3802:, is particularly popular, as it represents a convenient access point for composers conditioned on standard Western 12 EDO pitch and notation practices who are also interested in microtonality. Because 24 EDO contains all the pitches of 12 EDO, musicians employ the additional colors without losing any tactics available in 12 tone harmony. That 24 is a multiple of 12 also makes 24 EDO easy to achieve instrumentally by employing two traditional 12 EDO instruments tuned a quarter-tone apart, such as two pianos, which also allows each performer (or one performer playing a different piano with each hand) to read familiar 12 tone notation. Various composers, including
4858:
877:
342:
3724:
7959:
34:
7954:
61:
4283:
3645:
466:
3698:
3374:
3600:"Thai instruments of fixed pitch are tuned to an equidistant system of seven pitches per octave ... As in Western traditional music, however, all pitches of the tuning system are not used in one mode (often referred to as 'scale'); in the Thai system five of the seven are used in principal pitches in any mode, thus establishing a pattern of nonequidistant intervals for the mode."
1826:
784:
343:
33:
5162:; the same pattern repeats through the sharp notes, then the double-sharps, and so on, indefinitely. But each octave of all-natural or all-sharp or all-double-sharp notes flattens by two commas with every transition from naturals to sharps, or single sharps to double sharps, etc. The pattern is also reverse-symmetric in the flats: Descending by
3818:(7), tuning the 7th harmonic (7:4) with less than half a cent of error. Although it is a meantone temperament, it is a very flat one, with four of its perfect fifths producing a major third 17 cents flat (equated with the 11:9 neutral third). 26 EDO has two minor thirds and two minor sixths and could be an alternate temperament for
1013:
2406:
450:). Each colored graph shows how much error occurs (in cents) on the nearest approximation of the corresponding just interval (the black line on the center). Two black curves surrounding the graph on both sides represent the maximum possible error, while the gray ones inside of them indicate the half of it.
2133:
3888:
gives slightly lower total combined errors of approximation to 3:2, 5:4, 6:5, and their inversions than 31 EDO does, despite having a slightly less accurate fit for 5:4. 34 EDO does not accurately approximate the seventh harmonic or ratios involving 7, and is not meantone since its fifth is
3909:
46 EDO provides major thirds and perfect fifths that are both slightly sharp of just, and many say that this gives major triads a characteristic bright sound. The prime harmonics up to 17 are all within 6 cents of accuracy, with 10:9 and 9:5 a fifth of a cent away from pure. As it is not a
6099:
From the flute for two thousand years of the production process, and the
Japanese shakuhachi remaining in the production of Sui and Tang Dynasties and the actual temperament, identification of people using the so-called 'Seven Laws' at least two thousand years of history; and decided that this law
3900:
is the next EDO with a better perfect fifth than 29 EDO and 12 EDO. Its classical major third is also more accurate, at only six cents flat. It is not a meantone temperament, so it distinguishes 10:9 and 9:8, along with the classic and
Pythagorean major thirds, unlike 31 EDO. It is
851:
in 1585. According to F.A. Kuttner, a critic of giving credit to Zhu, it is known that Zhu "presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that Stevin "offered a mathematical definition of equal temperament plus a
3849:
is the lowest number of equal divisions of the octave whose perfect fifth is closer to just than in 12 EDO, in which the fifth is 1.5 cents sharp instead of 2 cents flat. Its classic major third is roughly as inaccurate as 12 EDO, but is tuned 14 cents flat rather than
3774:
is one of the most accurate EDOs to represent superpyth temperament (where 7:4 and 16:9 are the same interval) and is near the optimal generator for porcupine temperament. The fifths are so sharp that the major and minor thirds we get from stacking fifths will be the supermajor third (9/7) and
3786:
is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics (3:2, 5:4, 7:4, 11:8) within 20 cents. However, it does approximate some ratios between them (such as the 6:5 minor third) very well, making it attractive to microtonalists seeking unusual harmonic
1820:
3877:. 31 EDO does not have as accurate a perfect fifth as 12 EDO (like 19 EDO), but its major thirds and minor sixths are less than 1 cent away from just. It also provides good matches for harmonics up to 11, of which the seventh harmonic is particularly accurate.
3325:
for violas and cellos), which suggests that their semitone ratio is slightly higher than in conventional 12 tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval comprises seven steps, each tone is in the ratio of
1995:
2253:
3758:
meantone, it has a slightly flatter perfect fifth (at 695 cents), but its minor third and major sixth are less than one-fifth of a cent away from just, with the lowest EDO that produces a better minor third and major sixth than 19 EDO being 232 EDO. Its
1614:
1488:
4609:
Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into
5166:
the pattern reciprocally sharpens notes by two commas with every transition from natural notes to flattened notes, or flats to double flats, etc. If left unmodified, the two grave fifths in each block of all-natural notes, or all-sharps, or all-flat notes, are
3353:
rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves. During actual play, however, violinists choose pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio.
2258:
240:
and all other notes are defined as some multiple of semitones away from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz; it has varied considerably and generally risen over the past few hundred years.
3850:
14 cents sharp. It also tunes the 7th, 11th, and 13th harmonics flat by roughly the same amount, allowing 29 EDO to match intervals such as 7:5, 11:7, and 13:11 very accurately. Cutting all 29 intervals in half produces
3604:
2000:
867:
Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications". Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor.
3511:
the tempered perfect fifth is 686 cents wide (at the bottom of the tuning continuum), and marks the endpoint on the tuning continuum, at which the minor second expands to be as wide as the major second (at 171 cents
1239:
4002:
approximates all intervals within 6.25 cents, which is barely distinguishable. As an eightfold multiple of 12, it can be used fully like the common 12 EDO. It has been advocated by several composers, especially
862:
I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve
3966:
is a duplication of 29 EDO, which it contains as an embedded temperament. Like 29 EDO it can match intervals such as 7:4, 7:5, 11:7, and 13:11 very accurately, as well as better approximating just thirds and
4252:
4172:
3567:
6290:
4212:
4038:(3), so 2, 5, 12, 41, 53, 306, 665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to an integer number of octaves, are better approximations of 2, 5, 12, 41, 53, 306, 665 and 15601
3615:
A South
American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175 cent seven-tone equal temperament, which stretches the octave slightly, as with instrumental gamelan music.
1705:
76:
3990:
whatsoever). As it is a multiple of 12, 72 EDO can be considered an extension of 12 EDO, containing six copies of 12 EDO starting on different pitches, three copies of 24 EDO, and two copies of
3448:
3436:
4055:) is the sequence of divisions of octave that provides better and better approximations of the perfect fifth. Related sequences containing divisions approximating other just intervals are listed in a footnote.
3712:
3498:
the tempered perfect fifth is 720 cents wide (at the top of the tuning continuum), and marks the endpoint on the tuning continuum at which the width of the minor second shrinks to a width of 0 cents.
3424:
3403:
4758:
has become the most commonly used equal temperament. (Another reason is that 12 EDO is the smallest equal temperament to closely approximate 5 limit harmony, the next-smallest being 19 EDO.)
4343:
may not have their usual 12 EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let the number of steps in a semitone be
1099:
3563:, only slendro somewhat resembles five-tone equal temperament, while pelog is highly unequal; however, in 1972 Surjodiningrat, Sudarjana and Susanto analyze pelog as equivalent to 9-TET (133-cent steps
1883:
2141:
4083:
4752:
is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12
4099:
1700:
1494:
1368:
332:
instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Some wind instruments that can easily and spontaneously bend their tone, most notably
896:, but Zhu was the first person to mathematically solve 12 tone equal temperament, which he described in two books, published in 1580 and 1584. Needham also gives an extended account.
5281:, and made to connect at its far ends by slight adjustments to the size of one or several of the intervals, or left unmodified with occasional less-than-perfect fifths, flat by a comma.
978:
Plucked instrument players (lutenists and guitarists) generally favored equal temperament, while others were more divided. In the end, 12-tone equal temperament won out. This allowed
741:
7462:
641:
2401:{\displaystyle \quad x={\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}12\log _{2}\left({\frac {\ 550\ }{440}}\right){\Biggr )}={\frac {4}{\ 12\ }}={\frac {1}{\ 3\ }}~.}
750:, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the
6213:) â 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3, 9:8 and 16:9, 10:9 and 9:5, 16:15 and 15:8, 45:32 and 64:45, 27:20 and 40:27, 32:27 and 27:16, 81:64 and 128:81, 256:243 and 243:128
8296:
4120:
created three unusual equal temperaments after a thorough study of the properties of possible temperaments with step size between 30 and 120 cents. These were called
829:
12 tone equal temperament, which divides the octave into 12 intervals of equal size, is the musical system most widely used today, especially in
Western music.
594:
5350:
Some of the intermediate sizes of tones and semitones can also be generated in equal temperament systems, by modifying the sizes of the comma and semitones. One obtains
6955:
5246:
Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitones
2128:{\displaystyle \quad x={\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}\ 12\log _{2}\left({\frac {\ 660\ }{440}}\right)\ {\Biggr )}={\frac {7}{\ 12\ }}~.}
6081:
compares several equal temperaments in a graph with axes reversed from the axes in the first comparison of equal temperaments, and identical axes of the second.
4857:
8424:
4138:. They can be considered equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals. Their step sizes:
3982:
intervals well, providing near-just equivalents to the 3rd, 5th, 7th, and 11th harmonics. 72 EDO has been taught, written and performed in practice by
8282:
7780:
7336:
3830:
27 is the lowest number of equal divisions of the octave that uniquely represents all intervals involving the first eight harmonics. It tempers out the
1139:
6262:
6249:
6236:
6223:
6210:
6197:
6184:
6171:
6158:
6145:
6132:
6119:
4052:
8356:
8489:
8269:
2429:
In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents.
1815:{\displaystyle \quad \ x\ \equiv \ {\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}12\log _{2}\left({\frac {\ n\ }{a}}\right){\Biggr )}~.}
7457:
6303:
7362:
6800:
6733:
7398:
7239:
3806:, experimented with music for quarter-tone pianos. 24 EDO also approximates the 11th and 13th harmonics very well, unlike 12 EDO.
6524:
8499:
7883:
7555:
6975:
4300:
3662:
487:
6579:
6422:
Kuttner, Fritz A. (May 1975). "Prince Chu Tsai-YĂŒ's life and work: A re-evaluation of his contribution to equal temperament theory".
1044:
7280:
7258:
7015:
6928:
6855:
4322:
3684:
1990:{\displaystyle E_{660}=440\ {\mathsf {Hz}}\ \cdot \ 2^{\left({\frac {7}{\ 12\ }}\right)}\ \approx \ 659.255\ {\mathsf {Hz}}\ \quad }
513:
2248:{\displaystyle E_{550}=440\ {\mathsf {Hz}}\ \cdot \ 2^{\left({\frac {1}{\ 3\ }}\right)}\ \approx \ 554.365\ {\mathsf {Hz}}\ \quad }
858:
Kenneth
Robinson credits the invention of equal temperament to Zhu and provides textual quotations as evidence. In 1584 Zhu wrote:
282:
Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the
3704:'s notation system for 16 equal temperament: Intervals are notated similarly to those they approximate and there are fewer
8462:
8159:
4545:
The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the
4031:
7773:
7687:
6961:
4304:
3666:
1609:{\displaystyle P_{46}=440\ {\mathsf {Hz}}\ \cdot \ {\Bigl (}{\sqrt{2}}\ {\Bigr )}^{(46-49)}\approx 369.994\ {\mathsf {Hz}}\ }
1483:{\displaystyle P_{40}=440\ {\mathsf {Hz}}\ \cdot \ {\Bigl (}{\sqrt{2}}\ {\Bigr )}^{(40-49)}\approx 261.626\ {\mathsf {Hz}}\ }
491:
40:
7298:
3349:
to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a
1636:
876:
8647:
7589:
6763:
5753:, with three steps for the chromatic semitone, four steps for the diatonic semitone, and seven steps for the tone, where
2426:. The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.
47:
horizontally (open the image to view the full width), and each shaded rectangle is the width of one step in a scale. The
8861:
8349:
7632:
3941:
theory. It is not a meantone temperament, which put good thirds within easy reach by stacking fifths; instead, like all
5642:, with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone, where
8856:
8585:
8264:
7747:
7716:
7070:
6354:
7494:
1850:
is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see that
476:
6685:
8429:
8289:
8032:
7908:
7803:
6040:
7472:
5471:
and the result is seven-tone equal temperament. These two extremes are not included as "regular" diatonic tunings.
4293:
3655:
495:
480:
8414:
8241:
8229:
7766:
7488:
6424:
6352:
Varieschi, Gabriele U.; Gower, Christina M. (2010). "Intonation and compensation of fretted string instruments".
6332:
6580:"Quantifying ritual: Political cosmology, courtly music, and precision mathematics in seventeenth-century China"
8698:
7933:
8834:
8062:
4064:
699:
283:
6737:
6109:
OEIS sequences that contain divisions of the octave that provide improving approximations of just intervals:
3763:(at 505 cents), is seven cents sharper than just intonation's and five cents sharper than 12 EDO's.
1282:). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example,
677:
scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in
8668:
8494:
8419:
8342:
8191:
7873:
7568:
7548:
5090:
5065:
4862:
4766:
for the relationship results in exactly one equal temperament family, but the converse is not true: 47
8452:
6960:[Findings of new literatures concerning the hepta â equal temperament] (in Chinese). Archived from
3775:
subminor third (7/6). One step closer to each other are the classical major and minor thirds (5/4 and 6/5).
8690:
8653:
8643:
7888:
7699:
7677:
7639:
7155:
7095:
6045:
5745:
5695:
5668:
4642:, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24
4012:
3942:
3874:
3819:
798:
758:
where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to
293:
For tuning systems that divide the octave equally, but are not approximations of just intervals, the term
3114:
2985:
2856:
2665:
2536:
762:, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g.,
8686:
8008:
7838:
7622:
7424:
7384:
7380:
6324:
6020:
3701:
1294:
1113:
991:
979:
600:
543:
223:
4428:
also defines a unique family of one equal temperament and its multiples that fulfil this relationship.
837:
The two figures frequently credited with the achievement of exact calculation of equal temperament are
6659:
6200:) â 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3, 9:8 and 16:9, 10:9 and 9:5, 16:15 and 15:8, 45:32 and 64:45
941:
286:, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "
8694:
8639:
8621:
8616:
8611:
8606:
8601:
8596:
8591:
8576:
8571:
8566:
8561:
8556:
8551:
8318:
8141:
8136:
8131:
8126:
8121:
8116:
8111:
8101:
8096:
8091:
8086:
8081:
7813:
7008:
Toward a
Quarter-Tone Syntax: Analyses of selected works by Blackwood, Haba, Ives, and Wyschnegradsky
6816:
6468:
Robinson, Kenneth G.; Needham, Joseph (1962â2004). "Part 1: Physics". In
Needham, Joseph (ed.).
6373:
5938:
5800:
5664:
5634:
5584:
5575:
5508:
5477:
5325:
4020:
4016:
3999:
3975:
3963:
3918:
3897:
3885:
3862:
3851:
3846:
3783:
3771:
3736:
3477:
1026:
In 12 tone equal temperament, which divides the octave into 12 equal parts, the width of a
824:
325:
316:, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to
256:
245:
123:
5556:
19 steps. The imbedded 12 tone sub-system closely approximates the historically important
3723:
244:
Other equal temperaments divide the octave differently. For example, some music has been written in
8658:
8631:
8529:
8313:
8258:
7958:
7938:
5430:
5419:
5352:
4265:
1035:
961:
945:
539:
349:
313:
92:
6586:. Roger Hart Departments of History and Asian Studies, University of Texas, Austin. Archived from
6445:
A critical study of Chu Tsai-yĂŒ's contribution to the theory of equal temperament in
Chinese music
3937:(3). With its accurate cycle of fifths and multi-purpose comma step, 53 EDO has been used in
1335:
are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of
8723:
8509:
8504:
8235:
7953:
7863:
7693:
7541:
7390:
7222:
6389:
6363:
6338:
6328:
3930:
3866:
3799:
755:
560:
535:
122:
and
Western music in general, the most common tuning system since the 18th century has been
8728:
7823:
7188:
4004:
6847:
6540:
6472:. Science and Civilisation in China. Vol. 4. Cambridge, UK: University Press. p. 221.
