3791:, 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
4847:
64:
3592:
866:
4240:
4160:
3555:
4200:
3436:
3424:
3700:
3412:
3391:
4071:
4087:
331:
3713:
23:
7948:
7943:
50:
4272:
3687:
3634:
455:
3363:
1815:
332:
773:
3589:"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."
1002:
5151:; 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
3807:(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
22:
2395:
439:). 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.
2122:
3877:
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
3898:
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
6088:
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
3889:
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
840:
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
3838:
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
3763:
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
3775:
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
1809:
3866:. 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.
3314:
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
1984:
2242:
3747:
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
1603:
1477:
4598:
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
5155:
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
3342:
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.
2247:
229:
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.
3839:
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
3593:
1989:
856:
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.
3500:
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
1228:
3991:
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
851:
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
3955:
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
4241:
4161:
3556:
6279:
4201:
4027:(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
3604:
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.
1694:
65:
3979:
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
3437:
3425:
4044:) 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.
3701:
3487:
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.
3413:
3392:
4747:
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.)
4332:
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
1088:
3552:, 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
1872:
2130:
4072:
4741:
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
4088:
1689:
1483:
1357:
321:
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
885:, 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.
5270:, 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.
967:
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
730:
7451:
630:
2390:{\displaystyle \quad x={\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}12\log _{2}\left({\frac {\ 550\ }{440}}\right){\Biggr )}={\frac {4}{\ 12\ }}={\frac {1}{\ 3\ }}~.}
739:, 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
6202:) 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
4109:
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
818:
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.
583:
5339:
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
6944:
5235:
Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitones
2117:{\displaystyle \quad x={\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}\ 12\log _{2}\left({\frac {\ 660\ }{440}}\right)\ {\Biggr )}={\frac {7}{\ 12\ }}~.}
6070:
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.
5734:
5684:
4001:
4127:. 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:
3971:
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
8285:
7325:
3819:
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
6648:
1128:
930:
2418:
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.
1804:{\displaystyle \quad \ x\ \equiv \ {\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}12\log _{2}\left({\frac {\ n\ }{a}}\right){\Biggr )}~.}
7446:
6292:
6722:
8413:
3795:, experimented with music for quarter-tone pianos. 24 EDO also approximates the 11th and 13th harmonics very well, unlike 12 EDO.
6513:
8271:
7769:
6964:
6568:
6411:
Kuttner, Fritz A. (May 1975). "Prince Chu Tsai-YĂŒ's life and work: A re-evaluation of his contribution to equal temperament theory".
6251:
6238:
6225:
6212:
6199:
6186:
6173:
6160:
6147:
6134:
6121:
6108:
4041:
1033:
1979:{\displaystyle E_{660}=440\ {\mathsf {Hz}}\ \cdot \ 2^{\left({\frac {7}{\ 12\ }}\right)}\ \approx \ 659.255\ {\mathsf {Hz}}\ \quad }
8345:
2237:{\displaystyle E_{550}=440\ {\mathsf {Hz}}\ \cdot \ 2^{\left({\frac {1}{\ 3\ }}\right)}\ \approx \ 554.365\ {\mathsf {Hz}}\ \quad }
847:
Kenneth
Robinson credits the invention of equal temperament to Zhu and provides textual quotations as evidence. In 1584 Zhu wrote:
8478:
8258:
271:
Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the
3693:'s notation system for 16 equal temperament: Intervals are notated similarly to those they approximate and there are fewer
4846:
4534:
The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the
7351:
6950:
6789:
1598:{\displaystyle P_{46}=440\ {\mathsf {Hz}}\ \cdot \ {\Bigl (}{\sqrt{2}}\ {\Bigr )}^{(46-49)}\approx 369.994\ {\mathsf {Hz}}\ }
1472:{\displaystyle P_{40}=440\ {\mathsf {Hz}}\ \cdot \ {\Bigl (}{\sqrt{2}}\ {\Bigr )}^{(40-49)}\approx 261.626\ {\mathsf {Hz}}\ }
7287:
3338:
to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a
1625:
7387:
7228:
8488:
7872:
7544:
6752:
5742:, with three steps for the chromatic semitone, four steps for the diatonic semitone, and seven steps for the tone, where
2415:. The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.
36:
horizontally (open the image to view the full width), and each shaded rectangle is the width of one step in a scale. The
3930:
theory. It is not a meantone temperament, which put good thirds within easy reach by stacking fifths; instead, like all
5631:, with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone, where
4289:
3651:
476:
7059:
7483:
7269:
7247:
7004:
6917:
6844:
4311:
3673:
1839:
is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see that
502:
6674:
484:
7461:
5460:
and the result is seven-tone equal temperament. These two extremes are not included as "regular" diatonic tunings.
8451:
8148:
7477:
6341:
Varieschi, Gabriele U.; Gower, Christina M. (2010). "Intonation and compensation of fretted string instruments".
4020:
6569:"Quantifying ritual: Political cosmology, courtly music, and precision mathematics in seventeenth-century China"
7762:
7676:
4293:
3655:
480:
29:
688:
8636:
7578:
6726:
6098:
OEIS sequences that contain divisions of the octave that provide improving approximations of just intervals:
3752:(at 505 cents), is seven cents sharper than just intonation's and five cents sharper than 12 EDO's.
1271:). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example,
666:
scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in
4755:
for the relationship results in exactly one equal temperament family, but the converse is not true: 47
8850:
8338:
7621:
6949:[Findings of new literatures concerning the hepta â equal temperament] (in Chinese). Archived from
3764:
subminor third (7/6). One step closer to each other are the classical major and minor thirds (5/4 and 6/5).
7084:
4631:, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24
747:
where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to
282:
For tuning systems that divide the octave equally, but are not approximations of just intervals, the term
8845:
8574:
8253:
7736:
7705:
6343:
751:, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g.,
7373:
589:
8418:
8278:
8021:
7897:
7792:
6029:
4417:
also defines a unique family of one equal temperament and its multiples that fulfil this relationship.
826:
The two figures frequently credited with the achievement of exact calculation of equal temperament are
6189:) 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
275:, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "
8403:
8230:
8218:
7755:
6997:
Toward a
Quarter-Tone Syntax: Analyses of selected works by Blackwood, Haba, Ives, and Wyschnegradsky
6457:
Robinson, Kenneth G.; Needham, Joseph (1962â2004). "Part 1: Physics". In
Needham, Joseph (ed.).
6413:
6321:
1015:
In 12 tone equal temperament, which divides the octave into 12 equal parts, the width of a
865:
305:, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to
5545:
19 steps. The imbedded 12 tone sub-system closely approximates the historically important
736:
233:
Other equal temperaments divide the octave differently. For example, some music has been written in
8687:
7922:
465:
6575:. Roger Hart Departments of History and Asian Studies, University of Texas, Austin. Archived from
6434:
A critical study of Chu Tsai-yĂŒ's contribution to the theory of equal temperament in
Chinese music
3926:(3). With its accurate cycle of fifths and multi-purpose comma step, 53 EDO has been used in
1324:
are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of
8823:
8657:
8483:
8408:
8331:
8180:
8051:
7862:
7557:
7537:
5079:
5054:
4851:
4282:
4053:
3644:
549:
469:
272:
111:
and
Western music in general, the most common tuning system since the 18th century has been
7177:
8679:
8642:
8632:
8441:
7877:
7688:
7666:
7628:
7144:
6836:
6529:
6461:. Science and Civilisation in China. Vol. 4. Cambridge, UK: University Press. p. 221.
6442:
Chu-Tsaiyu the first formulator of the mathematics of "equal temperament" anywhere in the world
6034:
5657:
3931:
3863:
3808:
787:
7033:
85:
8675:
7997:
7827:
7611:
7467:
7413:
7369:
6313:
6009:
4909:
is implicit as the size ratio between the greater and lesser tones: Expressed as frequencies
4871:
can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps
3690:
3103:
2974:
2845:
2654:
2525:
1283:
1102:
980:
968:
532:
212:
6828:
6514:"The significance of the discovery of the musical equal temperament in the cultural history"
8683:
8628:
8610:
8605:
8600:
8595:
8590:
8585:
8580:
8565:
8560:
8555:
8550:
8545:
8540:
8307:
8130:
8125:
8120:
8115:
8110:
8105:
8100:
8090:
8085:
8080:
8075:
8070:
7802:
7407:â A foundational work on acoustics and the perception of sound. Especially the material in
7140:"Isomorphic controllers and dynamic tuning: Invariant fingerings across a tuning continuum"
6974:
6805:
6362:
5927:
5789:
5653:
5623:
5573:
5564:
5497:
5466:
5314:
4009:
4005:
3988:
3964:
3952:
3907:
3886:
3874:
3851:
3840:
3835:
3772:
3760:
3725:
3466:
841:
somewhat less precise computation of the corresponding numerical values in 1585 or later."
813:
314:
245:
234:
112:
6576:
515:
In an equal temperament, the distance between two adjacent steps of the scale is the same
8:
8647:
8620:
8518:
8302:
8247:
7947:
7927:
5419:
5408:
5341:
4627:
If there are notes outside the circle of fifths, one must then multiply these results by
4544:, the half-sharps and half-flats are not in the circle of fifths generated starting from
4254:
1024:
950:
934:
528:
338:
302:
81:
6366:
4348:
of a whole tone, while keeping the notes in the right order (meaning that, for example,
3878:
sharp instead of flat. It enables the 600 cent tritone, since 34 is an even number.
786:
Please expand the section to include this information. Further details may exist on the
670:. The basic step in cents for any equal temperament can be found by taking the width of
8712:
8498:
8493:
8224:
7942:
7852:
7682:
7530:
7489:
7379:
7288:"The gamelan pelog scale of Central Java as an example of a non-harmonic musical scale"
7211:
7208:
The
Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century
6378:
6352:
6327:
6317:
5656:. The imbedded 12 tone sub-system closely approximates the historically important
4960:
The notes in a regular diatonic tuning are connected in a "spiral of fifths" that does
3919:
3855:
3788:
744:
524:
7812:
6176:) 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
3993:
8691:
8652:
8466:
8312:
8195:
7832:
7393:
7383:
7347:
7339:
7318:
7265:
7243:
7224:
7173:
7163:
7000:
6913:
6840:
6829:
6785:
6521:
6002:
5304:
146:
8717:
7842:
6436:. Sinologica Coloniensia. Vol. 9. Wiesbaden, DE: Franz Steiner Verlag. p.
