DOCSLIB.ORG

Notation Through History

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Notation through History Neume Names 9th to 10th Centuries 11th to 13th Centuries Modern Notation Virga Punctum Podatus (Pes) Clivis Scandicus Climacus Torculus Porrectus Scandicus flexus Porrectus flexus Torculus Respinus Pes subpunctis Notation Symbols through History Greek Acutus Gravis Accents Neumes 6th to 13th Virga Virga centuries Jacans Punctum Mensur Maxim Longa Brevis Semibrevi Minim Semimini Fusa Semifus al a (Long) (Breve) s a ma a Notatio (Duple (Semibrev (Mini x e) m) n Long) th 13 century 14th century 15th to 17th centuri es Modern 17th to Double Whole Half Quarter Eighth Sixteent th Notatio 20 Whole- Note Note Note Note h Note centuri Note n es The Anatomy of a Note Notation English French German Italian Spanish Note note Note nota nota testa or Head tête de la note Notenkopf testina or oval capocchia Hals or asta, or Stem queue plica Notenhals gamba coda Fahne or Flag crochet uncinata or corchete Fähnchen bandiera Beam barre Balken barra barra Dot point Punkt punto puntillo punktierte nota con Dotted note pointée nota puntata Note puntillo Note Notation American British French German Italian Spanish Double Cuadrada or Double- Doppelganze or whole Breve Breve Doble ronde Doppelganzenote note Redonda Whole Ganze or Semibreve Ronde Semibreve Redonda note Ganzenote Halbe or Minima or Half note Minim Blanche Blanca Halbenote Bianca Quarter Viertel or Semiminima or Crotchet Noire Negra note Viertelnote Nera Eighth Quaver Croche Achtel or Achtelnote Croma Corchea note Sechzehntel or Sixteenth Double- Sechzehntelnote Semiquaver Semicroma Semicorchea note croche Sechzehntel or Sechzehntelnote Thirty- Quadruple- Zweiunddreissigstel or second Demisemiquaver Biscroma Fusa croche Zweiunddreissigstelnote note Sixty- Octuple Vierundsechzigstel or fourth Hemidemisemiquaver Semibiscroma Semifusa croche Vierundsechzigstelnote note One hundred Cent- Hundert und and vingt- achtundzwanzigstel or Centoventottavo Semihemidemisemiquaver Garrapatea twenty- huitième Hundert und (nota) eighth (note) achtundzwanzigstelnote note Notation American British French German Italian Spanish Double Double- Pausas de Breve rest Doppelganze Pause Pausa di breve whole rest pause Cuadrada Whole Pausa di Pausas de Semibreve rest Pause Ganze Pause rest semibreve Redonda Demi- Pausa di Pausas de Half rest Minim rest Halbe Pause pause minima Blanca Quarter Pausa di Pausas de Crotchet rest Soupir Viertelpause rest semiminima Negra Eighth Demi- Pausas de Quaver rest Achtelpause Pausa di croma rest soupir Corchea Sixteenth Quart de Pausa di Pausas de Semiquaver rest Sechzehntelpause rest soupir semicroma Semicorchea Thirty- Huitième Pausa di Pausas de second Demisemiquaver rest Zweiunddreissigstelpause de soupir biscroma Fusa rest Sixty- Seizième Pausa di Pausas de Hemidemisemiquaver rest Vierundsechzigstelpause fourth rest de soupir semibiscroma Semifusa One hundred Trente et Semihemidemisemiquaver Hundert und Pausa di Pausas de and deuxième rest achtundzwanzigstelpause centoventottavo Garrapatea twenty- de soupir eighth rest Table of Key Signatures: Sharp Key Signatures (Major) Sharp Key Signatures (Minor) Flat Key Signatures (Major) Flat Key Signatures (Minor) The circle of fifths: MUSICAL SCALES: One Octave Scales from c to c' SCALE STEPS (IN SEMITONES OR HALF-STEPS) SCALE NAME 1 2 3 4 5 6 7 8 9 10 11 12 13 Chromatic Scale c c# d d# e f f# g g# a a# b c' Major Scale c d e f g a b c' Minor, Natural c d eb f g ab bb c' Minor, Melodic Ascending c d eb f g a b c' Descending c d eb f g ab bb c' Minor, Harmonic c d eb f g ab b c' Whole Tone c d e f# g# a# c' Gypsy c d eb f# g ab b c' Pentatonic c d f g a c' Octatonic c c# d# e f# g a a# c' Blues c d eb e f gb g a bb b c' Table of Major Scales: C major G F Major Major D Bb Major Major A Eb Major Major E Ab Major Major B Db Major Major F# Gb Major Major C# Cb Major Major Table of Minor Scales: Natural Minor & Harmonic Minor Melodic Minor E D Minor Minor B G Minor Minor F# C Minor Minor C# F Minor Minor G# Bb Minor Minor D# Eb Minor Minor A# Ab Minor Minor Scales - Scale Syllables: Note: A hand symbol Key to Languages Italian English French German C C# D D# E F F# G G# A A# B C do di re ri mi fa fi sol si la li ti do do di re ri mi fa fi so si la li ti do ut di ré ri mi fa fi sol si la li ti ut do di re ri mi fa fi sol si la li ti do C B Bb A Ab G Gb F E Eb D Db C do ti te la le sol se fa mi me re ra do do ti te la le so se fa mi me re ra do ut ti te la le sol se fa mi me ré ra ut do ti te la le sol se fa mi me re ra do Table of Accents Pressure Percussive Accents Combined Accents Accent Strong Accent - Strong & Strong & Strong & generally meant for Staccato Legato Staccatissimo Strong attacks at loud Accents - Accents - Accents - dynamic levels of very very strongest Accents forte or louder percussive