6453:
Chu-Tsaiyu the first formulator of the mathematics of "equal temperament" anywhere in the world
8702:
8663:
8477:
8323:
8206:
7853:
7843:
7808:
7404:
7394:
7358:
7276:
7254:
7235:
7184:
7174:
7044:
7011:
6924:
6851:
6796:
6532:
6013:
5315:
157:
7478:
4920:
is implicit as the size ratio between the greater and lesser tones: Expressed as frequencies
4882:
can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps
8682:
8546:
8514:
8434:
8399:
8211:
8164:
8013:
7923:
7878:
7848:
7789:
7682:
7604:
7412:
7268:
7164:
7146:
6839:
6525:"The significance of the discovery of the musical equal temperament in the cultural history"
6381:
6050:
6030:
5278:
4976:
4546:
3243:
3181:
3052:
2923:
2794:
2732:
2603:
2479:
937:
747:
551:
527:
321:
267:
227:
7418:â A foundational work on acoustics and the perception of sound. Especially the material in
7151:"Isomorphic controllers and dynamic tuning: Invariant fingerings across a tuning continuum"
6985:
852:
somewhat less precise computation of the corresponding numerical values in 1585 or later."
8804:
8444:
8151:
7893:
7599:
6671:
6587:
6008:
5990:
5206:
4356:
4092:
4039:
3987:
3979:
3922:
2423:
678:
526:
In an equal temperament, the distance between two adjacent steps of the scale is the same
317:
309:
119:
100:
4638:
If there are notes outside the circle of fifths, one must then multiply these results by
4555:, the half-sharps and half-flats are not in the circle of fifths generated starting from
6377:
4359:
of a whole tone, while keeping the notes in the right order (meaning that, for example,
3889:
sharp instead of flat. It enables the 600 cent tritone, since 34 is an even number.
797:
Please expand the section to include this information. Further details may exist on the
681:. The basic step in cents for any equal temperament can be found by taking the width of
60:
8482:
8457:
8404:
8365:
8275:
7988:
7943:
7833:
7709:
7649:
7500:
7299:"The gamelan pelog scale of Central Java as an example of a non-harmonic musical scale"
7219:
The
Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century
6055:
5986:
5667:. The imbedded 12 tone sub-system closely approximates the historically important
5236:
5224:
5172:
5163:
4971:
The notes in a regular diatonic tuning are connected in a "spiral of fifths" that does
4889:
3870:
3835:
3831:
3760:
3619:
3548:
1031:
219:
96:
6187:) â 4:3 and 3:2, 5:4 and 8:5, 6:5 and 5:3, 7:4 and 8:7, 16:11 and 11:8, 16:13 and 13:8
8850:
8777:
8746:
8519:
8472:
8379:
8186:
7928:
7903:
7858:
7731:
7350:
7329:
6840:
5994:
5168:
5096:
5002:
4068:
3938:
3926:
3552:
287:
188:
108:
48:
6447:. Sinologica Coloniensia. Vol. 9. Wiesbaden, DE: Franz Steiner Verlag. p.
6393:
673:, which divide the octave into 1200 equal intervals (each called a cent). This
8741:
8581:
8384:
8106:
8003:
7998:
7993:
7978:
7973:
7868:
7828:
7818:
7726:
7704:
7667:
7594:
7099:
6872:
6759:
6035:
5706:
If the chromatic semitone is three-fourths the size of the diatonic semitone, i.e.
4950:
4117:
3945:, the very consonant thirds are represented by a Pythagorean diminished fourth (C-F
3803:
3795:
3587:
xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from
3544:
983:
951:
848:
670:
7120:
6090:'Hepta-equal temperament' in our folk music has always been a controversial issue.
1293:(the reference pitch) is the 49th key from the left end of a piano (tuned to
899:
Zhu obtained his result by dividing the length of string and pipe successively by
746:
In musical analysis, material belonging to an equal temperament is often given an
7505:
7306:
4355:
There is exactly one family of equal temperaments that fixes the semitone to any
1234:{\displaystyle \ P_{n}=P_{a}\ \cdot \ {\Bigl (}\ {\sqrt{2\ }}\ {\Bigr )}^{n-a}\ }
685:
above in cents (usually the octave, which is 1200 cents wide), called below
8829:
8824:
8814:
8409:
8389:
8057:
8052:
8042:
7983:
5595:
If the chromatic semitone is two-thirds the size of the diatonic semitone, i.e.
5488:
If the diatonic semitone is set double the size of the chromatic semitone, i.e.
5054:
4282:
4134:
4122:
4027:
3697:
3644:
3540:
759:
547:
465:
103:
by dividing an octave (or other interval) into steps such that the ratio of the
52:
7349:. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave.
3373:
8819:
8788:
8047:
7898:
7721:
7672:
7533:
7452:
7292:
Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta
7169:
7150:
6769:
6025:
5314:, with the others expanded to still fill out the octave), and both semitones (
5294:
5290:
5194:
5183:
4909:
4901:
4128:
3983:
3705:
3584:
3555:. It is now accepted that of the two primary tuning systems in gamelan music,
3522:
7178:
6536:
5219:
s c Îș s c s s c Îș s c s c Îș s
3986:
and his students (whose atonal inclinations typically avoid any reference to
928:
Zhu created several instruments tuned to his system, including bamboo pipes.
8809:
8763:
8201:
7918:
7644:
7564:
7442:
7408:
6879:(in French). Association pour la Recherche et le DĂ©veloppement de la Musique
6554:
6486:
3921:
has only had occasional use, but is better at approximating the traditional
3363:
995:
987:
880:
838:
751:
674:
156:), which divides the octave into 12 parts, all of which are equal on a
112:
104:
7074:
4091:), and split into 13 equal parts. This provides a very close match to
4011:
Other equal divisions of the octave that have found occasional use include
783:
7386:
On the Sensations of Tone as a Physiological Basis for the Theory of Music
1825:
1012:
17:
8196:
7627:
6693:
1309:
1027:
936:
Some of the first Europeans to advocate equal temperament were lutenists
754:
of a multiplication reduces it to addition. Furthermore, by applying the
333:
184:
7438:
4842:). Taking each semitone results in a different choice of perfect fifth.
3955:, allowing its fifth to be reached by a stack of six minor thirds (6:5).
8783:
8733:
5324:) the same size, then twelve equal semitones, two per tone, result. In
4998:
4307: in this section. Unsourced material may be challenged and removed.
4076:
3952:
3669: in this section. Unsourced material may be challenged and removed.
3556:
3526:
6385:
4263:
Alpha and beta may be heard on the title track of Carlos's 1986 album
3951:), reached by stacking eight perfect fourths. It also tempers out the
925:
such that after 12 divisions (an octave), the length was halved.
8751:
8467:
8394:
7913:
4404:
are in ascending order if they preserve their usual relationships to
4072:
44:
7758:
7323:
8768:
8756:
8334:
7447:
7354:
6368:
4865:
continuum, which include many notable "equal temperament" tunings.
4856:
3722:
3560:
1254:
875:
531:
237:
7422:, pages 430â556, (pdf pages 451â577) (see also wiki article
5811:
If the chromatic semitone is made the same size as three commas,
4095:
ratios consisting only of odd numbers. Each step is 146.3 cents (
887:
Chinese theorists had previously come up with approximations for
8773:
4042:
twelfths/fifths than in any equal temperament with fewer tones.
999:
767:
542:
of intervals would not sound evenly spaced and would not permit
329:
8338:
7762:
7537:
5460:
gets larger (and absorbs the space formerly used for the comma
4654:). (One must take the small semitone for this purpose: 19
3925:
consonances than 12, 19 or 31 EDO. Its extremely accurate
1094:{\displaystyle {\sqrt{2\ }}=2^{\tfrac {1}{12}}\approx 1.059463}
530:. Because the perceived identity of an interval depends on its
7232:
How Equal Temperament Ruined Harmony (and why you should care)
6896:
5775:
43. The imbedded 12 tone sub-system closely approximates
5449:
reduce to zero with the octave size kept fixed, the result is
5441:
There are two extreme cases that bracket this framework: When
4276:
3638:
1620:
Converting frequencies to their equal temperament counterparts
777:
459:
107:
of any adjacent pair of notes is the same. This system yields
7489:
All existing 18th century quotes on J.S. Bach and temperament
4067:
consists of the ratio 3:1 (1902 cents) conventionally a
43:
A comparison of some equal temperaments. The graph spans one
3369:
Five-, seven-, and nine-tone temperaments in ethnomusicology
6722:] (in Italian) (reprint ed.). Geneva, CH: Minkoff.
6257:
6244:
6231:
6218:
6205:
6192:
6179:
6166:
6153:
6140:
6127:
6114:
4047:
795:
about the general formulas for the equal-tempered interval.
336:, use tuning similar to string ensembles and vocal groups.
6341:(reprint ed.). New York, NY: Dover. pp. 493â511.
4804:, which are not complements of each other like in 19
7393:(reprint ed.). Whitefish, MT: Kellinger Publishing.
7290:
Surjodiningrat, W.; Sudarjana, P.J.; Susanto, A. (1972).
6174:) â 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7, 16:11 and 11:8
3309:
Violins, violas, and cellos are tuned in perfect fifths (
1266:
is the frequency of a reference pitch. The indes numbers
160:, with a ratio equal to the 12th root of 2, (
5339:, is exactly half the size of the same-size whole tones
5289:
An equal temperament can be created if the sizes of the
5262:, with some fixed proportion for each type of semitone.
1002:(at least its piano component) to develop and flourish.
27:
Musical tuning system with constant ratios between notes
7501:
Well Temperaments, based on the Werckmeister Definition
6668:
The Fronimo ... Dialogue on the art of a good beginning
6664:
Il Fronimo ... Dialogo sopra l'arte del bene intavolare
4045:
1, 2, 3, 5, 7, 12, 29, 41, 53, 200, ... (sequence
72:, one full octave ascending, notated only with sharps.
5989:. It is an exceedingly close approximation to 5-limit
4075:(that is, a perfect twelfth), called in this theory a
1841:
is the frequency of a pitch in equal temperament, and
1829:
Comparison of intervals in 12-TET with just intonation
1695:{\displaystyle \ E_{n}=E_{a}\ \cdot \ 2^{\ x}\ \quad }
1073:
320:
for acoustic reasons. Other instruments, such as some
312:, which can adjust the tuning of all notes except for
7453:
Huygens-Fokker Foundation Centre for Microtonal Music
2261:
2144:
2003:
1886:
1708:
1639:
1497:
1371:
1142:
1047:
702:
603:
563:
6627:
6607:
5277:
can be repeatedly appended to itself into a greater
3901:
more accurate in the 13 limit than 31 EDO.
3854:, which allows for lower errors for some just tones.
3547:(1966) their tuning varies widely, and according to
1624:
To convert a frequency (in Hz) to its equal 12
167:â 1.05946 ). That resulting smallest interval,
8797:
8714:
8677:
8630:
8537:
8528:
8443:
8372:
8306:
8297:
Twelve Microtonal Etudes for Electronic Music Media
8250:
8220:
8177:
8150:
8072:
8031:
8022:
7966:
7796:
7740:
7658:
7613:
7575:
7495:
Rosetta Revisited: Bach's Very Ordinary Temperament
1034:of the interval between two adjacent notes, is the
7335:
2400:
2247:
2127:
1989:
1814:
1694:
1608:
1482:
1233:
1093:
770:encoding standard uses integer note designations.
735:
661:
635:
588:
370:on each main interval of small prime limits (red:
64:12 tone equal temperament chromatic scale on
7049:Tonalsoft Encyclopedia of Microtonal Music Theory
5941:of one comma each. The comma size / step size is
4606:and the semitone and tone are the same interval.
3933:, as 53 is the denominator of a convergent to log
2345:
2296:
2293:
2093:
2038:
2035:
1801:
1752:
1749:
1561:
1538:
1435:
1412:
1211:
1180:
7073:. xenoharmonic (microtonal wiki). Archived from
5064:The three in-tune fifths are interrupted by the
1630:counterpart, the following formula can be used:
774:General formulas for the equal-tempered interval
7448:Xenharmonic wiki on EDOs vs. Equal Temperaments
7294:. Jogjakarta, IN: Gadjah Mada University Press.
6622:The Shorter Science & Civilisation in China
5464:), eventually the steps are all the same size,
3910:meantone system, it distinguishes 10:9 and 9:8.
1253:represents the pitch, or frequency (usually in
6897:Surjodiningrat, Sudarjana & Susanto (1972)
6736:. Appalachian State University. Archived from
5369:tend to zero, with the octave kept fixed, and
5305:) are altered to be the same (say, by setting
1877:have the following frequencies, respectively:
1274:are the labels assigned to the desired pitch (
1104:This interval is divided into 100 cents.
8350:
7774:
7549:
7475:. (2008) Latina, Il Levante Libreria Editrice
7420:Appendix XX: Additions by the translator
7275:(2nd ed.). London, UK: Springer-Verlag.
6463:
6461:
5993:and Pythagorean tuning, and is the basis for
3727:Comparison of equal temperaments from 9 to 25
3517:5 tone and 9 tone equal temperament
842:
111:steps perceived as equal in size, due to the
8:
8425:List of intervals in 5-limit just intonation
6919:Morton, David (1980). May, Elizabeth (ed.).
6405:
6403:
5171:: Each of the grave fifths out of tune by a
4900:) must be smaller than either of the tones (
4896:diatonic tuning, each of the two semitones (
3814:26 is the denominator of a convergent to log
290:" in that system, into 13 equal parts.
7473:Enharmonic instruments and music, 1470â1900
7326:(Report). 8096295 – via academia.edu.
7324:From galaxy to galaxy: Music of the spheres
7204:Boiles, J. (1969). "Terpehua though-song".
6481:
6479:
6252:) â 6:5 and 5:3, 7:5 and 10:7, 7:6 and 12:7
4549:. (This is not true in general; in 24
4273:Proportions between semitone and whole tone
494:. Unsourced material may be challenged and
352:A comparison of equal temperaments between
8534:
8357:
8343:
8335:
8028:
7781:
7767:
7759:
7556:
7542:
7534:
7234:. New York, NY: W.W.Norton & Company.
7121:"Three Asymmetric Divisions of the Octave"
6331:"The History of Musical Pitch in Europe".
6074:
6072:
4772:has two different semitones, where one is
4059:Equal temperaments of non-octave intervals
654:(typically the octave, which is 2:1) into
7465:A supplement to Mr. Chambers's cyclopĂŠdia
7332:. 269108386 – via researchgate.net.
