Just intonation - Knowledge

4125: 4093: 4079: 779: 2564:. For instance, multiplying the frequency of a note by 2 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: 3794: 3812: 4111: 3828: 1451: 4053: 3576:(added-tone and extended) chords usually require intervals beyond common 5 limit ratios in order to sound harmonious (for instance, the previous chord could be tuned to 8:10:12:13:18, using the A note from the 13th harmonic), which implies even more keys or frets. 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. 3955: 3972: 1154: 6089: 4068:

estimating pitch height. This may be contrasted with the more abstract use of ratios for representing pitches in which the amount by which two pitches differ and the "direction" of change may not be immediately obvious to most musicians. One caveat is the requirement for performers to learn and internalize a (large) number of new graphical symbols. However, the use of unique symbols reduces harmonic ambiguity and the potential confusion arising from indicating only cent deviations.

356: 1137: 4016:, incorporating Ellis' invention of cents, and continuing Johnston's step into "Extended JI", Sabat and Schweinitz propose unique symbols (accidentals) for each prime dimension of harmonic space. In particular, the conventional flats, naturals and sharps define a Pythagorean series of perfect fifths. The Pythagorean pitches are then paired with new symbols that commatically alter them to represent various other partials of the harmonic series 3677: 3476: 179: 3930: 3922: 6058: 3906: 281: 25: 3898: 121: 4045: 66: 3758:

to adjust the pitch by pushing the hand in deeper to flatten the note, or pulling it out to sharpen the note while playing. Some natural horns also may adjust the tuning with the hand in the bell, and valved cornets, trumpets, Flugelhorns, Saxhorns, Wagner tubas, and tubas have overall and valve-by-valve tuning slides, like valved horns.

3809:(1877); the starting note is presumed Pythagorean; a “+” is placed between if the next note is a just major third up, a “−” if it is a just minor third, among others; finally, subscript numbers are placed on the second note to indicate how many syntonic commas (81:80) to lower by. For example, the Pythagorean major third on C is C+E ( 3845:(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. For example, the Pythagorean major third on C is C−E while the just major third is C−E. An extension of this Pythagorean-based notation to higher primes is the 4126: 478:. Since 5-limit has been the most prevalent just intonation used in western music, western musicians have subsequently tended to consider this scale to be the only version of just intonation. In principle, there are an infinite number of possible "just intonations," since the harmonic series is infinite. 3780:

on how complex the ratios may become. 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

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Trombones have a slide that allows arbitrary tuning during performance. French horns can be tuned by shortening or lengthening the main tuning slide on the back of the instrument, with each individual rotary or piston slide for each rotary or piston valve, and by using the right hand inside the bell

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The unfretted stringed instruments such as those from the violin family (the violin, the viola, and 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

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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

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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. The primary reason for its use is that it is extremely easy to tune, as its

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In Western musical practice, bowed instruments such as violins, violas, cellos, and double basses are tuned using pure fifths or fourths. In contrast, keyboard instruments are rarely tuned using only pure intervals—the desire for different keys to have identical intervals in Western music makes this

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when the approximately equivalent flat note is substituted for a sharp note not available in the scale, or vice versa. 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.

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less than a Pythagorean perfect fifth 3:2. To write a perfect fifth, Johnston introduces a pair of symbols, + and − again, to represent this comma. 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

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In trying to get a more just system for instruments that is more adaptable like the human voice and fretless instruments, the tuning trade-offs between more consonant harmony versus easy transposability (between different keys) have traditionally been too complicated to solve mechanically, though

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The great advantage of such notation systems is that they allow the natural harmonic series to be precisely notated. At the same time, they provide some degree of practicality through their extension of staff notation, as traditionally trained performers may draw on their intuition for roughly

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Since the advent of personal computing, there have been more attempts to solve the perceived problem by trying to algorithmically solve what many professional musicians have learned through practice and intuition. Four of the main problems are that consonance cannot be perfect for some complex

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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). Most complex

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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

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chords, chords can have internal consistency but clash with the overall direction of the piece, and naively adjusting the tuning only taking into account chords in isolation can lead to a drift where the end of the piece is noticeably higher or lower in overall pitch rather than centered.

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scales. It is also possible to make diatonic scales that do not use fourths or fifths (3 limit), but use 5 and 7 limit intervals only. Thus, the notion of limit is a helpful distinction, but certainly does not tell us everything there is to know about a particular scale.

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often analyze solutions chord by chord instead of taking in the global context of the whole piece like it's theorized human players do. Since 2017, there has been research to address these problems algorithmically through dynamically adapted just intonation and machine learning.

415:. For example, in the diagram, if the notes G3 and C4 (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just 4094: 3963: 4080: 3990: 745: 4134:

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

4064:, "arrow") is a system of arrow-like accidentals that indicate prime-number comma alterations to tones in a Pythagorean series. It is used to notate both just intonation and equal temperaments. The size of the symbol indicates the size of the alteration. 2118:

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.

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are very small intervals that result from minute differences between pairs of just intervals. For example, the (5 limit) 5:4 ratio is different from the Pythagorean (3 limit) major third (81:64) by a difference of 81:80, called the

4649: 1145: 2720:, close to half an octave, above the tonic C, which is a discordant 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 are reasons to avoid it. 364: 3964: 3991: 753:

is a measure of interval size. It is logarithmic in the musical frequency ratios. The octave is divided into 1200 steps, 100 cents for each semitone. Cents are often used to describe how much a just interval deviates from

3813: 4020:. To facilitate quick estimation of pitches, cents indications may be added (e.g. downward deviations below and upward deviations above the respective accidental). A typically used convention is that cent deviations refer to the 2539:

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

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implied by the flat, natural, or sharp. A complete legend and fonts for the notation (see samples) are open source and available from the Plainsound Music Edition website. For example, the Pythagorean major third on C is

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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

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Flattening D by a comma 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.

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is a 3 limit interval because both numerator and denominator are multiples of 3 and 2. It is possible to have a scale that uses 5 limit intervals but not 2 limit intervals, i.e. no octaves, such as

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https://www.researchgate.net/publication/317591114_Playing_Music_in_Just_Intonation_-_A_Dynamically_Adapting_Tuning_Scheme&ved=2ahUKEwjU7p3_tL6EAxXJbPUHHXrRCTcQFnoECEoQAQ&usg=AOvVaw2j0eIzdqpVVVysw57bGaIe

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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.

4720: 4012: 3630:, 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 2050:. 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 3988: 4112: 2990:

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 = 25, 5 =

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by later writers, but may have been analyzed by other early Greeks or other early cultures as well. The oldest known description of the Pythagorean tuning system appears in Babylonian artifacts.

2054:) 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 2723:

The following chart shows one way to obtain a 12 tone scale by removing one note for each pair of enharmonic notes. In this method one discards the first column of the table (labeled "

3552:(or keys on a piano) 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 3579:

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

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is included in 5 limit, because it has 5 in the denominator. If a scale uses an interval of 21:20, it is a 7 limit just intonation, since 21 is a multiple of 7. The interval

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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.

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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

3961: 490:. The limit refers to the highest prime number fraction included in the intervals of a scale. All the intervals of any 3 limit just intonation will be multiples of 3. So 3885:. His notation "begins with the 16th-century Italian definitions of intervals and continues from there." Johnston notation is based on a diatonic C Major scale tuned in JI 3640:“Don’t be scared if your intonation differs from that of the piano. It is the piano that is out of tune. The piano with its tempered scale is a compromise in intonation." - 2478:

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:

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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.

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worked in Berlin to develop a different accidental-based method, the Extended Helmholtz-Ellis JI Pitch Notation. Following the method of notation suggested by

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of an octave apart, so mechanical and performance considerations have made this approach extremely rare. And the problem of how to tune complex chords such as

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A 12 tone scale is obtained by removing one note for each couple of enharmonic notes. This can be done in four ways that have in common the removal of G

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Note the use of the “+” between just major thirds, “−” between just minor thirds, “|” between Pythagorean minor thirds, and “±” between perfect fifths.

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Fig. 4: Comparison of Helmholtz-Ellis JI Pitch Notation and Johnston Notation. Unaltered naturals in Helmholtz-Ellis may be omitted if desired.

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is 400 cents in 12 TET, but the 5th harmonic, 5:4 is 386.314 cents. Thus, the just major third deviates by −13.686 cents.

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

5644: 5200: 4170: 3934:), and further prime-number extensions to create an accidental based exact JI notation for what he has named "Extended Just Intonation" ( 3593:

goes even further, and uses the tempo of beat patterns produced by some dissonant intervals as an integral part of several movements.

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Many singers (especially barbershop quartets) and fretless instrument players naturally aim for a more just intonation when playing:

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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

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pitches, indicated in a legend or directly in the score, allowing performers to readily use electronic tuning devices if desired.