percussive percussive and shorter while attack duration retaining possible with than full an notated duration of exaggerated notation short duration Normal Accent - Legato Normal & moderately sharp Accent - Legato and Legato Normal & Legato & Normal & attack that can be this can be Staccatissimo Accents - Staccatissimo Staccato Staccato used at any dynamic used at any - stressed and moderately Accents - Accents - Accents - level from dynamic quite short percussive moderately stressed moderately pianissimo to level and is with full percussive Medium and percussive fortissimo a slight note with short Accents moderately and short stress duration note duration short, without a separated noticable from next attack and note held to the full duration of the note Staccato Staccatissimo Light Accent - - an Accents short and exaggerated separated short duration of the note from the following note Table of Clefs | G-Clefs | F-Clefs | C-Clefs | Neutral Clefs | G-Clefs Notation English French German Italian Chiave di Treble clef Clef de sol G-Schlussel violino French Violin clef G-Octave Clefs Octave treble clef Octave treble clef Vocal tenor clef Double treble clef F-Clefs Notation English French German Italian Bass clef Clef de fa F-Schlussel Chiave di basso Baritone clef Sub-bass clef F-Octave Clefs Octave bass clef Octave bass clef C-Clefs Notation English French German Italian Baritone clef Tenor clef Clef d'ut C-Schlussel Chiave di tenore Alto clef Mezzo-soprano clef Soprano clef Neutral Clefs Notation English French German Italian Neutral clef percussion clef indefinite pitch clef Neutral clef percussion clef indefinite pitch clef Table of Dynamic Markings Term Abbreviation Definition Piano p Soft volume Pianissimo pp Pianississimo ppp Very soft volume Pianissississimo pppp Pianississississimo ppppp Extremely soft volume Pianissississississimo pppppp As soft as possible Mezzo-forte mf Medium loud volume Mezzo-piano mp Medium soft volume Fortissississimo ffff As loud as possible Fortississimo fff Extremely loud volume Fortissimo ff Very loud volume Forte f Loud volume Begin the note loudly with a very Forzando or Forzato fz or ffz strong accent sf or sff or Sforzando Loud and accented sfff sfz or sffz or Sforzato Loud and accented sfffz Combined Markings Abbreviation Definition Pianoforte pf Soft then immediately loud Loud and accented then Fortepiano fp immediately soft Begin the note very loudly, Fortissimo piano ffp instantly diminishing to a much softer volume. Begin the note moderately Mezzoforte piano mfp loudly, instantly diminishing to a much softer volume. Begin the note loudly, with a marked and sudden emphasis, Sforzando piano sfp instantly diminishing to a much softer volume. Dynamic Transitions Abbreviation Definition Crescendo cresc. Gradually increase volume or Decrescendo or decr. or Gradually decrease volume decresc. Diminuendo or dim. Gradually decrease volume Meter Basics | Meter Basics | Simple Meter | Compound Meter | METER NOTATION Meter is a recurring pattern of stresses or accents that provide the pulse or beat of music. Meter is notated at the beginning of a composition with a time signature. Time signatures are always notated with two numbers, one on top of the other, much like a fraction in math. The top number denotes the number of beats (or number of pulses) in each measure. The bottom number denotes the note value that receives the beat. The note values that can receive beats include double whole note, whole note (1), half note (2), quarter note (4), eighth note (8), sixteenth note (16), thirty-second note (32), sixty-fourth note (64), and one hundred and twenty-eighth note (128). An example of 4/4 meter means that there are four (4) beats in each measure and the quarter note (4) receives the beat. Thus, there are four quarter notes in each measure or a strong accent every four quarter notes.The table below shows an explanation of several common meters and their notations. Name Notation Meaning Rhythmic Notation Two-four There are 2 beats per measure Two quarter notes per measure meter or (top number) and Two-four the quarter note (bottom time number) gets one beat. Three-four There are 3 beats per measure Three quarter notes per measure meter or (top number) and Three-four the quarter note (bottom time number) gets one beat. Four-four There are 4 beats per measure Four quarter notes per measure meter or (top number) and Four-four the quarter note (bottom time number) gets one beat. Five-four There are 5 beats per measure Five quarter notes per measure meter or (top number) and Five-four the quarter note (bottom time number) gets one beat. Six-eight There are 6 beats per measure Six eighth notes per measure meter or (top number) and Six-eight the eighth note (bottom time number) gets one beat.
Recommended publications
  • The 17-Tone Puzzle — and the Neo-Medieval Key That Unlocks It