7168:
6367:
6161:) â 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7
6135:) â 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3
5518:with one step for the chromatic semitone
5136:, and then restarting in the sharps with
4416:to a proper fraction in the relationship
4323:Learn how and when to remove this message
4026:2, 5, 12, 41, 53, 306, 665 and 15601 are
3685:Learn how and when to remove this message
2374:
2353:
2344:
2343:
2321:
2308:
2295:
2294:
2269:
2260:
2232:
2231:
2192:
2187:
2165:
2164:
2149:
2143:
2101:
2092:
2091:
2066:
2053:
2037:
2036:
2011:
2002:
1974:
1973:
1934:
1929:
1907:
1906:
1891:
1885:
1800:
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1777:
1764:
1751:
1750:
1725:
1707:
1679:
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1638:
1594:
1593:
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1559:
1548:
1543:
1537:
1536:
1518:
1517:
1502:
1496:
1468:
1467:
1440:
1434:
1433:
1422:
1417:
1411:
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1392:
1391:
1376:
1370:
1216:
1210:
1209:
1198:
1188:
1179:
1178:
1163:
1150:
1141:
1072:
1058:
1048:
1046:
982:, new styles of symmetrical tonality and
855:The developments occurred independently.
712:
701:
623:
613:
602:
571:
562:
554:in an equal-tempered scale is the ratio:
514:Learn how and when to remove this message
195:, without qualification, generally means
7351:The International Conference SIGMAP-2008
7347:Approximation of 5-limit just intonation
7030:
6529:Journal of Xinghai Conservatory of Music
6286:
6078:
5854:that makes the lesser tone eight commas
5840:) the diatonic the same as five commas,
5456:a 5 tone equal temperament. As the
3696:
3456:steps, respectively, are fairly common.
3389:Five- and seven-tone equal temperament (
3372:
2433:
1824:
1011:
841:(also romanized as Chu-Tsaiyu. Chinese:
766:is 0 regardless of octave register. The
736:{\displaystyle \ c={\frac {\ w\ }{n}}\ }
59:
7305:. Neuroscience of Music. Archived from
6923:. Musics of Many Cultures. p. 70.
6846:(2nd ed.). Da Capo Press. p.
6510:
6409:
6279:
6068:
5250:, or into the five chromatic semitones
5099:"), followed by another perfect fifth,
4585:and the semitone becomes a unison, and
4348:, and the number of steps in a tone be
3735:Many instruments have been built using
6943:
6907:
6873:"Le tempérament égal à quintes justes"
6762:(30 June 2009) . Rasch, Rudolf (ed.).
6498:Complete Compendium of Music and Pitch
5522:, two steps for the diatonic semitone
3305:Seven-tone equal division of the fifth
2422:closely approximate some intervals in
2236:
2233:
2169:
2166:
1978:
1975:
1911:
1908:
1598:
1595:
1522:
1519:
1472:
1469:
1396:
1393:
51:ratios are separated in rows by their
7439:An Introduction to Historical Tunings
7345:Khramov, Mykhaylo (26â29 July 2008).
6100:system associated with the flute law.
4892:or "rotation" of it). To be called a
1133:, the following formula may be used:
7:
6768:. The Diapason Press. Archived from
4305:adding citations to reliable sources
3873:and represents a standardization of
3667:adding citations to reliable sources
492:adding citations to reliable sources
183:the width of an octave, is called a
7253:. Michigan State University Press.
6982:About "Seven- equal- tuning System"
6624:(abridgemed ed.). p. 385.
3480:'s valid tuning range, as shown in
975:), published posthumously in 1884.
6984:] (in Chinese). Archived from
6780:– via diapason.xentonic.org.
6628:Robinson & Needham (1962â2004)
6626:â reduced version of the original
6608:Robinson & Needham (1962â2004)
5285:Morphing diatonic tunings into EDO
4733:The smallest of these families is
4063:The equal-tempered version of the
3929:make it equivalent to an extended
636:{\displaystyle \ r={\sqrt{p\ }}\ }
25:
6734:"Spinacino 1507a: Thematic Index"
6304:"Perceptual Foundations of Sound"
5215:can be broken up into a sequence
3351:ratio of â 517:258 or â 2.00388:1
954:was the first to develop 12
948:, all of whom wrote music in it.
8500:Ptolemy's intense diatonic scale
7957:
7952:
6821:Musicalische paradoxal-Discourse
6765:Van de Spiegheling der singconst
6670:] (in Italian). Venice, IT:
6646:Abacus and Practical Mathematics
5534:, and the total number of steps
4281:
3643:
1108:Calculating absolute frequencies
966:van de Spiegheling der singconst
883:'s equal temperament pitch pipes
782:
534:, this scale in even steps is a
464:
341:
32:
7273:Tuning, Timbre, Spectrum, Scale
7149:; Plamondon, J. (Winter 2007).
6686:"Resound â corruption of music"
6470:Physics and Physical Technology
4292:needs additional citations for
3654:needs additional citations for
2412:Comparison with just intonation
2262:
2244:
2004:
1986:
1709:
1691:
218:is usually tuned relative to a
7688:Emancipation of the dissonance
6825:Paradoxical Musical Discussion
6523:Cho, Gene J. (February 2010).
4871:12 tone equal temperament
1579:
1567:
1453:
1441:
990:such as that written with the
550:.) Specifically, the smallest
128:12 tone equal temperament
1:
7590:Mode of limited transposition
7006:Skinner, Myles Leigh (2007).
5227:of it) of diatonic semitones
4698:has two semitones, one being
4660:has two semitones, one being
3481:
969:
819:Twelve-tone equal temperament
669:Scales are often measured in
662:Twelve-tone equal temperament
78:Play ascending and descending
8463:Harry Partch's 43-tone scale
8160:Harry Partch's 43-tone scale
7458:A.Orlandini: Music Acoustics
6566:Fusion of Music and Calendar
6148:) â 3:2 and 4:3, 5:4 and 8:5
5526:, three steps for the tones
5361:in the limit as the size of
4989:Starting on the subdominant
295:equal division of the octave
152:, informally abbreviated as
115:changes in pitch frequency.
8283:Sonata for Microtonal Piano
7748:List of atonal compositions
7717:Quartal and quintal harmony
7217:Cho, Gene Jinsiong (2003).
6714:Gorzanis, Giacomo (1982) .
6355:American Journal of Physics
5875:and the greater tone nine,
5115:, and another grave fifth,
4746:and in particular, 12
4624:and the perfect fifth into
1278:) and the reference pitch (
1257:), you are trying to find.
589:{\displaystyle \ r^{n}=p\ }
8878:
8430:List of meantone intervals
8290:Suite for Microtonal Piano
7096:"convergents(log2(3), 10)"
6793:Lutes, Viols, Temperaments
6443:Robinson, Kenneth (1980).
6041:List of meantone intervals
5265:The sequence of intervals
5053:âeach a composite of some
3635:Various equal temperaments
3476:mark the endpoints of the
3361:
1111:
822:
8420:List of musical intervals
8415:Consonance and dissonance
8242:Huygens-Fokker Foundation
8230:Boston Microtonal Society
7950:
7170:10.1162/comj.2007.31.4.15
6334:On the Sensations of Tone
6291:fig. 4.6, p. 58
5057:of the smaller intervals
4714:tone and the other being
4676:tone and the other being
4563:.) The extreme cases are
2462:Cents in just intonation
2459:Just intonation interval
2448:Decimal value in 12
2416:The intervals of 12
843:
7479:Fractal Microtonal Music
7340:– via Google docs.
7249:Jorgensen, Owen (1991).
7230:Duffin, Ross W. (2007).
6976:
6956:
6641:
6561:
6493:
5825:(in cents, in frequency
5279:spiral of 12 fifths
4863:regular diatonic tunings
4851:Regular diatonic tunings
4762:Each choice of fraction
3579:7-tone equal temperament
3358:Other equal temperaments
964:, which he described in
538:of multiplications. (An
8192:Otonality and Utonality
7322:Stewart, P.J. (2006) .
5395:is of course, the case
4869:The diatonic tuning in
3943:schismatic temperaments
3622:has traditionally used
3596:. According to Morton,
2440:Exact value in 12
1127:, of a note in 12
1118:To find the frequency,
912:and for pipe length by
689:, and dividing it into
236:, is tuned to 440
222:of 440 Hz, called
7889:Claus-Steffen Mahnkopf
7700:Polymodal chromaticism
7678:Dissonant counterpoint
7640:Second Viennese School
7156:Computer Music Journal
6977:äžćčłććŸ"çè°--ć
Œćæ§ćŒććæČçŹć¶äœäžèœŹè°
6838:Partch, Harry (1979).
6046:Diatonic and chromatic
6017:(the physics of music)
5231:, chromatic semitones
4866:
4846:Related tuning systems
4788:tone and the other is
3875:quarter-comma meantone
3739:tuning. Equivalent to
3728:
3720:
3432:), with 240 cent
3386:
2402:
2249:
2129:
1991:
1830:
1816:
1696:
1610:
1484:
1235:
1095:
1023:
1016:One octave of 12
884:
793:is missing information
737:
637:
590:
84:
8687:Temperament ordinaire
8065:(BohlenâPierce scale)
8009:Tui St. George Tucker
7623:Twelve-tone technique
7425:On Sensations of Tone
6921:The Music of Thailand
6817:Werckmeister, Andreas
6716:Intabolatura di liuto
6610:, p. 220 ff
6021:Music and mathematics
4860:
4692:. Similarly, 31
3726:
3700:
3376:
2403:
2250:
2130:
1992:
1828:
1817:
1697:
1611:
1485:
1236:
1114:Piano key frequencies
1096:
1015:
980:enharmonic modulation
879:
738:
638:
591:
63:
8490:List of compositions
8319:Generalized keyboard
7814:Easley Blackwood Jr.
7493:Dominic Eckersley: "
7471:Barbieri, Patrizio.
7357:. pp. 181â184.
7269:Sethares, William A.
6620:Ronan, Colin (ed.).
6302:O'Donnell, Michael.
5995:Turkish music theory
4301:improve this article
3964:58 equal temperament
3663:improve this article
3551:(2000) they contain
3478:syntonic temperament
2259:
2142:
2001:
1884:
1706:
1637:
1495:
1369:
1140:
1045:
825:12 equal temperament
700:
601:
561:
488:improve this section
226:, meaning one note,
124:12 equal temperament
8862:Chinese discoveries
8314:Enharmonic keyboard
8265:quarter tone pieces
8259:Beauty in the Beast
7939:Ivan Wyschnegradsky
7463:"Temperament" from
7045:"Equal-temperament"
7043:Monzo, Joe (2005).
6648:]. p. 389.
6568:] (in Chinese).
6500:] (in Chinese).
6378:2010AmJPh..78...47V
4431:For example, where
4412:). That is, fixing
4266:Beauty in the Beast
4065:BohlenâPierce scale
3525:(1949), Indonesian
2255:where in this case
1997:where in this case
1036:twelfth root of two
962:twelfth root of two
946:Francesco Spinacino
540:arithmetic sequence
284:BohlenâPierce scale
93:musical temperament
8857:Equal temperaments
8724:Chinese musicology
8510:Scale of harmonics
8505:Pythagorean tuning
8453:EulerâFokker genus
8236:Genesis of a Music
7864:Christiaan Huygens
7694:Klangfarbenmelodie
7309:on 27 January 2005
7223:Edwin Mellen Press
6842:Genesis of a Music
6827:] (in German).
6642:ćłæ±ç ăç çźäžćźçšæ°ćŠă 389饔
6265:) â 11:8 and 16:11
5001:) there are three
4975:close (unlike the
4867:
4248:(35.1 cents)
4208:(63.8 cents)
4168:(78.0 cents)
3978:approximates many
3931:Pythagorean tuning
3867:Christiaan Huygens
3820:barbershop harmony
3800:quarter-tone scale
3729:
3721:
3444:and 171 cent
3387:
2398:
2245:
2125:
1987:
1831:
1812:
1692:
1606:
1480:
1231:
1091:
1082:
1024:
885:
756:modular arithmetic
733:
650:divides the ratio
633:
586:
536:geometric sequence
456:General properties
99:that approximates
85:
8844:
8843:
8710:
8709:
8332:
8331:
8324:Modernism (music)
8173:
8172:
8074:Equal temperament
7844:Brian Ferneyhough
7756:
7755:
7585:Equal temperament
7530:
7526:
7521:
7517:
7513:
7509:
7364:978-989-8111-60-9
6980:[abstract of
6802:978-0-521-28883-5
6690:Philresound.co.uk
6674:. pp. 80â89.
6584:uts.cc.utexas.edu
6543:on 15 March 2012.
6386:10.1119/1.3226563
6014:Musical acoustics
5178:Since the comma,
5095:means "flat by a
4648:, six in 72
4335:In this section,
4333:
4332:
4325:
3865:was advocated by
3695:
3694:
3687:
3553:stretched octaves
3539:but according to
3377:Approximation of
3300:
3299:
2394:
2390:
2388:
2382:
2369:
2367:
2361:
2337:
2332:
2326:
2289:
2285:
2283:
2277:
2243:
2230:
2224:
2218:
2208:
2206:
2200:
2182:
2176:
2163:
2121:
2117:
2115:
2109:
2090:
2082:
2077:
2071:
2045:
2031:
2027:
2025:
2019:
1985:
1972:
1966:
1960:
1950:
1948:
1942:
1924:
1918:
1905:
1808:
1793:
1788:
1782:
1745:
1741:
1739:
1733:
1724:
1718:
1712:
1702:where in general
1690:
1682:
1674:
1668:
1642:
1605:
1592:
1557:
1553:
1535:
1529:
1516:
1479:
1466:
1431:
1427:
1409:
1403:
1390:
1230:
1207:
1203:
1196:
1187:
1177:
1171:
1145:
1081:
1063:
1056:
992:12-tone technique
816:
815:
732:
728:
723:
717:
705:
632:
628:
621:
606:
585:
566:
524:
523:
516:
209:In modern times,
193:equal temperament
189:Western countries
187:or half step. In
158:logarithmic scale
89:equal temperament
16:(Redirected from
8869:
8683:Well temperament
8669:Regular diatonic
8535:
8515:Tonality diamond
8359:
8352:
8345:
8336:
8212:Tonality diamond
8035:repeating scales
8029:
8014:Nicola Vicentino
7961:
7956:
7924:Nicola Vicentino
7849:Michael Finnissy
7790:Microtonal music
7783:
7776:
7769:
7760:
7683:Dynamic tonality
7605:Whole tone scale
7558:
7551:
7544:
7535:
7528:
7524:
7519:
7515:
7511:
7507:
7416:
7413:Internet Archive
7400:978-1-41917893-1
7389:. Translated by
7368:
7341:
7339:
7333:
7327:
7318:
7316:
7314:
7295:
7286:
7264:
7245:
7241:978-0-39306227-4
7226:
7221:. Lewiston, NY:
7213:
7191:
7182:
7172:
7142:
7136:
7135:
7133:
7132:
7116:
7110:
7109:
7107:
7106:
7092:
7086:
7085:
7083:
7082:
7067:
7061:
7060:
7058:
7056:
7040:
7034:
7028:
7022:
7021:
7003:
6997:
6996:
6994:
6993:
6972:
6966:
6965:
6957:æć
ł"äžćčłććŸ"æ°æçźèäœçćç°
6952:
6946:
6941:
6935:
6934:
6916:
6910:
6905:
6899:
6894:
6888:
6887:
6885:
6884:
6877:aredem.online.fr
6871:Cordier, Serge.