5265: 4165: 3022:, the just diatonic scale described above is used, though there are different possibilities, for instance for the sixth pitch ( 2122:

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

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Fig. 3: Just harmonic seventh chord (4:5:6:7:8) on C in Johnston notation. The size of the 7th is 968.826 cents:

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

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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):

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

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have proven a valuable tool for composers wanting to experiment with just intonation. They can be easily retuned with a

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Non-Western music, particularly that built on pentatonic scales, is largely tuned using just intonation. In China, the

427:, in which all intervals other than octaves consist of irrational-number frequency ratios. Acoustic pianos are usually 222: 4932:, ed. (2005). "The Extended Helmholtz-Ellis JI Pitch Notation: eine Notationsmethode für die natürlichen Intervalle". 3620:. Many commercial synthesizers provide the ability to use built-in just intonation scales or to create them manually. 4941: 4677: 4602: 4574: 4542: 4517: 4321: 3724: 3523: 342: 324: 262: 160: 102: 52: 3706: 3505: 306: 5637: 3698: 3497: 554:

regardless of its tuning method, for purposes of discussing tuning systems musicologists may distinguish between a

196: 38: 3152:. According to some musicians, one has a scale of a given 12 pitches and ten in addition (the tonic, shadja ( 5238: 3462:(This is just one example of explaining a 22 Śhruti scale of tones. There are many different explanations.) 782:

Just (black) major and parallel minor triad, compared to its equal temperament (gray) approximations, within the

740:{\displaystyle \ \left({\tfrac {\ 32\ }{27}}\right)\div \left({\tfrac {\ 7\ }{6}}\right)={\tfrac {\ 64\ }{63}}~.} 229: 3793: 3589:

by Terry Riley use a combination of very consonant and dissonant intervals for musical effect. In "Revelation",

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This scale is "asymmetric" in the sense that going up from the tonic two semitones we multiply the frequency by

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Other wind instruments, although built to a certain scale, can be micro-tuned to a certain extent by using the

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is obtained by multiplying the corresponding factors. For instance, the base ratio for the lower-left cell is

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An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. The

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is occasionally used to distinguish the 5:4 ratio from major thirds created using other tuning methods.

542:, or 3 limit tuning, allows ratios including the numbers 2 and 3 and their powers, such as 3:2, a 211: 5125: 3606:) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes 5630: 5361: 3590: 4056:

Fig. 5: Just harmonic thirteenth chord (4:5:6:7:9:11:13) on G in Sagittal notation (with mnemonics)

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An A-major scale, followed by three major triads, and then a progression of fifths in just intonation.

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there are audible examples of the difference in sound between equal temperament and just intonation.

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ratios for a given letter name or swara, we have a difference of 81:80 (22 cents), which is the

2098: 432: 1953:) to the left and six to the right. Each step consists of a multiplication of the previous pitch by 138: 5474: 4819: 4749: 3958:

Fig. 2: Staff notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C using Johnston notation

3687: 3486: 4725: 6188: 5961: 5797: 5610: 5444: 5270: 5195: 5118: 4427: 4374:(Reprint of 1st ed.). London, UK & New York: Oxford University Press. pp. 172–173. 4180: 4003: 3877: 3691: 3490: 3148:

Some accounts of Indian intonation system cite a given 12 swaras being divided into 22

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within a piece, or playing a repertoire of pieces in different keys, impractical to impossible.

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Note that the powers of 2 used in the second step may be interpreted as ascending or descending

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impractical. Some instruments of fixed pitch, such as electric pianos, are commonly tuned using

6070: 5867: 5753: 5466: 5429: 5419: 5228: 4266: 2713: 1450: 615:, the ratio of 64:63, is a 7 limit interval which is the distance between the Pythagorean 2556:) is multiplied by 2, and the resulting ratio is 64:45, which is a number between 1:1 and 2:1. 5462: 4715: 4241: 4221: 4007: 3976: 3806: 412: 236: 5717: 5671: 4741: 1153: 6183: 5981: 5976: 5470: 5415: 5397: 5392: 5387: 5382: 5377: 5372: 5367: 5352: 5347: 5342: 5337: 5332: 5327: 5039: 4231: 3581: 3010:. A scale with 25, 35 or even more pitches can be obtained by combining these base ratios. 881: 756: 636: 563: 475: 44: 5016: 3942:). For example, the Pythagorean major third on C is C-E+ while the just major third is C-E 2455:

The factors listed in the first row and column are powers of 3 and 5, respectively (e.g.,

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thirds (5:4) above F C G respectively. Johnston introduces new symbols for the septimal (

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ensembles naturally tend toward just intonation because of the comfort of its stability.

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Some fixed just intonation scales and systems, such as the diatonic scale above, produce

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Intervals of the C major scale in just intonation compared with equal temperament (blue)

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Genesis of a music : an account of a creative work, its roots and its fulfillments

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beginning and ending on the third note) – 16:15, 9:8, 10:9, 9:8, 16:15, 9:8, and 10:9.

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encompasses ratios additionally using the number 5 and its powers, such as 5:4, a

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When tuned in just intonation, many fixed-pitch instruments cannot be played in a new

6161: 5860: 5478: 5439: 5314: 5253: 5043: 4937: 4827: 4753: 4683: 4673: 4598: 4570: 4538: 4513: 4469: 4439: 4404: 4375: 4327: 4317: 3842: 2113: 2089: 2047: 1974: 1131: 889: 573: 567: 455: 424: 5504: 5098: 4892:(Summer 1991). "Ben Johnston's Extended Just Intonation: A guide for interpreters". 4846: 4590: 4558: 4505: 3954: 884:) and several of his own discovery / invention, including many interval patterns in 6205: 6151: 5991: 5914: 5899: 5810: 5458: 5322: 5290: 5210: 5175: 5076: 4903: 4565:

Music and the Power of Sound: The influence of tuning and interval on consciousness

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and about 0.8 Hz. In the just intonation triad, this roughness is absent. The

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there have been attempts throughout history with various drawbacks, including the

3459:(25:24). One can see the symmetry, looking at it from the tonic, then the octave. 2962:, while going down from the tonic two semitones we do not divide the frequency by 6156: 5941: 5919: 5722: 5580: 4929: 1164:

The 5-limit diatonic major scale is tuned in such a way that major triads on the

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makes the difference between equal temperament and just intonation more obvious.

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The phrase "just intonation" is used both to refer to one specific version of a

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ratio. This is mentioned by Schenker in reference to the teaching of Bruckner.

1472:, about 680 cents, noticeably smaller than the 702 cents of the pure 1189: 893: 612: 608: 589: 439: 416: 389: 381: 5035: 4438:. Urbana and Chicago, Illinois: University of Illinois Press. pp. 77–88. 550:. Although the interval from C to G is called a perfect fifth for purposes of 6235: 6220: 6141: 6001: 5836: 5748: 5712: 5553: 5522: 5295: 5248: 5155: 5081: 4958: 4889: 4269:(May 1994). "The Babylonian musical notation and the Hurrian melodic texts". 3921: 3777: 3537: 2712:, according to a convention which was valid even for C-based Pythagorean and 2097:

building block, the perfect fifth, is the simplest and consequently the most

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Ptolemy describes a variety of other just intonations derived from history (

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described a 5-limit diatonic scale in his influential text on music theory

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22 Note Just Intonation Keyboard Software with 12 Indian Instrument Sounds

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A selection of Just Intonation works edited by the Just Intonation Network

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On the Sensations of Tone as a Physiological Basis for the Theory of Music

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On the Sensations of Tone as a Physiological Basis for the Theory of Music

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Fig. 1: Legend of the Helmholtz-Ellis accidentals within the 23-limit

849:, and 60, Ptolemy quantified the tuning of what would later be called the 811:, which he called "intense diatonic". Given ratios of string lengths 120, 5986: 5951: 5936: 5841: 5779: 5696: 5691: 5605: 5600: 5590: 5185: 5165: 4431: 4144: 3882: 3613: 3597: 1750:

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

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positions. The dots on its soundboard indicate the harmonic positions:

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in this file may be more noticeable after listening to the above file.

