Just intonation

Harmonic series, partials 1–5 numbered About this soundPlay .

In music, just intonation (sometimes abbreviated as JI) or pure intonation is the tuning of musical intervals as (small) whole number ratios of frequencies. Any interval tuned in this way is called a just interval. Just intervals and chords are aggregates of harmonic series partials and may be seen as sharing a (lower) implied fundamental. For example, a tone with a frequency of 300 Hz and another with a frequency of 200 Hz are both multiples of 100 Hz (100 × 3 and 100 × 2 respectively). Their interval is, therefore, an aggregate of the second and third partials of the harmonic series of an implied fundamental frequency 100 Hz.

Without context, "just intonation" typically refers to 5-limit just intonation, where ratios only contain powers of the prime numbers 2, 3, and 5. American composer Ben Johnston proposed the term extended just intonation for composition involving ratios that contain prime numbers beyond 5, i.e. 7, 11, 13, etc..

Just intonation may be contrasted and compared with standard 12-tone equal temperament, which dominates Western instruments of fixed pitch (e.g., piano or organ) and default MIDI tuning on electronic keyboards. In equal temperament, all intervals are defined as an integer power of the basic step – the equal-tempered semitone, whose ratio is (100 cents) – so two notes separated by the same number of steps share the same frequency ratio. Except for the doubling of frequencies (one or more octaves), all intervals are, in fact, irrational and may not be expressed as a ratio of whole numbers. Just intonation, on the other hand, suggests many microtonally differentiated sizes of intervals, which stem from different regions of the harmonic series. For example, the major third has three standard tunings in 7-limit just intonation – 9:7 (435.08 cents), 81:64 (407.82 cents), and 5:4 (386.31 cents).

History [ edit ]

Pythagorean tuning, the first tuning system to be theoretically elaborated,[1] is a system in which all tones are generated using ratios of prime numbers 2 and 3 as well as their powers. The most basic of these is the ratio 3:2 itself, called the perfect fifth. Pythagorean tuning is, in this sense, a spiral of cycling fifths. The justly tuned perfect fifth with the ratio 3:2 (701.96 cents wide), however, is not equivalent to the modern equal-tempered perfect fifth on the piano with ratio (700.00 cents wide). Rather, it is larger than the equal-tempered fifth by the small interval of a twelfth of the Pythagorean comma (1.96 cents). A stack of 12 justly tuned perfect fifths, therefore, does not arrive at the same pitch class it began with. This new pitch class is one full Pythagorean comma "higher" than the starting pitch class, demonstrating how a continuous spiral of Pythagorean perfect fifths will generate an infinite collection of unique pitch classes within a frequency range.

In Pythagorean tuning, the most consonant intervals are the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third (81:64) and minor third (32:27) are complex and comparably much more dissonant than the smoother sounding intervals with simpler ratios obtained from a tuning system than introduces powers of the prime number 5.[2] The 5-limit major and minor thirds have ratios 5:4 and 6:5 respectively. The difference between the Pythagorean major third and the 5-limit major third – sometimes referred to as the Ptolemaic major third – is known as the syntonic comma and has the ratio of 81:80 (21.51 cents).

During the second century AD, Claudius Ptolemy described a 5-limit diatonic scale in his influential text on music theory Harmonics, which he called "tense diatonic".[3] Given ratios of string lengths 120, 112 ​12, 100, 90, 80, 75, 66 ​23, and 60,[3] Ptolemy quantified the consonant tuning of what would today be called the major scale beginning and ending on the mediant – 16:15, 9:8, 10:9, 9:8, 16:15, 9:8, and 10:9.

The guqin has a musical scale based on harmonic overtone positions. The dots on its soundboard indicate the harmonic positions: ​18, ​16, ​15, ​14, ​13, ​25, ​12, ​35, ​23, ​34, ​45, ​56, ​78.[4]

Diatonic scale [ edit ]

Primary triads in C About this soundPlay .
Just tuned diatonic scale derivation.[5]

The prominent notes of a given scale may be tuned so that their frequencies form (relatively) small whole number ratios.

The 5-limit diatonic major scale is tuned in such a way that major triads on the tonic, subdominant, and dominant are tuned in the proportion 4:5:6, and minor triads on the mediant and submediant are tuned in the proportion 10:12:15. Because of the two sizes of wholetone – 9:8 (major wholetone) and 10:9 (minor wholetone) – the supertonic must be microtonally lowered by a syntonic comma to form a pure minor triad.

