 ∑ 7 ∑ 9 bœ™ ‰™ ‰™ 5 &16 16 B J 16 16 16B J 16 13:12 4:3 4 œ™ œ œ™ œ 7 Bœœœœœ œ œ 5 J J ∑ 6 Bœ œœœœœ 7 J J ‰ 9 ‰™ Bœœœœ œ œ 5 ¢&16 16 16 16 16 16 38 8:7 Bœ Bœ œ œ œ œ Bœ 1 ° 5 œ 7 œ™ 3 œ™ 7 œ™ R œ™ 6 5 &16 µ ‰ 16 µJ Œ 4 Œ µ 16 µJ J 16 ∑ 8 bœ œ œ bœ 2 5 Bœœ 7 Bœ 3 Bœ 7 6 5 &16 ‰™ 16 ‰™ 4 Œ Œ 16 ∑ 16 ∑ 8 ‰™ 39:32 3 5 ∑ 7 ‰™ bœ™ ≈ 3 bœ™ ≈œ™ ≈ œ™ ≈ 7 Bœœœœœœœ 6 Bœœœœœœ 5 &16 16 B J 4B J J J 16 16 8

4 œ™ œ 5 J J 7 Bœœœ‰™ œ 3 ‰™ Bœ‰™ œ Œ 7 ∑ 6 ∑ 5 ¢&16 16 R 4 R R 16 16 8 43 1 ° 5 5 8 8 13 7 &8 ∑ 16 ∑ 16 ∑ 8 ‰ œŒ Œ œ‰ 16 ∑ 16 16:13 J J 13:8 œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ 2 5 Bœœ 5 8 8 J J J 13B 7 &8 Œ™ ‰ 16 ∑ 16 ∑ 8 Œ ‰ ‰ ‰ 16 16 14:13 3 5 œ™ 5 8 œœ8 13 7 &8 Œ 16 ∑ 16 ‰™ ‰™ B 8 Œ ‰ œÓ 16 ∑ 16 4:3 13:12 J 4 5 5 œ™œ 8 8 13 7 &8 ∑ 16 J J 16Bœœœœœœ‰ 8 ∑ 16 ∑ 16 18:13 5 8 13 7 5 Œ™ ‰ 165 ∑ 168 ∑ 8 œ‰ Œ ‰ œŒ 16 ∑ 16 ¢&8 µœœ J J

90 13:7 8:7 8:5 48 13:10 Bœ œ œ bœ œœœœœ bœ œ 1 ° 7 ∑ 13 ∑ 10 µœ™ 7 ‰ B 9 ≈ J J Œ &16 16:9 16 16 J 16 16 bœ 2 &167 ‰™ 1613 ∑ 1016 ∑ 167 ∑ 169 ∑ 7:4 3:2 œ 3 7 bœ™ Œ 13 ∑ 10 ∑ 7 ∑ 9 ‰™ ‰ &16B J 16 16 16 16 24:13 13:12 4:3 3:2 4 œ™ œ 7 ∑ 13 œœœœœœœœœœœœœ 10 ∑ 7 Bœœœœœœœ 9 J J ¢&16 16B 16 16 16 œ

53 œ 1 ° & ∑ 1611 ‰™ Œ ‰ J 1016 ∑ 167 ∑ 42 ∑ 1613 16:13

2 œœ œ œ œœ œ œ œœ œ œ œ & ∑ 1611 ‰™ œ Œ 1016B 167 B Œ 42 ∑ 1613 14:13 7:4 œ œ 3 & Bœ œœ ‰™ œ œ œ 1611 Bbœ™ Œ J ‰ 1016 ∑ 167 ≈Bœ œ Œ 42 Œ 1613 24:13 J 24:13 œ 4 & Bœ œœ œ œ œœ œ œ 1611 ∑ 1016 ∑ 167 ‰™ Bœœ œ œ 42 œ 1613 20:13 18:13 20:13 5 ‰™ ‰™ 11 ‰™ œ Œ 10 ‰™ ‰ 7 ∑ 2 ∑ 13 ¢& µœ œ œ 16 16 µœ œ µœ œœ 16 4 16

58 13:7 13:10 8:7 Bœ œ Bœ œ œ Bœ 1 ° 13 Bœœœ 2 7 6 œ™ R 5 3 &16 ‰™ ‰ ‰ ≈ 4 ∑ 16 Œ 16 µJ ≈ ≈ 16 ∑ 4 13:9 16:9 œœ œ œ œ œ œ 2 13 ‰™ B ‰ ‰™ ‰™ 2 Œ™ bœ 7 B bœ ‰™ 6 ‰™ ≈B 5 bœB œ 3 &16 4 J 16 16 16 J 4 3 13 ∑ 2 ∑ 7 Œ Bœ bœ 6 œ≈≈Bœ œ bœ 5 œ ‰™ 3 &16 4 16 B 16 R B 16 J 4 13:12 œ 4 &1613Bœœœœœœœœœœœ œ œ 42 Œ 167 ∑ 166 ∑ 165 ∑ 43 5:4 9:8 18:13 5 13 ∑ 2 Œ 7 œ ‰™ 6 ∑ 5 ‰ œ 3 ¢&16 4 œ 16 µœœ 16 16 µœ4

91 63 Bœ œ Bœ œ 1 ° Bœœ œ œ Bœœœ 6 œ 4 5 &43 ≈ ≈ µ ™ ™ 169 ≈ ‰ ≈ 4 Œ Œ µ Œ Œ Œ 4 ∑ 4

2 Bœ œ Bœ œ 6 4 5 &43 R ≈≈R Œ Œ 169 ≈ ≈ bœ R ≈‰ 4 Œ œ Œ œ bœ œ 4 ∑ 4

3 6 4 5 &43 ∑ 169 ∑ 4 ∑ 4 Œ Œ Œ Bbœ 4

4 6 4 5 &43 ∑ 169 ∑ 4 ∑ 4 Œ Œ œ Œ 4

5 3 œ 9 ≈ ≈ 6 Ó™ Ó™ 4 œ Ó 5 ¢&4 µœœµœ œœœœ 16 µœµœ œ œ œœœ 4 4 œ 4 67 bœ bœ œ bœ w 1 ° 5 œ 3 4 œ &4 Œ Œ Œ µ 4 Œ ∑ ∑ 4 Œ Œ µ

2 5 3 4 &4 Œ Œ bœ œ Œ 4 ∑ ∑ ∑ 4bœ Œ Ó w

3 5 3 œ 4 œ w &4 œ Œ Œ Ó 4 ∑ Œ Bbœ œ Œ Œ 4 Œ Œ Bbœ

4 5 3 œ œ 4 œ &4 ∑ 4 Œ Œ Œ Œ ∑ 4 Œ Œ Œ w

5 5 ∑ 3 ∑ ∑ Œ œ 4 œ Ó w ¢&4 4 œ 4 œ

92 2.5! Brass instruments in 7, 11 and 13-limit just intonation ! composition

Specifically selected or purpose-built instruments can provide an excellent resource for composers using extended just intonation. Doing so takes most of the tuning complexities away from the performers. However part of the point of the new repertoire accompanying this thesis is to give high quality performers the opportunity to expand their aural and kinaesthetic skills.

All scenarios detailed in this chapter can be played on standard brass instruments, whether they have a slide or valves. Scenarios One to Five, scored for natural instruments, can be easily adapted using the appropriate harmonics of standard brass. Scenario Six could not be so easily adapted. A solution is given on pages 95-98. The following discussion will deal with adaptations for trumpet, horn and trombone for each of the six scenarios.

Scenario One could be played on three trumpets in B!. Player 1 would tune the instrument to standard pitch in 12tone equal tempered relation to A=440 and play the entire part with no valves. Player 2 would move the first valve slide out far enough to play an A!, on first valve, matching the first trumpet’s 7th harmonic, then play the entire part on first valve. Player 3 would move both the second and third valve slides out far enough to play a very low G!, with valves two and three, matching the second trumpet’s 7th harmonic, then play the entire

93 part with valves two and three. Three horns could play the same music an octave lower than the trumpets with the valves tuned in the same way as described above. Three trombones could also play the same music an octave lower than the bugles/trumpets. The first trombone would play the entire part in first position; the second trombone in a low third position and the third trombone in a very low fifth position or a high sixth position.

Scenario Two could be played on trumpets by adjusting the main tuning slide of each instrument to match the required ratio. However, main tuning slides may not lower the pitch far enough for the 11th harmonic ratios, in which case a valve slide could be extended in conjunction with the main slide. For example the high A! could be played as the 8th harmonic on a C trumpet with first valve.

Both the main tuning slide and the first valve slide could be extended sufficiently to lower the instrument’s B" to A!. Horns could make adjustments in the same way. A simple solution for horns would be to transpose the trumpet version down an ninth and use the same tuning slide organisation as described above. B" to A! would become A" to G!. Trombones could easily play at the same pitch as the horns. Of course the slide provides the player with an advantage: great flexibility to reproduce exact ratios, but it also demands a high level of aural awareness required to hear the desired ratios and play them accurately.

On standard valved brass the tuning could be done before performance

94 allowing the players to concentrate on accurately producing the correct harmonic. When the instruments are played with a clear, centred tone quality performers can expect the harmonics to have the correct intonation. Trombone players would need sufficient aural training or access to an accurate sound file to practice with.

The process for playing Scenario Three on standard brass is no different to that just described regarding Scenario Two.

Scenario Four could be easily adapted for four standard horns, simply allocating the trutruca parts to the horns. An adaptation for trumpets or trombones using the same harmonics, 6 to 12, as the trutruca or horns would make the tessitura too high for comfort. The process described for Scenarios

One to Three would suffice though more than four parts would be required to fulfil the pitch requirements of Scenario Four.

Scenario Five could be adapted for standard horns using the same arrangement of tube lengths (tonalities) as the alphorns. Trumpets or trombones would not be well suited to play the work. The range in each part is too wide to for them as they rarely play higher than the 10th harmonic let alone the 18th required in some parts.

Scenario Six was scored for finger hole horns with the holes in each instrument precisely placed to achieve the desired ratios. The performers would not need

95 to concentrate on tuning but simply trust that the instruments were configured well. A transcription for standard brass would involve separate issues for slide instruments to valve instruments. Trombone players would need an excellent aural sense as the instrument’s harmonics would not provide much assistance with the tuning. The provision of an accurate sound file and practice with it would be essential for a successful rendition. One possible arrangement of the ratios for is shown in Example 10. Here, each trombone part has the same ratios as each finger hole horn part in Example 9. All the pitches have been transposed down a major ninth. The ratios and cent values are shown above each pitch, with the slide positions shown in bold type underneath and each harmonic below them. Although it is the convention amongst trombone players to show slide positions above pitches, they are presented below to allow the ratios and cent values to remain above the notes as they are in Example 9 (p 88). This makes for an easier comparison between the two examples.

96 !"#$%&' 1:1 13:7 8:5 13:10 8:7 ()*#&' 0 1072 814 454 231 bœ œ bœ 1 °B Bbœ µœ !"#$%&'()#*#(+), 1 2 5 3.5 3 !"#$%&'()* 8 8 8 6 5

!"#$%&' 1:1 16:9 13:8 13:9 16:13 ()*#&' 0 969 841 637 359 bœ Bbœ œ œ 2 B B B Bœ !"#$%&'()#*#(+), 1 1 2 4 V:2 !"#$%&'()* 8 7 7 7 7

!"#$%&' 3:2 39:32 14:13 1:1 7:4 ()*#&' 702 342 128 0 969 œ Bœ bœ 3 ? bœ Bbœ !"#$%&'()#*#(+), 1 1 V:2 1 3 !"#$%&'()* 6 5 6 4 4

!"#$%&' 4:3 13:12 1:1 24:13 3:2 ()*#&' 498 139 0 1061 702 bœ œ 4 ? B bœ Bœ œ !"#$%&'()#*#(+), 3 3 1 2 1 !"#$%&'()* 6 5 4 4 3

!"#$%&' 9:8 1:1 20:13 18:13 5:4 ()*#&' 204 0 746 563 386 œ bœ 5 ? µœ Bœ œ ¢!"#$%&'()#*#(+),V:2 1 V:4 2 4 !"#$%&'()* 6 4 5 3 3

Example 10. Trombone tuning plan for Scenario Six score. Trombone parts 1 and 4 can be played on Bb trombones, whilst the other parts require Bb/F trombones. Slide positions marked as "V" indicate the use of the F valve.

97 An adaptation of the same work for standard trumpet would require a different arrangement of the pitches. Even the careful placement of valve slides would not be sufficient to transcribe each part directly from the trombone setup. The

first note in the first part could be produced an octave higher on a B! trumpet without valves as the 8th harmonic. The second note could be played on second valve with the slide extended far enough and be played as the 8th harmonic. The third note would need to be played with valves two and three as the 8th harmonic yet the pitch would be too flat. This is because the second valve would already be further out than normal and the third valve slide would not be able to compensate enough. The other trumpet parts would have similar issues. However, it would be possible to devise an organisation of all the twenty pitches for a transcription to trumpets or horns. Careful arrangements of valve slides enable the production of many just intonation pitches using the harmonics of the chosen instruments to great advantage. The processes involved are described in many of the sections in Chapter Three, especially

3.3.1, 3.3.2 and 3.3.3.

98 2.6! Standard brass: observations regarding performance in ! non-! standard tunings

It is worth making a few observations concerning the use of standard brass in non-standard repertoire before Chapter Three, which discusses the use of brass in the new repertoire accompanying this thesis.

• Harmonics and tessitura

Players of standard brass, other than the horn, can usually produce up to the

11th harmonic on tube lengths a third or fourth below the open tube. The number of easily producible harmonics decreases with each shorter tube length.

Players can usually produce up to the 16th or 17th harmonics on the F horn and longer tube lengths and up to the 13th harmonic on the short B! horn. The use of non-standard harmonics requires more mental effort by the player and is more physically demanding because the lips gravitate to a more familiar harmonic.

• Valve slide settings

Main tuning slides and valve slides will only allow a little upward pitch movements of between 10 and 20 cents. Instruments are built to be played in relation to A=440 Hertz or a little higher. Thus when a tuning slide is pushed in as far as possible the pitch will be only a little higher than the standard expected pitch. The tuning slide when extended fully may be able to lower the pitch by 30 cents or so, yet not as far as a quartertone.

99 • Embouchure lipping

This technique is common to many ancient traditions and second nature to brass players. Lipping the pitch away from its natural centre results in a loss of resonance and carrying power. The limitations are that lipping a pitch upwards by more than 10 cents is extremely tiring as is lipping down by more than 20 cents.

• Timbral

Modern brass have a rich palette of tone colours yet the tonal qualities of some world instruments cannot be emulated. It has been demonstrated that standard brass can fulfil the tuning function of any world instrument, however the specific tonal qualities of some world instruments may be preferred by composers for some musical situations.

•! Audience expectations

Playing a non-standard instrument is a visual cue to the audience that an unusual timbre and unusual musical result may ensue. When using standard instruments to produce non-standard music a program note about emulating tone colours of other instruments or that unusual tunings will be played can provide the audience with a suitable impression of what they will hear.

Assisting the audience in this way may also address concerns of some performers that practical expertise in the realisation of non-standard tunings will be judged according to normal parameters.

100 Chapter 3! A new repertoire: French horn and close relatives

Most of the repertoire composed for this DCA was written for contemporary standard western instruments. Of the sixteen compositions, thirteen involve brass, three of which are unaccompanied solos for horn (my own instrument), a trombone and a 19-tone four-valve B! trumpet. The other ten works are ensemble pieces which include brass together with standard orchestral winds

(flute, clarinet, bassoon) and strings (violin, viola, cello, double bass) and a microtunable MIDI keyboard (Korg N364). Three compositions in the folio do not involve brass instruments yet are included for reasons of intonation. Eight compositions also include a vocalist singing microtones using standard vocal technique. Additionally three works include a purpose-built microtonal, diamond-shaped metallophone called the xenophone, designed and built by

Christiaan van der Vyver. All compositions were performed or recorded.

This chapter specifically examines three ways of composing for brass in just intonation which are a feature of the companion folio of compositions. In eight of these, performers are responsible for implementing the tuning requirements, in five others some suggestions regarding brass instrumental technique are given by the composer while in the remaining three detailed instructions are given to facilitate tuning implementation. While tuning requirements were specified in compositions involving other instruments (flute, clarinet, bassoon,

101 strings, keyboard, vocals) each performer was responsible for the tuning implementation on their respective instrument.

The repertoire has a range of tuning challenges for the performers. In most cases, compositions based on higher tuning limits were more difficult for performers than those with lower limits. This was mostly because performers were less familiar with certain harmonics and intervals associated with these harmonics, particularly the 11th and 13th.

In general greater assistance I gave as a composer to the performer regarding the tuning requirements made microtonal works easier to perform. With increased performability came an opportunity to compose with a richer harmonic language resulting in larger pitch worlds, sure in the knowledge that performers would be able to successfully contend with the added tuning complexity. An example of this process is the writing for trumpet in A Hundred

Valleys I (3.3.1). The four-valve trumpet was considered a multi-crook instrument with potentially twelve separate harmonic series. A composition process was triggered by observing the possible pitch ratios that were selectable using various valve combinations. The ratios of all the possible harmonics were calculated and pitches chosen to explore the musical possibilities.

Compositions in this chapter are detailed in order of tuning limit and pitch organisation. There are two works in 5-limit, five in 7-limit, three in 11-limit and

102 six in 13-limit just intonation. Two of these juxtapose just intonation with 12- tone equal temperament. Pitch in each composition is organised in one of two ways, scalar or chordal. The first method draws from a set of pitches conveniently organised in scalar form. The second method uses pitches organised in chords and chord progressions, melody being secondary.

Ten of the compositions draw pitch material from one or more of a set of 165, nine-note scales that I compiled in 2004; the full set is shown in Appendix A.

These scales are linear, i.e. defined by a sequence of pitch intervals. They are also octave limited. The nine-note scales have 165 permutations. These permutations are now illustrated below by showing the first, second, third and

final scales. Example 11 has an interval sequence of eight semitone steps followed by a major third, to the octave.

Example 11. Scale number 1 from the set of 165 nine-note scales.

The second scale has an interval sequence of seven semitone steps, followed by a whole tone and a minor third to the octave. This is shown in Example 12.

103 Example 12. Scale number 2 from the set of 165 nine-note scales.

The third scale has seven semitone steps followed by a minor third and a whole tone to the octave. Example 13 illustrates this.

Example 13. Scale number 3 from the set of 165 nine-note scales.

The process of permutation involves shifting larger intervals downwards towards the lowest step in the scale. This continues until the final scale which is the inversion of the first scale. In the final scale the first interval is a major third, followed by eight semitones steps as shown below in Example 14.

Example 14. Scale number 165 from the set of 165 nine-note scales.

The above scales shown above, along with the full set of 165 permutations

(Appendix A) imply equal divisions of the octave in which all semitones are assumed to be 100 cents, whole tones 200 cents and so on. It is possible to

104 convert each linear scale into a rational scale by using methods for approximating equal intervals (tones, semitones, quartertones) surveyed in

Chalmers (1993: 4; also 9: 161-203). One of these involves sub-dividing a larger interval into three non-equal rational intervals (Chalmers 1993: 4-5, 47). This would open up an infinite number of permutations in just intonation where there are only 165 in equal temperament. However, I contained these by using rational intervals that were no higher than 13-limit.

I used eighteen of the 165 scales, one scale per composition for six works and as many as five scales in one work. For seven works, scale numbers match the number of a chapter in Tao Te King the classic book of Lao Tzu (1983). For instance in the composition Seek the Unseen, scale number 12, used text from chapter 12; the text in chapter 72 inspires the musical content in Self-balance and scale number 72 supplies the pitch material and so on. In all eighteen cases rational intervals are used, although in the marimba part in Self-balance the pitch material includes the scale in its equal tempered form; in Five Tastes the scale was a mixture of rational and equal tempered intervals. In one work, We are

Sufficient, there are two scales. The text from chapter 17 informed much of the rhythmic material and was linked to scale number 17. The other scale in the same work is scale 141, which is the intervallic inversion of scale 17.

105 A significant issue that arises in converting a scale based on equal tempered intervals to rational scales is one of notation. In a scale where the semitone is the smallest interval notating the ratio 11:8 is beyond the scope of standard notation. This ratio is neither a perfect fourth as notated by Benjamin Britten in the “Prologue” and “Epilogue” of his Serenade for tenor, horn and strings (1944) or augmented fourth as notated by Eugene Bozza in the movement “La Chasse” from his Suite pour Quatre Cors en Fa (1952). Using quartertone symbols goes someway towards a notational solution. However it introduces many more than the 165 permutations of scales with nine pitches per octave that are possible within 12 equal tempered tuning. Even within 24 equal tempered tuning the number of possible permutations is increased but the number increases astronomically where actual tuning is based on ratios derived from the harmonic series. In order to reach a sensible compromise I have used symbols normally used in 24 equal division of the octave notation as a way to indicate any non-standard intervals a player uses in performance. Cents and

HEWM notation (Monzo 2005) are used when more precision is necessary.

Details of HEWM notation are given in Appendix C.

Early in 2007 I began a second process of pitch selection derived from harmonic singing. With this process I found a directly approachable method of involvement with harmonic series, subharmonic series and chordal systems born out of the interaction between interlinked series.

106 Six of the compositions have pitch material drawn from a set of intersecting harmonic series called “Harmonic Series Interval Set-13” (HSIS-13). The pitch set arose out of the personal exploration of harmonic singing using harmonics in a comfortable pitch range, mostly 4 to 13.

The process for deriving pitch material is described as follows. Begin by singing a fundamental and its 5th harmonic. Retain the pitch of the 5th harmonic and raise the fundamental by a pure major third, the higher pitch is now the 4th harmonic to the new fundamental.

Example 15 illustrates this process using note names and ratios. The cent values are shown in italics below each note, the ratios below them and the harmonic series number in bold type below. When the original fundamental is a C the 5th harmonic is an E two octaves and a major third higher, ratio 1:5. The new fundamental is an E two octaves below the retained harmonic, ratio 1:4. The higher pitch is now the 4th harmonic of E. All the harmonics from 4 to 13 of the

E series are notated and ratios calculated with reference to the original fundamental, namely C.

107 Example 15. Harmonic series notated from C plus the series starting on E.

The harmonic singing practice then returned to the original fundamental though this time with its 6th harmonic, thus C and G, ratio 1:6. The lower note was raised to form a ratio of 1:5 with the higher note. The 6th harmonic of C became the 5th harmonic of E!. The ratio between the two pitches was 1:5. The harmonics 4 to 13 of the E! series were notated as seen in Example 16.

108 ! Example 16. Harmonic series notated from C plus the series starting on E!.

The process returned to the original fundamental, again with its 6th harmonic and this time the lower note was raised to form a ratio of 1:4 with the higher note. The 6th harmonic of C became the 4th harmonic of G. Harmonics 4 to 13 of the G series were notated as seen in Example 17.

! Example 17. Harmonic series notated from C plus the series starting on G.

The practice continued with C, its 7th harmonic and all the associated variables, then C with its 8th harmonic, etc through to the 13th harmonic. The full set is shown in staff notation as Appendix B. The entire set was extended through to the 16th harmonic, is called HSIS-16 and is used in 3.3.3.

109 The commentary on each composition includes a list of all pitches within a one octave span. This is followed by a short description of the work. Under the sub- heading “Tuning Requirements” are details of the pitch specifications. The commentaries are completed with an examination of the method and strategy used to bring the work to performance. A brief diary giving performance or recording details concludes each section.

110 3.1! Extended just intonation compositions: Group A

Five of the eight works in this category involve brass instruments:

Prajnaparamita, Koto Sudoku (extended version), Three Treasures, Eluding Sight and Blavatsky’s Monologue. The other three, Five Tastes, Seek the Unseen and

Towards the Inaudible do not have any brass. They were all composed with the expectation that particular performers had the intonational capability to make the necessary tuning adjustments that deviate from standard tuning.

111 3.1.1.! 5-limit scale-based just intonation

• 12 pitches - harmonised Lydian scale

• Interaction of melody with a chord sequence, each chord in ratios 4:5:6

• Nine-note scale for the melody

• One fixed pitch instrument, one vocalist, one brass

Pitch List

The tuning aspects of this work, Prajnaparamita, demonstrate a method of achieving low limit just intonation on the horn in a work with relatively few tuning complexities.

Tuning Requirements

The nine main scale notes forming the basis of the melody are 1:1; 135:128; 9:8;

5:4; 45:32; 3:2; 27:16; 225:128; 15:8.

112 The keyboard part has seven major triads with the roots as ratios 1:1; 9:8; 5:4;

45:32; 3:2; 27:16; 15:8. The combined pitches of the seven triads are the twelve notes in the Pitch List above.

Tuning Implementation

A successful implementation of the melody for the horn player involves playing selected harmonics with five tube lengths: Horn in E, Horn in C!, Horn in B",

Horn in A and Horn in G!. Playing the F horn with the second valve (indicated as F:2) provides the length of a horn in E and its harmonic series, which is used for notes one and two of the melody, first encountered in bar seven, as well as the ninth note of the melody (bar 11, note two). This can be seen in Example 18.

Example 18. Horn tuning for the main melody, bars 7-13, notated for horn in F. Ratios and cent values of the pitches are shown above the notes, fingerings and harmonics below each note.

The F horn with valves two and three (F:2,3) provides the C! harmonic series, used for bar 8. The third valve of the F side was shifted higher than normal so

113 that the combination of second and third valves would provide the ratio 27:16 as a 9:8 ratio above 3:2 and the correct tuning for ratio 135:128. The B! horn without valves (B!:0) produces the B! series, used in bar 9, note two. The B! horn with valve two (B!:2) produces the A series, used for bar 11, note one and bar 17. B! horn with valve one (B!:1) produces the G" series, used for bars 10 and 12. The first valve slide of the B! horn was shifted further out than normal so that its 12th harmonic would match the E horn 15th harmonic. The valve slide setting is shown below and contrasted with a standard setting.

! !

! !

Illustration 7. Normal setting of valve slides above: F side left, B! side right. Valve slide setting for Prajnaparamita below: F side left, B! side right.

114 The adjustment of valve slides is one of the three main techniques for tuning the horn, the others being lipping and right hand movements. Valve slide settings can be adjusted to raise the pitch by up to 10 cents or so and lower the pitch by

30 cents or more. Such adjustments are an effective way to set tunings. Any success in this regard will minimise lipping. This is important for physical endurance as any lip adjustment away from a harmonic’s centre requires additional flexibility and strength.

Additional harmonics and tube lengths are required for the final section of the work, such as playing the B! horn with valves two and three: the 4th harmonic of that series in bar 114; the 2nd harmonic in bar 117 and the 3rd harmonic in bar 126. There are ten similar instances from bar 115 in which the horn player needs to determine a useful fingering.

It is possible to adjust each semitone in 1 cent increments with the Korg electronic keyboard (N364). Each semitone once tuned affected each octave. The tuning included two G"‘s a syntonic comma apart, therefore the higher one was mapped to the ‘A’ key. The keyboard player quickly adjusted to the transposition after the initial aural confusion. The key map of the twelve pitches is shown in Example 19.

115 Example 19. Keyboard tuning in Prajnaparamita showing the variation of each ‘key’ from 12-tone equal temperament.

The vocal part has the same melody as the horn. Implementation involved practicing in unison with the keyboard and in relation to the chords. The vocalist owned the keyboard, so had easy access to such practice.

This work, Prajnaparamita was written early in 2006 and has been performed twice. The debut took place at the University of New South Wales on 17 January

2007 for a Theosophical Society Convention. The second performance was at the University of Wollongong, Faculty of Creative Arts Theatre on 26 April 2007 in a concert wholly of my works. The musicians for both performances were

Wendy Dixon, soprano; Michael H Dixon, horn; Phillipa Cook, keyboard.

116 3.1.2.! 7-limit scale-based just intonation mixed with 12tet

• 13 pitches - interaction of 12tet and non-standard tuning

• Pitch beatings

• Nine note scale from three pentatonics

• One fixed pitch instrument, two flexibly pitched instruments, no brass

Pitch List

This work, Five Tastes, explores pitch beatings from the interaction of small intervals between just intonation and 12-tone equal tempered pitches and between two just intonation pitches. Each instrument, clarinet, cello and piano, has a pentatonic scale. The composition was inspired by a chapter in the

Chinese classic Tao Te Ching which refers to five tastes: sweet, salty, sour, bitter and acid. Each taste is represented by a dyad or chord.

117 Tuning Requirements

Example 20 shows three pentatonic scales, one for each instrument. The tonic of each scale is indicated as 1:1 for the clarinet and cello, 0 cents for the piano. The clarinet ratios are in relation to its tonic F, which is a perfect unison with the piano. The cello ratios are in relation to its tonic G, which is a perfect unison with the piano C.

Example 20. Original tuning parameters for Five Tastes. The ratios for the clarinet and cello pentatonics are shown, relative to their tonic, above each pitch. For each note the number of cents from C concert is shown underneath.

The other pitches for the clarinet player are D, which is tuned low to the piano;

B! is tuned high to the cello and both D! and A! are tuned high to standard pitch. The other pitches for the cello player are D and A, which are tuned high

118 to the piano; B! is tuned low to the clarinet and E is tuned high to standard pitch.

Tuning Implementation

A computer generated sound file of the full score was given to the performers with their parts. The tunings were organised from a tuning crib through the software Li’l Miss’ Scale Oven (2005). During a rehearsal in 2009, with myself present, the performers revealed they had not used the sound file. Instead they suggested a file of each pentatonic would be most useful. I suggested we settle on relative nearness and closeness for the ‘off-unisons’. This seemed successful.

Before rehearsals for the 2010 performance the clarinet and cello players received a sound file of their precisely tuned pentatonic scales. In the 2010 performance the pitch beatings sounded more effective than in the 2009 performance. This was perhaps due to greater tuning precision from the clarinet and cello players in addition to a more resonant acoustic.

This work, Five Tastes was written early in 2008 and was performed at the

University of Western Sydney Aurora Festival on 17 April 2008 by the ensemble

“Charisma” – Ros Dunlop; clarinet, Julia Ryder, cello; David Miller, piano and at the Sydney Conservatorium of Music on 29 August 2010 by the same performers.

119 3.1.3.! 11-limit scale-based just intonation featuring the koto

• 9 pitches - exploration of a small pitch set

• Pointillistic melodic treatment

• Nine-note scale

• One fixed pitch instrument (vocalist and two brass in extended version)

Pitch List

This work, Koto Sudoku demonstrates a practical way to achieve a higher limit just intonation pitch set with relatively few tuning complexities. The composition works well as a solo koto work with many interesting intervallic relations. The extended version came about because the other musicians were already preparing music to be performed in the same concert and it seemed a good opportunity to participate in an 11-limit sound world.

Each of the nine scale degrees corresponds to the nine numbers in the game sudoku. In Section One the sudoku grid, as it appeared in a newspaper is simply stated in 9/8 time. Subsequent sections of the music represent the working out of the game, adding numbers to fill in the empty spaces.

120 Additional rests in the music allow for a continuance of the pointillistic, spacious musical feel, therefore the time signature does not remain in 9/8.

Tuning Requirements

The nine ratios in this work have a group of three pairs of 11:8 intervals as the core. They are 1:1 + 11:8; 3:2 + 33:32 and 9:8 + 99:64. Of these 1:1, 3:2, 9:8 can be viewed as part of a chain of 3:2 intervals, as can 11:8, 33:32, 99:64. The three other pitches in the work are added to these two chains. The ratio 27:16 is the next pitch after 9:8. Table 4 shows 27:16 to the right of 9:8. The ratios 11:6 and

11:9 are next after 11:8 and are seen to the left of 11:8.

Table 4. Pitch relations in Koto Sudoku.

11:9 11:6 11:8 33:32 99:64

247c 1049c 551c 53c 755c

1:1 3:2 9:8 27:16

0c 702c 204c 906c

Tuning Implementation

All thirteen strings of the koto were tuned with the help of a computer generated sound patch written by Terumi Narushima in Pure Data. She used the sound patch to check the tuning during her practice sessions. I attended a couple of her practice sessions during which we tuned the strings aurally, then

121 compared the result to the some of the harmonics on the horn and adjusted as necessary. The 8th and 11th harmonics of the horn in D, F horn plus the third valve (F:3), provided the tuning for 1:1 and 11:8 on the koto. Likewise the 8th and 11th harmonics of the horn in E (F:2) provided the tuning for 9:8 and 99:64; the same harmonics of the horn in A (B!:2) providing 3:2 and 33:32. The other pitches could be easily tuned aurally as they are pure fifths. We also followed this procedure prior to the performance.

Koto players have a convention of assigning a number to each string. The performer suggested that it would help her reading of the score to do this.

Example 21 shows the strings with the assigned ratios.

! Example 21. Arrangement of the koto strings for Koto Sudoku. The precise ratios are shown above each pitch. For each note the number of cents from the tonic D is shown below the ratios.

The tuning for the soprano and the two horn players was reasonably straightforward though one note was added to make pitching easier in the first horn part. Originally the first entry for this part began on the 11th harmonic of the D horn. Adding the 8th harmonic provided a commonly used note as the

122 pitching basis and kept the 11:8 interval between the two horns. This can be seen in Example 22 below.

Example 22. Horn entry in Koto Sudoku.

Two rehearsals with all the players seemed adequate.

This work, Koto Sudoku, was written 2006 and was performed at the University of Wollongong, Faculty of Creative Arts on 19 October 2006 by Terumi

Narushima, koto; Wendy Dixon soprano; Michael H Dixon & Lisa Wynne-

Allen, horns.

123 3.1.4.! 11-limit scale-based just intonation featuring a microtonal ! metallaphone (xenophone)

• 9 pitches - harmonised as dyads

• Simple melodic treatment

• Nine-note scale

• One fixed pitch instrument and vocalist

Pitch List

This composition, Seek the Unseen, does not have any brass instruments yet is included because it explores a different group of nine 11-limit pitches from the previous group (3.1.3). The xenophone is a microtonal metallaphone called the xenophone (Illustration 8). The instrument was designed and built by

Christiaan van der Vyver. The design included the pitch arrangement of 53 ratios in a range of three octaves. Thirteen are doubled at the unison or octave to make a total of 66 metal bars. The nine ratios in this work were chosen to provide a smooth melodic progression. The pitches are all in the lowest range of the xenophone to blend with the soprano.

124 !!

!!Illustration 8. Xenophone. Permission granted by Christiaan van der Vyver in an email.

Tuning Requirements

The requirements are quite straightforward. The xenophone player's only pitch concern is to accurately strike the required bar. In contrast, the vocalist has to gain an aural awareness of the pitches and physical memory of the voice placement for each pitch where it occurs in the music.

125 The melodic ostinato is harmonised by using many of the possible dyads.

Example 23 shows the first group.

Example 23. Dyads harmonising the melody in Seek the Unseen.

Tuning Implementation

A sound file with the precise tunings of the full score was supplied to the soprano. It was the most important tool for learning the music. The strategy was no different to the manner for learning compositions in standard tuning: immersion in the soundscape of the work and practicing the part numerous times along with the sound file. However, more hours of work were required due to the unfamiliar tuning. Rehearsals with the xenophone were then straightforward due to the experience with the sound file.

As the player of the xenophone I had to spend many hours practicing the instrument to develop sufficient experience and physical memory of the

126 particular movements required. The only practical experience with percussion instruments I had previously was six months learning side drum technique with a practice pad and three years learning tabla. However, I did not ask a percussionist to prepare the part for two reasons. Firstly the layout of the xenophone is so different to any standard keyboard percussion that even an experienced player would require significant time to develop a physical memory of where the pitches lay. Also, access to the instrument was limited to times that there were no lectures in the room it was housed, further inhibiting practice availability. Secondly a musician would have to learn the relationship of pitches on the xenophone to any notation. Before making the decision to perform the xenophone part I had already gained practical experience whilst exploring the instrument and selecting pitches for composition. It seemed expedient to perform this and the other two works involving the xenophone.

This work was written early in 2007 and was performed at the University of

Wollongong, Faculty of Creative Arts on 26 April 2007 by Wendy Dixon soprano and Michael H Dixon, xenophone.

127 3.1.5.! 13-limit scale-based just intonation featuring the xenophone and koto

• 20 pitches - exploration of 13:8 intervals

• Pitch beatings

• Two nine-note scales

• Two fixed pitch instruments

Pitch List

This work, Towards the Inaudible, explores a soundscape in 13-limit just intonation.

Tuning Requirements

The xenophone has eight ratios with 13 in the numerator or denominator. Six of them are used in the composition and are paired with other pitches to provide

128 13:8 intervals. They are 1:1 + 13:8; 1:1 + 16:13; 8:5 + 13:10; 8:7 + 13:7; 16:9 + 13:9.

The ratio 24:13 is paired with the 3:2 from the koto pitch set. The koto has a set of five pairs of the interval 13:8. They are 3:2 + 39:32; 5:4 + 20:13; 7:4 + 14:13;

9:8 + 18:13; 4:3 + 13:12.

The two groups of pitches are seen below in Example 24. Each is a nine-note scales with two variants of one pitch spelling. The xenophone has the close pitches 24:13 and 13:7 both spelt as B! and the koto has 14:13 and 13:12 both spelt as D!. The two groups can also viewed as bi-tonal sets.

! Example 24. Two groups of ten pitches in Towards the Inaudible.

129 Tuning Implementation

Terumi Narushima wrote a Pure Data file of the ratio pitches to assist with the tuning of the strings. As per any composition for the koto the importance is to prepare the strings with the appropriate tuning prior to performance. There may be some shifting of the tuning during performance, though this risk was minimised due to the piece being relatively short.

This work was written early in 2007 and was performed at the University of

Wollongong, Faculty of Creative Arts on 26 April 2007 by Terumi Narushima, koto and Michael H Dixon, xenophone.

130 3.1.6.! 7-limit chord-based just intonation

• 34 pitches chosen from harmonics 1 to 13 in 28 linked harmonic series

• Ratios assigned to vowels - the text determines the chord progressions

• Each ratio in a chord can change in function

• One vocalist and four flexibly pitched instruments, one is brass

Pitch List

This work, Three Treasures, is the first composition I had written using interweaving harmonies in an extended way. It is the first in a group of three that use the same compositional procedures: selecting pitches from the same set of harmonic series intervals; mapping pitches to vowels; using the tonic chord

131 as a pivot in chord progressions. The larger pitch set is titled Harmonic Series

Interval Set-13 (HSIS-13) and can be seen in full as Appendix B. The pitch group for this work has no higher prime numbered ratio than seven, therefore it is a 7- limit just intonation composition. The other two works that use the same procedures are 3.1.7 in 11-limit and 3.1.8 in 13-limit.

Tuning Requirements

There are a large number of ratio pitches organised in chord progressions, each chord based on the ratios 4:5:6:7:9. A ratio pitch is used as the common tone between chords and all chords have one of the ratios from the tonic chord. The chord changes are largely determined by the pitch changes in the soprano line.

When the soprano line moves to a pitch that does not have a tone in common with the previous chord, the tonic chord is used as a pivot. The soprano’s pitches are determined by the vowel in the text because which is assigned to one of fifteen ratios as shown in Example 25.

Each vowel ratio has more than one chord associated with it to allow some choice in a progression to any upcoming vowel/pitch/chord. Example 26 shows the soprano pitch, ratio 45:28 for vowel ĭ, set to the chord with root 9:7 in bar 1 and the same soprano pitch set to the chord with root 10:7 in bar 13.

132 Ratios: 21:20 15:14 35:32 7:6 6:5 27:20 7:5 Cents: 84 119 155 267 316 520 583 bw & bw #w Bw Bbw bw w Vowel: !! ! aw ah "" a "

10:7 35:24 63:40 45:28 27:16 12:7 27:14 63:32 617 653 786 821 906 933 1137 1173 bw nw w ˜w Bw & #w Bw #w µ # $r % & ' ( a

Example 25. Vowel map in Three Treasures. The precise ratios are shown above each pitch. The number of cents for each note from the tonic G is shown under the ratios. Vowel types are shown underneath the pitches as phonetics.

b.1 Ratios: 45:28 b.13 45:28 Cents (to G):821 821 Soprano #˙ #˙ & 2 Ó Ó

9:8 45:28 10:7 204 821 617 Flute 2 w & 2 #˙ #˙ 27:14 5:4 1137 386 Violin & 2 ˜w w 9:7 25:14 435 1004 Horn & 2 Ó µw 9:7 10:7 #˙ 435 617 Violoncello ? 2 2 µw #w

Example 26. The ratio 45:28 set to the vowel ! with two harmonisations.

133 The use of common notes between chords is shown below in Example 27. Three functions for the note A in the flute can be seen. In the first chord it is a 9:4 interval to 1:1 in the cello, in the second chord it is a 7:4 interval to 9:7 in the horn and cello, in the third it is a 5:4 interval to 9:5 in the violin.

Example 27. Chord changes bars 1 to 2, Three Treasures.

134 Tuning Implementation

A tuning crib was prepared for the quintet yet I made a mistake recording it from the software and the players received a blank CD. Fortunately all the musicians had worked on my just intonation compositions in the recent past, in particular on the 13-limit work (3.1.8) and the singer also on the 11-limit work

(3.1.7) so were familiar with the musical expectations. They found the tuning in this 7-limit work much easier to achieve than those in 11-limit and 13-limit. I attribute this to it having much more familiar intervals to work with.

The rehearsal process was assisted in three ways. I was present at all rehearsals and could help clarify chord functions. I could suggest intonation adjustments to any musician. Lastly I could demonstrate pitches on the horn with a high degree of reliability due my familiarity with the intervals and because a number of the pitches were directly available from various harmonic series on the horn.

For two rehearsals and the performance a conductor was employed. This assisted by making the rhythmic and ensemble aspects much easier to manage.

Each musician received a full score and an individual part. By 2009 I decided it was more important for musicians to have the assistance of cent values to show relative tuning. I had been using an adaptation of Joseph Monzo’s HEWM notation (Monzo 2005). The acronym stands for Helmholtz-Ellis-Wolf-Monzo.

This notation clearly shows pitch relations yet still leaves musicians guessing as

135 to whether a pitch is higher or lower than another with the same spelling.

Details of HEWM notation are given in Appendix C.

In rehearsals we practiced chord changes to assist our aural awareness of the sound world. Observing the changes of function of any pitch allowed us to quite easily perform in this particular 7-limit sound world. The composition was also one of the least difficult to sing in the folio due mainly to many pitches being pre-empted by one of the instruments.

This work, Three Treasures, was written in 2009 and performed at St. Stephen’s

Church Hall, Newtown on 20 June 2009 by Wendy Dixon soprano; Janine

Grantham, flute; Rachel Westwood, violin; Michael H Dixon, horn and Matthew

Farrell, cello.

136 3.1.7.! 11-limit chord-based just intonation

• 55 pitches chosen from harmonics 1 to 13 in 28 linked harmonic series

• Ratios assigned to vowels - the text determines the chord progressions

• Each ratio in a chord can change in function

• One vocalist and three flexibly pitched instruments, all brass

Pitch List (on next page)

137 Ratios: 1:1 49:48 33:32 21:20 15:14 12:11 35:32 11:10 10:9 Cents (from Eb): 0 36 53 85 119 151 155 165 182 bbb & ˙ Bb˙ B˙ b˙ n˙ B˙ B˙ B˙ n˙

9:8 8:7 7:6 6:5 77:64 49:40 27:22 5:4 81:64 204 231 267 316 320 352 355 386 408 bb & b ˙ µ˙ Bb˙ b˙ b˙ B˙ B˙ n˙ ˙

9:7 21:16 4:3 27:20 15:11 11:8 25:18 7:5 45:32 435 471 498 520 537 551 568 583 590 b &b b µ˙ Bb˙ b˙ ˙ B˙ B˙ B˙ ∫˙ n˙

10:7 81:56 35:24 3:2 55:36 49:32 99:64 25:16 11:7 617 639 653 702 733 738 755 772 782 b &b b n˙ µ˙ Bb˙ b˙ B˙ Bb˙ B˙ n˙ n˙

45:28 18:11 33:20 5:3 27:16 12:7 55:32 7:4 99:56 821 853 867 884 906 933 937 969 986 b &b b n˙ B˙ B˙ n˙ ˙ µ˙ µ˙ Bb˙ #˙

25:14 9:5 11:6 15:8 21:11 77:40 27:14 35:18 55:28 63:32 1003 1018 1049 1088 1120 1134 1137 1151 1168 1173 b b˙ ˙ n˙ ˙ b˙ ˙ b˙ ˙ b˙ &b b #˙ B B µ B µ B

138 This work, Eluding Sight, uses the same procedures as the previous work (3.1.6 in 7-limit) and the next (3.1.8 in 13-limit): selecting pitches from the same set,

HSIS-13; mapping pitches to vowels; using the tonic chord as a pivot for chord progressions. The pitch group for this work has no higher prime numbered ratio than eleven, therefore it is an 11-limit just intonation composition. This composition includes chords with ratios 6:7:9:11 as a feature of the sound world.

Example 28 shows the feature chord in bar 5 on the first beat. The pitch with ratio 6 is in the horn, 7 in the trombone, 9 in the trumpet and 11 in the soprano.

Example 28. Feature chord with ratios 6:7:9:11. Bar 5, Eluding Sight.

139 Tuning Requirements

The vocal part uses thirteen ratios which are mapped to vowels, shown below in Example 29. Sometimes the pitches are played by one of the instruments before they are sung and other times they are not. Therefore the reliance on aural and physical memory is much more important than the previous work

(3.1.6.).

! Example 29. Vowel map in Eluding Sight. The precise ratios are shown above ! each pitch. The number of cents from the tonic E! is shown under the ratios ! for each note. Vowel types are shown underneath as phonetics.

Tuning Implementation

Significant aural awareness was required by the musicians to bring this score to fruition. The same Pitch List shown on page was given to each performer along with performance notes, a full score and an individual part. The performers found the Pitch List useful in comparing pitches to each other and determining the degree of tuning adjustments needed. The performers suggested the

140 performance notes could emphasise the aural aspects of learning the work in contrast with the notational aspects as well as the usefulness of the sound file in bringing awareness of the soundscape. A tuning crib was made, applied to the score then recorded and used by all four performers. The soprano had requested a half speed version of her line only. She used this to increase accuracy in reproduction. I also practiced with a sound file of the horn line only as well as practicing the other parts with the full score sound file. Additionally I listened to the sound file of the full score numerous times to achieve a strong immersion in the soundscape. One problem with the sound file was that the trumpet and trombone parts were on the same MIDI channel therefore they could not have separate lines for a half speed version.

The main intervals used in the work are heard in the Recitative section though it is not until the Aria that the main compositional procedures are used and the chord progressions dominate the soundscape. The feature chord, ratios 6:7:9:11, is first heard in bar 1 of the Aria, though minus the root. Example 30 shows this chord and illustrates a chord progression that was one of the most difficult for the brass players to enact in the composition. The trombone plays the 7 ratio, the trumpet plays the 9 ratio and the horn plays the 11 ratio and carries it over to bar two. Here the pitch A! is treated as the root to a new chord, though in first inversion.

141 ! Example 30. Chord progression, bars 1-2 in Eluding Sight. The precise ratios are shown above each pitch. The number of cents from the tonic E! is shown under the ratios for each note.

During rehearsals we all noted that chords ‘zinged’ when the intonation was true. We had one rehearsal without the soprano to work on the brass sound. At the first full rehearsal we felt that both pitch and ensemble issues were much easier to negotiate with the complete ensemble. We also found that rehearsing at half speed impeded the musical flow too much whereas two-thirds speed worked well for hearing the tuning of the chords and progressions. In each rehearsal one hour was long enough for our collective concentration.

Regarding the notation the performers found that the key signature was confusing for pitch recognition especially as the notated E" is really in relation to uninflected E rather than the E! in the key signature. They also suggested that marking accidentals more frequently than once a bar would assist in reading

142 the parts. The quartertone symbols were felt to be confusing at times by the musicians due to the large range they cover of around 60 cents. All such notational symbols represent a significant pitch range of 60 cents in the folio, from 30 cents above to 30 cents below the 12-tone equal tempered 0 cents. I have not resolved this issue even after considering other symbolic notation additions.