1949:). Starting from C, they are obtained by moving six steps (around the 1136: 5831: 5758: 5585: 5539: 5053: 4860: 4767: 3838: 3569: 865: 396: 5030: 3676: 3475: 309:. Statements consisting only of original research should be removed. 178: 6078: 6043: 5956: 5883: 5791: 5738: 5681: 4987: 4907: 4128:

Equal temperament and just intonation compared with square waveform

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Originally a system of notation to describe scales was devised by

3026:), and further modifications may be made to all pitches excepting 6178: 6074: 6057: 5894: 5686: 5559: 5509: 5071: 1177: 804: 4982: 4044: 5527: 5243: 5170: 5066: 4206: 3549: 3447:

or the praman in Indian music theory. These notes are known as

2561: 2051: 4430:(2006) . "A notation system for extended Just Intonation". In 4399:. Mathematical World. Vol. 28. Providence, Rhode Island: 3738:

accompaniment (with its attendant constraints on pitch), most

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symbols and prime-factor-power vectors described in Monzo's

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The Chrysalis Foundation – Just Intonation: Two Definitions

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The Music of James Tenney, Volume 1: Contexts and Paradigms

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are tuned in the proportion 4:5:6, and minor triads on the

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is discarded). This twelve-tone scale is fairly close to

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Xenharmonikôn: An Informal Journal of Experimental Music

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Mikrotöne und Mehr – Auf György Ligetis Hamburger Pfaden

4625: 1458:

In this example the interval from D up to A would be a

4997:. Vol. 18. pp. 1–2 – via Sagittal.org. 4316:. Cambridge: Cambridge University Press. p. 350. 3865:, amongst others, preferred to combine JI ratios with 712: 685: 654: 403:

tuned in this way is said to be pure, and is called a

3666: 2985: 2978:. For two methods that give "symmetric" scales, see 645: 141:. Please help to ensure that disputed statements are 4500: 4498: 4352:. Table 2: Relative positions of studs on the 3754:so that the pitches differ from equal temperament. 203:. Unsourced material may be challenged and removed. 75:

may be too technical for most readers to understand

5046:temperaments; a video featuring Pachelbel's canon. 4802:Monzo. "Helmholtz / Ellis / Wolf / Monzo system". 4665: 4562: 739: 486:Just intonations are categorized by the notion of 5013:works using just intonation by American composers 4583: 4551: 6233: 4959:"The Extended Helmholtz Ellis JI Pitch Notation" 4928: 4739: 4495: 446:, as well to a whole class of tunings which use 4370:Campbell, Murray & Greated, Clive (2001) . 4369: 1945:The ratios are computed with respect to C (the 4114:Equal temperament and just intonation compared 6028: 5868: 5638: 5126: 5022:Dante Rosati's 21 Tone Just Intonation guitar 5011:Art of the States: microtonal/just intonation 4850:(University of Illinois Press, 2021), 288-89. 4797: 4795: 4766: 4708: 4436:'Maximum Clarity' and Other Writings on Music 4422: 4420: 4394: 5201:List of intervals in 5-limit just intonation 4171:List of intervals in 5-limit just intonation 5093:Enharmonic instruments and music, 1470–1900 4595:Introduction to the Study of Musical Scales 3705:. Unsourced material may be challenged and 3504:. Unsourced material may be challenged and 2092:, but it does not offer much advantage for 53:Learn how and when to remove these messages 6035: 6021: 5875: 5861: 5645: 5631: 5133: 5119: 4792: 4784:: CS1 maint: location missing publisher ( 4700:: CS1 maint: location missing publisher ( 4417: 435:, and thus with no pure intervals at all. 5652: 4714: 4365: 4363: 3725:Learn how and when to remove this message 3524:Learn how and when to remove this message 3465: 2004:notes at both ends of this sequence is a 343:Learn how and when to remove this message 325:Learn how and when to remove this message 263:Learn how and when to remove this message 161:Learn how and when to remove this message 103:Learn how and when to remove this message 87:, without removing the technical details. 6042: 4988:"Sagittal: A Microtonal Notation System" 4888: 4589: 4557: 4504: 4426: 4346:"Qin tunings, some theoretical concepts" 4051: 4043: 3970: 3953: 3792: 1449: 1152: 1135: 777: 354: 137:Relevant discussion may be found on the 5082:Why does Just Intonation sound so good? 4818: 4672:(second, enlarged ed.). New York. 4532: 4166:List of compositions in just intonation 3847:Helmholtz / Ellis / Wolf / Monzo system 558:created using the 3:2 ratio and a 6234: 4748:. Cambridge University Press. p. 4660: 4569:(Rep Sub ed.). Inner Traditions. 4458: 4360: 4308: 2101:interval after the octave and unison. 359:Harmonic series, partials 1–5 numbered 16:Musical tuning based on pure intervals 6016: 5856: 5626: 5114: 4859: 4620: 4618: 4616: 4614: 4597:. Oriental Book Reprint Corporation. 1753: 1157:Just tuned diatonic scale derivation. 85:make it understandable to non-experts 4304: 4302: 4265: 3771: 3703:adding citations to reliable sources 3670: 3502:adding citations to reliable sources 3469: 2980:Five-limit tuning: twelve-tone scale 2107: 1745: 1199:) on C is shown in the table below: 274: 201:adding citations to reliable sources 172: 114: 59: 18: 5140: 3837:). A similar system was devised by 3821:) while the just major third is C+E 13: 5072:Medieval Music and Arts Foundation 4654: 4611: 4537:. India: BPI PVT Ltd. p. 23. 4510:The Ragas of Northern Indian Music 3667:Singing and scale-free instruments 2986:Extension of the twelve-tone scale 1195:The 5-limit diatonic major scale ( 1188:must be microtonally lowered by a 14: 6253: 5004: 4956: 4801: 4512:. London: Barrie & Rockliff. 4372:The Musician's Guide to Acoustics 4299: 4071: 4031:while the just major third is C-E 3788: 1125: 896:, and even an 11-limit diatonic. 562:using some other system, such as 34:This article has multiple issues. 6087: 6056: 5276:Ptolemy's intense diatonic scale 5077:Music Novatory – Just Intonation 4772:Das mathematisch-reine Tonsystem 3986: 3959: 3928: 3920: 3904: 3896: 3675: 3624:used a system on her 1986 album 3474: 3156:), and the pure fifth, pancham ( 3013: 1197:Ptolemy's intense diatonic scale 1141: 1118:. Indian music has an extensive 360: 279: 177: 119: 64: 23: 4976: 4950: 4922: 4882: 4853: 4838: 4812: 4733: 4643: 4526: 1122:for tuning in just intonation. 188:needs additional citations for 42:or discuss these issues on the 5822:part relation with additional 5063:Tellus Audio Cassette Magazine 4936:. Hamburg: von Bockel Verlag. 4744:Music: A Mathematical Offering 4482: 4452: 4388: 4338: 4259: 4060:Sagittal notation (from Latin 2482:For each cell of the table, a 803:During the second century AD, 592:and higher systems use higher 481: 1: 5706:Dynamic intonation adjustment 4535:Nad: Understanding Raga music 4401:American Mathematical Society 4253: 3768:or adjustments to fingering. 903:has a musical scale based on 5882: 5820:Physical just-intoned string 5239:Harry Partch's 43-tone scale 3746:are a good example of this. 1973:(ascending fifth), or their 1192:to form a pure minor triad. 792:has been attributed to both 7: 5095:. (2008) Latina, Il Levante 4986:; Keenan, David C. (2006). 4865:"Just Intonation Explained" 4724:. Longmans, Green. p. 4151: 3585:by La Monte Young and 2474:Colors indicate couples of 440:5-limit diatonic intonation 305:the claims made and adding 10: 6258: 5732:Just intonation in one key 5672:Fretless string instrument 5206:List of meantone intervals 4964:. Plainsound Music Edition 4176:List of meantone intervals 2716:scales. Note that it is a 2111: 1757: 1266:Harmonic of Fundamental F 1129: 773: 444:Ptolemy's intense diatonic 6242:Just tuning and intervals 6127: 6096: 6085: 6063: 6054: 5905:Consonance and dissonance 5890: 5819: 5790: 5767: 5731: 5718:Fretted string instrument 5705: 5661: 5573: 5490: 5453: 5406: 5313: 5304: 5219: 5196:List of musical intervals 5191:Consonance and dissonance 5148: 5050:Just Intonation Explained 4895:Perspectives of New Music 4824:The Music of James Tenney 4740:Benson, David J. (2007). 4533:Bagchee, Sandeep (1998). 4038: 4017: 3948: 3939: 3935: 3886: 3781:those prime numbers ≤ 7 ( 3776:Composers often impose a 2869: 2826: 2782: 2746: 2339: 2257: 2179: 1738: 1735: 1732: 1729: 1726: 1723: 1720: 1712: 1709: 1706: 1703: 1700: 1697: 1694: 1686: 1683: 1680: 1677: 1674: 1671: 1668: 1665: 1662: 1656: 1613: 1610: 1607: 1604: 1601: 1598: 1595: 1592: 1584: 1581: 1578: 1575: 1572: 1569: 1566: 1563: 1556:Harmonic of Fundamental B 1550: 1547: 1544: 1541: 1538: 1535: 1532: 1529: 1521: 1518: 1515: 1512: 1509: 1506: 1503: 1500: 1494: 1443: 1440: 1437: 1434: 1431: 1428: 1425: 1417: 1414: 1411: 1408: 1405: 1402: 1399: 1391: 1388: 1385: 1382: 1379: 1376: 1373: 1370: 1367: 1361: 1318: 1315: 1312: 1309: 1306: 1303: 1300: 1297: 1289: 1286: 1283: 1280: 1277: 1274: 1271: 1268: 1260: 1257: 1254: 1251: 1248: 1245: 1242: 1239: 1231: 1228: 1225: 1222: 1219: 1216: 1213: 1210: 1204: 5105:Plainsound Music Edition 4139:can be heard at about 4 3876:Beginning in the 1960s, 3658:Software solutions like 600:(e.g. 11, 13, 17, etc.) 395:(such as 3:2 or 4:3) of 5962:Otonality and utonality 5768:Retunable to a just key 4805:Tonalsoft Encyclopaedia 4181:List of pitch intervals 4041:for "combined" symbol) 4004:Wolfgang von Schweinitz 3855:Tonalsoft Encyclopaedia 584:. The specialized term 462:" tunings of the early 5754:Long-string instrument 4716:Helmholtz, Hermann von 4395:Wright, David (2009). 