Just intonation

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Harmonic series, partials 1–5 numbered About this sound Play .

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. The two notes in any just interval are members of the same harmonic series.Template:Efn Frequency ratios involving large integers such as 1024:927 are not generally said to be justly tuned. "Just intonation is the tuning system of the later ancient Greek modes as codified by Ptolemy; it was the aesthetic ideal of the Renaissance theorists; and it is the tuning practice of a great many musical cultures worldwide, both ancient and modern."[1]

Just intonation can be contrasted and compared with equal temperament, which dominates Western instruments of fixed pitch and default MIDI tuning. In equal temperament, all notes are defined as multiples of the same basic interval. Two notes separated by the same number of steps always have exactly the same frequency ratio. However, except for doubled frequencies (octaves), no other 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.


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. By listening to the above file, and then listening to this one, one might be able to hear a slight buzzing in this 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 temperament 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 and just temperaments more obvious.



Harmonic intervals come naturally to horns, vibrating strings, and in human singing voices.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

Recorded history

Pythagorean tuning, perhaps the first tuning system to be theorized in the West,[2] 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 cycle 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 wolf fifth at the end of the cycle, to obtain its closure.

Quarter-comma meantone obtained a more consonant tuning of the major and minor thirds, but when limited to twelve keys (see split keys), the system does not close, leaving a very dissonant diminished sixth between the first and last tones of the cycle of fifths.

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. In late Middle Ages, musicians realized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made consonant. 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 intonated ratio of

(81:64) x (80:81) = 5:4

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

(32:27) x (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 intonated 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 5-limit just intonation.

The Guqin has a musical scale based on harmonic overtone positions. The dots on its soundboard indicate the harmonic positions: 1/8, 1/6, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 5/6, 7/8.[3]

Modern practice

Primary forms of the just tone row from Ben Johnston's String Quartet No. 7, mov. 2[4] About this sound Play  and About this sound Play hexachords . Each permutation contains a just chromatic scale, however, transformations (transposition and inversion) produce pitches outside of the primary row form, as already occurs in the inversion of P0.

Today, despite the dominance of repertoire composed under equal-tempered systems and the prominence of the piano in musical training, musicians often approach just intonation either by accident or design because it is much easier to find (and hear) a point of stability than a point of calculated instability. A cappella groups that depend on close harmonies, such as barbershop quartets, usually use just intonation by design.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Bagpipes, tuned correctly, also use just intonation. There are several conventionally used instruments which, while not associated specifically with just intonation, can handle it quite well, including the trombone and the violin family of instruments.

Diatonic scale


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.[5]

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[5][6][7] (Ptolemy's intense diatonic scale[8]):

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
Natural 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. Others approaches are possible (see 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.

Twelve tone scale

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

Pythagorean tuning


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 GTemplate:Music DTemplate:Music ATemplate:Music ETemplate:Music BTemplate:Music F C G D A E B FTemplate:Music
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 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 multiple 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 FTemplate:Music-DTemplate:Music if GTemplate:Music is discarded, or B-GTemplate:Music if FTemplate:Music 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 "3-limit" tuning system, because the ratios are obtained by using only powers of n, where n is at most 3.

Quarter-comma meantone


The quarter-comma meantone tuning system uses a similar sequence of fifths to produce a twelve tone scale. However, it flattens the fifths by about 5.38 cents with respect to their just intonation, in order to generate justly tuned major thirds (with interval ratio 5:4).

Although this tuning system is based on a just ratio (5:4), it cannot be considered a just intonation system, because most of its intervals are irrational numbers (i.e. they cannot be expressed as fractions of integers). For instance:

Five-limit tuning


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
1 note
1/5 note

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 x 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:

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):

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 GTemplate:Music, according to a convention which was valid even for C-based Pythagorean and 1/4-comma meantone scales. We show here only one of the possible strategies (the others are discussed in Five-limit tuning). It consists of discarding the first column of the table (labeled "1/9"). The resulting 12-tone scale is shown below:

Asymmetric scale
Factor 1/3 1 3 9
5 A
1 F
1/5 DTemplate:Music

Extension of the twelve tone scale

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 (see Five-limit tuning for further details).

Indian scales

In Indian music, the just diatonic scale described above is used, though there are different possibilities, for instance for the 6th 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 Śrutis.[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 DTemplate:Music DTemplate:Music D D ETemplate:Music ETemplate:Music E E F F FTemplate:Music
Cents 0 90 112 182 204 294 316 386 408 498 520 590
Note FTemplate:Music G ATemplate:Music ATemplate:Music A A BTemplate:Music BTemplate:Music B B C
Cents 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

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 F:D, and still worse, a minor tone next to a fourth giving 40:27 for A:D. Moving D down to 10/9 alleviates these difficulties but creates new ones: G:D becomes 27:20, and B:G becomes 27:16.