The brass players required lipping, valve slide adjustments and non-standard slide positions to produce the required intonation. Only a few pitches were easily played on available harmonics. To some extent the brass players had to

find the pitch in the same way that singers do, by hearing the intervals and physically remembering where to place the pitch. Suitable horn fingerings were discovered through experimentation and playing in unison with the sound file.

I suggested some fingerings to the trumpet player. These involved approaching a pitch from a fingering nominally a semitone above, then adjusting the pitch lower using 1st or 3rd valve slide trigger adjustments. I also suggested a few slide positions to the trombone player along similar lines to those for the trumpet. My practice of all the parts assisted the overall process as I gained a strong awareness of all the harmonics used by the brass players. For my own part I added fingerings below every note as a guide to finding the correct intonation. Some notes required the addition of lip and hand adjustments.

Additionally I worked out horn fingerings to play the pitches in the soprano

143 part, partly for my own learning but also for use in rehearsals to quickly provide the correct pitch.

This work, Eluding Sight, was written 2008 and was performed twice in the same event at the Wendy Dixon Studio, George Street, Haymarket, Sydney on

17 May 2008 by Wendy Dixon soprano; Joshua Clark, trumpet; Michael H

Dixon, horn and Greg van der Struik, trombone. The work was performed at the beginning and end of the concert. We thought it was a good opportunity to allow audience and performers the possibility of more deeply experiencing the sound world on a second hearing.

144 3.1.8.! 13-limit chord-based just intonation

• 84 pitches chosen from harmonics 1 to 13 in 28 linked harmonic series

• Ratios assigned to vowels - the text determines the chord progressions

• Each ratio in a chord can change in function

• One vocalist and six flexibly pitched instruments, one is brass

Pitch List (on next page)

145 Ratios: 1:1 65:64 49:48 33:32 25:24 117:112 21:20 77:72 15:14 Cents (from D): 0 27 36 53 71 76 84 116 119 & ˙ ˙ Bb˙ µ˙ #˙ #˙ b˙ b˙ #˙ 13:12 12:11 35:32 10:9 143:128 9:8 91:80 8:7 55:48 139 151 155 182 192 204 223 231 236

& B˙ B˙ B˙ n˙ ˙ ˙ b˙ µ˙ µ˙ 7:6 33:28 6:5 77:64 39:32 11:9 49:40 27:22 99:80 267 284 316 320 342 347 351 355 369

& B˙ #˙ n˙ ˙ µ˙ µ˙ Bb˙ µ˙ µ˙ 5:4 121:96 81:64 9:7 21:16 169:128 4:3 27:20 65:48 386 401 408 435 471 481 498 520 525

& #˙ µ˙ #˙ ˜˙ B˙ n˙ ˙ ˙ ˙ 15:11 11:8 25:18 7:5 45:32 91:64 10:7 13:9 81:56 537 551 569 583 590 609 617 637 639

& µ˙ µ˙ µ˙ #˙ #˙ b˙ b˙ B˙ ˜˙ 35:24 117:80 143:96 3:2 55:36 49:32 99:64 25:16 11:7 653 658 690 702 734 738 755 772 782 & B˙ B˙ n˙ ˙ µ˙ Bb˙ µ˙ #˙ #˙ 63:40 8:5 77:48 45:28 13:8 18:11 33:20 5:3 786 814 818 821 841 853 867 884 & b˙ b˙ b˙ #˙ B˙ B˙ B˙ n˙

121:72 27:16 12:7 55:32 7:4 99:56 39:22 25:14 899 906 933 938 969 986 991 1004 & ˙ ˙ µ˙ µ˙ B˙ #˙ n˙ ˙

9:5 65:36 117:64 11:6 13:7 15:8 91:48 1018 1023 1044 1049 1072 1088 1107 b˙ & ˙ ˙ µ˙ ˙ #˙ #˙

21:11 77:40 27:14 35:18 55:28 63:32 143:72 1119 1134 1137 1151 1169 1173 1188 & b˙ B˙ ˜˙ B˙ ˜˙ n˙ ˙

146 This work, Blavatsky’s Monologue, uses the same procedures as the previous two works (3.1.6 in 7-limit & 3.1.7 in 11-limit): selecting pitches from the same set,

HSIS-13; mapping pitches to vowels; using the tonic chord as a pivot for progression. The pitch group for this work has no higher prime numbered ratio than thirteen, therefore it is a 13-limit just intonation composition.

Tuning Requirements

A total of eighty-four ratios are used, all belonging to the set called HSIS-13.

Some ratios are very near to each other, yet each has a function within a particular chord. The tonic chord with ratios 1:3:5:7:9:11:13 is seen below in

Example 31. The following two chords are simply extensions adding ratios 15:8 and 21:16. All the other pitches in the composition have a direct chordal relationship with at least one of ratios in the tonic chord.

147 13:8 b.86 841 Horn ...... & Ó Œ ‰ Bœ œ œ œ œ œ œ 11:8 551 Violin ° & Ó Œ µœ œ œ œ œ œ œ 9:8 ...... 7:4 204 969 ...... Viola B Ó Bœ œ œ œ œ œ ˙ 5:4 3:2 386 702 Violoncello ? #œ œ œ œ œ œ œ ˙ Œ œ œ œ œ œ œ œ ˙ 1:1 ...... Contrabass 0 (E string tuned ? down to D) ¢ ˙ ˙™ ˙ 15:8 b.87 1088 Fl. ‰ #œ. œ. œ. œ. œ. œ. œ. Ó & J ™

Hn. Ó Œ ‰ œ. œ. œ. œ. œ. œ. œ. & BJ 21:16 471 ...... Vln. ° Bœ œ œ œ œ œ œ & ˙ œ™ J

Vla...... B œ Bœ œ œ œ œ œ ˙™

Vc. ? Œ Ó ¢ œ. œ. œ. œ. œ. œ. œ.

Example 31. Tonic chord bar 86, Blavatsky’s Monologue. The precise ratios are shown above each pitch. The number of cents from the tonic D is shown under the ratios for each note.

148 Tuning Implementation

All the musicians required a significant aural awareness of harmonic series relations. All the musicians used the sound files of the full score and their individual parts with a playback file at half speed. The soprano was supplied with a score and sound file in April 2008, whilst the instrumentalists received theirs in August 2008. I listened to a sound file of the full score between forty and fifty times over many months to increase awareness of the sound world. I also practiced singing and lip buzzing along with the half speed version of the horn line some twenty times to increase my accuracy when playing the horn. I checked all the parts by playing along with the half speed sound files of individual lines. From this process I corrected ten or more errors in the sound

file as well as some notational errors. I then re-recorded the corrected sound

files on October 1 and gave them to the performers at the first rehearsal. Some of the errors were conveyed to me by the performers at the October 1 rehearsal, giving me an indication of the level of study and committment from them.

The musicians found the Pitch List useful for determining relative distance between pitches. Quartertone tone symbols were deemed partially useful especially when I emphasised that those symbols were approximate. Some musicians wondered if the three-quartertone symbol was appropriate for 7th

149 harmonics. They felt the tuning of 7th harmonics was similar to some standard musical situations.

What follows is the documentation of tuning actions and preparations that were specific to the horn. The harmonics 1 to 13 from the D, G and A series on the horn were directly used in the composition and therefore available as tuning benchmarks for all musicians. Practice with the sound file was essential to

finding suitable fingerings and was helpful in developing lip muscle memory for many passages. Further preparation included the practice of harmonic series to develop muscle memory and aural sensitivity. These slurs are found in

Appendix D, Section 2. After the October 2008 performance I developed further harmonic series exercises, Appendix D, Sections 1 & 3, and practiced them from

January 2009. This practice ensured that my performance in June 2009 was more secure. Lip muscle memory can ensure the pitch is played accurately even when the aural perception is confused. For example, the difference between the 12th and 13th harmonics is significant but not always easy to pick unassisted without clear knowledge of the context. Example 32 shows the horn entering on the 13th harmonic in bar 22. This is a difficult pitch to pick aurally in this instance due to the relation to the previous chord. In bar 21 the chord root is the 11th harmonic of the tonic D. The flute has the 13:8 interval to that root, and this flute pitch becomes 11:8 to the new root, which is the horn pitch.

150 143:128 Aria b.21 192 Flute ˙ w &2 Ó 23 Ó mp 13:8 841 Horn &2 ∑ 23Bw Ó mp 3:2 11:8 55:32 117:64 39:32 13:8 3:2 702 551 938 1044 342 841 702 Soprano: > j 2 3 µœ œfi Blavatksy &2 ˙ œ µœ 2 Œ µœ œ B œ µ 5:4 µœ œ 11:8 55:32 117:64 39:32 551 938 1044 342 µœ œ µœ µ˙ ˙ Violin ° 2 Ó µ 3 Ó &2 2 5:4 3:2 11:8 77:64 39:32 5:4 702 551 320 342 386 . Viola œ µœ ˙ w œ #œ B 2 23 µ 5:4 7:4 33:32 65:64 1:1 386 969 53 27 0 Bœ. µ˙ µw œ nœ Violoncello ? 2#œ 23 1:1 5:4 11:8 13:8 3:2 0 386 551 841 702 . Contrabass ? 2 œ #œ µ˙ 3 Ó Ó ¢ 2 2 Bœ œ

Example 32. Horn entry on the 13th harmonic as the root of the chord, bars 21-22, Aria in Blavatsky’s Monologue. The precise ratios are shown above each pitch. The number of cents from the tonic D is shown under the ratios for each note.

151 The tuning practice for the other performers is more difficult than for the horn player because they do not have the range of harmonic series pitches available to use as tuning benchmarks. This only further demonstrates the usefulness of the horn as a resource in just intonation works.

This work, Blavatsky’s Monologue, was written 2008 and was performed three times on two dates. The premiere and second performance were at the

Theosophical Society, 484 Kent Street, Sydney on 26 October 2008 by Wendy

Dixon soprano; Janine Grantham, flute; Rachel Westwood, violin; Robert Harris, viola; Matthew Farrell, cello; Dorit Herskovits, double bass; Michael H Dixon, horn. The second occasion was at St. Stephen’s Church Hall, Newtown on June

20, 2009 with the same performers and the addition of Peter Rankine, conductor.

The 2008 event began with a short spoken introduction followed by a performance of the work. Next came a longer talk detailing technical aspects of the composition along with demonstrations by the performers of some chords.

The work was then performed for a second time to allow audience and performers the opportunity to more deeply experience the complexity of the sound world.

152 3.2. Extended just intonation compositions: Group B

Five works come under this heading all including one or more brass instruments: Door of Many Mysteries; Self-balance; Stillness of Unceasing Activity;

Strive Not, and We are Sufficient. All these have linear scales as a main feature or a starting point for the composition process, from the set of 165, 9-note scales

(Appendix A). All horn parts included fingering suggestions and the solo trombone score included slide positions for the main pitches. However, the performers still had to do much of the fine-tuning. The horn players required lipping and right hand technique at times to adjust the tuning. One valve slide adjustment was recommended, this for the third horn player in 3.2.5.

Knowledge of the capabilities of the trombone was specific enough to know which harmonic was being used at any time yet not specific enough to know how far the player must adjust the slide from standard positions to achieve the desired tuning.

153 3.2.1.! 5-limit scale-based just intonation featuring the xenophone with brass

• 16 pitches - familiar tuning for the brass parts

• Tri-tonality, linked by ratios to 1:1 in the trombone

• Nine-note scales

• One fixed pitch instrument

Pitch List

This work, Strive Not, explores polytonality within a 5-limit tuning. The tuning was chosen for its familiarity to the brass players. Each instrument has its own tonic and 9-note scale, though the B! of the trombone could be viewed and heard as the overreaching tonic. The Pitch List has C indicated as the tonic, ratio

1:1 to provide a common reference for the notation.

Tuning Requirements

The xenophone and trombone have exactly the same arrangement of tones and semitones in their scales though the intervals between ratios is not the same.

154 For example the tonic for the xenophone is D! and the ratio to its third note E! is

10:9, whilst the tonic for the trombone is B! and the ratio to its third note C is

9:8. The horn has two ratios assigned to its third note. The interval from the first note is both 10:9 and 9:8. Example 33 shows the three scales.

! Example 33. Three ratio scales in Strive Not. The precise ratios are shown above each pitch. The number of cents from the notational reference C is shown under the ratios for each note.

Tuning Implementation

Both brass players have to play extensively in pure fifths to each other. The expectation to play beatless, perfect fifths is common practice, though not always easy to achieve as there is no room for tuning error. Many brass players are also used to tuning beatless, major thirds therefore the requirements for this composition seemed like normal practice for the players. The most unusual aspect for the performers was achieving the off-unison Gs such as in bar four.

155 The tuning of this was assisted by having the syntonic comma interval played in the xenophone, albeit just over an octave apart. Additionally the 5th harmonic of the F horn with first valve (E! horn) convincingly provides the lower G, the

40:27 interval to the notational tonic C, 1:1. This is shown in Example 34.

Example 34. Off-unsion, pitch beating, in bar 4 of Strive Not.

Fingerings were supplied to the horn player to take advantage of various harmonics that produce the desired tuning.

This work was written early in 2007 and was performed at the University of

Wollongong, Faculty of Creative Arts on 26 April 2007 by Lisa Wynne-Allen, horn; Greg van der Struik, trombone and Michael H Dixon, xenophone.

156 3.2.2.! 7-limit scale-based just intonation with multi-ratio expression

• 27 pitches - three variants for each of eight pitch spellings

• Simple harmonisation

• Five symmetrical nine-note scales

• One vocalist and two flexibly pitched instruments - both brass

Pitch List

In this work, Door of Many Mysteries, simple harmonies underpin microtonal pitch movement. The work was a starting point for the DCA and the focus of study to delve deeper into the work of harmonics.

157 In the set of 165, nine-note scale there are five that are symmetrical in the interval sequence, ascending and descending. The five scales are shown in

Example 35.

Example 35. Five symmetrical 9-note scales.

Each of the five is used in a different section. For example the scale used in the soprano part for Section 6 has the nine note names E, F, F!, G, G!, C, C!, D!, D.

Tuning Requirements

All the note names used in the composition except E 1:1, A 4:3 and B 3:2 have three ratios associated with them. For each note name there is a 3-limit, 5-limit and 7-limit ratio. This multi-ratio expression of pitch spellings is used throughout the work. The most extensive example is in bars 49 to 56. The soprano descends through all 24 ratios underpinned by harmonies in the horns.

Example 36 illustrates this.

158 b.64 27:14 243:128 15:8 9:5 16:9 7:4 12:7 81:64 5:3 8:5 128:81 14:9 1137 1110 1088 1018 996 969 933 906 884 814 792 765 Soprano & #œ #œ #œ œ œ œ #œ œ œ œ œ œ

81:64 9:8 1:1 3:2 9:8 1:1 906 , 204 0 , 702 204 , 0 , Horn 1 ° #œ œ œ Œ œ & Œ #œ Œ œ Œ #œ #œ 3:2 4:3 1:1 3:2 4:3 16:9 702 , 498 0 , 702 498 , 996 , ? ? ? Horn 2 & Œ œ Œ & œ œ Œ œ Œ & ¢ œ œ ˙

b.68 9:7 81:64 5:4 6:5 32:27 7:6 8:7 9:8 10:9 16:15 256:243 28:27 435 408 386 316 294 267 231 204 182 112 90 63

Sop. & #œ #œ #œ œ œ œ #œ œ œ œ œ œ

9:8 4:3 1:1 32:27 204 , 498 , 0 , 294 , Hn.1 ° Œ Œ & #œ ˙ œ Œ Œ Œ œ Œ #œ 1:1 3:2 16:9 3:2 16:9 4:3 128:81 32:27 0 702 996 702 996 498 792 294 , , , œ , Hn.2 ? ? œ Œ & Œ Œ œ œ ¢ ˙ œ œ œ

Example 36. Descending line of 24 ratios in Door of Many Mysteries.

159 Tuning Implementation

This is the first score during the candidature that was presented to performers with a sound file of the specific tuning. Unfortunately this sound file is now lost. It was prepared on a Kurzweil K2000 keyboard and recorded onto a Sony

Mini Disc. Neither the Mini Disc player or the disc itself are now useable. This is also the first composition performed during this DCA that has a large pitch world, in this case twenty-seven pitches. A composition with nine to twelve pitches could be considered a small pitch world. Listening to the sound increased familiarity with the sound world of the work and improved the quality of pitch reproduction. The recording session in late 2007, a year after the live performance, felt much easier for all three performers than the original rehearsals. This was also partly due to having performed additional works in the interim and gaining more comfort and confidence with just intonation performance in general.

Prior to rehearsals I prepared fingerings for the two horn parts. Some pitches required additional adjustment, done by lipping and right hand movement.

Adjustments required sufficient aural awareness to find the correct tuning. This process was aided by my participation at rehearsals as I could quickly and effectively clarify the tuning by comment or demonstration. The participation or presence of someone immersed in the pitch world of a just intonation

160 composition is as great an asset as an accurate sound file. After listening to the sound file I changed some pitches to clarify the harmony. I also changed some pitches during the three rehearsals to provide better aural cues for the soprano.

This work was written 2006 and was performed at the University of

Wollongong, Faculty of Creative Arts on 19 October 2006 by Wendy Dixon soprano; Michael H Dixon & Lisa Wynne-Allen, horns. It was also performed by the same performers at the University of New South Wales on 17 January

2007 and recorded at the Music Facility of ASCHAM School in November 2007.

161 3.2.3.! 7-limit scale-based just intonation, solo trombone

• 14 pitches - 8:7 interval featured

• Fanfare of three arpeggios each with ratios 4:5:6:7

• Nine-note scale

• One flexibly pitched instrument - brass

Pitch List

This work, Stillness of Unceasing Activity, uses harmonics that are in the normal range of the trombone as well as intervals associated with them, lower in the range. The composition came about after a discussion with trombonist Greg van der Struik who has often sought out new works by Australian composers.

Tuning Requirements

The composition has two pitch features. The first is the opening fanfare with harmonics 4, 5, 6, 7 starting on the first position of the B! trombone. This is followed in the next beat with the same harmonics starting on a low third

162 position, making the lower A! a perfect octave below the 7th harmonic of the B! series. The whole tone between the first B! and the first note of beat two is the ratio 8:7. The same harmonics are then played another 8:7 below, making the

G! in beat three a perfect octave below the G! of beat two. This requires using a slide position somewhat closer to six than five. Ten ratio pitches are used in the opening bar. Example 37 illustrates this with each beat shown on a separate line to see the detail of slide position, ratios and cents clearly.

Example 37. Harmonics 4, 5, 6, 7 of three fundamentals.

The second pitch feature in the work is the use of a 9-note scale in each musical section. The scales are not precisely the same and section two (Verse Two in the score) has one note missing. The three versions of scale are seen in Example 38.

163 Example 38. 9-note scale with three tonics. The seventh note of the G! scale is omitted in the composition.

Tuning Implementation

The primary need for the player is to present harmonics 4-7 in their natural state without adjusting the slide position. Familiarity with the intervals between harmonics in the first bar of the work (Example 37) provides most of the aural knowledge required. The 8:7 interval between first pitches of each opening arpeggio is featured throughout the work. In terms of setting the overall tuning, the musician who premiered the work played the B! pitch, ratio

8:7, in first position relative to A = 440 Hz, as per standard practice. This set the

B! at approximately 467 Hz and meant that A! 1:1 was approximately 409 Hz rather than 415 Hz when played in standard equal tempered relation to A 440

Hz. The alternative would be to tune the entire trombone higher than usual.

164 The performer found the computer generated sound file helpful in clarifying the tuning and the timing of the metric modulations, which are a feature of the opening fanfare. I attended a rehearsal on 18th April 2007 to further assist and clarify these matters.

The score has quartertone symbols to indicate notes far outside standard tuning parameters and HEWM notation (Appendix C) to indicate precise tuning and the relationships between pitches. There is one notational error in the score. The ratio 49:48 is 36 cents above 1:1, A! yet is notated as B" in the score. Its harmonic relation is a narrow minor third, ratio 7:6 above G!, therefore should be notated as B!". This looks clumsy so it is shown as A# in both the Pitch List and Example

38 above. The error is retained in the folio accompanying this thesis because that is the score the trombone player worked from. The pitch is correct on the sound file.

This work was written in 2006 and has been performed three times. The premiere was at the University of Wollongong, Faculty of Creative Arts on 26

April 2007 performed by Greg van der Struik. The second performance was at the Bundanoon Memorial Hall, Bundanoon on 18 August 2007. The third was at the Wendy Dixon Studio, Haymarket, Sydney on 17 May 2008.

165 3.2.4.! 7-limit scale-based just intonation, polytonal treatment

• 34 pitches - use of easily available horn harmonics

• Polytonality; pitch beatings

• Nine-note scales

• Flexibly pitched instruments - four brass

Pitch List

This work, We are Sufficient, demonstrates the use of a number of available 7th harmonics in three contexts: chordal; pitch ‘beating’ and polytonality. The pitch range for each horn player is within standard orchestration limits.

166 Tuning Requirements

Harmonics 2 to 9 are the main pitches in each horn part. The fourth horn uses the D harmonic series, the third horn uses the F! harmonic series, the second horn uses the A harmonic series and the first horn the E harmonic series. All four series can be viewed as harmonics of D, thus the four harmonic series tonics provide a ratio chord 4:5:6:9. Each horn part is not restricted to one harmonic series except in the final section of the music.

Two nine-note scales are used for melodic material which mirror each other in interval succession. They are shown in Example 39 without specific tunings because the ratios change according to the musical context.

Example 39. Interval scales in We are Sufficient.

Tuning Implementation

The third horn player is instructed to adjust the third valve slide of the B" horn lower than normal. This ensures that the 4th harmonic of the F! tube length, played on the B" horn with valves 2 and 3, matches the 5th harmonic of the D

167 horn. Illustration 9 shows the valve slide setting contrasted with the normal setting (B! side shown only).

! !

Illustration 9. Normal setting of valve slides, B! horn shown on the left. Valve slide setting for We are Sufficient, 3rd valve of the F side removed for clarity.

In the opening phrase of We are Sufficient horns two and four play the following sequence of harmonics: 3, 2, 5, 7. Horn three uses the same sequence for the first two notes one octave higher in the range, therefore harmonics 6 and 4, then the next two notes as harmonics 5 and 7. The first horn plays a mixture of octaves though essentially the same sequence: harmonics 12, 8, 10, 7. The relation between the horns in the opening is the chord with ratios 2:3:5:9. This is clearly seen in Example 40 bar two, first beat.

168 b.1 Ratios: 27:16 9:8 45:32 63:32 Cents (to D): 906 204 590 1173 Horn 1 ° 2 5 4 & 4 Œ ‰ œ 8 œ #œ. 4 Œ Œ Fingerings: F:2J ˙ Harmonics: 12 8 10 7 9:8 3:2 15:8 21:16 204 702 1088 471 #œ. Horn 2 ? 2 œ 5 4 Œ 4 Œ ‰ J 8 œ 4 & ˙ Œ Bb:2 3 2 5 7

15:8 5:4 25:16 35:32 1088 386 773 155 #œ Horn 3 ? 2 5#œ #œ. 4 4 Œ ‰ J 8 4 Œ & Œ Bb:2,3 ˙ 6 4 5 7

3:2 1:1 5:4 7:4 702 0 386 969 ˙ Horn 4 ? 2 j 5 #œ. 4 4 Œ ‰ œ 8 4 Œ Œ ¢ F:1,2 œ 3 2 5 7

Example 40. Harmonic series relations between the horns, bars 1 to 3 in We are Sufficient. Ratios and cents from the tonic D are shown above the notes. Fingerings and harmonics for each horn are shown underneath.

169 Following the opening phrase the third horn has a three note melodic fragment played on harmonics 8, 7 and 8, a gesture copied in the other horn parts in other keys. The indicated fingerings ensure that the players present the required tuning of the intervals. Example 41 illustrates this.

Additionally the music features a number of ‘beatings’ between close pitches, mostly in pairs though occasionally with three horns. Example 42 shows the small pitch difference expected between horns one and two.

170 Ratios: 5:4 35:32 5:4 b.4 Cents (to D): 386 155 386 Horn 3 ° 2 5 j 4 2 & 4 Œ #œ 8 œ #œ 4#˙ Ó 4 Fingering: Bb:2,3 - œ 1:1 0 ˙ œ ˙ Horn 4 ? 42 85 Œ. 4 ˙ 42 ¢ F:1,2

9:8 63:32 9:8 204 1173 204 Hn 1 ° 2 Œ 5 j 4 Ó & 4 œ 8 œ œ œ 4 ˙ F:2 - 1:1 0 Hn 2 ? 42 ∑ 85 ∑ 4 Ó ˙ F:1,2 1:1 7:4 0 969 ˙ œ ˙ œ- . Hn 4 œ ? 42 85 Œ. 4 J ¢ F:1,2 Example 41. Melodic fragments with harmonics 8, 7, 8 in three keys. Ratios and cents from the tonic D are shown above the notes. Fingerings are shown underneath.

Ratios: 5:3 4:3 32:27 28:27 Cents (to D): b.35 884 498 294 63 U Hn 1 ° 7 Œ Œ œ ‰ ‰ ‰ j & 8 Fingerings: Bb:1,2 F:0œ œ bœ 4:3 21:20 'beating' 498 85 hns 1 & 2 Hn 2 7 U & 8 Œ Œ j bœ. ˙ Bb:œ1,2 œ Bb:1,3 5:4 386 Hn 3 7 U & 8 Œ #œ œ ‰ Œ. Ó Bb:2,3 1:1 0 U Hn 4 ? 87 Œ œ œ ‰ Œ. Ó ¢ F:1,2 Example 42. Pitch beatings between horns 1 and 2.

171 A sound file of the full score was provided to the performers though not listened to by them, however I found it useful to check that the pitches and notation in the score were correct. The fingerings supplied to the performers were relied on more than aural knowledge yet the tuning worked well. For the

2006 performance the parts were notated with HEWM markings (Appendix C).

This notation was understood by one of the performers. A lattice style pitch chart was also supplied. During the rehearsals on April 16 and 17, with myself present, five minutes or so were spent on the opening three bars to gain familiarity with the sound of the 7th harmonic in each part and in the chordal context. One player queried a fingering, which was simply a typographical error. All players were comfortable with the tuning requirements. I agreed to conduct the performance to allow the performers to concentrate more on the tuning and less on ensemble issues.

For the 2010 performance players were supplied with published parts that had cent values assigned to every pitch. I left the HEWM markings (Appendix C) in the score, which could be used as a reference. Two rehearsals were ample.

This work, We are Sufficient was written 2006 and has been performed three times. The first performance was at the University of Wollongong, Faculty of

Creative Arts on 26 April 2007 by Heath Parkinson, Kate Herman, Alex Love and Francesco Lo Surdo, horns; Michael H Dixon, conductor. The same group

172 performed it for a horn class at the Sydney Conservatorium of Music on 30

April 2007. A third performance took place for the International Horn Society

Symposium at the Queensland Conservatorium of Music, Griffith University on

21 July 2010 with players Michael H Dixon, Neil Favell, Graham Nichols and

Francesco Lo Surdo.

173 3.2.5.! 13-limit scale-based just intonation combined with 12tet

• 24 pitches - interaction of 12tet and non-standard tuning

• Pitch beatings; bi-tonality

• Nine-note scales - multi-crook use of horn

• One fixed pitch instrument - one brass

Pitch List

This work, Self-balance, uses a large range of harmonics in the horn part from four separate tube lengths as well as some right hand technique to alter pitch and tone colour. Pitch beatings are exploited between equal tempered pitches and some horn harmonics.

174 Tuning Requirements

The horn is considered a multi-crook instrument for this work. Four valve settings are used: those for horn in A (B! horn, valve 2); horn in A! (B! horn, valve 1); horn in E (F horn, valve 2) and horn in E! (F horn, valve 1). Harmonics

1 to 13 are required on the A! horn; harmonics 2 to 14 on the E! horn; harmonics

4 to 16 on the E horn and harmonics 5 to 13 on the A horn.

Tuning Implementation

The horn player requires an aural and kinaesthetic awareness of harmonics 1 to

16. The player needs to use four tube lengths (crooks or valve settings).

Example 43 illustrates ‘crook’ changes at bars 94 and 95.

Right hand technique is required to lower some pitches. This lowering also has an accompanying tone colour change, which adds to the mood of the final section. This technique can be seen in the same example (43) at bars 97, 99 and

100.

Additionally pitch beatings occur between the horn and the marimba. These

‘off unisons’ are integral to the work and can be achieved by simply by not tempering certain harmonics as shown in the same example (43), bar 100.

175 93 Horn œ µœ™ & #œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ µœ™ ™ #E:œ 9 + œ10 œ µE: 10 + 11 E: 13 14 A: 13 œ œ œ œ œ œ b œ œ œ nœ œ œ œ œ œnœ œ œ &b b Marimba ? œ œ œ œ œ œ œ œ œ bbb ∑

95 half stop open Hn. { œ µœ™ & µœ™ ™ 4 Œ 43 42 45 E: 13 14 A: 13 œ œ E:8 7 œ6 µ˙7 ™ ˙6 œ œ œ b œ œ œnœ œ œ &b b 4 ∑ 43 ∑ 42 Œ 45 Mar. nœ œ ? œ œ œ œ œ œ œ œ œ 4 3 2 5 bbb 4 ∑ 4 ∑ 4 nœ nœ 4 {99 half stop open Hn. 5 Œ 4 & 4 œ 4 6œ µœ7 7 œ6 b 5 4 &b b4 #œ œ œ œ #œ œ œ œ 4 Mar. nœ œ ? 5 #œ œ œ œ #œ œ œ œ 4 bbb4 nœ nœ 4 don't adjust tuning 100 { half stop open 'beating' with marimba Hn. & 4 œ µœ6 #˙5 b &b b4#œ œ œ œ #œ œ œ œ Mar. #œ œ œ œ ? 4 #œ œ œ œ bbb4

Example 43. Multi crook use. The horn length changes from E horn to A horn in bars 94 and 95. Right hand pitch and colour change in bar 97, 99 and 100. { Pitch 'beating' with the marimba in bar 100.

176 The notation of the horn part in this work is slightly different to the other scores in the portfolio. The fingerings as such are not given for pitches, rather the horn length and sometimes the harmonic series number are supplied. Some quartertone symbols are used in the notation to indicate pitches that are too ambiguous when notated with chromatic symbols. These aspects are also illustrated in Example 43 (p176).

This work was written 2006 and has been performed twice. The premiere took place at College Chapel, University of Sydney, 30 November 2006 performed by

Dominic Harvey, horn and Mary Broughton, marimba. The second performance was by Angharad Johnson, horn and Veronica Walshaw, marimba at the ANU

School of Music on 10 November 2009.

177 178 3.3. Extended just intonation compositions: Group C

Standard brass valved instruments can be regarded as multi-crooked instruments. Further, the adjustment of valve slides can provide harmonic series ratios between tube lengths. The pitch of each harmonic from each series can be calculated and related to a central reference pitch (tonic, ratio 1:1).

Furthermore fine-tuning by lipping and hand technique can be anticipated and indicated in the score.

There are three compositions in this group all involving highly specific instructions for practical realisation. The two versions of A Hundred Valleys require the most complete use of brass in this portfolio. However, Seven Small

Wheels Revolving is the culmination of writing for the horn as a multi-crook instrument due to the extent of preparations done prior to composing the score.

The ratios between tube lengths were calculated and the ratios of harmonics 1 to 16 of each series charted. The composition was designed to be played without lip or right hand adjustments, yet the work has the most extensive use of microtonal pitch movement in the folio.

179 3.3.1.! 13-limit chord-based just intonation, solo trumpet

• 58 ratios - selected from HSIS-13

• Multi-crook treatment of a 19-tone four-valve trumpet

• Arpeggiated chords treated melodically; ratios can change in function

• Flexibly pitched instrument - one brass

Pitch List (on next page)

180 Ratio: 1:1 225:224 169:168 25:24 13:12 243:224 9:8 81:70 65:56 Cents: 0 8 10 70 139 141 204 253 258 & b˙ b˙ b˙ n˙ µ˙ Bb˙ n˙ µ˙ µ˙

7:6 169:144 6:5 135:112 169:140 175:144 5:4 91:72 81:64 267 277 316 323 326 338 386 405 408 & Bb˙ #˙ b˙ b˙ b˙ B˙ n˙ ˙ ˙

13:10 125:96 4:3 169:126 27:20 65:48 1215:896 25:18 39:28 454 457 498 508 520 525 527 569 574 & µ˙ µ˙ b˙ b˙ b˙ b˙ b˙ B˙ n˙

45:32 169:120 36:25 13:9 81:56 325:224 35:24 3:2 590 593 631 637 639 644 653 702

& ˙ b˙ µ˙ B˙ µ˙ µ˙ B˙ n˙

169:112 65:42 25:16 8:5 45:28 169:105 13:8 5:3 712 756 773 814 821 824 841 884

& ˙ µ˙ #˙ b˙ b˙ b˙ B˙ n˙

845:504 27:16 26:15 243:140 125:72 338:189 9:5 65:36 895 906 952 955 955 1006 1018 1023

& ˙ ˙ µ˙ µ˙ µ˙ b˙ b˙ b˙

405:224 11:6 13:7 15:8 40:21 48:25 27:14 325:168 1025 1049 1072 1088 1116 1129 1137 1142 & b˙ B˙ n˙ ˙ ˙ µ˙ µ˙ µ˙

181 In 2007 Trumpet player Stephen Altoft made a request to the global microtonal community for compositions in 19-tone equal temperament. This work, A

Hundred Valleys I, is an exploration of some just intonation possibilities for a trumpet configured to play in 19-tones per octave (19tet). It seemed reasonable to expect that pitch could be manipulated in the same way that is done with a standard trumpet: by using a movable third valve slide, and often the first, to lower certain notes and by lipping. In this way many pitches outside the 19 per octave are attainable.

Tuning Requirements

The 19-tone trumpet is configured so that depressing the shortest valve slide, the fourth, lowers the pitch from the main tube by 63 cents. This contrasts with

50 cents lower from a quartertone trumpet. Depressing the second valve lowers the pitch by 126 cents in contrast with 100 cents from a quartertone or standard trumpet. The combination of second and fourth valves lowers the pitch by 189 cents in contrast with 150 cents for a quartertone trumpet. On a 19-tone trumpet the combination of valves two and four lowers pitch by the same amount as the

first valve whereas the first valve on both a quartertone and standard trumpet lowers the pitch by 200 cents. However, once a combination of valves is used the distance is not exactly as stated, though the pitch can be manipulated by applying the movable tuning slides and by lipping technique. It was assumed

182 that movable valve slides would compensate for the sharpness of valve combinations including all 4 valves. The full details of the 19-tone trumpet are available at the website pages Microtonal Trumpet Project (n.d.). The cent values of all the available harmonics in range from low written F below the treble stave to high D above the treble stave were calculated then compared to pitches from

HSIS-13 (Appendix B). Pitches were selected that deviated no more than 8 cents up or down from the harmonics available on a 19tet trumpet in B! tonic.

Tuning Implementation

The performer was provided with a sound file at approximately the desired tempo then a half speed version later in 2008 to assist in preparations for recording the work. The prescribed fingerings were also of considerable use to the performer to accurately reproduce the intonation.

The score was notated within 19tet parameters with the addition of quartertone symbols to indicate that a natural 7th harmonic should be used. Example 44 illustrates this notation (bars 1 to 5). The fingerings required for selected tube lengths are indicated below the notes, each fingering remaining till countermanded by the next. The particular harmonics are shown below the

fingerings.

183 Example 44 Notation, fingerings and harmonics in the first phrase of A Hundred Valleys I.

This work was written 2008 and recorded to CD in 2009 by Stephen Altoft, trumpet, pending release in 2011.

184 3.3.2.! 13-limit chord-based just intonation, solo horn

• 60 ratios - selected from HSIS-13

• Complete use of horn tuning techniques - valve slide, lip, right hand

• Arpeggiated chords treated melodically; ratios can change in function

• Flexibly pitched instrument - one brass

Pitch List (on next page)

The intention in composing this solo horn work, A Hundred Valleys II, was to work further with the musical material of A Hundred Valleys I (3.3.1) and extend performance opportunities for the composition.

185 Ratio: 1:1 225:224 169:168 25:24 13:12 243:224 9:8 81:70 65:56 Cents: 0 8 10 70 139 141 204 253 258

& ˙ ˙ ˙ #˙ B˙ ˜˙ n˙ µ˙ µ˙

7:6 169:144 6:5 135:112 169:140 175:144 5:4 91:72 81:64 267 277 316 323 326 338 386 405 408 & Bb˙ b˙ b˙ b˙ b˙ B˙ n˙ ˙ ˙

13:10 125:96 4:3 169:126 27:20 65:48 1215:896 11:8 25:18 454 457 498 508 520 525 527 551 569 & B˙ µ˙ n˙ ˙ ˙ ˙ ˙ µ˙ #˙

39:28 45:32 169:120 36:25 13:9 81:56 325:224 35:24 3:2 574 590 593 631 637 639 644 653 702 b˙ b˙ ˙ ˙ ˙ n˙ & #˙ #˙ B ˜˙ B B

169:112 65:42 25:16 8:5 45:28 169:105 13:8 5:3 712 756 773 814 821 824 841 884 ˙ ˙ #˙ b˙ b˙ b˙ ˙ n˙ & µ B

845:504 27:16 26:15 243:140 125:72 338:189 9:5 65:36 895 906 952 955 955 1006 1018 1023 ˙ ˙ B˙ ˙ ˙ n˙ ˙ ˙ & µ µ

405:224 11:6 13:7 15:8 40:21 48:25 27:14 325:168 1025 1049 1072 1088 1116 1129 1137 1142 ˙ µ˙ #˙ #˙ #˙ b˙ ˜˙ ˜˙ &

186 Tuning Requirements

In the composition there are 16 pitches in lowest octave from the 1st harmonic,

G; 16 in the second octave from the 2nd harmonic; 31 in the third octave from the

4th harmonic; 36 in the 4th octave from the 8th harmonic. The highest note is the

16th harmonic of G.

Tuning Implementation

The tuning is achieved through three methods: valve slide adjustments at the outset; right hand technique to shift pitches up and down from their natural pitch focus; lipping (embouchure manipulation) for further fine-tuning. The movement of the right hand covering the bell will lower pitch. The closer this comes to a full closing the more muffled the tone quality becomes.

Straightening the right hand or removing it from the bell will raise the pitch.

Only a small amount of straightening is possible though this can make an appreciable difference. It is noted that the shape of the right hand in the bell differs from player to player. Lipping can retain a clear tonal focus in the sound up to a point but thereafter will affect the tone quality. Lower pitches are easier to manipulate and retain tonal focus than higher pitches, so the amount of required lipping varies over the range. Changes in tone colour are inevitable in certain passages of this work simply because of the extreme lowering of pitches with the right hand and lips.

187 All the pitches are in just intonation and are close to the natural harmonics arising out of each tube length (crook). Some slides need adjustments from standard settings to achieve optimal intonation. The adjustments are shown in

Illustration 10 and detailed in Table 5 (both on page 189).

! ! Illustration 10. Valve slide settings for A Hundred Valleys II. F side on the left. B! side on the right.

Table 5. Tuning of valve slides in detail for A Hundred Valleys II.

Written D (concert G) is the central pitch of the piece, 1/1. The G horn harmonic series notes will work best played on B! horn, valves 1 and 2 (written as B!:1,2 in the score).

1. Tune the main slide lower than normal. 2. Push B! horn, 1st valve all the way in.!! 1. and 2. compensate each other to equal the CD pitch. 3. Pull B! horn, 3rd valve most of the way out. Check the tuning by playing the 7th note of the piece with both B!:1,2 and B!:2,3. (The 13th harmonic of B!:1,2 equals the 14th harmonic of B!:2,3.)!! ! B!:2,3 fundamental = ratio 13:7 to written D. 4. Push F horn, 2nd valve all the way in. Check the tuning by playing the 4th note of line three with F:2 then the 1st note of Section 3 with B!:1,2. Adjust F:2 as required. (The 6th harmonic of F:2 equals the 5th harmonic of B!:1,2.)!! ! F:2 fundamental = ratio 5:3 to written D. 5. Pull F horn, 3rd valve most of the way out. 6. Push F horn, 1st valve all the way in. Check the tuning for 5. and 6. by playing the 1st note of line 5 with F:1,3 then the 2nd note of the piece using B!:1,2. Adjust F horn slides as required. (F:1,3 harmonics 6 and 9 equal B!:1,2 harmonics 4 and 6.)! !! F:1,3 fundamental = ratio 4:3 to written D.

188 Prescribed fingerings are included along with suggestions to lip up or down which are marked as high (h.) low (l.) then normal (norm.) as required. They apply until rescinded, as do lipping instructions. Shifting two fingers in the right hand from the wall of the horn and across the palm will lower the pitch enough to achieve the desired intonation in some arpeggios. A couple of passages require moving the wrist across the bell as well as lipping to get the pitch low enough. This results in a change of tone colour, an interesting contrast. The right hand is straightened to raise the pitch for one passage.

Usually the right hand is held in a curved shape. The movement of the right hand reduces the amount of lipping required, a welcome technique as any upwards adjustment by the lips adds considerably to the stress on the muscles and reduction of blood flow, in turn affecting endurance, flexibility and tone focus.

Some of the fingerings were decided at the outset of composition. Some others were arrived at after extensive practice with the sound file. Some changes were made after working on a fingering chart and noticing some alternatives. Prior to the 2008 performance I practiced the horn part, playing along with the sound

file some 40 to 50 times as well as singing and lip buzzing along with it whilst travelling in the car. One year later I adjusted some fingerings because I found

189 some passages easier than before. This was due to practicing and performing on the natural, valveless horn for other projects bringing about more skill in pitch manipulation.

The music has a series of arpeggios linked to each other by intervals in the harmonic series. For instance the eleventh note of the piece is a 5th harmonic below the sixth note of the piece. The tuning of the valve slides makes many of the linked arpeggios easy to achieve with the desired intonation. The arpeggios

I found more difficult to achieve were the ones with right hand and lip adjustments, in particular bars 42 to 43 and bar 54. The passage from bars 20 to

24 was awkward to achieve due to changing from a straight right hand position to bend the pitches up, back to a normal hand position whilst pitching a number of 13th harmonics. These harmonics are difficult to pitch due to their relative unfamiliarity and made more difficult due to the changes of fundamentals.

Example 45 shows the difficult arpeggios. The notation indicates cents above pitches and fingerings and harmonics below.

190 19 702 204 590 323 8 821 1025 œ 6 œ#œ ? #œ œ œ ? œ 8 ? 6 & 8 œ™ & bœ™ ™ bœ™ & 8 œ bœ™ & œ ‰ 8 Bb:1,2 Bb:2,3 F:1,2 Bb:2,3 F:1(high) J J 12 9 12 3 12 12 10 2 3 2 9 2 9 œ3

1025 841 637 139 139 590 1023 23 821 1025 œ œ œ 6 œ bœ 7 B Bœ œ Bœ œ Bœ 8 & 8 bœ œ œ 8 B B 8 F:1(high) F:1(normal) Bb:2,3 F:0 F:1,3 F:0 F:1,3 F:0 8 9 10 12 14 18 18 13 13 13 13 13 13

25 277 10 895 508 8 œ œ ? ? 5 12 & 8 œ & œ œ 8 Œ & 8 F:1,3 œ™ œ™ 14 12 10 2 8 10 2

42 639 253 955 1137 639 12 ? ? & 8 œ µœ œ & µœ ˜œ œ µœ µœ™ ˜œ™ & F:1,2,3˜ Bb:0(v.low) Bb:0(v.low) F:1,2,3˜ Bb:0(v.low) Bb:0(v.low) F:1,2,3 9 2 6 12 9 6 2 3

43 639 141 408 906 204 0 3 œ œ œ#œ œ œ#œ 10 #œ ? 17 & 4 œ˜ œ œ˜ ˜ ˜ œ œ˜ 8 œ œ™ Œ Œ & 8 ˜ ˜ ˜ ˜ ˜ B :1,2 œ F:1,2,3 F:1 F:1,2,3 F:1 Bb:1,2 F:1 F:1,2,3 F:1 b F:1,2 Bb:1,2 (high) (high) 9 11 9 11 10 11 9 11 10 9 6 2

54 884 1116 756 525 258 17 #œ j 8 œ ? #œ ? œ œ & œ œ &#œ œ œ µœ œ µ œ µ Œ & F:1,2J F:0 µœ #œ µ Bb:1,2 F:2 (high) (norm.) F:1,3 F:1,2(h.)µœF:1(v.low) 8 8 2 6 8 12 8 8 7 8 7 4 4 3 2 3

Example 45. Arpeggios with the most practical difficulty: bars 19-26; 42-43; 54 in A Hundred Valleys II.

191 This composition, A Hundred Valleys II, written in 2008, was performed by the composer and recorded for the Microtonal Workshop “Investigating listening and performance strategies for microtonal composition” at Australian National

University, Canberra School of Music, 15 December 2008. The work was re- recorded at the Australian Institute of Music for the same microtonal project 19

& 21 December 2009. A performance was included in a lecture at the

International Horn Society Symposium, Griffith University, Queensland

Conservatorium of Music on 21 July 2010 by Michael H Dixon.

The two recordings were used by David Gunawan and Associate Professor

Emery Schubert, with the assistance of Jonathan Jayanthakumar, to analyse the accuracy of my performance in relation to the ideal just intonation. The sound

file I had prepared from a tuning crib was the measure for comparison.

Preliminary results indicate similar accuracy to that measured and reported in the paper ‘Noteview: A Computer Program for the Analysis of single-line musical performances’ and the mean difference in accuracy between two performances of Twinkle, Twinkle Little Star was 7 cents (Gunawan & Schubert

2010).

192 3.3.3.! 13-limit chord-based just intonation

• 50 ratios - selected from HSIS-16

• Multi-crook - 12 tube lengths required from horn

• Each ratio can change in chord function

• Flexibly pitched instruments - one brass

Pitch List (next page)

This work, Seven Small Wheels Revolving, is a culmination of writing for the horn as a multi-crook instrument in this folio. Part of the purpose in writing this work was to create a microtonal piece that could be played with minimal right- hand or lip adjustments ensuring all the pitches could be produced with a clear tone. There would be no necessity for tone colour changes resulting from bending harmonics to fit the intonation requirements.

193 Ratio: 1:1 121:120 65:64 49:48 33:32 25:24 135:128 16:15 15:14 Cents: 0 14 27 36 53 70 92 112 119

& ˙ ˙ ˙ Bb˙ µ˙ #˙ #˙ b˙ #˙ 35:32 11:10 10:9 9:8 55:48 65:56 7:6 77:64 35:28 155 165 182 204 236 258 267 275 284

& ˜˙ ˜˙ ˙ ˙ µ˙ µ˙ Bb˙ #˙ #˙ 39:32 11:9 99:80 5:4 143:112 165:128 21:16 4:3 342 347 365 386 423 440 471 498 & B˙ B˙ B˙ n˙ ˙ µ˙ Bb˙ b˙

65:48 11:8 45:32 10:7 35:24 3:2 121:80 195:128 525 551 590 617 653 702 716 729 & b˙ B˙ n˙ ˙ B˙ n˙ ˙ µ˙

14:9 25:16 11:7 77:48 105:64 5:3 27:16 26:15 765 773 782 818 857 884 906 952 & Bb˙ #˙ #˙ b˙ B˙ n˙ ˙ B˙

7:4 16:9 25:14 143:80 28:15 15:8 91:48 55:28 969 996 1004 1006 1081 1088 1107 1169 b˙ b˙ #˙ b˙ b˙ n˙ ˙ ˙ & B µ

194 Tuning Requirements

A large pitch world was selected from forty fundamentals using harmonics 4 to

16, each harmonic series having one pitch in common with an originating series. The originating series was notated with C as the fundamental for notational simplicity, though any key note would be valid. This pitch world is referred to as HSIS-16. This is an extension of the scheme HSIS-13 (Chapter 3, p106-109) used for five other compositions in the portfolio. Both are 13-limit just intonation schemes though HSIS-16 has harmonics 14, 15 and 16 in addition.

