Just intonation

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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 5limit 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 12tone 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 equaltempered 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 7limit just intonation – 9:7 (435.08 cents), 81:64 (407.82 cents), and 5:4 (386.31 cents).
Contents
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 equaltempered perfect fifth on the piano with ratio (700.00 cents wide). Rather, it is larger than the equaltempered 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 5limit major and minor thirds have ratios 5:4 and 6:5 respectively. The difference between the Pythagorean major third and the 5limit 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 5limit diatonic scale in his influential text on music theory Harmonics, which he called "tense diatonic".^{[3]} Given ratios of string lengths 120, 112 ^{1}⁄_{2}, 100, 90, 80, 75, 66 ^{2}⁄_{3}, 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: ^{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}.^{[4]}
Diatonic scale [ edit ]
The prominent notes of a given scale may be tuned so that their frequencies form (relatively) small whole number ratios.
The 5limit 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.
5limit 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.
Twelvetone scale [ edit ]
There are several ways to create a just tuning of the twelvetone scale.
Pythagorean tuning [ edit ]
Pythagorean tuning can produce a twelvetone 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 3^{12} / 2^{19} = 531441 / 524288, or about 23 cents, known as the Pythagorean comma. To produce a twelvetone 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 BG♭ if F♯ is discarded). This twelvetone 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 "threelimit" tuning system, because the ratios can be expressed as a product of integer powers of only whole numbers less than or equal to 3.
Fivelimit tuning [ edit ]
A twelvetone 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 fivelimit tuning.
To build such a twelvetone scale (using C as the base note), we may start by constructing a table containing fifteen pitches:
Factor  ^{1}⁄_{9}  ^{1}⁄_{3}  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  
^{1}⁄_{5}  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:
 For each cell of the table, a base ratio is obtained by multiplying the corresponding factors. For instance, the base ratio for the lowerleft cell is 1/9 × 1/5 = 1/45.
 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 2^{6}, 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 2^{6} 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 12tone 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 Cbased Pythagorean and quartercomma 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 fivelimit tuning. It consists of discarding the first column of the table (labeled "^{1}⁄_{9}"). The resulting 12tone 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 twelvetone 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 5^{2} = 25, 5^{−2} = 1/25, 3^{3} = 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 fivelimit 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 
F♯  F♯  G  A♭  A♭  A  A  B♭  B♭  B  B  C 
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 22Śruti scale of tones. There are many different explanations.)
Practical difficulties [ edit ]
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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 DF, and still worse, a minor tone next to a fourth giving 40:27 for DA. Moving D down to 10:9 alleviates these difficulties but creates new ones: DG becomes 27:20, and DB 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 AC can be played as 6:5 while AD 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 CEGAD, in typical 5limit 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 outoftune wolf interval). However the frets may be removed entirely—this, unfortunately, makes intune 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 WellTuned 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 fixedpitch 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 Eflat) 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 builtin 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 scalefree instruments [ edit ]
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The human voice is among the most pitchflexible 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 microtuned if equal temperament is required.
Other wind instruments, although built to a certain scale, can be microtuned 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 7limit 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 (5limit) 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 5limit 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 ]
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 (Play (help·info)) while the just major third is C+E_{1} (Play (help·info)). 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 CE^{0} while the just major third is CE^{−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 fiveline 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 16thcentury 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 ( & ), undecimal (↑ & ↓), tridecimal ( & ), and further primenumber 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 CE+ while the just major third is CE♮.
In 2000–2004, Marc Sabat and Wolfgang von Schweinitz worked in Berlin to develop a different accidentalbased method, the Extended HelmholtzEllis 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 CGDAEBF♯ 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 CE♮ while the just major third is CE♮↓.
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 threesegment arrow" (⤊/⤋) to indicate the unidecimal diesis (ł/ in HelmholtzEllis or ↑/↓ in Johnston's notation).^{[24]}
Audio examples [ edit ]
 Just intonation (help·info) An Amajor scale, followed by three major triads, and then a progression of fifths in just intonation.
 Equal temperament (help·info) An Amajor 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.