6382:
662:, which divide the octave into 1200 equal intervals (each called a cent). This
8671:
8535:
8503:
8423:
8388:
8200:
8153:
8002:
7912:
7837:
7797:
7778:
7671:
7593:
7401:
7257:
7153:
7135:
6861:
6370:
6039:
6019:
5695:
If the chromatic semitone is three-fourths the size of the diatonic semitone, i.e.
5267:
4965:
4535:
3934:, the very consonant thirds are represented by a Pythagorean diminished fourth (C-F
3712:
3576:
xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from
3232:
3170:
3041:
2912:
2783:
2721:
2592:
2468:
926:
540:
516:
310:
256:
216:
7867:
7109:
6079:'Hepta-equal temperament' in our folk music has always been a controversial issue.
1282:(the reference pitch) is the 49th key from the left end of a piano (tuned to
888:
Zhu obtained his result by dividing the length of string and pipe successively by
735:
In musical analysis, material belonging to an equal temperament is often given an
8793:
8433:
8140:
7882:
7588:
7494:
7295:
6660:
5997:
5979:
5195:
4345:
4344:
There is exactly one family of equal temperaments that fixes the semitone to any
4081:
4028:
3976:
3968:
3911:
2412:
1608:
1223:{\displaystyle \ P_{n}=P_{a}\ \cdot \ {\Bigl (}\ {\sqrt{2\ }}\ {\Bigr )}^{n-a}\ }
674:
above in cents (usually the octave, which is 1200 cents wide), called below
667:
306:
298:
108:
89:
5584:
If the chromatic semitone is two-thirds the size of the diatonic semitone, i.e.
5477:
If the diatonic semitone is set double the size of the chromatic semitone, i.e.
92:
by dividing an octave (or other interval) into steps such that the ratio of the
8471:
8446:
8393:
8354:
8264:
7977:
7932:
7822:
7698:
7638:
7338:. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave.
6044:
5975:
5225:
5213:
5161:
5152:
4878:
3859:
3824:
3820:
3749:
3608:
3537:
3357:
1020:
208:
7522:
7441:
7281:
Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta
7158:
7139:
6758:
5303:, with the others expanded to still fill out the octave), and both semitones (
3544:. It is now accepted that of the two primary tuning systems in gamelan music,
8839:
8766:
8735:
8508:
8461:
8368:
8175:
7917:
7892:
7720:
7167:
6525:
5983:
5208:
s c Îș s c s s c Îș s c s c Îș s
5157:
5085:
4991:
4057:
3975:
and his students (whose atonal inclinations typically avoid any reference to
3927:
3915:
3541:
917:
Zhu created several instruments tuned to his system, including bamboo pipes.
276:
177:
97:
37:
7847:
7397:
6868:(in French). Association pour la Recherche et le DĂ©veloppement de la Musique
3910:
has only had occasional use, but is better at approximating the traditional
145:), which divides the octave into 12 parts, all of which are equal on a
8730:
8570:
8373:
8095:
7992:
7987:
7982:
7967:
7962:
7857:
7817:
7807:
7715:
7693:
7656:
7583:
7088:
7063:
6748:
6024:
4939:
4106:
4080:), and split into 13 equal parts. This provides a very close match to
4000:
Other equal divisions of the octave that have found occasional use include
3792:
3784:
3533:
972:
940:
837:
659:
7375:
On the Sensations of Tone as a Physiological Basis for the Theory of Music
49:
8818:
8813:
8803:
8398:
8378:
8046:
8041:
8031:
7972:
6682:
5043:
4123:
4111:
4016:
3529:
925:
Some of the first Europeans to advocate equal temperament were lutenists
748:
743:
of a multiplication reduces it to addition. Furthermore, by applying the
536:
41:
7427:
4831:). Taking each semitone results in a different choice of perfect fifth.
3944:, allowing its fifth to be reached by a stack of six minor thirds (6:5).
8808:
8777:
8036:
7887:
7710:
7661:
6014:
5313:) the same size, then twelve equal semitones, two per tone, result. In
5283:
5279:
5183:
5172:
4898:
4890:
4296: in this section. Unsourced material may be challenged and removed.
4117:
3972:
3694:
3658: in this section. Unsourced material may be challenged and removed.
3573:
3511:
6374:
4252:
Alpha and beta may be heard on the title track of Carlos's 1986 album
3940:), reached by stacking eight perfect fourths. It also tempers out the
914:
such that after 12 divisions (an octave), the length was halved.
8798:
8752:
8190:
7907:
7633:
7553:
7431:
6543:
6475:
4393:
are in ascending order if they preserve their usual relationships to
3352:
984:
976:
869:
827:
740:
663:
101:
93:
4271:
3686:
3633:
454:
8185:
7747:
7616:
7312:
3362:
1298:
1016:
762:
322:
173:
8323:
7436:
6357:
4854:
continuum, which include many notable "equal temperament" tunings.
8772:
8722:
7411:, pages 430â556, (pdf pages 451â577) (see also wiki article
5800:
If the chromatic semitone is made the same size as three commas,
4987:
4084:
ratios consisting only of odd numbers. Each step is 146.3 cents (
4065:
3941:
3545:
3515:
876:
Chinese theorists had previously come up with approximations for
4031:
twelfths/fifths than in any equal temperament with fewer tones.
772:
531:
of intervals would not sound evenly spaced and would not permit
8740:
8456:
8383:
7902:
5449:
gets larger (and absorbs the space formerly used for the comma
4643:). (One must take the small semitone for this purpose: 19
4061:
3914:
consonances than 12, 19 or 31 EDO. Its extremely accurate
1814:
1083:{\displaystyle {\sqrt{2\ }}=2^{\tfrac {1}{12}}\approx 1.059463}
1001:
519:. Because the perceived identity of an interval depends on its
33:
7278:
7221:
How Equal Temperament Ruined Harmony (and why you should care)
6885:
5764:
43. The imbedded 12 tone sub-system closely approximates
5438:
reduce to zero with the octave size kept fixed, the result is
5430:
There are two extreme cases that bracket this framework: When
4261:
1609:
Converting frequencies to their equal temperament counterparts
96:
of any adjacent pair of notes is the same. This system yields
8757:
8745:
7478:
All existing 18th century quotes on J.S. Bach and temperament
7343:
4056:
consists of the ratio 3:1 (1902 cents) conventionally a
4047:
3549:
1243:
520:
226:
32:
A comparison of some equal temperaments. The graph spans one
3358:
Five-, seven-, and nine-tone temperaments in ethnomusicology
8762:
7127:
6711:] (in Italian) (reprint ed.). Geneva, CH: Minkoff.
6246:
6233:
6220:
6207:
6194:
6181:
6168:
6155:
6142:
6129:
6116:
6103:
4036:
988:
784:
about the general formulas for the equal-tempered interval.
756:
325:, use tuning similar to string ensembles and vocal groups.
318:
6330:(reprint ed.). New York, NY: Dover. pp. 493â511.
4793:, which are not complements of each other like in 19
7382:(reprint ed.). Whitefish, MT: Kellinger Publishing.
7279:
Surjodiningrat, W.; Sudarjana, P.J.; Susanto, A. (1972).
6163:) â 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7, 16:11 and 11:8
3298:
Violins, violas, and cellos are tuned in perfect fifths (
3293:
1255:
is the frequency of a reference pitch. The indes numbers
149:, with a ratio equal to the 12th root of 2, (
5328:, is exactly half the size of the same-size whole tones
5278:
An equal temperament can be created if the sizes of the
5251:, with some fixed proportion for each type of semitone.
991:(at least its piano component) to develop and flourish.
16:
Musical tuning system with constant ratios between notes
7490:
Well Temperaments, based on the Werckmeister Definition
6657:
The Fronimo ... Dialogue on the art of a good beginning
6653:
Il Fronimo ... Dialogo sopra l'arte del bene intavolare
4034:
1, 2, 3, 5, 7, 12, 29, 41, 53, 200, ... (sequence
61:, one full octave ascending, notated only with sharps.
7133:
5978:. It is an exceedingly close approximation to 5-limit
4064:(that is, a perfect twelfth), called in this theory a
3505:
1830:
is the frequency of a pitch in equal temperament, and
1818:
Comparison of intervals in 12-TET with just intonation
1684:{\displaystyle \ E_{n}=E_{a}\ \cdot \ 2^{\ x}\ \quad }
1062:
309:
for acoustic reasons. Other instruments, such as some
301:, which can adjust the tuning of all notes except for
7442:
Huygens-Fokker Foundation Centre for Microtonal Music
5273:
2250:
2133:
1992:
1875:
1697:
1628:
1486:
1360:
1131:
1036:
691:
592:
552:
6616:
6596:
6456:
5266:
can be repeatedly appended to itself into a greater
3890:
more accurate in the 13 limit than 31 EDO.
3843:, which allows for lower errors for some just tones.
3536:(1966) their tuning varies widely, and according to
1613:
To convert a frequency (in Hz) to its equal 12
156:â 1.05946 ). That resulting smallest interval,
8286:
Twelve Microtonal Etudes for Electronic Music Media
7484:
Rosetta Revisited: Bach's Very Ordinary Temperament
1096:
1023:of the interval between two adjacent notes, is the
7324:
2400:
2389:
2236:
2116:
1978:
1803:
1683:
1597:
1471:
1222:
1082:
759:encoding standard uses integer note designations.
724:
650:
624:
577:
359:on each main interval of small prime limits (red:
53:12 tone equal temperament chromatic scale on
7038:Tonalsoft Encyclopedia of Microtonal Music Theory
5930:of one comma each. The comma size / step size is
4595:and the semitone and tone are the same interval.
3922:, as 53 is the denominator of a convergent to log
2334:
2285:
2282:
2082:
2027:
2024:
1790:
1741:
1738:
1550:
1527:
1424:
1401:
1200:
1169:
8837:
7062:. xenoharmonic (microtonal wiki). Archived from
5053:The three in-tune fifths are interrupted by the
1619:counterpart, the following formula can be used:
807:
763:General formulas for the equal-tempered interval
7552:
7437:Xenharmonic wiki on EDOs vs. Equal Temperaments
7283:. Jogjakarta, IN: Gadjah Mada University Press.
6611:The Shorter Science & Civilisation in China
5453:), eventually the steps are all the same size,
3899:meantone system, it distinguishes 10:9 and 9:8.
