Just intonation

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In music, just intonation (sometimes abbreviated as JI) or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a pure or just interval. Pure intervals are important in music because they correspond to the harmonic series, a vibrational pattern found in physical objects which correlates to human perception. The two notes in any just interval can be seen as two notes in the harmonic series of a lower implied fundamental.

Without context, "just intonation" often means 5-limit just intonation, in which the ratios of notes have no prime factor greater than 5. Frequency ratios containing large prime factors much larger than 5 or a lot of prime factors, such as 29 : 23 or 3^12 : 2^19 are not generally said to be justly tuned, because the implied fundamental from which they form a harmonic series would be too low. However, frequency ratios (relative to an initial tonic) can have a lot of prime factors due to extensive modulation (change of the tonic), while still being considered justly tuned, provided the ratio with the current tonic remains simple. In rare cases, ratios are used to very closely approximate an EDO tuning system, requiring just intonation to occasionally be distinguished from rational intonation.[1][clarification needed]

Just intonation can be contrasted and compared with 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 multiples of the same basic interval, or more precisely, the intervals are ratios which are integer powers of the smallest step ratio, so two notes separated by the same number of steps always have exactly the same frequency ratio. However, except for doubling of frequencies (one or more octaves), no intervals are exact ratios of small integers. Each just interval differs a different amount from its analogous equally-tempered interval.

Justly tuned intervals can be written as either ratios, with a colon (for example, 3:2), or as fractions, with a solidus (3/2). For example, two tones, one at 300 hertz (cycles per second) and the other at 200 hertz, are both multiples of 100 Hz and as such members of the harmonic series built on 100 Hz. Thus 3:2, known as a perfect fifth, may be defined as the musical interval (the ratio) between the second and third harmonics of any fundamental pitch.

Audio examples[edit | edit source]

  • About this sound Just intonation  An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.
  • About this sound Equal 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 sound Equal 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 sound Equal 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.

History[edit | edit source]

Pythagorean tuning, perhaps the first tuning system to be theorized in the West,[2][not in citation given] is a system in which all tones can be found using powers of the ratio 3:2, an interval known as a perfect fifth. It is easier to think of this system as a circle of fifths. Because a series of 12 fifths with ratio 3:2 does not reach the same pitch class it began with, this system uses a diminished sixth at the end of the cycle, to obtain its closure.[citation needed]

In Pythagorean tuning, the only highly consonant intervals were the perfect fifth and its inversion, the perfect fourth. The Pythagorean major third (81:64) and minor third (32:27) were dissonant, and this prevented musicians from using triads and chords, forcing them for centuries to write music with relatively simple texture.[citation needed] For instance, if one decreases by a syntonic comma (81:80) the frequency of E, C-E (a major third), and E-G (a minor third) become just. Namely, C-E is flattened to a justly tuned ratio of

(81:64) × (80:81) = 5:4

and at the same time E-G is widened to the just ratio of

(32:27) × (81:80) = 6:5

The drawback is that the fifths A-E and E-B, by flattening E, become almost as dissonant as the Pythagorean wolf fifth. But the fifth C-G stays consonant, since only E has been flattened (C-E × E-G = (5:4) × (6:5) = 3:2), and can be used together with C-E to produce a C-major triad (C-E-G).

By generalizing this simple rationale, Gioseffo Zarlino, in the late sixteenth century, created the first justly tuned 7-tone (diatonic) scale, which contained pure perfect fifths (3:2), pure major thirds, and pure minor thirds:

F → A → C → E → G → B → D

This is a sequence of just major thirds (M3, ratio 5:4) and just minor thirds (m3, ratio 6:5), starting from F:

F + M3 + m3 + M3 + m3 + M3 + m3

Since M3 + m3 = P5 (perfect fifth), i.e., (5:4) × (6:5) = 3:2, this is exactly equivalent to the diatonic scale obtained in five-limit just intonation.

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.[3]

Diatonic scale[edit | edit source]

It is possible to tune the familiar diatonic scale or chromatic scale in just intonation in many ways, all of which make certain chords purely tuned and as consonant and stable as possible, and the other chords not accommodated and considerably less stable.

Primary triads in C About this sound Play .
Just tuned diatonic scale derivation.[4]

The prominent notes of a given scale are tuned so that their frequencies form ratios of relatively small integers. For example, in the key of G major, the ratio of the frequencies of the notes G to D (a perfect fifth) is 3:2, while that of G to C (a perfect fourth) is 4:3. Three basic intervals can be used to construct any interval involving the prime numbers 2, 3, and 5 (known as 5-limit just intonation):

which combine to form:

  • 6:5 = Ts (minor third)
  • 5:4 = Tt (major third)
  • 4:3 = Tts (perfect fourth)
  • 3:2 = TTts (perfect fifth)
  • 2:1 = TTTttss (octave)

A just diatonic scale may be derived as follows. Suppose we insist that the chords F-A-C, C-E-G, and G-B-D be just major triads (then A-C-E and E-G-B are just minor triads, but D-F-A is not).