4314:Greek musical writings 4057: 4049: 3995: 3977:48.77 cents lower 3968: 3798: 3587:The Harp of New Albion 3466:Practical difficulties 2714:quarter-comma meantone 1455: 1158: 1150: 786: 741: 448:whole number intervals 369: 5463:Temperament ordinaire 5061:web published on the 4397:Mathematics and Music 4242:Superparticular ratio 4222:Music and mathematics 4055: 4047: 3974: 3957: 3796: 1489:. For example, on A: 1453: 1156: 1139: 1120:theoretical framework 781: 742: 358: 5982:Schenkerian analysis 5977:Progressive tonality 5775:Keyboard instruments 5266:List of compositions 5091:Barbieri, Patrizio. 4844:Wannamaker, Robert, 4468:. pp. 165, 73. 4403:. pp. 140–141. 4232:Pythagorean interval 3869:deviations from the 3699:improve this section 3632:Secrets of Synthesis 3582:The Well-Tuned Piano 3498:improve this section 2076:is discarded, or B–G 1963:(descending fifth), 643: 639:, 7:6 , since 637:septimal minor third 476:Meantone temperament 197:improve this article 130:factual accuracy is 5087:The Wilson Archives 5065:project archive at 4902:(2): 121, 106–137. 4488:Heinrich Schenker, 4285:10.1093/ml/75.2.161 4272:Music & Letters 3744:Barbershop quartets 3627:Beauty in the Beast 1140:Primary triads in C 853:(equivalent to the 830:, 100, 90, 80, 75, 6064:Division and ratio 5920:Secondary function 5677:Pedal steel guitar 5500:Chinese musicology 5286:Scale of harmonics 5281:Pythagorean tuning 5229:Euler–Fokker genus 4465:Genesis of a Music 4058: 4050: 3996: 3985:tuned 9:5 above C. 3969: 3799: 3544:One can have more 1765:Pythagorean tuning 1760:Pythagorean tuning 1754:Pythagorean tuning 1456: 1159: 1151: 790:Pythagorean tuning 787: 737: 729: 702: 671: 540:Pythagorean tuning 456:equal temperaments 370: 290:possibly contains 6229: 6228: 6010: 6009: 5915:Diatonic function 5850: 5849: 5620: 5619: 5486: 5485: 5101:Libreria Editrice 4759:978-0-521-85387-3 4475:978-0-306-80106-8 4445:978-0-252-03098-7 4410:978-0-8218-4873-9 4381:978-0-19-816505-7 4096:Equal temperament 3992: 3965: 3785:2, 3, 5, and 7). 3772:Western composers 3735: 3734: 3727: 3534: 3533: 3526: 3437: 3436: 3144: 3143: 3000:, 3 = 27, or 3 = 2942: 2941: 2747:Asymmetric scale 2451: 2450: 2114:Five-limit tuning 2108:Five-limit tuning 2090:equal temperament 2048:Pythagorean comma 1943: 1942: 1746:Twelve-tone scale 1743: 1742: 1448: 1447: 1147: 1132:Five-limit tuning 766:For example, the 733: 728: 723: 717: 701: 696: 690: 670: 665: 659: 648: 568:equal temperament 450:derived from the 431:with the octaves 425:equal temperament 386:musical intervals 366: 353: 352: 345: 335: 334: 327: 292:original research 273: 272: 265: 247: 212:"Just intonation" 171: 170: 163: 113: 112: 105: 57: 6249: 6206:Musical interval 6119: 6118: 6116: 6115: 6112: 6109: 6091: 6090: 6060: 6037: 6030: 6023: 6014: 6013: 5992:Tonality diamond 5900:Circle of fifths 5877: 5870: 5863: 5854: 5853: 5723:Wind instruments 5647: 5640: 5633: 5624: 5623: 5459:Well temperament 5445:Regular diatonic 5311: 5310: 5291:Tonality diamond 5135: 5128: 5121: 5112: 5111: 4999: 4998: 4992: 4984:Secor, George D. 4980: 4974: 4973: 4971: 4969: 4963: 4954: 4948: 4947: 4930:Stahnke, Manfred 4926: 4920: 4919: 4886: 4880: 4879: 4877: 4875: 4857: 4851: 4842: 4836: 4835: 4816: 4810: 4809: 4808:. tonalsoft.com. 4799: 4790: 4789: 4783: 4775: 4763: 4747: 4737: 4731: 4729: 4712: 4706: 4705: 4699: 4691: 4671: 4658: 4652: 4647: 4641: 4640: 4638: 4637: 4626:"22 shruti" 4622: 4609: 4608: 4587: 4581: 4580: 4568: 4555: 4549: 4548: 4530: 4524: 4523: 4502: 4493: 4490:Free Composition 4486: 4480: 4479: 4456: 4450: 4449: 4424: 4415: 4414: 4392: 4386: 4385: 4367: 4358: 4357: 4342: 4336: 4335: 4306: 4297: 4296: 4263: 4247:Whole-tone scale 4227:Musical interval 4212:Microtonal music 4202:Electronic tuner 4197:Dynamic tonality 4133: 4132: 4131: 4129: 4119: 4118: 4117: 4115: 4101: 4100: 4099: 4097: 4087: 4086: 4085: 4083: 4036: 4035: 4030: 4029: 3994: 3993: 3984: 3983: 3967: 3966: 3947: 3946: 3933: 3932: 3925: 3924: 3917: 3913: 3909: 3908: 3901: 3900: 3836: 3835: 3834: 3832: 3820: 3819: 3818: 3816: 3805:and modified by 3730: 3723: 3719: 3716: 3710: 3679: 3671: 3605: 3604: 3591:Michael Harrison 3574: 3567: 3565: 3564: 3561: 3558: 3529: 3522: 3518: 3515: 3509: 3478: 3470: 3276: 3275: 3268: 3267: 3254: 3253: 3246: 3245: 3235: 3234: 3227: 3226: 3207: 3206: 3199: 3198: 3185: 3184: 3177: 3176: 3163: 3162: 3039: 3038: 3009: 3008: 3004: 2999: 2998: 2994: 2977: 2975: 2974: 2971: 2968: 2961: 2959: 2958: 2955: 2952: 2924: 2923: 2914: 2913: 2904: 2903: 2894: 2893: 2884: 2882: 2881: 2878: 2875: 2841: 2839: 2808: 2807: 2769: 2767: 2766: 2763: 2760: 2744: 2743: 2738: 2736: 2735: 2732: 2729: 2718:diminished fifth 2711: 2710: 2701: 2699: 2698: 2695: 2692: 2685: 2683: 2682: 2679: 2676: 2669: 2667: 2666: 2663: 2660: 2653: 2651: 2650: 2647: 2644: 2630: 2628: 2627: 2624: 2621: 2614: 2612: 2611: 2608: 2605: 2598: 2596: 2595: 2592: 2589: 2582: 2580: 2579: 2576: 2573: 2555: 2553: 2552: 2549: 2546: 2536: 2534: 2532: 2531: 2528: 2525: 2518: 2516: 2515: 2512: 2509: 2502: 2500: 2499: 2496: 2493: 2473: 2471: 2469: 2468: 2465: 2462: 2404: 2403: 2394: 2393: 2384: 2383: 2374: 2373: 2364: 2363: 2354: 2352: 2351: 2348: 2345: 2282: 2280: 2268: 2267: 2210: 2209: 2164: 2162: 2161: 2158: 2155: 2147: 2145: 2144: 2141: 2138: 2127: 2126: 2087: 2086: 2081: 2080: 2075: 2074: 2069: 2068: 2063: 2062: 2041: 2040: 2038: 2037: 2034: 2031: 2024: 2022: 2021: 2018: 2015: 1996: 1995: 1991: 1986: 1985: 1981: 1972: 1971: 1967: 1962: 1961: 1957: 1951:circle of fifths 1851: 1850: 1822: 1821: 1814: 1813: 1806: 1805: 1798: 1797: 1790: 1789: 1779: 1778: 1561: 1560: 1492: 1491: 1481: 1480: 1476: 1471: 1470: 1466: 1202: 1201: 1149: 1148: 1117: 1115: 1114: 1111: 1108: 1101: 1099: 1098: 1095: 1092: 1085: 1083: 1082: 1079: 1076: 1069: 1067: 1066: 1063: 1060: 1053: 1051: 1050: 1047: 1044: 1037: 1035: 1034: 1031: 1028: 1021: 1019: 1018: 1015: 1012: 1005: 1003: 1002: 999: 996: 989: 987: 986: 983: 980: 973: 971: 970: 967: 964: 957: 955: 954: 951: 948: 941: 939: 938: 935: 932: 925: 923: 922: 919: 916: 848: 846: 845: 842: 839: 835: 829: 827: 826: 823: 820: 816: 805:Claudius Ptolemy 784:chromatic circle 765: 762: 761: 746: 744: 743: 738: 731: 730: 724: 721: 715: 713: 707: 703: 697: 694: 688: 686: 676: 672: 666: 663: 657: 655: 646: 634: 632: 631: 628: 625: 596:partials in the 521: 519: 518: 515: 512: 505: 503: 502: 499: 496: 472:Well temperament 433:slightly widened 368: 367: 348: 341: 330: 323: 319: 316: 310: 307:inline citations 283: 282: 275: 268: 261: 257: 254: 248: 246: 205: 181: 173: 166: 159: 155: 152: 