5-limit diatonic major scale on C[5][6][7] (Ptolemy's intense diatonic scale):[8]

Note Name C D E F G A B C
Ratio from C 1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1
Harmonic of Fundamental F 24 27 30 32 36 40 45 48
Cents 0 204 386 498 702 884 1088 1200
Step Name   T t s T t T s  
Ratio 9:8 10:9 16:15 9:8 10:9 9:8 16:15
Cents 204 182 112 204 182 204 112

For a justly tuned harmonic minor scale, the mediant is tuned 6:5 and the submediant is tuned 8:5. Natural minor would include a tuning of 9:5 for the subtonic.

Twelve-tone scale [ edit ]

There are several ways to create a just tuning of the twelve-tone scale.

Pythagorean tuning [ edit ]

Pythagorean tuning can produce a twelve-tone scale, but it does so by involving ratios of very large numbers, corresponding to natural harmonics very high in the harmonic series that do not occur widely in physical phenomena. This tuning uses ratios involving only powers of 3 and 2, creating a sequence of just fifths or fourths, as follows:

Note G D A E B F C G D A E B F
Ratio 1024:729 256:243 128:81 32:27 16:9 4:3 1:1 3:2 9:8 27:16 81:64 243:128 729:512
Cents 588 90 792 294 996 498 0 702 204 906 408 1110 612

The ratios are computed with respect to C (the base note). Starting from C, they are obtained by moving six steps (around the circle of fifths) to the left and six to the right. Each step consists of a multiplication of the previous pitch by 2/3 (descending fifth), 3/2 (ascending fifth), or their inversions (3/4 or 4/3).

Between the enharmonic notes at both ends of this sequence is a pitch ratio of 312 / 219 = 531441 / 524288, or about 23 cents, known as the Pythagorean comma. To produce a twelve-tone scale, one of them is arbitrarily discarded. The twelve remaining notes are repeated by increasing or decreasing their frequencies by a power of 2 (the size of one or more octaves) to build scales with multiple octaves (such as the keyboard of a piano). A drawback of Pythagorean tuning is that one of the twelve fifths in this scale is badly tuned and hence unusable (the wolf fifth, either F-D if G is discarded, or B-G if F is discarded). This twelve-tone scale is fairly close to equal temperament, but it does not offer much advantage for tonal harmony because only the perfect intervals (fourth, fifth, and octave) are simple enough to sound pure. Major thirds, for instance, receive the rather unstable interval of 81:64, sharp of the preferred 5:4 by an 81:80 ratio.[9] The primary reason for its use is that it is extremely easy to tune, as its building block, the perfect fifth, is the simplest and consequently the most consonant interval after the octave and unison.

Pythagorean tuning may be regarded as a "three-limit" tuning system, because the ratios can be expressed as a product of integer powers of only whole numbers less than or equal to 3.

Five-limit tuning [ edit ]

A twelve-tone scale can also be created by compounding harmonics up to the fifth. Namely, by multiplying the frequency of a given reference note (the base note) by powers of 2, 3, or 5, or a combination of them. This method is called five-limit tuning.

To build such a twelve-tone scale (using C as the base note), we may start by constructing a table containing fifteen pitches:

Factor 19 13 1 3 9
5 D A E B F note
10:9 5:3 5:4 15:8 45:32 ratio
182 884 386 1088 590 cents
1 B F C G D note
16:9 4:3 1:1 3:2 9:8 ratio
996 498 0 702 204 cents
15 G D A E B note
64:45 16:15 8:5 6:5 9:5 ratio
610 112 814 316 1018 cents

The factors listed in the first row and column are powers of 3 and 5, respectively (e.g., 1/9 = 3−2). Colors indicate couples of enharmonic notes with almost identical pitch. The ratios are all expressed relative to C in the centre of this diagram (the base note for this scale). They are computed in two steps:

  1. For each cell of the table, a base ratio is obtained by multiplying the corresponding factors. For instance, the base ratio for the lower-left cell is 1/9 × 1/5 = 1/45.
  2. The base ratio is then multiplied by a negative or positive power of 2, as large as needed to bring it within the range of the octave starting from C (from 1:1 to 2:1). For instance, the base ratio for the lower left cell (1/45) is multiplied by 26, and the resulting ratio is 64:45, which is a number between 1:1 and 2:1.