6868:
6862:
6861:
6845:
6835:
6829:
6828:
6813:
6807:
6806:
6788:
6782:
6781:
6779:
6777:
6756:
6750:
6749:
6747:
6745:
6730:
6724:
6723:
6711:
6705:
6704:
6702:
6701:
6692:. Archived from
6682:
6676:
6675:
6656:
6650:
6649:
6637:
6631:
6625:
6617:
6611:
6605:
6599:
6598:
6596:
6595:
6576:
6570:
6569:
6551:
6545:
6544:
6539:. Archived from
6520:
6514:
6508:
6502:
6501:
6483:
6474:
6473:
6465:
6456:
6455:
6450:
6440:
6434:
6433:
6419:
6413:
6407:
6398:
6397:
6371:
6349:
6343:
6342:
6337:. Translated by
6321:
6315:
6314:
6312:
6310:
6299:
6293:
6284:
6268:
6260:
6247:
6234:
6221:
6208:
6195:
6182:
6169:
6156:
6143:
6130:
6117:
6107:
6101:
6097:
6091:
6088:
6082:
6076:
6051:Electronic tuner
6031:Microtonal music
5984:
5982:
5977:
5975:
5966:
5964:
5963:
5961:
5960:
5957:
5954:
5936:
5931:
5916:
5900:
5898:
5874:
5872:
5853:
5851:
5839:
5837:
5824:
5822:
5808:
5806:
5805:
5793:
5791:
5789:
5788:
5785:
5782:
5774:
5772:
5768:
5751:
5750:
5743:
5741:
5733:
5731:
5727:
5725:
5724:
5721:
5718:
5703:
5701:
5700:
5686:
5684:
5682:
5681:
5678:
5675:
5663:
5661:
5657:
5640:
5639:
5632:
5630:
5622:
5620:
5616:
5614:
5613:
5610:
5607:
5592:
5590:
5589:
5574:
5572:
5570:
5569:
5566:
5563:
5555:
5553:
5549:
5533:
5529:
5525:
5521:
5517:
5514:
5513:
5505:
5503:
5494:
5493:
5485:
5483:
5482:
5470:
5468:
5463:
5459:
5455:
5453:
5448:
5444:
5438:
5436:
5435:
5427:
5425:
5424:
5413:
5411:
5402:
5400:
5394:
5393:
5392:
5385:
5381:
5378:in the limit as
5377:
5376:
5375:
5368:
5364:
5360:
5358:
5357:
5346:
5342:
5338:
5335:, the semitone,
5333:
5332:
5331:
5323:
5318:
5313:
5312:
5304:
5300:
5276:
5272:
5268:
5261:
5257:
5253:
5249:
5245:
5243:
5234:
5230:
5222:
5220:
5214:
5213:
5204:
5202:
5192:
5190:
5181:
5169:"wolf" intervals
5161:
5160:
5156:
5155:
5148:
5147:
5143:
5142:
5135:
5134:
5130:
5129:
5122:
5121:
5114:
5113:
5106:
5105:
5088:
5087:
5082:
5081:
5074:
5073:
5063:
5061:
5052:
5051:
5044:
5043:
5036:
5035:
5028:
5027:
5020:
5019:
5012:
5011:
4996:
4995:
4988:
4986:
4985:
4977:circle of fifths
4967:
4966:
4948:
4946:
4945:
4943:
4942:
4937:
4934:
4919:
4915:
4907:
4899:
4887:
4886:
4881:
4879:
4878:
4841:
4839:
4838:
4835:
4832:
4825:
4823:
4822:
4819:
4816:
4809:
4808:
4803:
4801:
4800:
4797:
4794:
4787:
4785:
4784:
4781:
4778:
4771:
4770:
4765:
4757:
4756:
4751:
4750:
4745:
4743:
4742:
4738:
4729:
4727:
4726:
4723:
4720:
4713:
4711:
4710:
4707:
4704:
4697:
4696:
4691:
4689:
4688:
4685:
4682:
4675:
4673:
4672:
4669:
4666:
4659:
4658:
4653:
4652:
4647:
4646:
4641:
4637:
4635:
4623:
4621:
4605:
4604:
4596:
4595:
4594:
4590:
4584:
4583:
4575:
4573:
4572:
4568:
4562:
4561:
4554:
4553:
4547:circle of fifths
4544:
4542:
4541:
4539:
4538:
4535:
4532:
4519:
4518:
4517:
4513:
4507:
4505:
4504:
4502:
4501:
4498:
4495:
4482:
4481:
4480:
4476:
4471:
4469:
4468:
4466:
4465:
4462:
4459:
4446:
4445:
4444:
4440:
4434:
4427:
4426:
4415:
4411:
4410:
4403:
4402:
4398:
4397:
4390:
4389:
4382:
4381:
4374:
4373:
4366:
4365:
4351:
4347:
4328:
4321:
4317:
4314:
4308:
4285:
4277:
4259:
4258:
4257:
4255:
4247:
4246:
4245:
4244:
4242:
4241:
4238:
4235:
4219:
4218:
4217:
4215:
4207:
4206:
4205:
4204:
4202:
4201:
4198:
4195:
4179:
4178:
4177:
4175:
4167:
4166:
4165:
4164:
4162:
4161:
4158:
4155:
4113:
4112:
4106:
4105:
4104:
4102:
4090:
4089:
4088:
4086:
4050:
3950:
3949:
3757:
3755:
3753:
3752:
3749:
3746:
3719:
3718:
3717:
3715:
3702:Easley Blackwood
3690:
3683:
3679:
3676:
3670:
3647:
3639:
3630:
3629:
3628:
3611:
3610:
3609:
3607:
3595:
3594:
3593:
3574:
3573:
3572:
3570:
3538:
3536:
3535:
3510:
3508:
3507:
3497:
3495:
3494:
3475:
3474:
3473:
3466:
3465:
3464:
3455:
3454:
3453:
3451:
3443:
3442:
3441:
3439:
3431:
3430:
3429:
3427:
3418:
3417:
3410:
3409:
3408:
3406:
3398:
3397:
3396:
3385:
3384:
3383:
3352:
3348:
3347:
3346:
3345:
3343:
3342:
3339:
3336:
3324:
3323:
3317:for violins and
3316:
3315:
3290:
3286:
3284:
3283:
3280:
3277:
3266:
3261:
3257:
3250:
3249:
3231:
3227:
3225:
3224:
3221:
3218:
3204:
3203:
3202:
3195:
3188:
3187:
3169:
3165:
3163:
3162:
3159:
3156:
3142:
3141:
3140:
3133:
3126:
3125:
3121:
3120:
3102:
3098:
3096:
3095:
3092:
3089:
3075:
3074:
3073:
3066:
3059:
3058:
3040:
3036:
3034:
3033:
3030:
3027:
3013:
3012:
3011:
3004:
2997:
2996:
2992:
2991:
2973:
2969:
2967:
2966:
2963:
2960:
2946:
2945:
2944:
2937:
2930:
2929:
2911:
2907:
2905:
2904:
2901:
2898:
2884:
2883:
2882:
2875:
2868:
2867:
2863:
2862:
2844:
2840:
2838:
2837:
2834:
2831:
2817:
2816:
2815:
2808:
2801:
2800:
2793:Perfect fourth (
2782:
2778:
2776:
2775:
2772:
2769:
2755:
2754:
2753:
2746:
2739:
2738:
2720:
2716:
2714:
2713:
2710:
2707:
2693:
2692:
2691:
2684:
2677:
2676:
2672:
2671:
2653:
2649:
2647:
2646:
2643:
2640:
2626:
2625:
2624:
2617:
2610:
2609:
2591:
2587:
2585:
2584:
2581:
2578:
2564:
2563:
2562:
2555:
2548:
2547:
2543:
2542:
2524:
2520:
2518:
2517:
2514:
2511:
2497:
2493:
2486:
2485:
2470:
2469:
2453:
2452:
2445:
2444:
2434:
2421:
2420:
2407:
2405:
2404:
2399:
2392:
2391:
2389:
2386:
2380:
2375:
2370:
2368:
2365:
2359:
2354:
2349:
2348:
2342:
2338:
2333:
2330:
2324:
2322:
2313:
2312:
2300:
2299:
2287:
2286:
2284:
2281:
2275:
2270:
2254:
2252:
2251:
2246:
2241:
2240:
2239:
2228:
2222:
2216:
2215:
2214:
2213:
2209:
2207:
2204:
2198:
2193:
2180:
2174:
2173:
2172:
2161:
2154:
2153:
2134:
2132:
2131:
2126:
2119:
2118:
2116:
2113:
2107:
2102:
2097:
2096:
2088:
2087:
2083:
2078:
2075:
2069:
2067:
2058:
2057:
2043:
2042:
2041:
2029:
2028:
2026:
2023:
2017:
2012:
1996:
1994:
1993:
1988:
1983:
1982:
1981:
1970:
1964:
1958:
1957:
1956:
1955:
1951:
1949:
1946:
1940:
1935:
1922:
1916:
1915:
1914:
1903:
1896:
1895:
1873:
1872:
1868:
1867:
1857:
1856:
1849:
1840:
1821:
1819:
1818:
1813:
1806:
1805:
1804:
1798:
1794:
1789:
1786:
1780:
1778:
1769:
1768:
1756:
1755:
1743:
1742:
1740:
1737:
1731:
1726:
1722:
1716:
1710:
1701:
1699:
1698:
1693:
1688:
1687:
1686:
1680:
1672:
1666:
1665:
1664:
1652:
1651:
1640:
1629:
1628:
1615:
1613:
1612:
1607:
1603:
1602:
1601:
1590:
1583:
1582:
1565:
1564:
1555:
1554:
1552:
1544:
1542:
1541:
1533:
1527:
1526:
1525:
1514:
1507:
1506:
1489:
1487:
1486:
1481:
1477:
1476:
1475:
1464:
1457:
1456:
1439:
1438:
1429:
1428:
1426:
1418:
1416:
1415:
1407:
1401:
1400:
1399:
1388:
1381:
1380:
1358:
1357:
1353:
1352:
1342:
1341:
1331:
1330:
1326:
1325:
1317:
1316:
1304:
1303:
1289:
1288:
1281:
1277:
1273:
1269:
1265:
1252:
1244:In this formula
1240:
1238:
1237:
1232:
1228:
1227:
1226:
1215:
1214:
1205:
1204:
1202:
1197:
1194:
1189:
1185:
1184:
1183:
1175:
1169:
1168:
1167:
1155:
1154:
1143:
1132:
1131:
1126:
1100:
1098:
1097:
1092:
1084:
1083:
1074:
1064:
1062:
1057:
1054:
1049:
1021:
1020:
974:
971:
959:
958:
942:Giacomo Gorzanis
938:Vincenzo Galilei
924:
922:
920:
919:
911:
909:
907:
906:
895:
894:
893:
846:
845:
811:
808:
802:
786:
778:
765:
748:integer notation
742:
740:
739:
734:
730:
729:
724:
721:
715:
713:
703:
692:
688:
684:
657:
653:
649:
646:where the ratio
642:
640:
639:
634:
630:
629:
627:
622:
619:
614:
604:
595:
593:
592:
587:
583:
576:
575:
564:
519:
512:
508:
505:
499:
468:
460:
449:
447:
446:
443:
440:
433:
431:
430:
427:
424:
417:
415:
414:
411:
408:
401:
399:
398:
395:
392:
385:
383:
382:
379:
376:
369:
368:
367:
360:
359:
358:
345:
310:string ensembles
303:
302:
279:
277:
276:
268:Arab tone system
264:
263:
262:
253:
252:
251:
234:
233:
217:
216:
215:
204:
203:
202:
182:
180:
179:
176:
173:
166:
165:
150:
149:
148:
139:
138:
137:
83:
82:
81:
79:
71:
70:
36:
21:
8877:
8876:
8872:
8871:
8870:
8868:
8867:
8866:
8847:
8846:
8845:
8840:
8837:(BohlenâPierce)
8805:833 cents scale
8793:
8716:
8706:
8673:
8626:
8524:
8445:Just intonation
8439:
8368:
8366:Musical tunings
8363:
8333:
8328:
8302:
8246:
8222:
8216:
8179:
8169:
8165:Double diatonic
8152:Just intonation
8146:
8068:
8034:
8024:
8018:
7962:
7948:
7894:Joel Mandelbaum
7824:JuliĂĄn Carrillo
7804:Richard Barrett
7792:
7787:
7757:
7752:
7736:
7660:
7654:
7615:
7609:
7600:Octatonic scale
7577:
7571:
7562:
7435:
7417:
7401:
7379:
7376:
7374:Further reading
7371:
7365:
7344:
7334:
7328:
7321:
7312:
7310:
7297:
7289:
7283:
7267:
7261:
7248:
7242:
7229:
7216:
7206:Ethnomusicology
7203:
7199:
7194:
7144:
7143:
7139:
7130:
7128:
7125:wendycarlos.com
7119:Carlos, Wendy.
7118:
7117:
7113:
7104:
7102:
7094:
7093:
7089:
7080:
7078:
7069:
7068:
7064:
7054:
7052:
7042:
7041:
7037:
7031:Sethares (2005)
7029:
7025:
7018:
7005:
7004:
7000:
6991:
6989:
6978:
6974:
6973:
6969:
6958:
6954:
6953:
6949:
6942:
6938:
6931:
6918:
6917:
6913:
6906:
6902:
6895:
6891:
6882:
6880:
6870:
6869:
6865:
6858:
6837:
6836:
6832:
6815:
6814:
6810:
6803:
6791:Lindley, Mark.