    The 17-Tone Puzzle — and the Neo-Medieval Key That Unlocks It

    The 17-tone Puzzle — And the Neo-medieval Key That Unlocks It by George Secor A Grave Misunderstanding The 17 division of the octave has to be one of the most misunderstood alternative tuning systems available to the microtonal experimenter. In comparison with divisions such as 19, 22, and 31, it has two major advantages: not only are its fifths better in tune, but it is also more manageable, considering its very reasonable number of tones per octave. A third advantage becomes apparent immediately upon hearing diatonic melodies played in it, one note at a time: 17 is wonderful for melody, outshining both the twelve-tone equal temperament (12-ET) and the Pythagorean tuning in this respect. The most serious problem becomes apparent when we discover that diatonic harmony in this system sounds highly dissonant, considerably more so than is the case with either 12-ET or the Pythagorean tuning, on which we were hoping to improve. Without any further thought, most experimenters thus consign the 17-tone system to the discard pile, confident in the knowledge that there are, after all, much better alternatives available. My own thinking about 17 started in exactly this way. In 1976, having been a microtonal experimenter for thirteen years, I went on record, dismissing 17-ET in only a couple of sentences: The 17-tone equal temperament is of questionable harmonic utility. If you try it, I doubt you’ll stay with it for long.1 Since that time I have become aware of some things which have caused me to change my opinion completely.
  • MTO 20.2: Wild, Vicentino's 31-Tone Compositional Theory

