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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. -
Pedal 32 Contra Diaphone C. Bomb 32 Contra Tibia
PEDAL 8 VOX HUMANA (S) 8 TRUMPET 32 CONTRA DIAPHONE C. BOMB 8 VOX HUMANA 8 STYLE D TRUMPET 32 CONTRA TIBIA CLAUSA 4 OCTAVE 8 TUBA HORN 16 BOMBARDE 4 OCTAVE HORN 8 OPEN DIAPASON 16 DOUBLE ENGLISH HORN 4 PICCOLO 8 HORN DIAPASON 16 OPHICLEIDE 4 SOLO STRING 2 RKS 8 SOLO TIBIA CLAUSA 16 DIAPHONE 4 VIOL 2 RKS 8 TIBIA CLAUSA 16 DIAPHONIC HORN 4 GAMBETTE 2 RKS 8 CLARINET 16 SOLO TIBIA CLAUSA 4 LIEBLICH FLUTE 8 KINURA CLARION 4 16 BASS CLARINET 4 CONCERT FLUTE 8 ORCHESTRAL OBOE 16 CONTRA GAMBA 2 RKS 4 VOX HUMANA (M) 8 MUSETTE FRENCH HORN 16 OBOE HORN 2 2/3 TWELFTH 8 KRUMET 16 BOURDON 2 PICCOLO 8 SAXOPHONE 16 GEMSHORN 2 RKS OCTAVE 8 SOLO STRING 2 RKS 8 TUBA MIRABILIS SOLO ON ACCOMP. 8 VIOLIN 2 RKS 8 ENGLISH HORN BOMBARDE 8 GAMBA 2 RKS 8 TUBA HORN MIDI ON ACCOMP. 8 QUINTADENA 8 OPEN DIAPASON 8 PIANO OPEN DIAP. 8 LIEBLICH FLUTE 8 HORN DIAPASON HARP SUB MIXTURE IV 8 VOX HUMANA (S) 8 SOLO TIBIA CLAUSA HARP SCHARF IV 8 VOX HUMANA 8 TIBIA CLAUSA SOLO CHRYSOGLOTT 4 SOLO PICCOLO 8 TIBIA CLAUSA PIZZ. CHRYSOGLOTT MIXTURE III 4 PICCOLO 8 CLARINET SNARE DRUM 2 2/3 SOLO TWELFTH 8 CELLO 2 RKS CASTANETS 2 SOLO PICCOLO 8 FLUTE TAMBOURINE 2 PICCOLO ACCOMP. TO PEDAL WOOD BLOCK 1 3/5 SOLO TIERCE GREAT TO PEDAL TOM-TOM 1 1/3 SOLO LARIGOT SOLO TO PEDAL CHOKE CYMBAL SUB OCTAVE MIDI ON PEDAL TAP CYMBAL UNISON OFF 16 PIANO OPEN DIAP. -
The Lost Harmonic Law of the Bible
The Lost Harmonic Law of the Bible Jay Kappraff New Jersey Institute of Technology Newark, NJ 07102 Email: [email protected] Abstract The ethnomusicologist Ernest McClain has shown that metaphors based on the musical scale appear throughout the great sacred and philosophical works of the ancient world. This paper will present an introduction to McClain’s harmonic system and how it sheds light on the Old Testament. 1. Introduction Forty years ago the ethnomusicologist Ernest McClain began to study musical metaphors that appeared in the great sacred and philosophical works of the ancient world. These included the Rg Veda, the dialogues of Plato, and most recently, the Old and New Testaments. I have described his harmonic system and referred to many of his papers and books in my book, Beyond Measure (World Scientific; 2001). Apart from its value in providing new meaning to ancient texts, McClain’s harmonic analysis provides valuable insight into musical theory and mathematics both ancient and modern. 2. Musical Fundamentals Figure 1. Tone circle as a Single-wheeled Chariot of the Sun (Rg Veda) Figure 2. The piano has 88 keys spanning seven octaves and twelve musical fifths. The chromatic musical scale has twelve tones, or semitone intervals, which may be pictured on the face of a clock or along the zodiac referred to in the Rg Veda as the “Single-wheeled Chariot of the Sun.” shown in Fig. 1, with the fundamental tone placed atop the tone circle and associated in ancient sacred texts with “Deity.” The tones are denoted by the first seven letters of the alphabet augmented and diminished by and sharps ( ) and flats (b). -
Introduction to GNU Octave