146: 143:reliably sourced 123: 122: 115: 108: 101: 97: 94: 88: 68: 67: 60: 49: 27: 26: 19: 6257: 6256: 6252: 6251: 6250: 6248: 6247: 6246: 6232: 6231: 6230: 6225: 6196:Just intonation 6123: 6113: 6110: 6107: 6106: 6104: 6103: 6092: 6088: 6083: 6061: 6050: 6041: 6011: 6006: 5942:Major and minor 5932:Just intonation 5886: 5881: 5851: 5846: 5821: 5815: 5798:overtone series 5795: 5792:Flageolet tones 5786: 5763: 5727: 5701: 5663:Just intonation 5657: 5651: 5621: 5616: 5613:(Bohlen–Pierce) 5581:833 cents scale 5569: 5492: 5482: 5449: 5402: 5300: 5221:Just intonation 5215: 5144: 5142:Musical tunings 5139: 5036:Just intonation 5027:Just Intonation 5007: 5002: 4990: 4981: 4977: 4967: 4965: 4961: 4955: 4951: 4944: 4927: 4923: 4887: 4883: 4873: 4871: 4858: 4854: 4843: 4839: 4817: 4813: 4800: 4793: 4777: 4776: 4764: 4760: 4738: 4734: 4713: 4709: 4693: 4692: 4680: 4659: 4655: 4648: 4644: 4635: 4633: 4624: 4623: 4612: 4605: 4591:Daniélou, Alain 4588: 4584: 4577: 4559:Daniélou, Alain 4556: 4552: 4545: 4531: 4527: 4520: 4506:Daniélou, Alain 4503: 4496: 4487: 4483: 4476: 4457: 4453: 4446: 4425: 4418: 4411: 4393: 4389: 4382: 4368: 4361: 4344: 4343: 4339: 4324: 4307: 4300: 4264: 4260: 4256: 4251: 4185: 4154: 4145:square waveform 4127: 4124: 4123: 4113: 4110: 4109: 4095: 4092: 4091: 4082:Just intonation 4081: 4078: 4077: 4074: 4033: 4032: 4027: 4026: 4010:in his classic 3987: 3981: 3980: 3960: 3944: 3943: 3927: 3919: 3918:), tridecimal ( 3915: 3911: 3903: 3895: 3830: 3827: 3826: 3824: 3814: 3811: 3810: 3791: 3774: 3731: 3720: 3714: 3711: 3696: 3680: 3669: 3602: 3601: 3572: 3562: 3559: 3556: 3555: 3553: 3530: 3519: 3513: 3510: 3495: 3479: 3468: 3273: 3272: 3265: 3264: 3251: 3250: 3243: 3242: 3232: 3231: 3224: 3223: 3204: 3203: 3196: 3195: 3182: 3181: 3174: 3173: 3016: 3006: 3002: 3001: 2996: 2992: 2991: 2988: 2972: 2969: 2966: 2965: 2963: 2956: 2953: 2950: 2949: 2947: 2921: 2920: 2911: 2910: 2901: 2900: 2891: 2890: 2879: 2876: 2873: 2872: 2870: 2837: 2835: 2805: 2804: 2764: 2761: 2758: 2757: 2755: 2733: 2730: 2727: 2726: 2724: 2708: 2707: 2696: 2693: 2690: 2689: 2687: 2680: 2677: 2674: 2673: 2671: 2664: 2661: 2658: 2657: 2655: 2648: 2645: 2642: 2641: 2639: 2625: 2622: 2619: 2618: 2616: 2609: 2606: 2603: 2602: 2600: 2593: 2590: 2587: 2586: 2584: 2577: 2574: 2571: 2570: 2568: 2550: 2547: 2544: 2543: 2541: 2529: 2526: 2523: 2522: 2520: 2513: 2510: 2507: 2506: 2504: 2497: 2494: 2491: 2490: 2488: 2487: 2466: 2463: 2460: 2459: 2457: 2456: 2401: 2400: 2391: 2390: 2381: 2380: 2371: 2370: 2361: 2360: 2349: 2346: 2343: 2342: 2340: 2278: 2276: 2265: 2264: 2207: 2206: 2159: 2156: 2153: 2152: 2150: 2142: 2139: 2136: 2135: 2133: 2116: 2110: 2084: 2083: 2078: 2077: 2072: 2071: 2066: 2065: 2060: 2059: 2046:, known as the 2035: 2032: 2029: 2028: 2026: 2019: 2016: 2013: 2012: 2010: 2009: 1993: 1989: 1988: 1983: 1979: 1978: 1969: 1965: 1964: 1959: 1955: 1954: 1848: 1847: 1819: 1818: 1811: 1810: 1803: 1802: 1795: 1794: 1787: 1786: 1762: 1756: 1748: 1558: 1557: 1478: 1474: 1473: 1468: 1464: 1463: 1462:with the ratio 1142: 1134: 1128: 1112: 1109: 1106: 1105: 1103: 1096: 1093: 1090: 1089: 1087: 1080: 1077: 1074: 1073: 1071: 1064: 1061: 1058: 1057: 1055: 1048: 1045: 1042: 1041: 1039: 1032: 1029: 1026: 1025: 1023: 1016: 1013: 1010: 1009: 1007: 1000: 997: 994: 993: 991: 984: 981: 978: 977: 975: 968: 965: 962: 961: 959: 952: 949: 946: 945: 943: 936: 933: 930: 929: 927: 920: 917: 914: 913: 911: 843: 840: 837: 836: 833: 831: 824: 821: 818: 817: 814: 812: 776: 759: 758: 755: 714: 711: 687: 684: 680: 656: 653: 649: 644: 641: 640: 629: 626: 623: 622: 620: 598:overtone series 516: 513: 510: 509: 507: 500: 497: 494: 493: 491: 484: 452:harmonic series 409:harmonic series 378:pure intonation 374:just intonation 361: 349: 338: 337: 336: 331: 320: 314: 311: 296: 284: 280: 269: 258: 252: 249: 206: 204: 194: 182: 167: 156: 150: 147: 136: 128:This article's 124: 120: 109: 98: 92: 89: 81:help improve it 78: 69: 65: 28: 24: 17: 12: 11: 5: 6255: 6245: 6244: 6227: 6226: 6224: 6223: 6218: 6213: 6208: 6203: 6198: 6193: 6192: 6191: 6181: 6176: 6175: 6174: 6164: 6159: 6154: 6149: 6144: 6139: 6134: 6128: 6125: 6124: 6122: 6121: 6100: 6098: 6094: 6093: 6086: 6084: 6082: 6081: 6067: 6065: 6062: 6055: 6052: 6051: 6040: 6039: 6032: 6025: 6017: 6008: 6007: 6005: 6004: 5999: 5994: 5989: 5984: 5979: 5974: 5969: 5964: 5959: 5954: 5949: 5944: 5939: 5934: 5929: 5924: 5923: 5922: 5912: 5910:Diatonic scale 5907: 5902: 5897: 5891: 5888: 5887: 5880: 5879: 5872: 5865: 5857: 5848: 5847: 5845: 5844: 5839: 5834: 5828: 5826: 5817: 5816: 5814: 5813: 5808: 5802: 5800: 5794:(harmonics) or 5788: 5787: 5785: 5784: 5783: 5782: 5771: 5769: 5765: 5764: 5762: 5761: 5756: 5751: 5746: 5741: 5735: 5733: 5729: 5728: 5726: 5725: 5720: 5715: 5709: 5707: 5703: 5702: 5700: 5699: 5694: 5689: 5684: 5679: 5674: 5668: 5666: 5659: 5658: 5650: 5649: 5642: 5635: 5627: 5618: 5617: 5615: 5614: 5608: 5603: 5598: 5593: 5588: 5583: 5577: 5575: 5571: 5570: 5568: 5567: 5562: 5557: 5547: 5542: 5537: 5536: 5535: 5530: 5525: 5520: 5512: 5507: 5502: 5496: 5494: 5488: 5487: 5484: 5483: 5457: 5455: 5451: 5450: 5448: 5447: 5442: 5437: 5432: 5427: 5412: 5410: 5404: 5403: 5401: 5400: 5395: 5390: 5385: 5380: 5375: 5370: 5365: 5355: 5350: 5345: 5340: 5335: 5330: 5325: 5319: 5317: 5308: 5302: 5301: 5299: 5298: 5293: 5288: 5283: 5278: 5273: 5268: 5263: 5262: 5261: 5256: 5246: 5241: 5236: 5234:Harmonic scale 5231: 5225: 5223: 5217: 5216: 5214: 5213: 5208: 5203: 5198: 5193: 5188: 5183: 5181:Interval ratio 5178: 5173: 5168: 5163: 5158: 5152: 5150: 5146: 5145: 5138: 5137: 5130: 5123: 5115: 5109: 5108: 5102: 5096: 5089: 5084: 5079: 5074: 5069: 5056: 5047: 5038:compared with 5033: 5024: 5019: 5014: 5006: 5005:External links 5003: 5001: 5000: 4975: 4949: 4942: 4921: 4908:10.2307/833435 4890:Fonville, John 4881: 4852: 4837: 4822:, ed. (1984). 4820:Garland, Peter 4811: 4791: 4758: 4732: 4707: 4678: 4653: 4642: 4610: 4603: 4582: 4575: 4550: 4543: 4525: 4518: 4494: 4481: 4474: 4451: 4444: 4416: 4409: 4387: 4380: 4359: 4337: 4322: 4310:Barker, Andrew 4298: 4279:(2): 161–179. 