    MTO 20.2: Wild, Vicentino's 31-Tone Compositional Theory

    Volume 20, Number 2, June 2014 Copyright © 2014 Society for Music Theory Genus, Species and Mode in Vicentino’s 31-tone Compositional Theory Jonathan Wild NOTE: The examples for the (text-only) PDF version of this item are available online at: http://www.mtosmt.org/issues/mto.14.20.2/mto.14.20.2.wild.php KEYWORDS: Vicentino, enharmonicism, chromaticism, sixteenth century, tuning, genus, species, mode ABSTRACT: This article explores the pitch structures developed by Nicola Vicentino in his 1555 treatise L’Antica musica ridotta alla moderna prattica . I examine the rationale for his background gamut of 31 pitch classes, and document the relationships among his accounts of the genera, species, and modes, and between his and earlier accounts. Specially recorded and retuned audio examples illustrate some of the surviving enharmonic and chromatic musical passages. Received February 2014 Table of Contents Introduction [1] Tuning [4] The Archicembalo [8] Genus [10] Enharmonic division of the whole tone [13] Species [15] Mode [28] Composing in the genera [32] Conclusion [35] Introduction [1] In his treatise of 1555, L’Antica musica ridotta alla moderna prattica (henceforth L’Antica musica ), the theorist and composer Nicola Vicentino describes a tuning system comprising thirty-one tones to the octave, and presents several excerpts from compositions intended to be sung in that tuning. (1) The rich compositional theory he develops in the treatise, in concert with the few surviving musical passages, offers a tantalizing glimpse of an alternative pathway for musical development, one whose radically augmented pitch materials make possible a vast range of novel melodic gestures and harmonic successions.
  • Pietro Aaron on Musica Plana: a Translation and Commentary on Book I of the Libri Tres De Institutione Harmonica (1516)

    Pietro Aaron on Musica Plana: a Translation and Commentary on Book I of the Libri Tres De Institutione Harmonica (1516)

    Pietro Aaron on musica plana: A Translation and Commentary on Book I of the Libri tres de institutione harmonica (1516) Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Matthew Joseph Bester, B.A., M.A. Graduate Program in Music The Ohio State University 2013 Dissertation Committee: Graeme M. Boone, Advisor Charles Atkinson Burdette Green Copyright by Matthew Joseph Bester 2013 Abstract Historians of music theory long have recognized the importance of the sixteenth- century Florentine theorist Pietro Aaron for his influential vernacular treatises on practical matters concerning polyphony, most notably his Toscanello in musica (Venice, 1523) and his Trattato della natura et cognitione de tutti gli tuoni di canto figurato (Venice, 1525). Less often discussed is Aaron’s treatment of plainsong, the most complete statement of which occurs in the opening book of his first published treatise, the Libri tres de institutione harmonica (Bologna, 1516). The present dissertation aims to assess and contextualize Aaron’s perspective on the subject with a translation and commentary on the first book of the De institutione harmonica. The extensive commentary endeavors to situate Aaron’s treatment of plainsong more concretely within the history of music theory, with particular focus on some of the most prominent treatises that were circulating in the decades prior to the publication of the De institutione harmonica. This includes works by such well-known theorists as Marchetto da Padova, Johannes Tinctoris, and Franchinus Gaffurius, but equally significant are certain lesser-known practical works on the topic of plainsong from around the turn of the century, some of which are in the vernacular Italian, including Bonaventura da Brescia’s Breviloquium musicale (1497), the anonymous Compendium musices (1499), and the anonymous Quaestiones et solutiones (c.1500).
  • Musical Techniques

    Musical Techniques

    Musical Techniques Musical Techniques Frequencies and Harmony Dominique Paret Serge Sibony First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2017 The rights of Dominique Paret and Serge Sibony to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2016960997 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-058-4 Contents Preface ........................................... xiii Introduction ........................................ xv Part 1. Laying the Foundations ............................ 1 Introduction to Part 1 .................................. 3 Chapter 1. Sounds, Creation and Generation of Notes ................................... 5 1.1. Physical and physiological notions of a sound .................. 5 1.1.1. Auditory apparatus ............................... 5 1.1.2. Physical concepts of a sound .......................... 7 1.1.3.
  • Plainsound Etudes