Note that the powers of 2 used in the second step may be interpreted as ascending or descending octaves. For instance, multiplying the frequency of a note by 26 means increasing it by 6 octaves. Moreover, each row of the table may be considered to be a sequence of fifths (ascending to the right), and each column a sequence of major thirds (ascending upward). For instance, in the first row of the table, there is an ascending fifth from D and A, and another one (followed by a descending octave) from A to E. This suggests an alternative but equivalent method for computing the same ratios. For instance, one can obtain A, starting from C, by moving one cell to the left and one upward in the table, which means descending by a fifth and ascending by a major third:

2/3 × 5/4 = 10/12 = 5/6.

Since this is below C, one needs to move up by an octave to end up within the desired range of ratios (from 1:1 to 2:1):

5/6 × 2/1 = 10/6 = 5/3.

A 12-tone scale is obtained by removing one note for each couple of enharmonic notes. This can be done in at least three ways, which have in common the removal of G, according to a convention which was valid even for C-based Pythagorean and quarter-comma meantone scales. Note that it is a diminished fifth, close to half an octave, above the tonic C, which is a disharmonic interval; also its ratio has the largest values in its numerator and denominator of all tones in the scale, which make it least harmonious: all reasons to avoid it.

This is only one possible strategy of five-limit tuning. It consists of discarding the first column of the table (labeled "​19"). The resulting 12-tone scale is shown below:

Asymmetric scale
Factor 1/3 1 3 9
5 A E B F
5:3 5:4 15:8 45:32
1 F C G D
4:3 1:1 3:2 9:8
1/5 D A E B
16:15 8:5 6:5 9:5

Extension of the twelve-tone scale [ edit ]

The table above uses only low powers of 3 and 5 to build the base ratios. However, it can be easily extended by using higher positive and negative powers of the same numbers, such as 52 = 25, 5−2 = 1/25, 33 = 27, or 3−3 = 1/27. A scale with 25, 35 or even more pitches can be obtained by combining these base ratios, as in five-limit tuning.

Indian scales [ edit ]

In Indian music, the just diatonic scale described above is used, though there are different possibilities, for instance for the sixth pitch (Dha), and further modifications may be made to all pitches excepting Sa and Pa.[10]

Note Sa Re Ga Ma Pa Dha Ni Sa
Ratio 1:1 9:8 5:4 4:3 3:2 5:3 or 27:16 15:8 2:1
Cents 0 204 386 498 702 884 or 906 1088 1200

Some accounts of Indian intonation system cite a given 22 Shrutis.[11][12] According to some musicians, one has a scale of a given 12 pitches and ten in addition (the tonic, Shadja (Sa), and the pure fifth, Pancham (Pa), are inviolate):

Note C D D D D E E E E F F
Ratio 1:1 256:243 16:15 10:9 9:8 32:27 6:5 5:4 81:64 4:3 27:20
Cents 0 90 112 182 204 294 316 386 408 498 520
45:32 729:512 3:2 128:81 8:5 5:3 27:16 16:9 9:5 15:8 243:128 2:1
590 612 702 792 814 884 906 996 1018 1088 1110 1200

Where we have two ratios for a given letter name, we have a difference of 81:80 (or 22 cents), which is known as the syntonic comma.[9] One can see the symmetry, looking at it from the tonic, then the octave.

(This is just one example of explaining a 22-Śruti scale of tones. There are many different explanations.)

Practical difficulties [ edit ]

Some fixed just intonation scales and systems, such as the diatonic scale above, produce wolf intervals. The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32:27 for D-F, and still worse, a minor tone next to a fourth giving 40:27 for D-A. Moving D down to 10:9 alleviates these difficulties but creates new ones: D-G becomes 27:20, and D-B becomes 27:16. This fundamental problem arises in any system of tuning using a limited number of notes.

One can have more frets on a guitar to handle both As, 9:8 with respect to G and 10:9 with respect to G so that A-C can be played as 6:5 while A-D can still be played as 3:2. 9:8 and 10:9 are less than 1/53 of an octave apart, so mechanical and performance considerations have made this approach extremely rare. And the problem of how to tune chords such as C-E-G-A-D, in typical 5-limit just intonation, is left unresolved (for instance, A could be 4:3 below D (making it 9:8, if G is 1) or 4:3 above E (making it 10:9, if G is 1) but not both at the same time, so one of the fourths in the chord will have to be an out-of-tune wolf interval). However the frets may be removed entirely—this, unfortunately, makes in-tune fingering of many chords exceedingly difficult, due to the construction and mechanics of the human hand—and the tuning of most complex chords in just intonation is generally ambiguous.