From HSIS-16 a smaller set of fundamentals was chosen. The ratios being achievable by adjusting the valve slides of a standard double horn. The standard sharpness of valve combinations was assumed as a starting point for the calculations. The fundamentals are shown in Table 6 on the next page.

195 Table 6. Valve slide settings in Seven Small Wheels Revolving.

Valve(s) Cents from Cents Ideal cents Ideal ratio Adjustment

(fingering) F horn positive to F horn to F horn hand or lip

F:0 0 0 0 1:1 0

F:2 -112 1088 1088 15:8 0

F:1 -204 996 996 16:9 0

F:1,2 -306 894 ------

F:3 -316 884 884 5:3 0

F:2,3 -413 787 782 11:7 -5

F:1,3 -495 705 702 3:2 -3

F:1,2,3 -592 608 617 10:7 +9

B!:0 498 498 498 4:3 0 B!:2 386 386 386 5:4 0

B!:1 267 267 267 7:6 0

B!:3 212 212 ------

B!:1,2 161 161 165 11:10 +4

B!:2,3 112 112 112 16:15 0

B!:1,3 9 9 14 121:120 +5

B!:1,2,3 -84 1116 ------0

196 189 pitches are available to the player with this setup, none requiring more than a 9-cent lip or hand adjustment. Only a few pitches are in the very low and very high range, the rest fall within the standard recital range. There are 16 pitches in the lowest octave; 28 in the octave preceding written middle C; 49 in the octave starting at middle C; 80 in the octave above that and 15 from high C upwards.

Many of these pitches are available from more than one tube length. For instance the written E (bottom line of the treble clef), ratio 5:4 (386 cents from F) be played on four tube lengths: those with fingerings F:1,2,3 7th harmonic; F:3

6th harmonic; F:0 5th harmonic and B!:2 4th harmonic. This convergence of harmonics offers chord changing possibilities which are used in Sections 2, 3, and 4 of the composition. For example the bassoon C in bars 23 and 24 functions as both the 3rd harmonic of the F series and the 4th harmonic of the C series. The E in bars 25 to 27 is the 5th harmonic of the C series, the 4th harmonic of the E series and the 3rd harmonic of the A series. The horn part harmonises the bassoon part with a number of harmonics from each series as illustrated in

Example 46.

197 b.23 Ratios: 3:2 Cents: 702 Bassoon ? 5 4 -˙ ˙™ -˙ ˙™ 5:4 3:2 7:4 3:2 5:4 21:16 9:8 27:16 3:2 21:16 386 702 969 702 386 471 204 906 702 471 Horn & 5 4 œ Bbœ œ œ œ Fingerings: F:0œ (F horn) œ µF:1,3œ (Cœ horn) µœ Harmonics: 5 6 7 6 5 7 6 9 8 7

15:8 b.25 1088 Bsn. ? ˙- ˙™ ˙- ˙™

27:16 33:32 15:8 33:32 75:64 165:128 135:128 105:64 15:8 906 53 1088 53 275 439 92 857 1088 Hn. & #œ œ œ µœ œ µœ µ #œ Bœ œ <µF:1,3>œ (C horn) F:2 (E horn) 7 9 11 10 11 10 11 9 7 8

b.27 1:1 0 Bsn. ? ˙- ˙ œ

35:32 25:16 5:4 155 773 386 Hn. & Bœ #œ Bb!" (A horn) ˙™ 7 5 4

Example 46. Converging harmonics and chord changes in Seven Small Wheels Revolving.

198 A slightly different process using microtonal shifts occurs in Sections Five, Six and Seven. For instance in bar 62 the microtonal shift in the horn part from ratio

33:28 to 7:6 is set up harmonically by the chord sequence preceding it from bar

60. The horn note in bar 60 is the 6th harmonic of the bassoon A! and the 7th harmonic of F in bar 61. The horn moves to the 11th harmonic which is also the

14th harmonic of the bassoon’s C" in bar 62. The horn’s pitch, ratio 33:28 is therefore the 3rd harmonic shifting to ratio 7:6 the 2nd harmonic. Example 47 illustrates this process.

Example 47. Microtonal shift in the horn part, bar 62, Seven Small Wheels Revolving.

199 The bassoon player plays eighteen pitches per octave, most of which appear in all three octaves in the range. They are presented in Table 7 which shows the note names and ratios, cent values in brackets.

! Table 7. Bassoon pitches in Seven Small Wheels Revolving.

F 1:1 (0c) G! 16:15 (112c) G" 11:10 (165c) G 10:9 (182c)

G 9:8 (204c) A! 7:6 (267c) G 75:64 (274c) G 5:4 (386c)

B! 21:16 (471c) B! 4:3 (498c) B 10:7 (617c) C 3:2 (702c)

C! 11:7 (787c) D 5:3 (884c) E! 7:4 (969c) E! 16:9 (996c)

E 15:8 (1088c) F" 55:28 (1169c)

Twelve of them are octaves of the horn’s fundamentals, the other six assist with the harmony. For example the G! 75:64 (274) is the 5th harmonic of the horn’s E series.

Tuning implementation

The horn player requires a knowledge of the harmonic series and be able to navigate a large pitch world. The horn player must make tuning adjustments: by altering valve slides, and make fine adjustments small right hand movements or lipping. The valve slide settings are shown in Illustration 11 and Table 8. The horn player must follow the prescribed fingerings assiduously.

200 Practice in unison with a sound file made from a computer generated tuning crib greatly assisted preparations.

Once the tuning of the valve slides is completed, all the performance information is supplied in the score: fingerings and small adjustments, ratios and cents for each pitch and courtesy arrows to indicate a small pitch movement when the pitches had the same note name (staff position).

! !

Illustration 11. Valve slide settings for Seven Small Wheels Revolving F side settings on left, B! settings on right.

201 ! Table 8. Valve slide settings for Seven Small Wheels Revolving.

F:0 = concert F at standard pitch relative to A440Hz.

F:2 16th harmonic = F:0 15th harmonic

F:1 9th harmonic = F:0 8th harmonic

F:3 3rd harmonic = F:0 5th harmonic

F:2,3 14th harmonic = F:0 11th harmonic

F;1,3 8th harmonic = F:0 6th harmonic

F:1,2,3 7th harmonic = F:0 5th harmonic

B!:0 6th harmonic = F:0 8th harmonic

B!:2 6th harmonic = F:2 8th harmonic

B!:1 8th harmonic = B!:0 7th harmonic

B!:2,3 5th harmonic = B!:0 4th harmonic

B!:1,2 10th harmonic = F:0 11th harmonic

! Pitches tuned using F:2,3 require a small downward shift of 5 cents by lip or hand to meet the desired ratios. ! Pitches tuned using F:1,3 require a small downward shift of 3 cents by lip or hand to meet the desired ratios. ! Pitches tuned using F:1,2,3 require an upward shift of 9 cents by lip or hand to meet the desired ratios. ! Pitches tuned using B!:1,2 require a small upward shift of 5 cents by lip or hand to meet the desired ratios.

This work was written 2010. It was recorded at the University of Western

Sydney Music Department on 17 April 2010 by Sophia Rhee and Michael H

Dixon. It was also performed by Matthew Farrell, cello and Michael H Dixon at the International Horn Society Symposium, Griffith University, Queensland

Conservatorium of Music on 21 July 2010.

202 Epilogue

The horn is the focus of the creative work undertaken for this DCA. My role has been to unlock some of the microtonal potential inherent in the instrument with particular reference to extended just intonation, to explore its xenharmonic qualities. The standard double horn with its 16 separate harmonic series can be viewed as a multi-crook instrument. Even when the valve slides are tuned in

12-tone equal temperament there are a large number of microtonal intervals available, especially in the range between C (written) in the middle of the treble staff to the C above. It was the constant encounter with harmonics as a performer that prompted the compositional study in just intonation that is the subject of the companion folio of works. While it would have been possible to study the harmonics in the context of equal tempered tuning systems based on divisions of the octave other than twelve, such an approach is beyond the scope of this thesis. George Secor, designer of the electronic keyboard known as the

Scalatron and himself a horn player, has outlined schemes for playing the horn using a number of non-twelve equal temperaments in the journal

Xenharmonikȏn (1976). By contrast this thesis has been focussed on harmonics available in the horn and used this as a basis for exploration of just intonation tuning.

203 A vast pitch world is available using just intonation when the valve slides are tuned in relation to each other using just ratios. A significant musical and scientific study of just intervals playable on the horn, tuba and trombone was undertaken by Marc Sabat and Robin Hayward (2006). Their study is of a “23- limit subset of harmonic space” in which A440Hz is the central pitch for tuning to fit well with standard string instruments (Sabat, Hayward 2006: 23). Of particular interest is the proportional tuning of the horn valve slides.41 In this folio, four of the compositions also require specific valve slide adjustments, a technique I have often used when playing baroque and classical works in standard orchestral and ensemble situations. It seemed natural to use the same techniques for a new repertoire. Rearranging valve slides made it possible to use a number of key centres which feature in the folio of compositions. In some of the compositions pitches are also tuned by lipping and right hand adjustments, techniques which open the door to a vast pitch world even without the use of valve slide adjustments. All works in the folio fit into a prime-limit category no higher than 13-limit drawing on the harmonic vocabulary familiar to most horn players. The 13th harmonic is playable on any length of tube in the standard double horn whereas higher primes are not.

Even when a player is familiar with these techniques additional assistance is required to navigate the world of extended just intonation. Players’ adaptive training and experience with intonation in standard music provide an obvious

204 starting point to extend aural awareness necessary to perform music in extended just intonation. In order to assist players composers need to provide tuning cues over and above those given in standard notation. In this folio I have offered four levels of assistance: performance notes, notation, sound files or assistance in person at rehearsals. The clarity and accuracy of such assistance along with the willingness, ability and contribution of particular performers determined the quality of the musical realisation of the compositions. More assistance becomes necessary as the just intonation prime-limit increases. It is easier for players to achieve accurate intonation within 3-limit and 5-limit because most performers are already familiar with intervals that fall within these limits. 7-limit is slightly more difficult than 3-limit and 5-limit yet less difficult than 11-limit or 13-limit. This is because lower limit intervals are aurally more familiar and generally easier to manipulate.

The works in my folio that I consider best integrate composing and performing are the three in Section 3.3. Seven Small Wheels Revolving is the only composition in which all combinations of harmonics and tube lengths are pre-set. The work is designed to be played with one timbre throughout. This is the same approach used for the trumpet in A Hundred Valleys I but different to the approach in A

Hundred Valleys II. The latter incorporates all the pitch manipulation techniques available to a player of the standard double horn. Pitches that require significant lip and hand manipulations can be chosen for their timbral qualities.

205 It seems that Johannes Brahms wrote for the horn in such a way in his Horn Trio

Opus 40 (1865). His writing for natural horn requires many pitches to be produced with a variety of adjustments to the standard right hand position, adjustments which result in timbral changes. The timbre and the pitch work beautifully together so that the work is enhanced when playing with natural horn technique. Conversely, a great deal of quality is missed when played entirely with valves. The same connection between timbre and pitch is evident in A Hundred Valleys II. However the significant hand and lipping techniques it requires arose incidentally because the work was evolved from ideas that originated in the composition A Hundred Valleys I for solo trumpet. Further composition beyond this DCA will incorporate all the technical resources available to the horn player for work in extended just intonation fully integrated with an exploration of timbral qualities.

Most of the techniques used in horn playing are used not only by players of other brass instruments but also by players using a wide variety of lip vibrated wind instruments or labrosones similar to those described in Chapters One and

Two. Many of these labrosones have an ancient tradition which includes techniques such as lipping to adjust pitch (e.g. cornett, 1.2: p 17) and the use of the right hand inserted into the bell (conch, 1.1: p 8). Drawing on the concept of prime-limits developed by Harry Partch (1974) I developed a greater understanding of the use of horn harmonics. A further classification of

206 labrosones by available harmonics provided a new framework for exploring their use in extended just intonation composition (Chapter 1). This allowed me to think of my instrument as a member of a family with many distant relatives.

Links between the horn and members of this extended family is demonstrated by the musical scenarios represented in Chapter Two.

The companion folio presented with this CD represents my work as a composer-performer; newly composed work has been inspired and nurtured through an intimate knowledge of the horn and playing technique which in turn has been extended as it inevitably undertakes new technical challenges presented by new repertoire.

The microtonal playing technique on which the folio has foundations is built on the mastery of horn repertoire in both ensemble - in the fields of symphony, opera, ballet, chamber music - and solo - in concertos and recitals. My understanding of microtonal playing technique has been especially informed by my experience with the natural horn as played in music of the baroque and classical period. Working with the horn in recording studios - playing for film and television scores and performing jazz and non-Western music including ragas on the horn accompanied by tabla - has also played a part in developing this technique. I was compelled to find new pathways for horn playing beyond the constraints or safety of standard concert repertoire.

207 My impetus to compose has always been associated with horn playing but composing always took second place to playing. This all changed in 1990 when

I first read the article entitled “An Introduction to “In Tune” the Scale of Just

Intonation”, by trombonist Jack Holland (1982). The article kindled my interest in the harmonic series and its musical possibilities in the context of my own instrument. From this point onwards I found myself composing music in ways that would inevitably draw on knowledge of the horn I had acquired as a player over twenty years.42

Composing music in extended just intonation has made it possible to create music of great beauty using the technical resources of standard instruments.

Through writing music that extends these resources and playing it as a member of the ensemble composed for, I have found a new satisfaction that comes from playing chamber music. The musical realisation of new repertoire based on extended just intonation has fostered a new level of engagement with other musicians who have played in the ensembles for which I have composed.

Rehearsal of such repertoire tends to be focused on the art of intonation which draws on the technical resources of my playing rather than the notation used to convey my directions as a composer. A contribution both as a performer and a composer has contributed enormously in developing an instrumental repertoire based extended just intonation.

208 My venture into extended just intonation composition has opened a door to the world of microtonal music for standard western instrumentalists. Microtonal music is no longer the enclave of contemporary composers like Partch who became involved in designing and constructing special instruments required for its performance even though Partch's contribution has played a vital role in renewing the interest of composers and instrument-builders alike. But the ongoing development and propagation of microtonal composition is more likely if microtonality is included as part of the standard training regime of modern instrumentalists. Without such inclusion microtonal music will remain marginalised despite new instrument design, electronic or otherwise. As a composer-performer I see myself as a role model for performing musicians who will hopefully develop and propagate such a repertoire in the future.

209 End Notes

1 From a discussion with Kirsten Barry – principal period oboe player in the Australian Brandenburg Orchestra.

2 Discussion with John Foster - director of Australian Baroque Brass - 14 Jan 2010.

3 These are sometimes known as rizhok, or rih in the Ukraine - see Mizynec, V 1987.

4 Another instrument of about 70 centimetres in length is also called the lituus but is most probably of Celtic origin (Meucci 90). A further use of the name lituus comes in the motet "O Jesu Christ, meins lebens licht" by J. S. Bach. However the instrument with the ten harmonics required from it is clearly different to the ancient Roman one. A recent recreation is 2.7 metres long (Ghosh, Palap for BBC News, 30 May 2009).

5 This pitch occurs in mov.IV, bar 4.

6 Scott Kinmont - Associate Principal Trombone with the Sydney Symphony Orchestra - does so as do many players in the United States.

7 Discussion with Scott Kinmont, January 2010.

8 Thanks to Joshua Clark - Principal Trumpet Australian Opera and Ballet Orchestra - for this information.

9 Discussion with Adam Malone - Trumpet player in the Police Band and freelance musician - 2010.

10 Conversation with Adam Malone, 2010.

11 The scores are easily accessible from the website IMSLP, details in the References.

12 This pitch occurs in the fourth bar after figure 11.

13 This pitch occurs in Movement 4, bar 133.

14 This pitch occurs in Movement 4, bar 80.

15 This pitch occurs in Act 2, the sixth bar after figure 240.

16 This pitch occurs in Act 1, bar 136.

17 This pitch occurs in Act 3, bar 986.

18 This pitch occurs in bars 9 to 11.

211 19 This pitch occurs in the Vorspiel to Act 1, three bars before figure K.

20 This occurs in Part 5, Songe d’une nuit du Sabbat, bar 484.

21 Discussion with Angus Charlton - tuba player Army Band and Canberra Symphony Orchestra.

22 The descant horn in B! soprano is available from some manufacturers, even a double horn in B! alto and B! soprano.

23 Thanks to Matt van Emmerick - freelance musician and winner of the Australasian Open Brass Band Championships 2009 - for bringing this to my attention, March 2009.

24 Barry Tuckwell also performed this work in the presence of the composer. His recording and two recordings of Dennis Brain are included in the Reference List.

25 I have performed this song as per Example 1.5. in two separate concert events in the 1990’s.

26 Anders Nykvist - member of Klangforum Wien - confirmed his use of a quartertone instrument in an email, 4 Feb 2011.

27 The first occurrence is three bars before Fig.3 and the fifth bar after Fig.3; the second occurrence is the third bar of Fig.10; the third occurrence is Fig.13.

28 This occurs in mov.1, bar 70. The scores for the Zelenka works I have access to are photocopies. I have not been able to source the scores elsewhere, therefore I have included a notated excerpt for this work.

29 This occurs in mov.1, bars 70, 72, 169 & 173; also in Aria 2, bar 10. Capriccio V in G has a high written E in mov.1 bar 25 , played as the 20th harmonic.

30 18th harmonics in bars 4, 5, 19, 20 and later in the movement; also in BWV 213, mov.1 bars 89, 95, 129, 152.

31 These are in the Arie “Unsre Stärke heisst zu schwach” bar 16 and the repeat of the section.

32 This occur in the closing Choral, bar 3.

33 These occur in duets 1, 3, 6 and 7.

34 These occur in Symphony no.51 in mov.2, bar 6 and in the Divertimento in bar 70.

35 This occurs in mov.4, var.IV, bar 70.

36 This occurs in mov.1, bar 58.

37 This occurs in two places: mov.1, two bars before letter H; mov.3, 5 bars before the end of the work.

38 This occurs in the Scherzo, second repeated section, end of bar 7 and the fifth bar after Fig.E.

212 39 This occurs in the Finale, Fig.45.

40 International Horn Society Symposium in Brisbane, Australia on 21 July 2010.

41 In one of their tunings the 2nd slide is tuned in the ratio 15:16 to the open tube; the 1st valve to the ratio 15:17; the 1st and 2nd combined produce the ratio 15:18; the 3rd to the ratio 15:19; 2nd and 3rd to 15:20; 1st and 3rd to 15:21 and all three to 15:22.

42 I began playing the trumpet in 1970 and the horn in 1972.

213 REFERENCES

24 Microtonal Studies Donald Bousted 2006, CD, The Microtonal Trumpet, Freiburg.

Agrell, J 2008, ‘Ontogeny Recapitulates Phylogeny, or: Only Natural’, The Horn Call: Journal of the International Horn Sociey, vol.XXXIX, no.1, pp85-87.

Albertyn, E 2005, ‘Indigenous African Horns: Ixilongo, Makondere, and African Horn Bands', The Horn Call: Journal of the International Horn Society, vol.XXXV, no.3, pp48-52.

Altoft, S 2006, 24 Microtonal Studies Donald Bousted, CD liner notes, The Microtonal Trumpet, Freiburg.

Bach J S 1851/1725, Cantata ‘Woe schön leuchtet der Morgenstern BWV 1, Bach-Gesellschaft Ausgabe, Breitkopf & Härtel, online IMSLP, accessed 22 January 2011, http://imslp.org/wiki/List_of_compositions_by_Johann_Sebastian_Bach,_by_BWV_number

Bach J S 1852b/1735, Cantata ‘Wär Gott nicht mit uns diese Zeit’ BWV 14, Bach-Gesellschaft Ausgabe, Breitkopf & Härtel, online IMSLP, accessed 22 January 2011, http://imslp.org/wiki/List_of_compositions_by_Johann_Sebastian_Bach,_by_BWV_number

Bach J S 1857/1715, Cantata ‘Der Himmel lacht! die Erde jubiliert’ BWV 31, Bach-Gesellschaft Ausgabe, Breitkopf & Härtel, online IMSLP, accessed 22 January 2011, http://imslp.org/wiki/List_of_compositions_by_Johann_Sebastian_Bach,_by_BWV_number

Bach J S 1860/1723, Cantata ‘Schauet doch und sehet’ BWV 46, Bach-Gesellschaft Ausgabe, Breitkopf & Härtel, online IMSLP, accessed 22 January 2011, http://imslp.org/wiki/List_of_compositions_by_Johann_Sebastian_Bach,_by_BWV_number

Bach J S 1868/1724, Cantata ‘Halt im Gedächtnis Jesum Christ’ BWV 67, Bach-Gesellschaft Ausgabe, Breitkopf & Härtel, online IMSLP, accessed 22 January 2011, http://imslp.org/wiki/List_of_compositions_by_Johann_Sebastian_Bach,_by_BWV_number

Bach J S 1870/1723, Cantata ‘Du sollt Gott, deinem Herren, lieben BWV 77, Bach-Gesellschaft Ausgabe, Breitkopf & Härtel, online IMSLP, accessed 22 January 2011, http://imslp.org/wiki/List_of_compositions_by_Johann_Sebastian_Bach,_by_BWV_number

Bach J S 1887/1715, Cantata ‘Ach! ich sehe, itzt, de ich zur Hochzeit gehe’ BWV 162, Bach- Gesellschaft Ausgabe, Breitkopf & Härtel, online IMSLP, accessed 22 January 2011, http://imslp.org/wiki/List_of_compositions_by_Johann_Sebastian_Bach,_by_BWV_number

Baines, A 1993, Brass Instruments: Their History and Development, Dover Publications, New York.

Baines, A, Dickey, B 2007, ‘Cornett’, accessed 15 July 2009, Grove Music Online, Oxford Music Online database.

Baines, A C, Herbert, T 2007, ‘Bugle’, accessed 15 July 2009, Grove Music Online, Oxford Music Online database.

Baines, A C, Meyers, A 2007, ‘Cornet’, accessed 15 July 2009, Grove Music Online, Oxford Music Online database.

215 Baines, A C, Meyers, A, Herbert, T 2007, ‘Trombone’, accessed 15 July 2009, Grove Music Online, Oxford Music Online database.

Battan S M 1980, ‘Alois H!aba's Neue Harmonielehre des diatonischen, chromatischen, Viertel-, Drittel-, Sechstel-, und Zwölftel Tonsystems’, PhD thesis, Eastman School of Music, University of Rochester.

Bate, P 1966, The Trumpet and Trombone: An Outline of their History, Development and Construction, W W Norton, London.

Bate, P, Tarr, E 2007 ‘Valve’, accessed 15 July 2009, Grove Music Online, Oxford Music Online database.

Beer, A 1965, Sonorum Speculum, reprinted in ‘The Development of 31-tone music’, http://www.huygens-fokker.org/docs/beerart.html

Beethoven, L n.d.a/1808, Symphony V C moll op.67, musical score, Universal Edition, Wien.

Beethoven, L n.d.b/1808, Symphony VI (Pastorale) F dur op.68, musical score, Universal Edition, Wien.

Beethoven, L 1999/1794, Sextets for strings and winds op.81b, musical score, Dover Publications, Mineola.

Bendinelli, C 1975/1614, The Entire Art of Trumpet Playing, E H Tarr (trans.), The Brass Press Editions Bim, Vaurmarens Switzerland.

Berg, A 1954/1915, Drei Orchester Stücken op.6, musical score, Universal Edition, Wien.

Berg, A 1978/1937, Lulu, F Cerha (ed.), musical score, Universal Edition, Wien.

Berg, A 1955/1922, Wozzeck op.7, H E Apostel (rev.), musical score, Universal Edition, Wien.

Berlioz, H 1972/1830, Symphony fantastique op.14, musical score, Ernst Eulenberg, London.

Between Two Notes (excerpt) 2006, F Strauss (director), documentary, accessed 17 January 2011, uploaded 16 December 2007, http://dotsub.com/view/53521250-2051-4e55-a154-a1a59d702538

Bevan, C 1978, The Tuba Family, Faber, London.

Bevan, C 1997, ‘The Low Brass’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp143-156.

Bevan, C 1994, ‘Trombone’, The New Grove Dictionary of Jazz, B Kernfeld (ed.), St. Martin’s Press, New York, pp1219-1220.

Blaauw, M n.d., Instruments, accessed 17 January 2011, http://www.marcoblaauw.com/en/instruments

Bousted, D 2009, Duo-1 for Quarter-tone Trumpet and Quarter-tone Horn, accessed 19 March 2010, live performance 7 March 2009, UK MicroFest3, microtonalprojects, www.youtube.com

Bozza, E 1952, Suite pour Quatre Cors en Fa, Alphonse Leduc, Paris.

216 Brahms, J 1996/1876, Symphony no.1 c-moll op.68, musical score, R Pascall (ed.) G Henle Verlag, München.

Brahms, J 1991/1877, Symphony no.2 D-dur op.73, musical score, H Gel (ed.), Breitkopf & Härtel, Wiesbaden.

Brahms, J n.d./185?, Horn Trio op.40, International Music Company, New York.

Britten, B 1960, A Midsummer Night’s Dream op.64, musical score, Boosey & Hawkes, London.

Britten, B 1944, Serenade op.31 (1943) for Tenor, Horn and Strings, Winthrop Rogers Edition, Hawkes & Son, London.

Britten, B 1944, ‘Serenade’ in The British Music Collection, Boyd Neel String Orchestra, Benjamin Britten (conductor), Peter Pears (tenor), Dennis Brain (horn), Decca, London.

Britten, B 1953, ‘Serenade’ in Britten: Serenade, Les Illuminations, Nocturne / Pears, Brain, Britten, New Symphony Orchestra, Eugene Goosens (conductor), Peter Pears (tenor), Dennis Brain (horn), Decca, London.

Britten, B 1963, ‘Serenade’ in Britten: Serenade, Les Illuminations / Pears, Britten, London Symphony Orchestra, Benjamin Britten (conductor), Peter Pears (tenor), Barry Tuckwell (horn), Decca, London.

Brown, H M, Palmer, F 2001, ‘Aerophone’, The New Grove Dictionary of Music and Musicians, 2nd edition, S Sadie (ed.), vol.1, Oxford University Press, Oxford, pp447-9.

Catler, J 2003, The Nature of Music, FreeNote Music, New York.

Chalmers, J 1993, Divisions of the Tetrachord, Frog Peak Music, Hanover NH.

Clark, M 2005, Sound of the Silk Road: Musical Instruments of Asia, Boston Museum of Fine Arts Publications, Boston.

Concierto para Trompa “Ab origine” Salvador Brotons, accessed 15 May 2010, live performance 11 October 2009, http://www.youtube.com

Cuecas de una Misma Tierra, 2009, accessed 29 Aug 2009, Las Araucanas,! http://www.youtube.com

Darreg, I 1982, ‘Progress Report: New Musical Instruments and Tuning Systems’, Interval, vol.4, no.1, pp13-14.

Davidson, R 2002, Tibrogargan Round, live performance 29 July 2002, Brisbane Powerhouse, Brisbane Australia.

Dauprat, L F 1994/1824, Méthode de Cor Alto et Cor Basse: Complete English Translation of the First Edition 1824 Published by Zetter, V Roth (ed.), Birdalone Music, Bloomington Indiana.

Davies, H 2007, ‘Instrumental Modifications and Extended Performance Techniques: Brass’, accessed 15 July 2009, Grove Music Online, Oxford Music Online database.

Dempster, S 2002, Twenty Tantalizing Titillating Trombones Tooting Turbine Tonics, live performance 29 July 2002, Brisbane Powerhouse, Brisbane Australia.

217 Dickey, B 1997, ‘The Cornett’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp51-67.

Dixon, M H 2003, Chemchok Heruka, musical score manuscript, held by M Dixon, Sydney, NSW.

Dixon, M H 2001, Just intonation: a study through its notation. Mmus thesis, Griffith University.

DjeDje, J C 1998, ‘West Africa: An Introduction’, Garland Encyclopedia of World Music, R Stone (ed.), vol.1, pp442-470.

Domnich, H 1985/1807, Méthode de Premier et de Second Cor 1807, D Poulsen (trans.), Hans Pizka Edition, Kirchheim Germany.

Don Ellis Autumn 2007/1968, Wounded Bird Records, Sony BMG Music Entertainment, New York.

Dudgeon, R 1997, ‘Keyed Brass’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp131-142.

Duvernoy, F 1802, Méthode pour le Cor: Suivie de Duo et de Trio, accessed 19 March 2010, facsimile, Mme Le Roi, Imprimerie du Conservatoire de Musique, Paris, http://www.free-scores.com

Dournon, G 2000, Handbook for the Collection of Traditional Music and Musical Instruments, UNESCO Publishing, Paris, France.

Dysart, P 2007, Music harmony and donuts, accessed 20 October 2010, http://members.cox.net/dysartp/traditionalTonnetz.html

Electric Bath: The Don Ellis Orchestra 1998/1967, Sony Music Studios, Columbia Recording Studios Hollywood Los Angeles.

Ellis, D 1975, Quarter-Tones: A Text with Musical Examples, Exercises and Etudes, Plainview, New York.

Evans, R 1997, ‘The Post-classical Horn’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp207-216.

Farkas, P 1956, The Art of French Horn Playing, Summy-Birchard Company, Illinois.

Farmer, H G 1957, ‘The Music of Islam’, The New Oxford History of Music, E Wellesz (ed.), vol.1, Oxford University Press, London, pp421-477.

Faust, R 1977 Prelude, The Music of Randall E. Faust, Macomb Illinois.

Faust, R 1996, Harmonielehre: Variations for Horn, The Music of Randall E. Faust, Macomb, Illinois.

Fitzpatrick, H 1970, The Horn and Horn-playing: And the Austro-Bohemian Tradition from 1680-1830, Oxford University Press, London.

Foster, J 2010, The Natural Trumpet and other Related Instruments, Kookaburra Music, Sydney Australia.

Franz, O 1942, Complete method for the French Horn, W Gebhardt (rev.), Cundy-Bettoney, New York.

218 Flintoff, B 2004, Taonga Poro Singing Treasures: The Musical Instruments of the Maori, Craig Potton Publishing, Nelson New Zealand.

Gann, K 1994, ‘Downtown Beats for the 1990s: Rhys Chatham, Mikel Rouse, Michael Gordon, Larry Polansky, Ben Neill’, Contemporary Music Review, vol.10, no.1, pp33-49.

Goertzen, C 1998, ‘Lithuania’, Garland Encyclopedia of World Music, T Rice, J Porter & C Goertzen (eds), vol.8, Garland Publishing Inc, New York, pp509-515.

Gregory, R 1961, The Horn: A Guide to the Modern Instrument, Faber & Faber, London.

Gunawan D & Schubert E 2010, ‘Noteview: A Computer Program for the Analysis of Single-line Musical Performances’ Acoustics Australia, vol.38, no.1, pp25-30.

Handel, G F 1962/1749, Music for the Royal Fireworks HWV 351, H F Redlich (ed.), musical score, Hallische Handel Ausgabe, Serie IV: Bd.13, Bärenreiter-Verlag, Kassel.

Harrison, Lou. 1971. Lou Harrison’s Music Primer: Various items about Music to 1970, C. F. Peters Corporation, New York.

Harrison, L 2006/1978, Main Bersama-sama for Gamelan Degung and French Horn, accessed 26 October 2010, musical score, J Diamond (ed.), American Gamelan Institute, http://www.gamelan.org/composers/harrison/harrison_MainBersama-sama.pdf

Harvey, D 1999, Have You Practiced Today: A Practice Schedule for the Canberra School of Music Horn Student, manuscript, held by Michael H Dixon, Sydney, NSW.

Haydn, J 1985/1762, Concerto per il Corno di Caccia 1762 Hoboken VII d:3, M Ohmiya (ed.), musical score, Joseph Haydn Werke, G Henle Verlag, München.

Haydn, J 2004/1767, Divertimento in Es für Horn, Violine und Violoncello 1767 Hoboken IV: 5, A Friesenhagen (ed.), musical score, G Henle Verlag, München.

Haydn, J 1964/1765, Sinfonie in D 1765 Hoboken I: 31, H Walter (ed.), musical score, G Henle Verlag, München-Duisburg.

Haydn, J 1951/1772, Symphony no. 51 c1772/4, H Schultz (ed.), musical score, series 1 vol.5, The Haydn Society Inc, Boston.

Hayward, R 2006, Monophones for Double Horn and 6-valved F Tuba, accessed 19 March 2010, musical score, Plainsound Music Edition, http://www.robinhayward.de/compositions.html

Heim, B 1990, ‘Practical Tuning, Temperament, and Conditioning for Hornsits and other Wind Instrumentalists: Understanding and Attaining Intonational Flexibility in Musical Performance’, MMus thesis, The Graduate School, The University of Tulsa.

Helffer, M 2001, ‘Tibetan Music: Monastic Music: Notations: Aerophones’, The New Grove Dictionary of Music and Musicians, 2nd edition, S Sadie (ed.), vol.25, Oxford University Press, Oxford, pp177-8.

Herbert, T, Wallace, J (eds) 1997, ‘Glossary’, The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp273-284

Herbert, T 1997, ‘’Sackbutt’: The Early Trombone’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp68-83.

219 Hiebert, T 1997, ‘The Horn in the Baroque and Classical Periods’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp103-114.

Hill, D 1990, Thoughtful Wanderings for Natural Horn and Percussion, musical score, International Horn Society Online Library, http://www.hornsociety.org

Hill, D 1983, Extended Techniques for the Horn: A Practical Handbook for Composers and Performers, Florida: Studio 224.

Hodge, R 2005, The Brass Book: A Book of Graded Studies, Exercises and Scales, Clefs ‘n’ Keys Music, Mittagong Australia.

Holland, J 1982, ‘An Introduction to “In Tune” the Scale of Just Intonation’, The Brass Bulletin vol.40, pp58-64.

Honingh, Aline 2009, ‘Compactness in the Euler-lattice: A parsimonious pitch spelling model’ Musicae Scientiae, Vol XIII, n° 1, pp117-138.

Hunt, A 2008, H-system music theory, accessed 23 October 2010, http://www.h-pi.com/theory/foreword.html

‘I wouldn’t mind dying, (but I gotta go by myself) by Reverend I B Ware with Wife and Son’ 1993/1928, in Guitar Evangelists (1928-1951): Complete Recorded Works in Chronological Order, CD, Document Records, Vienna.

Irons, E 1966/1938, Twenty-seven Groups of Exercises for Cornet and Trumpet, Southern Music, San Antonio Texas.

Johnston, Ben 2006, Maximum Clarity and other writings on music, University of Illinois Press, Urbana and Chicago.

Jones, F 2010, ‘The Legacy of the Alphorn’, The Horn Call: Journal of the International Horn Society, vol.XL, no.2, pp41-46.

Kaiser, J 2010, Gear, accessed 23 October 2010, http://www.jeffkaiser.com/gear.html

Karaška, A 1984, ‘Ožragis’, The New Grove Dictionary of Musical Instruments, S Sadie (ed.), vol.2, Macmillan Press Limited, Londond, p978.

Kaufmann, W 1975, Tibetan Buddhist Chant: Musical Notations and Interpretations of a Song Book by the Bkah Brgyud Pa and Sa Skya Pa Sects, Indiana University Press, Bloomington Indiana.

Keenan, D 2004, Sagittal, accessed 23 October 2010, http://www.sagittal.org/

Knopoff, S 2007, ‘Didjeridu’, accessed 19 May 2010, Grove Music Online, Oxford Music Online database.

Larr 2009, Van Laar Quartertone Trumpet, accessed 23 May 2009, uploaded 12 March 2009, http://www.youtube.com

Lawrance, P 1992, Graded Lip Flexibilites, musical score, Brass Wind Publications, Rutland UK, Re-published in Australian Music Examinations Board Orchestral Horn Series, Allans Publishing, Melbourne.

220 Leuba, C 2004/1962, A Study of Musical Intonation, Prospect Publications, Vancouver Canada.

Ligeti, G 2002, Hamburgisches Konzert für Horn Solo und Kammerorchester, musical score, Schott, Mainz.

Ligeti, G 1953, Sechs Bagatelles für Bläserquintett, musical score, Schott, Mainz.

Ligeti, G 1984, Trio für Violin, Horn und Klavier (1982), musical score, Schott, Mainz.

LOCANA Plays the Music of Michael Hugh Dixon 2003, CD, Self-published, Brisbane Australia.

McDonald, H 2005, Horn Warmups, musical score manuscript, held by M H Dixon, Sydney, Australia.

McKinnon, J W 2001, ‘Lituus’, The New Grove Dictionary of Music and Musicians, vol.15, Oxford University Press, Oxford, p11.

Maalouf, I n.d., Ibrahim Maalouf Official Website, accessed 23 October 2010, http:www.ibrahimmaalouf.com

Macek, V V 1977/?, Serenata in Dis for Wind Octet, LP, Supraphon “Domovina” Studio, Prague.

Mandelbaum J n.d., Wind Quintet no.2, accessed 29 October 2008, notated music of horn part, http://www.youtube.com originally from http://www.afmm.com, no longer available online.

Mapuche and Mack Yidhaky, accessed 28 August 2009, uploaded 15 December 2006, http://www.youtube.com

Marcuse S, 1975, A Survey of Musical Instruments, Harper & Row, New York.

Martin, F 1941, Ballad for Trombone and Piano, musical score, Universal Edition, New York.

Mehta, R n.d., rajeshmehta, accessed 23 October 2010 http://www.rajesh-mehta.com/instruments

Merewether, R 1978, The Horn, the Horn, Paxman Musical Instruments Ltd, London.

Meucci, R 1989, ‘Roman Military Instruments and the Lituus’, The Galpin Society Journal, ! vol. 42, pp85-97.

Microtonalprojects 2010, Microtonal Trumpet and Horn, accessed 18 January 2011, http://wn.com/microtonalprojects

Microtonal Trumpet Project, n.d., accessed 28 August 2009, http://www.microtonaltrumpet.com/index.htm

Mizynec, V 1987, Folk Instruments of Ukraine, Bayda Books, Melbourne.

Mnozil Brass 2009, Mnozil Brass Alphorn, accessed 28 Ausgust 2009, http://www.youtube.com

Montagu, J 2007, ‘Shofar’, The New Grove Dictionary of Music and Musicians Grove, Oxford University Press, Oxford, vol.23, pp274-275.

Monzo, J 2005, ‘HEWM (Helmholtz-Ellis-Wolf-Monzo)’, Tonalsoft Encyclopedia of Microtonal Music Theory,

221 Morley-Pegge, R, Bate, P, Weston, S, Myers, A 2007, ‘Ophicleide’, accessed 19 May 2010, Grove Music Online, Oxford Music Online database.

Mortenson, G 2008, ‘Ibrahim Maalouf: A New Voice/A Different Way’, accessed PDF 18 January 2011, International Trumpet Guild Journal, http://www.trumpetguild.org/pdf/2008journal/200806Maalouf.pdf , pp12-21.

Moscow Art Trio, Sydney Festival, live performance, 29 January 2009, Spiegeltent Hyde Park Sydney.

Mozart, W A n.d./1786, Twelve Pieces, K. 487 for Two Horns: Facsimile of the First Publication 1802, musical score, Thompson Edition Inc, South Carolina.

Mozart, W A 1991/17?, Sinfonia Concertante in Es für Flöte, Oboe, Horn, Fagott und Orchester KV Anh. I/9 (297b), R Levin (recon.), musical score, Bärenreiter, Kassel.

Musgrave, T 1971, Horn Concerto, musical score, J & W Chester Ltd, London.

Musica Hecha con Clarin Cajamarquino, Cajon y Flauta, accessed 28 August 2009, uploaded 2 Aug 2007, http://www.youtube.com

Music Tradicional del Peru, accessed 28 August 2009, uploaded 20 December 2008, http://www.youtube.com

Myers, A 1997, ‘How Brass Instruments Work’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp19-23

Myers, A 1997, ‘Design, Technology and Manufacture since 1800’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp115–30

Mysleviček, J 1962/1770, Octet II for Winds, musical score, Editions Supraphon, Prague.

Nguillatun, accessed 28 August 2009, uploaded 16 Aug 2009, Musica y Video de Cecil Gonzalez, , http://www.youtube.com

Nketia, J 1962, ‘The Hocket-Technique in African Music’, Journal of the International Folk Music Council, Vol. 14, International Council for Traditional Music, pp44-52.

Oldham, G, Campbell, M, Greated, C 2007, ‘Harmonics’, accessed 15 July 2009, Grove Music Online, Oxford Music Online database.

Olsen, D 1998, ‘The Distribution, Symbolism, and Use of Musical Instruments’, D Olsen & A D, Sheehy (eds), Garland Encyclopedia of World Music, vol.2, Garland Publishing Inc., New York, pp28-42.

Partch, H 1974, Genesis of a Music, Da Capo Press, New York.

Perhson, J 1987, Harmonic Etude, musical score, Seesaw Music, New Jersey.

Polansky, L 1993 ‘Horn’, including musical score, Xenharmonikȏn 14, J H Chalmers & E M Wilson (eds), Spring 1993, pp80-90

222 Polk, K 1997, ‘Brass Instruments in Art Music in the Middle Ages’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp38-50.

Poulenc, Francis 1963, Histiore de Barbar le Petit Éléphant pour Récitant et Orchestre, J Francaix (orch.), musical score Chester Music, London.

Reck, D 2005, Review of Drumming and Chanting in God's Own Country: The Temple Music of Kerala in South India 2003, Yearbook for Traditional Music, vol.37, pp181-182.

Reportaje al Clarin Cajamarquino Patrimonio del Peru, accessed 28 August 2009, uploaded 9 August 2008, http://www.youtube.com

Rossini, G n.d./1828, Rendez-vous de Chasse Fantasy for 4 horns in D, musical score, E Leloir (ed.), Simrock, London.

Russian Horn Capella 2010, accessed 1 May 2010, http://www.horncapella.com

Rycroft, D K 2001, ‘Side-blown Trumpet’, The New Grove Dictionary of Music and Musicians Grove, Oxford University Press, Oxford, vol.23, p353.

Sabat, M, Hayward, R 2006, Towards an Expanded Definition of Consonance: Tuneable Intervals on Horn, Tuba and Trombone, accessed 23 October 2010, Plainsound Music Edition, http://adagio.calarts.edu/~msabat/ms/MSwork.html

Sabat, M 2010/2006, Hairy, Hippy, Happy, accessed 23 October 2010, musical score and recording, performed by Anna Robinson, horn; Matt Barbier, trombone; Luke Storm, tuba, http://adagio.calarts.edu/~msabat/ms/MSwork.html

Sabat, M 2005, Wonderful Scatter: Music scenery with Tuba, accessed 23 October 2010, live recording played by Robin Hayward, http://adagio.calarts.edu/~msabat/ms/MSwork.html

Sabat, M, Schweinitz, W 2005, The Extended Helmholtz-Ellis JI Pitch Notation, accessed 23 October 2010, Plainsound Music Edition http://adagio.calarts.edu/~msabat/ms/pdfs/notation.pdf

Sachs, C 1964, Real-Lexicon der Musikinstrumente: Zugleich ein Polyglosser fur das gesante Instrumentengebeit, 2nd edition, Dover Publications, New York.

Sarkissian, M 1997, ‘Lip-vibrated Instruments of the Ancient and Non-western World’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, Cambridge University Press, Cambridge, pp5-18.

Sarkissian, M, Tarr, E 2007, ‘Trumpet’, accessed 19 May 2010, Grove Music Online, Oxford Music Online database.

Schumann, R 1968/1849, Konzerstück for Vier Hörner und Orchestra op.68, musical score, piano reduction, International Music Company, New York.

Schumann, R c1882/1850, Symphony no. 3 (Rhenish) op.97, A Dörffel (ed.), musical score, C F Peters, Leipzig, online IMSLP, accessed 22 January 2011, http://imslp.org/wiki/List_of_compositions_by_Robert Schumann,_by_BWV_number

Scott, J E 1957, ‘Roman Music’, The New Oxford History of Music, E Wellesz (ed.), vol.1, Oxford University Press, London, pp404-420.

223 Secor, G 1976, ‘An Approach to the Construction of Microtonal Valved Brass Instruments - the French Horn, Xenharmonikȏn 5, J H Chalmers (ed.), Houston, Texas, 3 pages.

Sentā & Hakubutsukan 1989, The JVC/Simthsonian Folkways Video Anthology of World Music and Dance, JVC in collaboration with the National Museum of Ethnology (Osaka), Round Records, Cambridge Mass.

Shostakovich, D 1994/1937, Symphony no. 5 op.47, musical score, Dover Publications Inc., New York.

Simon, A 1998, ‘Music in Sudan’, Garland Encyclopedia of World Music, R Stone (ed.), vol.1, pp549-573.

Simon, A 2007, ‘Sudan’, accessed 19 May 2010, Grove Music Online, Oxford Music Online database.

Smithers, D 1973, The Music and History of the Baroque Trumpet before 1721, Syracuse University Press, Syracuse, USA.

Steele-Perkins, C 2001, The Trumpet, Yehudi Menuhin Music Guides, Kahn & Averill, London,.

Stimmhorn 2010, Press Information, accessed 23 October 2010, http://www.stimmhorn.ch/docs/Doku_E-F.pdf , pp1-32.

Stockhausen, M 2011, Instrumente, accessed 18 January 2011, http://www.markusstockhausen.de/en/instrumente

Strauss, R 1938, Daphne: Bükolische Trajödie in Einem Aüfzug op.82, musical score, B. Schotts Söhne, Mainz.

Strauss, R 1996/1917, Die Frau Ohne Schatten, Verlag Dr Richard Strauss, musical score, BmbH & Co. KG, Wien.

Strauss, R 1906, Symphonia Domestica, musical score, Bote & G Beck, Berlin.

Stravinsky, I 1965/1913, Le Sacre du Printemps, musical score, Dover, New York.

Tarr, E 1988, The Trumpet, translated from Die Trompete (1977) by S Plank, S & E Tarr, Portland Oregon.

Tarr, E 1997, ‘The Trumpet before 1800’, in T Herbert & J Wallace (eds), The Cambridge Companion to Brass Instruments, ed. Cambridge University Press, Cambridge, pp84-102.

Tarr, E 2001, ‘Trumpet’, Grove Dictionary of Music and Musicians, S Sadie (ed.), Oxford University Press, Oxford.

Tarr, E H 2007, Revolution: Music of a Golden Age Vol.2, CD Liner Notes, Tubicum Records, Sydney.

Tarr, E 2007, ‘Trumpet’, accessed 15 July 2009, Grove Music Online, Oxford Music Online database.

Tenney, J 1995, Spectrum II for Wind Quintet, self-published available through Frog Peak Music, Hanover NH.

224 Thrasher, A R 1984, ‘Laba’, The New Grove Dictionary of Musical Instruments, S Sadie (ed.), vol.2, Macmillan Press Limited, London, p493.

Tuckwell, B 1978, Playing the Horn: A Practical Guide, Oxford University Press, London.

Tzu, Lao 1983/1922, Tao Teh King, I Mears (trans.), Theosophical Publishing House, London.

Underground Overlays from the Cistern Chapel 1995, CD, New Albion Records, San Fransisco.

Wagner, R 1976/1868, Die Meistersinger von Nürnberg, musical score, Dover Publications, New York.

Weber, C M c1975/1815, Concertino für Horn und Orchester op.45, musical score, Edition Peters, Leipzig.

Widman, J 1973, Air für Horn solo, musical score, Schott, Mainz.

Wilby, P 1999, Dove Descending: Sonata in Open Tones, musical score, Novello, London, Brass Band music series – General editor, Bram Gay.