    Plainsound Etudes

    Wolfgang von Schweinitz Plainsound Etudes FOR VIOLA SOLO Three Just Intonation Studies based on a flexible non-tempered 11-limit 31-tone scale op. 58 b 2015 for Andrew McIntosh and all violists with an interest in the sound and performance practice of microtonal just intonation PLAINSOUND MUSIC EDITION This score is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License ACCIDENTALS for microtonal just intonation EXTENDED HELMHOLTZ-ELLIS JI PITCH NOTATION The exact intonation of each pitch is written out by means of the following harmonically defined accidentals: Pythagorean series of perfect fifths, based on the open strings E e n v V ( … c g d a e … ) lowers / raises the pitch by a syntonic comma dmuU Ffow 81 : 80 = circa 21.5 cents lowers / raises the pitch by two syntonic commas cltT Ggpx circa 43 cents lowers / raises the pitch by a septimal comma < > 64 : 63 = circa 27.3 cents lowers / raises the pitch by two septimal commas • ¶ circa 54.5 cents (not used in this score) raises / lowers the pitch by an 11-limit undecimal quarter-tone 4 5 33 : 32 = circa 53.3 cents These 'Helmholtz-Ellis' accidentals for just intonation were designed in collaboration with Marc Sabat. The attached arrows for pitch alterations by a syntonic comma are transcriptions of the notation used by Hermann von Helmholtz in his book "Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik" (1863). – The annotated English translation "On the Sensations of Tone as a Physiological Basis for the Theory of Music" (published 1875/1885) was made by Alexander J.
  • MTO 12.3: Duffin, Just Intonation in Renaissance Theory and Practice

    MTO 12.3: Duffin, Just Intonation in Renaissance Theory and Practice

    Volume 12, Number 3, October 2006 Copyright © 2006 Society for Music Theory Ross W. Duffin ABSTRACT: Just intonation has a reputation as a chimerical, theoretical system that simply cannot work in practice. This is based on the assessment of most modern authorities and supported by misgivings expressed during the Renaissance when the practice was supposedly at its height. Looming large among such misgivings are tuning puzzles printed by the 16th-century mathematician, Giovanni Battista Benedetti. However, Renaissance music theorists are so unanimous in advocating the simple acoustical ratios of Just intonation that it seems clear that some reconciliation must have occurred between the theory and practice of it. This article explores the basic theory of Just intonation as well as problematic passages used to deny its practicability, and proposes solutions that attempt to satisfy both the theory and the ear. Ultimately, a resource is offered to help modern performers approach this valuable art. Received June 2006 Introduction | Theoretical Background | Benedetti's Puzzles | Problematic Passages | Is Just Tuning Possible? A New Approach | Problem Spots in the Exercises | Rehearsal Usage | Conclusion Introduction The idea . that one can understand the ratios of musical consonances without experiencing them with the senses is wrong. Nor can one know the theory of music without being versed in its practice. [1] So begins the first of two letters sent by the mathematician Giovanni Battista Benedetti to the composer Cipriano de Rore in 1563. Subsequently publishing the letters in 1585,(1) Benedetti was attempting to demonstrate that adhering to principles of Just intonation, as championed most famously by Gioseffo Zarlino,(2) would, in certain cases, cause the pitch of the ensemble to migrate.
  • Temperament in Bach's Well-Tempered Clavier

    Temperament in Bach's Well-Tempered Clavier

    TEMPERAMENT IN BACH’S WELL-TEMPERED CLAVIER A historical survey and a new evaluation according to dissonance theory Sergio Martínez Ruiz Universitat Autònoma de Barcelona Facultat de Filosofia i Lletres Departament d’Art i de Musicologia Doctorate in Musicology Supervisor: Jordi Ballester i Gibert July 2011 INDEX FOREWORD..............................................................................................................................................5 ACKNOWLEDGEMENTS.......................................................................................................................7 INTRODUCTION......................................................................................................................................7 OBJECTIVES.............................................................................................................................................8 A DECORATIVE SCROLL IN BACH’S MANUSCRIPT ....................................................................9 A HISTORICAL SURVEY .....................................................................................................................11 CARL PHILIPP EMANUEL BACH (1753)...................................................................................................11 ROBERT HALFORD MACDOWALL BOSANQUET (1876) ...........................................................................14 JAMES MURRAY BARBOUR (1947) .........................................................................................................15 HERBERT KELLETAT