Introduction to GNU Octave Hubert Selhofer, revised by Marcel Oliver updated to current Octave version by Thomas L. Scofield 2008/08/16 line 1 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 8 6 4 2 -8 -6 0 -4 -2 -2 0 -4 2 4 -6 6 8 -8 Contents 1 Basics 2 1.1 What is Octave? ........................... 2 1.2 Help! . 2 1.3 Input conventions . 3 1.4 Variables and standard operations . 3 2 Vector and matrix operations 4 2.1 Vectors . 4 2.2 Matrices . 4 1 2.3 Basic matrix arithmetic . 5 2.4 Element-wise operations . 5 2.5 Indexing and slicing . 6 2.6 Solving linear systems of equations . 7 2.7 Inverses, decompositions, eigenvalues . 7 2.8 Testing for zero elements . 8 3 Control structures 8 3.1 Functions . 8 3.2 Global variables . 9 3.3 Loops . 9 3.4 Branching . 9 3.5 Functions of functions . 10 3.6 Efficiency considerations . 10 3.7 Input and output . 11 4 Graphics 11 4.1 2D graphics . 11 4.2 3D graphics: . 12 4.3 Commands for 2D and 3D graphics . 13 5 Exercises 13 5.1 Linear algebra . 13 5.2 Timing . 14 5.3 Stability functions of BDF-integrators . 14 5.4 3D plot . 15 5.5 Hilbert matrix . 15 5.6 Least square fit of a straight line . 16 5.7 Trapezoidal rule . 16 1 Basics 1.1 What is Octave? Octave is an interactive programming language specifically suited for vectoriz- able numerical calculations. -
Intervals and Transposition
CHAPTER 3 Intervals and Transposition Interval Augmented and Simple Intervals TOPICS Octave Diminished Intervals Tuning Systems Unison Enharmonic Intervals Melodic Intervals Perfect, Major, and Minor Tritone Harmonic Intervals Intervals Inversion of Intervals Transposition Consonance and Dissonance Compound Intervals IMPORTANT Tone combinations are classifi ed in music with names that identify the pitch relationships. CONCEPTS Learning to recognize these combinations by both eye and ear is a skill fundamental to basic musicianship. Although many different tone combinations occur in music, the most basic pairing of pitches is the interval. An interval is the relationship in pitch between two tones. Intervals are named by the Intervals number of diatonic notes (notes with different letter names) that can be contained within them. For example, the whole step G to A contains only two diatonic notes (G and A) and is called a second. Figure 3.1 & ww w w Second 1 – 2 The following fi gure shows all the numbers within an octave used to identify intervals: Figure 3.2 w w & w w w w 1ww w2w w3 w4 w5 w6 w7 w8 Notice that the interval numbers shown in Figure 3.2 correspond to the scale degree numbers for the major scale. 55 3711_ben01877_Ch03pp55-72.indd 55 4/10/08 3:57:29 PM The term octave refers to the number 8, its interval number. Figure 3.3 w œ œ w & œ œ œ œ Octavew =2345678=œ1 œ w8 The interval numbered “1” (two notes of the same pitch) is called a unison. Figure 3.4 & 1 =w Unisonw The intervals that include the tonic (keynote) and the fourth and fi fth scale degrees of a Perfect, Major, and major scale are called perfect. -
Wurlitzer's Style 216 Was His Creation
Style 216 Wurlitzer in the Rialto Th eatre, Pasadena. B 'henct & Kaufmann Archn-es The Rare Breed by Tom Delay The late summer of 1925 saw the first gans, often with only an 8' manual Tibia In addition to keeping the same basic shipment of a scarce two-manual, ten-rank Clausa. As late as 1928, Wurlitzer pro list of ranks , he specified the ubiquitou s style of Wurlitzer theatre pipe organ. ducecl a style 210 2/9 with only an 8' Tibia English (post) Hom added to the Solo, and Much has been said over the years about - no 4' Piccolo , let alone Tibia Twelfth moved the Orchestral Oboe to the Solo the style 216 Wurlitzer, but not since the or Piccolo 2'. As an organist for Fox West from the Main . early 1960s has much been seen in print Coast Theatres he carried some degree of For an organ designed in 1928, it had a about what makes a style 216 such a dif weight and persuaded the Wurlitzer Com swprisingly modem stop layout. However , ferent breed. At that time, Gordon Kibbee pany to respecify their style 210: by today 's concert standards the Tibia included the 216 in his excellent series on MAIN SOLO Clausa 's appearance at only 8'-4' might Wurlitzer style specifications. (THEATRE Flute 16-2 Vox Humana 8 seem unthinkable , it was a major improve ORGAN, Fall 1960, page 9) Viol 'd Orchestre 8-2 Tuba Hom 16-4 ment towards the present day expectations Unknown, except by reputation , outside Open Diapason 16-4 Tibia Clausa 8 of a theatre organ. -