Wolfe, J n.d.a, Brass instrument (lip reed) acoustics: an introduction, accessed 15 July 2009, http://www.phys.unsw.edu.au/jw/brassacoustics.html

Wolfe, J n.d.b, Didjeridu harmonics, accessed 15 July 2009, http://www.phys.unsw.edu.au/jw/???harmonics.html

Wolfe, J n.d.c, How harmonic are harmonics?, accessed 15 July 2009, http://www.phys.unsw.edu.au/jw/harmonics.html

Yamamoto, H c1996, The JVC Smithsonian Folkways Video Anthology of Music and Dance Africa, JVC Company of America/Multicultural Media, Barre Vt.

Zelenka, The Orchestral Works 1977, CD, Arkiv, Bern.

225 TUNING THE HORN

APPENDIX A

165, 9-note scales

Michael Hugh Dixon 165, 9-note scales

Total number of scales used in DCA compositions and relation to chapter number of Lao Tzu’s Tao Teh King:

Title Quantity Scale number Chapter (from the Tao Teh King)

Door of Many Mysteries 5 35, 49, 69, 96, 131 1

Five Tastes 1 67

Koto Sudoku 1 51 51

Prajnaparamita 1 75

Seek the Unseen 1 12 12

Self-balance 1 72 72

Stillness of Unceasing Activity 1 136 48

Strive Not 3 73 73

Towards the Inaudible 2 112, 140

We are Sufficient 2 17, 141 17

Total = 18

Appendix A - i 165, nine-note scales

Michael H Dixon 2004 1. 2. œ bœ œ œ ? œ bœ nœ bœ nœ œ #œ œ bœ nœ bœ nœ œ #œ

3. 4. œ bœ œ œ ? œ bœ nœ bœ nœ œ #œ œ bœ nœ bœ nœ œ #œ

5. 6. bœ nœ bœ bœ ? œ bœ nœ bœ nœ œ #œ œ bœ nœ bœ nœ œ #œ

7. 8. bœ œ œ bœ ? œ bœ nœ bœ nœ œ #œ œ bœ nœ bœ nœ œ #œ

9. 10. œ œ bœ nœ ? œ bœ nœ bœ nœ œ #œ œ bœ nœ bœ nœ œ #œ

11. 12. bœ nœ bœ bœ ? œ bœ nœ bœ nœ œ œ œ bœ nœ bœ nœ œ œ

13. 14. bœ œ œ bœ ? œ bœ nœ bœ nœ œ œ œ bœ nœ bœ nœ œ œ

15. 16. œ œ bœ nœ ? œ bœ nœ bœ nœ œ œ œ bœ nœ bœ nœ œ œ

17. 18. bœ nœ bœ bœ nœ œ ? œ bœ nœ bœ nœ œ œ bœ nœ bœ nœ œ

19. 20. bœ bœ nœ œ bœ nœ ? œ bœ nœ bœ nœ œ œ bœ nœ bœ nœ œ

21. 22. bœ nœ bœ bœ ? œ bœ nœ bœ nœ #œ œ œ bœ nœ bœ nœ #œ œ

Appendix A - p.1 23. 24. œ bœ œ œ œ bœ ? œ bœ nœ bœ nœ #œ œ bœ nœ bœ nœ #œ

25. 26. œ œ œ œ bœ nœ ? œ bœ nœ bœ nœ #œ œ bœ nœ bœ nœ #œ

27. 28. #œ œ bœ #œ œ œ ? œ bœ nœ bœ nœ #œ œ bœ nœ bœ nœ #œ

29. 30. #œ #œ œ œ bœ nœ ? œ bœ nœ bœ nœ #œ œ bœ nœ bœ nœ #œ

31. 32. bœ nœ bœ bœ nœ œ ? œ bœ nœ bœ nœ œ œ bœ nœ bœ nœ œ

33. 34. bœ bœ nœ œ bœ nœ ? œ bœ nœ bœ nœ œ œ bœ nœ bœ nœ œ

35. S bœ nœ bœ nœ ? œ bœ nœ bœ nœ

36. 37. bœ nœ bœ bœ ? œ bœ nœ bœ œ #œ œ œ bœ nœ bœ œ #œ œ

38. 39. œ bœ œ œ bœ ? œ bœ nœ bœ œ #œ œ bœ nœ bœ œ #œ œ

40. 41. œ œ œ bœ nœ ? œ bœ nœ bœ œ #œ œ bœ nœ bœ œ #œ œ

42. 43. œ bœ œ œ ? œ bœ nœ bœ œ #œ #œ œ bœ nœ bœ œ #œ #œ

Appendix A - p.2 44. 45. #œ #œ œ œ bœ nœ ? œ bœ nœ bœ œ #œ œ bœ nœ bœ œ #œ

46. 47. œ bœ nœ bœ œ bœ nœ œ ? œ bœ nœ bœ œ œ bœ nœ bœ œ

48. œ bœ bœ nœ ? œ bœ nœ bœ œ

49. S œ œ bœ nœ ? œ bœ nœ bœ œ

50. 51. bœ nœ bœ nœ œ #œ ? œ bœ nœ bœ œ œ #œ œ #œ #œ œ #œ

52. 53. œ œ #œ œ ? œ #œ œ #œ #œ œ #œ œ #œ œ #œ #œ œ #œ

54. 55. œ #œ œ œ #œ œ ? œ #œ œ #œ #œ œ œ #œ œ #œ #œ #œ

56. 57. bœ nœ bœ nœ bœ nœ ? œ bœ nœ bœ œ œ bœ nœ œ œ #œ œ

58. 59. œ bœ bœ bœ œ ? œ bœ nœ œ œ #œ œ bœ nœ œ œ #œ œ

60. 61. œ œ bœ œ œ ? œ bœ nœ œ œ #œ œ bœ nœ œ œ #œ œ

62. 63. bœ nœ œ #œ ? œ bœ nœ œ œ #œ œ œ bœ nœ œ œ #œ #œ

Appendix A - p.3 64. 65. #œ œ œ #œ #œ œ ? œ bœ nœ œ œ #œ œ bœ nœ œ œ #œ

66. 67. œ #œ œ œ bœ nœ bœ ? œ bœ nœ œ œ #œ œ bœ nœ œ œ

68. œ bœ nœ œ ? œ bœ nœ œ œ

69. S œ bœ bœ nœ ? œ bœ nœ œ œ

70. 71. œ bœ nœ bœ nœ bœ nœ ? œ bœ nœ œ œ œ œ bœ nœ œ œ

72. 73. bœ nœ bœ bœ nœ œ ? œ bœ nœ œ #œ œ œ bœ nœ œ #œ œ

74. 75. bœ bœ nœ œ bœ nœ ? œ bœ nœ œ #œ œ œ bœ nœ œ #œ œ

76. 77. œ #œ œ œ #œ œ ? œ bœ nœ œ #œ #œ œ bœ nœ œ œ #œ

78. 79. œ bœ nœ bœ bœ nœ œ ? œ bœ nœ œ #œ œ bœ nœ œ #œ œ

80. 81. œ bœ bœ nœ œ bœ nœ ? œ bœ nœ œ #œ œ bœ nœ œ #œ œ

82. 83. œ #œ œ œ #œ œ ? œ bœ nœ œ #œ #œ œ bœ nœ œ œ #œ

Appendix A - p.4 84. 85. œ #œ œ #œ œ œ #œ œ ? œ bœ nœ #œ œ bœ bœ nœ œ #œ

86. 87. œ bœ bœ œ #œ œ ? œ bœ bœ nœ œ #œ œ bœ bœ nœ œ #œ

88 89. œ œ bœ œ œ œ ? œ bœ bœ nœ œ #œ œ bœ bœ nœ œ #œ

90. 91. œ bœ nœ #œ œ #œ ? œ bœ bœ nœ œ #œ œ bœ bœ nœ œ #œ

92. 93. œ œ #œ œ ? œ bœ bœ nœ œ #œ #œ œ bœ bœ nœ œ #œ #œ

94. 95. œ #œ œ bœ nœ bœ ? œ bœ bœ nœ œ #œ œ bœ bœ nœ œ œ

96. S bœ nœ œ ? œ bœ bœ nœ œ œ

97. 98. bœ bœ nœ œ bœ nœ ? œ bœ bœ nœ œ œ œ bœ bœ nœ œ œ

99. 100. bœ nœ bœ nœ œ #œ ? œ bœ bœ nœ œ œ bœ bœ nœ #œ œ #œ

101. 102. œ #œ œ œ bœ bœ nœ ? œ bœ bœ nœ #œ œ bœ bœ nœ #œ œ

103. 104. œ bœ nœ œ #œ œ ? œ bœ bœ nœ #œ œ œ bœ bœ nœ #œ #œ

Appendix A - p.5 105. 106. œ #œ œ #œ œ œ #œ œ #œ ? œ bœ bœ nœ œ bœ bœ œ #œ

107. 108. œ #œ œ œ œ bœ bœ nœ ? œ bœ bœ œ #œ œ bœ bœ œ #œ

109. 110. œ œ bœ nœ #œ œ #œ œ ? œ bœ bœ œ #œ œ bœ bœ œ #œ

111. 112. œ #œ œ #œ œ bœ nœ bœ nœ bœ nœ ? œ bœ bœ œ œ bœ bœ

113. 114. œ #œ œ œ ? œ bœ œ œ #œ œ #œ œ bœ œ œ #œ œ #œ

115. 116. bœ bœ nœ œ #œ œ ? œ bœ œ œ #œ œ œ bœ œ œ #œ œ

117. 118. œ #œ œ œ #œ œ ? œ bœ œ œ #œ #œ œ bœ œ œ œ #œ

119. 120. œ #œ œ œ #œ œ ? œ bœ œ #œ œ #œ œ bœ œ #œ œ #œ

121. 122. œ #œ œ bœ bœ ? œ œ bœ nœ œ #œ œ œ bœ nœ œ #œ œ

123. 124. œ bœ œ œ bœ ? œ œ bœ nœ œ #œ œ œ bœ nœ œ #œ œ

125. 126. œ œ bœ nœ ? œ œ bœ nœ œ #œ œ œ œ bœ nœ œ #œ œ

Appendix A - p.6 127. 128. #œ œ #œ #œ œ œ ? œ œ bœ nœ œ #œ œ œ bœ nœ œ #œ

129. 130. #œ #œ œ œ #œ œ ? œ œ bœ nœ œ #œ œ œ bœ nœ œ #œ

131. S œ bœ nœ bœ ? œ œ bœ nœ œ

132. 133. œ bœ nœ œ œ #œ #œ œ ? œ œ bœ nœ œ œ œ bœ nœ œ

134. 135. œ #œ œ bœ nœ bœ nœ ? œ œ bœ nœ œ œ œ œ bœ nœ œ

136. 137. œ #œ œ œ ? œ œ bœ nœ #œ œ #œ œ œ bœ nœ #œ œ #œ

138. 139. bœ bœ nœ œ bœ nœ ? œ œ bœ nœ #œ œ œ œ bœ nœ #œ œ

140. 141. œ #œ œ bœ nœ bœ nœ ? œ œ bœ nœ #œ #œ œ œ bœ nœ œ

142. 143. œ bœ nœ bœ bœ nœ œ ? œ œ bœ œ #œ œ œ bœ œ #œ œ

144. 145. œ bœ bœ nœ œ #œ œ ? œ œ bœ œ #œ œ œ bœ œ #œ œ

146. 147. œ #œ œ œ #œ œ ? œ œ bœ œ #œ #œ œ œ bœ œ œ #œ

Appendix A - p.7 148. 149. bœ nœ bœ nœ bœ nœ œ #œ œ #œ ? œ œ bœ œ œ œ œ #œ

150. 151. œ #œ œ œ œ #œ #œ œ ? œ œ œ œ #œ œ œ œ œ #œ

152. 153. œ œ #œ œ #œ œ #œ œ ? œ œ œ œ #œ œ œ œ œ #œ

154. 155. œ #œ œ œ #œ œ ? œ œ œ œ œ #œ œ œ œ #œ œ #œ

156. 157. œ #œ œ œ bœ ? œ œ œ #œ œ #œ œ bœ nœ œ #œ œ #œ

158. 159. œ œ #œ œ ? œ bœ nœ œ #œ œ #œ œ bœ nœ œ #œ œ #œ

160. 161. œ #œ œ œ #œ œ ? œ bœ nœ œ #œ œ œ bœ nœ œ #œ #œ

162. 163. œ bœ nœ bœ nœ œ #œ œ ? œ bœ nœ œ œ bœ nœ #œ œ #œ

164. 165. œ #œ œ œ #œ œ ? œ bœ œ #œ œ #œ œ œ œ #œ œ #œ

Appendix A - p.8 TUNING THE HORN

APPENDIX B

HSIS-13 (Harmonic Series Interval Set - 13)

Michael Hugh Dixon Harmonic Series Interval Set - 13 (HSIS - 13)

This Appendix shows a collection of 46 harmonic series. The notation for the collection was devised as a practical memory aid while I was learning harmonic singing. All series are shown with harmonics four to thirteen though in some instances an octave harmonic like the 4th is omitted when it is easily deduced.

The first series is notated with C in the bass clef as the fundamental, which is easy to read and was close to the pitch I sang. This first series is referred to as the originating series. Each of the subsequent 45 series was chosen by using a pitch in the originating series and making it a different harmonic, ie. changing the fundamental. These series are called

Derivation 1 to 45.

At the beginning of each derivation two pitches show how the series was derived from the original series. This relation is shown as a ratio expressing the total interval rather than a ratio octave reduced, which is the standard just intonation practice. For example, in Derivation 1 the bass clef C and treble clef E are expressed as the ratio 5:1. The next two pitches show E as the fundamental with harmonics 4 to 13 expressed as ratios to the original series C (1:1). The distance from C is shown in cents.

The format clearly shows the relationship of each derivation which starts on each harmonic of the original. Derivation 1 comes from the original 5th

Appendix B - i harmonic. Derivations 2 & 3 come from the original 6th harmonic.

Derivations 4, 5, & 6 come from the original 7th harmonic. Derivations 7, 8,

9 & 10 come from the original 8th harmonic. Derivations 11 to 15 come from the original 9th harmonic. Derivations 16 to 21 come from the original 10th harmonic. Derivations 22 to 28 come from the original 11th harmonic. Derivations 29 to 36 come from the original 12th harmonic.

Derivations 37 to 45 come from the original 13th harmonic.

Appendix B - ii Harmonic Series Interval Set - 13 (HSIS-13) Michael H Dixon 2007

bw “” ° w w B w µw w Bw & w w w Original Harmonics: 4 5 6 7 8 9 10 11 12 13 Series Ratios: (4:1) 5:4 3:2 7:4 (8:1) 9:8 5:4 11:8 3:2 13:8 Cents: 0 386 702 969 0 204 386 551 702 841 ? w ¢ 1

“” w µw ° w w w #w #w µw & #w w w4 5 6 7 8 9 10 11 12 13 (5:1) 5:4 25:16 15:8 35:32 5:4 45:32 5:4 55:32 15:8 65:64 Derivation 386 386 772 1088 155 386 590 772 937 1088 27 1 ? w w ¢ 1

“” bw Bw ° w bw bw w w Bw & w bw 5 6 7 8 9 10 11 12 13 (6:1) 3:2 9:5 21:20 6:5 27:20 3:2 33:20 9:5 39:20 Derivation 702 702 1018 85 316 520 702 867 1018 1157 2 ? w bw ¢ 1

” “ w µw w Bw ° w Bw w w & w w w 4 5 6 7 8 9 10 11 12 13 (6:1) 3:2 15:8 9:8 21:16 3:2 27:16 15:8 33:32 15:8 39:32 Derivation 702 702 1088 204 471 702 906 1088 53 1088 342 3 w ? w ¢ 1

“” Bbw w bw Bbw bw ° bw bw Bbw Bbw Bw B & 969 B969 B653w B969 36 267 471 653 818 969 1108 (7:1) 7:4 35:24 7:4 49:48 7:6 21:16 35:24 77:48 7:4 91:48 Derivation 6 5 6 7 8 9 10 11 12 13 4 ? w Bbw ¢ 1

Appendix B - p.1 “” bw ∫w Bbw bw bw Bbw Bw ° bw bw Bbw & 969 B 969 85 352 583 787 969 1134 85 224 (7:1) 7:4 21:20 49:40 7:5 63:40 7:4 77:40 21:20 91:80 Derivation 5 5 6 7 8 9 5 11 12 13 ? w bw ¢ 1 “” w bw Bw bw Bbw Bbw Bbw Bw B ° bw Bw Bw & 969 B 969 155 471 738 969 1173 155 320 471 610 (7:1) 7:4 35:32 21:16 49:32 7:4 63:32 35:32 77:64 21:16 91:64 Derivation 6 4 5 6 7 8 9 10 11 6 13 Bbw ? w ¢ 1 “” w #w #w µw w ° w w w µw µw & 0 0 #617w µ933 0 231 435 617 782 933 1072 Derivation (8:1) 1:1 10:7 12:7 1:1 8:7 9:7 10:7 11:7 12:7 13:7 7 7 5 6 7 8 9 10 11 12 13 ? w µw ¢ 1 “” w w w w w Bw w B ° w w w Bbw & 0 0 884 0 267 498 702 884 1049 0 139 Derivation (8:1) 1:1 5:3 1:1 7:6 4:3 3:2 5:3 11:6 1:1 13:12 8 6 5 6 7 8 9 10 11 12 13 ? w w ¢ 1 “” w w bw bw w Bw bw B ° w bw bw & 0 0 316 583 814 1018 0 165 316 455 Derivation (8:1) 1:1 6:5 7:5 8:5 9:5 1:1 11:10 6:5 13:10 9 5 6 7 8 9 10 11 12 13 bw ? w ¢ 1 ” “ w w w Bw w Bbw w w µ ° w w w & 0 0 386 702 969 0 204 386 551 702 841 (8:1) 1:1 5:4 3:2 7:4 1:1 9:8 5:4 11:8 3:2 13:8 Derivation 4 5 6 7 8 9 10 11 12 13 10 w ? w ¢ 1

Appendix B - p.2 “” w w ° w w w #w µw w B & w Bw 204 204 #590w 906 1173 204 408 590 755 906 1045 (9:1) 9:8 45:32 27:16 63:32 9:8 81:64 45:32 99:64 27:16 117:64 Derivation 8 5 6 7 8 9 5 11 12 13 11 ? w w ¢ 1 “” w w #w ° w w w ˜w #w #w µ & #w µw µ 204 204 821 1137 204 933 639 821 986 1137 76 (9:1) 9:8 45:28 27:14 9:8 9:7 81:56 45:28 99:56 27:14 117:112 Derivation 7 5 6 7 8 9 10 11 12 13 12 ? w µw ¢ 1 “” w w µw w Bw ° w w Bw w w & w 204 204 1088 204 471 702 906 1088 53 204 342 (9:1) 9:8 15:8 9:8 21:16 3:2 27:16 15:8 33:32 9:8 39:32 Derivation 6 5 6 7 8 9 10 11 12 13 13 ? w w ¢ 1 ” “ w w w bw w w Bw B ° w w bw & 204 204 520 787 1018 22 204 369 520 659 (9:1) 9:8 27:20 63:40 9:5 81:80 9:8 99:80 27:20 117:80 Derivation 5 6 7 8 9 10 11 12 13 14 bw ? w ¢ 1 ” w w w “ w w w #w µ ° #w w B & w 204 204 590 906 1173 204 408 590 755 906 (9:1) 9:8 45:32 27:16 63:32 9:8 81:64 45:32 99:64 27:16 Derivation 4 5 6 7 8 9 10 11 12 15 ? w w ¢ 1 ” w “ bw ° w w w µw µw w & w Bw 386 386 µ568w 884 1151 182 386 568 733 884 1023 (10:1) 5:4 25:18 5:3 35:18 10:9 5:4 25:18 55:36 5:3 65:36 Derivation 9 5 6 7 8 9 10 11 12 13 16 ? w w ¢ 1

Appendix B - p.3 w “” w µw ° w w #w #w µw & #w w 386 386w 772 1088 155 386 590 772 937 1088 27 (10:1) 5:4 25:16 15:8 35:32 5:4 45:32 5:4 55:32 15:8 65:64 Derivation 4 5 6 7 8 9 10 11 12 13 17 ? w w ¢ 1

w “” w #w µw ° w w #w #w #w µ & #w #w 386 386 1003 119 386 617 821 772 1168 119 258 (10:1) 5:4 25:14 15:14 5:4 10:7 45:28 5:4 55:28 15:14 65:56 Derivation 7 5 6 7 8 9 10 11 12 13 18 ? w #w ¢ 1 “” w w #w µw w µw ° w w Bw w & #w 386 386 70 386 653 386 1088 70 235 386 525 (10:1) 5:4 25:24 54 35:24 5:4 15:8 25:24 55:48 54 65:48 Derivation 6 5 6 7 8 9 10 11 6 13 19 w ? w ¢ 1

w ◊ÿ ° w w µw w Bw & w Bbw w 386 386w 702 969 0 204 386 551 702 841 (10:1) 5:4 3:2 7:4 (8:1) 9:8 5:4 11:8 3:2 13:8 Derivation 5 6 7 8 9 10 11 12 13 20 w ? w ¢ 1

w ◊ÿ w w ° w #w #w µw µ & #w w w 386 386w 772 1088 155 386 590 772 937 1088 27 (10:1) 5:4 25:16 15:8 35:32 5:4 45:32 5:4 55:32 15:8 65:64 Derivation 4 5 6 7 8 9 10 11 12 13 21 w ? w ¢ 1 µw “” w ° µw w w µw w Bw & w Bw B B 551 551 µ551w B867 1134 165 369 551 716 867 1006 (11:1) 11:8 11:8 33:20 77:40 11:10 99:80 11:8 121:80 33:20 143:80 Derivation 10 5 6 7 8 9 10 11 12 13 22 ? w Bw ¢ 1

Appendix B - p.4 w µ “” w w ° µw bw w µw µw w B & w Bw B 551 551 µ733 1049 116 347 551 733 898 1049 1188 (11:1) 11:8 55:36 11:6 77:72 11:9 11:8 55:36 121:72 11:6 143:72 Derivation 9 5 6 7 8 9 10 11 12 13 23 ? w Bw ¢ 1 w µ “” w µw w ° µw bw µw µw µw & w µw 551 551 µ937 53 320 551 755 937 1102 53 192 (11:1) 11:8 55:32 33:32 77:64 11:8 99:64 55:32 121:64 33:32 143:128 Derivation 8 5 6 7 8 9 10 11 12 13 24 ? w µw ¢ 1 w ” µ “ w ˜w #w w ° µw #w µw µw #w µ & µw 551 551 1168 284 551 782 986 1168 133 284 423 (11:1) 11:8 55:28 33:28 11:8 11:7 99:56 55:28 121:112 33:28 143:112 Derivation 7 5 6 7 8 9 10 11 12 13 25 ? w #w ¢ 1 w “” w w µ w µw µw w µ ° µw w µw bw B & µ 551 551 235 551 818 1049 53 235 401 551 690 (11:1) 11:8 55:48 11:8 77:48 11:6 33:32 55:48 121:96 11:8 143:96 Derivation 6 5 6 7 8 9 10 11 12 13 26 Bw ? w ¢ 1 w µ ◊ÿ w ° w w µw w Bw & w Bw B B 551 µ551w B867 1134 165 369 551 716 867 1006 (11:1) 11:8 33:20 77:40 11:10 99:80 11:8 121:80 33:20 143:80 Derivation 5 6 7 8 9 10 11 12 13 27 ? w Bw ¢ 1 w µ ◊ÿ w µw w ° bw µw µw µw & w µw 551 µ551w µ937 53 320 551 755 937 1102 53 192 (11:1) 11:8 55:32 33:32 77:64 11:8 99:64 55:32 121:64 33:32 143:128 Derivation 4 5 6 7 8 9 10 11 12 13 28 ? w µw ¢ 1

Appendix B - p.5 w “” bw ° w w w µw w Bw & w bw B B 702 702 µ537w B853 1120 151 355 537 702 853 992 (12:1) 3:2 15:11 18:11 21:11 12:11 27:22 15:11 3:2 18:11 39:22 Derivation 11 5 6 7 8 9 10 11 12 13 29 ? w Bw ¢ 1 w “” bw Bw ° w bw bw w w Bw & bw 702 702 1018 85 316 520 702 867 1018 1157 (12:1) 3:2 9:5 21:20 6:5 27:20 3:2 33:20 9:5 39:20 Derivation 10 6 7 8 9 10 11 12 13 30 ? w bw ¢ 1 w “” w w w w w w Bw B ° w w Bbw & 702 702 884 0 267 498 702 884 1049 0 139 Derivation (12:1) 3:2 5:3 1:1 7:6 4:3 3:2 5:3 11:6 1:1 13:12 31 9 5 6 7 8 9 10 11 12 13 ? w w ¢ 1 w “” w w w w w w µw B ° w w Bw & 702 702 1088 204 471 702 906 1088 53 1088 342 Derivation (12:1) 3:2 15:8 9:8 21:16 3:2 27:16 15:8 33:32 15:8 39:32 32 8 5 6 7 8 9 10 11 12 13 ? w w ¢ 1 w “” w #w w µw #w #w µ ° w #w µw w µ & 702 702 119 435 702 933 1137 119 53 435 342 Derivation (12:1) 3:2 15:14 9:7 3:2 12:7 27:14 15:14 33:32 9:7 39:32 33 7 5 6 7 8 9 10 11 12 13 w ? w µ ¢ 1 w ◊ÿ ° w w µw w Bw & w w Bbw w 702 702 386w 702 969 0 204 386 551 702 841 (12:1) 3:2 5:4 3:2 7:4 1:1 9:8 5:4 11:8 3:2 13:8 Derivation 6 5 6 7 8 9 10 11 12 13 34 ? w ¢ 1

Appendix B - p.6 w ◊ÿ bw Bw ° bw bw w w Bw & w bw 702 702 1018 85 316 520 702 867 1018 1157 (12:1) 3:2 9:5 21:20 6:5 27:20 3:2 33:20 9:5 39:20 Derivation 5 6 7 8 9 10 11 12 13 35 ? w bw ¢ 1 ÿ w ◊ w w w w w µw B ° w w Bw & 702 702w 1088 204 471 702 906 1088 53 1088 342 Derivation (12:1) 3:2 15:8 9:8 21:16 3:2 27:16 15:8 33:32 15:8 39:32 36 4 5 6 7 8 9 10 11 12 13 ? w w ¢ 1 Bw ” B“w w bw Bw bw ° w bw Bw Bw & 841 841 525w B841 1108 139 342 525 690 841 980 Derivation (13:1) 13:8 65:48 13:8 91:46 13:12 39:32 65:48 143:96 13:8 169:96 37 12 5 6 7 8 9 10 11 12 13 ? w Bw ¢ 1 Bw ” B“w w Bw bw bw ° bw Bbw bw w & 841 841 676w 992 59 290 494 676 841 992 1131 Derivation (13:1) 13:8 65:44 39:22 91:88 13:11 117:88 65:44 13:8 39:22 169:88 38 11 5 6 7 8 9 10 11 12 13 ? w bw ¢ 1 w B “” bw Bw w w Bw bw Bw ° Bw w B B & 841 841 B841w 1157 224 841 659 841 1006 1157 96 Derivation (13:1) 13:8 13:8 39:20 91:80 13:10 117:80 13:8 143:80 39:20 169:80 39 10 5 6 7 8 9 10 11 12 13 ? w Bw ¢ 1 Bw “” w bw Bw w Bw Bbw w B ° bw Bw bw B & 841 841 B1021 139 406 637 841 1021 1188 139 278 Derivation (13:1) 13:8 65:36 13:12 91:72 13:9 13:8 65:36 143:72 13:12 169:144 40 9 5 6 7 8 9 10 11 12 13 ? w Bw ¢ 1

Appendix B - p.7 Bw “” w w w w Bw w w B ° B w Bw bw B & 841 841 27 342 610 841 1045 27 192 342 482 (13:1) 13:8 65:64 39:32 91:64 13:8 117:64 65:64 143:128 39:32 169:128 Derivation 8 5 6 7 8 9 10 11 12 13 41 w ? w B ¢ 1

w ” B “ w w µw w Bw Bw Bw µw µ ° µw µw & 841 841 258 574 841 1072 76 258 423 574 713 (13:1) 13:8 65:56 39:28 13:8 13:7 117:112 65:56 143:112 39:28 169:112 Derivation 7 5 6 7 8 9 10 11 12 13 42 Bw ? w ¢ 1

Bw ÿ ◊ w bw Bw bw ° w w bw Bw Bw & 841 B841 525w B841 1108 139 342 525 690 841 980 (13:1) 13:8 65:48 13:8 91:46 13:12 39:32 65:48 143:96 13:8 169:96 6 5 6 7 8 9 10 11 12 13 Derivation 43 Bw ? w ¢ 1

w B ◊ÿ bw w w Bw bw Bw ° Bw w B B & 841 B841w 1157 224 841 659 841 1006 1157 96 (13:1) 13:8 39:20 91:80 13:10 117:80 13:8 143:80 39:20 169:80 5 6 7 8 9 10 11 12 13 Derivation 44 Bw ? w ¢ 1

Bw ◊ÿ w w w Bw w w B ° w Bw bw B & 841 B841w 27 342 610 841 1045 27 192 342 482 (13:1) 13:8 65:64 39:32 91:64 13:8 117:64 65:64 143:128 39:32 169:128 4 5 6 7 8 9 10 11 12 13 Derivation 45 Bw ? w ¢ 1 Appendix B - p.8 TUNING THE HORN

APPENDIX C

HEWM NOTATION (Helmholtz-Ellis-Wolf-Monzo) HEWM Notation

Helmholtz-Ellis-Wolf-Monzo

For many works in the accompanying folio I used a modification of the notation system HEWM. It has its roots in just intonation lattices that were developed by

Euler in the eighteenth century, Oettingen then Helmholtz and Ellis in the nineteenth century then Reimann, Fokker, Johnston, Tenney, Wolf, Monzo and others in the twentieth century (Monzo 2005; Dysart 2007; Honingh 2009). The web encyclopedia Tonalsoft: Encyclopedia of Microtonal Music Theory has full details of the HEWM system. An outline of the system as I used it for 13-limit just intonation is given below.

Any modification to standard notation or any new notation requires significant study by performers. Pitches on a staff can be considered as having precise relations to each other though for many musicians they are a only a guide to the intonation. Precision is gained aurally according to context. I my folio I have considered the pitches on the staff as a guide only. I relied on the experience and skill of the performers, their time spent learning the music with an accurate sound file, their time spent with me confirming or suggesting changes to their intonation, their time spent studying the HEWM markings in the score and their time studying the Pitch Lists. I did not expect the performers to make any calculations or to understand the language of just intonation ratios. The use of

Appendix C - p.1 cents helped to quickly see distance between pitches though HEWM notation clearly shows the relationship of a pitch to a central reference.

I studied notation in depth in my Master’s thesis Just intonation: a study through its notation (2001) and found HEWM of most use to me at that stage. I acknowledge the various just intonation notations used by composers and performers including those by Partch (1974), Harrison (1971) and Johnston from the 1960’s (2006). Other staff notations using additional symbols are the sagittal notation (Keenan 2004) which can be adapted for just intonation and the recent 23-limit system of Sabat and Schweinitz (2005). From my experience performing with symbolic microtonal notation on both standard 5-line staves and non-standard staves such as Aaron Hunt’s 25-line staff (2008) I have found that notation systems that require new symbols are not as easy to read compared with standard symbols on standard staves where microtonal deviation is expressed numerically, using either cents or ratios. Ultimately familiarity with microtonal intonation is an aural art that can only be developed by a performer through constant practice. Non-standard symbols place extra demands on the performer. For notating microtonal music, in the long term it seems more prudent to adopt a notation system like HEWM which employs the best of both standard and non-standard notation systems; it provides a microtonal specification for a pitch on a standard staff - more appropriate for

Appendix C - p.2 performance - using two numbers to specify fine tuning - more appropriate for rehearsal.

HEWM notation employs two numbers connected with a pitch on a standard staff.

The larger numbers indicate the harmonic each pitch is based on.

The smaller numbers show the placement or distance from the central pitch, ratio 1:1.

The central pitch does not need a notational number though sometimes I indicated it with T for tonic.

rd The pitch a pure perfect fifth from 1:1 is indicated as 31, the 3 harmonic interval and only once removed from 1:1 (the distance of a 3rd to a 2nd harmonic, pure perfect fifth).

th The pitch a pure major third from 1:1 is indicated as 51, the 5 harmonic interval and only once removed from 1:1 (the distance of a 5th to a 4th harmonic, pure major third).

th The pitch a pure septimal seventh from 1:1 is indicated as 71, the 7 harmonic interval and only once removed from 1:1 (the distance of a 7th to a 4th harmonic, pure septimal seventh).

Appendix C - p.3 th The pitch a pure neutral fourth from 1:1 is is indicated as 111, the 11 harmonic interval and only once removed from 1:1 (the distance of an 11th to an 8th harmonic, pure neutral fourth).

th The pitch a neutral sixth from 1:1 is is indicated as 131, the 13 harmonic interval and only once removed from 1:1 (the distance of a 13th to an 8th harmonic, neutral sixth).

When a pitch is twice removed from 1:1 by the same interval it is indicated with as a small 2. Pitches can be thrice removed and so on. For example the figure below shows a line of 3rd harmonics. The pitches to the left of 1:1 (the note C for convenience) can be viewed as subharmonics and are indicated with a minus sign (-). In the table the number of cents (1200 in an octave, 100 for an equal- tempered semitone) is shown to clarify relative distance).

3-4 3-3 3-2 3-1 T 3 1 3 2 3 3 3 4

A! E! B! F C G D A E

792c 294c 996c 498c 0c 702c 204c 906c 408c

This process works exactly the same for any prime numbered harmonic.

Pitches can be a combination of prime numbered harmonics such as the 15th harmonic (ratio 15:8) which is a combination of the 3rd and 5th harmonics. This is indicated as 3151.

Appendix C - p.4 Pitches can be a combination of harmonics and subharmonics such as the ratio

45:28 which is a 3rd harmonic twice removed from 1:1, a 5th harmonic once

th removed and a 7 subharmonic once removed. This is indicated as 32517-1.

HEWM notation indicates chordal context clearly. An example of pitch movement from a chord with ratios 6:7:9:11 to the same set of ratios beginning

th on the 11 harmonic will suffice. This is indicated as 31 : 71 : 32 : 111. for the first chord and 111 : 111 71 : 111 32 : 112 for the second chord.

In the some of the scores in the folio of compositions the root of a chord is indicated using bold type and other parts of the chord in normal type.

Appendix C - p.5 TUNING THE HORN

APPENDIX D

A Method for Playing the Horn

in Extended Just Intonation

Michael H Dixon Table of Contents

Section 1 Progressive Establishment of Resources 1 Set 1 Orientation 10 Set 2 Exploring the 7th harmonic 13 Set 3 Visiting the 9th harmonic 21 Set 4 The 10th harmonic 27 Set 5 Acquanting ourselves with the 11th harmonic 32 Set 6 The 12th harmonic 39 Set 7 Discovering the 13th harmonic 43 Set 8 The 14th harmonic 48 Set 9 The 15th harmonic 53 Set 10 The 16th harmonic 58

Section 2 Harmonic Series Slurs 63 Set 1 Light Slurs 64 Set 2 Harmonics 4 to 6 & 7 69 Set 3 Harmonics 4 to 6, 7 & 9 74 Set 4 Harmonics 4 to 6, 7, 9 & 11 79 Set 5 Harmonics 4 to 6, 7, 9, 11 & 13 84 Set 6 Harmonics 4 to 6, 7, 9, 11, 13 & 15 92

Section 3 Wider Intervals 101 Set 1 From the 7th harmonic 102 Set 2 From the 11th harmonic 109 Set 3 From the 13th harmonic 117

Method: Appendix D - i Harmonics

Section One – Progressive Establishment of Resources

A double horn has sixteen separate harmonic series. Each tube length has a fundamental which is referred to by note name in sounding pitch. The convention for horn writing is to notate all pitches a perfect fifth higher than they sound, the written C equals sounding F. The F horn has a tube length of approximately 3.75 metres, its fundamental being one octave below the bass staff.

!!!

This fundamental and the harmonics arising from it are played without valves and indicated to the player with the fingering (valve choice) F:0. All fingerings will be indicated in bold type.

Cent values for each note are shown above each pitch, the fingering below each pitch and the harmonic series number below that. This format is used throughout the Method. Cent values provide a useful visual comparison between pitches. There are 100 cents in an equal tempered semitone and 1200

Method: Appendix D - p.1 cents in an octave. Written C can be indicated as 0 cents or 1200 cents, though usually the former. Cents will be abbreviated throughout: 0 cents as 0c.

A second tube length is the E horn, played on the F horn with valve two (F:2).

When the valve slide is tuned an equal tempered semitone below the open F horn, the cent value for the horn in E fundamental, written B, is 1100 cents. It is visually more helpful to make pitch comparisons with positive numbers, hence the B is indicated as 1100 cents rather than -100.

A third tube length is the E! horn, played on the F horn with valve one (F:1).

This written B! is 1000c from written C.

! The series a minor third lower than the open F horn has two distinct

! versions: with valve three (F:3) and with valves one and two in

! combination (F:1,2). The pitch difference between each series is about 10

! cents depending on valve slide set up. This amount is significant enough

! for many harmonic contexts to warrant the consideration of them as

! separate.

A fourth tube length is the higher version of the D horn, played on the F horn with valves one and two (!"#$%). This written A is 910c from written C.

A fifth tube length is the lower version of the D horn, played on the F horn with valve three (!"&). This written A is either 900c or 885c from written C. When the

Method: Appendix D - p.2 third valve by itself is set for an equal tempered written A (900c) the written A! is high (815c). When the third valve in combination with the second is set to produce an equal tempered written A! the written A is low (885c). The author uses the latter as a default setting as it only takes a small movement of two or three fingers in the right hand to shift the A with valves one and two down by

10 cents rather than shifting the low A on third valve up by 15 cents.

A sixth tube length is the D! horn, played on the F horn with valves two and three (!"#$%). This written A! is either 815c or 800c from written C as explained above.

3rd valve slide set for D horn!! ! 3rd valve slide set for D! horn in 12tet relation to A=440Hz.!! ! in 12tet relation to A=440Hz.

A seventh tube length is the C horn, played on the F horn with valves one and three (!"&$%). This written G is either 715c or 730c from written C, depending on

Method: Appendix D - p.3 which third valve setting is used. It is notated throughout as either G or G half sharp.

An eighth tube length is the B horn, played on the F horn with valves one, two and three (!"#$%$&). This written F! is either 738c or 753c from written C, depending on which third valve setting is used. It is notated throughout as

F".The relations between these eight tube lengths are repeated a perfect fourth higher on the B# horn. There is a microtonal overlap between the two sets. The difference between the open F horn (F:0) and B# horn with valves one and three

(B#:1,3) is about 15 or 30 cents depending on the valve slide set up. The difference between F horn with second valve (F:2) and B# horn with all three valves (B#:1,2,3) is even greater being either 38 cents or about 53 cents, the latter being slightly more than an equal tempered quartertone (Gregory 1961; Heim

1990: 113).

Harmonics 1 to 16 in each of the sixteen tube lengths are shown in staff notation on pages seven to nine with both the high and low settings of the third valve slide for all affected series.

Method: Appendix D - p.4 $%&'() !"" * ++, *,- ."/ 0"!0 0/0 1++ ˙ ˙ ˙ ˙ ˙ B˙ Bb˙ n˙ # b˙ ˙ ˙ B b ? ˙ & ˙ ˙ B ˙ ˙ !"#$%&"#$' (b') "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() /"" 00"* .+, 0,- ,"/ -!0 /0 *++ ˙ #˙ ˙ ˙ #˙ #˙ µ˙ ˙ µ˙ B A ? ˙ B˙ ˙ & ˙ #˙ ˙ !"#$%&"#$' ˙ (b'* "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 1"" 0""* ,+, ,- !"/ +!0 00/0 0++ b˙ n˙ b˙ b˙ ˙ ˙ B˙ b˙ B˙ B Ab ? b˙ Bb˙ b˙ & b˙ ˙ b˙ !"#$%&"#$'b˙ (b'+ "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() *0" -0* !-, 00.- /0/ .,0 0"!0 -+ ˙ ˙ #˙ ˙ ˙ ˙ #˙ µ˙ ˙ µ B G ? ˙ & ˙ ˙ ˙ ˙ #˙ !"#$%&"#$' ˙ (b'+,* "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() *"" -"* !+, 00,- /"/ .!0 0"/0 ++ ˙ #˙ ˙ ˙ ˙ #˙ µ˙ ˙ µ˙ B G ? ˙ & ˙ B˙ ˙ ˙ #˙ !"#$%&"#$' ˙ (b'- ./"$/ 0%11"#$2 "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 0+! ++. !.0 00!/ 1+- .1, 0"*, .1 ˙ #˙ ˙ ˙ ˙ #˙ µ˙ ˙ ˙ B G ? ˙ & ˙ B˙ ˙ ˙ #˙ !"#$%&"#$' ˙ (b'- .345 0%11"#$2 "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 00! +0. !"0 0"+/ 10- ,,, -!, 1 b˙ b˙ ˙ ˙ b˙ B˙ Bb˙ n˙ Gb ? b˙ b˙ b˙ b˙ B & b˙ ˙ b˙ !"#$%&"#$'b˙ (b'*,-./"$/ 0%11"#$2 "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 0"" +"* /+, 0",- 1"/ ,!0 -/0 00++ b˙ n˙ b˙ b˙ b˙ ˙ B˙ b˙ B˙ B Gb ? b˙ & Bb˙ b˙ b˙ ˙ b˙ !"#$%&"#$'b˙ (b'*,-.345 0%11"#$2 "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

Method: Appendix D - p.5 $%&'() 1" .1* /0, --- *1/ !+0 +.0 000+ ˙ ˙ ˙ b˙ b˙ n˙ ˙ F ? µ˙ ˙ µ˙ ˙ µ µ ˙ & ˙ µ˙ b˙ !"#$%&"#$' ˙ (b'+,- ./"$/ 0%11"#$2 "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 0! .0. /"0 -+/ *0- !,, +!, 00"1 ˙ ˙ ˙ b˙ n˙ ˙ F ? ˙ ˙ ˙ ˙ µ˙ B ˙ & ˙ ˙ b˙ !"#$%&"#$' ˙ (b'+,- .345 0%11"#$2 "#$%&'()*+ ! * 1 / ! , . + - 0" 00 0* 01 0/ 0! 0,

$%&'() " ."* 1+, -,- *"/ !!0 +/0 0"++ ˙ ˙ ˙ Bb˙ n˙ ˙ F ? ˙ ˙ ˙ ˙ µ˙ B ˙ & ˙ ˙ Bb˙ !"#$%&"#$' ˙ !') "#$%&'()*+ ! , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 00!1 ,!! 11- -** 0!. !"/ .-/ 0"/0 ˙ #˙ ˙ ˙ µ˙ E ? ˙ µ ˙ ˙ #˙ ˜˙ ˜ µ˙ ˜ & ˙ ˙ µ˙ ˜ ˜ !"#$%&"#$' (b'+,*,- ./"$/ 0%11"#$2˜˙ ˜ "#$%&'()*+µ!˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 001+ ,/" 1*/ -". 0/* /+- ..- 0"*, ˙ #˙ ˙ #˙ µ˙ E ? ˙ µ ˙ #˙ #˙ ˜˙ µ˙ ˜ & ˙ ˙ µ˙ ˜ !"#$%&"#$' (b'+,*,-.345 0%11"#$2#˙ ˜ "#$%&'()*+µ!˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 00"" ,"* *+, +,- 0"/ /!0 ./0 -++ ˙ ˙ ˙ #˙ ˙ E ? #˙ #˙ µ˙ #˙ µ B ˙ & #˙ B˙ ˙ #˙ !"#$%&"#$' !'* #˙ "#$%&'()*+ !˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() 0""" !"* 0+, .,- / 1!0 ,/0 +++ b˙ ˙ b˙ n˙ b˙ Eb ? ˙ ˙ ˙ B˙ ˙ B B b˙ & ˙ Bb˙ b˙ !"#$%&"#$' !'+ ˙ "#$%&'()*+ b!˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() -0" /0* -, ,.- 000/ *,0 !!0 .-+ ˙ #˙ ˙ D ? ˙ #˙ ˙ ˙ µ˙ ˙ ˙ & ˙ ˙ ˙ µ !"#$%&"#$' !'+,* #˙ ˙ "#$%&'()*+ !˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

Method: Appendix D - p.6 $%&'() -"" /"* +, ,,- 00"/ *!0 !/0 .++ ˙ ˙ ˙ #˙ ˙ D ? ˙ #˙ µ˙ ˙ µ B ˙ & B˙ ˙ ˙ !"#$%&"#$' !'- ./"$/ 0%11"#$2 #˙ ˙ "#$%&'()*+ !˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() ++! 1+. .0 ,!/ 0"+- *1, !*, ..1 ˙ ˙ ˙ #˙ ˙ D ? ˙ #˙ µ˙ ˙ B ˙ & B˙ ˙ ˙ !"#$%&"#$' !'- .345 0%11"#$2 #˙ ˙ "#$%&'()*+ !˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() +0! 10. 0 !+/ 0"0- 0,, /!, ."1 b˙ n˙ b˙ Db ? b˙ ˙ ˙ b˙ B˙ ? b˙ b˙ & b˙ b˙ b˙ B !"#$%&"#$' !'*,- ./"$/ 0%11"#$2 ˙ b˙ "#$%&'()*+ b!˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() +"" 1"* 00+, !,- 0""/ 0!0 //0 ,++ b˙ n˙ b˙ Db ? b˙ b˙ ˙ ˙ b˙ B˙ B b˙ & Bb˙ b˙ b˙ B !"#$%&"#$' !'*,- .345 0%11"#$2 ˙ b˙ "#$%&'()*+ b!˙ , - . / 0 1 2 3 !4 !! !, !- !. !/ !0

$%&'() .1" *1* 000, /-- -1/ +0 1.0 ,0+ ˙ #˙ ˙ C ? µ˙ #˙ µ˙ ˙ µ ˙ µ˙ & ˙ µ˙ µ˙ ˙ !"#$%&"#$' µ!'+,-./"$/ 0%11"#$2 ˙ µ˙ "#$%&'()*+ µ!˙ , - . ! , . + - 0" 00 0* 01 0/ 0! 0,

$%&'() .0! *0. 00"0 /+/ -0- ,, 1!, ,"1 ˙ #˙ ˙ C ? ˙ ˙ ˙ ˙ B˙ ˙ ˙ & ˙ ˙ ˙ µ !"#$%&"#$' !'+,- .345 0%11"#$2 ˙ ˙ "#$%&'()*+ !˙ , - . ! , . + - 0" 00 0* 01 0/ 0! 0,

$%&'() ,!1 0!! 0"1- /** +!. / *-/ !/0 ˙ ˙ ˙ B ? ˜˙ & #˙ ˜˙ ˙ ˜ ˜ ˙ ˜˙ ˙ ˜˙ ˜˙ #˙ !"#$%&"#$' !'+,*,-˜ ./"$/ 0%11"#$2#˙ ˜˙ "#$%&'()*+˜!˙ , - . ! , . + - 0" 00 0* 01 0/ 0! 0,

$%&'() ,1+ 0/" 0"*/ /". +/* 00+- *.- !*, ˙ #˙ ˙ B ? ˜˙ & #˙ ˜˙ ˙ ˜ ˙ ˜˙ ˙ ˜˙ ˜˙ #˙ !"#$%&"#$' ˜!'+,*,-.345 0%11"#$2#˙ ˜˙ "#$%&'()*+ ˜!˙ , - . ! , . + - 0" 00 0* 01 0/ 0! 0,

Method: Appendix D - p.7 The use of ½ flats & sharps and ¾ flats & sharps in the notation of the harmonic series above and the exercises that follow may seem unusual. These indicate pitches 30 to 70 cents different to twelve-tone equal temperament (12tet). The use of a ½ flat (!) indicates a pitch 30 to 70 cents lower than an uninflected note, a ¾ flat (") between 30 to 70 cents lower than a regular flat; ½ sharp (#) indicates a pitch between 30 to 70 cents higher than an uninflected note, a ¾ sharp ($) 30 to 70 cents higher than a regular sharp. Therefore there are a number of enharmonic equivalents such as B! and A$ that indicate the same pitch range, likewise B" and A#, B# and C! and so on. Use simply depends on context.