6790:
6789:
6785:
6775:
6773:
6772:on 17 July 2011
6758:
6757:
6753:
6743:
6741:
6740:on 25 July 2011
6732:
6731:
6727:
6720:Lute tabulation
6713:
6712:
6708:
6699:
6697:
6684:
6683:
6679:
6672:Girolamo Scotto
6658:
6657:
6653:
6643:
6639:
6638:
6634:
6619:
6618:
6614:
6606:
6602:
6593:
6591:
6578:
6577:
6573:
6563:
6559:LÇ lĂŹ rĂłng tĆng
6553:
6552:
6548:
6522:
6521:
6517:
6509:
6505:
6495:
6491:YuĂš lÇ quĂĄn shĆ«
6485:
6484:
6477:
6467:
6466:
6459:
6448:
6442:
6441:
6437:
6425:Ethnomusicology
6421:
6420:
6416:
6408:
6401:
6351:
6350:
6346:
6323:
6322:
6318:
6308:
6306:
6301:
6300:
6296:
6287:Sethares (2005)
6285:
6281:
6277:
6272:
6271:
6256:
6243:
6239:) â 6:5 and 5:3
6230:
6226:) â 5:4 and 8:5
6217:
6204:
6191:
6178:
6165:
6152:
6139:
6126:
6113:
6108:
6104:
6098:
6094:
6089:
6085:
6079:Sethares (2005)
6077:
6070:
6065:
6060:
6016:
6009:Just intonation
6004:
5991:just intonation
5980:
5978:
5970:
5968:
5958:
5955:
5952:
5951:
5949:
5944:
5942:
5918:
5903:
5902:
5878:
5876:
5856:
5855:
5843:
5841:
5828:
5826:
5814:
5812:
5803:
5802:
5799:
5786:
5783:
5780:
5779:
5777:
5776:
5770:
5755:
5754:
5748:
5747:
5736:
5735:
5722:
5719:
5716:
5715:
5713:
5708:
5707:
5698:
5697:
5694:
5679:
5676:
5673:
5672:
5670:
5669:
5659:
5644:
5643:
5637:
5636:
5625:
5624:
5611:
5608:
5605:
5604:
5602:
5597:
5596:
5587:
5586:
5583:
5576:meantone system
5567:
5564:
5561:
5560:
5558:
5557:
5551:
5536:
5535:
5531:
5527:
5523:
5519:
5511:
5510:
5507:
5498:
5496:
5495:(in cents) and
5491:
5489:
5480:
5479:
5476:
5466:
5465:
5461:
5457:
5451:
5450:
5446:
5442:
5433:
5432:
5429:
5422:
5421:
5418:
5406:
5404:
5398:
5396:
5390:
5389:
5387:
5383:
5379:
5373:
5372:
5370:
5366:
5362:
5355:
5354:
5351:
5344:
5340:
5336:
5329:
5328:
5326:
5321:
5316:
5307:
5306:
5302:
5298:
5287:
5274:
5270:
5266:
5259:
5255:
5254:, or into both
5251:
5247:
5241:
5239:
5232:
5228:
5218:
5216:
5211:
5210:
5200:
5198:
5188:
5186:
5179:
5158:
5157:
5151:
5150:
5145:
5144:
5138:
5137:
5132:
5131:
5125:
5124:
5117:
5116:
5109:
5108:
5101:
5100:
5085:
5084:
5077:
5076:
5069:
5068:
5059:
5058:
5047:
5046:
5039:
5038:
5031:
5030:
5023:
5022:
5015:
5014:
5007:
5006:
4991:
4990:
4983:
4982:
4980:
4954:
4953:
4938:
4935:
4930:
4929:
4927:
4922:
4921:
4917:
4913:
4905:
4897:
4884:
4883:
4876:
4875:
4873:
4853:
4848:
4836:
4833:
4830:
4829:
4827:
4820:
4817:
4814:
4813:
4811:
4806:
4805:
4798:
4795:
4792:
4791:
4789:
4782:
4779:
4776:
4775:
4773:
4768:
4767:
4763:
4754:
4753:
4748:
4747:
4740:
4739:
4736:
4734:
4724:
4721:
4718:
4717:
4715:
4708:
4705:
4702:
4701:
4699:
4694:
4693:
4686:
4683:
4680:
4679:
4677:
4670:
4667:
4664:
4663:
4661:
4656:
4655:
4650:
4649:
4644:
4643:
4639:
4626:
4625:
4612:
4611:
4599:
4598:
4592:
4591:
4588:
4586:
4578:
4577:
4570:
4569:
4566:
4564:
4557:
4556:
4551:
4550:
4536:
4533:
4530:
4529:
4527:
4522:
4521:
4515:
4514:
4511:
4509:
4499:
4496:
4493:
4492:
4490:
4485:
4484:
4478:
4477:
4474:
4472:
4463:
4460:
4457:
4456:
4454:
4449:
4448:
4442:
4441:
4438:
4436:
4435:is an integer,
4432:
4418:
4417:
4413:
4406:
4405:
4400:
4399:
4393:
4392:
4385:
4384:
4377:
4376:
4369:
4368:
4361:
4360:
4357:proper fraction
4349:
4345:
4329:
4318:
4312:
4309:
4298:
4286:
4275:
4253:
4251:
4250:
4249:
4239:
4236:
4233:
4232:
4230:
4229:
4228:
4226:
4213:
4211:
4210:
4209:
4199:
4196:
4193:
4192:
4190:
4189:
4188:
4186:
4173:
4171:
4170:
4169:
4159:
4156:
4153:
4152:
4150:
4149:
4148:
4146:
4110:
4108:
4100:
4098:
4097:
4096:
4084:
4082:
4081:
4080:
4061:
4046:
4037:
4005:JuliĂĄn Carrillo
3988:just intonation
3980:just intonation
3947:
3946:
3936:
3817:
3750:
3747:
3744:
3743:
3741:
3740:
3713:
3711:
3710:
3709:
3691:
3680:
3674:
3671:
3660:
3648:
3637:
3626:
3625:
3623:
3605:
3603:
3602:
3601:
3591:
3590:
3588:
3581:
3568:
3566:
3565:
3564:
3533:
3532:
3530:
3519:
3505:
3504:
3502:
3492:
3491:
3489:
3471:
3470:
3468:
3462:
3461:
3459:
3449:
3447:
3446:
3445:
3437:
3435:
3434:
3433:
3425:
3423:
3422:
3421:
3415:
3414:
3404:
3402:
3401:
3400:
3394:
3393:
3391:
3381:
3380:
3378:
3371:
3366:
3360:
3350:
3340:
3337:
3334:
3333:
3331:
3330:
3329:
3327:
3319:
3318:
3311:
3310:
3307:
3288:
3281:
3278:
3275:
3274:
3272:
3264:
3259:
3255:
3245:
3244:
3229:
3222:
3219:
3216:
3215:
3213:
3200:
3199:
3197:
3193:
3183:
3182:
3180:Major seventh (
3167:
3160:
3157:
3154:
3153:
3151:
3138:
3137:
3135:
3131:
3123:
3122:
3116:
3115:
3113:Minor seventh (
3100:
3093:
3090:
3087:
3086:
3084:
3071:
3070:
3068:
3064:
3054:
3053:
3038:
3031:
3028:
3025:
3024:
3022:
3009:
3008:
3006:
3002:
2994:
2993:
2987:
2986:
2971:
2964:
2961:
2958:
2957:
2955:
2942:
2941:
2939:
2935:
2925:
2924:
2922:Perfect fifth (
2909:
2902:
2899:
2896:
2895:
2893:
2880:
2879:
2877:
2873:
2865:
2864:
2858:
2857:
2842:
2835:
2832:
2829:
2828:
2826:
2813:
2812:
2810:
2806:
2796:
2795:
2780:
2773:
2770:
2767:
2766:
2764:
2751:
2750:
2748:
2744:
2734:
2733:
2718:
2711:
2708:
2705:
2704:
2702:
2689:
2688:
2686:
2682:
2674:
2673:
2667:
2666:
2651:
2644:
2641:
2638:
2637:
2635:
2622:
2621:
2619:
2615:
2605:
2604:
2589:
2582:
2579:
2576:
2575:
2573:
2560:
2559:
2557:
2553:
2545:
2544:
2538:
2537:
2522:
2515:
2512:
2509:
2508:
2506:
2495:
2491:
2481:
2480:
2472:
2467:
2466:
2450:
2449:
2442:
2441:
2424:just intonation
2418:
2417:
2414:
2379:
2358:
2323:
2317:
2304:
2274:
2257:
2256:
2197:
2188:
2183:
2145:
2140:
2139:
2106:
2068:
2062:
2049:
2016:
1999:
1998:
1939:
1930:
1925:
1887:
1882:
1881:
1876:
1870:
1869:
1863:
1862:
1860:
1852:
1851:
1847:
1842:
1838:
1833:
1779:
1773:
1760:
1730:
1704:
1703:
1675:
1656:
1643:
1635:
1634:
1626:
1625:
1622:
1558:
1498:
1493:
1492:
1432:
1372:
1367:
1366:
1361:
1355:
1354:
1348:
1347:
1345:
1337:
1336:
1334:
1328:
1327:
1321:
1320:
1312:
1311:
1307:
1299:
1298:
1292:
1284:
1283:
1279:
1275:
1271:
1267:
1263:
1258:
1250:
1245:
1208:
1190:
1159:
1146:
1138:
1137:
1129:
1128:
1124:
1119:
1116:
1110:
1068:
1050:
1043:
1042:
1032:frequency ratio
1018:
1017:
1008:
972:
956:
955:
934:
917:
915:
914:
913:
904:
902:
901:
900:
891:
890:
888:
874:
835:
827:
821:
812:
806:
803:
796:
787:
776:
763:
714:
698:
697:
690:
686:
682:
679:ethnomusicology
655:
651:
647:
615:
599:
598:
567:
559:
558:
520:
509:
503:
500:
485:
469:
458:
453:
452:
451:
444:
441:
438:
437:
435:
428:
425:
422:
421:
419:
412:
409:
406:
405:
403:
396:
393:
390:
389:
387:
380:
377:
374:
373:
371:
365:
364:
362:
356:
355:
353:
351:
346:
318:just intonation
300:
299:
274:
273:
271:
260:
259:
257:
249:
248:
246:
229:
228:
213:
212:
210:
200:
199:
197:
177:
174:
171:
170:
168:
163:
161:
146:
145:
143:
135:
134:
132:
126:(also known as
120:classical music
77:
75:
74:
73:
66:
65:
58:
57:
56:
42:
37:
28:
23:
22:
15:
12:
11:
5:
8875:
8873:
8865:
8864:
8859:
8849:
8848:
8842:
8841:
8839:
8838:
8832:
8827:
8822:
8817:
8812:
8807:
8801:
8799:
8795:
8794:
8792:
8791:
8786:
8781:
8771:
8766:
8761:
8760:
8759:
8754:
8749:
8744:
8736:
8731:
8726:
8720:
8718:
8712:
8711:
8708:
8707:
8681:
8679:
8675:
8674:
8672:
8671:
8666:
8661:
8656:
8651:
8636:
8634:
8628:
8627:
8625:
8624:
8619:
8614:
8609:
8604:
8599:
8594:
8589:
8579:
8574:
8569:
8564:
8559:
8554:
8549:
8543:
8541:
8532:
8526:
8525:
8523:
8522:
8517:
8512:
8507:
8502:
8497:
8492:
8487:
8486:
8485:
8480:
8470:
8465:
8460:
8458:Harmonic scale
8455:
8449:
8447:
8441:
8440:
8438:
8437:
8432:
8427:
8422:
8417:
8412:
8407:
8405:Interval ratio
8402:
8397:
8392:
8387:
8382:
8376:
8374:
8370:
8369:
8364:
8362:
8361:
8354:
8347:
8339:
8330:
8329:
8327:
8326:
8321:
8316:
8310:
8308:
8304:
8303:
8301:
8300:
8293:
8286:
8279:
8272:
8267:
8262:
8254:
8252:
8248:
8247:
8245:
8244:
8239:
8232:
8226:
8224:
8218:
8217:
8215:
8214:
8209:
8207:Xenharmonicity
8204:
8199:
8194:
8189:
8183:
8181:
8175:
8174:
8171:
8170:
8168:
8167:
8162:
8156:
8154:
8148:
8147:
8145:
8144:
8139:
8134:
8129:
8124:
8119:
8114:
8109:
8104:
8099:
8094:
8089:
8084:
8078:
8076:
8070:
8069:
8067:
8066:
8060:
8055:
8050:
8045:
8039:
8037:
8026:
8020:
8019:
8017:
8016:
8011:
8006:
8001:
7996:
7991:
7989:Adriaan Fokker
7986:
7981:
7976:
7970:
7968:
7964:
7963:
7951:
7949:
7947:
7946:
7944:La Monte Young
7941:
7936:
7931:
7926:
7921:
7916:
7911:
7909:John Schneider
7906:
7901:
7896:
7891:
7886:
7881:
7876:
7871:
7866:
7861:
7856:
7854:BjĂžrn Fongaard
7851:
7846:
7841:
7836:
7834:Mildred Couper
7831:
7826:
7821:
7816:
7811:
7806:
7800:
7798:
7794:
7793:
7788:
7786:
7785:
7778:
7771:
7763:
7754:
7753:
7751:
7750:
7744:
7742:
7738:
7737:
7735:
7734:
7729:
7724:
7719:
7714:
7713:
7712:
7710:Distance model
7702:
7697:
7690:
7685:
7680:
7675:
7670:
7664:
7662:
7656:
7655:
7653:
7652:
7650:Spectral music
7647:
7642:
7637:
7636:
7635:
7630:
7619:
7617:
7611:
7610:
7608:
7607:
7602:
7597:
7592:
7587:
7581:
7579:
7573:
7572:
7563:
7561:
7560:
7553:
7546:
7538:
7532:
7531:
7503:
7498:
7491:
7486:
7476:
7469:
7460:
7455:
7450:
7445:
7434:
7433:External links
7431:
7430:
7429:
7415:(archive.org).
7399:
7375:
7372:
7370:
7369:
7363:
7342:
7319:
7287:
7281:
7265:
7259:
7246:
7240:
7227:
7214:
7200:
7198:
7195:
7193:
7192:
7147:Sethares, W.A.
7137:
7127:. Serendip LLC
7111:
7087:
7062:
7035:
7023:
7016:
7010:. p. 55.
6998:
6967:
6964:on 2007-10-27.
6947:
6936:
6929:
6911:
6900:
6889:
6863:
6856:
6830:
6808:
6801:
6783:
6751:
6725:
6706:
6677:
6651:
6640:Hanson, Lau.