Instruments of the Orchestra
INSTRUMENTS OF THE ORCHESTRA String Family WHAT: Wooden, hollow-bodied instruments strung with metal strings across a bridge. WHERE: Find this family in the front of the orchestra and along the right side. HOW: Sound is produced by a vibrating string that is bowed with a bow made of horse tail hair. The air then resonates in the hollow body. Other playing techniques include pizzicato (plucking the strings), col legno (playing with the wooden part of the bow), and double-stopping (bowing two strings at once). WHY: Composers use these instruments for their singing quality and depth of sound. HOW MANY: There are four sizes of stringed instruments: violin, viola, cello and bass. A total of forty-four are used in full orchestras. The string family is the largest family in the orchestra, accounting for over half of the total number of musicians on stage. The string instruments all have carved, hollow, wooden bodies with four strings running from top to bottom. The instruments have basically the same shape but vary in size, from the smaller VIOLINS and VIOLAS, which are played by being held firmly under the chin and either bowed or plucked, to the larger CELLOS and BASSES, which stand on the floor, supported by a long rod called an end pin. The cello is always played in a seated position, while the bass is so large that a musician must stand or sit on a very high stool in order to play it. These stringed instruments developed from an older instrument called the viol, which had six strings. -
PEDAL ORGAN Feet Pipes
Henry Willis III 1927, 2005 H&H PEDAL ORGAN Feet Pipes Open Bass 16 32 Open Diapason * 16 32 Bordun 16 32 Lieblich Bordun (from Swell) 16 - Principal * 8 32 Flute * 8 32 Fifteenth 4 32 Mixture * (19.22.26.29) I V 128 Ophicleide (from Tuba Minor) 16 12 Trombone * 16 32 CHOIR ORGAN Quintaton 16 61 Violoncello * (old bass) 8 61 Orchestral Flute 8 61 Dulciana 8 61 Unda Maris (bass from Dulciana) 8 49 Concert Flute 4 61 Nazard * 2 2/3 61 Harmonic Piccolo 2 61 Tierce * 1 3/5 61 Corno di Bassetto 8 61 Cor Anglais 8 61 Tremolo Tuba Minor 8 61 Tuba Magna * (horizontal, unenclosed) 8 61 GREAT ORGAN Double Open Diapason 16 61 Open Diapason No. 1 * (old bass) 8 61 Open Diapason No. 2 * (old bass) 8 61 Claribel Flute * (bass from Stopped Diapason) 8 61 Stopped Diapason * (wood) 8 61 Principal * 4 61 Chimney Flute * 4 61 Fifteenth * 2 61 Full Mixture * (15.19.22.26) IV 244 Sharp Mixture * (26.29.33) III 183 Trumpet * 8 61 SWELL ORGAN Lieblich Bordun 16 61 Geigen Diapason 8 61 Rohr Flute 8 61 Echo Viole 8 61 Voix Célestes (tenor c) 8 61 Geigen Principal 4 61 Flûte Triangulaire 4 61 Flageolet 2 61 Sesquialtera (12.17) II 122 Mixture (15.19.22) III 183 Oboe 8 61 Waldhorn 16 61 Trumpet 8 61 Clarion 4 61 Tremolo New stops are indicated by *. Couplers I Choir to Pedal XII Choir to Great II Choir Octave to Pedal XIII Choir Octave to Great III Great to Pedal XIV Choir Sub Octave to Great IV Swell to Pedal XV Swell to Great V Swell Octave to Pedal XVI Swell Octave to Great XVII Swell Sub Octave to Great VI Choir Octave VII Choir Sub Octave XVIII Swell Octave -