Some composers deliberately use these wolf intervals and other dissonant intervals as a way to expand the tone color palette of a piece of music. For example, the extended piano pieces The Well-Tuned Piano by LaMonte Young and The Harp Of New Albion by Terry Riley use a combination of very consonant and dissonant intervals for musical effect. In "Revelation", Michael Harrison goes even further, and uses the tempo of beat patterns produced by some dissonant intervals as an integral part of several movements.

For many fixed-pitch instruments tuned in just intonation, one cannot change keys without retuning the instrument. For instance, if a piano is tuned in just intonation intervals and a minimum of wolf intervals for the key of G, then only one other key (typically E-flat) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes modulation within a piece, or playing a repertoire of pieces in different keys, impractical to impossible.

Synthesizers have proven a valuable tool for composers wanting to experiment with just intonation. They can be easily retuned with a microtuner. Many commercial synthesizers provide the ability to use built-in just intonation scales or to create them manually. Wendy Carlos used a system on her 1986 album Beauty in the Beast, where one electronic keyboard was used to play the notes, and another used to instantly set the root note to which all intervals were tuned, which allowed for modulation. On her 1987 lecture album Secrets of Synthesis there are audible examples of the difference in sound between equal temperament and just intonation.

Singing and scale-free instruments [ edit ]

The human voice is among the most pitch-flexible instruments in common use. Pitch can be varied with no restraints and adjusted in the midst of performance, without needing to retune. Although the explicit use of just intonation fell out of favour concurrently with the increasing use of instrumental accompaniment (with its attendant constraints on pitch), most a cappella ensembles naturally tend toward just intonation because of the comfort of its stability. Barbershop quartets are a good example of this.

The unfretted stringed instruments from the violin family (the violin, the viola, the cello and the double bass) are quite flexible in the way pitches can be adjusted. Stringed instruments that are not playing with fixed pitch instruments tend to adjust the pitch of key notes such as thirds and leading tones so that the pitches differ from equal temperament.

Trombones have a slide that allows arbitrary tuning during performance.

Wind instruments with valves are biased towards natural tuning and must be micro-tuned if equal temperament is required.

Other wind instruments, although built to a certain scale, can be micro-tuned to a certain extent by using the embouchure or adjustments to fingering.

Western composers [ edit ]

Composers often impose a limit on how complex the ratios may become.[13] For example, a composer who chooses to write in 7-limit just intonation will not employ ratios that use powers of prime numbers larger than 7. Under this scheme, ratios like 11:7 and 13:6 would not be permitted, because 11 and 13 cannot be expressed as powers of those prime numbers ≤ 7 (i.e. 2, 3, 5, and 7).

Though just intonation in its simplest form (5-limit) may seem to suggest a necessarily tonal logic, it need not be the case. Some music of Kraig Grady and Daniel James Wolf uses just intonation scales designed by Erv Wilson explicitly for a consonant form of atonality, and many of Ben Johnston's early works, like the Sonata for Microtonal Piano and String Quartet No. 2, use serialism to achieve a more atonal result.

Alternatively, composers such as La Monte Young, Ben Johnston, James Tenney, Marc Sabat, Wolfgang von Schweinitz, Chiyoko Szlavnics, Catherine Lamb, Kristofer Svensson, and Thomas Nicholson have sought a new kind tonality and harmony – one based on the perception and experience of sound, which not only allows for the more familiar consonant structures, but also extends them beyond the 5-limit into a nuanced and diverse network of relationships between tones.[14]

Yuri Landman devised a just intonation musical scale from an atonal prepared guitar playing technique based on adding a third bridge under the strings. When this bridge is positioned at nodal positions of the guitar strings' harmonic series, the volume of the instrument increases and the overtone becomes clear, having a consonant relation to the complementary opposed string part creating a harmonic multiphonic tone.[15]

Staff notation [ edit ]

Legend of the HE Accidentals
Ex. 1: Legend of the Helmholtz-Ellis Accidentals
Pythagorean diatonic scale on C About this soundPlay . Johnston's notation.
Just intonation diatonic scale on C About this soundPlay . Johnston's notation (Pythagorean major scale in Helmholtz-Ellis notation).
Just intonation diatonic scale on C. Helmholtz-Ellis notation.
Just harmonic seventh chord (4:5:6:7:8) on C About this soundPlay . The 7th is 968.826 cents, a septimal quarter tone lower than B.