4257: 4255: 4252: 4250: 4249: 4244: 4239: 4237:Regular number 4234: 4229: 4224: 4219: 4214: 4209: 4204: 4199: 4193: 4192: 4191: 4189: 4188:Article topics 4184: 4183: 4178: 4173: 4168: 4162: 4161: 4160: 4158: 4153: 4150: 4149: 4148: 4121: 4107: 4089: 4073: 4072:Audio examples 4070: 4022:tempered pitch 3998:In 2000–2004, 3910:), undecimal ( 3891:syntonic comma 3871:equal tempered 3822: 3790: 3789:Staff notation 3787: 3773: 3770: 3750:as thirds and 3733: 3732: 3683: 3681: 3674: 3668: 3665: 3660:Hermode Tuning 3538:wolf intervals 3532: 3531: 3482: 3480: 3473: 3467: 3464: 3455:(256:243) and 3445:syntonic comma 3439:Where we have 3435: 3434: 3431: 3428: 3425: 3422: 3419: 3416: 3413: 3410: 3407: 3404: 3401: 3398: 3395: 3392: 3389: 3386: 3383: 3380: 3377: 3374: 3371: 3368: 3365: 3361: 3360: 3357: 3354: 3351: 3348: 3345: 3342: 3339: 3336: 3333: 3330: 3327: 3324: 3321: 3318: 3315: 3312: 3309: 3306: 3303: 3300: 3297: 3294: 3291: 3287: 3286: 3283: 3280: 3277: 3269: 3261: 3258: 3255: 3247: 3239: 3236: 3228: 3220: 3217: 3214: 3211: 3208: 3200: 3192: 3189: 3186: 3178: 3170: 3167: 3146: 3145: 3142: 3141: 3138: 3135: 3132: 3129: 3126: 3123: 3120: 3117: 3113: 3112: 3109: 3106: 3103: 3100: 3097: 3094: 3091: 3088: 3084: 3083: 3078: 3073: 3068: 3063: 3058: 3053: 3048: 3043: 3015: 3012: 2987: 2984: 2944: 2943: 2940: 2939: 2936: 2933: 2930: 2926: 2925: 2915: 2905: 2895: 2885: 2867: 2866: 2863: 2860: 2857: 2853: 2852: 2847: 2842: 2833: 2828: 2824: 2823: 2820: 2817: 2814: 2810: 2809: 2799: 2794: 2789: 2784: 2780: 2779: 2776: 2773: 2770: 2753: 2749: 2748: 2704: 2703: 2633: 2632: 2558: 2557: 2537: 2453: 2452: 2449: 2448: 2445: 2442: 2439: 2436: 2433: 2429: 2428: 2425: 2422: 2419: 2416: 2413: 2409: 2408: 2405: 2395: 2385: 2375: 2365: 2355: 2337: 2336: 2333: 2330: 2327: 2324: 2321: 2317: 2316: 2313: 2310: 2307: 2304: 2301: 2297: 2296: 2293: 2288: 2283: 2274: 2269: 2259: 2255: 2254: 2251: 2248: 2245: 2242: 2239: 2235: 2234: 2231: 2228: 2225: 2222: 2219: 2215: 2214: 2211: 2201: 2196: 2191: 2186: 2181: 2177: 2176: 2174: 2171: 2168: 2165: 2148: 2131: 2112:Main article: 2109: 2106: 2042:, or about 23 1941: 1940: 1937: 1934: 1931: 1928: 1925: 1922: 1919: 1916: 1913: 1910: 1907: 1904: 1901: 1897: 1896: 1893: 1890: 1887: 1884: 1881: 1878: 1875: 1872: 1869: 1866: 1863: 1860: 1857: 1853: 1852: 1844: 1841: 1838: 1835: 1832: 1829: 1826: 1823: 1815: 1807: 1799: 1791: 1783: 1776:, as follows: 1758:Main article: 1755: 1752: 1747: 1744: 1741: 1740: 1737: 1734: 1731: 1728: 1725: 1722: 1719: 1715: 1714: 1711: 1708: 1705: 1702: 1699: 1696: 1693: 1689: 1688: 1685: 1682: 1679: 1676: 1673: 1670: 1667: 1664: 1661: 1658: 1654: 1653: 1651: 1649: 1647: 1645: 1643: 1641: 1639: 1637: 1635: 1633: 1631: 1629: 1627: 1625: 1623: 1621: 1619: 1616: 1615: 1612: 1609: 1606: 1603: 1600: 1597: 1594: 1591: 1587: 1586: 1583: 1580: 1577: 1574: 1571: 1568: 1565: 1562: 1553: 1552: 1549: 1546: 1543: 1540: 1537: 1534: 1531: 1528: 1524: 1523: 1520: 1517: 1514: 1511: 1508: 1505: 1502: 1499: 1496: 1446: 1445: 1442: 1439: 1436: 1433: 1430: 1427: 1424: 1420: 1419: 1416: 1413: 1410: 1407: 1404: 1401: 1398: 1394: 1393: 1390: 1387: 1384: 1381: 1378: 1375: 1372: 1369: 1366: 1363: 1359: 1358: 1356: 1354: 1352: 1350: 1348: 1346: 1344: 1342: 1340: 1338: 1336: 1334: 1332: 1330: 1328: 1326: 1324: 1321: 1320: 1317: 1314: 1311: 1308: 1305: 1302: 1299: 1296: 1292: 1291: 1288: 1285: 1282: 1279: 1276: 1273: 1270: 1267: 1263: 1262: 1259: 1256: 1253: 1250: 1247: 1244: 1241: 1238: 1234: 1233: 1230: 1227: 1224: 1221: 1218: 1215: 1212: 1209: 1206: 1190:syntonic comma 1130:Main article: 1127: 1126:Diatonic scale 1124: 851:Phrygian scale 775: 772: 736: 727: 720: 710: 706: 700: 693: 683: 679: 675: 669: 662: 652: 613:septimal comma 609:syntonic comma 580:, and 15:8, a 574:5-limit tuning 560:tempered fifth 552:music analysis 483: 480: 411:of an implied 351: 350: 333: 332: 287: 285: 278: 271: 270: 185: 183: 176: 169: 168: 127: 125: 118: 111: 110: 72: 70: 63: 58: 32: 31: 29: 22: 15: 9: 6: 4: 3: 2: 6254: 6243: 6240: 6239: 6237: 6222: 6219: 6217: 6214: 6212: 6209: 6207: 6204: 6202: 6199: 6197: 6194: 6190: 6187: 6186: 6185: 6182: 6180: 6177: 6173: 6170: 6169: 6168: 6165: 6163: 6160: 6158: 6155: 6153: 6150: 6148: 6145: 6143: 6140: 6138: 6135: 6133: 6130: 6129: 6126: 6102: 6101: 6099: 6095: 6080: 6076: 6072: 6069: 6068: 6066: 6059: 6053: 6049: 6045: 6038: 6033: 6031: 6026: 6024: 6019: 6018: 6015: 6003: 6002:Voice leading 6000: 5998: 5995: 5993: 5990: 5988: 5985: 5983: 5980: 5978: 5975: 5973: 5970: 5968: 5965: 5963: 5960: 5958: 5955: 5953: 5950: 5948: 5945: 5943: 5940: 5938: 5935: 5933: 5930: 5928: 5925: 5921: 5918: 5917: 5916: 5913: 5911: 5908: 5906: 5903: 5901: 5898: 5896: 5893: 5892: 5889: 5885: 5878: 5873: 5871: 5866: 5864: 5859: 5858: 5855: 5843: 5840: 5838: 5835: 5833: 5830: 5829: 5827: 5825: 5818: 5812: 5809: 5807: 5804: 5803: 5801: 5799: 5793: 5789: 5781: 5778: 5777: 5776: 5773: 5772: 5770: 5766: 5760: 5757: 5755: 5752: 5750: 5749:Tromba marina 5747: 5745: 5742: 5740: 5737: 5736: 5734: 5730: 5724: 5721: 5719: 5716: 5714: 5711: 5710: 5708: 5704: 5698: 5695: 5693: 5690: 5688: 5685: 5683: 5680: 5678: 5675: 5673: 5670: 5669: 5667: 5664: 5660: 5656: 5648: 5643: 5641: 5636: 5634: 5629: 5628: 5625: 5612: 5609: 5607: 5604: 5602: 5599: 5597: 5594: 5592: 5589: 5587: 5584: 5582: 5579: 5578: 5576: 5572: 5566: 5563: 5561: 5558: 5555: 5554:Carnatic raga 5551: 5548: 5546: 5543: 5541: 5538: 5534: 5531: 5529: 5526: 5524: 5523:Turkish makam 5521: 5519: 5516: 5515: 5513: 5511: 5508: 5506: 5503: 5501: 5498: 5497: 5495: 5489: 5480: 5476: 5472: 5468: 5464: 5460: 5456: 5452: 5446: 5443: 5441: 5438: 5436: 5433: 5431: 5428: 5425: 5421: 5420:quarter-comma 5417: 5414: 5413: 5411: 5409: 5405: 5399: 5396: 5394: 5391: 5389: 5386: 5384: 5381: 5379: 5376: 5374: 5371: 5369: 5366: 5363: 5359: 5356: 5354: 5351: 5349: 5346: 5344: 5341: 5339: 5336: 5334: 5331: 5329: 5326: 5324: 5321: 5320: 5318: 5316: 5312: 5309: 5307: 5303: 5297: 5296:Tonality flux 5294: 5292: 5289: 5287: 5284: 5282: 5279: 5277: 5274: 5272: 5269: 5267: 5264: 5260: 5257: 5255: 5252: 5251: 5250: 5247: 5245: 5242: 5240: 5237: 5235: 5232: 5230: 5227: 5226: 5224: 5222: 5218: 5212: 5209: 5207: 5204: 5202: 5199: 5197: 5194: 5192: 5189: 5187: 5184: 5182: 5179: 5177: 5174: 5172: 5169: 5167: 5164: 5162: 5159: 5157: 5154: 5153: 5151: 5147: 5143: 5136: 5131: 5129: 5124: 5122: 5117: 5116: 5113: 5106: 5103: 5100: 5097: 5094: 5090: 5088: 5085: 5083: 5080: 5078: 5075: 5073: 5070: 5068: 5064: 5060: 5057: 5055: 5051: 5048: 5045: 5041: 5037: 5034: 5032: 5031:Mark Nowitzky 5028: 5025: 5023: 5020: 5018: 5015: 5012: 5009: 5008: 4996: 4989: 4985: 4979: 4960: 4957:Sabat, Marc. 4953: 4945: 4943:3-932696-62-X 4939: 4935: 4931: 4925: 4917: 4913: 4909: 4905: 4901: 4897: 4896: 4891: 4885: 4870: 4866: 4862: 4856: 4849: 4848: 4841: 4833: 4829: 4825: 4821: 4815: 4807: 4806: 4798: 4796: 4787: 4781: 4773: 4769: 4768:Eitz, Carl A. 4761: 4755: 4751: 4746: 4745: 4736: 4727: 4723: 4722: 4717: 4711: 4703: 4697: 4689: 4685: 4681: 4679:0-306-71597-X 4675: 4670: 4669: 4663: 4662:Partch, Harry 4657: 4651: 4646: 4631: 4627: 4621: 4619: 4617: 4615: 4606: 4604:81-7069-098-6 4600: 4596: 4592: 4586: 4578: 4576:0-89281-336-9 4572: 4567: 4566: 4560: 4554: 4546: 4544:81-86982-07-8 4540: 4536: 4529: 4521: 4519:0-214-15689-3 4515: 4511: 4507: 4501: 4499: 4491: 4485: 4477: 4471: 4467: 4466: 4461: 4460:Partch, Harry 4455: 4447: 4441: 4437: 4433: 4429: 4428:Johnston, Ben 4423: 4421: 4412: 4406: 4402: 4398: 4391: 4383: 4377: 4373: 4366: 4364: 4355: 4351: 4347: 4341: 4333: 4329: 4325: 4323:0-521-23593-6 4319: 4315: 4311: 4305: 4303: 4294: 4290: 4286: 4282: 4278: 4274: 4273: 4268: 4262: 4258: 4248: 4245: 4243: 4240: 4238: 4235: 4233: 4230: 4228: 4225: 4223: 4220: 4218: 4215: 4213: 4210: 4208: 4205: 4203: 4200: 4198: 4195: 