The Unexpected Number Theory and Algebra of Musical Tuning Systems Or, Several Ways to Compute the Numbers 5,7,12,19,22,31,41,53, and 72
The Unexpected Number Theory and Algebra of Musical Tuning Systems or, Several Ways to Compute the Numbers 5,7,12,19,22,31,41,53, and 72 Matthew Hawthorn \Music is the pleasure the human soul experiences from counting without being aware that it is counting." -Gottfried Wilhelm von Leibniz (1646-1716) \All musicians are subconsciously mathematicians." -Thelonius Monk (1917-1982) 1 Physics In order to have music, we must have sound. In order to have sound, we must have something vibrating. Wherever there is something virbrating, there is the wave equation, be it in 1, 2, or more dimensions. The solutions to the wave equation for any given object (string, reed, metal bar, drumhead, vocal cords, etc.) with given boundary conditions can be expressed as a superposition of discrete partials, modes of vibration of which there are generally infinitely many, each with a characteristic frequency. The partials and their frequencies can be found as eigenvectors, resp. eigenvalues of the Laplace operator acting on the space of displacement functions on the object. Taken together, these frequen- cies comprise the spectrum of the object, and their relative intensities determine what in musical terms we call timbre. Something very nice occurs when our object is roughly one-dimensional (e.g. a string): the partial frequencies become harmonic. This is where, aptly, the better part of harmony traditionally takes place. For a spectrum to be harmonic means that it is comprised of a fundamental frequency, say f, and all whole number multiples of that frequency: f; 2f; 3f; 4f; : : : It is here also that number theory slips in the back door. -
RTOS Organ Specs As Built
Rochester’s RKO Palace 4/21 Wurlitzer - Opus 1951 12/25/1928 Original specification by house organist Tom Grierson The instrument known today as the RTOS-GRIERSON 4/23 Wurlitzer began life as a 4-manual, 21-rank Special, Opus 1951. It was shipped from North Tonawanda on September 12, 1928 to the new 2916-seat Keith-Albee Palace Theatre (later renamed RKO Palace) on Clinton Avenue North at Mortimer Street in Rochester, NY. It was premiered on Christmas Day of that year by Rochester organist Tom Grierson who is said to have participated significantly in the specification of the instrument. Over the years Wurlitzer historians have often commented that Tom’s British roots are strongly reflected in its design. While Opus1951 is often said to be similar to a Publix #1, there were a number of differences. Besides having one more rank in the solo chamber and a 15 hp blower, other significant variations include: Main chamber: • The Tuba horn (16 ´ - 4 ´) was not placed in the main, but in the solo chamber instead. • The Dulciana and Solo String I ranks were not included in this instrument. • A 15” wp Gamba (16 ´ - 4 ´), Violin (8 ´ - 4 ´) and Violin Celeste (8 ´ - 4 ´ tc) were included. • The Diaphonic Diapason (16 ´ - 8 ´) is 73 notes and has no 4 ´ octave. Solo Chamber: • There was no Vox Humana in the solo. • There is a Horn Diapason (8 ´) in the solo • The Tuba Mirabilis is extended to 16 ´ (wood) Bombarde. • Piano, Sleigh Bells, Vibraphone mechanism and second Xylophone were not included. Console: • No “master” swell pedal. -
Theatre Owner's Manual