Some additions to the notation serve as aids throughout:

1. Above the stave, cent values provide a relative reference.

2. Some pitches have two sets of cent values to show the difference

between alternative settings of 3rd valve slides and combinations

involving 3rd valve slides.

3. Fingerings appear in bold type directly below the stave at the

beginning of each line and remain active until altered. All valve

combinations are shown horizontally to avoid colliding with the

information below them.

Method: Appendix D - p.8 4. The ubiquitous harmonic series numbers, always shown in italics,

may appear unnecessary yet they assist in developing awareness of

which harmonic is in use at any time. Such knowledge is of great

benefit when preparing music with more intricate writing.

Once familiar with the layout of this Method, players may find it useful to select exercises from each section that deal primarily with certain harmonics, or pertain to particular key centres. Playing a few exercises on a regular basis will greatly assist in preparing for extended just intonation compositions. The full set of exercises would take far too long to complete daily. Section One: 1½ hours; Section Two 45 minutes; Section Three 3 hours.

Players would benefit from practicing without electronic tuners or drone makers. Such aids will provide greater value once players feel comfortable with the natural harmonics. The practice of these first exercises with a steady, focused and resonant tone will bring to life the subtle tuning that is inherent in the instrument.

Method: Appendix D - p.9 SET I!! Orientation

The exercises in Set 1 establish the valve setting parameters in the familiar context of major triads played on harmonics 4, 5 and 6 in each of the 16 series.

These harmonics will often sound perfectly in tune. The intonation is optimised when the pitches are well balanced in volume with the 4th and 6th harmonics mostly equal (or the 4th favoured slightly) and the 5th softer. Working with these harmonics will encourage the player to feel and hear the subtle differences between series before engaging with higher harmonics. Knowledge gained will assist in making fingering choices for some of the more complex situations dealt with in new repertoire.

Internally saying the harmonic series number whilst playing will develop a habit that will assist the player navigate between harmonics in more difficult circumstances.

Vary the dynamics. Start at a different level each session. Perhaps begin the first group pp, the next p, then mp, mf, f, ff then repeat. On the next practice session play through the same set beginning p and so on.

Method: Appendix D - p.10 h!"!#$%!&' ()*+,- ' ./0 ./0 1'2 ' 1'2 ∑ ∑ & ˙ ˙ ˙ ˙ !$%&'($%&)" !"#˙ ˙ "#$%&'()*+ ! , , - ! -

()*+,- ' ./0 ./0 1'2 ' 1'2 & ∑ #˙ Ó Ó #˙ !$%&'($%&)" ˙!"* #˙ #˙ ˙ "#$%&'()*+ ! , , - ! -

()*+,- ' ./0 ./0 1'2 ' 1'2 & ∑ ˙ ∑ ˙ !$%&'($%&)"b˙!"+ ˙ ˙ b˙ "#$%&'()*+ ! , , - ! -

()*+,- 3' .40 .40 132 3' 132 ∑ ∑ & ˙ ˙ !$%&'($%&)" !"+,*˙ #˙ #˙ ˙ "#$%&'()*+ ! , , - ! - ()*+,- ' ./0 ./0 1'2 ' 1'2 533/67 5.137 5.137 50/17 533/67 50/17 ∑ ∑ & ˙ ˙ !$%&'($%&)" !"-˙ #˙ #˙ ˙ "#$%&'()*+ ! , , - ! - ()*+,- 36 &'3 &'3 131 36 131 5'7 5./07 5./07 51'27 5'7 51'27 ∑ ∑ & b˙ b˙ !$%&'($%&)"b!"*,-˙ ˙ ˙ b˙ "#$%&'()*+ ! , , - ! - ()*+,- .' &30 &30 1.2 .' 1.2 5367 5&'37 5&'37 51317 5367 51317 & ∑ ∑ ˙ ˙ <µ>˙ <µ>˙ !$%&'($%&)"<µ>!"+,-˙ <µ>˙ "#$%&'()*+ ! , , - ! - ()*+,- 6. &.4 &.4 166 6. 166 5./7 5&2&7 5&2&7 51&'7 5./7 51&'7 & ∑ ∑ ˙ ˙ ˜˙ ˜˙ !$%&'($%&)" ˜!"+,*,-˙ ˜ ˜ ˜˙ "#$%&'()*+ ! , , - ! -

Method: Appendix D - p.11 !" ,"- ,"-3')41 /!! !" /!! &'()*+ #"$% #,.,% #,.,% #/,0% #"$% #/,0% ∑ ∑ & ˙ ˙ ˜˙ ˜˙ '()*+,()*-" µ!˙b"#$%$&˜ ˜ µ˙ "#$%&'()*+ ! , , - ! - &'()*+ "0 ,12 ,12 /". "0 /". #1!% #,01% #,01% #/1/% #1!% #/1/% ∑ ∑ & ˙ ˙ <µ>˙ <µ>˙ '()*+,()*-" <µ>!˙b"#$& <µ>˙ "#$%&'()*+ ! , , - ! - &'()*+ 1! ,01 ,01 /1/ 1! /1/ #0% #"$2% #"$2% #/0.% #0% #/0.% ∑ b˙ ∑ b˙ & b˙ ˙ ˙ b˙ '()*+,()*-" !b"%$& "#$%&'()*+ ! , , - ! - &'()*+ 0 "$2 "$2 /0. 0 /0. #11$!% #"/1% #"/1% #2$/% #11$!% #2$/% ∑ ˙ ∑ ˙ & ˙ #˙ #˙ ˙ '()*+,()*-" !b"& "#$%&'()*+ ! , , - ! -

&'()*+ 10 "-2 "-2 /1. 10 /1. ∑ ˙ ∑ ˙ & ˙ #˙ #˙ ˙ '()*+,()*-" !b"#% "#$%&'()*+ ! , , - ! -

&'()*+ 0 "$2 "$2 /0. 0 /0. ∑ b˙ ∑ b˙ & b˙ ˙ ˙ b˙ '()*+,()*-" !b"# "#$%&'()*+ ! , , - ! -

&'()*+ 0 "$2 "$2 /0. 0 /0. ∑ ˙ ∑ ˙ & ˙ #˙ #˙ ˙ '()*+,()*-" !b"% "#$%&'()*+ ! , , - ! -

&'()*+ 0 "$2 "$2 /0. 0 /0. ˙ ˙ & ˙ ˙ ∑ ˙ ∑ ˙ '()*+,()*-" !b". "#$%&'()*+ ! , , - ! -

Method: Appendix D - p.12 SET 2! Exploring the 7th harmonic

The ratios of just intonation are the ratios of the harmonics. In this Method, extended just intonation refers to music using prime numbered harmonics above the 5th. Prime numbered harmonics are specified because non-primes are either octaves of lower harmonics or directly derived from lower harmonics.

The principles below describe these relations.

1. The 2nd harmonic is but an octave above the 1st, as is the 4th above the

2nd, 8th above the 4th and so on. The 6th harmonic is an octave above

the 3rd, as is the 12th to the 6th; the 10th to the 5th, etc.

2. The other non-prime harmonics are derived from the intervals

between lower harmonics. The 9th harmonic is a factor of the 3rd. The

interval between the 1st and 3rd harmonics is a perfect twelfth as is

the interval between the 3rd and 9th harmonics. The same interval

occurs between the 9th and 27th harmonics.

3. The principle above at number 2 applies to other prime numbered

harmonics. The interval between the 1st and 5th harmonics is the

same as between the 5th and 25th harmonics. The interval between the

1st and 7th harmonics is the same as between the 7th and 49th

harmonics. This principle applies to all prime-numbered harmonics.

Method: Appendix D - p.13 4. Other odd numbered harmonics are derived from prime numbered

harmonics. The 15th harmonic is a factor of the 3rd and 5th harmonics.

The 21st harmonic is a factor of the 3rd and 7th harmonics and so on.

The staff notation of the harmonic series from numbers 1 to 32 may help clarify these points.

The 7th harmonic marks the beginning of extended just intonation in this

Method. Although a number of musicians have argued for the use of the 7th harmonic in standard repertoire this has not been widely practiced. Eighteenth century violinist Tartini used 7th harmonic intervals for a range of harmonic contexts (Beer 1965). Horn player Christopher Leuba suggested using this

Method: Appendix D - p.14 harmonic for dominant seventh chords in A study in intonation (1962/2004).

I occasionally have success with such an approach yet also find that the dominant seventh is well served when played as a discord to the octave of the root, a pure major second below octave. Many examples from the compositions of Beethoven work well this way. Richard Merewether, horn player and instrument designer, argued for the use of the narrow minor third interval in doleful cadences such as the end of the Second Movement of Beethoven’s Eroica

Symphony. In his pamphlet The Horn he referred to the ratio 7:6, the interval between the 6th and 7th harmonics (Merewether 1978).

In standard repertoire the 7th harmonic is quite easily recognised as being ‘in tune’ as long as the player can clearly hear the pitch or pitches to tune with. The louder it is played relative to other pitches the more likely it will sound sour in which case one may make a better intonation choice playing a sharper variation.

Many horn players will be familiar with the 7th harmonic in harmonic slurring practice. Well-known compositions include Rossini’s Le Rendez-vouz de Chasse in which the written B! is played untempered (n.d.). The Serenade for tenor, horn and strings (1943) by Britten requires the 7th harmonic as well as its octave, the 14th.

In his Six Bagatelles for Wind Quintet (1953) Ligeti asks for an ostinato figure to be played on the natural harmonics of the F horn that includes the 7th. Two

Method: Appendix D - p.15 works by Randall Faust use the natural 7th harmonic: Prelude (1977) and

Harmonielehre (1996). Additionally anyone who has played or heard an alphorn, cor de chasse or the south American instruments trutruca and clarin will notice the quality of the 7th harmonic. Thea Musgrave asked for 7th harmonics to be used in a different function: to approximate ¼ tones in her Concerto for Horn

(1971).

In the following exercises cent values show the variation between tuning slide placement for the tuning alternatives involving the 3rd valve slide. However, the notation does not always show the alternatives. For example the 7th harmonic of

F:2,3 can be notated as both a regular flat and a three quarter flat. In the exercise it is shown as G(!) rather than placing both a flat and a three quarter flat alongside each other.

Method: Appendix D - p.16 h!"!#$%!&'! ()*+,- ' ./0 1'2 303 ./0 1'2 303 ./0 303 & ˙ Bb˙ Ó Ó ˙ Bb˙ Ó Ó Bb˙ Ó Ó !$%&'($%&)" œ!"# œ œ ˙ "#$%&'()*+ ! , - . , - . , .

' 303 1'2 303 ' 303 ' 303 ? & Bb˙ Ó Ó Bb˙ Ó Ó ˙ &Bb˙ Ó Ó ˙ Bb˙ ˙ ! . ˙/ . 0 . 1 .

()*+,- ' ./0 1'2 303 ./0 1'2 303 ./0 303

& #˙ B˙ Ó Ó #˙ B˙ Ó Ó B˙ Ó Ó !$%&'($%&)" !"*œ #œ #œ #˙ "#$%&'()*+ ! , - . , - . , .

' 303 1'2 303 ' 303 ' 303 ? & B˙ Ó Ó B˙ Ó Ó ˙ & B˙ Ó Ó ˙ B˙ ˙ ! . #/˙ . 0 . 1 .

()*+,- ' ./0 1'2 303 ./0 1'2 303 ./0 303

& ˙ Bb˙ Ó Ó ˙ Bb˙ Ó Ó Bb˙ Ó Ó !$%&'($%&)"b!"+œ œ œ ˙ "#$%&'()*+ ! , - . , - . , .

' 303 1'2 303 ' 303 ' 303 ? & Bb˙ Ó Ó Bb˙ Ó Ó b˙ &Bb˙ Ó Ó b˙ Bb˙ b˙ ! . ˙/ . 0 . 1 .

()*+,- 4' .30 142 313 .30 142 313 .30 313

& ˙ ˙ Ó Ó ˙ ˙ Ó Ó ˙ Ó Ó !$%&'($%&)" !"+,*œ#œ #œ #˙ "#$%&'()*+ ! , - . , - . , .

4' 313 142 313 4' 313 4' 313 ? ? & ˙ Ó Ó ˙ & ˙ Ó Ó ˙ & ˙ Ó Ó ˙ ˙ ˙! . / . 0 . 1 .

Method: Appendix D - p.17 ! -$. /!0 1.1 -$. /!0 1.1 -$. 1.1 '()*+, "##$%& "-/#& ".$/& "1%2& "-/#& ".$/& "1%2& "-/#& "1%2& ∑ Ó Ó Ó Ó & ˙ B˙ ˙ B˙ B˙ !$%&'($%&)" !"#œ #œ #œ #˙ "#$%&'()*+ ! , - . , - . , . ! 1.1 /!0 1.1 ! 1.1 ! 1.1 "##$%& "1%2& ".$/& "1%2& "##$%& "1%2& "##$%& "1%2& ? ˙ ? & B˙ Ó Ó & B˙ Ó Ó ˙ & B˙ Ó Ó ˙ B˙ ˙ ! . / . 0 . 1 . #% 2!# /#/ 1$2 2!# /#/ 1$2 2!# 1$2 '()*+, "!& "-$.& "/!0& "1.1& "-$.& "/!0& "1.1& "-$.& "1.1& & b˙ ˙ Ó Ó b˙ ˙ Ó Ó ˙ Ó Ó !$%&'($%&)"b!"*+#œ œ œ ˙ "#$%&'()*+ ! , - . , - . , . #% 1$2 /#/ 1$2 #% 1$2 #% 1$2 "!& "1.1& "/!0& "1.1& "!& "1.1& "!& "1.1& ? b˙ ? & ˙ Ó Ó &˙ Ó Ó b˙ &˙ Ó Ó b˙ ˙ b˙! . / . 0 . 1 .

-! 2#. /-0 111 2#. /-0 111 2#. 111 '()*+, "#%& "2!#& "/#/& "1$2& "2!#& "/#/& "1$2& "2!#& "1$2& & Ó Ó Ó Ó Ó Ó œ <µ>˙ ˙ œ <µ>˙ ˙ ˙ ˙ !$%&'($%&)"<µ>!",+#œ "#$%&'()*+ ! , - . , - . , . -! 111 /-0 111 -! 111 -! 111 "#%& "1$2& "/#/& "1$2& "#%& "1$2& "#%& "1$2& ? ? Ó Ó <µ>˙ & Ó Ó & Ó Ó & ˙ ˙ <µ>˙ ˙ <µ>˙ ˙ <µ>˙! . / . 0 . 1 .

%- 2-1 /%% #!00 2-1 /%% #!00 2-1 #!00 '()*+, "-$& "202& "/2!& "#!!/& "202& "/2!& "#!!/& "202& "#!!/& & ˙ Ó Ó ˙ Ó Ó ˙ Ó Ó œ ˜˙ œ ˜˙ ˙ !$%&'($%&)"˜!",+*+#œ ˜ ˜ ˜ "#$%&'()*+ ! , - . , - . , . %- #!00 /%% #!00 %- #!00 %- #!00 "-$& "#!!/& "/2!& "#!!/& "-$& "#!!/& "-$& "#!!/& ? ? & ˙ Ó Ó ˜˙ & ˙ Ó Ó & ˙ Ó Ó ˙ ˙ / . ˜˙0 . ˜1 . ˜˙! .

Method: Appendix D - p.18 !" ,"- /!! 10.. ,"- /!! 10.. ,"- 10.. &'()*+ #"$% #,.,% #/,0% #100/% #,.,% #/,0% #100/% #,.,% #100/% Ó Ó Ó Ó Ó Ó & œ ˜˙ ˙ œ ˜˙ ˙ ˙ ˙ '()*+,()*-"!µbœ"#$%$&<˜> <˜> <˜> "#$%&'()*+ ! , - . , - . , . !" 10.. /!! 10.. !" 10.. !" 10.. #"$% #100/% #/,0% #100/% #"$% #100/% #"$% #100/% ? ˜˙ ? & ˙ Ó Ó & ˙ Ó Ó µ˙ & ˙ Ó Ó µ˙ ˙ / . 0 . 1 . µ˙! .

"0 ,12 /". --- ,12 /". --- ,12 --- &'()*+ #1!% #,01% #/1/% #-$,% #,01% #/1/% #-$,% #,01% #-$,% b˙ Ó Ó b˙ Ó Ó b˙ Ó Ó & œ <µ>˙ œ <µ>˙ ˙ '()*+,()*-"<µ>!œb"#$& "#$%&'()*+ ! , - . , - . , . "0 --- /". --- "0 --- "0 --- #1!% #-$,% #/1/% #-$,% #1!% #-$,% #1!% #-$,% ? & b˙ Ó Ó b˙ Ó Ó <µ>˙ &b˙ Ó Ó <µ>˙ b˙ <µ>˙ ! . <µ>/˙ . 0 . 1 .

1! ,01 /1/ -$, ,01 /1/ -$, ,01 -$, &'()*+ #0% #"$2% #/0.% #-2-% #"$2% #/0.% #-2-% #"$2% #-2-% ˙ ˙ ˙ & œ b˙ Ó Ó œ b˙ Ó Ó ˙ Ó Ó '()*+,()*-" b!œb"%$& "#$%&'()*+ ! , - . , - . , .

1! -$, /1/ -$, 1! -$, 1! -$, #0% #-2-% #/0.% #-2-% #0% #-2-% #0% #-2-% ? ˙ Ó Ó ˙ Ó Ó b˙ &˙ Ó Ó b˙ ˙ & b˙ ! . b˙/ . 0 . 1 .

0 "$2 /0. -2- "$2 /0. -2- "$2 -2- &'()*+ #11$!% #"/1% #2$/% #-!,% #"/1% #2$/% #-!,% #"/1% #-!,% ˙ B˙ Ó Ó ˙ B˙ Ó Ó B˙ Ó Ó & œ #œ #œ #˙ '()*+,()*-" !b"& "#$%&'()*+ ! , - . , - . , . 0 -2- /0. -2- 0 -2- 0 -2- #11$!% #-!,% #2$/% #-!,% #11$!% #-!,% #11$!% #-!,% ? B˙ Ó Ó B˙ Ó Ó ˙ &B˙ Ó Ó ˙ B˙ & ˙ ! . /˙ . 0 . 1 .

Method: Appendix D - p.19 #$%&'( !" )*+ ,!- *,* )*+ ,!- *,* )*+ *,* ˙ ˙ Ó Ó ˙ ˙ Ó Ó ˙ Ó Ó & œ #œ #œ #˙ &'()*+'()," !b"#$% "#$%&'()*+ ! , - . , - . , .

!" *,* ,!- *,* !" *,* !" *,* ˙ Ó Ó ˙ Ó Ó ? ˙ ˙ Ó Ó ˙ ˙ & ˙ & ! . /˙ . 0 . 1 .

#$%&'( " ).+ ,"- *+* ).+ ,"- *+* ).+ *+* b˙ Bb˙ Ó Ó b˙ Bb˙ Ó Ó Bb˙ Ó Ó & bœ œ œ ˙ &'()*+'()," !b"# "#$%&'()*+ ! , - . , - . , .

" *+* ,"- *+* " *+* " *+* Bb˙ Ó Ó Bb˙ Ó Ó ? b˙ Bb˙ Ó Ó b˙ Bb˙ & b˙ & ! . b˙/ . 0 . 1 .

#$%&'( " ).+ ,"- *+* ).+ ,"- *+* ).+ *+* ˙ B˙ Ó Ó ˙ B˙ Ó Ó B˙ Ó Ó & œ #œ #œ #˙ &'()*+'()," !b"% "#$%&'()*+ ! , - . , - . , .

" *+* ,"- *+* " *+* " *+* B˙ Ó Ó B˙ Ó Ó ? ˙ B˙ Ó Ó ˙ B˙ & ˙ & ! . /˙ . 0 . 1 .

#$%&'( " ).+ ,"- *+* ).+ ,"- *+* ).+ *+* Bb˙ Bb˙ Bb˙ & œ œ ˙ Ó Ó œ ˙ Ó Ó ˙ Ó Ó &'()*+'()," !b"- "#$%&'()*+ ! , - . , - . , .

" *+* ,"- *+* " *+* " *+* Bb˙ Bb˙ ? ˙ Bb˙ ˙ Bb˙ & ˙ Ó Ó Ó Ó & Ó Ó ! . ˙/ . 0 . 1 .

Method: Appendix D - p.20 SET 3! Visiting the 9th harmonic

Set 3 uses the 9th harmonic and the intervals leading up to it plus the step above. In a gradual exploration up the harmonic series the steps and intervals to the 8th harmonic provide the next logical group, however the interval between the 8th and 7th harmonics appears at the end of each group in Set 2 therefore does not require a set of exercises specifically for its study.

Players with any valveless horn of any tube length use the 9th harmonic for the second degree of the C scale. Sometimes a player may adjust it flatter than it naturally occurs for melodic contexts as happens when using Meantone temperament, which has a smooth stepwise movement from scale degrees

1 to 2 to 3.

In some just intonation contexts the 9th harmonic needs no adjustment. Even though many of the intervals presented here in Set 3 need little elucidation, practice will consolidate awareness of harmonic series use and provide another point of stabilisation among unfamiliar intervals.

Method: Appendix D - p.21 "#$%&' ! ()* +!, -*- ! ,!. ()* +!, -*- ! ,!. +!, -*- ! ,!. ˙ ˙ ˙ & œBbœ ˙ ∑ œBbœ ˙ ∑ œBbœ ˙ ∑ !$%&'($%&)" !"#œ œ œ "#$%&'()*+ ! , - . / 0 , - . / 0 - . / 0 -*- ! ,!. -*- ,!. +!, ,!. ()* ,!. ! ,!. ()* ,!. ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ & Bbœ ∑ Bb˙ ∑ ˙ ∑ ˙ ∑ ∑ .J / 0 . 0 - 0 , 0 !˙ 0 12 0

"#$%&' ! ()* +!, -*- ! ,!. ()* +!, -*- ! ,!. +!, -*- ! ,!. & #œ Bœ ˙ #˙ ∑ #œ Bœ ˙ #˙ ∑ #œ Bœ ˙ #˙ ∑ !$%&'($%&)"!"*œ#œ #œ "#$%&'()*+ ! , - . / 0 , - . / 0 - . / 0

-*- ! ,!. -*- ,!. +!, ,!. ()* ,!. ! ,!. ()* ,!. j #˙ & Bœ ˙ #˙ ∑ B˙ #˙ ∑ #˙ #˙ ∑ #˙ ∑ #˙ ∑ #˙ . / 0 . 0 - 0 #,˙ 0 !˙ 0 12 0

"#$%&' ! ()* +!, -*- ! ,!. ()* +!, -*- ! ,!. +!, -*- ! ,!. n˙ ˙ ˙ & œBbœb˙ ∑ œBbœb˙ ∑ œBbœb˙ ∑ !$%&'($%&)"b!"+œnœ œ "#$%&'()*+ ! , - . / 0 , - . / 0 - . / 0

-*- ! ,!. -*- ,!. +!, ,!. ()* ,!. ! ,!. ()* ,!. jb˙ ˙ ∑ ˙ ∑ ˙ ∑ ˙ ∑ ˙ ∑ ˙ ˙ & Bbœ Bb˙ ˙ ˙ . / 0 . 0 - 0 , 0 b!˙ 0 12 0

"#$%&' /! (-* +/, -+- /! ,/. (-* +/, -+- /! ,/. +/, -+- /! ,/.

& œ œ ˙ ˙ ∑ œ œ ˙ ˙ ∑ œ œ ˙ ˙ ∑ !$%&'($%&)" !"+,*œ#œ #œ "#$%&'()*+ ! , - . / 0 , - . / 0 - . / 0 -+- /! ,/. -+- ,/. +/, ,/. (-* ,/. /! ,/. (-* ,/. j ˙ ∑ ˙ ∑ ˙ ∑ ˙ ∑ ˙ ∑ #˙ ˙ & œ ˙ ˙ ˙ . / 0 . 0 - 0 #,˙ 0 !˙ 0 12 0

Method: Appendix D - p.22 ! -$. /!0 1.1 ! 0!2 -$. /!0 1.1 ! 0!2 '()*+, "##$%& "-/#& ".$/& "1%2& "##$%& "#$1& "-/#& ".$/& "1%2& "##$%& "#$1& ˙ ∑ ˙ ∑ & œ œ ˙ œ œ ˙ !$%&'($%&)" !"#œ #œ #œ "#$%&'()*+ ! , - . / 0 , - . / 0 /!0 1.1 ! 0!2 1.1 ! 0!2 1.1 0!2 ".$/& "1%2& "##$%& "#$1& "1%2& "##$%& "#$1& "1%2& "#$1& j & œ ˙ ˙ ∑ œ ˙ ˙ ∑ ˙ ˙ ∑ -œ . / 0 . / 0 . 0

/!0 0!2 -$. 0!2 ! 0!2 -$. 0!2 ".$/& "#$1& "-/#& "#$1& "##$%& "#$1& "-/#& "#$1& ˙ ∑ ˙ ∑ ˙ ∑ #˙ ˙ & ˙ - 0 #,˙ 0 !˙ 0 12 0

#% 2!# /#/ 1$2 #% 0#1 2!# /#/ 1$2 #% 0#1 '()*+, "!& "-$.& "/!0& "1.1& "!& "0!2& "-$.& "/!0& "1.1& "!& "0!2&

& bœœ b˙ b˙ ∑ bœœ b˙ b˙ ∑ !$%&'($%&)"b!"*+#œ nœ œ "#$%&'()*+ ! , - . / 0 , - . / 0 /#/ 1$2 #% 0#1 1$2 #% 0#1 1$2 0#1 "/!0& "1.1& "!& "0!2& "1.1& "!& "0!2& "1.1& "0!2& j & bœ b˙ b˙ ∑ bœ b˙ b˙ ∑ b˙ b˙ ∑ b-œ. / 0 . / 0 . 0 /#/ 0#1 2!# 0#1 #% 0#1 2!# 0#1 "/!0& "0!2& "-$.& "0!2& "!& "0!2& "-$.& "0!2& & b˙ ∑ b˙ ∑ b˙ ∑ ˙ b˙ b˙ ˙ - 0 , 0 b!˙ 0 12 0 -! 2#. /-0 111 -! 0-% 2#. /-0 111 -! 0-% '()*+, "#%& "2!#& "/#/& "1$2& "#%& "0#1& "2!#& "/#/& "1$2& "#%& "0#1& & œ ˙ <µ>˙ ∑ œ ˙ <µ>˙ ∑ nœ <µ>œ <µ> œ <µ>œ <µ> !$%&'($%&)"<µ>!",+#œ "#$%&'()*+ ! , - . / 0 , - . / 0

/-0 111 -! 0-% 111 -! 0-% 111 0-% "/#/& "1$2& "#%& "0#1& "1$2& "#%& "0#1& "1$2& "0#1& j & œ ˙ <µ>˙ ∑ œ ˙ <µ>˙ ∑ ˙ <µ>˙ ∑ <µ>-œ . <µ>/ 0 . <µ>/ 0 . 0 /-0 0-% 2#. 0-% -! 0-% 2#. 0-% "/#/& "0#1& "2!#& "0#1& "#%& "0#1& "2!#& "0#1& & <µ>˙ ∑ <µ>˙ ∑ <µ>˙ ∑ ˙ <µ>˙ <µ>˙ ˙ - 0 , 0 <µ>!˙ 0 12 0

Method: Appendix D - p.23 !" ,"- /!! 10.. !" .!/ ,"- /!! 10.. !" .!/ &'()*+ #"$% #,.,% #/,0% #100/% #"$% #.,.% #,.,% #/,0% #100/% #"$% #.,.% ∑ j ∑ & œ ˙ ˜˙ œ ˙ ˜˙ œ ˜œ ˜ œ ˜œ ˜ !'()*+'(),"˜!"#$%$&œ<˜> <˜> "#$%&'()*+ ! , - . / 0 , - . / 0 /!! 10.. !" .!/ 10.. !" .!/ 10.. .!/ #/,0% #100/% #"$% #.,.% #100/% #"$% #.,.% #100/% #.,.% ∑ j ∑ ∑ & œ ˜˙ ˜˙ œ ˜˙ ˜˙ ˙ ˜˙ ˜-œ . / 0 . / 0 . 0 /!! .!/ ,"- .!/ !" .!/ ,"- .!/ #/,0% #.,.% #,.,% #.,.% #"$% #.,.% #,.,% #.,.%

& ˜˙ ∑ ˜˙ ∑ ˜˙ ∑ <˜>˙ ˜˙ ˜˙ ˙ - 0 <˜>, 0 ˜!˙ 0 12 0 !" ,"- /!! 10.. !" .!/ ,"- /!! 10.. !" .!/ &'()*+ #"$% #,.,% #/,0% #100/% #"$% #.,.% #,.,% #/,0% #100/% #"$% #.,.% ˙ ˜˙ ∑ ˙ ˜˙ ∑ & œ ˜œ nœ µ œ ˜œ œ µ !'()*+'(),"µ-œb<"#$%$&˜> <˜> "#$%&'()*+ ! , - . / 0 , - . / 0 /!! 10.. !" .!/ 10.. !" .!/ 10.. .!/ #/,0% #100/% #"$% #.,.% #100/% #"$% #.,.% #100/% #.,.% j & œ µ˙ ˜˙ ∑ œ µ˙ ˜˙ ∑ ˙ ˜˙ ∑ ˜-œ . / 0 . / 0 . 0

/!! .!/ ,"- .!/ !" .!/ ,"- .!/ #/,0% #.,.% #,.,% #.,.% #"$% #.,.% #,.,% #.,.% ˜˙ ∑ ˜˙ ∑ ˜˙ ∑ <˜>˙ ˜˙ & ˜˙ ˙ - 0 <˜>, 0 µ!˙ 0 12 0 "0 ,12 /". --- "0 ."! ,12 /". --- "0 ."! &'()*+ #1!% #,01% #/1/% #-$,% #1!% #.1-% #,01% #/1/% #-$,% #1!% #.1-% ˙ <µ>˙ ˙ <µ>˙ & œ <µ>œ bœ<µ> ∑ œ <µ>œ bœ<µ> ∑ !'()*+'(),"<µ>-œb"#$& "#$%&'()*+ ! , - . / 0 , - . / 0 /". --- "0 ."! --- "0 ."! --- ."! #/1/% #-$,% #1!% #.1-% #-$,% #1!% #.1-% #-$,% #.1-% & œ bœ<µ>˙ <µ>˙ ∑ bœ<µ>˙ <µ>˙ ∑ b˙ <µ>˙ ∑ <µ>- . / 0 .J / 0 . 0

/". ."! ,12 ."! "0 ."! ,12 ."! #/1/% #.1-% #,01% #.1-% #1!% #.1-% #,01% #.1-% <µ>˙ ∑ <µ>˙ ∑ <µ>˙ ∑ ˙ <µ>˙ & <µ>˙ ˙ - 0 , 0 <µ>!˙ 0 12 0

Method: Appendix D - p.24 !" ,$! 0!0 2., !" 1!2 ,$! 0!0 2., !" 1!2 &'()*+ #$% #-./% #0$1% #2/2% #$% #1$,% #-./% #0$1% #2/2% #$% #1$,% b˙ b˙ bœœ b˙ ∑ bœœ b˙ ∑ & bœ œ œ &'()*+'()," !b"#$% "#$%&'()*+ ! , - . / 0 , - . / 0 0!0 2., !" 1!2 2., !" 1!2 2., 1!2 #0$1% #2/2% #$% #1$,% #2/2% #$% #1$,% #2/2% #1$,% b˙ b˙ b˙ & bœœ b˙ ∑ œ b˙ ∑ ˙ ∑ - . / 0 .J / 0 . 0

0!0 1!2 ,$! 1!2 !" 1!2 ,$! 1!2 #0$1% #1$,% #-./% #1$,% #$% #1$,% #-./% #1$,% b˙ b˙ b˙ ˙ b˙ & b˙ ∑ ∑ ∑ - 0 ,˙ 0 b!˙ 0 12 0

$ -./ 0$1 2/2 $ 1$, -./ 0$1 2/2 $ 1$, &'()*+ #!!."% #-0!% #/.0% #2",% #!!."% #!.2% #-0!% #/.0% #2",% #!!."% #!.2% œ ˙ n˙ ∑ œ ˙ ˙ ∑ & œ #œ œ #œ œ &'()*+'(),"!b"% "#$%&'()*+ ! , - . / 0 , - . / 0 2/2 $ 1$, 2/2 1$, 0$1 2/2 $ 1$, #2",% #!!."% #!.2% #2",% #!.2% #/.0% #2",% #!!."% #!.2% œ ˙ ˙ ∑ œ ˙ ˙ ∑ ˙ ˙ ∑ & œ J - . / 0 . / 0 . 0

0$1 1$, -./ 1$, $ 1$, -./ 1$, #/.0% #!.2% #-0!% #!.2% #!!."% #!.2% #-0!% #!.2% ˙ ˙ ˙ #˙ ˙ & ˙ ∑ ∑ ∑ - 0 #,˙ 0 !˙ 0 12 0

&'()*+ !$ -2/ 0!1 202 !$ 1!, -2/ 0!1 202 !$ 1!, œ ˙ ˙ œ ˙ ˙ & #œ œ ∑ #œ œ ∑ &'()*+'(),"!œb"-$# "#$%&'()*+ ! , - . / 0 , - . / 0

0!1 202 !$ 1!, 202 !$ 1!, 202 1!, ˙ ˙ ˙ ˙ ˙ & œ œ ∑ œ ∑ ˙ ∑ - . / 0 .J / 0 . 0

0!1 1!, -2/ 1!, !$ 1!, -2/ 1!, ˙ ˙ ˙ #˙ ˙ & ˙ ∑ ∑ ∑ - 0 #,˙ 0 !˙ 0 12 0

Method: Appendix D - p.25 "#$%&' ! ()* +!, -*- ! ,!. ()* +!, -*- ! ,!. +!, -*- ! ,!. bœ Bbœ b˙ n˙ ∑ bœBbœb˙ ˙ ∑ bœBbœb˙ ˙ ∑ & bœ œ œ $%&'()%&'*"!b"# - . / 0 "#$%&'()*+ ! , - . / 0 , - . / 0

-*- ! ,!. -*- ,!. +!, ,!. ()* ,!. ! ,!. ()* ,!. bœb˙ ˙ b˙ ˙ ˙ ˙ ˙ ˙ ˙ & B ∑ B ∑ b˙ ∑ ˙ ∑ ∑ .J / 0 . 0 - 0 , 0 b!˙ 0 12 0

"#$%&' ! ()* +!, -*- ! ,!. ()* +!, -*- ! ,!. +!, -*- ! ,!. œ Bœ ˙ #˙ ∑ œ Bœ ˙ #˙ ∑ œ Bœ ˙ #˙ ∑ & œ #œ #œ $%&'()%&'*"!b"+ - . / 0 "#$%&'()*+ ! , - . / 0 , - . / 0

-*- ! ,!. -*- ,!. +!, ,!. ()* ,!. ! ,!. ()* ,!. ˙ #˙ #˙ #˙ #˙ #˙ #˙ #˙ & Bœ ∑ B˙ ∑ ˙ ∑ #˙ ∑ ∑ .J / 0 . 0 - 0 , 0 !˙ 0 12 0

"#$%&' ! ()* +!, -*- ! ,!. ()* +!, -*- ! ,!. +!, -*- ! ,!. Bbœn˙ n˙ Bbœ ˙ ˙ Bbœ ˙ ˙ & nœ œ œ ∑ œ œ ∑ œ ∑ $%&'()%&'*"!b", - . / 0 "#$%&'()*+ ! , - . / 0 , - . / 0

-*- ! ,!. -*- ,!. +!, ,!. ()* ,!. ! ,!. ()* ,!. bœ ˙ ˙ b˙ ˙ ˙ ˙ ˙ ˙ ˙ & B J ∑ B ∑ ˙ ∑ ˙ ∑ ∑ . / 0 . 0 - 0 , 0 !˙ 0 12 0

Method: Appendix D - p.26 SET 4! To the 10th harmonic

Most horn players are quite familiar with the 10th harmonic. Even if they have not used baroque or classical horns many will use the 10th harmonics for high written A and G! - played as B":0 and B":2 respectively. However, the interval between the 7th and 10th harmonics will feel unfamiliar. As the intervals surrounding the 7th harmonic begin to sound and feel normal they will still need much practice to allow the body to have a sense of conviction in playing them. More straightforward intervals such as the two sizes of whole tones from harmonics 9 to 10 and 8 to 9 help to develop sensitivity to subtle differences.

This set uses the 6th harmonic as the starting point.

Method: Appendix D - p.27 h!"!#$%!&' *+,-./ (') 010 ' )'& 231 010 ' )'& 231 ˙ ˙ ˙ ˙ & œ Bbœ œ ∑ Bbœ œ ∑ !$%&'($%&)" !"# "#$%&'()*+ ! , - . /0 , - . /0

' )'& 231 ' 231 010 231 œ ˙ ˙ ∑ ˙ ˙ ∑ b˙ ˙ & J B - . /0 - /0 , /0

*+,-./ (') 010 ' )'& 231 010 ' )'& 231 #˙ #˙ & #œ Bœ œ#˙ ∑ Bœ œ#˙ ∑ !$%&'($%&)" !"* "#$%&'()*+ ! , - . /0 , - . /0

' )'& 231 ' 231 010 231 #˙ #˙ #˙ & œ#˙ ∑ ˙ ∑ B˙ -J . /0 - /0 , /0

*+,-./ (') 010 ' )'& 231 010 ' )'& 231 ˙ ˙ & œ Bbœ bœ ˙ ∑ Bbœ bœ ˙ ∑ !$%&'($%&)" !"+ "#$%&'()*+ ! , - . /0 , - . /0

' )'& 231 ' 231 010 231 ˙ ˙ ˙ & bœ ˙ ∑ b˙ ∑ Bb˙ -J . /0 - /0 , /0

*+,-./ (4) 0(0 4' )4& 201 0(0 4' )4& 201 #˙ #˙ & Bœ œ ˙ ∑ Bœ œ ˙ ∑ !$%&'($%&)" !"+,*œ "#$%&'()*+ ! , - . /0 , - . /0

4' )4& 201 4' 201 0(0 201 j #˙ #˙ #˙ & œ ˙ ∑ ˙ ∑ B˙ - . /0 - /0 , /0

Method: Appendix - p.28 !"# .%. " #"0 2&% .%. " #"0 2&% ()*+,- $%&!' $./0' $11&/' $1&.' $2!1' $./0' $11&/'$1&.'$2!1' #˙ #˙ & œ œ ˙ ∑ œ œ ˙ ∑ !$%&'($%&)" !"#œ "#$%&'()*+ ! , - . /0 , - . /0

" #"0 2&% " 2&% .%. 2&% $11&/' $1&.' $2!1' $11&/' $2!1' $./0' $2!1' j #˙ #˙ #˙ & œ ˙ ∑ ˙ ∑ ˙ - . /0 - /0 , /0

!1! .&0 1/ #1. 0"1 .&0 1/ #1. 0"1 ()*+,- $!"#' $.%.' $"' $#"0' $2&%' $.%.' $"' $#"0' $2&%' b˙ ˙ ∑ b˙ ˙ ∑ & bœ œ bœ œ bœ !$%&'($%&)" !"*+# "#$%&'()*+ ! , - . /0 , - . /0 1/ #1. 0"1 1/ 0"1 .&0 0"1 $"' $#"0' $2&%' $"' $2&%' $.%.' $2&%' j ˙ ˙ ˙ & bœ b˙ ∑ b˙ ∑ ˙ - . /0 - /0 , /0

!2# ... 2" #2/ 01% ... 2" #2/ 01% ()*+,- $!1!' $.&0' $1/' $#1.' $0"1' $.&0' $1/' $#1.' $0"1'

& œ <µ>œ <µ>˙ ˙ ∑ œ <µ>œ <µ>˙ ˙ ∑ !$%&'($%&)" <µ>!",+#œ "#$%&'()*+ ! , - . /0 , - . /0 2" #2/ 01% #2/ 01% 2" 01% $1/' $#1.' $0"1' $#1.' $0"1' $1/' $0"1' j & <µ>œ <µ>˙ ˙ ∑ <µ>˙ ˙ ∑ <µ>˙ ˙ - . /0 . /0 - /0

!// 1"## /2 #/! 02. 1"## /2 #/! 02. ()*+,- $!0"' $1""!' $2&' $#0#' $0#0' $1""!' $2&' $#0#' $0#0' ∑ ∑ & œ ˜œ ˜˙ <˜>˙ œ ˜œ ˜˙ <˜>˙ !$%&'($%&)"˜!",+*+#œ "#$%&'()*+ ! , - . /0 , - . /0 /2 #/! 02. #/! 02. /2 02. $2&' $#0#' $0#0' $#0#' $0#0' $2&' $0#0' j ∑ ∑ & ˜œ ˜˙ <˜>˙ ˜˙ <˜>˙ ˜˙ <˜>˙ - . /0 . /0 - /0

Method: Appendix - p.29 !"" -%.. "/ ."! $/1 -%.. "/ ."! $/1 '()*+, #!$%& #-%%!& #/0& #.$.& #$.$& #-%%!& #/0& #.$.& #$.$& ˙ ˙ & œ œ µœ ˜˙ <˜> ∑ œ µœ ˜˙ <˜> ∑ '()*+,()*-" !˜b"#$%$& "#$%&'()*+ ! , - . /0 , - . /0

"/ ."! $/1 ."! $/1 "/ $/1 #/0& #.$.& #$.$& #.$.& #$.$& #/0& #$.$& ˙ ˙ ˙ & µœ ˜˙ <˜> ∑ ˜˙ <˜> ∑ µ˙ <˜> -J . /0 . /0 - /0

!/. 111 /% ./" $-2 111 /% ./" $-2 '()*+, #!-!& #10$& #-"& #.-1& #$%-& #10$& #-"& #.-1& #$%-& œ <µ>˙ ˙ œ <µ>˙ ˙ & <µ>œ bœ <µ> ∑ bœ <µ> ∑ '()*+,()*-" !b"#$& "#$%&'()*+ ! , - . /0 , - . /0

/% ./" $-2 ./" $-2 /% $-2 #-"& #.-1& #$%-& #.-1& #$%-& #-"& #$%-& œ <µ>˙ ˙ <µ>˙ ˙ ˙ ˙ & <µ>J ∑ ∑ <µ> - . /0 . /0 - /0

!-! 10$ -" .-1 $%- 10$ -" .-1 $%- '()*+, #!%.& #121& #%& #.%$& #/02& #121& #%& #.%$& #/02& bœ b˙ ˙ bœ b˙ ˙ & bœ œ ∑ œ ∑ '()*+,()*-" !b"%$& "#$%&'()*+ ! , - . /0 , - . /0

-" .-1 $%- -" $%- 10$ $%- #%& #.%$& #/02& #%& #/02& #121& #/02& bœ b˙ ˙ b˙ ˙ b ˙ ˙ & J ∑ ∑ - . /0 - /0 , /0

!%. 121 % .%$ /02 121 % .%$ /02 '()*+, #20!& #1"$& #--0"& #-01& #/!-& #1"$& #--0"& #-01& #/!-& œ ˙ #˙ œ ˙ #˙ & œ œ ∑ œ ∑ '()*+,()*-" !b"& "#$%&'()*+ ! , - . /0 , - . /0 % .%$ /02 % /02 121 /02 #--0"& #-01& #/!-& #--0"& #/!-& #1"$& #/!-& œ ˙ #˙ ˙ #˙ #˙ & J ∑ ∑ ˙ - . /0 - /0 , /0

Method: Appendix - p.30 $%&'() !"# *!* "+ #", -*. *!* "+ #", -*. œ ˙ #˙ œ ˙ #˙ & œ Bœ ∑ Bœ ∑ &'()*+'()," !b"#$% "#$%&'()*+ ! , - . /0 , - . /0

"+ #", -*. "+ -*. *!* -*. œ ˙ #˙ ˙ #˙ #˙ & J ∑ ∑ B˙ - . /0 - /0 , /0

$%&'() !+# *.* + #+, -/. *.* + #+, -/. bœ bœ ˙ ˙ bœ bœ ˙ ˙ & bœB ∑ B ∑ &'()*+'()," !b"# "#$%&'()*+ ! , - . /0 , - . /0

+ #+, -/. + -/. *.* -/. bœ ˙ ˙ b˙ ˙ b˙ ˙ & J ∑ ∑ B - . /0 - /0 , /0

$%&'() !+# *.* + #+, -/. *.* + #+, -/. œ œ #˙ #˙ œ œ #˙ #˙ & œ B ∑ B ∑ &'()*+'()," !b"% "#$%&'()*+ ! , - . /0 , - . /0

+ #+, -/. + -/. *.* -/. œ#˙ #˙ ˙ #˙ ˙ #˙ & J ∑ ∑ B - . /0 - /0 , /0

$%&'() !+# *.* + #+, -/. *.* + #+, -/. bœ œ ˙ ˙ bœ œ ˙ ˙ & œB ∑ B ∑ &'()*+'()," !b"- "#$%&'()*+ ! , - . /0 , - . /0

+ #+, -/. + -/. *.* -/. œ ˙ ˙ ˙ ˙ b˙ ˙ & J ∑ ∑ B - . /0 - /0 , /0

Method: Appendix - p.31 SET 5! Acquainting ourselves with the 11th harmonic

Most horn players have a little familiarity with the 11th harmonic from listening to Benjamin Britten’s Serenade for tenor, horn and strings (1943); music for cor de chasse; music for alphorns and perhaps György Ligeti’s Trio for violin, horn and piano (1982) and Hamburg Concerto (2002). Many will have played some of this music adding practical knowledge. Some horn players will have played the instruments cor de chasse and alphorn. Few realise that the 11th harmonic can be used to play some early blues such as I wouldn’t mind dying but I gotta go by myself sung by Rev I. B. Ware with wife and son. This tune is printed in Chapter

One (p 37). Douglas Hill, in his book Extended Techniques for Horn (1983) suggests using 11th harmonics (along with 13ths and 7ths) to present quartertone passages. 11th harmonics are virtually indistinguishable from quartertones though both 13th and 7th harmonics would require some adjustments to be precise.

This set uses the 8th harmonic as the starting point.