6632:
6612:
6600:
6571:
6546:
6515:
6511:Kuttner (1975)
6503:
6475:
6457:
6435:
6414:
6410:Kuttner (1975)
6399:
6344:
6316:
6294:
6278:
6276:
6273:
6270:
6269:
6267:
6266:
6253:
6240:
6227:
6214:
6201:
6188:
6175:
6162:
6149:
6136:
6123:
6102:
6092:
6083:
6067:
6066:
6064:
6061:
6059:
6058:
6056:Musical tuning
6053:
6048:
6043:
6038:
6033:
6028:
6023:
6018:
6011:
6005:
6003:
6000:
5999:
5998:
5987:syntonic comma
5809:
5796:
5795:
5744:the result is
5704:
5691:
5690:
5633:the result is
5593:
5580:
5579:
5506:the result is
5486:
5473:
5472:
5439:
5414:For instance:
5386:tend to zero;
5286:
5283:
5225:circular shift
5182:, expands the
5173:diatonic comma
5003:perfect fifths
4890:circular shift
4861:Figure 1: The
4852:
4849:
4847:
4844:
4331:
4330:
4289:
4287:
4280:
4274:
4271:
4261:
4260:
4220:
4180:
4060:
4057:
4035:
4009:
4008:
3997:
3993:
3992:
3973:
3969:
3968:
3961:
3957:
3956:
3934:
3927:perfect fifths
3916:
3912:
3911:
3907:
3903:
3902:
3895:
3891:
3890:
3883:
3879:
3878:
3871:Adriaan Fokker
3860:
3856:
3855:
3844:
3840:
3839:
3836:syntonic comma
3832:septimal comma
3828:
3824:
3823:
3815:
3812:
3808:
3807:
3793:
3789:
3788:
3781:
3777:
3776:
3769:
3765:
3764:
3761:perfect fourth
3733:
3693:
3692:
3651:
3649:
3642:
3636:
3633:
3613:
3612:
3580:
3577:
3518:
3515:
3514:
3513:
3499:
3370:
3367:
3359:
3356:
3306:
3303:
3302:
3301:
3298:
3297:
3294:
3291:
3270:
3267:
3262:
3253:
3239:
3238:
3235:
3232:
3211:
3208:
3205:
3191:
3177:
3176:
3173:
3170:
3149:
3146:
3143:
3129:
3110:
3109:
3106:
3103:
3082:
3079:
3076:
3062:
3048:
3047:
3044:
3041:
3020:
3017:
3014:
3000:
2981:
2980:
2977:
2974:
2953:
2950:
2947:
2933:
2919:
2918:
2915:
2912:
2891:
2888:
2885:
2871:
2852:
2851:
2848:
2845:
2824:
2821:
2818:
2804:
2790:
2789:
2786:
2783:
2762:
2759:
2756:
2742:
2728:
2727:
2724:
2721:
2700:
2697:
2694:
2680:
2661:
2660:
2657:
2654:
2633:
2630:
2627:
2613:
2602:Major second (
2599:
2598:
2595:
2592:
2571:
2568:
2565:
2551:
2535:Minor second (
2532:
2531:
2528:
2525:
2504:
2501:
2498:
2489:
2475:
2474:
2463:
2460:
2457:
2454:
2446:
2438:
2437:Interval Name
2413:
2410:
2409:
2408:
2397:
2385:
2378:
2373:
2364:
2357:
2352:
2347:
2341:
2336:
2329:
2320:
2316:
2311:
2307:
2303:
2298:
2292:
2280:
2273:
2268:
2265:
2238:
2235:
2227:
2221:
2212:
2203:
2196:
2191:
2186:
2179:
2171:
2168:
2160:
2157:
2152:
2148:
2136:
2135:
2124:
2112:
2105:
2100:
2095:
2086:
2081:
2074:
2065:
2061:
2056:
2052:
2048:
2040:
2034:
2022:
2015:
2010:
2007:
1980:
1977:
1969:
1963:
1954:
1945:
1938:
1933:
1928:
1921:
1913:
1910:
1902:
1899:
1894:
1890:
1874:
1858:
1845:
1836:
1823:
1822:
1811:
1803:
1797:
1792:
1785:
1776:
1772:
1767:
1763:
1759:
1754:
1748:
1736:
1729:
1721:
1715:
1685:
1678:
1671:
1663:
1659:
1655:
1650:
1646:
1621:
1618:
1617:
1616:
1600:
1597:
1589:
1586:
1581:
1578:
1575:
1572:
1569:
1563:
1551:
1547:
1540:
1532:
1524:
1521:
1513:
1510:
1505:
1501:
1490:
1474:
1471:
1463:
1460:
1455:
1452:
1449:
1446:
1443:
1437:
1425:
1421:
1414:
1406:
1398:
1395:
1387:
1384:
1379:
1375:
1359:
1343:
1332:
1305:
1290:
1261:
1248:
1242:
1241:
1225:
1222:
1219:
1213:
1201:
1193:
1182:
1174:
1166:
1162:
1158:
1153:
1149:
1122:
1109:
1106:
1102:
1101:
1090:
1087:
1080:
1077:
1071:
1067:
1061:
1053:
1022:on a monochord
1007:
1004:
933:
930:
873:
870:
865:
864:
847:) in 1584 and
834:
831:
823:Main article:
820:
817:
814:
813:
790:
788:
781:
775:
772:
744:
743:
727:
720:
711:
708:
658:equal parts. (
644:
643:
626:
618:
612:
609:
596:
582:
579:
574:
570:
522:
521:
472:
470:
463:
457:
454:
348:
347:
340:
339:
338:
220:standard pitch
101:just intervals
39:
38:
31:
30:
29:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8874:
8863:
8860:
8858:
8855:
8854:
8852:
8836:
8833:
8831:
8828:
8826:
8823:
8821:
8818:
8816:
8813:
8811:
8808:
8806:
8803:
8802:
8800:
8796:
8790:
8787:
8785:
8782:
8779:
8778:Carnatic raga
8775:
8772:
8770:
8767:
8765:
8762:
8758:
8755:
8753:
8750:
8748:
8747:Turkish makam
8745:
8743:
8740:
8739:
8737:
8735:
8732:
8730:
8727:
8725:
8722:
8721:
8719:
8713:
8704:
8700:
8696:
8692:
8688:
8684:
8680:
8676:
8670:
8667:
8665:
8662:
8660:
8657:
8655:
8652:
8649:
8645:
8644:quarter-comma
8641:
8638:
8637:
8635:
8633:
8629:
8623:
8620:
8618:
8615:
8613:
8610:
8608:
8605:
8603:
8600:
8598:
8595:
8593:
8590:
8587:
8583:
8580:
8578:
8575:
8573:
8570:
8568:
8565:
8563:
8560:
8558:
8555:
8553:
8550:
8548:
8545:
8544:
8542:
8540:
8536:
8533:
8531:
8527:
8521:
8520:Tonality flux
8518:
8516:
8513:
8511:
8508:
8506:
8503:
8501:
8498:
8496:
8493:
8491:
8488:
8484:
8481:
8479:
8476:
8475:
8474:
8471:
8469:
8466:
8464:
8461:
8459:
8456:
8454:
8451:
8450:
8448:
8446:
8442:
8436:
8433:
8431:
8428:
8426:
8423:
8421:
8418:
8416:
8413:
8411:
8408:
8406:
8403:
8401:
8398:
8396:
8393:
8391:
8388:
8386:
8383:
8381:
8378:
8377:
8375:
8371:
8367:
8360:
8355:
8353:
8348:
8346:
8341:
8340:
8337:
8325:
8322:
8320:
8317:
8315:
8312:
8311:
8309:
8305:
8299:
8298:
8294:
8292:
8291:
8287:
8285:
8284:
8280:
8278:
8277:
8273:
8271:
8268:
8266:
8263:
8261:
8260:
8256:
8255:
8253:
8249:
8243:
8240:
8238:
8237:
8233:
8231:
8228:
8227:
8225:
8219:
8213:
8210:
8208:
8205:
8203:
8200:
8198:
8195:
8193:
8190:
8188:
8185:
8184:
8182:
8176:
8166:
8163:
8161:
8158:
8157:
8155:
8153:
8149:
8143:
8140:
8138:
8135:
8133:
8130:
8128:
8125:
8123:
8120:
8118:
8115:
8113:
8110:
8108:
8105:
8103:
8100:
8098:
8095:
8093:
8090:
8088:
8085:
8083:
8080:
8079:
8077:
8075:
8071:
8064:
8061:
8059:
8056:
8054:
8051:
8049:
8046:
8044:
8041:
8040:
8038:
8036:
8030:
8027:
8021:
8015:
8012:
8010:
8007:
8005:
8002:
8000:
7997:
7995:
7992:
7990:
7987:
7985:
7982:
7980:
7977:
7975:
7972:
7971:
7969:
7965:
7960:
7955:
7945:
7942:
7940:
7937:
7935:
7934:Elaine Walker
7932:
7930:
7929:Claude Vivier
7927:
7925:
7922:
7920:
7917:
7915:
7912:
7910:
7907:
7905:
7904:Roger Redgate
7902:
7900:
7897:
7895:
7892:
7890:
7887:
7885:
7884:Stu Mackenzie
7882:
7880:
7879:György Ligeti
7877:
7875:
7872:
7870:
7867:
7865:
7862:
7860:
7857:
7855:
7852:
7850:
7847:
7845:
7842:
7840:
7837:
7835:
7832:
7830:
7827:
7825:
7822:
7820:
7817:
7815:
7812:
7810:
7807:
7805:
7802:
7801:
7799:
7795:
7791:
7784:
7779:
7777:
7772:
7770:
7765:
7764:
7761:
7749:
7746:
7745:
7743:
7739:
7733:
7732:Unified field
7730:
7728:
7725:
7723:
7720:
7718:
7715:
7711:
7708:
7707:
7706:
7703:
7701:
7698:
7696:
7695:
7691:
7689:
7686:
7684:
7681:
7679:
7676:
7674:
7671:
7669:
7666:
7665:
7663:
7657:
7651:
7648:
7646:
7643:
7641:
7638:
7634:
7631:
7629:
7626:
7625:
7624:
7621:
7620:
7618:
7612:
7606:
7603:
7601:
7598:
7596:
7593:
7591:
7588:
7586:
7583:
7582:
7580:
7574:
7570:
7569:post-tonality
7566:
7559:
7554:
7552:
7547:
7545:
7540:
7539:
7536:
7522:
7504:
7502:
7499:
7496:
7492:
7490:
7487:
7484:
7480:
7477:
7474:
7470:
7468:
7466:
7461:
7459:
7456:
7454:
7451:
7449:
7446:
7444:
7440:
7437:
7436:
7432:
7427:
7426:
7421:
7414:
7410:
7406:
7402:
7396:
7392:
7388:
7387:
7382:
7381:Helmholtz, H.
7378:
7377:
7373:
7366:
7360:
7356:
7352:
7348:
7343:
7338:
7337:"Alt. link 2"
7331:
7330:"Alt. link 1"
7325:
7320:
7308:
7304:
7300:
7293:
7288:
7284:
7282:1-85233-797-4
7278:
7274:
7270:
7266:
7262:
7260:0-87013-290-3
7256:
7252:
7247:
7243:
7237:
7233:
7228:
7224:
7220:
7215:
7211:
7207:
7202:
7201:
7196:
7190:
7186:
7180:
7176:
7171:
7166:
7162:
7158:
7157:
7152:
7148:
7141:
7138:
7126:
7122:
7115:
7112:
7101:
7097:
7091:
7088:
7077:on 2015-11-18
7076:
7072:
7066:
7063:
7050:
7046:
7039:
7036:
7032:
7027:
7024:
7019:
7017:9780542998478
7013:
7009:
7002:
6999:
6988:on 2007-09-30
6987:
6983:
6979:
6971:
6968:
6963:
6959:
6951:
6948:
6945:
6944:Boiles (1969)
6940:
6937:
6932:
6930:0-520-04778-8
6926:
6922:
6915:
6912:
6909:
6908:Morton (1980)
6904:
6901:
6898:
6893:
6890:
6878:
6874:
6867:
6864:
6859:
6857:0-306-80106-X
6853:
6849:
6844:
6843:
6834:
6831:
6826:
6822:
6818:
6812:
6809:
6804:
6798:
6794:
6787:
6784:
6771:
6767:
6766:
6761:
6760:Stevin, Simon
6755:
6752:
6739:
6735:
6729:
6726:
6721:
6717:
6710:
6707:
6696:on 2012-03-24
6695:
6691:
6687:
6681:
6678:
6673:
6669:
6665:
6661:
6655:
6652:
6647:
6636:
6633:
6629:
6623:
6616:
6613:
6609:
6604:
6601:
6590:on 2012-03-05
6589:
6585:
6581:
6575:
6572:
6567:
6560:
6556:
6550:
6547:
6542:
6538:
6534:
6530:
6526:
6519:
6516:
6513:, p. 200
6512:
6507:
6504:
6499:
6492:
6488:
6482:
6480:
6476:
6471:
6464:
6462:
6458:
6454:
6446:
6439:
6436:
6432:(2): 163â206.
6431:
6427:
6426:
6418:
6415:
6412:, p. 163
6411:
6406:
6404:
6400:
6395:
6391:
6387:
6383:
6379:
6375:
6370:
6365:
6361:
6357:
6356:
6348:
6345:
6340:
6336:
6335:
6330:
6326:
6325:Helmholtz, H.
6320:
6317:
6305:
6298:
6295:
6292:
6288:
6283:
6280:
6274:
6264:
6259:
6254:
6251:
6246:
6241:
6238:
6233:
6228:
6225:
6220:
6215:
6212:
6207:
6202:
6199:
6194:
6189:
6186:
6181:
6176:
6173:
6168:
6163:
6160:
6155:
6150:
6147:
6142:
6137:
6134:
6129:
6124:
6121:
6116:
6111:
6110:
6106:
6103:
6096:
6093:
6087:
6084:
6080:
6075:
6073:
6069:
6062:
6057:
6054:
6052:
6049:
6047:
6044:
6042:
6039:
6037:
6034:
6032:
6029:
6027:
6024:
6022:
6019:
6015:
6012:
6010:
6007:
6006:
6001:
5996:
5992:
5988:
5973:
5947:
5940:
5939:53 steps
5935:
5930:
5926:
5922:
5915:
5911:
5907:
5897:
5893:
5889:
5885:
5881:
5871:
5867:
5863:
5859:
5850:
5846:
5835:
5831:
5821:
5817:
5810:
5807:
5798:
5797:
5767:
5763:
5759:
5752:
5739:
5730:
5711:
5705:
5702:
5693:
5692:
5688:
5666:
5665:31 steps
5656:
5652:
5648:
5641:
5628:
5619:
5600:
5594:
5591:
5582:
5581:
5577:
5548:
5544:
5540:
5515:
5501:
5487:
5484:
5475:
5474:
5467:t t t t t t t
5440:
5437:
5426:
5417:
5416:
5415:
5409:
5359:
5348:
5334:
5319:
5310:
5296:
5292:
5284:
5282:
5280:
5263:
5238:
5226:
5212:T t s T t T s
5208:
5196:
5185:
5176:
5174:
5170:
5165:
5154:
5141:
5128:
5120:
5112:
5104:
5098:
5094:
5093:
5080:
5072:
5067:
5056:
5050:
5042:
5034:
5026:
5018:
5010:
5004:
5000:
4994:
4978:
4974:
4969:
4965:
4961:
4957:
4952:
4941:
4933:
4925:
4916:). The comma
4911:
4903:
4895:
4891:
4885:T t s T t T s
4872:
4864:
4859:
4855:
4850:
4845:
4843:
4760:
4731:
4634:
4630:
4620:
4616:
4607:
4602:
4581:
4560:
4548:
4525:
4488:
4452:
4429:
4425:
4421:
4409:
4396:
4388:
4380:
4372:
4364:
4358:
4353:
4342:
4338:
4327:
4324:
4316:
4306:
4302:
4296:
4295:
4290:This section
4288:
4284:
4279:
4278:
4272:
4270:
4268:
4267:
4256:
4224:
4221:
4216:
4184:
4181:
4176:
4144:
4141:
4140:
4139:
4137:
4136:
4131:
4130:
4125:
4124:
4119:
4115:
4103:
4094:
4087:
4078:
4074:
4070:
4069:perfect fifth
4066:
4058:
4056:
4054:
4049:
4043:
4041:
4033:
4029:
4024:
4022:
4018:
4014:
4006:
4001:
3998:
3995:
3994:
3989:
3985:
3981:
3977:
3974:
3971:
3970:
3965:
3962:
3959:
3958:
3954:
3944:
3940:
3939:Turkish music
3932:
3928:
3924:
3920:
3917:
3914:
3913:
3908:
3905:
3904:
3899:
3896:
3893:
3892:
3887:
3884:
3881:
3880:
3876:
3872:
3868:
3864:
3861:
3858:
3857:
3853:
3848:
3845:
3842:
3841:
3837:
3833:
3829:
3826:
3825:
3821:
3813:
3810:
3809:
3805:
3801:
3797:
3794:
3791:
3790:
3785:
3782:
3779:
3778:
3773:
3770:
3767:
3766:
3762:
3738:
3734:
3731:
3730:
3725:
3716:
3708:equivalents.