TH-202/TH-302 Theatre Models IMPORTANT! Organs which contain GeniSys™ technology no longer include the GeniSys™ Controller Guide within the model specific Owner’s Manual. The correct GeniSys™ Controller Guide must be downloaded and/or printed separately. Please check the CODE version of the software installed within the organ to determine which version of the GeniSys™ Controller Guide is required. The CODE version is briefly displayed within the GeniSys™ Controller’s LCD display when the organ starts up. Copyright © 2016 Allen Organ Company All Rights Reserved AOC P/N 033-00221-1 Revised 10/2016 ALLEN ORGAN COMPANY For more than sixty years--practically the entire history of electronic organs-- Allen Organ Company has built the finest organs that technology would allow. In 1939, Allen built and marketed the world’s first electronic oscillator organ. The tone generators for this instrument used two hundred forty-four vacuum tubes, contained about five thousand components, and weighed nearly three hundred pounds. Even with all this equipment, the specification included relatively few stops. By 1959, Allen had replaced vacuum tubes in oscillator organs with transistors. Thousands of transistorized instruments were built, including some of the largest, most sophisticated oscillator organs ever designed. Only a radical technological breakthrough could improve upon the performance of Allen’s oscillator organs. Such a breakthrough came in conjunction with the United States Space Program in the form of highly advanced digital microcircuits. In 1971, Allen produced and sold the world’s first musical instrument utilizing digitally sampled voices! Your organ is significantly advanced since the first generation Allen digital instrument. -
Frequency Ratios and the Perception of Tone Patterns
Psychonomic Bulletin & Review 1994, 1 (2), 191-201 Frequency ratios and the perception of tone patterns E. GLENN SCHELLENBERG University of Windsor, Windsor, Ontario, Canada and SANDRA E. TREHUB University of Toronto, Mississauga, Ontario, Canada We quantified the relative simplicity of frequency ratios and reanalyzed data from several studies on the perception of simultaneous and sequential tones. Simplicity offrequency ratios accounted for judgments of consonance and dissonance and for judgments of similarity across a wide range of tasks and listeners. It also accounted for the relative ease of discriminating tone patterns by musically experienced and inexperienced listeners. These findings confirm the generality ofpre vious suggestions of perceptual processing advantages for pairs of tones related by simple fre quency ratios. Since the time of Pythagoras, the relative simplicity of monics of a single complex tone. Currently, the degree the frequency relations between tones has been consid of perceived consonance is believed to result from both ered fundamental to consonance (pleasantness) and dis sensory and experiential factors. Whereas sensory con sonance (unpleasantness) in music. Most naturally OCCUf sonance is constant across musical styles and cultures, mu ring tones (e.g., the sounds of speech or music) are sical consonance presumably results from learning what complex, consisting of multiple pure-tone (sine wave) sounds pleasant in a particular musical style. components. Terhardt (1974, 1978, 1984) has suggested Helmholtz (1885/1954) proposed that the consonance that relations between different tones may be influenced of two simultaneous complex tones is a function of the by relations between components of a single complex tone. ratio between their fundamental frequencies-the simpler For single complex tones, ineluding those of speech and the ratio, the more harmonics the tones have in common.