1242:represents the pitch, or frequency (usually in
6886:Surjodiningrat, Sudarjana & Susanto (1972)
6725:. Appalachian State University. Archived from
6340:
5358:tend to zero, with the octave kept fixed, and
5294:) are altered to be the same (say, by setting
1866:have the following frequencies, respectively:
1263:are the labels assigned to the desired pitch (
1093:This interval is divided into 100 cents.
8339:
7763:
7538:
7464:. (2008) Latina, Il Levante Libreria Editrice
7409:Appendix XX: Additions by the translator
7264:(2nd ed.). London, UK: Springer-Verlag.
6452:
6450:
5982:and Pythagorean tuning, and is the basis for
3716:Comparison of equal temperaments from 9 to 25
3623:
3506:5 tone and 9 tone equal temperament
831:
100:steps perceived as equal in size, due to the
8414:List of intervals in 5-limit just intonation
6908:Morton, David (1980). May, Elizabeth (ed.).
6804:
6394:
6392:
5160:: Each of the grave fifths out of tune by a
4889:) must be smaller than either of the tones (
4885:diatonic tuning, each of the two semitones (
3803:26 is the denominator of a convergent to log
279:" in that system, into 13 equal parts.
7462:Enharmonic instruments and music, 1470â1900
7315:(Report). 8096295 – via academia.edu.
7313:From galaxy to galaxy: Music of the spheres
7193:Boiles, J. (1969). "Terpehua though-song".
6470:
6468:
6241:) â 6:5 and 5:3, 7:5 and 10:7, 7:6 and 12:7
4839:
4538:. (This is not true in general; in 24
4262:Proportions between semitone and whole tone
3567:
3346:
483:. Unsourced material may be challenged and
341:A comparison of equal temperaments between
8346:
8332:
7770:
7756:
7545:
7531:
7223:. New York, NY: W.W.Norton & Company.
7110:"Three Asymmetric Divisions of the Octave"
6493:
6447:
6320:"The History of Musical Pitch in Europe".
6312:
6063:
6061:
4761:has two different semitones, where one is
4048:Equal temperaments of non-octave intervals
643:(typically the octave, which is 2:1) into
7454:A supplement to Mr. Chambers's cyclopĂŠdia
7368:
7321:. 269108386 – via researchgate.net.
7237:
7157:
6389:
6356:
6290:
6150:) â 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7
6124:) â 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3
5507:with one step for the chromatic semitone
5125:, and then restarting in the sharps with
4834:
4405:to a proper fraction in the relationship
4312:Learn how and when to remove this message
4015:2, 5, 12, 41, 53, 306, 665 and 15601 are
3674:Learn how and when to remove this message
971:, new styles of symmetrical tonality and
844:The developments occurred independently.
543:in an equal-tempered scale is the ratio:
503:Learn how and when to remove this message
184:, without qualification, generally means
7340:The International Conference SIGMAP-2008
7336:Approximation of 5-limit just intonation
7256:
7019:
6702:
6536:
6518:Journal of Xinghai Conservatory of Music
6465:
6431:
6275:
6067:
5843:that makes the lesser tone eight commas
5829:) the diatonic the same as five commas,
5445:a 5 tone equal temperament. As the
4845:
3711:
3685:
3445:steps, respectively, are fairly common.
3378:Five- and seven-tone equal temperament (
3361:
1813:
1000:
864:
830:(also romanized as Chu-Tsaiyu. Chinese:
755:is 0 regardless of octave register. The
725:{\displaystyle \ c={\frac {\ w\ }{n}}\ }
48:
7333:
7310:
7294:. Neuroscience of Music. Archived from
6994:
6912:. Musics of Many Cultures. p. 70.
6835:(2nd ed.). Da Capo Press. p.
6647:
6499:
6425:
6410:
6398:
6058:
5239:, or into the five chromatic semitones
5088:"), followed by another perfect fifth,
4574:and the semitone becomes a unison, and
4337:, and the number of steps in a tone be
3724:Many instruments have been built using
8838:
7218:
7192:
6932:
6907:
6896:
6862:"Le tempérament égal à quintes justes"
6826:
6751:(30 June 2009) . Rasch, Rudolf (ed.).
6747:
6487:Complete Compendium of Music and Pitch
5511:, two steps for the diatonic semitone
3294:Seven-tone equal division of the fifth
2411:closely approximate some intervals in
2225:
2222:
2158:
2155:
1967:
1964:
1900:
1897:
1587:
1584:
1511:
1508:
1461:
1458:
1385:
1382:
40:ratios are separated in rows by their
8327:
7751:
7526:
7428:An Introduction to Historical Tunings
7334:Khramov, Mykhaylo (26â29 July 2008).
7031:
6089:system associated with the flute law.
4881:or "rotation" of it). To be called a
1122:, the following formula may be used:
444:
7777:
6757:. The Diapason Press. Archived from
4294:adding citations to reliable sources
4265:
3862:and represents a standardization of
3656:adding citations to reliable sources
3627:
766:
481:adding citations to reliable sources
448:
172:the width of an octave, is called a
8353:
7242:. Michigan State University Press.
7205:
6971:About "Seven- equal- tuning System"
6859:
6779:
6613:(abridgemed ed.). p. 385.
6542:
6511:
6505:
6474:
3469:'s valid tuning range, as shown in
964:), published posthumously in 1884.
13:
7362:
7107:
6973:] (in Chinese). Archived from
6769:– via diapason.xentonic.org.
6628:
6617:Robinson & Needham (1962â2004)
6615:â reduced version of the original
6597:Robinson & Needham (1962â2004)
5274:Morphing diatonic tunings into EDO
4722:The smallest of these families is
4052:The equal-tempered version of the
3918:make it equivalent to an extended
625:{\displaystyle \ r={\sqrt{p\ }}\ }
14:
8862:
7421:
6723:"Spinacino 1507a: Thematic Index"
6608:
6293:"Perceptual Foundations of Sound"
5204:can be broken up into a sequence
3340:ratio of â 517:258 or â 2.00388:1
943:was the first to develop 12
937:, all of whom wrote music in it.
8489:Ptolemy's intense diatonic scale
7946:
7941:
6810:Musicalische paradoxal-Discourse
6754:Van de Spiegheling der singconst
6659:] (in Italian). Venice, IT:
6635:Abacus and Practical Mathematics
5523:, and the total number of steps
4270:
3632:
1097:Calculating absolute frequencies
955:van de Spiegheling der singconst
872:'s equal temperament pitch pipes
771:
523:, this scale in even steps is a
453:
330:
21:
7262:Tuning, Timbre, Spectrum, Scale
7138:; Plamondon, J. (Winter 2007).
7101:
7077:
7052:
7025:
7013:
6988:
6957:
6937:
6926:
6901:
6890:
6879:
6853:
6820:
6798:
6773:
6741:
6715:
6696:
6675:"Resound â corruption of music"
6667:
6641:
6622:
6602:
6590:
6561:
6459:Physics and Physical Technology
6092:
6082:
6073:
4281:needs additional citations for
3643:needs additional citations for
2401:Comparison with just intonation
2251:
2233:
1993:
1975:
1698:
1680:
207:is usually tuned relative to a
7677:Emancipation of the dissonance
6814:Paradoxical Musical Discussion
6512:Cho, Gene J. (February 2010).
6404:
6334:
6306:
6284:
6269:
4860:12 tone equal temperament
1568:
1556:
1442:
1430:
994:
979:such as that written with the
539:.) Specifically, the smallest
117:12 tone equal temperament
1:
7579:Mode of limited transposition
6995:Skinner, Myles Leigh (2007).
6263:
5216:of it) of diatonic semitones
4687:has two semitones, one being
4649:has two semitones, one being
3470:
958:
808:Twelve-tone equal temperament
658:Scales are often measured in
651:Twelve-tone equal temperament
67:Play ascending and descending
8452:Harry Partch's 43-tone scale
8149:Harry Partch's 43-tone scale
7447:A.Orlandini: Music Acoustics
6555:Fusion of Music and Calendar
6137:) â 3:2 and 4:3, 5:4 and 8:5
6051:
5515:, three steps for the tones
5350:in the limit as the size of
4978:Starting on the subdominant
284:equal division of the octave
141:, informally abbreviated as
104:changes in pitch frequency.
7:
8272:Sonata for Microtonal Piano
7737:List of atonal compositions
7706:Quartal and quintal harmony
7206:Cho, Gene Jinsiong (2003).
6703:Gorzanis, Giacomo (1982) .
6344:American Journal of Physics
5990:
5864:and the greater tone nine,
5104:, and another grave fifth,
4735:and in particular, 12
4613:and the perfect fifth into
1267:) and the reference pitch (
1246:), you are trying to find.
578:{\displaystyle \ r^{n}=p\ }
10:
8867:
8419:List of meantone intervals
8279:Suite for Microtonal Piano
7185:
7085:"convergents(log2(3), 10)"
6782:Lutes, Viols, Temperaments
6432:Robinson, Kenneth (1980).
6030:List of meantone intervals
5254:The sequence of intervals
5042:âeach a composite of some
3624:Various equal temperaments
3465:mark the endpoints of the
3350:
1100:
821:
811:
8786:
8703:
8666:
8619:
8526:
8517:
8432:
8409:List of musical intervals
8404:Consonance and dissonance
8361:
8295:
8239:
8231:Huygens-Fokker Foundation
8219:Boston Microtonal Society
8209:
8166:
8139:
8061:
8020:
8011:
7955:
7939:
7785:
7729:
7647:
7602:
7564:
7159:10.1162/comj.2007.31.4.15
6323:On the Sensations of Tone
6280:fig. 4.6, p. 58
5046:of the smaller intervals
4703:tone and the other being
4665:tone and the other being
4552:.) The extreme cases are
2451:Cents in just intonation
2448:Just intonation interval
2437:Decimal value in 12
2405:The intervals of 12
920:
832:
7468:Fractal Microtonal Music
7329:– via Google docs.
7238:Jorgensen, Owen (1991).
7219:Duffin, Ross W. (2007).
6965:
6945:
6630:
6550:
6482:
5814:(in cents, in frequency
5268:spiral of 12 fifths
4852:regular diatonic tunings
4840:Regular diatonic tunings
4751:Each choice of fraction
3568:7-tone equal temperament
3347:Other equal temperaments
953:, which he described in
860:
527:of multiplications. (An
8181:Otonality and Utonality
7311:Stewart, P.J. (2006) .