3707:
3703:
3699:
3689:
3686:
3678:
3668:
3664:
3658:
3657:
3652:This section
3650:
3646:
3641:
3640:
3634:
3632:
3621:
3620:Chinese music
3617:
3608:
3599:
3598:
3597:
3586:
3578:
3576:
3571:
3562:
3558:
3554:
3550:
3546:
3542:
3529:are tuned to
3528:
3524:
3521:According to
3516:
3500:
3487:
3486:
3485:
3483:
3482:Figure 1
3479:
3457:
3452:
3440:
3428:
3420:
3407:
3399:
3375:
3368:
3365:
3357:
3355:
3322:
3314:
3304:
3295:
3292:
3271:
3268:
3263:
3254:
3251:
3248:
3241:
3240:
3236:
3233:
3212:
3209:
3206:
3192:
3189:
3186:
3179:
3178:
3174:
3171:
3150:
3147:
3144:
3130:
3127:
3119:
3112:
3111:
3107:
3104:
3083:
3080:
3077:
3063:
3060:
3057:
3051:Major sixth (
3050:
3049:
3045:
3042:
3021:
3018:
3015:
3001:
2998:
2990:
2984:Minor sixth (
2983:
2982:
2978:
2975:
2954:
2951:
2948:
2934:
2931:
2928:
2921:
2920:
2916:
2913:
2892:
2889:
2886:
2872:
2869:
2861:
2854:
2853:
2849:
2846:
2825:
2822:
2819:
2805:
2802:
2799:
2792:
2791:
2787:
2784:
2763:
2760:
2757:
2743:
2740:
2737:
2731:Major third (
2730:
2729:
2725:
2722:
2701:
2698:
2695:
2681:
2678:
2670:
2664:Minor third (
2663:
2662:
2658:
2655:
2634:
2631:
2628:
2614:
2611:
2608:
2601:
2600:
2596:
2593:
2572:
2569:
2566:
2552:
2549:
2541:
2534:
2533:
2529:
2526:
2505:
2502:
2499:
2490:
2487:
2484:
2477:
2476:
2473:tuning error
2464:
2461:
2458:
2455:
2447:
2439:
2436:
2435:
2432:
2431:
2430:
2427:
2425:
2411:
2395:
2383:
2376:
2371:
2362:
2355:
2350:
2339:
2334:
2327:
2318:
2314:
2309:
2305:
2301:
2290:
2278:
2271:
2266:
2263:
2225:
2219:
2210:
2201:
2194:
2189:
2184:
2177:
2158:
2155:
2150:
2146:
2138:
2137:
2122:
2110:
2103:
2098:
2084:
2079:
2072:
2063:
2059:
2054:
2050:
2046:
2032:
2020:
2013:
2008:
2005:
1967:
1961:
1952:
1943:
1936:
1931:
1926:
1919:
1900:
1897:
1892:
1888:
1880:
1879:
1878:
1866:
1855:
1848:
1839:
1827:
1809:
1795:
1790:
1783:
1774:
1770:
1765:
1761:
1757:
1746:
1734:
1727:
1719:
1713:
1683:
1676:
1669:
1661:
1657:
1653:
1648:
1644:
1633:
1632:
1631:
1619:
1587:
1584:
1576:
1573:
1570:
1549:
1545:
1530:
1511:
1508:
1503:
1499:
1491:
1461:
1458:
1450:
1447:
1444:
1423:
1419:
1404:
1385:
1382:
1377:
1373:
1365:
1364:
1363:
1351:
1340:
1324:
1318:
1315:
1302:
1296:
1287:
1264:
1256:
1251:
1223:
1220:
1217:
1199:
1191:
1172:
1164:
1160:
1156:
1151:
1147:
1136:
1135:
1134:
1125:
1115:
1107:
1105:
1088:
1085:
1078:
1075:
1069:
1065:
1059:
1051:
1041:
1040:
1039:
1037:
1033:
1029:
1014:
1010:
1005:
1003:
1001:
997:
993:
989:
985:
981:
976:
967:
963:
960:based on the
953:
949:
947:
943:
939:
931:
929:
926:
897:
882:
878:
871:
869:
861:
860:
859:
856:
853:
850:
840:
832:
830:
826:
818:
810:
807:February 2019
800:
794:
791:This section
789:
785:
780:
779:
773:
771:
769:
761:
760:pitch classes
757:
753:
749:
725:
718:
709:
706:
696:
695:
694:
680:
676:
672:
667:
665:
663:
624:
616:
610:
607:
597:
580:
577:
572:
568:
557:
556:
555:
553:
549:
546:to different
545:
544:transposition
541:
537:
533:
529:
518:
515:
507:
497:
493:
489:
483:
482:
478:
473:This section
471:
467:
462:
461:
455:
350:
344:
337:
335:
331:
327:
323:
319:
315:
311:
306:
305:can be used.
304:
296:
291:
289:
288:pseudo-octave
285:
280:
269:
265:
254:
242:
239:
235:
232:
225:
221:
207:
205:
194:
190:
186:
159:
155:
154:12 equal
151:
140:
129:
125:
121:
116:
114:
110:
106:
102:
98:
97:tuning system
94:
90:
80:
69:
62:
54:
50:
49:just interval
46:
41:
35:
19:
8835:Lambda scale
8742:Arabic maqam
8699:Werckmeister
8538:
8530:Temperaments
8307:Other topics
8295:
8288:
8281:
8274:
8257:
8251:Compositions
8234:
8223:publications
8178:Concepts and
8073:
8063:Lambda scale
8004:Harry Partch
7999:Yuri Landman
7994:Lou Harrison
7979:Wendy Carlos
7974:Glenn Branca
7874:Ben Johnston
7869:Charles Ives
7829:Franklin Cox
7819:Heinz Bohlen
7741:Compositions
7727:Tone cluster
7705:Polytonality
7692:
7668:Chromaticism
7661:and concepts
7595:Mystic chord
7584:
7512:ARDINALITIES
7482:
7464:
7423:
7419:
7411:– via
7385:
7346:
7311:. Retrieved
7307:the original
7302:
7296:As cited by
7291:
7272:
7250:
7231:
7218:
7209:
7205:
7163:(4): 15â32.
7160:
7154:
7140:
7129:. Retrieved
7124:
7114:
7103:. Retrieved
7100:WolframAlpha
7090:
7079:. Retrieved
7075:the original
7065:
7053:. Retrieved
7048:
7038:
7033:, p. 58
7026:
7007:
7001:
6990:. Retrieved
6986:the original
6981:
6970:
6962:the original
6950:
6939:
6920:
6914:
6903:
6892:
6881:. Retrieved
6876:
6866:
6841:
6833:
6824:
6820:
6811:
6792:
6786:
6774:. Retrieved
6770:the original
6764:
6754:
6742:. Retrieved
6738:the original
6728:
6719:
6715:
6709:
6698:. Retrieved
6694:the original
6689:
6680:
6667:
6663:
6654:
6645:
6635:
6621:
6615:
6603:
6592:. Retrieved
6588:the original
6583:
6574:
6565:
6558:
6549:
6541:the original
6528:
6518:
6506:
6497:
6490:
6469:
6452:
6444:
6438:
6429:
6423:
6417:
6362:(1): 47â55.
6359:
6353:
6347:
6333:
6319:
6307:. Retrieved
6297:
6282:
6105:
6095:
6086:
6036:Piano tuning
5971:
5967:exactly, or
5945:
5933:
5928:
5924:
5920:
5913:
5909:
5905:
5895:
5891:
5887:
5883:
5879:
5869:
5865:
5861:
5857:
5848:
5844:
5833:
5829:
5819:
5815:
5765:
5761:
5757:
5737:
5728:
5709:
5654:
5650:
5646:
5626:
5617:
5598:
5546:
5542:
5538:
5499:
5407:
5349:
5308:
5288:
5264:
5195:greater tone
5177:
5152:
5139:
5126:
5118:
5110:
5102:
5091:
5078:
5070:
5048:
5040:
5032:
5024:
5016:
5008:
4992:
4972:
4970:
4963:
4959:
4955:
4939:
4931:
4923:
4902:greater tone
4893:
4870:
4868:
4854:
4761:
4732:
4632:
4628:
4618:
4614:
4608:
4600:
4579:
4558:
4523:
4486:
4450:
4430:
4423:
4419:
4407:
4394:
4386:
4378:
4370:
4362:
4354:
4340:
4336:
4334:
4319:
4310:
4299:Please help
4294:verification
4291:
4264:
4262:
4222:
4182:
4142:
4133:
4127:
4121:
4118:Wendy Carlos
4116:
4093:justly tuned
4062:
4044:
4028:denominators
4025:
4010:
3991:36 EDO.
3834:but not the
3804:Charles Ives
3681:
3672:
3661:Please help
3656:verification
3653:
3618:
3614:
3582:
3520:
3458:
3412:
3390:
3388:
3320:
3312:
3308:
3246:
3184:
3117:
3055:
2988:
2926:
2859:
2797:
2735:
2668:
2606:
2539:
2482:
2428:
2415:
1864:
1853:
1843:
1834:
1832:
1623:
1349:
1338:
1322:
1313:
1300:
1285:
1259:
1246:
1243:
1120:
1117:
1103:
1025:
1009:
988:atonal music
984:polytonality
977:
965:
952:Simon Stevin
950:
935:
927:
898:
886:
866:
857:
854:
849:Simon Stevin
836:
828:
804:
792:
745:
668:
659:
645:
525:
510:
501:
486:Please help
474:
314:open strings
307:
298:
294:
292:
281:
266:, while the
243:
230:
208:
196:
192:
153:
142:
131:
127:
117:
88:
86:
67:
53:prime limits
8830:Delta scale
8825:Gamma scale
8815:Alpha scale
8717:non-Western
8715:Traditional
8410:Pitch class
8390:Millioctave
8373:Measurement
8270:just pieces
8058:Delta scale
8053:Gamma scale
8043:Alpha scale
8033:Non-octave-
8023:Tunings and
7984:Ivor Darreg
7809:BĂ©la BartĂłk
7391:Ellis, A.J.
7145:Milne, A.;
7055:26 February
7051:. Joe Monzo
6660:Galilei, V.
6339:Ellis, A.J.
6329:Ellis, A.J.
5771:21 + 14 + 8
5660:15 + 10 + 6
5295:minor tones
5184:lesser tone
5066:grave fifth
5055:permutation
4910:lesser tone
4313:August 2017
4032:convergents
4021:17 EDO
4017:15 EDO
4013:13 EDO
4000:96 EDO
3996:96 EDO
3976:72 EDO
3972:72 EDO
3960:58 EDO
3919:53 EDO
3915:53 EDO
3906:46 EDO
3894:41 EDO
3886:34 EDO
3882:34 EDO
3863:31 EDO
3859:31 EDO
3843:29 EDO
3827:27 EDO
3811:26 EDO
3796:24 EDO
3792:24 EDO
3784:23 EDO
3780:23 EDO
3772:22 EDO
3768:22 EDO
3737:19 EDO
3732:19 EDO
3543:(1966) and
1295:440 Hz
1030:, i.e. the
1006:Mathematics
973: 1605
863:operations.
675:logarithmic
113:logarithmic
105:frequencies
8851:Categories
8820:Beta scale
8798:Non-octave
8789:Tetrachord
8691:Kirnberger
8654:Schismatic
8221:Groups and
8180:techniques
8048:Beta scale
7899:Joe Maneri
7859:Alois HĂĄba
7839:John Eaton
7722:Tone Clock
7673:Cyclic set
7659:Techniques
7614:Genres and
7576:Scales and
7483:Jim Kukula
7131:2016-09-01
7105:2014-06-18
7081:2014-06-18
6992:2007-06-25
6883:2010-06-02
6700:2012-03-20
6594:2012-03-20
6555:Zhu, Zaiyu
6487:Zhu, Zaiyu
6275:References
6255:(sequence
6242:(sequence
6229:(sequence
6216:(sequence
6203:(sequence
6190:(sequence
6177:(sequence
6164:(sequence
6151:(sequence
6138:(sequence
6125:(sequence
6112:(sequence
6026:Microtuner
4341:whole tone
3984:Joe Maneri
3787:territory.
3706:enharmonic
3675:March 2020
3362:See also:
3168:1.77777...
3101:1.66666...
2910:1.42222...
2843:1.33333...
2590:1.06666...
1112:See also:
921:â 1.029302
908:â 1.059463
418:, yellow:
402:, indigo:
308:Unfretted
18:Fifth tone
8810:A12 scale
8764:Octoechos
8729:ShĂ-Ăšr-lÇ
8678:Irregular
8495:Otonality
8435:Microtone
8202:Sonido 13
7967:Inventors
7919:Ezra Sims
7797:Composers
7645:Serialism
7565:Atonality
7443:Kyle Gann
7383:(2005) .
7303:telia.com
7189:1531-5169
7179:0148-9267
6537:1000-4270
6369:0906.0127
6063:Footnotes
5904: 3
5794:meantone.
5756: 3
5645: 3
5552:9 + 6 + 4
5537: 3
5452:t t t t t
5201:T = s c Îș
5193:into the
5005:in a rowâ
4888:(or some
4030:of first
3364:Sonido 13
3234:1088.270
3207:1.887749
3145:1.781797
3078:1.681793
3016:1.587401
2949:1.498307
2887:1.414214
2855:Tritone (
2758:1.259921
2696:1.189207
2629:1.122462
2567:1.059463
2456:Pitch in
2315:
2220:≈
2178:⋅
2060:
1962:≈
1920:⋅
1771:
1720:≡
1670:⋅
1585:≈
1574:−
1531:⋅
1459:≈
1448:−
1405:⋅
1221:−
1173:⋅
1086:≈
996:serialism
881:Zhu Zaiyu
839:Zhu Zaiyu
799:talk page
752:logarithm
504:June 2011
475:does not
386:, green:
334:trombones
191:the term
8695:Vallotti
8648:septimal
8640:Meantone
8400:Interval
8197:Semitone
7409:71425252
7271:(2005).
7212:: 42â47.
7183:Online:
7071:"665edo"
6819:(1707).
6776:20 March
6662:(1584).
6557:(1580).
6489:(1584).
6394:20827087
6309:11 March
6002:See also
5981:â 21.506
5974:= 22.642
5965: Âą
5773:=
5687:meantone
5662:=
5554:=
5159:♯
5146:♯
5133:♯
4999:key of C
4997:(in the
4597:, where
4401:♯
4337:semitone
4071:plus an
3948:♭
3527:gamelans
3293:1200.00
3242:Octave (
3124:♭
2995:♭
2866:♭
2820:1.33484
2675:♭
2546:♭
2478:Unison (
2465:12
1871:♯
1356:♯
1329:♯
1089:1.059463
1028:semitone
552:interval
528:interval
434:, cyan:
326:keyboard
185:semitone
8784:Slendro
8734:Dastgah
8659:Miracle
8622:96-tone
8617:72-tone
8612:58-tone
8607:53-tone
8602:41-tone
8597:34-tone
8592:31-tone
8582:24-tone
8577:23-tone
8572:22-tone
8567:19-tone
8562:17-tone
8557:15-tone
8552:12-tone
8483:7-limit
8478:5-limit
7616:schools
7197:Sources
6744:14 June
6374:Bibcode
6261:in the
6258:A061416
6248:in the
6245:A060529
6235:in the
6232:A061919
6222:in the
6219:A061918
6209:in the
6206:A061921
6196:in the
6193:A061920
6183:in the
6180:A060233
6170:in the
6167:A060527
6157:in the
6154:A060526
6144:in the
6141:A060525
6131:in the
6128:A054540
6122:) â 3:2
6118:in the
6115:A060528
5979:
5969:
5962:
5950:
5943:
5877:
5842:
5838:
5827:
5823:
5813:
5790:
5778:
5726:
5714:
5683:
5671:
5615:
5603:
5571:
5559:
5497:
5492:s = 2 c
5490:
5405:
5401:
5397:
5240:
5217:
5209:octave
5199:
5189:t = s c
5187:
5164:fourths
5086:T t t s
5060:T T t s
4944:
4928:
4894:regular
4840:
4828:
4824:
4812:
4802:
4790:
4786:
4774:
4728:
4716:
4712:
4700:
4690:
4678:
4674:
4662:
4540:
4528:
4503:
4491:
4467:
4455:
4243:
4231:
4227:√
4203:
4191:
4187:√
4163:
4151:
4147:√
4109:√
4077:tritave
4051:in the
4048:A060528
3967:sixths.
3953:kleisma
3754:
3742:
3557:slendro
3344:
3332:
3328:√
3321:C G D A
3313:G D A E
3285:
3273:
3237:+11.73
3226:
3214:
3198:√
3172:996.09
3164:
3152:
3136:√
3108:+15.64
3105:884.36
3097:
3085:
3069:√
3046:-13.69
3043:813.69
3035:
3023:
3007:√
2976:701.96
2968:
2956:
2940:√
2914:609.78
2906:
2894:
2878:√
2847:498.04
2839:
2827:
2811:√
2788:+13.69
2785:386.31
2777:
2765:
2749:√
2726:-15.64
2723:315.64
2715:
2703:
2687:√
2656:203.91
2648:
2636:
2620:√
2597:-11.73
2594:111.73
2586:
2574:
2558:√
2519:
2507:
2226:554.365
1968:659.255
1588:369.994
1462:261.626
1319:), and
1310:middle
1297:), and
916:√
903:√
833:History
693:parts:
496:removed
481:sources
448:
436:
432:
420:
416:
404:
400:
388:
384:
372:
330:fretted
181:
169:
162:√
8752:Mugham
8738:Maqam
8632:Linear
8586:pieces
8547:6-tone
8468:Hexany
8395:Savart
8276:Mother
8025:scales
7914:Sevish
7578:tuning
7508:AVORED
7467:(1753)
7407:
7397:
7361:
7313:19 May
7279:
7257:
7251:Tuning
7238:
7187:
7177:
7014:
6927:
6854:
6799:
6535:
6392:
5901:Hence
5273:, and
5237:commas
5235:, and
5223:(or a
5037:, and
4949:or as
4908:, and
4636:steps.