Method: Appendix D - p.32 h!"!#$%!&' ()*+,- ' .'& /01 223 .'& /01 223 .'& 223 œ ˙ µ˙ œ ˙ µ˙ ˙ µ˙ & œ ∑ J ∑ ∑ !$%&'($%&)" !"# "#$%&'()*+ ! , -. -- , -. -- , --

' 223 414 223 5'. 223 µ˙ µ˙ µ˙ & ˙ ∑ Bb˙ ∑ ˙ ∑ ! -- / -- 0 --

/01 223 ' 223 5'. 223 ' 223 5'. 223 µ˙ µ˙ ? ˙ µ˙ ? µ˙ ˙ µ˙ & ˙ ∑ ∑ & ∑ ˙ & ∑ 1 -- 2˙ -- 3 -- 4 -- -4 --

()*+,- ' .'& /01 223 .'& /01 223 .'& 223 #˙ ˙ #˙ ˙ ˙ & œ#œ µ ∑ #œ µ ∑ #˙ µ ∑ !$%&'($%&)" !"* J "#$%&'()*+ ! , -. -- , -. -- , --

' 223 414 223 5'. 223 /01 223 µ˙ µ˙ µ˙ µ˙ & ˙ ∑ B˙ ∑ #˙ ∑ ∑ ! -- / -- 0 -- #1˙ --

' 223 5'. 223 ' 223 5'. 223 ˙ ? #˙ ˙ ? ˙ #˙ ˙ & µ ∑ &µ ∑ ˙ &µ ∑ µ 2˙ -- 3 -- 4 -- -4 --

()*+,- ' .'& /01 223 .'& /01 223 .'& 223 ˙ ˙ ˙ ˙ ˙ & bœ œ B ∑ œ B ∑ ˙ B ∑ !$%&'($%&)" !"+ J "#$%&'()*+ ! , -. -- , -. -- , -- ' 223 414 223 5'. 223 B˙ B˙ B˙ & b˙ ∑ Bb˙ ∑ ˙ ∑ ! -- / -- 0 --

/01 223 ' 223 5'. 223 ' 223 5'. 223 ˙ ˙ ? ˙ ˙ ? ˙ ˙ ˙ & B ∑ B ∑ &B ∑ b˙ & B ∑ B ˙1 -- b2˙ -- 3 -- 4 -- -4 --

Method: Appendix D - p.33 #$%&'( !" )!* +,- .-! )!* +,- .-! )!* .-! ˙ ˙ ˙ & œ œ #˙ µ ∑ œ#˙ µ ∑ ˙ µ ∑ !&'()*&'(+" !"#$% J "#$%&'()*+ ! , -. -- , -. -- , --

!" .-! ,/, .-! /!) .-! ˙ ˙ ˙ & ˙ µ ∑ ˙ µ ∑ µ ∑ ! -- B/ -- 0˙ --

+,- .-! !" .-! /!) .-! !" .-! /!) .-! ˙ ˙ ? ˙ ? ˙ ˙ ˙ & µ ∑ µ ∑ ˙ & µ ∑ ˙ & µ ∑ µ #˙1 -- 2˙ -- 3 -- 4 -- -4 --

" )"* +1- ..! )"* +1- ..! )"* ..! #$%&'( 0!!1.2 0!1,2 0+/!2 0.+-2 0!1,2 0+/!2 0.+-2 0!1,2 0.+-2 & œ œ #˙ µ˙ ∑ œ #˙ µ˙ ∑ ˙ µ˙ ∑ !&'()*&'(+" !", J "#$%&'()*+ ! , -. -- , -. -- , --

" ..! ,-, ..! /") ..! 0!!1.2 0.+-2 0,.*2 0.+-2 0-1/2 0.+-2 ˙ ˙ ˙ & ˙ µ ∑ ˙ µ ∑ µ ∑ ! -- B/ -- 0˙ --

+1- ..! " ..! /") ..! " ..! /") ..! 0+/!2 0.+-2 0!!1.2 0.+-2 0-1/2 0.+-2 0!!1.2 0.+-2 0-1/2 0.+-2 ˙ ˙ ? ˙ ˙ ? ˙ ˙ ˙ & µ ∑ µ ∑ & µ ∑ ˙ & µ ∑ µ #1˙ -- 2˙ -- 3 -- 4 -- -4 --

!. )!, *"! .-- )!, *"! .-- )!, .-- #$%&'( 0"2 0)"*2 0+1-2 0..!2 0)"*2 0+1-2 0..!2 0)"*2 0..!2 ˙ ˙ ˙ & bœ bœ ˙ B ∑ bœ ˙ B ∑ b˙ B ∑ !&'()*&'(" !"%$, J "#$%&'()*+ ! , -. -- , -. -- , --

!. .-- ,1* .-- /!/ .-- 0"2 0..!2 0,-,2 0..!2 0/")2 0..!2 ˙ ˙ ˙ & b˙ B ∑ b ˙ B ∑ B ∑ ! -- / -- b0˙ --

Method: Appendix D - p.34 !"# )'' #) )'' *#* )'' #) )'' *#* )'' $%&'( $))#( $"( $))#( $*"+( $))#( $"( $))#( $*"+( $))#( ˙ ˙ ? b˙ ˙ ? ˙ b˙ ˙ & B ∑ B ∑ &B ∑ b˙ &B ∑ B !˙ "" b #˙ "" $ "" % "" "% ""

%" +%! !#' )&# +%! !#' )&# +%! )&# ,-./01 $#)($+#2( $!"#( $)''( $+#2( $!"#( $)''( $+#2( $)''( j & œ<µ>œ ˙ <#>˙ ∑ <µ>œ ˙ <#>˙ ∑ <µ>˙ <#>˙ ∑ !&'()*&'(+" !"#$% '()*+,-./0 & 1 "2 "" 1 "2 "" 1 ""

%" )&# 222 )&# *%+ )&# !#' )&# $#)( $)''( $2&!( $)''( $*#*( $)''( $!"#( $)''( # ˙ # ˙ # ˙ # ˙ ? & ˙ < > ∑ ˙ < > ∑ < > ∑ < > ∑ & "" 3 "" <µ>4˙ "" !˙ ""

%" )&# *%+ )&# %" )&# *%+ )&# $#)( $)''( $*#*( $)''( $#)( $)''( $*#*( $)''( ? ˙ ? ? ˙ &<#>˙ ∑ <µ>˙ &<#>˙ ∑ &<#>˙ ∑ <µ> <#>˙ # "" $ "" %˙ "" "% ""

)% +)* !%2 '"! +)* !%2 '"! +)* '"! ,-./01 $%&($+!+( $!+!( $)&2( $+!+( $!+!( $)&2( $+!+( $)&2( j & œ ˜œ <˜>˙ #˙ ∑ ˜œ <˜>˙ #˙ ∑ ˜˙ #˙ ∑ !&'()*&'(+" ˜!"#$,$% '()*+,-./0 & 1 "2 "" 1 "2 "" 1 ""

)% '"! #"++ '"! *)) '"! !%2 '"! $%&( $)&2( $#""*( $)&2( $*!"( $)&2( $!+!( $)&2( ? & #˙ ∑ #˙ ∑ #˙ ∑ #˙ ∑ ˜˙ ˙ ˙ & "" 3 "" ˜4 "" <˜>!˙ ""

)% '"! *)) '"! )% '"! *)) '"! $%&( $)&2( $*!"( $)&2( $%&( $)&2( $*!"( $)&2( ? ˙ ? ? ˜ &#˙ ∑ ˜˙ &#˙ ∑ &#˙ ∑ ˜˙ #˙ % "" $ "" ˜%˙ "" "% ""

Method: Appendix D - p.35 !" ,!- ."/ 01. ,!- ."/ 01. ,!- 01. &'()*+ #"$%#,.,% #.,.% #!$/% #,.,% #.,.% #!$/% #,.,% #!$/% œ œ <˜>˙ #˙ ∑ œ <˜>˙ #˙ ∑ ˙ #˙ ∑ & µ ˜ ˜J ˜ '()*+,()*-" !b"#$%$& "#$%&'()*+ ! , -. -- , -. -- , --

!" 01. 21," 01. -!! 01. ."/ 01. #"$% #!$/% #211$% #!$/% #-.1% #!$/% #.,.% #!$/% #˙ #˙ #˙ #˙ & ˙ ∑ ˙ ∑ ˙ ∑ ∑ ! -- / -- ˜0 -- <˜>1˙ --

!" 01. -!! 01. !" 01. -!! 01. #"$% #!$/% #-.1% #!$/% #"$% #!$/% #-.1% #!$/% #˙ ? ˜˙ #˙ ? #˙ ˜˙ #˙ & ∑ & ∑ µ˙ & ∑ µ˙2 -- 3 -- 4 -- -4 --

"1 ,". .20 !$2 ,". .20 !$2 ,". !$2 &'()*+ #2!% #,2/% #.12% #!00% #,2/% #.12% #!00% #,2/% #!00% œ œ ˙ <#>˙ œ ˙ <#>˙ ˙ <#>˙ & <µ> ∑ <µ>J ∑ <µ> ∑ '()*+,()*-" !b"#$& "#$%&'()*+ ! , -. -- , -. -- , --

"1 !$2 /// !$2 -", !$2 .20 !$2 #2!% #!00% #/$.% #!00% #-2-% #!00% #.12% #!00% ˙ <#>˙ <#>˙ <#>˙ <#>˙ & ∑ b˙ ∑ <µ>˙ ∑ ∑ ! -- / -- 0 -- 1˙ --

"1 !$2 -", !$2 "1 !$2 -", !$2 #2!% #!00% #-2-% #!00% #2!% #!00% #-2-% #!00% <#>˙ ? ˙ <#>˙ ? <#>˙ <µ>˙ <#>˙ & ∑ <µ> & ∑ ˙ & ∑ 2˙ -- 3 -- 4 -- -4 --

2! ,2/ .12 !00 ,2/ .12 !00 ,2/ !00 &'()*+ #1% #,1.% #"$0% #!!2% #,1.% #"$0% #!!2% #,1.% #!!2% bœ bœ ˙ B˙ bœ ˙ B˙ b˙ B˙ & ∑ J ∑ ∑ '()*+,()*" !b"%$& "#$%&'()*+ ! , -. -- , -. -- , -- 2! !00 /$. !00 -2- !00 .12 !00 #1% #!!2% #/0/% #!!2% #-1,% #!!2% #"$0% #!!2% b˙ B˙ B˙ B˙ B˙ & ∑ ˙ ∑ b˙ ∑ ˙ ∑ ! -- / -- 0 -- 1 --

Method: Appendix D - p.36 !" "&& '!' "&& !" "&& '!' "&& #$% #""!% #'$(% #""!% #$% #""!% #'$(% #""!% B˙ ? b˙ B˙ ? B˙ b˙ B˙ & ∑ & ∑ b˙ & ∑ b!˙ "" # "" $ "" "$ ""

$ ($0 2)& ""! ($0 2)& ""! ($0 ""! *+,-./ #!!)"% #!)1% #2'!% #"2&% #!)1% #2'!% #"2&% #!)1% #"2&% œ œ #˙ µ˙ œ #˙ µ˙ ˙ µ˙ & ∑ J ∑ ∑ $%&'()%&'*" !b"# &'()*+,-./ % 0 "1 "" 0 "1 "" 0 ""

$ ""! 1&1 ""! '$( ""! #!!)"% #"2&% #1"0% #"2&% #&)'% #"2&% ˙ µ˙ µ˙ µ˙ & ∑ B˙ ∑ ˙ ∑ % "" 2 "" 3 ""

2)& ""! $ ""! '$( ""! $ ""! '$( ""! #2'!% #"2&% #!!)"% #"2&% #&)'% #"2&% #!!)"% #"2&% #&)'% #"2&% µ˙ µ˙ ? ˙ µ˙ ? µ˙ ˙ µ˙ & #˙ ∑ ∑ & ∑ ˙ & ∑ 4 "" !˙ "" # "" $ "" "$ ""

*+,-./ !$ (!0 21& "&! (!0 21& "&! (!0 "&! œ œ #˙ µ˙ œ#˙ µ˙ ˙ µ˙ & ∑ J ∑ $%&'()%&'*" !b"+,- &'()*+,-./ % 0 "1 "" 0 "1 "" 0 ""

!$ "&! 1'1 "&! '!( "&! 21& "&! ˙ ˙ ˙ ˙ ˙ µ ˙ µ µ µ & ∑ ∑ B ∑ ˙ ∑ #˙ % "" 2 "" 3 "" 4 ""

!$ "&! '!( "&! !$ "&! '!( "&! µ˙ ? ˙ µ˙ ? µ˙ ˙ µ˙ & ∑ ∑ & ∑ ˙ & ∑ !˙ "" # "" $ "" "$ ""

Method: Appendix D - p.37 "#$%&' ! (!) *+, --. (!) *+, --. (!) --. bœ œ ˙ B ˙ œ ˙ B ˙ ˙ B ˙ & ∑ J ∑ $%&'()%&'*" !b"# "#$%&'()*+ ! , -. -- , -. -- , -- ! --. /,/ --. 0!( --. *+, --. b˙ B ˙ B ˙ B ˙ B ˙ & ∑ ∑ Bb˙ ∑ b˙ ∑ ˙ ! -- / -- 0 -- 1 --

! --. 0!( --. ! --. 0!( --. B ˙ B ˙ B ˙ b˙ B ˙ ∑ ∑ ? b˙ ∑ & ∑ b˙ & 2 -- b 3˙ -- 4 -- -4 --

"#$%&' ! (!) *+, --. (!) *+, --. (!) --. ! --. œ#œ #˙ µ˙ #œ#˙ µ˙ #˙ µ˙ ˙ µ˙ & ∑ J ∑ ∑ ∑ $%&'()%&'*" !b"+ ! -- "#$%&'()*+ ! , -. -- , -. -- , -- /,/ --. 0!( --. *+, --. ˙ ˙ ˙ ˙ µ µ µ & B ∑ ˙ ∑ #˙ ∑ / -- 0 -- 1 --

! --. 0!( --. ! --. 0!( --. µ˙ µ˙ µ˙ ˙ µ˙ ∑ ∑ ? ˙ ∑ & ˙ & 2 -- 3˙ -- 4 -- -4 --

"#$%&' ! (!) *+, --. (!) *+, --. (!) --. œ œ ˙ B ˙ œ ˙ B ˙ ˙ B ˙ & ∑ J ∑ ∑ $%&'()%&'*" !b", "#$%&'()*+ ! , -. -- , -. -- , -- ! --. /,/ --. 0!( --. *+, --. ˙ B ˙ b˙ B ˙ B ˙ B ˙ & ∑ B ∑ ˙ ∑ ˙ ∑ ! -- / -- 0 -- 1 --

! --. 0!( --. ! --. 0!( --. ˙ ˙ ˙ ˙ ˙ B B ? ˙ B B & ˙ ∑ ∑ & ∑ 2 -- 3˙ -- 4 -- -4 --

Method: Appendix D - p.38 SET 6! The 12th harmonic

The sound of the 12th harmonic is very familiar as it is simply the octave above the 6th harmonic, in turn an octave above the 3rd harmonic. However, many of the 12th harmonics on the double horn are avoided other than the high notes from written A!!to C above the treble staff, played on the B! horn. The notes descending from the high G at the top of the treble staff down to C" can be played as 12th harmonics on the F side though are usually played on the B!! horn. The G is usually played as the 9th harmonic of the B!!horn (Bb:0), the F"!as the 9th harmonic of the A horn (Bb:2), the F as the 8th harmonic of the Bb!horn

(Bb:0) and so on. Exponents of baroque and classical horns use the 12th harmonic extensively, however some intervals associated with it are largely unfamiliar such as 11:12 and 7:12. This set of exercises provides experience with such intervals.

Retain a longer rest between items if desired.

Method: Appendix D - p.39 h!"!#$%!&' ()*+,- ' .'& /01 223 4'. /01 4'. .'& 4'. ' 4'. 515 4'. 0&3 4'. ˙ œ œ µ˙ ˙ ˙ ˙ ˙ ˙ ˙ ˙ B ˙ & œ Ó Ó Ó ˙ Ó Bb˙ Ó !$%&'($%&)" !"# "#$%&'()*+ ! , -. -- -/ -. -/ , -/ ! -/ 0 -/ -1 -/

()*+,- ' .'& /01 223 4'. /01 4'. .'& 4'. ' 4'. 515 4'. 0&3 4'. #œ µ˙ #˙ #˙ #˙ #˙ #˙ #˙ µ˙ #˙ & œ#œ Ó Ó #˙ Ó ˙ Ó B˙ Ó !$%&'($%&)" !"* "#$%&'()*+ ! , -. -- -/ -. -/ , -/ ! -/ 0 -/ -1 -/

()*+,- ' .'& /01 223 4'. /01 4'. .'& 4'. ' 4'. 515 4'. 0&3 4'. œ B˙ ˙ ˙ ˙ ˙ ˙ ˙ B˙ ˙ & bœ œ Ó Ó ˙ Ó b˙ Ó Bb˙ Ó !$%&'($%&)" !"+ "#$%&'()*+ ! , -. -- -/ -. -/ , -/ ! -/ 0 -/ -1 -/

()*+,- 3' .3& /51 213 43. /51 43. .3& 43. 3' 43. 545 43. 023 43. #œ µ˙ ˙ #˙ ˙ ˙ ˙ ˙ µ˙ ˙ & œ œ Ó Ó ˙ Ó ˙ Ó B˙ Ó !$%&'($%&)" !"+,* "#$%&'()*+ ! , -. -- -/ -. -/ , -/ ! -/ 0 -/ -1 -/

' .'& /01 223 4'. /01 4'. ()*+,- 633027 63057 6/437 62/17 61047 6/437 61047 ˙ ˙ ˙ & œ œ #œ µ Ó #˙ Ó !$%&'($%&)" !"- "#$%&'()*+ ! , -. -- -/ -. -/ .'& 4'. ' 4'. 515 4'. 0&3 4'. 63057 61047 633027 61047 652&7 61047 60.17 61047 ˙ ˙ ˙ µ˙ ˙ & ˙ Ó ˙ Ó B˙ Ó , -/ ! -/ 0 -/ -1 -/

32 .35 &'3 211 434 &'3 434 .35 434 ()*+,- 6'7 6.'&7 6/017 62237 64'.7 6/017 64'.7 6.'&7 64'.7 ˙ b˙ b˙ b˙ & bœ bœ œ B Ó ˙ Ó b˙ Ó !$%&'($%&" !"*,- "#$%&'()*+ ! , -. -- -/ -. -/ , -/ 32 434 50& 434 021 434 6'7 64'.7 65157 64'.7 60&37 64'.7 b˙ b˙ ˙ b˙ & b˙ Ó b ˙ Ó ! -/ 0 -/ -1 -/

Method: Appendix D - p.40 !" -!. .$0 %1$ 2!- .$0 2!- '()*+, #$%& #-$/& #."$& #%00& #2$2& #."$& #2$2& ˙ ˙ & œ <µ>œ œ <#>˙ <µ> Ó ˙ <µ> Ó !&'()*&'(+" !"#$% "#$%&'()*+ ! , -. -- -/ -. -/

-!. 2!- !" 2!- /// 2!- 12$ 2!- #-$/& #2$2& #$%& #2$2& #/1.& #2$2& #1%0& #2$2& ˙ ˙ ˙ ˙ ˙ & <µ>˙ <µ> Ó ˙ <µ> Ó <µ> Ó <µ> , -/ ! -/ 0˙ -/ -1 -/

%! -%2 .!/ 0". 2%% .!/ 2%% -%2 2%% '()*+, #!1& #-.-& #.-.& #%1/& #2."& #.-.& #2."& #-.-& #2."& ˙ ˙ ˙ & œ ˜œ<˜>œ #˙ ˜ Ó <˜>˙ ˜ Ó ˜˙ ˜ Ó !&'()*&'(+"˜!"#$,$% "#$%&'()*+ ! , -. -- -/ -. -/ , -/

%! 2%% $"-- 2%% 1/. 2%% #!1& #2."& #$""2& #2."& #12/& #2."& #˙ & ˙ ˜˙ Ó ˜˙ Ó ˜˙ ˜! -/ 0˙ -/ -1 -/

%! -%2 .!/ 0". 2%% .!/ 2%% '()*+, #!1& #-.-& #.-.& #%1/& #2."& #.-.& #2."& œœ n˙ B˙ ˙ B˙ & Bœ B Ó Ó !&'()*&'(+" -b"#$,$% "#$%&'()*+ ! , -. -- -/ -. -/ -%2 2%% %! 2%% $"-- 2%% 1/. 2%% #-.-& #2."& #!1& #2."& #$""2& #2."& #12/& #2."& b˙ ˙ B˙ B˙ B˙ B˙ & B Ó B˙ Ó ˙ Ó , -/ ! -/ 0 -/ -1 -/

!" -!. .$0 %1$ 2!- .$0 2!- '()*+, #$%& #-$/& #."$& #%00& #2$2& #."$& #2$2& œ <#>˙ <µ>˙ ˙ <µ>˙ & œ <µ>œ Ó Ó !&'()*&'(+" -b"#$% "#$%&'()*+ ! , -. -- -/ -. -/ -!. 2!- !" 2!- /// 2!- 12$ 2!- #-$/& #2$2& #$%& #2$2& #/1.& #2$2& #1%0& #2$2& <µ>˙ <µ>˙ <µ>˙ ˙ <µ>˙ & <µ>˙ Ó ˙ Ó b˙ Ó , -/ ! -/ 0 -/ -1 -/

Method: Appendix D - p.41 !" ,!- .$! "11 2!2 .$! 2!2 ,!- 2!2 #$% #,$.% #/01% #""!% #2$,% #/01% #2$,% #,$.% #2$,% &'()*+ b˙ b˙ b˙ bœ bœ œ B˙ ˙ b˙ & Ó Ó Ó &'()*+'()" !b"#$% "#$%&'()*+ ! , -. -- -/ -. -/ , -/ !" 2!2 -0. 2!2 0"1 2!2 #$% #2$,% #-1-% #2$,% #0.!% #2$,% b˙ b˙ ˙ b˙ b˙ B & Ó ˙ Ó ! -/ 0 -/ -1 -/

$ ,$. /01 ""! 2$, /01 2$, &'()*+ #!!0"% #!0-% #/2!% #"/1% #102% #/2!% #102% ˙ ˙ œ œ #œ µ˙ #˙ & 158 Ó Ó &'()*+'()," !b"% "#$%&'()*+ ! , -. -- -/ -. -/ ,$. 2$, $ 2$, -1- 2$, 0.! 2$, #!0-% #102% #!!0"% #102% #-".% #102% #0,1% #102% ˙ ˙ ˙ ˙ ˙ ˙ ˙ B & Ó Ó B˙ Ó , -/ ! -/ 0 -/ -1 -/

&'()*+ !$ ,!. /-1 "1! 2!, /-1 2!, ,!. 2!, !$ 2!, -2- 2!, 0"! 2!, ˙ ˙ ˙ ˙ ˙ ˙ ˙ œ œ#œ µ˙ #˙ ˙ ˙ B & Ó Ó Ó Ó B˙ Ó &'()*+'()," !b"-$# "#$%&'()*+ ! , -. -- -/ -. -/ , -/ ! -/ 0 -/ -1 -/

$ ,$. /01 ""! 2$, /01 2$, ,$. 2$, $ 2$, -1- 2$, 0.! 2$, &'()*+ ˙ b˙ b˙ b˙ b˙ b˙ B ˙ b˙ bœ œ œB ˙ ˙ ˙ b˙ & Ó Ó Ó Ó B Ó &'()*+'()," !b"- "#$%&'()*+ ! , -. -- -/ -. -/ , -/ ! -/ 0 -/ -1 -/

$ ,$. /01 ""! 2$, /01 2$, ,$. 2$, $ 2$, -1- 2$, 0.! 2$, &'()*+ ˙ ˙ ˙ ˙ ˙ ˙ ˙ œ#œ#œ µ˙ #˙ #˙ ˙ µ & Ó Ó Ó Ó B˙ Ó &'()*+'()," !b"# "#$%&'()*+ ! , -. -- -/ -. -/ , -/ ! -/ 0 -/ -1 -/

$ ,$. /01 ""! 2$, /01 2$, ,$. 2$, $ 2$, -1- 2$, 0.! 2$, &'()*+ ˙ ˙ ˙ ˙ ˙ ˙ B ˙ ˙ œ œ œB ˙ ˙ ˙ b˙ & Ó Ó Ó Ó B Ó &'()*+'()," !b". "#$%&'()*+ ! , -. -- -/ -. -/ , -/ ! -/ 0 -/ -1 -/

Method: Appendix D - p.42 SET 7! Discovering the 13th harmonic

Players have left this harmonic virtually untouched other than in passing in some glissandi or when underpitching the 14th harmonic in the “Prologue” of the Britten Serenade for tenor, horn and strings (1943). Few composers have actually written for the horn’s 13th harmonic. A few exceptions being Joel

Mandelbaum and György Ligeti. The horn opens Mandelbaum’s Wind Quintet

Number 2 with a beautiful melody extending from the 6th to 16th harmonics in the F series. The melody is printed in Chapter One (p 45). Two of Ligeti’s compositions that feature the horn - Trio for horn, violin and piano (1982) and the

Hamburg Concerto (2002) - require natural harmonics including the 13th. Some other works that include the 13th harmonic are Joseph Perhson’s Harmonic Etude

(1987); Douglass Hill’s Thoughtful Wanderings for natural horn and percussion

(1990) and Jörg Widmann’s Air (1973).

To avoid numbers overlapping each other in the following exercises, the cents values are shown only for each minum (half-note).

The 13th harmonics are quite higher in the range on the shorter tube lengths.

Each player must determine which tube length to stop at. This could vary from day to day.

Method: Appendix D - p.43 h!"!#$%!&' ()*+,- .'/ 0&1 221 0&1 304 0&1 /'& 0&1 ˙ ˙ ˙ ˙ œ œµœ ˙ B µ˙ B ˙ B ˙ B & œBbœœ ∑ ∑ ∑ ∑ !$%&'($%&" !"#œœ "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14

' 0&1 545 0&1 .'/ 0&1 304 0&1 ' 0&1 545 0&1 B ˙ B ˙ B ˙ B ˙ B ˙ Bb˙ B ˙ ˙ ∑ b˙ ∑ ∑ ∑ ∑ & B ˙ ˙ / 14 . 14 - 14 , 14 !˙ 14 1! 14

()*+,- .'/ 0&1 221 0&1 304 0&1 /'& 0&1 #œµœ#˙ µ˙ µ˙ µ˙ #˙ µ˙ µ˙ & #œBœœ#œ ∑ ∑ ∑ #˙ ∑ !$%&'($%&" !")œ#œ "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14

' 0&1 545 0&1 .'/ 0&1 304 0&1 ' 0&1 545 0&1 µ˙ µ˙ µ˙ µ˙ µ˙ B ˙ µ˙ & ˙ ∑ B˙ ∑ #˙ ∑ ∑ ∑ / 14 . 14 - 14 #˙, 14 !˙ 14 1! 14

()*+,- .'/ 0&1 221 0&1 304 0&1 /'& 0&1 œBœ ˙ B˙ B˙ B˙ ˙ B˙ B˙ & œBbœbœœ ∑ ∑ ∑ ˙ ∑ !$%&'($%&" b!"*œœ "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14

' 0&1 545 0&1 .'/ 0&1 304 0&1 ' 0&1 545 0&1 B˙ B˙ B˙ B˙ B˙ Bb˙ B˙ & b˙ ∑ Bb˙ ∑ ˙ ∑ ∑ ∑ / 14 . 14 - 14 ˙, 14 b!˙ 14 1! 14

()*+,- .1/ 021 241 021 354 021 /1& 021 ˙ ˙ ˙ ˙ œ#œµœ ˙ µ ∑ µ˙ µ ∑ #˙ µ ∑ ˙ µ ∑ & œBœœ !$%&'($%&" !"*+)œ#œ "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14

1' 021 5.5 021 .1/ 021 354 021 1' 021 5.5 021 ˙ ˙ ˙ ˙ ˙ ˙ ˙ µ ∑ µ ∑ µ ∑ µ ∑ µ ∑ B µ & ˙ B˙ ˙ / 14 . 14 - 14 #˙, 14 !˙ 14 1! 14

Method: Appendix D - p.44 '() ,./ 00/ ,./ 1,+ ,./ )(. ,./ !"#$%& *+,'- *,)+- *01+- *,)+- *1'/- *,)+- */,2- *,)+- œ#œµœ ˙ µ˙ ∑ µ˙ µ˙ ∑ #˙ µ˙ ∑ ˙ µ˙ ∑ & œBœœ !$%&'($%&" !"#œ#œ "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14 ( ,./ 2+2 ,./ '() ,./ 1,+ ,./ ( ,./ 2+2 ,./ *//,0- *,)+- *20.- *,)+- *+,'- *,)+- *1'/- *,)+- *//,0- *,)+- *20.- *,)+- ˙ ˙ ˙ ˙ ˙ ˙ ˙ µ ∑ µ ∑ µ ∑ µ ∑ µ ∑ B µ & ˙ B˙ ˙ / 14 . 14 - 14 #˙, 14 !˙ 14 1! 14 '/' ,0+ 0++ ,0+ .(/ ,0+ )/2 ,0+ !"#$%& *'()- *,./- *00/- *,./- *1,+- *,./- *)(.- *,./- ˙ ˙ ˙ ˙ bœ œBœb˙ B ∑ B˙ B ∑ ˙ B ∑ b˙ B ∑ & bœBbœbœ !$%&'($%&" b!")*#œ œ "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14 /0 ,0+ 2,. ,0+ '/' ,0+ .(/ ,0+ /0 ,0+ 2,. ,0+ *(- *,./- *2+2- *,./- *'()- *,./- *1,+- *,./- *(- *,./- *2,2- *,./- ˙ ˙ ˙ ˙ ˙ b˙ ˙ B ∑ B ∑ B ∑ B ∑ B ∑ B B & b˙ Bb˙ b˙ / 14 . 14 - 14 ˙, 14 b!˙ 14 1! 14 '1) ,'/ 0,/ ,'/ ./+ ,'/ )1. ,'/ !"#$%& *'/'- *,0+- *0++- *,0+- *.(/- *,0+- *)/2- *,0+- ˙ ˙ ˙ ˙ ˙ & œ<µ>œ œ<µ>œ<µ> B ∑ <µ>˙ B ∑ ˙ B ∑ <µ>˙ B ∑ <µ>œ œ !$%&'($%&" !"+*#œ œ "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14 1( ,'/ 222 ,'/ '1) ,'/ ./+ ,'/ 1( ,'/ 222 ,'/ */0- *,0+- *2,.- *,0+- *'/'- *,0+- *.(/- *,0+- */0- *,0+- *2,.- *,0+- ˙ ˙ ˙ ˙ ˙ ˙ ˙ & ˙ B ∑ B ∑ B ∑ B ∑ B ∑ B ˙ <µ>˙ / 14 . 14 - 14 ˙, 14 !˙ 14 1! 14 '00 ,2. +(. ,2. .12 ,2. )0' ,2. !"#$%& *'.(- *,'2- *0,2- *,'2- *.).- *,'2- *).)- *,'2- 4 ˙ #˙ Ó #˙Ó #˙ Ó #˙Ó & 2 œ œ˜œ<˜>œ#œ˜ #˙ <˜>˙ ˜˙ œ˜œ ˜ !$%&'($%&"˜!"+*)*#œ<˜> "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14

01 ,2. /()) ,2. '00 ,2. .12 ,2. 01 ,2. /()) ,2. *1,- *,'2- */(('- *,'2- *'.(- *,'2- *.).- *,'2- *1,- *,'2- */(('- *,'2- #˙ #˙ #˙ #˙ ? ˙ #˙ ˙ #˙ & ˙ Ó Ó Ó Ó ˜ & Ó ˜/ 14 ˙ ˙ . 14 ˜- 14 <˜>˙, 14 ! 14 1! 14

Method: Appendix D - p.45 '(( -.* /+* -.* *0. -.* 1(' -.* !"#$%& )'*+, )-'., )(-., )-'., )*1*, )-'., )1*1, )-'., ˙ #˙ #˙ #˙ #˙ 4 œ œ˜œ<˜>œ#œ˜ Ó 3#˙ Ó ˙ Ó B˙ Ó & 2 œ˜œ µ 2 '()*+,()*" !µœb"#$%$&<˜> "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14 (0 -.* 2+11 -.* '(( -.* *0. -.* (0 -.* 2+11 -.* )0-, )-'., )2++', )-'., )'*+, )-'., )*1*, )-'., )0-, )-'., )2++', )-'., #˙ #˙ #˙ #˙ #˙ ˙ #˙ ˙ Ó ˙ Ó Ó Ó Ó 2 & µ ˜˙ ˙ 2 / 14 . 14 - 14 <˜>, 14 µ!˙ 14 1! 14 '01 -'2 (-2 -'2 *2/ -'2 10* -'2 !"#$%& )'2', )-(/, )(//, )-(/, )*+2, )-(/, )12., )-(/, ˙ B ˙ B ˙ B ˙ B ˙ bœœ<µ>œ œ<µ>œ<µ> Ó <µ>˙ Ó ˙ Ó <µ>˙ Ó & œ<µ>œ '()*+,()*" !œb"#$& "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14 0+ -'2 ... -'2 '01 -'2 *2/ -'2 0+ -'2 ... -'2 )2(, )-(/, ).-*, )-(/, )'2', )-(/, )*+2, )-(/, )2(, )-(/, ).-*, )-(/, B ˙ B ˙ B ˙ B ˙ B ˙ b˙ B ˙ ˙ Ó Ó Ó Ó Ó & b˙ <µ>˙ ˙ / 14 . 14 - 14 , 14 !˙ 14 1! 14

'2' -(/ (// -(/ *+2 -(/ 12. -(/ !"#$%& )'+1, )-*2, )((2, )-*2, )0-/, )-*2, )1+*, )-*2, ˙ b˙ b˙ b˙ b˙ bœbœ œBœB B B˙ B ˙ B b˙ B & œbœBbœ Ó Ó Ó Ó '()*+,()*" b!œb"%$& "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14 2( -(/ .-* -(/ '2' -(/ *+2 -(/ 2( -(/ .-* -(/ )+, )-*2, )./., )-*2, )'+1, )-*2, )0-/, )-*2, )+, )-*2, )./., )-*2, Bb˙ Bb˙ Bb˙ Bb˙ Bb˙ Bb˙ Bb˙ b˙ b˙ & Ó B Ó b˙ Ó ˙ Ó Ó / 14 . 14 - 14 , 14 b!˙ 14 1! 14

'+1 -*2 ((2 -*2 0-/ -*2 1+* -*2 !"#$%& )/-', )-1/, )(0/, )-1/, )0'2, )-1/, )2-., )-1/, ˙ B ˙ B ˙ B ˙ B ˙ œœœ#œ µœ µ˙ #˙ ˙ & #œœB Ó Ó Ó Ó '()*+,()*" !œb"& "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14 + -*2 ./. -*2 '+1 -*2 0-/ -*2 + -*2 ./. -*2 )22-(, )-1/, ).(*, )-1/, )/-', )-1/, )0'2, )-1/, )22-(, )-1/, ).(*, )-1/, B ˙ B ˙ B ˙ B ˙ B ˙ B ˙ B ˙ ˙ ˙ & Ó B Ó ˙ Ó #˙ Ó Ó / 14 . 14 - 14 , 14 !˙ 14 1! 14

Method: Appendix D - p.46 !"#$%& '() *+( +,( *+( -., *+( )(/ *+( ˙ ˙ ˙ ˙ ˙ œ œ#œµœ B µ˙ B #˙ B ˙ B & #œ œBœ ∑ ∑ ∑ ∑ &'()*+'()" !œb"#$% "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14

(0 *+( .'. *+( '() *+( -., *+( (0 *+( .'. *+( ˙ ˙ ˙ ˙ ˙ B ˙ ˙ ˙ B B B B B B & ∑ B˙ ∑ ˙ ∑ #˙ ∑ ∑ / 14 . 14 - 14 , 14 !˙ 14 1! 14

!"#$%& '0) */( ++( */( -*, */( )0/ */( œb˙ B ˙ ˙ B ˙ B ˙ B ˙ bœbœœ œB B ˙ ˙ & œbœB ∑ ∑ ∑ ∑ &'()*+'()" b!œb"# "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14 0 */( .,. */( '0) */( -*, */( 0 */( .,. */( B ˙ B ˙ B ˙ B ˙ B ˙ Bb˙ B ˙ b˙ ∑ Bb˙ ∑ b˙ ∑ ∑ ∑ & ˙ b˙ / 14 . 14 - 14 , 14 ! 14 1! 14

'0) */( ++( */( -*, */( )0/ */( !"#$%& œ ˙ µ˙ ˙ µ˙ µ˙ µ˙ œœ#œ#œµ µ #˙ #˙ & #œ œB ∑ ∑ ∑ ∑ &'()*+'()" !œb"% "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14

0 */( .,. */( '0) */( -*, */( 0 */( .,. */( µ˙ µ˙ µ˙ µ˙ µ˙ B ˙ µ˙ ˙ ˙ & ∑ B ∑ ˙ ∑ #˙ ∑ ∑ / 14 . 14 - 14 , 14 !˙ 14 1! 14

'0) */( ++( */( -*, */( )0/ */( !"#$%& œB œ ˙ B ˙ B ˙ B ˙ ˙ B ˙ B ˙ Bbœœœ ˙ & œœœ ∑ ∑ ∑ ∑ &'()*+'()" !b", "#$%&'()*+ ! , - . / 0 12 11 13 14 11 14 12 14 0 14

0 */( .,. */( '0) */( -*, */( 0 */( .,. */( B ˙ B ˙ B ˙ B ˙ B ˙ Bb˙ B ˙ ˙ Bb˙ ˙ & ∑ ∑ ∑ ˙ ∑ ˙ ∑ / 14 . 14 - 14 , 14 ! 14 1! 14

Method: Appendix D - p.47 SET 8! The 14th harmonic

The 14th harmonic, an octave above the 7th harmonic needs no introduction, however intervals associated with it will feel quite different to play than those with the 7th harmonic.

Occasionally loosen the lip muscles by descending further through the harmonics to the 2nd or 1st.

The exercises on short tube lengths are included for completeness, omit any as needed.

!"#$%&'$'()&*"+'+",#'-"#.""&'%&#"+/$(,'%0'1",%+"12

Method: Appendix D - p.48 !"!#$%!&' ()*+,- h .&/ 010 2'3 010 44/ 010 ˙ b˙ b˙ b˙ œœµœ œB B ˙ B µ˙ B & œ Ó Ó Ó !$%&'($%&" !"# "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

5.1 010 3'& 010 ' 010 /'.. 010 b˙ b˙ b˙ ˙ b˙ ˙ B ˙ B B B & Ó Ó ˙ Ó Ó -. -1 , -1 ! -1 -2 -1

()*+,- .&/ 010 2'3 010 44/ 010 ˙ ˙ ˙ #œ œ#œµ˙ B #˙ B ˙ B & œ#œ µ Ó Ó µ Ó !$%&'($%&" !") "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

5.1 010 3'& 010 ' 010 /'.. 010 ˙ ˙ ˙ #˙ ˙ #˙ B B B B & Ó #˙ Ó ˙ Ó Ó -. -1 , -1 ! -1 -2 -1

()*+,- .&/ 010 2'3 010 44/ 010 b˙ b˙ b˙ œ œ œB˙ B ˙ B ˙ B & bœ œ B Ó Ó B Ó !$%&'($%&" !"* "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

5.1 010 3'& 010 ' 010 /'.. 010 b˙ b˙ b˙ ˙ b˙ ˙ B B B B & Ó ˙ Ó b˙ Ó Ó -. -1 , -1 ! -1 -2 -1

()*+,- .4/ 020 2/3 020 41/ 020 œµ˙ B˙ ˙ B˙ B˙ & œ œ#œ µœ Ó Ó µ˙ Ó !$%&'($%&" !"*+) "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

501 020 3/& 020 /' 020 /'0. 020 B˙ B˙ B˙ #˙ B˙ & #˙ Ó ˙ Ó ˙ Ó Ó -. -1 , -1 ! -1 -2 -1

Method: Appendix - p.49 '() .,. 01+ .,. //) .,. !"#$%& *'+,- *./(- *,'0- *./(- */2,- *./(- œ œµ˙ B˙ ˙ B˙ ˙ B˙ & œ œ#œ µ Ó Ó µ Ó !$%&'($%&" !"# "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1 2', .,. +1( .,. 1 .,. )1'' .,. *20)- *./(- *)'.- *./(- *))'/- *./(- *)102- *./(- B˙ B˙ B˙ #˙ B˙ & #˙ Ó ˙ Ó ˙ Ó Ó -. -1 , -1 ! -1 -2 -1

'/, .'( 0)0 .'( /,, .'( !"#$%& *'()- *.,.- *01+- *.,.- *//)- *.,.- œbœB˙ ˙ b˙ ˙ ˙ ˙ & bœbœ œB Ó Ó B Ó !$%&'($%&" !")*# "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1 (1) .'( +). .'( )/ .'( ))12 .'( *2',- *.,.- *+1(- *.,.- *1- *.,.- *)1''- *.,.- ˙ ˙ ˙ ˙ ˙ & ˙ Ó b˙ Ó b˙ Ó Ó -. -1 , -1 ! -1 -2 -1

'0) ... 02+ ... /') ... !"#$%& *'/,- *.'(- *0)0- *.'(- */,,- *.'(- œB˙ ˙ ˙ ˙ ˙ & œ<µ>œ œ µœ<µ> Ó <µ> Ó <µ>˙ Ó !$%&'($%&" !"+*# "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1 (), ... +2( ... 21 ... )))' ... *(1)- *.'(- *+).- *.'(- *)/- *.'(- *))12- *.'(- ˙ ˙ ˙ #˙ ˙ & ˙ Ó <µ>˙ Ó ˙ Ó Ó -. -1 , -1 ! -1 -2 -1 '.( )1++ 0// )1++ ,1( )1++ !"#$%& *'0.- *)110- *0(1- *)110- */'.- *)110- #˙ ˙ ˙ ˙ & œ<˜>œ#œ˜œ Ó ˜˙ Ó #˙ Ó !$%&'($%&" ˜!"+*)*#œ˜ "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

(2. )1++ +/0 )1++ /2 )1++ ))() )1++ *(+(- *)110- *+(+- *)110- *2'- *)110- *))+,- *)110- ˙ Ó ˙ Ó ˙ Ó <˜>˙ #˙ Ó & <˜>˙ ˜˙ ˜˙ -. -1 , -1 ! -1 -2 -1

Method: Appendix - p.50 '() -.// +00 -.// 1.) -.// !"#$%& *'+(, *-..+, *+)., *-..+, *0'(, *-..+, ˙ ˙ ˙ œ#œ˜œ#˙ ˜˙ #˙ & µœ˜œ<˜> Ó Ó Ó '()*+,()*" !b"#$%$& "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1 )2( -.// /0+ -.// 02 -.// --)- -.// *)/), *-..+, */)/, *-..+, *2', *-..+, *--/1, *-..+, ˙ ˙ ˙ ˙ n˙ ˙ <˜> & <˜> Ó ˜˙ Ó µ˙ Ó Ó -. -1 , -1 ! -1 -2 -1 '+- ((( +2/ ((( 0'- ((( !"#$%& *'01, *('), *+-+, *('), *011, *('), ˙ b˙ b˙ b˙ œ œ<µ>œ<µ>œB <µ>˙ <µ>˙ & œ<µ> Ó Ó Ó '()*+,()*" !b"#$& "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1 )-1 ((( /2) ((( 2. ((( ---' ((( *).-, *('), */-(, *('), *-0, *('), *--.2, *('), b˙ b˙ b˙ ˙ ˙ ˙ ˙ & Ó <µ> Ó ˙ Ó Ó -. -1 , -1 ! -1 -2 -1 '01 (') +-+ (') 011 (') !"#$%& *')-, *(1(, *+./, *(1(, *00-, *(1(, bœ b˙ ˙ b˙ ˙ ˙ bœbœ œBœ B B˙ & Ó Ó Ó '()*+,()*" !b"%$& "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1 ).- (') /-( (') -0 (') --.2 (') *2'1, *(1(, */.), *(1(, *., *(1(, *-.'', *(1(, ˙ ˙ ˙ ˙ ˙ ˙ b˙ b˙ & Ó Ó Ó Ó -. -1 , -1 ! -1 -2 -1 ')- (1( +./ (1( 00- (1( !"#$%& *'/1, *(0), *1'+, *(0), *021, *(0), ˙ ˙ ˙ ˙ œ#œ µœ œB B ˙ B µ˙ B & œ Ó Ó Ó '()*+,()*" !b"& "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1 2'1 (1( /.) (1( . (1( -.'' (1( *2+-, *(0), *-'(, *(0), *--'0, *(0), *-.+2, *(0), ˙ ˙ ˙ #˙ ˙ #˙ B ˙ B B B & Ó Ó ˙ Ó Ó -. -1 , -1 ! -1 -2 -1

Method: Appendix - p.51 !"#$%& '() *+* +), *+* (-) *+* œ ˙ B ˙ ˙ B ˙ B ˙ œ œ#œ µœ B µ˙ & Ó Ó Ó &'()*+'()" !b"#$% "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

.*- *+* ,)/ *+* )0 *+* )0*' *+* B ˙ B ˙ B ˙ #˙ B ˙ #˙ ˙ ˙ & Ó Ó Ó Ó -. -1 , -1 ! -1 -2 -1

!"#$%& '/) *-* +0, *-* (() *-* ˙ Bb˙ Bb˙ Bb˙ bœ œ œB œbœB b˙ B ˙ & Ó Ó Ó &'()*+'()" !b"# "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

.'- *-* ,0/ *-* 0 *-* )0'' *-* Bb˙ Bb˙ Bb˙ ˙ Bb˙ ˙ ˙ b˙ & Ó Ó Ó Ó -. -1 , -1 ! -1 -2 -1

!"#$%& '/) *-* +0, *-* (() *-* œ ˙ B ˙ ˙ B ˙ B ˙ œ#œ#œµœ µ µ˙ & Ó Ó Ó &'()*+'()" !b"% "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

.'- *-* ,0/ *-* 0 *-* )0'' *-* B ˙ B ˙ B ˙ #˙ B ˙ #˙ #˙ ˙ & Ó Ó Ó Ó -. -1 , -1 ! -1 -2 -1

!"#$%& '/) *-* +0, *-* (() *-* ˙ Bb˙ Bb˙ Bb˙ œ œ œB œ œB ˙ B ˙ & Ó Ó Ó &'()*+'()" !b", "#$%&'()*+ ! , -. -- -/ -0 -1 -/ -1 -- -1

.'- *-* ,0/ *-* 0 *-* )0'' *-* Bb˙ Bb˙ Bb˙ ˙ Bb˙ ˙ ˙ ˙ & Ó Ó Ó Ó -. -1 , -1 ! -1 -2 -1

Method: Appendix - p.52 SET 9! The 15th harmonic

!"#$%$&'#()$'*(+,-(#."'*(./(+,$#(,&01.'$%(/&1$2$&0(&#($+(30.*"%-#(&(3"0-(1&4.0(50*(

/0.1(+,-(67+,(,&01.'$%8(9,-(#&1-($'+-0:&2(.%%"0#(;-+<--'(,&01.'$%#(=(>(?(&'*(@(

>(6A8(B'-(<&C(+.(&330.&%,(+,-(6D+,(,&01.'$%(<$+,."+(E.$'E(#.(,$E,($'(+,-(0&'E-($#(

+.("#-(+<.(2-'E+,#(./(+";-8(F.0(-G&132-(+,-(67+,(&'*(6D+,(,&01.'$%#(./(+,-(Hb(+";-(

2-'E+,(%&'( ;-( 32&C-*( .'(+,-(Ib( ,.0'( &#( ,&01.'$%#( @( &'*( 6A8( J-+( K($':.2:-#( +,-(

30&%+$%-(./($'+-0:&2#(6=L6DM(65L6D(&'*(66L6D(<,$%,(%&'(;-("#-/"2(/.0(1.0-(%.132-G(

4"#+($'+.'&+$.'(<.0N8

The exercises on short tube lengths are included for completeness, omit any as needed.