Then we obtain this scale[4][5][6] (Ptolemy's intense diatonic scale):[7]

Note Name C D E F G A B C
Ratio 1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1
Harmonic 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

The major thirds are correct, and two minor thirds are right, but D-F is a 32:27 semiditone (according to the harmonic numbers of those tones within the harmonic series). Other approaches are possible, such as five-limit tuning, but it is impossible to get all six above-mentioned chords correct. Concerning triads, the triads on I, IV, and V are 4:5:6, the triad on ii is 27:32:40, the triads on iii and vi are 10:12:15, and the triad on vii is 45:54:64.

For a justly tuned minor scale two of the steps must be switched to obtain optimal minor thirds, as in five-limit tuning. Another variant (see Template:Article section) uses the minor tone of 10/9 and gives the minor seventh a ratio of 16:9 (=1.778) instead of 9:5, which is closer to the equal temperament tuning of 1000 cents (a logarithmic unit of measure used for musical intervals, dividing the octave into 12 semitones of 100 cents each), i.e., a ratio of 210/12:1 = 1.782, but then major and minor thirds including this tone are out of just intonation.

Twelve-tone scale[edit | edit source]

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

Pythagorean tuning[edit | edit source]

The oldest known form of tuning, 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 difference in pitch of nearly 24 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.[8] 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 are obtained by using only powers of n, where n is at most 3.

Five-limit tuning[edit | edit source]

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, we may start by constructing a table containing fifteen pitches:

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

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 | edit source]

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 | edit source]

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.[9]

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.[10][11] 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.[8] 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 | edit source]

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.

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 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 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 unfretted stringed instruments[edit | edit source]

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.

Western composers[edit | edit source]

Music written in just intonation is most often tonal but need not be; 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 Ben Johnston's Sonata for Microtonal Piano (1964) uses serialism to achieve an atonal result. Composers often impose a limit on how complex the ratios used are: for example, a composer may write in "7-limit JI", meaning that no prime number larger than 7 features in the ratios they use. Under this scheme, the ratio 10:7, for example, would be permitted, but 11:7 would not be, as all non-prime numbers are octaves of, or mathematically and tonally related to, lower primes (example: 12 is a double octave of 3, while 9 is a square of 3). Yuri Landman derived a just intoned musical scale from an initially considered atonal prepared guitar playing technique based on adding a third bridge under the strings. When this bridge is positioned in the noded positions of the harmonic series the volume of the instrument increases and the overtone becomes clear and has a consonant relation to the complementary opposed string part creating a harmonic multiphonic tone.[12]

Staff notation[edit | edit source]

Legend of the HE Accidentals
Ex. 1: Legend of the HE Accidentals
Pythagorean diatonic scale on C About this sound Play . Johnston's notation.
Just intonation diatonic scale on C About this sound Play . 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 sound Play . 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.[13] For example, the Pythagorean major third on C is C+E (About this sound Play ) while the just major third is C+E1 (About this sound Play ). 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.[14] 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.[15] 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."[16]

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".[17] 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.[18] 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.[19] 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[20] using Johnston's notation About this sound Play 

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

Saggittal notation is based on notation of equal temperaments to 72-tet 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).[21]

See also[edit | edit source]

Notes[edit | edit source]

Sources[edit | edit source]

  1. "Rational Intonation". Tonalsoft. 
  2. 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. 
  3. "Qin Tunings, Some Theoretical Concepts". Table 2: Relative positions of studs on the qin. 
  4. 4.0 4.1 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.
  5. Wright, David (2009). Mathematics and Music, Mathematical World 28 (Providence, Rhode Island: American Mathematical Society) pp. 140–41. ISBN 978-0-8218-4873-9.
  6. 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.
  7. Partch, Harry (1979). Genesis of a Music, pp. 165 & 73. ISBN 978-0-306-80106-8.
  8. 8.0 8.1 Danielou, Alain (1968). The Ragas of Northern Indian Music. Barrie & Rockliff, London. ISBN 0-214-15689-3. 
  9. Bagchee, Sandeep. Nad: Understanding Raga Music. BPI (India) PVT Ltd. p. 23. ISBN 81-86982-07-8. 
  10. Danielou, Alain (1995). Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness. Inner Traditions; Rep Sub edition. ISBN 0892813369. 
  11. Danielou, Alain (1999). Introduction to the Study of Musical Scales. Oriental Book Reprint Corporation. ISBN 8170690986. 
  12. 3rd Bridge Helix Archived 2012-08-24 at the Wayback Machine. by Yuri Landman on
  13. 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.
  14. 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.
  15. Garland, Peter, ed. (1984). The Music of James Tenney. Soundings. Vol. 13. Santa Fe, New Mexico: Soundings Press. OCLC 11371167. 
  16. "Just Intonation Explained", Accessed February 2016.
  17. Johnston & Gilmore (2006), p.77-88.
  18. 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. 
  19. Sabat, Marc. "The Extended Helmholtz Ellis JI Pitch Notation" (PDF). Plainsound Music Edition. Retrieved March 11, 2014. 
  20. 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.
  21. 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 | edit source]


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