5384:is of course, the case
4858:The diatonic tuning in
3932:schismatic temperaments
3611:has traditionally used
3585:. According to Morton,
2429:Exact value in 12
1116:, of a note in 12
1107:To find the frequency,
901:and for pipe length by
678:, and dividing it into
225:, is tuned to 440
211:of 440 Hz, called
7878:Claus-Steffen Mahnkopf
7689:Polymodal chromaticism
7667:Dissonant counterpoint
7629:Second Viennese School
7145:Computer Music Journal
6966:äžćčłććŸ"çè°--ć
Œćæ§ćŒććæČçŹć¶äœäžèœŹè°
6827:Partch, Harry (1979).
6035:Diatonic and chromatic
6006:(the physics of music)
5220:, chromatic semitones
4855:
4835:Related tuning systems
4777:tone and the other is
3864:quarter-comma meantone
3728:tuning. Equivalent to
3717:
3709:
3421:), with 240 cent
3375:
2391:
2238:
2118:
1980:
1819:
1805:
1685:
1599:
1473:
1224:
1084:
1012:
1005:One octave of 12
873:
782:is missing information
726:
626:
579:
73:
8676:Temperament ordinaire
8054:(BohlenâPierce scale)
7998:Tui St. George Tucker
7612:Twelve-tone technique
7414:On Sensations of Tone
6910:The Music of Thailand
6806:Werckmeister, Andreas
6705:Intabolatura di liuto
6599:, p. 220 ff
6010:Music and mathematics
4849:
4681:. Similarly, 31
3715:
3689:
3365:
2392:
2239:
2119:
1981:
1817:
1806:
1686:
1600:
1474:
1225:
1103:Piano key frequencies
1085:
1004:
969:enharmonic modulation
868:
727:
627:
580:
52:
8479:List of compositions
8308:Generalized keyboard
7803:Easley Blackwood Jr.
7482:Dominic Eckersley: "
7460:Barbieri, Patrizio.
7346:. pp. 181â184.
7258:Sethares, William A.
6609:Ronan, Colin (ed.).
6291:O'Donnell, Michael.
5984:Turkish music theory
4290:improve this article
3953:58 equal temperament
3652:improve this article
3540:(2000) they contain
3467:syntonic temperament
2248:
2131:
1990:
1873:
1695:
1626:
1484:
1358:
1129:
1034:
814:12 equal temperament
689:
590:
550:
477:improve this section
215:, meaning one note,
113:12 equal temperament
8851:Chinese discoveries
8303:Enharmonic keyboard
8254:quarter tone pieces
8248:Beauty in the Beast
7928:Ivan Wyschnegradsky
7452:"Temperament" from
7034:"Equal-temperament"
7032:Monzo, Joe (2005).
6637:]. p. 389.
6557:] (in Chinese).
6489:] (in Chinese).
6367:2010AmJPh..78...47V
4420:For example, where
4401:). That is, fixing
4255:Beauty in the Beast
4054:BohlenâPierce scale
3514:(1949), Indonesian
2244:where in this case
1986:where in this case
1025:twelfth root of two
951:twelfth root of two
935:Francesco Spinacino
529:arithmetic sequence
273:BohlenâPierce scale
82:musical temperament
8846:Equal temperaments
8713:Chinese musicology
8499:Scale of harmonics
8494:Pythagorean tuning
8442:EulerâFokker genus
8225:Genesis of a Music
7853:Christiaan Huygens
7683:Klangfarbenmelodie
7298:on 27 January 2005
7212:Edwin Mellen Press
6831:Genesis of a Music
6816:] (in German).
6631:ćłæ±ç ăç çźäžćźçšæ°ćŠă 389饔
6254:) â 11:8 and 16:11
4990:) there are three
4964:close (unlike the
4856:
4237:(35.1 cents)
4197:(63.8 cents)
4157:(78.0 cents)
3967:approximates many
3920:Pythagorean tuning
3856:Christiaan Huygens
3809:barbershop harmony
3789:quarter-tone scale
3718:
3710:
3433:and 171 cent
3376:
2387:
2234:
2114:
1976:
1820:
1801:
1681:
1595:
1469:
1220:
1080:
1071:
1013:
874:
745:modular arithmetic
722:
639:divides the ratio
622:
575:
525:geometric sequence
445:General properties
88:that approximates
74:
8833:
8832:
8699:
8698:
8321:
8320:
8313:Modernism (music)
8162:
8161:
8063:Equal temperament
7833:Brian Ferneyhough
7745:
7744:
7574:Equal temperament
7519:
7515:
7510:
7506:
7502:
7498:
7353:978-989-8111-60-9
6969:[abstract of
6791:978-0-521-28883-5
6679:Philresound.co.uk
6663:. pp. 80â89.
6573:uts.cc.utexas.edu
6532:on 15 March 2012.
6375:10.1119/1.3226563
6003:Musical acoustics
5167:Since the comma,
5084:means "flat by a
4637:, six in 72
4324:In this section,
4322:
4321:
4314:
3854:was advocated by
3684:
3683:
3676:
3542:stretched octaves
3528:but according to
3366:Approximation of
3289:
3288:
2383:
2379:
2377:
2371:
2358:
2356:
2350:
2326:
2321:
2315:
2278:
2274:
2272:
2266:
2232:
2219:
2213:
2207:
2197:
2195:
2189:
2171:
2165:
2152:
2110:
2106:
2104:
2098:
2079:
2071:
2066:
2060:
2034:
2020:
2016:
2014:
2008:
1974:
1961:
1955:
1949:
1939:
1937:
1931:
1913:
1907:
1894:
1797:
1782:
1777:
1771:
1734:
1730:
1728:
1722:
1713:
1707:
1701:
1691:where in general
1679:
1671:
1663:
1657:
1631:
1594:
1581:
1546:
1542:
1524:
1518:
1505:
1468:
1455:
1420:
1416:
1398:
1392:
1379:
1219:
1196:
1192:
1185:
1176:
1166:
1160:
1134:
1070:
1052:
1045:
981:12-tone technique
805:
804:
721:
717:
712:
706:
694:
621:
617:
610:
595:
574:
555:
513:
512:
505:
198:In modern times,
182:equal temperament
178:Western countries
176:or half step. In
147:logarithmic scale
78:equal temperament
8858:
8672:Well temperament
8658:Regular diatonic
8524:
8523:
8504:Tonality diamond
8348:
8341:
8334:
8325:
8324:
8201:Tonality diamond
8024:repeating scales
8018:
8017:
8003:Nicola Vicentino
7950:
7945:
7913:Nicola Vicentino
7838:Michael Finnissy
7779:Microtonal music
7772:
7765:
7758:
7749:
7748:
7672:Dynamic tonality
7594:Whole tone scale
7547:
7540:
7533:
7524:
7523:
7517:
7513:
7508:
7504:
7500:
7496:
7405:
7402:Internet Archive
7389:978-1-41917893-1
7378:. Translated by
7357:
7330:
7328:
7322:
7316:
7307:
7305:
7303:
7284:
7275:
7253:
7234:
7230:978-0-39306227-4
7215:
7210:. Lewiston, NY:
7202:
7180:
7171:
7161:
7131:
7125:
7124:
7122:
7121:
7105:
7099:
7098:
7096:
7095:
7081:
7075:
7074:
7072:
7071:
7056:
7050:
7049:
7047:
7045:
7029:
7023:
7017:
7011:
7010:
6992:
6986:
6985:
6983:
6982:
6961:
6955:
6954:
6946:æć
ł"äžćčłććŸ"æ°æçźèäœçćç°
6941:
6935:
6930:
6924:
6923:
6905:
6899:
6894:
6888:
6883:
6877:
6876:
6874:
6873:
6866:aredem.online.fr
6860:Cordier, Serge.