One can have more frets on a guitar to handle both A's, 9/8 with respect to G and 10/9 with respect to G so that C:A can be played as 6:5 while D:A can still be played as 3:2. 9/8 and 10/9 are less than 1/53 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 farther, 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 can not change keys without retuning the instrument. For instance, a piano 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. Many commercial synthesizers provide the ability to use built-in just intonation scales or to program your own. 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.


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.

Western composers

Most composers don't specify how instruments are to be tuned, although historically most have assumed one tuning system which was common in their time; in the 20th century most composers assumed equal temperament would be used. However, a few have specified just intonation systems for some or all of their compositions, including John Luther Adams, Glenn Branca, Martin Bresnick, Wendy Carlos, Lawrence Chandler, Tony Conrad, Fabio Costa, Stuart Dempster, David B. Doty, Arnold Dreyblatt, Kyle Gann, Kraig Grady, Lou Harrison, Michael Harrison, Ben Johnston, Elodie Lauten, György Ligeti, Douglas Leedy, Pauline Oliveros, Harry Partch, Robert Rich, Terry Riley, Marc Sabat, Wolfgang von Schweinitz, Adam Silverman, James Tenney, Michael Waller, Daniel James Wolf, and La Monte Young. Eivind Groven's tuning system was schismatic temperament, which is capable of far closer approximations to just intonation consonances than 12-note equal temperament or even meantone temperament, but still alters the pure ratios of just intonation slightly in order to achieve a simpler and more flexible system than true just intonation.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

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.[13]

Staff notation

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 on C About this sound Play . 7th: 968.826 cents, a septimal quarter tone lower than BTemplate:Music.

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.[14] 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.[15] 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.[16] 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. 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 (Template:Music & Template:Music), undecimal (Template:Music & Template:Music), tridecimal (Template:Music & Template:Music), and further prime 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-ETemplate:Music.

In the years 2000–2004, Marc Sabat and Wolfgang von Schweinitz worked together 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 following 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 FTemplate:Music and so on. The advantage for musicians is that conventional reading of the basic fourths and fifths remains familiar. Such an approach has also been advocated by Daniel James Wolf. In the Sabat-Schweinitz design, syntonic commas are marked by arrows attached to the flat, natural or sharp sign, Septimal Commas using Giuseppe Tartini's symbol, and Undecimal Quartertones using the common practice quartertone signs (a single cross and backwards flat). 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-ETemplate:Music while the just major third is C-ETemplate:Music-arrow-down.

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 of such notation systems is that they allow the natural harmonic series to be precisely notated.

See also




  1. Gilmore, Bob (2006). "Introduction", "Maximum Clarity" and Other Writings On Music, p.xiv. ISBN 978-0-252-03098-7.
  2. The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts. See: {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  3. Template:Cite web
  4. Fonville, John (Summer, 1991). "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", p.127, Perspectives of New Music, Vol. 29, No. 2, pp. 106–137.
  5. 5.0 5.1 Murray Campbell, Clive Greated (1994). The Musician's Guide to Acoustics, p.172-73. ISBN 978-0-19-816505-7.
  6. Wright, David (2009). Mathematics and Music, p.140-41. ISBN 978-0-8218-4873-9.
  7. Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), "Maximum clarity" and Other Writings on Music, p.78. ISBN 978-0-252-03098-7.
  8. Partch, Harry (1979). Genesis of a Music, p.165&73. ISBN 978-0-306-80106-8.
  9. 9.0 9.1 {{#invoke:citation/CS1|citation |CitationClass=book }}
  10. {{#invoke:citation/CS1|citation |CitationClass=book }}
  11. {{#invoke:citation/CS1|citation |CitationClass=book }}
  12. {{#invoke:citation/CS1|citation |CitationClass=book }}
  13. 3rd Bridge Helix by Yuri Landman on furious.com
  14. 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.
  15. 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.
  16. {{#invoke:citation/CS1|citation |CitationClass=book }}
  17. Johnston & Gilmore (2006), p.77-88.
  18. {{#invoke:citation/CS1|citation |CitationClass=book }}
  19. Template:Cite web
  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.

External links

Template:Musical tuning Template:Instrument tunings Template:Tonality