O-+&$'(&(2.'E-0(0-#+(;-+<--'($'+-0:&2#($/(*-#$0-*8

Method: Appendix D - p.53 !"!#$%!&' ()*+,- h ./. 0'11 1&0 0'11 2'3 0'11 œ b˙ n˙ ˙ ˙ ˙ œ œµœ œB B B ˙ & œ Ó Ó Ó !$%&'($%&" !"# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2 440 0'11 51/ 0'11 3'& 0'11 ' 0'11 ' 0'11 ˙ ˙ ˙ ˙ ˙ ˙ µ˙ ˙ ˙ & Ó Ó Ó ˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

()*+,- ./. 0'11 1&0 0'11 2'3 0'11 ˙ #˙ #˙ #˙ #œ œ#œµœB µ˙ #˙ & œ#œ µ Ó Ó Ó !$%&'($%&" !") "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

440 0'11 51/ 0'11 3'& 0'11 ' 0'11 ' 0'11 ˙ ˙ #˙ #˙ #˙ #˙ #˙ & µ Ó #˙ Ó #˙ Ó ˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

()*+,- ./. 0'11 1&0 0'11 2'3 0'11 b˙ n˙ ˙ ˙ œ œ œBœB B˙ ˙ & bœ œ B Ó Ó Ó !$%&'($%&" !"* "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

440 0'11 51/ 0'11 3'& 0'11 ' 0'11 ' 0'11 ˙ ˙ ˙ ˙ b˙ ˙ ˙ ˙ & B Ó Ó ˙ Ó b˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

()*+,- .2. 0'.1 140 0'.1 203 0'.1 œ œµœB˙ #˙ µ˙ #˙ ˙ #˙ & œ œ#œµ Ó Ó Ó !$%&'($%&" !"*+) "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

4/0 0'.1 5./ 0'.1 30& 0'.1 0' 0'.1 0' 0'.1 #˙ #˙ #˙ #˙ ˙ #˙ & µ˙ Ó #˙ Ó ˙ Ó ˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

Method: Appendix D - p.54 '(' -.// /+- -.// 0.2 -.// !"#$%& )'*+, )-.01, )/2(, )-.01, )(/0, )-.01, œ œµœB˙ #˙ µ˙ #˙ ˙ #˙ & œ œ#œµ Ó Ó Ó !$%&'($%&" !"# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2 **- -.// 1/( -.// 2.+ -.// . -.// . -.// )*1(, )-.01, )10-, )-.01, )-/', )-.01, )--/*, )-.01, )--/*, )-.01, ˙ #˙ #˙ #˙ #˙ ˙ #˙ & µ Ó #˙ Ó ˙ Ó ˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2 '/+ --.1 /*( --.1 0-0 --.1 !"#$%& )'(', )-.//, )/+-, )-.//, )0.2, )-.//, bœBœBb˙ n˙ B˙ ˙ b˙ ˙ & bœbœ œBœ Ó Ó Ó !$%&'($%&" !")*# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

*(( --.1 +.- --.1 2-' --.1 -* --.1 -* --.1 )**-, )-.//, )1/(, )-.//, )2.+, )-.//, )., )-.//, )., )-.//, ˙ ˙ ˙ ˙ ˙ b˙ ˙ & B Ó ˙ Ó b˙ Ó b˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

''' ---/ /0- ---/ 012 ---/ !"#$%& )'/+, )--.1, )/*(, )--.1, )0-0, )--.1, œ œ ˙ #˙ ˙ #˙ ˙ #˙ & œ<µ>œ œ<#>œ<µ> B Ó B Ó <µ> Ó !$%&'($%&" !"+*# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

*/- ---/ +-( ---/ 21+ ---/ 1. ---/ 1. ---/ )*((, )--.1, )+.-, )--.1, )2-', )--.1, )-*, )--.1, )-*, )--.1, #˙ #˙ #˙ #˙ ˙ #˙ & <#>˙ Ó ˙ Ó <µ>˙ Ó ˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

!"#$%& -.22 --+- /'+ --+- 0** --+- )-..0, )--2(, )/0', )--2(, )0+., )--2(, #œ ˙ <˜>˙ #˙ <˜>˙ <˜>˙ & œ<˜>œ#œ˜œ Ó Ó ˜˙ Ó !$%&'($%&" ˜!"+*)*#œ˜ "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

(.+ --+- +1' --+- 2*0 --+- *1 --+- *1 --+- )*/', )--2(, )+2+, )--2(, )2++, )--2(, )1/, )--2(, )1/, )--2(, <˜>˙ <˜>˙ <˜>˙ <˜>˙ ˜˙ <˜>˙ & #˙ Ó <˜>˙ Ó ˜˙ Ó ˙ Ó Ó -- -2 -. -2 , -2 ˜! -2 -3 -2

Method: Appendix D - p.55 '()) ''-' /0- ''-' +11 ''-' !"#$%& *'((+, *'')., */+0, *'')., *+-(, *'')., ˙ ˙ ˙ ˙ œ#œ˜œ#œ <˜> #˙ <˜> ˜˙ <˜> & µœ˜œ<˜> Ó Ó Ó '()*+,()*" !b"#$%$& "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2 .(- ''-' -20 ''-' )1+ ''-' 12 ''-' 12 ''-' *1/0, *'')., *-)-, *'')., *)--, *'')., *2/, *'')., *2/, *'')., ˙ #˙ <˜>˙ <˜>˙ <˜>˙ <˜>˙ µ <˜>˙ & Ó <˜>˙ Ó ˜˙ Ó µ˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2 000 '''/ /+' '''/ +2) '''/ !"#$%& *0/-, *''(2, */1., *''(2, *+'+, *''(2, œb˙ n˙ ˙ ˙ ˙ œ œ<#>œ<µ>œB µ <µ>˙ & œ<µ> Ó Ó Ó '()*+,()*" !b"#$& "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2 1/' '''/ -'. '''/ )2- '''/ 2( '''/ 2( '''/ *1.., *''(2, *-(', *''(2, *)'0, *''(2, *'1, *''(2, *'1, *''(2, ˙ ˙ ˙ ˙ ˙ ˙ <#>˙ ˙ ˙ & Ó Ó <µ> Ó ˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2 0/- ''(2 /1. ''(2 +'+ ''(2 !"#$%& *0.0, *'(//, */-', *'(//, *+(), *'(//, ˙ ˙ ˙ ˙ bœbœ œBœbœµœB µ˙ b˙ & Ó Ó Ó '()*+,()*" !b"%$& "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2 1.. ''(2 -(' ''(2 )'0 ''(2 '1 ''(2 '1 ''(2 *11', *'(//, *2/., *'(//, *)(-, *'(//, *(, *'(//, *(, *'(//, ˙ ˙ ˙ ˙ b˙ ˙ B˙ ˙ b˙ & Ó Ó Ó b˙ Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2 0.0 '(// /-' '(// +() '(// !"#$%& *01-,*'(+2, */).,*'(+2, *./+, *'(+2, œ œB ˙ #˙ ˙ #˙ ˙ #˙ œ œ#œµœ B B & Ó Ó Ó '()*+,()*" !b"& "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2 11' '(// 2/. '(// )(- '(// ( '(// ( '(// *12., *'(+2, *2+', *'(+2, *'/0, *'(+2, *''/1, *'(+2, *''/1, *'(+2, #˙ #˙ #˙ #˙ ˙ #˙ µ˙ #˙ ˙ ˙ & Ó Ó Ó Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

Method: Appendix D - p.56 !"#$%& '(' )*'+ +,) )*'+ ()- )*'+ œ œB ˙ #˙ ˙ #˙ ˙ #˙ œ œ#œµœ B B & Ó Ó Ó &'()*+'()" !b"#$% "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

,.) )*'+ /'. )*'+ -)0 )*'+ )* )*'+ )* )*'+ #˙ #˙ #˙ #˙ ˙ #˙ µ˙ #˙ ˙ ˙ & Ó Ó Ó Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

!"#$%& '.' )*++ +0) )*++ (*- )*++ bœ œBb˙ n˙ ˙ ˙ b˙ ˙ bœ œ œB œ B B & Ó Ó Ó &'()*+'()" !b"# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

,,) )*++ /+. )*++ -*0 )*++ * )*++ * )*++ ˙ ˙ ˙ ˙ b˙ ˙ B ˙ ˙ ˙ b˙ & Ó Ó Ó Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

!"#$%& '.' )*++ +0) )*++ (*- )*++ œB ˙ #˙ ˙ #˙ #˙ œ#œ#œµœ œµ µ ˙ & Ó Ó Ó &'()*+'()" !b"% "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

,,) )*++ /+. )*++ -*0 )*++ * )*++ * )*++ #˙ #˙ #˙ #˙ ˙ #˙ µ˙ #˙ #˙ ˙ & Ó Ó Ó Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

'.' )*++ +0) )*++ (*- )*++ !"#$%& œBb˙ n˙ ˙ ˙ ˙ œ œ œB œ œB B ˙ & Ó Ó Ó &'()*+'()" !b", "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -0 -2 -/ -2

,,) )*++ /+. )*++ -*0 )*++ * )*++ * )*++ ˙ ˙ ˙ ˙ ˙ ˙ B ˙ ˙ ˙ ˙ & Ó Ó Ó Ó Ó -- -2 -. -2 , -2 ! -2 -3 -2

Method: Appendix D - p.57 SET 10! The 16th harmonic

Players rarely use the 16th harmonic on the double horn because the highest notes in normal ensemble and recital range can be played as 12th harmonics on the Bb side of the horn. Also players will often use a descant horn pitched in F or Eb to produce notes above high C, making those pitches playable as 12th harmonics. Although all players are familiar with the sound of the 16th harmonic, because it is an octave above the 8th harmonic, few will have practiced the interval between the 16th and 13th harmonics (16:13, the inversion of 13:8) or 16:11 (inversion of 11:8), both of which may be used in extended just intonation compositions. As Jon Catler points out in The Nature of Music (2003) an 8-note scale is completed with the 16th harmonic at the top of the sequence from the 8th harmonic. These eight notes would seem to be the 1st quasi-diatonic sequence from the harmonic series. The seven-note step-wise scale bounded by the 7th and 14th harmonics, is too unusual to sound diatonic to most ears, though does have an interesting collection of intervals.

The exercises on short tube lengths are included for completeness, omit any as needed. Retain a longer rest between intervals if desired.

Method: Appendix D - p.58 ()*+,- h!"!#$%!&' .'// ' 010 ' /&. ' œ bœn˙ ˙ b˙ ˙ ˙ ˙ œ œµœ œB B B B & œ Ó Ó Ó !$%&'($%&" !"# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 2'3 ' 44. ' 5/1 ' 3'& ' ' ' ˙ ˙ ˙ ˙ ˙ ˙ µ˙ ˙ & Ó Ó Ó ˙ Ó ˙ Ó -/ -3 -- -3 -. -3 , -3 ! -3

()*+,- .'// ' 010 ' /&. ' ˙ ˙ ˙ œ#œµœB œ#˙ B ˙ µ˙ & œ#œ#œµ Ó Ó Ó !$%&'($%&" !") "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3

2'3 ' 44. ' 5/1 ' 3'& ' ' ' ˙ ˙ ˙ ˙ ˙ #˙ ˙ #˙ & Ó µ Ó Ó #˙ Ó ˙ Ó -/ -3 -- -3 -. -3 , -3 ! -3

()*+,- .'// ' 010 ' /&. ' bœn˙ b˙ b˙ b˙ b˙ œ œ œBœB B B˙ & bœ œ B Ó Ó Ó !$%&'($%&" !"* "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3

2'3 ' 44. ' 5/1 ' 3'& ' ' ' b˙ b˙ b˙ b˙ b˙ ˙ ˙ ˙ & Ó B Ó Ó ˙ Ó b˙ Ó -/ -3 -- -3 -. -3 , -3 ! -3

()*+,- .'0/ .' 020 .' /4. .' 2.3 .' ˙ ˙ ˙ ˙ œ œµœBœ#˙ B˙ µ˙ ˙ & œ œ#œµ Ó Ó Ó Ó !$%&'($%&" !"*+) "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 -/ -3

41. .' 501 .' 3.& .' .' .' ˙ ˙ ˙ ˙ & µ˙ Ó #˙ Ó ˙ Ó ˙ Ó -- -3 -. -3 , -3 ! -3

Method: Appendix D - p.59 '()) ( /0/ ( )1' ( +(2 ( !"#$%& *'(+,-*'').- */.1- *'').- *)20- *'').- *0)+- *'').- ˙ ˙ ˙ ˙ œ œµœBœ#˙ B˙ µ˙ ˙ & œ œ#œµ Ó Ó Ó Ó !$%&'($%&" !"# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 -/ -3

..' ( ,)0 ( 2(1 ( ( ( *.,0- *'').- *,+'- *'').- *')/- *'').- *'').- *'').- ˙ ˙ ˙ ˙ ˙ & µ Ó #˙ Ó ˙ Ó ˙ Ó -- -3 -. -3 , -3 ! -3

''(, '. /)1 '. ).0 '. !"#$%& *'())- *(- */0/- *(- *)1'- *(- b˙ b˙ b˙ œbœBœBbœn˙ Bb˙ B˙ & bœbœ œB Ó Ó Ó !$%&'($%&" !")*# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3

+'+ '. .00 '. 1(' '. 2'/ '. '. '. *+(2- *(- *..'- *(- *,)0- *(- *2(1- *(- *(- *(- b˙ b˙ b˙ b˙ b˙ b˙ & Ó B˙ Ó ˙ Ó b˙ Ó b˙ Ó -/ -3 -- -3 -. -3 , -3 ! -3

!"#$%& '()) ,( /// ,( )+' ,( +,2 ,( *'(+,- *'.- */)1- *'.- *).0- *'.- *+'+- *'.- œBœ œ#˙ ˙ ˙ ˙ B˙ ˙ ˙ ˙ & œ<µ>œ œµœ<µ> Ó Ó Ó <µ> Ó !$%&'($%&" !"+*# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 -/ -3

.)' ,( 1'0 ,( 2,1 ,( ,( ,( *.00- *'.- *1('- *'.- *2'/- *'.- *'.- *'.- ˙ ˙ ˙ ˙ & <#>˙ Ó ˙ Ó <µ>˙ Ó ˙ Ó -- -3 -. -3 , -3 ! -3

''1' ., '(22 ., )/1 ., !"#$%& *''20- *,)- *'((+- *,)- *)+/- *,)- #œ œ<˜>˙ ˜˙ b˙ ˜˙ b˙ ˜˙ & œ<˜>œ#œ˜œ Ó Ó Ó !$%&'($%&" ˜!"+*)*#œ˜ "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 +.. ., 0(1 ., 1,/ ., 2.+ ., ., ., *+1(- *,)- *.)/- *,)- *121- *,)- *212- *,)- *,)- *,)- ˜˙ ˜˙ ˜˙ ˜˙ ˜˙ & ˜˙ Ó ˙ Ó ˙ Ó ˙ Ó Ó -/ -3 -- -3 -. -3 ˜, -3 ˜!˙ -3

Method: Appendix D - p.60 ''(' -. '0** -. /2( -. !"#$%& )''*+, )./, )'001, )./, )/12, )./, œ ˙ ˙ ˙ ˙ ˙ œ#œ˜œ#œ <˜> µ µ #˙ µ & µœ˜œ<˜> Ó Ó Ó '()*+,()*" !b"#$%$& "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 1-- -. +0( -. (.2 -. *-1 -. -. -. )1(0, )./, )-/2, )./, )(*(, )./, )*(*, )./, )./, )./, ˙ ˙ ˙ ˙ ˙ ˜˙ µ #˙ µ µ µ µ & Ó Ó <#>˙ Ó ˜˙ Ó µ˙ Ó -/ -3 -- -3 -. -3 , -3 ! -3

'0// .0 222 .0 /1' .0 !"#$%& )'01., )'-, )2/(, )'-, )/-+, )'-, œbœn˙ ˙ b˙ ˙ ˙ ˙ œ œµœ<µ>œB B & œ<µ> Ó Ó Ó '()*+,()*" !b"#$& "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 1.* .0 -/' .0 ('+ .0 *.( .0 .0 .0 )1'1, )'-, )-++, )'-, )(0', )'-, )*'2, )'-, )'-, )'-, ˙ ˙ ˙ ˙ ˙ <µ>˙ <#>˙ ˙ ˙ & Ó Ó Ó <µ> Ó ˙ Ó -/ -3 -- -3 -. -3 , -3 ! -3 ''0. '- 2/( '- /-+ '- !"#$%& )'0//, )0, )2+2, )0, )/(', )0, bœ bœn˙ b˙ b˙ b˙ b˙ b˙ bœ œBœbœB B B B & bœ Ó Ó Ó '()*+,()*" !b"%$& "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 1'1 '- -++ '- (0' '- *'2 '- '- '- )10*, )0, )--', )0, )./+, )0, )*0(, )0, )0, )0, b˙ b˙ b˙ b˙ b˙ b˙ B˙ ˙ b˙ b˙ & Ó Ó Ó Ó Ó -/ -3 -- -3 -. -3 , -3 ! -3 '0// 0 2+2 0 /(' 0 !"#$%& )'01., )''/-, )2-(, )''/-, )/*+, )''/-, œB œ#˙ ˙ B ˙ ˙ ˙ ˙ œ œ#œµœ œB B & Ó Ó Ó '()*+,()*" !b"& "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3 10* 0 --' 0 ./+ 0 *0( 0 0 0 )+/1, )''/-, )-.+, )''/-, ).1', )''/-, )'/2, )''/-, )''/-, )''/-, ˙ ˙ ˙ ˙ ˙ ˙ µ˙ #˙ ˙ ˙ & Ó Ó Ó Ó Ó -/ -3 -- -3 -. -3 , -3 ! -3

Method: Appendix D - p.61 !"#$%& '()* '( )+) '( *,' '( œ œB œ#˙ ˙ B ˙ ˙ ˙ ˙ œ œ#œµœ B B & Ó Ó Ó &'()*+'()" !b"#$% "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3

+'- '( ,.' '( /). '( -'0 '( '( '( ˙ ˙ ˙ ˙ ˙ ˙ µ˙ #˙ ˙ ˙ & Ó Ó Ó Ó Ó -/ -3 -- -3 -. -3 , -3 ! -3

!"#$%& '(** ( ).) ( *0' ( œBbœn˙ b˙ Bb˙ b˙ ˙ b˙ bœ œ œB œbœB B & Ó Ó Ó &'()*+'()" !b"# "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3

+(- ( ,,' ( /*. ( -(0 ( ( ( b˙ b˙ b˙ b˙ b˙ b˙ B ˙ ˙ ˙ b˙ & Ó Ó Ó Ó Ó -/ -3 -- -3 -. -3 , -3 ! -3

!"#$%& '(** ( ).) ( *0' ( œ œB œ#˙ ˙ B ˙ ˙ ˙ ˙ œ#œ#œµœ µ µ & Ó Ó Ó &'()*+'()" !b"% "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3

+(- ( ,,' ( /*. ( -(0 ( ( ( ˙ ˙ ˙ ˙ ˙ ˙ µ˙ #˙ #˙ ˙ & Ó Ó Ó Ó Ó -/ -3 -- -3 -. -3 , -3 ! -3

!"#$%& '(** ( ).) ( *0' ( œBbœn˙ ˙ Bb˙ ˙ ˙ ˙ œ œ œB œ œB B & Ó Ó Ó &'()*+'()" !b", "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3 -1 -3 -0 -3

+(- ( ,,' ( /*. ( -(0 ( ( ( ˙ ˙ ˙ ˙ ˙ ˙ B ˙ ˙ ˙ ˙ & Ó Ó Ó Ó Ó -/ -3 -- -3 -. -3 , -3 ! -3

Method: Appendix D - p.62 Section Two – Harmonic Series Slurs

Many authors of tutor books for the horn recommend practicing slurs between harmonics going as high as the 16th yet avoiding the 7th, 11th, 13th and even the

15th. They simply use standard arpeggio notes. Examples abound from Oscar

Franz’s (1881) to Philip Farkas (1956) and Barry Tuckwell (1983). Recently

Jeffrey Agrell drew attention to this and suggested slurring with all harmonics to allow the lip muscles to find their way around adjacent harmonics before learning to skip over particular ones (2008). Some unpublished warmups and teaching aids, such as warmups by Hector MacDonald (2005) and teaching aids produced by Dominic Harvey (1999) include the 7th harmonic. Additionally I have heard a number of professionals slurring through to the 8th harmonic with no omissions. Also any player working with the Australian Music Examinations

Board publications for Horn Grade Two and Three (Lawrance 1992) will have noticed that the slurring exercises use natural harmonics including the 7th and

Grade Four includes the 11th harmonic (Lawrance 1992).

The slurring exercises in this Section skip over many of the well known harmonics. Each set gradually prepares aural and physical awareness of the less used harmonics, the odd ones. The last exercise of Set 5 begins on the 11th harmonic, which is much easier to accomplish after practicing the early sets.

Method: Appendix D - p.63 SET 1!! Light Slurs

The following three exercises are useful in a warmup, played quasi glissando.

They also help firm up connections with harmonics from the 8th upwards. In particular the last exercise more aural and kinaesthetic familiarity with the 11th and 13th harmonics.

Method: Appendix D - p.64 #,,-.. j j & œ<˜>œ œ œ<˜>œ œ ‰ Œ œ<µ>œ œ œ œ œ œ œ œ ‰ Œ !'()*+'()"˜!"#$%$&œ˜ œ œ !"#$& "#$%&'()*+ ! / 01 ! / 01

j j & bœbœ œ œ œ œ œ œ œ ‰ Œ œ œ#œ œ œ œ œ œ œ ‰ Œ !"%$& !"& ! / 01 ! / 01

j œ œ & œ œ#œ œ œ œ œ œ œ ‰ Œ bœ œ œ œ œ œ œ ‰ Œ !"#$% !"# J ! / 01 ! / 01

#œ œ œ œ & œ#œ œ œ œ œ œ ‰ Œ µœ˜œ<˜> œ œ œ œ œ ‰ Œ !"% J J ,b"#$%$& ! / 01 ! / 01

œ œ œ œ œ œ œ œ œ œ<µ>œ œ œ œ œ œ œ œ & J ‰ Œ J ‰ Œ !"- ,b"#$& ! / 01 ! / 01

bœbœ œ œ œ œ œ œ œ œ œ#œ œ œ œ œ œ œ & J ‰ Œ J ‰ Œ ,b"%$& ,b"& ! / 01 ! / 01

œ œ#œ œ œ œ œ œ œ bœ œ œ œ œ œ œ œ œ & J ‰ Œ J ‰ Œ ,b"#$% ,b"# ! / 01 ! / 01

œ#œ#œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ & J ‰ Œ J ‰ Œ ,b"% ,b"- ! / 01 ! / 01

Method: Appendix D - p.65 j œ j & œ<˜>œ#œ˜œ œ œ œ ‰ Œ œ<µ>œ œ<#>œ<µ> œ œ œ œ ‰ Œ !'()*+'()" ˜!"#$%$&œ˜ œ !"#$& "#$%&'()*+ ! , -. -- -/ ! , -. -- -/

œbœ œ j œ œ œ j & bœbœ œB œ œ œ ‰ Œ œ œ#œ µ œ œ œ ‰ Œ !"%$& !"& ! , -. -- -/ ! , -. -- -/

œ œ œ j œ œ œ œ œ & œ œ#œ µ œ œ œ ‰ Œ bœ œ B œ œ ‰ Œ !"#$% !"# J ! , -. -- -/ ! , -. -- -/

œ#œ œ #œ˜œ œ & œ#œ#œµ œ œ œ ‰ Œ µœ˜œ<˜>œ œ œ œ ‰ Œ !"% J J ,b"#$%$& ! , -. -- -/ ! , -. -- -/

œ œ œ œ œ µœ œ œ œ œ œ œ œµœ<µ> œ œ œ œ & J ‰ Œ J ‰ Œ !"- ,b"#$& ! , -. -- -/ ! , -. -- -/

bœ œ bœbœ œ Bœ œ œ œ œ œ œ#œµœ œ œ œ œ & J ‰ Œ J ‰ Œ ,b"%$& ,b"& ! , -. -- -/ ! , -. -- -/

œ œbœ œ œ œ#œµœ œ œ œ œ bœ œ œB œ œ œ & J ‰ Œ J ‰ Œ ,b"#$% ,b"# ! , -. -- -/ ! , -. -- -/

œ œ œ œ œ#œ#œµœ œ œ œ œ œ œ œB œ œ œ & J ‰ Œ J ‰ Œ ,b"% ,b"- ! , -. -- -/ ! , -. -- -/

Method: Appendix D - p.66 #œ œ<˜>œ˜œ œ nœ œ & œ<˜>œ #œ ˜œ œ œ œ œ Œ !'()*+'()" ˜!"#$%$&œ˜ œ "#$%&'()*+ ! , -. -- -/ -0 -1 -2 -3

œ Bœ œ #œ œ˜œ nœ œ œ & œ<µ>œ œ<#>œ <µ> œ œ œ œ Œ !"#$& ! , -. -- -/ -0 -1 -2 -3

bœ œ bœ Bœ Bbœ nœ œBbœ œ œ œ & bœ bœ œ B œ œ œ Œ !"%$& ! , -. -- -/ -0 -1 -2 -3

œ œ œ µœ Bœ #œ œ Bœ œ œ œ & œ œ #œ µ œ œ œ Œ !"& ! , -. -- -/ -0 -1 -2 -3

œ œ œ µœ œ #œ œ nœ œ œ œ & œ œ #œ µ œ œ œ Œ !"#$% ! , -. -- -/ -0 -1 -2 -3

bœ œ œ Bœ Bbœ nœ œBbœ œ œ œ & bœ œ œ B œ œ œ Œ !"# ! , -. -- -/ -0 -1 -2 -3

œ #œ œ œ œ #œ œ #œ µœ B B œ œ œ œ & œ #œ µ œ œ Œ !"% ! , -. -- -/ -0 -1 -2 -3

œ œ œ #œ nœ œ #œ ˜œ #œ <˜> œ œ œ œ & µœ˜œ<˜> œ œ Œ ,b"#$%$& ! , -. -- -/ -0 -1 -2 -3

Method: Appendix D - p.67 bœ nœ œ œ bœ œ µœ œ Bœ B B œ œ œ œ & œ œ œ œ Œ !"# ! " #$ ## #% #& #' #( #)

bœ nœ œ œ bœ œ<#>œ <µ>œ Bœ œ œ œ œ & œ<µ>œ œ œ Œ $b"%&' ! " #$ ## #% #& #' #( #)

bœœ nœ bœ œœ œ bœ bœ œ Bœ BœB œ œ œ œ œ & Œ $b"(&' ! " #$ ## #% #& #' #( #)

œ œ Bœ #œ œ œ Bœ œ œ œ œ #œ µœ B œ œ œ œ & Œ $b"' ! " #$ ## #% #& #' #( #)

œ œ #œ œ œ nœ œ œ #œ µœ œ B œ œ œ œ & œ œ Œ $b"%&( ! " #$ ## #% #& #' #( #)

bœ œ Bbœ nœ bœ œBbœ œ œ bœ œ œ Bœ B œ œ œ œ & Œ $b"% ! " #$ ## #% #& #' #( #)

œ œ Bœ #œ œ œ nœ œ œ œ #œ #œ µœ µ œ œ œ œ & Œ $b"( ! " #$ ## #% #& #' #( #)

œ Bbœ nœ œ œBbœ œ œ œ œ Bœ œ B œ œ œ œ œ & Œ $b"# ! " #$ ## #% #& #' #( #)

Method: Appendix D - p.68 SET 2!! Harmonics 4 to 6 and 7

The exercises in this set have more extensive slurring. To assist the next sets begin with the stability of harmonics 4, 5, & 6 then skip the even numbers.

The notation of all sets here follows that used in Section One with no changes in cent values. All exercises assume that the valve slides are tuned relative to 12- tone equal temperament without specifying the placement of the 3rd valve slide.

That remains the choice of each player.

Vary the dynamics. Perhaps begin mp, do the next mf, then f, ff, pp, p, then start again.

Method: Appendix D - p.69 ? & œ ˜œ Ó & œ ˜œ œ œ ˜˙ !'()*+'()"˜!"#$%$&œ <˜> ˜œ "#$%&'()*+ ! , - . - , ! / 0

? & œ Ó & <µ>œ œ œ ˙ !"#$&œ œ œ œ ! , - . - , ! / 0

? bœ Ó & bœ Bbœ œ b˙ & b!"%$&œ œ œ œ ! , - . - , ! / 0

?œ & œ Bœ œ ˙ Ó & !"&œ #œ œ œ ! , - . - , ! / 0

?œ & œ œ œ ˙ Ó & !"#$%œ #œ œ œ ! , - . - , ! / 0

? œ & œ Bbœ œ b˙ Ó & b!"#œ œ œ œ ! , - . - , ! / 0

? #œ & #œ Bœ œ ˙ Ó & !"%œ #œ œ œ ! , - . - , ! / 0

? œ ˜œ Ó & & # œ ˜œ œ œ µ˙ µ,œb"#$%$&< > œ ! , - . - , ! / 0

Method: Appendix - p.70 bœ ? œ & œ œ B œ œ œ Ó & !"#œ œ œ ! " # $ # " ! % & '

?<µ>œ & œ <µ>œ bœ œ œ œ Ó & $œb"%&' œ œ ! " # $ # " ! % & '

?bœ & œ bœ œ œ œ bœ Ó & b$œb"(&' œ bœ ! " # $ # " ! % & '

œ ? œ & #œ œ B œ œ œ Ó & $œb"' œ œ ! " # $ # " ! % & '

œ ? œ & #œ œ œ œ œ Ó & $bœ"%&( œ ! " # $ # " ! % & œ'

bœ ?bœ & œ bœ B œ œ bœ Ó & b$œb"% œ bœ ! " # $ # " ! % & '

œ ? œ & #œ œ B œ œ œ Ó & $œb"( œ œ ! " # $ # " ! % & '

œ Bbœ œ ? œ & œ œ œ œ Ó $b"# œ œ ! " # $ # " ! % & '

Method: Appendix - p.71 Ó & œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ <˜>!"#$%$&œ ˜ œ œ œ œ œ ! " # " ! $%&

& œ œ œ œ œ Ó !"#$&œ <µ>œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ ! " # " ! $%&

Ó & bœœ œ œ œ œ œ œ œ œ œ œ œ œ œ !"%$&œ œ œ œ œ œ ! " # " ! $%&

& œ Bœ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó #!"&œ œ œ œ œ œ ! " # " ! $%&

& œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó #!"#$%œ œ œ œ œ œ ! " # " ! $%&

& Bbœ œ œ œ œ Ó !"#œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ ! " # " ! $%&

& Bœ œ œ œ œ Ó #!"%œ #œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ ! " # " ! $%&

& œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó <˜>'œb"#$%$&˜ œ œ œ œ #œ ! " # " ! $%&

Method: Appendix - p.72 & œ Bbœ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó !"#œ œ œ œ œ œ ! " # " ! $%&

& œ bœ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó $œb"%&'<µ> œ œ œ œ œ ! " # " ! $%&

bœ œ œ œ œ & œ bœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó $b"(&' ! " # " ! $%&

& #œ œ Bœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó $b"' ! " # " ! $%&

œ œ œ œ œ & #œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó $b"%&( ! " # " ! $%&

bœ œ œ œ œ & œ bœ B œ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó $b"% ! " # " ! $%&

œ œ œ œ œ & #œ œ B œ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó $b"( ! " # " ! $%&

bœ œ œ œ œ & œ œ B œ œ œ œ œ œ œ œ œ œ œ œ œ œ Ó $b"# ! " # " !

Method: Appendix - p.73 SET 3!!Harmonics 4 to 6, 7 and 9

? & œ œ Ó & œ œ ˜ œ œ ˜ ˜˙ !"#$%$&œ <˜>œ ˜ œ œ ˜ ! " # $ % $ # " ! & '

? & <µ>œ µœ Ó & <µ>œ œ œ œ ˙ !"#$&œ œ œ œ ! " # $ % $ # " ! & '

?bœ & bœ œ bœ œ œ b˙ Ó & b!"%$&œ œ œ œ ! " # $ % $ # " ! & '

œ ? œ Ó & œ Bœ œ œ ˙ & !"&œ #œ œ œ ! " # $ % $ # " ! & '

?œ & œ œ œ œ œ ˙ Ó & !"#$%œ #œ œ œ ! " # $ % $ # " ! & '

œ ?œ & œ Bbœ œ œ b˙ Ó & b!"#œ œ œ œ ! " # $ % $ # " ! & '

#œ ? #œ & #œ Bœ œ œ ˙ Ó & !"%œ #œ œ œ ! " # $ % $ # " ! & '

? œ ˜œ œ ˜œ Ó & & œ ˜œ œ œ µ˙ µ'œb"#$%$&<˜> œ ! " # $ % $ # " ! & '

Method: Appendix - p.74 œ ? œ & œ œ Bbœ œ œ œ œ Ó & !"#œ œ œ ! " # $ % $ # " ! & ' (

bœ œ œ ?<µ>œ & œ <µ>œ œ œ œ Ó & $œb"%&' œ œ ! " # $ % $ # " ! & ' (

bœ bœ œ ? bœ & œ bœ œ œ bœ Ó & $bbœ"(&' œ bœ ! " # $ % $ # " ! & ' (

œ œ œ ? œ & #œ œ B œ œ œ Ó & $œb"' œ œ ! " # $ % $ # " ! & ' (

œ ?œ & #œ œ œ œ œ œ œ Ó & $œb"%&( œ œ ! " # $ % $ # " ! & ' (

bœ œ œ ?bœ & œ bœ B œ œ bœ Ó & b$œb"% œ bœ ! " # $ % $ # " ! & ' (

œ #œ œ ? œ & #œ œ B œ œ œ Ó & $bœ"( œ œ ! " # $ % $ # " ! & ' (

œ œ Bbœ œ œ œ ? œ & œ œ œ œ Ó & $b"# œ ! " # $ % $ # " ! & ' (

Method: Appendix - p.75 & œ Ó <µ>œ Ó œ œ ˜ œ œ <µ>œ œ œ œ <˜>!"#$%$&œ ˜ œ !"#$&œ œ ! " # $ # " ! ! " # $ # " !

bœ œ & bœœ œ œ Ó œ Bœ œ œ Ó !"%$&œ œ #!"&œ œ ! " # $ # " ! ! " # $ # " !

œ & œ œ œ Ó œBbœ œ œ Ó #!"#$%œ œ œ œ !"#œ œ ! " # $ # " ! ! " # $ # " !

#œ œ & #œ Bœ œ œ Ó œ œ ˜ œ œ Ó #!"%œ œ <˜>'œb"#$%$&˜ œ ! " # $ # " ! ! " # $ # " !

œ <µ>œ & œBbœ œ œ Ó <µ>œ bœ œ œ Ó !"(œ œ 'bœ"#$& œ ! " # $ # " ! ! " # $ # " !

bœ bœ œ œ œ œ & œ bœ œ œ Ó #œ œ B œ œ Ó 'b"%$& 'b"& ! " # $ # " ! ! " # $ # " !

œ bœ œ œ & #œ œ œ œ œ œ Ó œ bœB œ œ Ó 'b"#$% 'b"# ! " # $ # " ! ! " # $ # " !

#œ bœ œ œ & #œ œ Bœ œ œ œ Ó œ œB œ œ Ó 'b"% 'b"( ! " # $ # " ! ! " # $ # " !

Method: Appendix - p.76 & œ ˜œ œ œ œ œ œ œ œ œ œ œ Œ ˜!"#$%$&œ œ œ œ œ ! " # " ! $%& ˜œ'

& œ <µ>œ œ œ œ œ œ œ œ œ œ œ Œ <µ>!"#$&œ œ œ œ œ ! " # " ! $%& <µ>œ'

& bœ bœ œ œ œ œ œ œ œ œ œ œ Œ b!"%$&œ œ œ œ œ ! " # " ! $%& bœ'

& œ œ œ œ œ œ œ œ œ œ œ œ Œ !"&œ B œ œ œ œ ! " # " ! $%& 'œ

& œ œ œ œ œ œ œ œ œ œ œ œ Œ !"#$%œ œ œ œ œ ! " # " ! $%& 'œ

& bœ œ œ œ œ œ œ œ œ œ œ œ Œ !"#œ B œ œ œ œ ! " # " ! $%& œ'

& œ #œ œ œ œ œ œ œ œ œ œ œ Œ #!"%œ B œ œ œ œ ! " # " ! $%& #œ'

œ œ œ œ & œ œ ˜ œ œ œ œ œ œ œ œ œ œ œ Œ ˜'b"#$%$& ! " # " ! $%& ˜œ'

Method: Appendix - p.77 œ œ œ œ & œ Bbœ œ œ œ œ œ œ œ œ œ œ œ Œ !"# ! " # " ! $%& œ'

& œ bœ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ Œ $b"%&' ! " # " ! $%& œ'

bœ œ œ œ & bœ œ œ œ œ œ œ œ œ œ œ œ œ Œ $b"(&' ! " # " ! $%& bœ'

œ œ œ œ & œ Bœ œ œ œ œ œ œ œ œ œ œ œ Œ $b"' ! " # " ! $%& 'œ

œ œ œ œ & œ œ œ œ œ œ œ œ œ œ œ œ œ Œ $b"%&( ! " # " ! $%& 'œ

bœ œ œ œ œ œ œ œ œ œ œ œ & bœ B œ œ œ œ Œ $b"% ! " # " ! $%& b'œ

œ #œ œ œ œ œ œ œ œ œ œ œ & œ B œ œ œ œ Œ $b"( ! " # " ! $%& œ'

bœ œ œ œ œ œ œ œ œ œ œ œ & œ B œ œ œ œ Œ $b"# œ ! " # " ! $%& '

Method: Appendix - p.78 SET 4!!Harmonics 4 to 6, 7, 9 and 11

? & ˜œ #œ œ ˜œ Ó & œ ˜œ œ œ œ œ ˜œ !'()*+'()"˜!"#$%$&œ<˜> œ "#$%&'()*+ ! , - . / 00 / . - , ! 1 2

? & <µ>œ <#>œ œ <µ>œ Ó & <µ>œ œ œ œ œ !"#$&œ œ œ œ ! , - . / 00 / . - , ! 1 2

Bœ ? bœ & bœœ bœ œ œ œ bœ Ó & b!"%$&œ œ œ œ ! , - . / 00 / . - , ! 1 2

œ µœ œ ? œ Ó & œ Bœ œ œ œ & !"&œ #œ œ œ ! , - . / 00 / . - , ! 1 2

œ µœ œ ? œ Ó & œ œ œ œ œ & !"#$%œ #œ œ œ ! , - . / 00 / . - , ! 1 2

Bœ ? œ & œ Bbœ œ œ œ œ bœ Ó & b!"#œ œ œ œ ! , - . / 00 / . - , ! 1 2

#œ µœ #œ ? #œ & #œ Bœ Bœ #œ œ Ó & !"%œ #œ #œ œ ! , - . / 00 / . - , ! 1 2

#œ ? ˜œ œ ˜œ Ó & & œ ˜œ œ œ œ œ µœ ,bµ"#$%$&œ<˜> œ ! , - . / 00 / . - , ! 1 2

Method: Appendix - p.79 œ bœ œ µ œ œ ? œ & œ œ B œ œ œ Œ & !"#œ œ œ ! " # $ % && % $ # " ! ' ( &

# œ bœ <µ>œ < > œ œ ?<µ>œ Œ & œ <µ>œ œ œ œ & $œb"%&' œ œ ! " # $ % && % $ # " ! ' ( &

bœ bœ Bœ œ œ ? & œ bœ œ œ bœ Œ & $bbœ"(&' œ bœ ! " # $ % && % $ # " ! bœ' ( &

œ œ µœ œ œ ? & #œ œ B œ œ œ Œ & $œb"' œ œ ! " # $ % && % $ # " ! œ' ( &

œ œ µœ œ œ ? & #œ œ B œ œ œ Œ & $œb"%&( œ œ ! " # $ % && % $ # " ! œ' ( &

œ bœ œ B œ œ ? & œ bœ B œ œ bœ Œ & b$œb"% œ bœ ! " # $ % && % $ # " ! bœ' ( &

œ œ #œ µ œ œ ? & #œ œ B œ œ œ Œ & $œb"( œ œ ! " # $ % && % $ # " ! œ' ( &

œ Bœ œ œ Bbœ œ œ ? œ & œ œ œ œ Œ & $b"# œ œ ! " # $ % && % $ # " ! ' ( &

Method: Appendix - p.80 & œ#œ œ Ó <µ>œ<#>œ œ Ó œ œ˜ œ œ <µ>œ œ œ œ !'()*+'()"<˜>!"#$%$&œ˜ œ !"#$&œ œ "#$%&'()*+ ! , - . // . - , ! ! , - . // . - , !

bœ Bœ œ Ó œ µœ œ Ó & bœœ œ œ œ Bœ œ œ !"%$&œ œ #!"&œ œ ! , - . // . - , ! ! , - . // . - , !

œ µœ œ Ó œ Bœ œ & œ Bœ œ œ œBbœ œ œ Ó #!"#$%œ œ !"#œ œ ! , - . // . - , ! ! , - . // . - , !

#œ µœ œ Ó ˜œ #œ œ Ó & #œ#œ Bœ œ œ œ œ˜œ œ œ œ œ !"% <˜,>b"#$%$& ! , - . // . - , ! ! , - . // . - , !

µœ <#>œ & œBbœ œ œ œ œ Ó œ bœ<µ>œ œ œ œ Ó !"-œ œ ,œb"#$&<µ> œ ! , - . // . - , ! ! , - . // . - , !

œ œ bœ bœ B œ œ œ œ µ œ œ & œ bœ œ œ Ó #œ œ B œ œ Ó ,b"%$& ,b"& ! , - . // . - , ! ! , - . // . - , !

œ œ œ œ µ œ œ bœ œB œ bœ & #œ œ B œ œ Ó œ bœB B bœ œ Ó ,b"#$% ,b"# ! , - . // . - , ! ! , - . // . - , !

œ œ#œ µœ #œ œ bœ œB œ bœ & #œ œ B B œ#œ Ó œ œB B œ œ Ó ,b"% ,b"- ! , - . // . - , ! ! , - . // . - , !

Method: Appendix - p.81 #œ # œ & œ ˜œ œ œ Œ œ<µ>œ< > œ œ Œ ˜!"#$%$&œ œ <µ>!"#$&œ œ ! " # $$ # " ! ! " # $$ # " ! ˜œ% <µ>œ%

& bœ bœ Bœ œ œ Œ œ œ µœ œ œ Œ !"%$&bœ œ !"&œ B œ ! " # $$ # " ! bœ% ! " # $$ # " ! œ%

œ Bœ & œ œ µ œ œ Œ Bbœ œ œBbœ Œ !"#$%œ B œ !"#œ œ ! " # $$ # " ! œ% ! " # $$ # " ! œ%

#œ µœ œ œ #œ œ & #œ Bœ œ œ Œ œ œ ˜ œ œ Œ !"% ˜'b"#$%$& ! " # $$ # " ! #œ% ! " # $$ # " ! ˜œ%

µœ <#>œ & œBbœ œ œ œ œ Œ œ bœ<µ>œ œ œ œ Œ !"( <'µ>b"#$& ! " # $$ # " ! œ% ! " # $$ # " ! <µ>œ%

bœ Bœ œ œ µœ œ & bœœ œ œ Œ œ Bœ œ œ Œ 'b"%$& 'b"& ! " # $$ # " ! bœ% ! " # $$ # " ! œ%

œ µœ œ bœ œ Bœ œ œ & œ œ œ œ Œ bœB œ Œ 'b"#$% 'b"# ! " # $$ # " ! œ% ! " # $$ # " ! bœ%

œ œ #œ µœ œ œ bœ œ B œ œ & œ B œ Œ œB œ Œ 'b"% 'b"( œ ! " # $$ # " ! œ% ! " # $$ # " ! %

Method: Appendix - p.82 & œ#œ œ œ œ œ œœœ Œ <µ>œ<#>œ œ œ œ œ œœœ Œ !"#$%$&œ˜ œ œ œ !"#$&œ œ œ œ ! " ## " ! $%& ! " ## " ! $%&

œ œ œ œ œ œ & bœ bœ B œ œ œ œ œœ œœ Œ œ œ µ œ œ œ œ œœ œœ Œ B!"& ! " ## " ! $%& ! " ## " ! $%&

œ œ œ œ œ œ & œ œ µ œ œ œ œ œœ œœ Œ Bbœ œ B œ œ œ œ œœ œœ Œ !"#$% !"# ! " ## " ! $%& ! " ## " ! $%&

µœ œ œ #œ œ œ & Bœ#œ œ œ œ œ œœ œœ Œ œ˜œ œ œ œ œ œœ œœ Œ !"% 'b"#$%$& ! " ## " ! $%& ! " ## " ! $%&

œ µœ œ œ œ œ œœœ œ µœ œ œ œ œ œœœ & Bbœ œ œ œ Œ Bbœ œ œ œ Œ !"( 'b"#$& ! " ## " ! $%& ! " ## " ! $%&

bœ Bœ œ œ œ œ œœœ œ µœ œ œ œ œ œœœ & Bbœ œ œ œ Œ Bœ œ œ œ Œ 'b"%$& 'b"& ! " ## " ! $%& ! " ## " ! $%&

œ µœ œ œ œ œ œœœ bœ œBœ œ œ œ œ œ œœœœœ & œ œ œ œ Œ B Œ 'b"#$% 'b"# ! " ## " ! $%& ! " ## " ! $%&

œ œ œ œ#œ µœ œ œ œ œ œ œœœœœ bœ œB œ œ œ œ œœ œœ & B Œ B Œ 'b"% 'b"( ! " ## " ! $%& ! " ## " ! $%&

Method: Appendix - p.83 SET 5!!Harmonics 4 to 6, 7, 9, 11 and 13

œ ? œ<˜>œ ˜ #œ ? œ ˜ & ˜œ #œ œ œ ˜œ Ó !'()*+'()" !"#$%$& œ œ œ ˜œ "#$%&'()*+ ! , - . / 00 01 00 / . ˜ - ˜œ, ! 1 2

<µ>œ ? œ œ Bœ ? œ & <µ>œ<#>œ œ œ <µ>œ Ó !"#$& œ œ œ œ ! , - . / 00 01 00 / . - œ, ! 1 2

œ ? bœ œ Bœ œ ? œ & bœ bœ B œ œ bœ bœ Ó !"%$& bœ œ œ ! , - . / 00 01 00 / . - , ! 1 2

#œ ? œ œ µœ œ ? œ œ & œ œ µ œ œ œ Ó !"& œ B œ œ ! , - . / 00 01 00 / . - , ! 1 2

#œ ? œ œ µœ œ ? œ œ & œ œ µ œ œ œ Ó & !"#$% œ œ œ ! , - . / 00 01 00 / . - , ! 1 2

œ œ œ Bœ B œ œ ? œ & œ Bbœ œ œ bœ Ó & b!"#œ œ œ ! , - . / 00 01 00 / . - , ! 1 2

#œ µœ µœ œ œ ? œ #œ & #œ Bœ œ œ œ Ó & !"%œ #œ œ ! , - . / 00 01 00 / . - , ! 1 2

#œ #œ œ ? œ œ ˜œ œ œ ˜œ Ó & & œ ˜œ œ œ µœ µ,œb<"#$%$&˜> ! , - . / 00 01 00 / . - , ! 1 2

!"#$%&'()**"+&,-(.(/(*012 œ Bœ œ œ bœ œ µ œ œ ? œ & œ œB œ œ œ Œ & !"#œ ! " # $ % && &' && % $ # " ! ' ( œ&

# œ Bœ œ œ bœ œ< > œ œ ? <µ>œ Œ & œ <µ>œ œ œ œ & $bœ"%&' ! " # $ % && &' && % $ # " ! ' ( œ&

Bbœ bœ bœ Bœ Bœ bœ œ ? bœ & œ bœ œ œ bœ Œ & b$œb"(&' œ bœ ! " # $ % && &' && % $ # " ! ' ( &

Bœ œ œ µœ œ œ œ ? œ & #œ œ B œ œ œ Œ & $œb"' œ œ ! " # $ % && &' && % $ # " ! ' ( &

Bœ œ œ µœ œ œ œ ?œ & #œ œ œ œ œ Œ & $œb"%&( œ œ ! " # $ % && &' && % $ # " ! ' ( &

œ Bœ œ bœ œ B œ œ ?bœ & œ bœB œ œ bœ Œ & $bbœ"% œ bœ ! " # $ % && &' && % $ # " ! ' ( &

œ µœ œ œ #œ µ œ œ ? œ & #œ œ B œ œ œ Œ & $œb"( œ œ ! " # $ % && &' && % $ # " ! ' ( &

œ Bœ œ œ bœ œ B œ œ ? œ & œ œB œ œ Œ $bœ"# œ œ ! " # $ % && &' && % $ # " ! ' ( &

!"#$%&'()**"+&,-(.(/(*012 #œ & œ #œ œ œ Œ œ œ ˜ œ œ <˜>!"#$%$&œ ˜ œ ! " # $ %% %& %% $ # " !