6857:
6851:
6850:
6834:
6824:
6818:
6817:
6802:
6796:
6795:
6777:
6771:
6770:
6768:
6766:
6745:
6739:
6738:
6736:
6734:
6719:
6713:
6712:
6700:
6694:
6693:
6691:
6690:
6681:. Archived from
6671:
6665:
6664:
6645:
6639:
6638:
6626:
6620:
6614:
6606:
6600:
6594:
6588:
6587:
6585:
6584:
6565:
6559:
6558:
6540:
6534:
6533:
6528:. Archived from
6509:
6503:
6497:
6491:
6490:
6472:
6463:
6462:
6454:
6445:
6444:
6439:
6429:
6423:
6422:
6408:
6402:
6396:
6387:
6386:
6360:
6338:
6332:
6331:
6326:. Translated by
6310:
6304:
6303:
6301:
6299:
6288:
6282:
6273:
6257:
6249:
6236:
6223:
6210:
6197:
6184:
6171:
6158:
6145:
6132:
6119:
6106:
6096:
6090:
6086:
6080:
6077:
6071:
6065:
6040:Electronic tuner
6020:Microtonal music
5973:
5971:
5966:
5964:
5955:
5953:
5952:
5950:
5949:
5946:
5943:
5925:
5920:
5905:
5889:
5887:
5863:
5861:
5842:
5840:
5828:
5826:
5813:
5811:
5797:
5795:
5794:
5782:
5780:
5778:
5777:
5774:
5771:
5763:
5761:
5757:
5740:
5739:
5732:
5730:
5722:
5720:
5716:
5714:
5713:
5710:
5707:
5692:
5690:
5689:
5675:
5673:
5671:
5670:
5667:
5664:
5652:
5650:
5646:
5629:
5628:
5621:
5619:
5611:
5609:
5605:
5603:
5602:
5599:
5596:
5581:
5579:
5578:
5563:
5561:
5559:
5558:
5555:
5552:
5544:
5542:
5538:
5522:
5518:
5514:
5510:
5506:
5503:
5502:
5494:
5492:
5483:
5482:
5474:
5472:
5471:
5459:
5457:
5452:
5448:
5444:
5442:
5437:
5433:
5427:
5425:
5424:
5416:
5414:
5413:
5402:
5400:
5391:
5389:
5383:
5382:
5381:
5374:
5370:
5367:in the limit as
5366:
5365:
5364:
5357:
5353:
5349:
5347:
5346:
5335:
5331:
5327:
5324:, the semitone,
5322:
5321:
5320:
5312:
5307:
5302:
5301:
5293:
5289:
5265:
5261:
5257:
5250:
5246:
5242:
5238:
5234:
5232:
5223:
5219:
5211:
5209:
5203:
5202:
5193:
5191:
5181:
5179:
5170:
5158:"wolf" intervals
5150:
5149:
5145:
5144:
5137:
5136:
5132:
5131:
5124:
5123:
5119:
5118:
5111:
5110:
5103:
5102:
5095:
5094:
5077:
5076:
5071:
5070:
5063:
5062:
5052:
5050:
5041:
5040:
5033:
5032:
5025:
5024:
5017:
5016:
5009:
5008:
5001:
5000:
4985:
4984:
4977:
4975:
4974:
4966:circle of fifths
4956:
4955:
4937:
4935:
4934:
4932:
4931:
4926:
4923:
4908:
4904:
4896:
4888:
4876:
4875:
4870:
4868:
4867:
4830:
4828:
4827:
4824:
4821:
4814:
4812:
4811:
4808:
4805:
4798:
4797:
4792:
4790:
4789:
4786:
4783:
4776:
4774:
4773:
4770:
4767:
4760:
4759:
4754:
4746:
4745:
4740:
4739:
4734:
4732:
4731:
4727:
4718:
4716:
4715:
4712:
4709:
4702:
4700:
4699:
4696:
4693:
4686:
4685:
4680:
4678:
4677:
4674:
4671:
4664:
4662:
4661:
4658:
4655:
4648:
4647:
4642:
4641:
4636:
4635:
4630:
4626:
4624:
4612:
4610:
4594:
4593:
4585:
4584:
4583:
4579:
4573:
4572:
4564:
4562:
4561:
4557:
4551:
4550:
4543:
4542:
4536:circle of fifths
4533:
4531:
4530:
4528:
4527:
4524:
4521:
4508:
4507:
4506:
4502:
4496:
4494:
4493:
4491:
4490:
4487:
4484:
4471:
4470:
4469:
4465:
4460:
4458:
4457:
4455:
4454:
4451:
4448:
4435:
4434:
4433:
4429:
4423:
4416:
4415:
4404:
4400:
4399:
4392:
4391:
4387:
4386:
4379:
4378:
4371:
4370:
4363:
4362:
4355:
4354:
4340:
4336:
4317:
4310:
4306:
4303:
4297:
4274:
4266:
4248:
4247:
4246:
4244:
4236:
4235:
4234:
4233:
4231:
4230:
4227:
4224:
4208:
4207:
4206:
4204:
4196:
4195:
4194:
4193:
4191:
4190:
4187:
4184:
4168:
4167:
4166:
4164:
4156:
4155:
4154:
4153:
4151:
4150:
4147:
4144:
4102:
4101:
4095:
4094:
4093:
4091:
4079:
4078:
4077:
4075:
4039:
3939:
3938:
3746:
3744:
3742:
3741:
3738:
3735:
3708:
3707:
3706:
3704:
3691:Easley Blackwood
3679:
3672:
3668:
3665:
3659:
3636:
3628:
3619:
3618:
3617:
3600:
3599:
3598:
3596:
3584:
3583:
3582:
3563:
3562:
3561:
3559:
3527:
3525:
3524:
3499:
3497:
3496:
3486:
3484:
3483:
3464:
3463:
3462:
3455:
3454:
3453:
3444:
3443:
3442:
3440:
3432:
3431:
3430:
3428:
3420:
3419:
3418:
3416:
3407:
3406:
3399:
3398:
3397:
3395:
3387:
3386:
3385:
3374:
3373:
3372:
3341:
3337:
3336:
3335:
3334:
3332:
3331:
3328:
3325:
3313:
3312:
3306:for violins and
3305:
3304:
3279:
3275:
3273:
3272:
3269:
3266:
3255:
3250:
3246:
3239:
3238:
3220:
3216:
3214:
3213:
3210:
3207:
3193:
3192:
3191:
3184:
3177:
3176:
3158:
3154:
3152:
3151:
3148:
3145:
3131:
3130:
3129:
3122:
3115:
3114:
3110:
3109:
3091:
3087:
3085:
3084:
3081:
3078:
3064:
3063:
3062:
3055:
3048:
3047:
3029:
3025:
3023:
3022:
3019:
3016:
3002:
3001:
3000:
2993:
2986:
2985:
2981:
2980:
2962:
2958:
2956:
2955:
2952:
2949:
2935:
2934:
2933:
2926:
2919:
2918:
2900:
2896:
2894:
2893:
2890:
2887:
2873:
2872:
2871:
2864:
2857:
2856:
2852:
2851:
2833:
2829:
2827:
2826:
2823:
2820:
2806:
2805:
2804:
2797:
2790:
2789:
2782:Perfect fourth (
2771:
2767:
2765:
2764:
2761:
2758:
2744:
2743:
2742:
2735:
2728:
2727:
2709:
2705:
2703:
2702:
2699:
2696:
2682:
2681:
2680:
2673:
2666:
2665:
2661:
2660:
2642:
2638:
2636:
2635:
2632:
2629:
2615:
2614:
2613:
2606:
2599:
2598:
2580:
2576:
2574:
2573:
2570:
2567:
2553:
2552:
2551:
2544:
2537:
2536:
2532:
2531:
2513:
2509:
2507:
2506:
2503:
2500:
2486:
2482:
2475:
2474:
2459:
2458:
2442:
2441:
2434:
2433:
2423:
2422:
2410:
2409:
2396:
2394:
2393:
2388:
2381:
2380:
2378:
2375:
2369:
2364:
2359:
2357:
2354:
2348:
2343:
2338:
2337:
2331:
2327:
2322:
2319:
2313:
2311:
2302:
2301:
2289:
2288:
2276:
2275:
2273:
2270:
2264:
2259:
2243:
2241:
2240:
2235:
2230:
2229:
2228:
2217:
2211:
2205:
2204:
2203:
2202:
2198:
2196:
2193:
2187:
2182:
2169:
2163:
2162:
2161:
2150:
2143:
2142:
2123:
2121:
2120:
2115:
2108:
2107:
2105:
2102:
2096:
2091:
2086:
2085:
2077:
2076:
2072:
2067:
2064:
2058:
2056:
2047:
2046:
2032:
2031:
2030:
2018:
2017:
2015:
2012:
2006:
2001:
1985:
1983:
1982:
1977:
1972:
1971:
1970:
1959:
1953:
1947:
1946:
1945:
1944:
1940:
1938:
1935:
1929:
1924:
1911:
1905:
1904:
1903:
1892:
1885:
1884:
1862:
1861:
1857:
1856:
1846:
1845:
1838:
1829:
1810:
1808:
1807:
1802:
1795:
1794:
1793:
1787:
1783:
1778:
1775:
1769:
1767:
1758:
1757:
1745:
1744:
1732:
1731:
1729:
1726:
1720:
1715:
1711:
1705:
1699:
1690:
1688:
1687:
1682:
1677:
1676:
1675:
1669:
1661:
1655:
1654:
1653:
1641:
1640:
1629:
1618:
1617:
1604:
1602:
1601:
1596:
1592:
1591:
1590:
1579:
1572:
1571:
1554:
1553:
1544:
1543:
1541:
1533:
1531:
1530:
1522:
1516:
1515:
1514:
1503:
1496:
1495:
1478:
1476:
1475:
1470:
1466:
1465:
1464:
1453:
1446:
1445:
1428:
1427:
1418:
1417:
1415:
1407:
1405:
1404:
1396:
1390:
1389:
1388:
1377:
1370:
1369:
1347:
1346:
1342:
1341:
1331:
1330:
1320:
1319:
1315:
1314:
1306:
1305:
1293:
1292:
1278:
1277:
1270:
1266:
1262:
1258:
1254:
1241:
1233:In this formula
1229:
1227:
1226:
1221:
1217:
1216:
1215:
1204:
1203:
1194:
1193:
1191:
1186:
1183:
1178:
1174:
1173:
1172:
1164:
1158:
1157:
1156:
1144:
1143:
1132:
1121:
1120:
1115:
1089:
1087:
1086:
1081:
1073:
1072:
1063:
1053:
1051:
1046:
1043:
1038:
1010:
1009:
963:
960:
948:
947:
931:Giacomo Gorzanis
927:Vincenzo Galilei
913:
911:
909:
908:
900:
898:
896:
895:
884:
883:
882:
835:
834:
800:
797:
791:
775:
767:
754:
737:integer notation
731:
729:
728:
723:
719:
718:
713:
710:
704:
702:
692:
681:
677:
673:
646:
642:
638:
635:where the ratio
631:
629:
628:
623:
619:
618:
616:
611:
608:
603:
593:
584:
582:
581:
576:
572:
565:
564:
553:
508:
501:
497:
494:
488:
457:
449:
438:
436:
435:
432:
429:
422:
420:
419:
416:
413:
406:
404:
403:
400:
397:
390:
388:
387:
384:
381:
374:
372:
371:
368:
365:
358:
357:
356:
349:
348:
347:
334:
299:string ensembles
292:
291:
268:
266:
265:
257:Arab tone system
253:
252:
251:
242:
241:
240:
223:
222:
206:
205:
204:
193:
192:
191:
171:
169:
168:
165:
162:
155:
154:
139:
138:
137:
128:
127:
126:
72:
71:
70:
68:
60:
59:
25:
8866:
8865:
8861:
8860:
8859:
8857:
8856:
8855:
8836:
8835:
8834:
8829:
8826:(BohlenâPierce)
8794:833 cents scale
8782:
8705:
8695:
8662:
8615:
8513:
8434:Just intonation
8428:
8357:
8355:Musical tunings
8352:
8322:
8317:
8291:
8235:
8211:
8205:
8168:
8158:
8154:Double diatonic
8141:Just intonation
8135:
8057:
8023:
8013:
8007:
7951:
7937:
7883:Joel Mandelbaum
7813:JuliĂĄn Carrillo
7793:Richard Barrett
7781:
7776:
7746:
7741:
7725:
7649:
7643:
7604:
7598:
7589:Octatonic scale
7566:
7560:
7551:
7424:
7406:
7390:
7365:
7363:Further reading
7360:
7354:
7323:
7317:
7301:
7299:
7286:
7272:
7250:
7231:
7195:Ethnomusicology
7188:
7183:
7132:
7128:
7119:
7117:
7114:wendycarlos.com
7108:Carlos, Wendy.
7106:
7102:
7093:
7091:
7083:
7082:
7078:
7069:
7067:
7058:
7057:
7053:
7043:
7041:
7030:
7026:
7020:Sethares (2005)
7018:
7014:
7007:
6993:
6989:
6980:
6978:
6967:
6963:
6962:
6958:
6947:
6943:
6942:
6938:
6931:
6927:
6920:
6906:
6902:
6895:
6891:
6884:
6880:
6871:
6869:
6858:
6854:
6847:
6825:
6821:
6803:
6799:
6792:
6780:Lindley, Mark.