Originally a system of notation to describe scales was devised by Hauptmann and modified by Helmholtz (1877) in which Pythagorean notes are started with and subscript numbers are added indicating how many commas (81:80, syntonic comma) to lower by.[16] For example, the Pythagorean major third on C is C+E (About this soundPlay ) while the just major third is C+E1 (About this soundPlay ). A similar system was devised by Carl Eitz and used in Barbour (1951) in which Pythagorean notes are started with and positive or negative superscript numbers are added indicating how many commas (81:80, syntonic comma) to adjust by.[17] For example, the Pythagorean major third on C is C-E0 while the just major third is C-E−1.

While these systems allow precise indication of intervals and pitches in print, more recently some composers have been developing notation methods for Just Intonation using the conventional five-line staff. James Tenney, amongst others, preferred to combine JI ratios with cents deviations from the equal tempered pitches, indicated in a legend or directly in the score, allowing performers to readily use electronic tuning devices if desired.[18] Beginning in the 1960s, Ben Johnston had proposed an alternative approach, redefining the understanding of conventional symbols (the seven "white" notes, the sharps and flats) and adding further accidentals, each designed to extend the notation into higher prime limits. His notation "begins with the 16th-century Italian definitions of intervals and continues from there."[19]

Johnston‘s method is based on a diatonic C Major scale tuned in JI, in which the interval between D (9:8 above C) and A (5:3 above C) is one syntonic comma less than a Pythagorean perfect fifth 3:2. To write a perfect fifth, Johnston introduces a pair of symbols representing this comma, + and −. Thus, a series of perfect fifths beginning with F would proceed C G D A+ E+ B+. The three conventional white notes A E B are tuned as Ptolemaic major thirds (5:4) above F C G respectively. Johnston introduces new symbols for the septimal (7 & 7 upside-down), undecimal ( & ), tridecimal (13 & 13 upside down), and further prime-number extensions to create an accidental based exact JI notation for what he has named "Extended Just Intonation".[20] For example, the Pythagorean major third on C is C-E+ while the just major third is C-E.

In 2000–2004, Marc Sabat and Wolfgang von Schweinitz worked in Berlin to develop a different accidental-based method, the Extended Helmholtz-Ellis JI Pitch Notation.[21] Following the method of notation suggested by Helmholtz in his classic "On the Sensations of Tone as a Physiological Basis for the Theory of Music", incorporating Ellis' invention of cents, and continuing Johnston's step into "Extended JI", Sabat and Schweinitz consider each prime dimension of harmonic space to be represented by a unique symbol. In particular they take the conventional flats, naturals and sharps as a Pythagorean series of perfect fifths. Thus, a series of perfect fifths beginning with F proceeds C-G-D-A-E-B-F and so on.

For higher primes, additional signs have been designed. To facilitate quick estimation of pitches, cents indications may be added (downward deviations below and upward deviations above the respective accidental). The convention used is that the cents written refer to the tempered pitch implied by the flat, natural, or sharp sign and the note name. A complete legend and fonts for the notation (see samples) are open source and available from Plainsound Music Edition.[22] For example, the Pythagorean major third on C is C-E while the just major third is C-E↓.

Staff notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C[23] using Johnston's notation About this soundPlay 

One of the great advantages[vague] of such notation systems is that they allow the natural harmonic series to be precisely notated.

Sagittal notation is based on notation of equal temperaments that may be used to approximate just intonation. For example, it uses "a simple three-segment arrow" (⤊/⤋) to indicate the unidecimal diesis (ł/half flat in Helmholtz-Ellis or / in Johnston's notation).[24]

Audio examples [ edit ]

  • About this soundJust intonation  An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.
  • About this soundEqual temperament  An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. The beating in this file may be more noticeable after listening to the above file.
  • About this soundEqual temperament and just intonation compared  A pair of major thirds, followed by a pair of full major chords. The first in each pair is in equal temperament; the second is in just intonation. Piano sound.
  • About this soundEqual temperament and just intonation compared with square waveform  A pair of major chords. The first is in equal temperament; the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just intonation between the two chords. In the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz. In the just intonation triad, this roughness is absent. The square waveform makes the difference between equal temperament and just intonation more obvious.