  • JI Primer Ed.3

    THE JUST INTONATION PPRIRIMMEERR An Introduction to the Theory and Practice of Just Intonation by David B. Doty Third Edtion, December 2002 ISBN 0-9726810-0-0 ©1993, 1994, 2002 The Just Intonation Network A Project of Other Music, Inc. 535 Stevenson Street San Francisco, CA 94103 Phone: (415) 864-8123 Fax: (415) 864-8726 www.justintonation.net [email protected] iii Contents The Harmonic Series and the Subharmonic Series Preface to the Third Edition (2002).................... iv ........................................................................... 28 Preface to the Second Edition (1994)................. iv Converting Ratios to a Harmonic or Introduction .....................................1 Subharmonic Series Segment ..................... 28 Identities...................................................... 30 What is Just Intonation? ...................................... 1 Prime Numbers, Primary Intervals, A Little History.................................................... 2 and Prime Limits ............................................... 30 Antiquity ....................................................... 2 What is a Chord? ............................................... 30 The Middle Ages and the Renaissance ......... 2 Interval Names .................................................. 32 The Common-Practice Period Notation............................................................. 32 and the Rise of Temperament ....................... 3 Anomalies.......................................................... 33 The End of
  • Tuning the Notes of the Piece Dsilton by Ear

    Tuning the Notes of the Piece Dsilton by Ear

    1 Tuning the Notes of the Piece Dsilton by Ear Excerpt of the ensemble Dsilton’s recital´ at the international symposium „Microtones - Small Is Beautiful“ of the International Ekmelic Music Society in Salzburg, June 30. 2017 Georg Vogel 2017 The tonal space of the pieces Dsilton /dsi ːlton/ and in extended versions also of Duschscha & Utcis Gesol in fbb is based on a transcription of a fieldrecording of a gyil-instrument, alternately spelled dzil. (Dzil Duett - Ghana: Ancient Ceremonies, Dance Music & Songs, Nonesuch Records 1979) The Instrument used for transcribing and exploring the tonality is the Clavinet, which can be tuned by ear very easily. The intervals found on the recording can be easily described by using whole-number-ratios of multiples of the simple prime-numbers 2, 3, 5 & 7. 1. Middle C The starting point for the basic grouping is a middle c. Its the m ost important note, the base of the grouping. It is tuned to 263,092 Hz, which is the minor third, t he c above a 440 Hz of 31-et. 2. Pure Fifths The ratio of the first intervalls to tune is based on the prime-numbers 2 & 3. 3 to 2, the ratio of 1.5, is used to tune a g above the c and an f below the c to have two purely tuned perfect fifths. # $ $ $ $ 4/3 9/8 3/2 3/2 Ill. 1 Whats a fifth? For intervalls this term referres to the fifth note of a scale, basically con - structed by stacking the ratio of 3/2 to create a tonal space of wholetones and semitones, used to visualise this grouping.
  • Just Tuning and the Unavoidable Discrepancies Paul F. Zweifel

    Just Tuning and the Unavoidable Discrepancies Paul F. Zweifel

    Just Tuning and the Unavoidable Discrepancies Paul F. Zweifel I. Introduction Although in the past just tuning has been more of a theoretical than a practical music concept, the advent of computerized instruments and synthesizers makes it quite possible that a piece of music could be programmed to be performed always in just tuning, perhaps with the tuning scheme modulating in concordance with modulations in the composition. In any event, a keyboard-driven synthesizer could easily be programmed to perform nonchromatic music in just tuning, without the necessity of manual retuning for different keys. For these reasons, I have thought it worthwhile to study the just-tuning scheme in some detail, with a view to improving it. I have chosen the scheme described in Thomas D. Rossing's book, Table 9.3, as the paradigm with which to begin my study. 1 To review briefly what is meant by the term "just tuning," the I, IV, and V chords are all required to obey the canonical frequency ratios of 1:5/4:3/2 = 4:5:6. (For convenience, the canonical-or IThomas D. Rossing, The Science ofSound, 2d ed. (Reading, MA: Addison-Wesley Pub. Co., 1990), 181. This table gives the frequency ratios of the principal twenty-two chromatic notes in a single octave relative to the lowest note, denoted C. The octave is denoted C'. To retune in another key, simply replace C with the relevant tonic note. 90 Indiana Theory Review Vol. 15/2 just-ratios for various intervals are given in Appendix C, Table 1.) Serendipitously, the iii and vi chords also come out justly, in the ratios 10: 12: 15.
  • Why the Kirnberger Kernel Is So Small