4576:where
4391:, and
4132:, and
4107:), or
4073:octave
4034:of log
3852:58 EDO
3798:, the
3549:Tenzer
3545:McPhee
3512:each).
3175:+3.91
2979:-1.96
2917:-9.78
2850:+1.96
2659:-3.91
2393:
2387:
2381:
2366:
2360:
2331:
2325:
2288:
2282:
2276:
2242:
2229:
2223:
2217:
2205:
2199:
2181:
2175:
2162:
2120:
2114:
2108:
2089:
2076:
2070:
2044:
2030:
2024:
2018:
1984:
1971:
1965:
1959:
1947:
1941:
1923:
1917:
1904:
1807:
1787:
1781:
1744:
1738:
1732:
1723:
1717:
1711:
1689:
1681:
1673:
1667:
1641:
1604:
1591:
1556:
1534:
1528:
1515:
1478:
1465:
1430:
1408:
1402:
1389:
1229:
1206:
1195:
1186:
1176:
1170:
1144:
1055:
998:, and
944:, and
932:Europe
731:
722:
716:
704:
664:below.
631:
620:
605:
584:
565:
328:, and
45:octave
8769:Pelog
8757:Muqam
8703:Young
8664:Magic
8539:Equal
8473:Limit
8380:Pitch
8187:Limit
7520:CALES
7355:Porto
6823:[
6718:[
6666:[
6644:[
6564:[
6496:[
6390:S2CID
6364:arXiv
5932:= 53
5927:+ 10
5923:+ 16
5792:comma
5734:with
5685:comma
5623:with
5573:comma
5399:s = c
5291:major
5097:comma
5092:grave
4951:cents
4622:steps
4520:sets
4483:sets
4447:sets
4223:gamma
4143:alpha
4135:gamma
4123:alpha
3756:comma
3561:pelog
3523:Kunst
3269:1200
3230:1.875
3210:1100
3148:1000
2652:1.125
2471:cents
2291:round
2033:round
1747:round
1255:hertz
872:China
671:cents
532:ratio
297:, or
270:uses
238:hertz
224:A 440
109:pitch
91:is a
8774:Raga
8385:Cent
7633:List
7567:and
7525:ETER
7523:by P
7405:OCLC
7395:ISBN
7359:ISBN
7315:2006
7277:ISBN
7255:ISBN
7236:ISBN
7185:ISSN
7175:ISSN
7057:2019
7012:ISBN
6925:ISBN
6852:ISBN
6797:ISBN
6778:2012
6746:2012
6562:ćŸæŠèé
6533:ISSN
6494:æšćŸć
šæž
6311:2017
6263:OEIS
6250:OEIS
6237:OEIS
6224:OEIS
6211:OEIS
6198:OEIS
6185:OEIS
6172:OEIS
6159:OEIS
6146:OEIS
6133:OEIS
6120:OEIS
5985:the
5953:1300
5937:for
5912:+ 2
5908:+ 2
5894:= 9
5868:= 8
5847:= 5
5818:= 3
5764:+ 2
5760:+ 2
5653:+ 2
5649:+ 2
5545:+ 2
5541:+ 2
5445:and
5428:and
5403:and
5382:and
5365:and
5320:and
5293:and
5258:and
5207:just
4874:(12
4826:and
4617:â 2
4508:and
4339:and
4254:Play
4214:Play
4183:beta
4174:Play
4129:beta
4101:play
4085:play
4053:OEIS
4040:just
4019:and
3923:just
3869:and
3714:Play
3606:Play
3585:Thai
3569:Play
3559:and
3541:Hood
3467:and
3450:Play
3438:Play
3426:Play
3413:{{7
3411:and
3405:Play
3201:2048
3081:900
3019:800
2952:700
2890:600
2823:500
2781:1.25
2761:400
2699:300
2632:200
2570:100
1861:and
1346:and
1270:and
1000:jazz
768:MIDI
660:See
548:keys
479:any
477:cite
361:and
322:wind
255:and
7628:Row
7529:UCH
7441:by
7165:doi
6848:134
6449:vii
6382:doi
5983:Âą ,
5919:27
5804:TET
5801:53
5749:TET
5746:43
5740:= 0
5699:TET
5696:43
5638:TET
5635:31
5629:= 0
5588:TET
5585:31
5530:=
5512:TET
5509:19
5502:= 0
5481:TET
5478:19
5434:TET
5423:TET
5410:= 0
5391:TET
5388:12
5374:TET
5356:TET
5330:TET
5327:12
5311:= 0
4984:TET
4981:12
4979:in
4973:not
4877:TET
4807:EDO
4769:EDO
4755:EDO
4749:EDO
4741:EDO
4735:12
4730:).
4695:EDO
4657:EDO
4651:EDO
4645:EDO
4603:= 1
4593:EDO
4582:= 0
4571:EDO
4552:EDO
4516:EDO
4510:31
4479:EDO
4473:19
4443:EDO
4420:q t
4303:by
3665:by
3627:TET
3592:TET
3575:).
3534:TET
3506:TET
3501:In
3493:TET
3488:In
3472:TET
3463:TET
3416:TET
3395:TET
3382:TET
3039:1.6
2972:1.5
2943:128
2719:1.2
2468:TET
2451:TET
2443:TET
2419:TET
2335:440
2328:550
2306:log
2159:440
2151:550
2080:440
2073:660
2051:log
1901:440
1893:660
1762:log
1627:TET
1512:440
1386:440
1130:TET
1019:TET
994:or
957:TET
892:TET
889:12
844:æ±èŒć
490:by
366:TET
363:60
357:TET
354:10
301:EDO
275:TET
272:24
261:TET
258:31
250:TET
247:19
214:TET
211:12
201:TET
198:12
144:12
141:or
136:TET
133:12
118:In
95:or
87:An
8853::
8701:,
8697:,
8693:,
8646:,
8142:96
8137:72
8132:58
8127:53
8122:41
8117:34
8112:31
8107:24
8102:23
8097:22
8092:19
8087:17
8082:15
7481:,
7403:.
7353:.
7301:.
7210:13
7208:.
7173:.
7161:31
7159:.
7153:.
7123:.
7098:.
7047:.
6875:.
6850:.
6795:.
6688:.
6582:.
6531:.
6527:.
6478:^
6460:^
6451:.
6430:19
6428:.
6402:^
6388:.
6380:.
6372:.
6360:78
6358:.
6327:;
6289:,
6071:^
5959:53
5948:=
5917:=
5890:+
5886:+
5882:=
5864:+
5860:=
5832:=
5787:5
5781:1
5769:=
5723:4
5717:3
5712:=
5680:4
5674:1
5658:=
5612:3
5606:2
5601:=
5568:3
5562:1
5550:=
5431:7
5420:5
5371:5
5353:7
5347:.
5343:=
5301:,
5269:,
5205:a
5197:,
5175:.
5083:=
5021:,
4987:).
4968:.
4962:â
4958:=
4926:=
4914:t
4912:,
4906:T
4904:,
4898:s
4837:3
4831:2
4821:3
4815:1
4799:9
4793:8
4783:7
4777:1
4725:5
4719:3
4709:5
4703:2
4687:3
4681:2
4671:3
4665:1
4631:â
4627:4
4613:7
4587:7
4565:5
4537:5
4531:2
4526:=
4489:=
4453:=
4437:12
4422:=
4383:,
4375:,
4367:,
4352:.
4269:.
4225::
4185::
4145::
4126:,
4114:.
4023:.
4015:,
3898:41
3847:29
3751:3
3745:1
3631:.
3624:7
3589:7
3583:A
3531:5
3503:7
3490:5
3484:.
3469:7
3460:5
3419:}}
3392:5
3379:7
3296:0
3287:=
3258:=
3252:)
3228:=
3217:15
3196:=
3190:)
3166:=
3155:16
3139:32
3134:=
3128:)
3099:=
3067:=
3061:)
3037:=
3005:=
2999:)
2970:=
2938:=
2932:)
2908:=
2903:45
2897:64
2876:=
2870:)
2841:=
2814:32
2809:=
2803:)
2779:=
2747:=
2741:)
2717:=
2685:=
2679:)
2650:=
2618:=
2612:)
2588:=
2583:15
2577:16
2556:=
2550:)
2530:0
2527:0
2521:=
2503:0
2500:1
2494:=
2488:)
2363:12
2302:12
2279:12
2111:12
2047:12
2021:12
1944:12
1758:12
1735:12
1577:49
1571:46
1550:12
1504:46
1451:49
1445:40
1424:12
1378:40
1362::
1200:12
1079:12
1060:12
1038::
986:,
970:c.
940:,
918:2
905:2
666:)
445:8
439:13
429:8
423:11
413:4
397:4
381:2
324:,
206:.
178:12
147:ET
130:,
8780:)
8776:(
8705:)
8689:(
8685:/
8650:)
8642:(
8588:)
8584:(
8358:e
8351:t
8344:v
7782:e
7775:t
7768:v
7557:e
7550:t
7543:v
7527:B
7518:S
7516:F
7514:O
7510:C
7506:F
7497:"
7485:.
7428:)
7367:.
7317:.
7285:.
7263:.
7244:.
7225:.
7181:.
7167::
7134:.
7108:.
7084:.
7059:.
7020:.
6995:.
6933:.
6886:.
6860:.
6805:.
6748:.
6703:.
6630:.
6597:.
6396:.
6384::
6376::
6366::
6313:.
5997:.
5976:Âą
5972:Îș
5956:/
5946:Îș
5934:Îș
5929:Îș
5925:Îș
5921:Îș
5914:s
5910:t
5906:T
5899:.
5896:Îș
5892:Îș
5888:c
5884:s
5880:T
5873:,
5870:Îș
5866:c
5862:s
5858:t
5852:,
5849:Îș
5845:s
5836:Âł
5834:Îș
5830:c
5820:Îș
5816:c
5784:/
5766:s
5762:t
5758:T
5742:,
5738:Îș
5732:,
5729:s
5720:/
5710:c
5689:.
5677:/
5655:s
5651:t
5647:T
5631:,
5627:Îș
5621:,
5618:s
5609:/
5599:c
5578:.
5565:/
5547:s
5543:t
5539:T
5532:t
5528:T
5524:s
5520:c
5516:,
5504:,
5500:Îș
5469:,
5462:Îș
5458:s
5454:,
5447:Îș
5443:s
5412:.
5408:Îș
5384:Îș
5380:s
5367:Îș
5363:c
5345:t
5341:T
5337:s
5322:c
5317:s
5309:Îș
5303:t
5299:T
5297:(
5275:Îș
5271:c
5267:s
5260:c
5256:s
5252:c
5248:s
5244:.
5242:Îș
5233:c
5229:s
5221:,
5203:,
5191:,
5180:Îș
5153:C
5149:â
5140:F
5127:F
5123:â
5119:B
5111:B
5107:â
5103:E
5089:(
5079:A
5075:â
5071:D
5062:.
5049:D
5045:â
5041:G
5033:G
5029:â
5025:C
5017:C
5013:â
5009:F
4993:F
4964:t
4960:T
4956:Îș
4947:,
4940:t
4936:/
4932:T
4924:Îș
4918:Îș
4880:)
4834:/
4818:/
4810:(
4796:/
4780:/
4764:q
4744:,
4737:k
4722:/
4706:/
4684:/
4668:/
4640:n
4633:s
4629:t
4619:s
4615:t
4601:q
4589:k
4580:q
4574:,
4567:k
4559:C
4543:.
4534:/
4524:q
4512:k
4506:,
4500:3
4497:/
4494:1
4487:q
4475:k
4470:,
4464:2
4461:/
4458:1
4451:q
4439:k
4433:k
4424:s
4414:q
4408:C
4395:F
4387:F
4379:E
4371:D
4363:C
4350:t
4346:s
4326:)
4320:(
4315:)
4311:(
4297:.
4240:2
4237:/
4234:3
4200:2
4197:/
4194:3
4160:2
4157:/
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4111:3
4079:(
4036:2
4007:.
3935:2
3838:.
3822:.
3816:2
3748:/
3688:)
3682:(
3677:)
3673:(
3659:.
3537:,
3509:,
3496:,
3341:2
3338:/
3335:3
3289:2
3282:1
3279:/
3276:2
3265:2
3260:2
3256:2
3247:C
3223:8
3220:/
3194:2
3185:B
3161:9
3158:/
3132:2
3118:B
3094:3
3091:/
3088:5
3072:8
3065:2
3056:A
3032:5
3029:/
3026:8
3010:4
3003:2
2989:A
2965:2
2962:/
2959:3
2936:2
2927:G
2900:/
2881:2
2874:2
2860:G
2836:3
2833:/
2830:4
2807:2
2798:F
2774:4
2771:/
2768:5
2752:2
2745:2
2736:E
2712:5
2709:/
2706:6
2690:2
2683:2
2669:E
2645:8
2642:/
2639:9
2623:2
2616:2
2607:D
2580:/
2561:2
2554:2
2540:D
2523:1
2516:1
2513:/
2510:1
2496:1
2492:2
2483:C
2396:.
2384:3
2377:1
2372:=
2356:4
2351:=
2346:)
2340:)
2319:(
2310:2
2297:(
2272:1
2267:=
2264:x
2237:z
2234:H
2211:)
2202:3
2195:1
2190:(
2185:2
2170:z
2167:H
2156:=
2147:E
2123:.
2104:7
2099:=
2094:)
2085:)
2064:(
2055:2
2039:(
2014:1
2009:=
2006:x
1979:z
1976:H
1953:)
1937:7
1932:(
1927:2
1912:z
1909:H
1898:=
1889:E
1875:5
1865:C
1859:5
1854:E
1846:a
1844:E
1837:n
1835:E
1810:.
1802:)
1796:)
1791:a
1784:n
1775:(
1766:2
1753:(
1728:1
1714:x
1684:x
1677:2
1662:a
1658:E
1654:=
1649:n
1645:E
1599:z
1596:H
1580:)
1568:(
1562:)
1546:2
1539:(
1523:z
1520:H
1509:=
1500:P
1473:z
1470:H
1454:)
1442:(
1436:)
1420:2
1413:(
1397:z
1394:H
1383:=
1374:P
1360:4
1350:F
1344:4
1339:C
1333:4
1323:F
1314:C
1308:(
1306:4
1301:C
1291:4
1286:A
1280:a
1276:n
1272:a
1268:n
1262:a
1260:P
1249:n
1247:P
1224:a
1218:n
1212:)
1192:2
1181:(
1165:a
1161:P
1157:=
1152:n
1148:P
1123:n
1121:P
1076:1
1070:2
1066:=
1052:2
968:(
923:,
910:,
809:)
805:(
801:.
764:c
726:n
719:w
710:=
707:c
691:n
687:w
683:p
656:n
652:p
648:r
625:n
617:p
611:=
608:r
581:p
578:=
573:n
569:r
517:)
511:(
506:)
502:(
498:.
484:.
442:/
426:/
410:/
407:7
394:/
391:5
378:/
375:3
278:.
231:A
175:/
172:1
164:2
68:C
55:.
20:)
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