4194: 4190: 4187: 4186: 4182: 4179: 4177: 4174: 4172: 4169: 4167: 4164: 4163: 4159: 4156: 4155: 4146: 4142: 4138: 4130: 4122: 4116: 4108: 4105: 4098: 4090: 4084: 4076: 4075: 4069: 4065: 4063: 4054: 4046: 4042: 4040: 4023: 4019: 4018:(Fig. 1) 4015: 4014: 4009: 4005: 4001: 3978: 3973: 3956: 3952: 3950: 3949:(Fig. 4) 3941: 3937: 3931: 3923: 3907: 3899: 3892: 3888: 3887:(Fig. 4) 3884: 3879: 3874: 3872: 3868: 3864: 3858: 3856: 3852: 3848: 3844: 3840: 3833: 3817: 3808: 3804: 3795: 3786: 3784: 3779: 3769: 3767: 3762: 3759: 3755: 3753: 3752:leading tones 3747: 3745: 3741: 3729: 3726: 3718: 3708: 3704: 3700: 3694: 3693: 3689: 3684:This section 3682: 3678: 3673: 3672: 3664: 3661: 3656: 3652: 3650: 3644: 3643: 3638: 3635: 3633: 3629: 3628: 3623: 3619: 3615: 3611: 3609: 3599: 3594: 3592: 3588: 3584: 3583: 3577: 3571: 3551: 3547: 3542: 3539: 3528: 3525: 3517: 3507: 3503: 3499: 3493: 3492: 3488: 3483:This section 3481: 3477: 3472: 3471: 3463: 3460: 3458: 3454: 3450: 3446: 3442: 3432: 3429: 3426: 3423: 3420: 3417: 3414: 3411: 3408: 3405: 3402: 3399: 3396: 3393: 3390: 3387: 3384: 3381: 3378: 3375: 3372: 3369: 3366: 3363: 3362: 3358: 3355: 3352: 3349: 3346: 3343: 3340: 3337: 3334: 3331: 3328: 3325: 3322: 3319: 3316: 3313: 3310: 3307: 3304: 3301: 3298: 3295: 3292: 3289: 3288: 3284: 3281: 3278: 3270: 3262: 3259: 3256: 3248: 3240: 3237: 3229: 3221: 3218: 3215: 3212: 3209: 3201: 3193: 3190: 3187: 3179: 3171: 3168: 3165: 3164: 3161: 3159: 3155: 3151: 3139: 3136: 3133: 3130: 3127: 3124: 3121: 3118: 3115: 3114: 3110: 3107: 3105:5:3 or 27:16 3104: 3101: 3098: 3095: 3092: 3089: 3086: 3085: 3082: 3079: 3077: 3074: 3072: 3069: 3067: 3064: 3062: 3059: 3057: 3054: 3052: 3049: 3047: 3044: 3041: 3040: 3037: 3036: 3035: 3033: 3029: 3025: 3021: 3014:Indian scales 3011: 2983: 2981: 2937: 2934: 2931: 2928: 2927: 2919: 2916: 2909: 2906: 2899: 2896: 2889: 2886: 2868: 2864: 2861: 2858: 2855: 2854: 2851: 2848: 2846: 2843: 2840: 2834: 2832: 2829: 2825: 2821: 2818: 2815: 2812: 2811: 2803: 2800: 2798: 2795: 2793: 2790: 2788: 2785: 2781: 2777: 2774: 2771: 2754: 2751: 2750: 2745: 2742: 2741: 2740: 2721: 2719: 2715: 2638: 2637: 2636: 2567: 2566: 2565: 2563: 2538: 2485: 2481: 2480: 2479: 2477: 2446: 2443: 2440: 2437: 2434: 2431: 2430: 2426: 2423: 2420: 2417: 2414: 2411: 2410: 2406: 2399: 2396: 2389: 2386: 2379: 2376: 2369: 2366: 2359: 2356: 2338: 2334: 2331: 2328: 2325: 2322: 2319: 2318: 2314: 2311: 2308: 2305: 2302: 2299: 2298: 2294: 2292: 2289: 2287: 2284: 2281: 2275: 2273: 2270: 2263: 2260: 2256: 2252: 2249: 2246: 2243: 2240: 2237: 2236: 2232: 2229: 2226: 2223: 2220: 2217: 2216: 2212: 2205: 2202: 2200: 2197: 2195: 2192: 2190: 2187: 2185: 2182: 2178: 2175: 2172: 2169: 2166: 2149: 2132: 2129: 2128: 2125: 2124: 2123: 2120: 2115: 2105: 2102: 2100: 2095: 2091: 2057: 2053: 2049: 2045: 2007: 2003: 1998: 1976: 1952: 1948: 1938: 1935: 1932: 1929: 1926: 1923: 1920: 1917: 1914: 1911: 1908: 1905: 1902: 1899: 1898: 1894: 1891: 1888: 1885: 1882: 1879: 1876: 1873: 1870: 1867: 1864: 1861: 1858: 1855: 1854: 1845: 1842: 1839: 1836: 1833: 1830: 1827: 1824: 1816: 1808: 1800: 1792: 1784: 1781: 1780: 1777: 1775: 1771: 1766: 1761: 1751: 1717: 1716: 1691: 1690: 1659: 1655: 1652: 1650: 1648: 1646: 1644: 1642: 1640: 1638: 1636: 1634: 1632: 1630: 1628: 1626: 1624: 1622: 1620: 1618: 1617: 1589: 1588: 1555: 1554: 1527:Ratio from A 1526: 1525: 1497: 1493: 1490: 1488: 1483: 1461: 1452: 1422: 1421: 1396: 1395: 1364: 1360: 1357: 1355: 1353: 1351: 1349: 1347: 1345: 1343: 1341: 1339: 1337: 1335: 1333: 1331: 1329: 1327: 1325: 1323: 1322: 1294: 1293: 1265: 1264: 1237:Ratio from C 1236: 1235: 1207: 1203: 1200: 1198: 1193: 1191: 1187: 1183: 1179: 1175: 1171: 1167: 1162: 1155: 1138: 1133: 1123: 1121: 909: 906: 902: 897: 895: 891: 887: 883: 879: 875: 871: 867: 863: 858: 856: 852: 810: 806: 801: 799: 795: 791: 785: 780: 771: 769: 763: 752: 747: 734: 725: 718: 708: 704: 698: 691: 681: 677: 673: 667: 660: 650: 638: 618: 614: 610: 605: 601: 599: 595: 591: 587: 586:perfect third 583: 582:major seventh 579: 575: 571: 569: 565: 561: 557: 556:perfect fifth 553: 549: 546:, and 9:4, a 545: 544:perfect fifth 541: 537: 534: 530: 526: 489: 479: 477: 473: 469: 465: 461: 457: 453: 449: 445: 441: 436: 434: 430: 426: 420: 418: 414: 410: 406: 405:just interval 402: 398: 394: 391: 387: 383: 379: 375: 357: 347: 344: 329: 326: 318: 308: 304: 300: 294: 293: 288:This article 286: 277: 276: 267: 264: 256: 245: 242: 238: 235: 231: 228: 224: 221: 217: 214: – 213: 209: 208:Find sources: 202: 198: 192: 191: 186:This article 184: 180: 175: 174: 165: 162: 154: 144: 140: 134: 133: 126: 117: 116: 107: 104: 96: 86: 82: 76: 73:This article 71: 62: 61: 56: 54: 47: 46: 41: 40: 35: 30: 21: 20: 6195: 5997:Tonicization 5972:Polytonality 5967:Parallel key 5931: 5927:Figured bass 5744:Natural horn 5662: 5611:Lambda scale 5518:Arabic maqam 5475:Werckmeister 5306:Temperaments 5220: 4994: 4978: 4966:. Retrieved 4952: 4933: 4924: 4899: 4893: 4884: 4872:. Retrieved 4869:KyleGann.com 4868: 4855: 4845: 4840: 4823: 4814: 4804: 4771: 4743: 4735: 4719: 4710: 4667: 4656: 4645: 4634:. Retrieved 4630:22shruti.com 4629: 4594: 4585: 4564: 4553: 4534: 4528: 4509: 4489: 4484: 4463: 4454: 4435: 4432:Gilmore, Bob 4396: 4390: 4371: 4353: 4349: 4340: 4313: 4276: 4270: 4261: 4066: 4061: 4059: 4021: 4011: 3997: 3883:prime limits 3878:Ben Johnston 3875: 3863:James Tenney 3859: 3854: 3846: 3841:and used in 3800: 3782: 3775: 3763: 3760: 3756: 3748: 3736: 3721: 3712: 3697:Please help 3685: 3657: 3653: 3649:archicembalo 3645: 3642:Pablo Casals 3639: 3636: 3631: 3625: 3622:Wendy Carlos 3614:Synthesizers 3612: 3595: 3586: 3580: 3578: 3573:(C→E→G→A→D), 3543: 3535: 3520: 3511: 3496:Please help 3484: 3461: 3456: 3452: 3448: 3440: 3438: 3157: 3153: 3147: 3080: 3075: 3070: 3065: 3060: 3055: 3050: 3045: 3031: 3027: 3023: 3020:Indian music 3017: 2989: 2945: 2917: 2907: 2897: 2887: 2849: 2844: 2836: 2830: 2801: 2796: 2791: 2786: 2722: 2705: 2634: 2559: 2483: 2454: 2444:1018 ¢ 2397: 2387: 2377: 2367: 2357: 2290: 2285: 2277: 2271: 2261: 2247:1088 ¢ 2203: 2198: 2193: 2188: 2183: 2121: 2117: 2103: 2000:Between the 1999: 1946: 1944: 1763: 1749: 1484: 1457: 1194: 1163: 1160: 898: 878:Eratosthenes 859: 808: 802: 798:Eratosthenes 788: 748: 602: 594:prime number 590:7 limit 585: 572: 559: 555: 538: 525:Wendy Carlos 485: 437: 421: 404: 390:whole number 377: 373: 371: 339: 321: 312: 289: 259: 250: 240: 233: 226: 219: 207: 195:Please help 190:verification 187: 157: 151:January 2021 148: 129: 99: 90: 74: 50: 43: 37: 36:Please help 33: 6184:Irreducible 6114:Denominator 5987:Sonata form 5952:Neotonality 5842:Moodswinger 5780:Split sharp 5697:Synthesizer 5692:Human voice 5653:Instrument 5606:Delta scale 5601:Gamma scale 5591:Alpha scale 5493:non-Western 5491:Traditional 5186:Pitch class 5166:Millioctave 5149:Measurement 4874:28 February 4632:(main page) 4350:silkqin.com 4267:West, M. 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