    Why the Kirnberger Kernel Is So Small

    Why the Kirnberger Kernel Is So Small ∗ Don N. Page † Theoretical Physics Institute Department of Physics, University of Alberta Room 238 CEB, 11322 – 89 Avenue Edmonton, Alberta, Canada T6G 2G7 (2009 July 29) Abstract Defining the musical interval of the Kirnberger kernel, or Kirn-kern, to be one-twelfth the atom of Kirnberger, or the difference between a grad and a schisma, its natural logarithm, k = (161/12) ln 2 − 7 ln 3 − ln 5, is extremely small, k ≈ 0.000 000 739 401. Here an explanation of this coincidence is given −1 −1 by showing that k = (1/6)(11 tanh [(3/23)/11] − 21 tanh [(3/23)/21]) ≈ (255)/(3 · 72112233) ≈ 0.000 000 739 322. arXiv:0907.5249v1 [physics.pop-ph] 30 Jul 2009 ∗Alberta-Thy-14-09, arXiv:0907.5249 †Internet address: [email protected] 1 Introduction The historical accident that the most common musical scale today divides the octave into 12 semitones is no doubt due to the fact that 3 is close to 219/12, so that a justly tuned perfect fifth (pitch ratio 3:2 or 3/2) is very close to seven semitones of equal temperament (pitch ratio 27/12). The difference has been defined by Andreas Werck- meister (1645–1706) to be the grad [1], which (on a logarithmic scale) is one-twelfth of the Pythagorean or ditonic comma that is the difference between twelve perfect fifths and seven octaves (pitch ratio (3/2)12/27 =312/219 = 531441/524 288 ≈ 1.013 643). Having divided up the octave into twelve semitones so that a pitch ratio of three may be closely approximated by nineteen semitones, it is a rather fortuitous “accident” that a pitch ratio of five may also be rather closely approximated by an integer number of semitones, in this case twenty-eight.
  • Rahn, Marchetto's Tuning

    Rahn, Marchetto's Tuning

    Volume 4, Number 6, November 1998 Copyright © 1998 Society for Music Theory Jay Rahn KEYWORDS: semitone, monochord, geometry, perception, beats, counterpoint, chant, pedagogy, scale ABSTRACT: In his historically momentous account of tuning, Marchetto of Padua (fl. 1305–19) proposed dividing the whole tone of received theory (9/8=204 cents) into 5 parts. This report develops determinate arithmetic and geometric realizations of Marchetto’s formulation—directly applicable to the medieval monochord, and sonically illustrated by digital files. The resulting intervals’ feasibility is compared with current findings in the psychology of interval perception. Conjectures are offered as to how and to whom Marchetto’s tuning was taught. A dual formulation of “nuanced,” heterogenous tonal systems is advanced to assess structural effects of Marchetto’s tuning on pieces and to suggest ways one could learn to perceive and sing his intervals nowadays. INTRODUCTION 1. OVERVIEW 2. GEOMETRIC CONSTRUCTIONS FOR THE MONOCHORD 3. FEASIBILITY OF DRAFTING MARCHETTO’S 9-FOLD DIVISION 4. INTERVAL PERCEPTION AND MARCHETTO’S DIESES 5. PEDAGOGICAL ASPECTS OF MARCHETTO’S FORMULATION 6. OTHER REALIZATIONS OF MARCHETTO’S DIESES 7. MARCHETTO’S CHROMATICISMS AND RECENT SCALE THEORY FOOTNOTES WORKS CITED 1 of 23 INTRODUCTION [0.0] Marchetto of Padua (fl. 1305–19) was arguably the most important European music theorist between Guido of Arezzo (fl. ~991/2–1033+) and Gioseffo Zarlino (1517–90). A leading experimental composer of his era, Marchetto taught at Padua’s Cathedral, a job that would involve training choirboys, leading the chorus in liturgical chant, performing his own part in written and/or improvised discant (counterpoint), and both performing and rehearsing/directing others in cutting-edge music of his own creation (Gallo 1974: 42–43; Vecchi 1954: 166–68).