<#>œ Bœ œ Œ & œ œ <µ>œ œ œ œ !"#$&œ <µ> œ ! " # $ %% %& %% $ # " !

œ bœ Bœ B œ œ Œ & bœ Bbœ œ œ !"%$&œ œ ! " # $ %% %& %% $ # " !

œ œ µœ µ œ œ Œ & œ Bœ œ œ #!"&œ œ ! " # $ %% %& %% $ # " !

œ œ µœ µ œ œ Œ & œ Bœ œ œ #!"#$%œ œ ! " # $ %% %& %% $ # " !

Bœ Bœ œ & œ œ œ œ Œ !"#œ œ œ œ ! " # $ %% %& %% $ # " !

œ bœ µœ µ œ œ & #œ Bœ œ œ Œ #!"%œ œ ! " # $ %% %& %% $ # " !

#œ œ #œ œ œ & œ œ ˜ œ œ Œ <˜>'œb"#$%$&˜ œ ! " # $ %% %& %% $ # " !

!"#$%&'()**"+&,-(.(/(*012 œ œ µœ B œ œ & œ Bbœ œ œ Œ !"#œ œ ! " # $ %% %& %% $ # " !

<#>œ Bœ œ & œ bœ œ œ œ œ Œ $œb"%&' œ ! " # $ %% %& %% $ # " !

Bbœ bœ bœ Bœ œ œ œ & œ bœ œ œ Œ $b"(&' ! " # $ %% %& %% $ # " !

Bœ œ œ µœ œ œ œ & #œ œ B œ œ Œ $b"' ! " # $ %% %& %% $ # " !

Bœ œ œ µœ œ œ œ & #œ œ œ œ Œ $b"%&( ! " # $ %% %& %% $ # " !

œ œ Bœ B œ œ & œ bœ Bbœ œ œ œ Œ $b"% ! " # $ %% %& %% $ # " !

µœ œ #œ µœ œ œ œ & #œ œ B œ œ Œ $b"( ! " # $ %% %& %% $ # " !

œ Bœ œ bœ œ B œ œ & œ œ B œ œ Œ $b"# ! " # $ %% %& %% $ # " !

!"#$%&'()**"+&,-(.(/(*012 #œ #œ œ & œ ˜œ œ œ Ó ˜!"#$%$&œ œ ! " # $$ $% $$ # " ! ˜œ%

# œ Bœ œ & œ œ < > œ œ Ó <µ>!"#$&œ œ ! " # $$ $% $$ # " ! <µ>œ%

Bœ & bœ bœ Bœ œ œ œ Ó !"%$&bœ œ ! " # $$ $% $$ # " ! bœ%

µœ & œ œ µœ œ œ œ Ó !"&œ B œ ! " # $$ $% $$ # " ! œ%

µœ & œ œ µœ œ œ œ Ó !"#$%œ œ ! " # $$ $% $$ # " ! œ%

Bœ Bœ œ & Bbœ œ œ œ Ó !"#œ œ ! " # $$ $% $$ # " ! œ%

œ µœ œ & Bœ #œ µ œ œ Ó #!"%œ œ ! " # $$ $% $$ # " ! #œ%

#œ œ #œ œ œ & œ œ ˜ œ œ Ó '˜b"#$%$& ! " # $$ $% $$ # " ! ˜œ%

!"#$%&'()**"+&,-(.(/(*011 œ œ µœ B œ œ & œ Bbœ œ œ Ó !"# ! " # $$ $% $$ # " ! œ%

<#>œ Bœ œ & œ bœ <µ>œ œ œ œ Ó <$µ>b"%&' ! " # $$ $% $$ # " ! <µ>œ%

bœ bœ Bœ B œ œ & bœ œ œ œ Ó $b"(&' ! " # $$ $% $$ # " ! bœ%

œ œ µœ B œ œ & œ Bœ œ œ Ó $b"' ! " # $$ $% $$ # " ! œ%

œ œ µœ B œ œ & œ œ œ œ Ó $b"%&( ! " # $$ $% $$ # " ! œ%

œ œ Bœ B œ œ & bœ Bbœ œ œ Ó $b"% ! " # $$ $% $$ # " ! bœ%

µœ œ #œ µœ œ œ œ & œ B œ Ó $b"( ! " # $$ $% $$ # " ! œ%

œ Bœ œ bœ œ B œ œ & œ B œ Ó $b"# œ ! " # $$ $% $$ # " ! !

!"#$%&'()**"+&,-(.(/(*012 #œ Bœ & œ #œ œ œ Œ <µ>œ<#>œ œ œ Œ #!"#$%$&œ ˜ œ !"#$&œ œ ! " ## #$ ## " ! ! " ## #$ ## " !

Bœ µœ & bœ bœ Bœ œ œ œ Œ œ œ µœ œ œ œ Œ B!"& ! " ## #$ ## " ! ! " ## #$ ## " !

œ µœ œ Bœ Bœ œ & œ œ µ œ œ Œ œ œ œ œ Œ !"#$% !"# ! " ## #$ ## " ! ! " ## #$ ## " !

µœ µœ œ µœ µœ œ & Bœ #œ œ œ Œ Bœ #œ œ œ Œ !"% 'b"#$%$& ! " ## #$ ## " ! ! " ## #$ ## " !

µœ Bœ œ µœ Bœ œ & Bbœ œ œ œ Œ bœ œ œ œ Œ !"( 'b"#$& ! " ## #$ ## " ! ! " ## #$ ## " !

bœ œ bœ BœB œ œ œ µœ B œ œ & œ œ Œ Bœ œ Œ 'b"%$& 'b"& ! " ## #$ ## " ! ! " ## #$ ## " !

œ Bœ œ µœ B œ œ bœ œ Bœ œ œ œ & Bœ œ Œ B Œ 'b"#$% 'b"# ! " ## #$ ## " ! ! " ## #$ ## " !

µœ œ Bœ œ œ #œ µœ œ œ œ bœ œ B œ œ & B Œ B Œ 'b"% 'b"( ! " ## #$ ## " ! ! " ## #$ ## " !

!"#$%&'()**"+&,-(.(/(*012 #œ œ Bœ œ & œ #œ œ œ œ œ œ Œ <µ>œ<#>œ œ œ œ œ œ Œ ˜!"#$%$& !"#$& ! "" "# "" ! $%& ! "" "# "" ! $%&

Bœ œ µœ œ & bœ Bœ œ œ œ œ œ Œ œ µœ µœ œ œ œ œ Œ !"%$& !"& ! "" "# "" ! $%& ! "" "# "" ! $%&

µœ œ œ Bœ œ œ œ œ & œ µœ µœ œ œ œ œ Œ œ B œ œ Œ !"#$% !"# ! "" "# "" ! $%& ! "" "# "" ! $%&

œ µœ œ œ œ œ #œ #œ œ œ œ œ & #œ µ œ œ Œ ˜œ œ œ Œ !"% 'b"#$%$& ! "" "# "" ! $%& ! "" "# "" ! $%&

µœ Bœ œ œ œ œ <#>œ Bœ œ œ œ œ & œ œ œ Œ <µ>œ œ œ Œ !"( 'b"#$& ! "" "# "" ! $%& ! "" "# "" ! $%&

bœ œ œ œ bœ BœB œ œ œ œ œ œ µœ B œ œ œ œ œ & Œ Œ 'b"%$& 'b"& ! "" "# "" ! $%& ! "" "# "" ! $%&

œ œ œ œ œ µœ B µœ œ œ œ œ œ Bœ B œ œ œ œ œ & Œ Œ 'b"#$% 'b"# ! "" "# "" ! $%& ! "" "# "" ! $%&

œ œ œ œ #œ µœ µ œ œ œ œ œ œ Bœ B œ œ œ œ œ & Œ Œ 'b"% 'b"( ! "" "# "" ! $%& ! "" "# "" ! $%&

!"#$%&'()**"+&,-(.(/(*012 SET 6!!Harmonics 4 to 6, 7, 9, 11, 13 and 15

œ ? œ<˜>œ˜ #œ œ œ ? œ ˜ & œ #œ <˜> œ œ œ Ó !'()*+'()" !"#$%$& œ˜ œ ˜ œ "#$%&'()*+ ! , - . / 00 01 0, 01 00 / . œ- œ, ! 1 ˜2

œ ? œ œ<µ> œ #œ œ ?œ & œ<#>œ B œ œ <µ>œ Ó !"#$& œ<µ> œ œ œ ! , - . / 00 01 0, 01 00 / . - œ, ! 1 2

bœ œ œ œ œ ? œ ? & bœ Bœ B œ œ bœ Ó !"%$& bœœ œ œ bœ ! , - . / 00 01 0, 01 00 / . - œ, ! 1 2

œ #œ œ #œ œ ? œ ? & œ µœ µ œ œ œ Ó !"& œ Bœ œ œ œ ! , - . / 00 01 0, 01 00 / . - œ, ! 1 2

œ #œ œ #œ œ ? œ ? & œ µœ µ œ œ œ Ó & !"#$% œ œ œ œ œ ! , - . / 00 01 0, 01 00 / . - œ, ! 1 2

œ œ œ œ œ Bœ B œ œ ? œ & œBbœ œ œ bœ Ó & b!"#œ œ œ ! , - . / 00 01 0, 01 00 / . - , ! 1 2

œ #œ œ œ #œ µœ µ œ œ ? #œ & #œ Bœ œ œ œ Ó & !"%œ #œ œ ! , - . / 00 01 0, 01 00 / . - , ! 1 2

#œ #œ<˜>œ œ œ ? œ œ˜œ œ œ ˜œ Ó & & œ˜œ œ œ µœ µ,œb<"#$%$&˜> ! , - . / 00 01 0, 01 00 / . - , ! 1 2

Method: Appendix D - p.92 œ Bœ œ œ œ œ bœ œ µ œ œ ? œ & œ œB œ œ œ Œ & !"#œ ! " # $ % && &' &" &' && % $ # " ! ' ( œ&

# œ Bœ œ œ œ œ bœ œ< > œ œ ? <µ>œ Œ & œ<µ>œ œ œ œ & $bœ"%&' ! " # $ % && &' &" &' && % $ # " ! ' ( œ&

Bbœ œ œ bœ bœ Bœ Bœ bœ œ ?bœ & œ bœ œ œ bœ Œ & b$œb"(&' œ bœ ! " # $ % && &' &" &' && % $ # " ! ' ( &

Bœ #œ œ œ œ µœ œ œ œ ? œ & #œ œ B œ œ œ Œ & $œb"' œ œ ! " # $ % && &' &" &' && % $ # " ! ' ( &

Bœ #œ œ œ œ µœ œ œ œ ? œ & #œ œ œ œ œ Œ & $œb"%&( œ œ ! " # $ % && &' &" &' && % $ # " ! ' ( &

œ Bœ œ œ œ bœ œ B œ œ ?bœ & œ bœB œ œ bœ Œ & $bbœ"% œ bœ ! " # $ % && &' &" &' && % $ # " ! ' ( &

œ µœ #œ œ œ œ #œ µ œ œ ? œ & #œ œ B œ œ œ Œ & $œb"( œ œ ! " # $ % && &' &" &' && % $ # " ! ' ( &

œ œ Bœ œ œ œ bœ œ B œ œ ? œ & œ œB œ œ Œ & $bœ"# œ œ ! " # $ % && &' &" &' && % $ # " ! ' ( &

Method: Appendix D - p.93 #œ œ œ & œ #œ <˜> œ œ Œ œ œ ˜ œ œ <˜>!"#$%$&œ ˜ œ ! " # $ %% %& %! %& %% $ # " !

#œ <#>œ Bœ œ œ Œ & œ œ <µ>œ œ œ œ !"#$&œ <µ> œ ! " # $ %% %& %! %& %% $ # " !

œ Bœ œ œ œ & Bbœ bœ B œ œ Œ !"%$&œ bœ œ œ ! " # $ %% %& %! %& %% $ # " !

œ µœ #œ œ œ & Bœ œ µ œ œ Œ #!"&œ œ œ œ ! " # $ %% %& %! %& %% $ # " !

µœ #œ œ & œ œ µœ œ œ œ Œ #!"#$%œ œ œ œ ! " # $ %% %& %! %& %% $ # " !

œ œ œ œ Bœ B œ œ & œ œ œ œ Œ !"#œ œ ! " # $ %% %& %! %& %% $ # " !

œ #œ œ bœ µœ µ œ œ & #œ Bœ œ œ Œ #!"%œ œ ! " # $ %% %& %! %& %% $ # " !

#œ œ œ œ #œ <˜> œ œ & œ œ ˜ œ œ Œ <˜>'œb"#$%$&˜ œ ! " # $ %% %& %! %& %% $ # " !

Method: Appendix D - p.94 œ œ œ œ µœ B œ œ & œ Bbœ œ œ Œ !"#œ œ ! " # $ %% %& %! %& %% $ # " !

œ <#>œ Bœ œ œ & œ bœ œ œ œ œ Œ $œb"%&' œ ! " # $ %% %& %! %& %% $ # " !

Bbœ œ œ bœ bœ Bœ œ œ œ & œ bœ œ œ Œ $b"(&' ! " # $ %% %& %! %& %% $ # " !

Bœ #œ œ œ œ µœ œ œ œ & #œ œ B œ œ Œ $b"' ! " # $ %% %& %! %& %% $ # " !

Bœ #œ œ œ œ µœ œ œ œ & #œ œ œ œ Œ $b"%&( ! " # $ %% %& %! %& %% $ # " !

œ œ œ œ Bœ B œ œ & œ bœ Bbœ œ œ œ Œ $b"% ! " # $ %% %& %! %& %% $ # " !

µœ #œ œ œ #œ µœ œ œ œ & #œ œ B œ œ Œ $b"( ! " # $ %% %& %! %& %% $ # " !

œ Bœ œ œ œ bœ œ B œ œ & œ œ B œ œ Œ $b"# ! " # $ %% %& %! %& %% $ # " !

Method: Appendix D - p.95 #œ #œ <˜>œ œ œ & œ ˜œ œ œ Ó ˜!"#$%$&œ œ ! " # $$ $% $& $% $$ # " ! ˜œ%

# œ Bœ #œ œ œ & œ œ < > œ œ Ó <µ>!"#$&œ œ ! " # $$ $% $& $% $$ # " ! <µ>œ%

œ œ œ bœ Bœ B œ œ Ó & !"%$&bœ œ œ œ ! " # $$ $% $& $% $$ # " ! bœ%

µœ #œ œ & œ œ µœ œ œ œ Ó !"&œ B œ ! " # $$ $% $& $% $$ # " ! œ%

µœ #œ œ & œ œ µœ œ œ œ Ó !"#$%œ œ ! " # $$ $% $& $% $$ # " ! œ%

œ Bœ Bœ œ œ & Bbœ œ œ œ Ó !"#œ œ ! " # $$ $% $& $% $$ # " ! œ%

œ µœ #œ œ œ & Bœ #œ µ œ œ Ó #!"%œ œ ! " # $$ $% $& $% $$ # " ! #œ%

#œ œ œ œ #œ <˜> œ œ & œ œ ˜ œ œ Ó '˜b"#$%$& ! " # $$ $% $& $% $$ # " ! ˜œ%

Method: Appendix D - p.96 œ œ œ œ µœ B œ œ & œ Bbœ œ œ Ó !"# ! " # $$ $% $& $% $$ # " ! œ%

œ <#>œ Bœ œ œ & œ bœ <µ>œ œ œ œ Ó <$µ>b"%&' ! " # $$ $% $& $% $$ # " ! <µ>œ%

bœ œ œ bœ Bœ B œ œ & bœ œ œ œ Ó $b"(&' ! " # $$ $% $& $% $$ # " ! bœ%

œ #œ œ œ µœ B œ œ & œ Bœ œ œ Ó $b"' ! " # $$ $% $& $% $$ # " ! œ%

œ #œ œ œ µœ B œ œ & œ œ œ œ Ó $b"%&( ! " # $$ $% $& $% $$ # " ! œ%

œ œ œ œ Bœ B œ œ & bœ Bbœ œ œ Ó $b"% ! " # $$ $% $& $% $$ # " ! bœ%

µœ #œ œ œ #œ µœ œ œ œ & œ B œ Ó $b"( ! " # $$ $% $& $% $$ # " ! œ%

œ Bœ œ œ œ bœ œ B œ œ & œ B œ Ó $b"# œ ! " # $$ $% $& $% $$ # " ! %

Method: Appendix D - p.97 #œ<˜>œ œ Bœ #œ œ & œ #œ œ œ Œ <µ>œ<#>œ œ œ Œ #!"#$%$&œ˜ œ !"#$&œ œ ! " ## #$ #% #$ ## " ! ! " ## #$ #% #$ ## " !

œ Bœ œ œ œ œ µœ #œ œ œ & bœ bœ B œ œ Œ œ œ µ œ œ Œ B!"& ! " ## #$ #% #$ ## " ! ! " ## #$ #% #$ ## " !

œ œ µœ #œ œ œ œ Bœ œ œ & œ œ µ œ œ Œ bœ œ B œ œ Œ !"#$% ! " ## #$ #% #$ ## " ! ! " ## #$ #% #$ ## " !

#œ œ µœ µœ œ œ #œ #œ<˜> œ œ & Bœ #œ œ œ Œ œ˜œ œ œ Œ !"% 'b"#$%$& ! " ## #$ #% #$ ## " ! ! " ## #$ #% #$ ## " !

œ œ œ œ œ œ œ µœ B œ œ œ<#>œ B œ œ & Bbœ œ Œ bœ µ œ Œ !"( 'b"#$& ! " ## #$ #% #$ ## " ! ! " ## #$ #% #$ ## " !

bœ œ œ œ#œ œ bœ BœB œ œ œ µœ B œ œ & œ œ Œ Bœ œ Œ 'b"%$& 'b"& ! " ## #$ #% #$ ## " ! ! " ## #$ #% #$ ## " !

œ #œ œ œ œ œ œ µœ B œ œ œ Bœ B œ œ & œ œ Œ Bbœ œ Œ 'b"#$% 'b"# ! " ## #$ #% #$ ## " ! ! " ## #$ #% #$ ## " !

œ µœ#œ œ œ œ Bœ œ œ œ œ #œ µ œ œ bœ œ B œ œ & B Œ B Œ 'b"% 'b"( ! " ## #$ #% #$ ## " ! ! " ## #$ #% #$ ## " !

Method: Appendix D - p.98 #œ<˜>œ œ Bœ #œ œ & œ #œ œ œ Œ <µ>œ<#>œ œ œ Œ ˜!"#$%$& !"#$& ! "" "# "$ "# "" ! ! "" "# "$ "# "" !

Bœ œ œ µœ #œ œ & bœ Bœ œ œ Œ œ µœ µœ œ Œ !"%$& !"& ! "" "# "$ "# "" ! ! "" "# "$ "# "" !

µœ #œ œ œ Bœ œ œ œ & œ µœ µœ œ Œ œ B œ Œ !"#$% !"# ! "" "# "$ "# "" ! ! "" "# "$ "# "" !

œ µœ #œ œ œ #œ #œ<˜>œ ˜œ œ & #œ µ œ Œ ˜œ œ Œ !"% 'b"#$%$& ! "" "# "$ "# "" ! ! "" "# "$ "# "" !

œ œ µœ Bœ œ œ <#>œ Bœ œ œ & œ œ Œ <µ>œ œ Œ !"( 'b"#$& ! "" "# "$ "# "" ! ! "" "# "$ "# "" ! %

bœ œ œ œ #œ œ bœ Bœ B œ œ œ µœ B œ œ & Œ Œ 'b"%$& 'b"& ! "" "# "$ "# "" ! ! "" "# "$ "# "" !

œ #œ œ œ œ œ œ µœ B œ œ œ Bœ B œ œ & Œ Œ 'b"#$% 'b"# ! "" "# "$ "# "" ! ! "" "# "$ "# "" !

œ #œ œ œ œ œ #œ µœ µ œ œ œ Bœ B œ œ & Œ Œ 'b"% 'b"( ! "" "# "$ "# "" ! ! "" "# "$ "# "" !

Method: Appendix D - p.99 #œ œ œ œ #œ œ & #œ <˜> #œ Œ <#>œ B œ Œ !"#$%$& !"#$& !! !" !# !" !! !! !" !# !" !!

Bœ œ œ µœ #œ œ & Bœ œ Œ µœ œ Œ !"%$& !"& !! !" !# !" !! !! !" !# !" !!

µœ #œ œ œ Bœ œ œ œ & µœ œ Œ B Œ !"#$% !"# !! !" !# !" !! !! !" !# !" !!

œ µœ #œ œ œ #œ #œ <˜>œ œ œ & µ Œ Œ !"% 'b"#$%$& !! !" !# !" !! !! !" !# !" !!

œ œ µœ Bœ œ œ <#>œ Bœ œ œ & Œ Œ !"( 'b"#$& !! !" !# !" !! !! !" !# !" !!

œ #œ Bœ Bbœ œ œ µœ Bœ œ œ & Œ Œ 'b"%$& 'b"& !! !" !# !" !! !! !" !# !" !!

#œ œ œ œ µœ Bœ œ œ Bœ B œ & Œ Œ 'b"#$% 'b"# !! !" !# !" !! !! !" !# !" !!

#œ œ œ œ µœ µœ œ œ Bœ B œ & Œ Œ 'b"% 'b"( !! !" !# !" !! !! !" !# !" !!

Method: Appendix D - p.100 Section Three – Wider Intervals

This section explores larger intervals within harmonic series. The least familiar harmonics 7, 11, 13 are the focus. Intervals from the 7th harmonic are in Set 1, from the 11th harmonic in Set 2 and from the 13th harmonic in Set 3.

Each of these sets begins with open F horn for orientation. There seems little need to print every exercise for every tube length. Players can refer to Section

One for clarity on the pitch differences between tube lengths if needed.

The inclusion of cent values may assist the understanding of relative distance.

This remains the same regardless of the starting point, shown here at the beginning of each set though disregarding any octave differences (a major 9th reads as 204 cents rather than 1404 (1200 + 204). The notation, to reflect these issues, does not vary from Sections One and Two.

As with the exercises in Sections 1 and 2, vary the dynamic from time to time also slur from time to time.

The aural and practical experience, obviously has more value than the notation.

However, reading the notation whilst playing may help to guide the ear and lips to the required pitch. Internally saying the harmonic number whilst playing can assist in a similar way. The important issue here is to ‘feel’ the intervals.

Method: Appendix D - p.101 SET 1! From the 7th harmonic

In Set 1 intervals begin by descending from the 7th harmonic, then ascending through to an octave interval. The study continues by introducing the wide semitone between the 14th and 15th harmonics rather than continuing the exercise from the 7th harmonic. Players can choose the intervals greater than an octave and include intervals to the lowest harmonics (3, 2, 1) as desired.

Set 1 begins with the F horn.

Repeat the exercise with all valves and valve combinations on the F horn side.

The relative distance between harmonics remains the same for all tube lengths.

The longest tube length is printed.

Continue with the B! horn. The exercises for all three valves are printed ,followed by the open B! horn.

Omit any exercises with higher harmonics as desired.

Alternate between tonguing and slurring.

Method: Appendix D - p.102 #$%&'( !"! )*+ !"! ,-" !"! *

& Bb˙ ˙ ˙ Ó Bb˙ ˙ Ó Bb˙ ˙ Ó !$%&'($%&" !"# ˙ ˙ "#$%&'()*+ ! , ! ! - ! ! . ! !"! .+** !"! +*/ !"! ,-" ˙ ˙ & Bb˙ ˙ ˙ Ó Bb˙ ˙ Ó Bb˙ ˙ Ó ! / ! ! 0 ! ! 12 !

!"! 00. !"! )*+ !"! -/. !"! !"! b˙ µ˙ ˙ B˙ B & Bb˙ ˙ Ó Bb˙ ˙ Ó Bb˙ ˙ Ó Bb˙ ˙ Ó ! 11 ! ! 13 ! ! 14 ! ! 1. !

!"! )*+ ,-" !"! )*+ *

& Bbœ œ œ œ ˙ Ó Bbœ œ œ ˙ Ó ! , - , ! ! , œ. , !

!"! ,-" * !"! * +*/ bœ ˙ Ó bœ œ œ œ ˙ Ó & B œ œ B ! - œ. - ! ! / 0 / !

!"! * ,-" !"! * 00. œ µœ & Bbœ œ œ ˙ Ó Bbœ œ œ ˙ Ó ! / 12 / ! ! / 11 / !

!"! * )*+ !"! * -/. œ Bœ & Bbœ œ œ ˙ Ó Bbœ œ œ ˙ Ó ! / 13 / ! ! / 14 / !

!"! +*/ ,-" !"! +*/ 00. œ µœ & Bbœ œ œ ˙ Ó Bbœ œ œ ˙ Ó ! 0 12 0 ! ! 0 11 0 !

Method: Appendix D - p.103 !"! #$% &$# !"! #$% '%( œ œ œ œ Bœ œ & Bbœ ˙ Ó Bbœ ˙ Ó ! " #$ " ! ! " #% " ! !"! )'" **( !"! )'" &$# œ µœ œ œ œ œ & Bbœ ˙ Ó Bbœ ˙ Ó ! #& ## #& ! ! #& #$ #& ! !"! )'" '%( !"! **( &$# œ Bœ œ µœ œ œ & Bbœ ˙ Ó Bbœ ˙ Ó ! #& #% #& ! ! ## #$ ## ! !"! **( '%( !"! &$# '%( µœ Bœ œ œ Bœ œ & Bbœ ˙ Ó Bbœ ˙ Ó ! ## #% ## ! ! #$ #% #$ !

& ˙ ˙ Ó ˙ ˙ Ó ˙ ˙ Ó !"#$%$&˜˙ ˙ ! ' ! ! <˜> ( ! ! ˜˙) !

& ˙ Ó ˜˙ Ó <˜>˙ Ó ˙ ˜ ˙ ˙! " ˙! ˙! #& ˙! ! * !

& #˙ Ó ˜˙ Ó #˙ Ó ˙! ## ˙! ˙! #$ ˙! ˙! #% ˙!

˙ ˙ ˙ n˙ ˙ ˜˙ ˙ & Ó <˜> Ó Ó ˙! #) ˙! #) #( #) #) #' #)

Method: Appendix D - p.104 & œ ˙ Ó œ ˙ Ó ˜œ œ œ ˜œ œ ! " <˜> # " ! ! " ˜œ$ " !

& œ ˙ Ó œ œ ˜œ œ ˙ Ó œ œ ! ˜% & % ! ! <˜> # ˜œ$ # !

& œ<˜>œ œ Ó œ #œ œ Ó œ! ˜% '( % ˙! œ! ˜% '' % ˙!

œ #œ & œ ˜ œ Ó œ œ Ó œ! ˜% ') % ˙! œ! ˜% '* % ˙!

& œ<˜>œ œ Ó œ #œ œ Ó œ! ˜& '( & ˙! œ! ˜& '' & ˙!

& œ ˜œ œ Ó œ #œ œ Ó œ! ˜& ') & ˙! œ! ˜& '* & ˙!

& <˜>œ #œ œ Ó <˜>œ ˜œ œ Ó œ! '( '' '( ˙! œ! '( ') '( ˙!

#œ & <˜>œ œ Ó #œ ˜œ œ Ó œ! '( '* '( ˙! œ! '' ') '' ˙!

#œ #œ & #œ œ Ó ˜œ #œ Ó œ! '' '* '' ˙! œ! ') '* '' ˙!

Method: Appendix D - p.105 & ˙ ˙ ˙ Ó ˙ ˙ Ó ˙ ˙ Ó !b"#$%˜ <˜>˙ ! " ! ! # ! ! µ˙$ !

˙ & ˙ µ˙ ˙ Ó ˙ ˜˙ ˙ Ó ˙ <˜> ˙ Ó ! '( ! ! % ! ! & !

#˙ ˜˙ #˙ & ˙ ˙ Ó ˙ ˙ Ó ˙ ˙ Ó ! '' ! ! ') ! ! '* !

˙ <˜>˙ µ˙ & ˙ ˙ Ó ˙ ˙ Ó ˙ ˙ Ó ! '$ ! ! '# ! ! '" !

œ ˙ Ó œ ˙ Ó & ˜œ œ œ ˜œ œ ! " <˜># " ! ! " µœ$ " !

œ ˙ Ó œ œ ˜œ œ ˙ Ó & œ œ µ ! <˜># µœ$ # ! ! % & % !

œ #œ & œ µœ<˜> œ ˙ Ó œ µœ œ ˙ Ó ! % '( % ! ! % '' % !

˜œ #œ & œ µœ œ ˙ Ó œ µœ œ ˙ Ó ! % ') % ! ! % '* % !

œ #œ & œ ˜œ<˜> œ ˙ Ó œ ˜œ œ ˙ Ó ! & '( & ! ! & '' & !

˜œ #œ & œ ˜œ œ ˙ Ó œ ˜œ œ ˙ Ó ! & ') & ! ! & '* & !

Method: Appendix D - p.106 œ #œ œ œ ˜œ œ & œ<˜> ˙ Ó œ<˜> ˙ Ó ! "# "" "# ! ! "# "$ "# !

œ #œ œ #œ ˜œ œ & œ<˜> ˙ Ó œ ˙ Ó ! "# "% "# ! ! "" "$ "" !

#œ #œ œ ˜œ #œ œ & œ ˙ Ó œ ˙ Ó ! "" "% "" ! ! "$ "% "$ !

b˙ ˙ b˙ ˙ b˙ ˙ & B ˙ Ó B ˙ Ó B Ó !b"# ˙ ! & ! ! ' ! ! ( !

b˙ ˙ ˙ b˙ ˙ ˙ b˙ ˙ ˙ & B Ó B Ó B Ó ! * ! ! "# ! ! ) ! ˙ ˙ B˙ b˙ B ˙ b˙ ˙ b˙ ˙ & B Ó B Ó B Ó ! "" ! ! "$ ! ! "% ! Bb˙ Bb˙ n˙ Bb˙ Bb˙ ˙ ˙ b˙ ˙ & B Ó Ó Ó ! "( ! "( "' "( "( "& "(

bœ ˙ bœ ˙ bœ ˙ & B œ œ œ Ó B œ œ Ó B œ œ Ó ! & ' & ! ! & œ( & ! ! ' œ( ' !

œ bœ œ œ œ ˙ bœ œ œ œ ˙ bœ œB œ ˙ & B Ó B Ó B Ó ! ) * ) ! ! ) "# ) ! ! ) "" ) !

Method: Appendix D - p.107 œ Bœ bœ œ œ ˙ bœ œ œ ˙ & B Ó B Ó ! " #$ " ! ! " #% " !

bœ œ œ œ ˙ bœ œ Bœ œ ˙ & B Ó B Ó ! & #' & ! ! & ## & !

œ Bœ bœ œ œ ˙ bœ œ œ ˙ & B Ó B Ó ! & #$ & ! ! & #% & !

œ œ bœ œ B œ ˙ bœ œ œ ˙ & B Ó B Ó ! #' ## #' ! ! #' #$ #' !

Bœ œ œ œ bœ œ œ ˙ bœ B ˙ & B Ó B Ó ! #' #% #' ! ! ## #$ ## !

œ Bœ œ œ Bœ œ bœ B ˙ bœ ˙ & B Ó B Ó ! ## #% ## ! ! #$ #% #$ !

Method: Appendix D - p.108 SET 2!!From the 11th harmonic

#$%&'( !!" )*+ !!" ,-. !!" - SET 2 From the 11th harmonic µ˙ ˙ ˙ µ˙ ˙ ˙ µ˙ ˙ & Ó Ó ˙ Ó !$%&'($%&" !"# "#$%&'()*+ !! !, !! !! - !! !! . !!

!!" /+/ !!" 0-, !!" )*+ µ˙ ˙ µ˙ ˙ µ˙ ˙ & Bb˙ Ó ˙ Ó Ó !! 0 !! !! 1˙ !! !! / !!

!!" - !!" 0-, !!" *." µ˙ ˙ µ˙ ˙ ˙ µ˙ B˙ ˙ & Ó Ó Ó !! 2˙ !! !! !3 !! !! !4 !!

!!" /+/ !!" "-** !!" - b˙ ˙ ˙ µ˙ B ˙ µ˙ ˙ µ˙ ˙ & Ó Ó Ó !! !2 !! !! !1 !! !! !0 !!

!!" )*+ ,-. !!" )*+ - !!" )*+ /+/ µœ œ œ œ ˙ µœ œ œ ˙ µœ œ œ ˙ & Ó œ Ó Bbœ Ó !! !, - !, !! !! !, . !, !! !! !, / !, !!

!!" )*+ 0-, !!" )*+ )*+ !!" )*+ - µœ œ œ ˙ µœ œ œ ˙ µœ œ œ ˙ & œ Ó œ Ó Ó !! !, 0 !, !! !! !, 1 !, !! !! !, œ2 !, !!

!!" ,-. - !!" ,-. /+/ !!" ,-. 0-, µœ œ œ ˙ µœ œ œ ˙ µœ œ œ ˙ & œ Ó Bbœ Ó œ Ó !! - . - !! !! - / - !! !! - 0 - !!

!!" ,-. )*+ !!" ,-. - !!" - /+/ µœ œ œ ˙ µœ œ œ ˙ µœ ˙ & Ó Ó œBbœ œ Ó !! - 1œ - !! !! - œ2 - !! !! . / . !!

Method: Appendix - p.109 !!" # $#% !!" # &'( !!" # # µœ ˙ µœ ˙ µœ ˙ & œ œ œ Ó œ œ Ó œ œ Ó !! " # " !! !! " $œ " !! !! " œ% " !!

!!" )() $#% !!" )() &'( !!" )() # µœ ˙ µœ ˙ µœ ˙ & Bbœ œ œ Ó Bbœ œ Ó Bbœ œ Ó !! & # & !! !! & $œ & !! !! & œ% & !!

!!" $#% &'( !!" $#% # !!" &'( # µœ ˙ µœ ˙ µœ ˙ & œ œ Ó œ œ Ó Ó !! # $œ # !! !! # œ% # !! !! $œ œ% $œ !!

!!" $#% '*" !!" $#% )() !!" $#% "#'' bœ œ µœ œ Bœ œ ˙ µœ œB œ ˙ µœ œ œ ˙ & Ó Ó Ó !! !' !( !' !! !! !' !% !' !! !! !' !$ !' !!

!!" $#% # !!" '*" )() !!" '*" "#'' œ bœ œ µœ œ œ ˙ µœBœB œ ˙ µœBœ œ ˙ & Ó Ó Ó !! !' !# !' !! !! !( !% !( !! !! !( !$ !( !!

!!" '*" # !!" )() "#'' œ bœ nœ bœ µœ Bœ œ ˙ µœ B B ˙ & Ó Ó !! !( !# !( !! !! !% !$ !% !!

!!" )() # !!" "#'' # bœ œ œ œ œ œ µœ B ˙ µœ ˙ & Ó Ó !! !% !# !% !! !! !$ !# !$ !!

Method: Appendix - p.110 & #˙ <˜>˙ ˙ Ó #˙ ˙ ˙ Ó !"#$%$& ˜ !! !" !! !! # !!

#˙ ˙ Ó #˙ ˙ Ó & ˜˙ ˙ !! $ !! !! % !!

& #˙ ˙ Ó #˙ ˙ Ó ˙ !! ˜ & !! !! <˜>˙' !!

& #˙ ˙ Ó #˙ ˜˙ ˙ Ó !! ˜˙( !! !! !) !!

#˙ ˙ & #˙ ˙ Ó #˙ ˙ Ó !! !* !! !! !( !!

˙ ˜˙ & #˙ <˜> ˙ Ó #˙ ˙ Ó !! !' !! !! !& !!

#œ ˙ Ó #œ ˙ Ó #œ ˙ Ó & <˜>œ˜œ œ <˜>œ˜œ œ <˜>œ œ œ !! !" # !" !! !! !" $ !" !! !! !" % !" !!

& #œ<˜>œ œ ˙ Ó #œ<˜>œ œ ˙ Ó #œ<˜>œ œ ˙ Ó ˜œ œ !! !" & !" !! !! !"<˜> ' !" !! !! !"˜œ( !" !!

Method: Appendix - p.111 #œ ˙ Ó #œ ˙ Ó #œ ˙ Ó & ˜œ˜œ œ ˜œ œ œ ˜œ œ œ !! " # " !! !! " $ " !! !! " ˜ % " !!

#œ ˙ Ó #œ ˙ Ó #œ ˙ Ó & ˜œ œ ˜œ œ œ œ œ œ ˜ !! " <˜> & " !! !! " ˜œ' " !! !! # $ # !!

& #œ œ œ ˙ Ó #œ œ œ ˙ Ó #œ œ œ ˙ Ó ˜ ˜œ ˜ œ ˜ !! # % # !! !! # <˜> & # !! !! # ˜œ' # !!

& #œ œ œ ˙ Ó #œ œ œ ˙ Ó #œ œ œ ˙ Ó ˜œ œ !! $ % $ !! !! $ <˜> & $ !! !! $ ˜œ' $ !!

& #œ ˙ Ó #œ ˙ Ó #œ ˙ Ó ˜œ œ ˜œ œ œ œ !! % ˜œ' % !! !! % ˜œ' % !! !!<˜> & ˜œ' & !!

#œ œ œ & #œ˜œ œ ˙ Ó #œ˜œ œ ˙ Ó #œ˜œ<˜> œ ˙ Ó !! !( !) !( !! !! !( !' !( !! !! !( !& !( !!

˜œ #œ œ œ #œ œ œ & #œ˜œ œ ˙ Ó #œ ˙ Ó #œ <˜> ˙ Ó !! !( !% !( !! !! !) !' !) !! !! !) !& !) !!

˜œ œ œ nœ & #œ #œ œ ˙ Ó #œ <˜> ˙ Ó !! !) !% !) !! !! !' !& !' !!

œ ˜œ œ œ ˜œ œ & #œ ˙ Ó #œ<˜> ˙ Ó !! !' !% !' !! !! !& !% !& !!

Method: Appendix - p.112 #˙ ˙ ˙ #˙ ˙ #˙ ˙ & <˜> Ó ˜˙ Ó µ˙ Ó !b"#$% !! !" !! !! # !! !! $ !!

#˙ ˙ Ó #˙ ˙ Ó #˙ ˙ Ó & ˙ ˜˙ ˙ !! % !! !! & !! !! <˜>' !!

#˙ ˙ #˙ ˜˙ ˙ #˙ #˙ ˙ & Ó Ó Ó !! µ˙( !! !! !) !! !! !* !!

˙ #˙ ˙ ˙ #˙ <˜>˙ ˙ #˙ µ ˙ & Ó Ó Ó !! !( !! !! !' !! !! !& !!

#œ œ œ ˙ #œ œ œ ˙ #œ œ œ ˙ & <˜> ˜œ Ó <˜> µœ Ó <˜> œ Ó !! !" # !" !! !! !" $ !" !! !! !" % !" !!

#œ<˜>œ œ ˙ Ó #œ<˜>œ œ ˙ Ó #œ<˜>œ œ ˙ Ó & ˜œ œ !! !" & !" !! !! !"<˜>' !" !! !! !" µœ( !" !!

#œ˜œ œ œ ˙ Ó #œ˜œ œ ˙ Ó #œ˜œ œ ˙ Ó & µ œ ˜œ !! # $ # !! !! # % # !! !! # & # !!

#œ˜œ œ ˙ Ó #œ˜œ œ ˙ Ó #œ œ œ ˙ Ó & œ µ œ !! #<˜>' # !! !! # µœ( # !! !! $ % $ !!

Method: Appendix - p.113 #œ œ œ ˙ Ó #œ œ œ ˙ Ó #œ œ œ ˙ Ó & µ ˜œ µ œ µ !! " # " !! !! "<˜>$ " !! !! " µœ% " !!

#œ ˙ Ó #œ ˙ Ó #œ ˙ Ó & œ˜œ œ œ œ œ œ œ !! & # & !! !! &<˜>$ & !! !! & µœ% & !!

#œ ˙ Ó #œ ˙ Ó #œ ˙ Ó & ˜œ œ ˜œ œ œ œ !! # µœ% # !! !! # µœ% # !! !!<˜>$ µœ% $ !!

#œ˜œ #œ œ ˙ #œ˜œ œ œ ˙ #œ˜œ<˜>œ œ ˙ & Ó Ó Ó !! !' !( !' !! !! !' !% !' !! !! !' !$ !' !!

œ #œ˜œ µ œ ˙ #œ #œ œ œ ˙ #œ #œ<˜>œ œ ˙ & Ó Ó Ó !! !' !# !' !! !! !( !% !( !! !! !( !$ !( !!

#œ #œ µœ œ ˙ #œ œ<˜>œ nœ ˙ #œ œ µœ œ ˙ & Ó Ó Ó !! !( !# !( !! !! !% !$ !% !! !! !% !# !% !!

œ #œ <˜>œ µ œ ˙ & Ó !! !$ !# !$ !!

Method: Appendix - p.114 B˙ ˙ ˙ B˙ ˙ ˙ B˙ ˙ ˙ & Ó Ó Ó $%&'()%&'" !b"# "#$%&'()*+ !! !, !! !! - !! !! . !!

˙ ˙ ˙ ˙ ˙ ˙ B b˙ B B & B Ó ˙ Ó ˙ Ó !! 0 !! !! 1 !! !! / !!

B˙ ˙ B˙ ˙ ˙ B˙ B˙ ˙ & Ó Ó Ó !! 2˙ !! !! !3 !! !! !4 !!

b˙ ˙ ˙ B˙ B ˙ B˙ ˙ B˙ ˙ & Ó Ó Ó !! !2 !! !! !1 !! !! !0 !!

œ ˙ œ ˙ œ ˙ B œ œ œ B œ œ œ B œ œ œ & Ó Ó µ Ó !! !, - !, !! !! !, . !, !! !! !, / !, !!

Bœ œ œ ˙ Bœ œ œ ˙ Bœ œ œ ˙ & œ Ó œ Ó Ó !! !, 0 !, !! !! !, 1 !, !! !! !, œ2 !, !!

œ ˙ œ ˙ œ ˙ B œ œ œ B œ bœ œ B œ œ & Ó B Ó œ Ó !! - . - !! !! - / - !! !! - 0 - !!

œ ˙ œ ˙ œ ˙ B œ œ B œ œ B œ bœ œ & œ Ó Ó B Ó !! - 1 - !! !! - œ2 - !! !! . / . !!

Method: Appendix - p.115 Bœ œ œ ˙ Bœ œ œ ˙ Bœ œ œ ˙ & œ Ó œ Ó Ó !! " # " !! !! " $ " !! !! " œ% " !!

œ ˙ œ ˙ œ ˙ B bœ œ B bœ œ B bœ œ & B œ Ó B œ Ó B Ó !! & # & !! !! & $ & !! !! & œ% & !!

Bœ ˙ Bœ ˙ Bœ ˙ & œ œ œ Ó œ œ Ó œ œ Ó !! # $ # !! !! # œ% # !! !! $ œ% $ !! bœ œ Bœ œ Bœ œ ˙ Bœ œB œ ˙ Bœ œ œ ˙ & Ó Ó Ó !! !' !( !' !! !! !' !% !' !! !! !' !$ !' !!

œ bœ œ Bœ œ œ ˙ BœBœB œ ˙ BœBœ œ ˙ & Ó Ó Ó !! !' !# !' !! !! !( !% !( !! !! !( !$ !( !!

œ bœ nœ bœ Bœ Bœ œ ˙ Bœ B B ˙ & Ó Ó !! !( !# !( !! !! !% !$ !% !! bœ œ œ œ œ œ Bœ B ˙ Bœ ˙ & Ó Ó !! !% !# !% !! !! !$ !# !$ !!

Method: Appendix - p.116 SET 3 From the 13th harmonic

$%&'() !"# *+, !"# --# !"# .!/ B˙ ˙ ˙ B˙ µ˙ ˙ B˙ ˙ ˙ & Ó Ó Ó !$%&'($%&" !"# !" !. !" #$%&'()*+, !" !- !" !" !! !"

!"# ,+" !"# + !"# 0/0 ˙ ˙ ˙ ˙ ˙ ˙ B ˙ B B & Ó ˙ Ó Bb˙ Ó !" / !" !" 0 !" !" 1 !"

!"# *+, !"# .!/ !"# + B˙ ˙ B˙ ˙ B˙ ˙ Ó Ó Ó & ˙ ˙ !" 2 !" !" 3 !" !" 4˙ !"

!"# 0/0 !"# #+!! !"# + B˙ Bb˙ ˙ B˙ ˙ ˙ B˙ ˙ ˙ & Ó Ó Ó !" !4 !" !" !3 !" !" !2 !"

!"# *+, --# !"# *+, .!/ !"# *+, ,+" œ ˙ œ ˙ œ ˙ B œ µœ œ B œ œ œ B œ œ œ & Ó Ó Ó !" !- !! !- !" !" !- !. !- !" !" !- / !- !"

!"# *+, + !"# *+, 0/0 !"# *+, *+, Bœ œ œ ˙ Bœ œ œ ˙ Bœ œ œ ˙ & œ Ó Bbœ Ó œ Ó !" !- 0 !- !" !" !- 1 !- !" !" !- 2 !- !"

!"# *+, .!/ !"# *+, + !"# --# .!/ Bœ œ œ ˙ Bœ œ œ ˙ Bœ µœ œ œ ˙ & Ó Ó Ó !" !- œ3 !- !" !" !- œ4 !- !" !" !! !. !! !"

!"# --# ,+" !"# --# + !"# --# 0/0 œ ˙ œ ˙ œ ˙ B µœ œ œ B µœ œ B µœ œ & Ó œ Ó Bbœ Ó !" !! / !! !" !" !! 0 !! !" !" !! 1 !! !"

Method: Appendix D - p.117 !"# $$# %&' !"# $$# (!) !"# $$# & Bœ µœ œ ˙ Bœ µœ œ ˙ Bœ µœ œ ˙ & œ Ó Ó Ó !" !! # !! !" !" !! œ$ !! !" !" !! œ% !! !"

!"# (!) '&" !"# (!) & !"# (!) *)* Bœ œ œ ˙ Bœ œ œ ˙ Bœ œ œ ˙ & œ Ó œ Ó Bbœ Ó !" !& ' !& !" !" !& ( !& !" !" !& ) !& !"

!"# (!) %&' !"# (!) (!) !"# (!) & Bœ œ œ ˙ Bœ œ œ ˙ Bœ œ œ ˙ & œ Ó Ó Ó !" !& # !& !" !" !& œ$ !& !" !" !& œ% !& !"

!"# '&" & !"# '&" *)* !"# '&" %&' Bœ œ œ ˙ Bœ œ œ ˙ Bœ œ œ ˙ & œ Ó Bbœ Ó œ Ó !" ' ( ' !" !" ' ) ' !" !" ' # ' !"

!"# '&" (!) !"# '&" & !"# & *)* œ ˙ œ ˙ œ ˙ B œ œ B œ œ B & Ó Ó œBbœ œ Ó !" ' œ$ ' !" !" ' œ% ' !" !" ( ) ( !"

!"# & %&' !"# & (!) !"# & & Bœ ˙ Bœ ˙ Bœ ˙ & œ œ œ Ó œ œ Ó œ œ Ó !" ( # ( !" !" ( $œ ( !" !" ( %œ ( !"

!"# *)* %&' !"# *)* (!) !"# *)* & Bœ ˙ Bœ ˙ Bœ ˙ & Bbœ œ œ Ó Bbœ œ Ó Bbœ œ Ó !" ) # ) !" !" ) $œ ) !" !" ) %œ ) !"

!"# %&' (!) !"# %&' & !"# (!) & Bœ ˙ Bœ ˙ Bœ ˙ & œ œ Ó œ œ Ó Ó !" # $œ # !" !" # %œ # !" !" $œ %œ $œ !"

Method: Appendix D - p.118