 Equal temperament and just intonation compared (help·info) 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.
 Equal temperament and just intonation compared with square waveform (help·info) 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 ]
 List of compositions in just intonation
 Mathematics of musical scales
 Microtonal music
 Microtuner
 Pythagorean interval
 List of intervals in 5limit just intonation
 List of meantone intervals
 List of musical intervals
 List of pitch intervals
 Wholetone scale
 Superparticular number
 Regular number
 Hexany
 Electronic tuner
Notes [ edit ]
Sources [ edit ]
 ^ 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.
 ^ 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.
 ^ ^{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)
 ^ "Qin Tunings, Some Theoretical Concepts". Table 2: Relative positions of studs on the qin.
 ^ ^{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 9780198165057.
 ^ Wright, David (2009). Mathematics and Music, Mathematical World 28 (Providence, Rhode Island: American Mathematical Society) pp. 140–41. ISBN 9780821848739.
 ^ 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 9780252030987.
 ^ Partch, Harry (1979). Genesis of a Music, pp. 165 & 73. ISBN 9780306801068.
 ^ ^{a} ^{b} Danielou, Alain (1968). The Ragas of Northern Indian Music. London: Barrie & Rockliff. ISBN 0214156893.
 ^ Bagchee, Sandeep. Nad: Understanding Raga Music. BPI (India) PVT Ltd. p. 23. ISBN 8186982078.
 ^ Danielou, Alain (1995). Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness (Rep Sub ed.). Inner Traditions. ISBN 0892813369.
 ^ Danielou, Alain (1999). Introduction to the Study of Musical Scales. Oriental Book Reprint Corporation. ISBN 8170690986.
 ^ 19011974, 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.
 ^ "Plainsound Music Edition".
 ^ 3rd Bridge HelixArchived 20120824 at the Wayback Machine by Yuri Landman on furious.com
 ^ 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.
 ^ Benson, David J. (2007). Music: A Mathematical Offering, p.172. ISBN 9780521853873. Cites Eitz, Carl A. (1891). Das mathematischreine Tonsystem. Leipzig.
 ^ Garland, Peter, ed. (1984). The Music of James Tenney. Soundings. Vol. 13. Santa Fe, New Mexico: Soundings Press. OCLC 11371167.
 ^ "Just Intonation Explained", KyleGann.com. Accessed February 2016.
 ^ Johnston & Gilmore (2006), p.7788.
 ^ Manfred Stahnke, ed. (2005). "The Extended HelmholtzEllis JI Pitch Notation: eine Notationsmethode für die natürlichen Intervalle". Mikrotöne und Mehr – Auf György Ligetis Hamburger Pfaden. Hamburg: von Bockel Verlag. ISBN 393269662X.
 ^ Sabat, Marc. "The Extended Helmholtz Ellis JI Pitch Notation" (PDF). Plainsound Music Edition. Retrieved March 11, 2014.
 ^ 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.
 ^ Secor, George D. and Keenan, David C. (2006). "Sagittal: A Microtonal Notation System", p.2, Sagittall.org. Originally printed in Xenharmonikôn: An Informal Journal of Experimental Music, Volume 18.
External links [ edit ]
 Art of the States: microtonal/just intonation works using just intonation by American composers
 The Chrysalis Foundation – Just Intonation: Two Definitions
 Dante Rosati's 21 Tone Just Intonation guitar
 Just Intonation by Mark Nowitzky
 Just intonation compared with meantone and 12equal temperaments; a video featuring Pachelbel's canon.
 Just Intonation Explained by Kyle Gann
 A selection of Just Intonation works edited by the Just Intonation Network web published on the Tellus Audio Cassette Magazine project archive at Ubuweb
 Medieval Music and Arts Foundation
 Music Novatory – Just Intonation
 Why does Just Intonation sound so good?
 The Wilson Archives
 Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900. (2008) Latina, Il Levante
 22 Note Just Intonation Keyboard Software with 12 Indian Instrument Sounds Libreria Editrice
 Plainsound Music Edition – JI music and research, information about the HelmholtzEllis JI Pitch Notation