See also [ edit ]

Notes [ edit ]

Sources [ edit ]

  1. ^ The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts. See: West, M.L. (May 1994). "The Babylonian Musical Notation and the Hurrian Melodic Texts". Music & Letters. 75 (2): 161–179. doi:10.1093/ml/75.2.161. JSTOR 737674.
  2. ^ Helmholtz, Hermann von (1954). Ellis, Alexander J. (ed.). On the Sensations of Tone as a Physiological Basis for the Theory of Music. New York: Dover. p. 435.
  3. ^ a b Greek musical writings. Barker, Andrew, 1943-. Cambridge: Cambridge University Press. 1984–1989. p. 350. ISBN 0521235936. OCLC 10022960. CS1 maint: others (link) CS1 maint: date format (link)
  4. ^ "Qin Tunings, Some Theoretical Concepts". Table 2: Relative positions of studs on the qin.
  5. ^ a b Murray Campbell and Clive Greated, The Musician's Guide to Acoustics (London and New York: Oxford University Press, 2001), pp. 172–73. Reprint of the first edition (London: Dent, 1987). ISBN 978-0-19-816505-7.
  6. ^ Wright, David (2009). Mathematics and Music, Mathematical World 28 (Providence, Rhode Island: American Mathematical Society) pp. 140–41. ISBN 978-0-8218-4873-9.
  7. ^ Johnston, Ben (2006), "A Notation System for Extended Just Intonation" (2003), in "Maximum Clarity" and Other Writings on Music, edited by Bob Gilmore (Urbana and Chicago: University of Illinois Press, 2006), p. 78. ISBN 978-0-252-03098-7.
  8. ^ Partch, Harry (1979). Genesis of a Music, pp. 165 & 73. ISBN 978-0-306-80106-8.
  9. ^ a b Danielou, Alain (1968). The Ragas of Northern Indian Music. London: Barrie & Rockliff. ISBN 0-214-15689-3.
  10. ^ Bagchee, Sandeep. Nad: Understanding Raga Music. BPI (India) PVT Ltd. p. 23. ISBN 81-86982-07-8.
  11. ^ Danielou, Alain (1995). Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness (Rep Sub ed.). Inner Traditions. ISBN 0892813369.
  12. ^ Danielou, Alain (1999). Introduction to the Study of Musical Scales. Oriental Book Reprint Corporation. ISBN 8170690986.
  13. ^ 1901-1974, Partch, Harry. Genesis of a music : an account of a creative work, its roots and its fulfillments (Second edition, enlarged ed.). New York. ISBN 030671597X. OCLC 624666.
  14. ^ "Plainsound Music Edition".
  15. ^ 3rd Bridge HelixArchived 2012-08-24 at the Wayback Machine by Yuri Landman on
  16. ^ Hermann von Helmholtz (1885). On the Sensations of Tone as a Physiological Basis for the Theory of Music, p.276. Longmans, Green. Note the use of the + between just major thirds, − between just minor thirds, | between Pythagorean minor thirds, and ± between perfect fifths.
  17. ^ Benson, David J. (2007). Music: A Mathematical Offering, p.172. ISBN 978-0-521-85387-3. Cites Eitz, Carl A. (1891). Das mathematisch-reine Tonsystem. Leipzig.
  18. ^ Garland, Peter, ed. (1984). The Music of James Tenney. Soundings. Vol. 13. Santa Fe, New Mexico: Soundings Press. OCLC 11371167.
  19. ^ "Just Intonation Explained", Accessed February 2016.
  20. ^ Johnston & Gilmore (2006), p.77-88.
  21. ^ Manfred Stahnke, ed. (2005). "The Extended Helmholtz-Ellis JI Pitch Notation: eine Notationsmethode für die natürlichen Intervalle". Mikrotöne und Mehr – Auf György Ligetis Hamburger Pfaden. Hamburg: von Bockel Verlag. ISBN 3-932696-62-X.
  22. ^ Sabat, Marc. "The Extended Helmholtz Ellis JI Pitch Notation" (PDF). Plainsound Music Edition. Retrieved March 11, 2014.
  23. ^ Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", p.121. Perspectives of New Music 29, no. 2 (Summer): 106–37.
  24. ^ Secor, George D. and Keenan, David C. (2006). "Sagittal: A Microtonal Notation System", p.2, Originally printed in Xenharmonikôn: An Informal Journal of Experimental Music, Volume 18.

External links [ edit ]

What is this?