6778:
6774:
6764:
6762:
6761:on 17 July 2011
6746:
6742:
6732:
6730:
6729:on 25 July 2011
6721:
6720:
6716:
6709:Lute tabulation
6701:
6697:
6688:
6686:
6673:
6672:
6668:
6661:Girolamo Scotto
6646:
6642:
6632:
6627:
6623:
6607:
6603:
6595:
6591:
6582:
6580:
6567:
6566:
6562:
6552:
6548:LÇ lĂŹ rĂłng tĆng
6541:
6537:
6510:
6506:
6498:
6494:
6484:
6480:YuĂš lÇ quĂĄn shĆ«
6473:
6466:
6455:
6448:
6437:
6430:
6426:
6414:Ethnomusicology
6409:
6405:
6397:
6390:
6339:
6335:
6311:
6307:
6297:
6295:
6289:
6285:
6276:Sethares (2005)
6274:
6270:
6266:
6261:
6260:
6245:
6232:
6228:) â 6:5 and 5:3
6219:
6215:) â 5:4 and 8:5
6206:
6193:
6180:
6167:
6154:
6141:
6128:
6115:
6102:
6097:
6093:
6087:
6083:
6078:
6074:
6068:Sethares (2005)
6066:
6059:
6054:
6049:
6005:
5998:Just intonation
5993:
5980:just intonation
5969:
5967:
5959:
5957:
5947:
5944:
5941:
5940:
5938:
5933:
5931:
5907:
5892:
5891:
5867:
5865:
5845:
5844:
5832:
5830:
5817:
5815:
5803:
5801:
5792:
5791:
5788:
5775:
5772:
5769:
5768:
5766:
5765:
5759:
5744:
5743:
5737:
5736:
5725:
5724:
5711:
5708:
5705:
5704:
5702:
5697:
5696:
5687:
5686:
5683:
5668:
5665:
5662:
5661:
5659:
5658:
5648:
5633:
5632:
5626:
5625:
5614:
5613:
5600:
5597:
5594:
5593:
5591:
5586:
5585:
5576:
5575:
5572:
5565:meantone system
5556:
5553:
5550:
5549:
5547:
5546:
5540:
5525:
5524:
5520:
5516:
5512:
5508:
5500:
5499:
5496:
5487:
5485:
5484:(in cents) and
5480:
5478:
5469:
5468:
5465:
5455:
5454:
5450:
5446:
5440:
5439:
5435:
5431:
5422:
5421:
5418:
5411:
5410:
5407:
5395:
5393:
5387:
5385:
5379:
5378:
5376:
5372:
5368:
5362:
5361:
5359:
5355:
5351:
5344:
5343:
5340:
5333:
5329:
5325:
5318:
5317:
5315:
5310:
5305:
5296:
5295:
5291:
5287:
5276:
5263:
5259:
5255:
5248:
5244:
5243:, or into both
5240:
5236:
5230:
5228:
5221:
5217:
5207:
5205:
5200:
5199:
5189:
5187:
5177:
5175:
5168:
5147:
5146:
5140:
5139:
5134:
5133:
5127:
5126:
5121:
5120:
5114:
5113:
5106:
5105:
5098:
5097:
5090:
5089:
5074:
5073:
5066:
5065:
5058:
5057:
5048:
5047:
5036:
5035:
5028:
5027:
5020:
5019:
5012:
5011:
5004:
5003:
4996:
4995:
4980:
4979:
4972:
4971:
4969:
4943:
4942:
4927:
4924:
4919:
4918:
4916:
4911:
4910:
4906:
4902:
4894:
4886:
4873:
4872:
4865:
4864:
4862:
4842:
4837:
4825:
4822:
4819:
4818:
4816:
4809:
4806:
4803:
4802:
4800:
4795:
4794:
4787:
4784:
4781:
4780:
4778:
4771:
4768:
4765:
4764:
4762:
4757:
4756:
4752:
4743:
4742:
4737:
4736:
4729:
4728:
4725:
4723:
4713:
4710:
4707:
4706:
4704:
4697:
4694:
4691:
4690:
4688:
4683:
4682:
4675:
4672:
4669:
4668:
4666:
4659:
4656:
4653:
4652:
4650:
4645:
4644:
4639:
4638:
4633:
4632:
4628:
4615:
4614:
4601:
4600:
4588:
4587:
4581:
4580:
4577:
4575:
4567:
4566:
4559:
4558:
4555:
4553:
4546:
4545:
4540:
4539:
4525:
4522:
4519:
4518:
4516:
4511:
4510:
4504:
4503:
4500:
4498:
4488:
4485:
4482:
4481:
4479:
4474:
4473:
4467:
4466:
4463:
4461:
4452:
4449:
4446:
4445:
4443:
4438:
4437:
4431:
4430:
4427:
4425:
4424:is an integer,
4421:
4407:
4406:
4402:
4395:
4394:
4389:
4388:
4382:
4381:
4374:
4373:
4366:
4365:
4358:
4357:
4350:
4349:
4346:proper fraction
4338:
4334:
4318:
4307:
4301:
4298:
4287:
4275:
4264:
4242:
4239:
4238:
4228:
4225:
4222:
4221:
4219:
4218:
4217:
4215:
4202:
4199:
4198:
4188:
4185:
4182:
4181:
4179:
4178:
4177:
4175:
4162:
4159:
4158:
4148:
4145:
4142:
4141:
4139:
4138:
4137:
4135:
4099:
4097:
4089:
4086:
4085:
4073:
4070:
4069:
4050:
4035:
4026:
3994:JuliĂĄn Carrillo
3977:just intonation
3969:just intonation
3936:
3935:
3925:
3806:
3739:
3736:
3733:
3732:
3730:
3729:
3702:
3699:
3698:
3680:
3669:
3663:
3660:
3649:
3637:
3626:
3615:
3614:
3612:
3594:
3591:
3590:
3580:
3579:
3577:
3570:
3557:
3554:
3553:
3522:
3521:
3519:
3508:
3494:
3493:
3491:
3481:
3480:
3478:
3460:
3459:
3457:
3451:
3450:
3448:
3438:
3435:
3434:
3426:
3423:
3422:
3414:
3411:
3410:
3404:
3403:
3393:
3390:
3389:
3383:
3382:
3380:
3370:
3369:
3367:
3360:
3355:
3349:
3339:
3329:
3326:
3323:
3322:
3320:
3319:
3318:
3316:
3308:
3307:
3300:
3299:
3296:
3277:
3270:
3267:
3264:
3263:
3261:
3253:
3248:
3244:
3234:
3233:
3218:
3211:
3208:
3205:
3204:
3202:
3189:
3188:
3186:
3182:
3172:
3171:
3169:Major seventh (
3156:
3149:
3146:
3143:
3142:
3140:
3127:
3126:
3124:
3120:
3112:
3111:
3105:
3104:
3102:Minor seventh (
3089:
3082:
3079:
3076:
3075:
3073:
3060:
3059:
3057:
3053:
3043:
3042:
3027:
3020:
3017:
3014:
3013:
3011:
2998:
2997:
2995:
2991:
2983:
2982:
2976:
2975:
2960:
2953:
2950:
2947:
2946:
2944:
2931:
2930:
2928:
2924:
2914:
2913:
2911:Perfect fifth (
2898:
2891:
2888:
2885:
2884:
2882:
2869:
2868:
2866:
2862:
2854:
2853:
2847:
2846:
2831:
2824:
2821:
2818:
2817:
2815:
2802:
2801:
2799:
2795:
2785:
2784:
2769:
2762:
2759:
2756:
2755:
2753:
2740:
2739:
2737:
2733:
2723:
2722:
2707:
2700:
2697:
2694:
2693:
2691:
2678:
2677:
2675:
2671:
2663:
2662:
2656:
2655:
2640:
2633:
2630:
2627:
2626:
2624:
2611:
2610:
2608:
2604:
2594:
2593:
2578:
2571:
2568:
2565:
2564:
2562:
2549:
2548:
2546:
2542:
2534:
2533:
2527:
2526:
2511:
2504:
2501:
2498:
2497:
2495:
2484:
2480:
2470:
2469:
2461:
2456:
2455:
2439:
2438:
2431:
2430:
2413:just intonation
2407:
2406:
2403:
2368:
2363:
2347:
2342:
2333:
2332:
2312:
2310:
2306:
2297:
2293:
2284:
2283:
2263:
2258:
2249:
2246:
2245:
2221:
2220:
2186:
2181:
2177:
2176:
2172:
2154:
2153:
2138:
2134:
2132:
2129:
2128:
2095:
2090:
2081:
2080:
2057:
2055:
2051:
2042:
2038:
2026:
2025:
2005:
2000:
1991:
1988:
1987:
1963:
1962:
1928:
1923:
1919:
1918:
1914:
1896:
1895:
1880:
1876:
1874:
1871:
1870:
1865:
1859:
1858:
1852:
1851:
1849:
1841:
1840:
1836:
1831:
1827:
1822:
1789:
1788:
1768:
1766:
1762:
1753:
1749:
1740:
1739:
1719:
1714:
1696:
1693:
1692:
1668:
1664:
1649:
1645:
1636:
1632:
1627:
1624:
1623:
1615:
1614:
1611:
1583:
1582:
1555:
1549:
1548:
1547:
1537:
1532:
1526:
1525:
1507:
1506:
1491:
1487:
1485:
1482:
1481:
1457:
1456:
1429:
1423:
1422:
1421:
1411:
1406:
1400:
1399:
1381:
1380:
1365:
1361:
1359:
1356:
1355:
1350:
1344:
1343:
1337:
1336:
1334:
1326:
1325:
1323:
1317:
1316:
1310:
1309:
1301:
1300:
1296:
1288:
1287:
1281:
1273:
1272:
1268:
1264:
1260:
1256:
1252:
1247:
1239:
1234:
1205:
1199:
1198:
1197:
1187:
1179:
1177:
1168:
1167:
1152:
1148:
1139:
1135:
1130:
1127:
1126:
1118:
1117:
1113:
1108:
1105:
1099:
1061:
1057:
1047:
1039:
1037:
1035:
1032:
1031:
1021:frequency ratio
1007:
1006:
997:
961:
945:
944:
923:
906:
904:
903:
902:
893:
891:
890:
889:
880:
879:
877:
863:
824:
816:
810:
801:
795:
792:
785:
776:
765:
752:
703:
701:
690:
687:
686:
679:
675:
671:
668:ethnomusicology
644:
640:
636:
612:
604:
602:
591:
588:
587:
560:
556:
551:
548:
547:
509:
498:
492:
489:
474:
458:
447:
442:
441:
440:
433:
430:
427:
426:
424:
417:
414:
411:
410:
408:
401:
398:
395:
394:
392:
385:
382:
379:
378:
376:
369:
366:
363:
362:
360:
354:
353:
351:
345:
344:
342:
340:
335:
307:just intonation
289:
288:
263:
262:
260:
249:
248:
246:
238:
237:
235:
218:
217:
202:
201:
199:
189:
188:
186:
166:
163:
160:
159:
157:
152:
150:
135:
134:
132:
124:
123:
121:
115:(also known as
109:classical music
66:
63:
62:
55:
54:
47:
46:
45:
31:
26:
17:
12:
11:
5:
8864:
8854:
8853:
8848:
8831:
8830:
8828:
8827:
8821:
8816:
8811:
8806:
8801:
8796:
8790:
8788:
8784:
8783:
8781:
8780:
8775:
8770:
8760:
8755:
8750:
8749:
8748:
8743:
8738:
8733:
8725:
8720:
8715:
8709:
8707:
8701:
8700:
8697:
8696:
8670:
8668:
8664:
8663:
8661:
8660:
8655:
8650:
8645:
8640:
8625:
8623:
8617:
8616:
8614:
8613:
8608:
8603:
8598:
8593:
8588:
8583:
8578:
8568:
8563:
8558:
8553:
8548:
8543:
8538:
8532:
8530:
8521:
8515:
8514:
8512:
8511:
8506:
8501:
8496:
8491:
8486:
8481:
8476:
8475:
8474:
8469:
8459:
8454:
8449:
8447:Harmonic scale
8444:
8438:
8436:
8430:
8429:
8427:
8426:
8421:
8416:
8411:
8406:
8401:
8396:
8394:Interval ratio
8391:
8386:
8381:
8376:
8371:
8365:
8363:
8359:
8358:
8351:
8350:
8343:
8336:
8328:
8319:
8318:
8316:
8315:
8310:
8305:
8299:
8297:
8293:
8292:
8290:
8289:
8282:
8275:
8268:
8261:
8256:
8251:
8243:
8241:
8237:
8236:
8234:
8233:
8228:
8221:
8215:
8213:
8207:
8206:
8204:
8203:
8198:
8196:Xenharmonicity
8193:
8188:
8183:
8178:
8172:
8170:
8164:
8163:
8160:
8159:
8157:
8156:
8151:
8145:
8143:
8137:
8136:
8134:
8133:
8128:
8123:
8118:
8113:
8108:
8103:
8098:
8093:
8088:
8083:
8078:
8073:
8067:
8065:
8059:
8058:
8056:
8055:
8049:
8044:
8039:
8034:
8028:
8026:
8015:
8009:
8008:
8006:
8005:
8000:
7995:
7990:
7985:
7980:
7978:Adriaan Fokker
7975:
7970:
7965:
7959:
7957:
7953:
7952:
7940:
7938:
7936:
7935:
7933:La Monte Young
7930:
7925:
7920:
7915:
7910:
7905:
7900:
7898:John Schneider
7895:
7890:
7885:
7880:
7875:
7870:
7865:
7860:
7855:
7850:
7845:
7843:BjĂžrn Fongaard
7840:
7835:
7830:
7825:
7823:Mildred Couper
7820:
7815:
7810:
7805:
7800:
7795:
7789:
7787:
7783:
7782:
7775:
7774:
7767:
7760:
7752:
7743:
7742:
7740:
7739:
7733:
7731:
7727:
7726:
7724:
7723:
7718:
7713:
7708:
7703:
7702:
7701:
7699:Distance model
7691:
7686:
7679:
7674:
7669:
7664:
7659:
7653:
7651:
7645:
7644:
7642:
7641:
7639:Spectral music
7636:
7631:
7626:
7625:
7624:
7619:
7608:
7606:
7600:
7599:
7597:
7596:
7591:
7586:
7581:
7576:
7570:
7568:
7562:
7561:
7550:
7549:
7542:
7535:
7527:
7521:
7520:
7492:
7487:
7480:
7475:
7465:
7458:
7449:
7444:
7439:
7434:
7423:
7422:External links
7420:
7419:
7418:
7404:(archive.org).
7388:
7364:
7361:
7359:
7358:
7352:
7331:
7308:
7276:
7270:
7254:
7248:
7235:
7229:
7216:
7203:
7189:
7187:
7184:
7182:
7181:
7136:Sethares, W.A.
7126:
7116:. Serendip LLC
7100:
7076:
7051:
7024:
7012:
7005:
6999:. p. 55.
6987:
6956:
6953:on 2007-10-27.
6936:
6925:
6918:
6900:
6889:
6878:
6852:
6845:
6819:
6797:
6790:
6772:
6740:
6714:
6695:
6666:
6640:
6629:Hanson, Lau.
6621:
6601:
6589:
6560:
6535:
6504:
6500:Kuttner (1975)
6492:
6464:
6446:
6424:
6403:
6399:Kuttner (1975)
6388:
6333:
6305:
6283:
6267:
6265:
6262:
6259:
6258:
6256:
6255:
6242:
6229:
6216:
6203:
6190:
6177:
6164:
6151:
6138:
6125:
6112:
6091:
6081:
6072:
6056:
6055:
6053:
6050:
6048:
6047:
6045:Musical tuning
6042:
6037:
6032:
6027:
6022:
6017:
6012:
6007:
6000:
5994:
5992:
5989:
5988:
5987:
5976:syntonic comma
5798:
5785:
5784:
5733:the result is
5693:
5680:
5679:
5622:the result is
5582:
5569:
5568:
5495:the result is
5475:
5462:
5461:
5428:
5403:For instance:
5375:tend to zero;
5275:
5272:
5214:circular shift
5171:, expands the
5162:diatonic comma
4992:perfect fifths
4879:circular shift
4850:Figure 1: The
4841:
4838:
4836:
4833:
4320:
4319:
4278:
4276:
4269:
4263:
4260:
4250:
4249:
4209:
4169:
4049:
4046:
4024:
3998:
3997:
3986:
3982:
3981:
3962:
3958:
3957:
3950:
3946:
3945:
3923:
3916:perfect fifths
3905:
3901:
3900:
3896:
3892:
3891:
3884:
3880:
3879:
3872:
3868:
3867:
3860:Adriaan Fokker
3849:
3845:
3844:
3833:
3829:
3828:
3825:syntonic comma
3821:septimal comma
3817:
3813:
3812:
3804:
3801:
3797:
3796:
3782:
3778:
3777:
3770:
3766:
3765:
3758:
3754:
3753:
3750:perfect fourth
3722:
3682:
3681:
3640:
3638:
3631:
3625:
3622:
3602:
3601:
3569:
3566:
3507:
3504:
3503:
3502:
3488:
3359:
3356:
3348:
3345:
3295:
3292:
3291:
3290:
3287:
3286:
3283:
3280:
3259:
3256:
3251:
3242:
3228:
3227:
3224:
3221:
3200:
3197:
3194:
3180:
3166:
3165:
3162:
3159:
3138:
3135:
3132:
3118:
3099:
3098:
3095:
3092:
3071:
3068:
3065:
3051:
3037:
3036:
3033:
3030:
3009:
3006:
3003:
2989:
2970:
2969:
2966:
2963:
2942:
2939:
2936:
2922:
2908:
2907:
2904:
2901:
2880:
2877:
2874:
2860:
2841:
2840:
2837:
2834:
2813:
2810:
2807:
2793:
2779:
2778:
2775:
2772:
2751:
2748:
2745:
2731:
2717:
2716:
2713:
2710:
2689:
2686:
2683:
2669:
2650:
2649:
2646:
2643:
2622:
2619:
2616:
2602:
2591:Major second (
2588:
2587:
2584:
2581:
2560:
2557:
2554:
2540:
2524:Minor second (
2521:
2520:
2517:
2514:
2493:
2490:
2487:
2478:
2464:
2463:
2452:
2449:
2446:
2443:
2435:
2427:
2426:Interval Name
2402:
2399:
2398:
2397:
2386:
2374:
2367:
2362:
2353:
2346:
2341:
2336:
2330:
2325:
2318:
2309:
2305:
2300:
2296:
2292:
2287:
2281:
2269:
2262:
2257:
2254:
2227:
2224:
2216:
2210:
2201:
2192:
2185:
2180:
2175:
2168:
2160:
2157:
2149:
2146:
2141:
2137:
2125:
2124:
2113:
2101:
2094:
2089:
2084:
2075:
2070:
2063:
2054:
2050:
2045:
2041:
2037:
2029:
2023:
2011:
2004:
1999:
1996:
1969:
1966:
1958:
1952:
1943:
1934:
1927:
1922:
1917:
1910:
1902:
1899:
1891:
1888:
1883:
1879:
1863:
1847:
1834:
1825:
1812:
1811:
1800:
1792:
1786:
1781:
1774:
1765:
1761:
1756:
1752:
1748:
1743:
1737:
1725:
1718:
1710:
1704:
1674:
1667:
1660:
1652:
1648:
1644:
1639:
1635:
1610:
1607:
1606:
1605:
1589:
1586:
1578:
1575:
1570:
1567:
1564:
1561:
1558:
1552:
1540:
1536:
1529:
1521:
1513:
1510:
1502:
1499:
1494:
1490:
1479:
1463:
1460:
1452:
1449:
1444:
1441:
1438:
1435:
1432:
1426:
1414:
1410:
1403:
1395:
1387:
1384:
1376:
1373:
1368:
1364:
1348:
1332:
1321:
1294:
1279:
1250:
1237:
1231:
1230:
1214:
1211:
1208:
1202:
1190:
1182:
1171:
1163:
1155:
1151:
1147:
1142:
1138:
1111:
1098:
1095:
1091:
1090:
1079:
1076:
1069:
1066:
1060:
1056:
1050:
1042:
1011:on a monochord
996:
993:
922:
919:
862:
859:
854:
853:
836:) in 1584 and
823:
820:
812:Main article:
809:
806:
803:
802:
779:
777:
770:
764:
761:
733:
732:
716:
709:
700:
697:
647:equal parts. (
633:
632:
615:
607:
601:
598:
585:
571:
568:
563:
559:
511:
510:
461:
459:
452:
446:
443:
337:
336:
329:
328:
327:
209:standard pitch
90:just intervals
28:
27:
20:
19:
18:
15:
9:
6:
4:
3:
2:
8863:
8852:
8849:
8847:
8844:
8843:
8841:
8825:
8822:
8820:
8817:
8815:
8812:
8810:
8807:
8805:
8802:
8800:
8797:
8795:
8792:
8791:
8789:
8785:
8779:
8776:
8774:
8771:
8768:
8767:Carnatic raga
8764:
8761:
8759:
8756:
8754:
8751:
8747:
8744:
8742:
8739:
8737:
8736:Turkish makam
8734:
8732:
8729:
8728:
8726:
8724:
8721:
8719:
8716:
8714:
8711:
8710:
8708:
8702:
8693:
8689:
8685:
8681:
8677:
8673:
8669:
8665:
8659:
8656:
8654:
8651:
8649:
8646:
8644:
8641:
8638:
8634:
8633:quarter-comma
8630:
8627:
8626:
8624:
8622:
8618:
8612:
8609:
8607:
8604:
8602:
8599:
8597:
8594:
8592:
8589:
8587:
8584:
8582:
8579:
8576:
8572:
8569:
8567:
8564:
8562:
8559:
8557:
8554:
8552:
8549:
8547:
8544:
8542:
8539:
8537:
8534:
8533:
8531:
8529:
8525:
8522:
8520:
8516:
8510:
8509:Tonality flux
8507:
8505:
8502:
8500:
8497:
8495:
8492:
8490:
8487:
8485:
8482:
8480:
8477:
8473:
8470:
8468:
8465:
8464:
8463:
8460:
8458:
8455:
8453:
8450:
8448:
8445:
8443:
8440:
8439:
8437:
8435:
8431:
8425:
8422:
8420:
8417:
8415:
8412:
8410:
8407:
8405:
8402:
8400:
8397:
8395:
8392:
8390:
8387:
8385:
8382:
8380:
8377:
8375:
8372:
8370:
8367:
8366:
8364:
8360:
8356:
8349:
8344:
8342:
8337:
8335:
8330:
8329:
8326:
8314:
8311:
8309:
8306:
8304:
8301:
8300:
8298:
8294:
8288:
8287:
8283:
8281:
8280:
8276:
8274:
8273:
8269:
8267:
8266:
8262:
8260:
8257:
8255:
8252:
8250:
8249:
8245:
8244:
8242:
8238:
8232:
8229:
8227:
8226:
8222:
8220:
8217:
8216:
8214:
8208:
8202:
8199:
8197:
8194:
8192:
8189:
8187:
8184:
8182:
8179:
8177:
8174:
8173:
8171:
8165:
8155:
8152:
8150:
8147:
8146:
8144:
8142:
8138:
8132:
8129:
8127:
8124:
8122:
8119:
8117:
8114:
8112:
8109:
8107:
8104:
8102:
8099:
8097:
8094:
8092:
8089:
8087:
8084:
8082:
8079:
8077:
8074:
8072:
8069:
8068:
8066:
8064:
8060:
8053:
8050:
8048:
8045:
8043:
8040:
8038:
8035:
8033:
8030:
8029:
8027:
8025:
8019:
8016:
8010:
8004:
8001:
7999:
7996:
7994:
7991:
7989:
7986:
7984:
7981:
7979:
7976:
7974:
7971:
7969:
7966:
7964:
7961:
7960:
7958:
7954:
7949:
7944:
7934:
7931:
7929:
7926:
7924:
7923:Elaine Walker
7921:
7919:
7918:Claude Vivier
7916:
7914:
7911:
7909:
7906:
7904:
7901:
7899:
7896:
7894:
7893:Roger Redgate
7891:
7889:
7886:
7884:
7881:
7879:
7876:
7874:
7873:Stu Mackenzie
7871:
7869:
7868:György Ligeti
7866:
7864:
7861:
7859:
7856:
7854:
7851:
7849:
7846:
7844:
7841:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7806:
7804:
7801:
7799:
7796:
7794:
7791:
7790:
7788:
7784:
7780:
7773:
7768:
7766:
7761:
7759:
7754:
7753:
7750:
7738:
7735:
7734:
7732:
7728:
7722:
7721:Unified field
7719:
7717:
7714:
7712:
7709:
7707:
7704:
7700:
7697:
7696:
7695:
7692:
7690:
7687:
7685:
7684:
7680:
7678:
7675:
7673:
7670:
7668:
7665:
7663:
7660:
7658:
7655:
7654:
7652:
7646:
7640:
7637:
7635:
7632:
7630:
7627:
7623:
7620:
7618:
7615:
7614:
7613:
7610:
7609:
7607:
7601:
7595:
7592:
7590:
7587:
7585:
7582:
7580:
7577:
7575:
7572:
7571:
7569:
7563:
7559:
7558:post-tonality
7555:
7548:
7543:
7541:
7536:
7534:
7529:
7528:
7525:
7511:
7493:
7491:
7488:
7485:
7481:
7479:
7476:
7473:
7469:
7466:
7463:
7459:
7457:
7455:
7450:
7448:
7445:
7443:
7440:
7438:
7435:
7433:
7429:
7426:
7425:
7416:
7415:
7410:
7403:
7399:
7395:
7391:
7385:
7381:
7377:
7376:
7371:
7370:Helmholtz, H.
7367:
7366:
7355:
7349:
7345:
7341:
7337:
7332:
7327:
7326:"Alt. link 2"
7320:
7319:"Alt. link 1"
7314:
7309:
7297:
7293:
7289:
7282:
7277:
7273:
7271:1-85233-797-4
7267:
7263:
7259:
7255:
7251:
7249:0-87013-290-3
7245:
7241:
7236:
7232:
7226:
7222:
7217:
7213:
7209:
7204:
7200:
7196:
7191:
7190:
7179:
7175:
7169:
7165:
7160:
7155:
7151:
7147:
7146:
7141:
7137:
7130:
7115:
7111:
7104:
7090:
7086:
7080:
7066:on 2015-11-18
7065:
7061:
7055:
7039:
7035:
7028:
7021:
7016:
7008:
7006:9780542998478
7002:
6998:
6991:
6977:on 2007-09-30
6976:
6972:
6968:
6960:
6952:
6948:
6940:
6934:
6933:Boiles (1969)
6929:
6921:
6919:0-520-04778-8
6915:
6911:
6904:
6898:
6897:Morton (1980)
6893:
6887:
6882:
6867:
6863:
6856:
6848:
6846:0-306-80106-X
6842:
6838:
6833:
6832:
6823:
6815:
6811:
6807:
6801:
6793:
6787:
6783:
6776:
6760:
6756:
6755:
6750:
6749:Stevin, Simon
6744:
6728:
6724:
6718:
6710:
6706:
6699:
6685:on 2012-03-24
6684:
6680:
6676:
6670:
6662:
6658:
6654:
6650:
6644:
6636:
6625:
6618:
6612:
6605:
6598:
6593:
6579:on 2012-03-05
6578:
6574:
6570:
6564:
6556:
6549:
6545:
6539:
6531:
6527:
6523:
6519:
6515:
6508:
6502:, p. 200
6501:
6496:
6488:
6481:
6477:
6471:
6469:
6460:
6453:
6451:
6443:
6435:
6428:
6421:(2): 163â206.
6420:
6416:
6415:
6407:
6401:, p. 163
6400:
6395:
6393:
6384:
6380:
6376:
6372:
6368:
6364:
6359:
6354:
6350:
6346:
6345:
6337:
6329:
6325:
6324:
6319:
6315:
6314:Helmholtz, H.
6309:
6294:
6287:
6281:
6277:
6272:
6268:
6253:
6248:
6243:
6240:
6235:
6230:
6227:
6222:
6217:
6214:
6209:
6204:
6201:
6196:
6191:
6188:
6183:
6178:
6175:
6170:
6165:
6162:
6157:
6152:
6149:
6144:
6139:
6136:
6131:
6126:
6123:
6118:
6113:
6110:
6105:
6100:
6099:
6095:
6085:
6076:
6069:
6064:
6062:
6057:
6046:
6043:
6041:
6038:
6036:
6033:
6031:
6028:
6026:
6023:
6021:
6018:
6016:
6013:
6011:
6008:
6004:
6001:
5999:
5996:
5995:
5985:
5981:
5977:
5962:
5936:
5929:
5928:53 steps
5924:
5919:
5915:
5911:
5904:
5900:
5896:
5886:
5882:
5878:
5874:
5870:
5860:
5856:
5852:
5848:
5839:
5835:
5824:
5820:
5810:
5806:
5799:
5796:
5787:
5786:
5756:
5752:
5748:
5741:
5728:
5719:
5700:
5694:
5691:
5682:
5681:
5677:
5655:
5654:31 steps
5645:
5641:
5637:
5630:
5617:
5608:
5589:
5583:
5580:
5571:
5570:
5566:
5537:
5533:
5529:
5504:
5490:
5476:
5473:
5464:
5463:
5456:t t t t t t t
5429:
5426:
5415:
5406:
5405:
5404:
5398:
5348:
5337:
5323:
5308:
5299:
5285:
5281:
5271:
5269:
5252:
5227:
5215:
5201:T t s T t T s
5197:
5185:
5174:
5165:
5163:
5159:
5154:
5143:
5130:
5117:
5109:
5101:
5093:
5087:
5083:
5082:
5069:
5061:
5056:
5045:
5039:
5031:
5023:
5015:
5007:
4999:
4993:
4989:
4983:
4967:
4963:
4958:
4954:
4950:
4946:
4941:
4930:
4922:
4914:
4905:). The comma
4900:
4892:
4884:
4880:
4874:T t s T t T s
4861:
4853:
4848:
4844:
4832:
4749:
4720:
4623:
4619:
4609:
4605:
4596:
4591:
4570:
4549:
4537:
4514:
4477:
4441:
4418:
4414:
4410:
4398:
4385:
4377:
4369:
4361:
4353:
4347:
4342:
4331:
4327:
4316:
4313:
4305:
4295:
4291:
4285:
4284:
4279:This section
4277:
4273:
4268:
4267:
4259:
4257:
4256:
4245:
4213:
4210:
4205:
4173:
4170:
4165:
4133:
4130:
4129:
4128:
4126:
4125:
4120:
4119:
4114:
4113:
4108:
4104:
4092:
4083:
4076:
4067:
4063:
4059:
4058:perfect fifth
4055:
4045:
4043:
4038:
4032:
4030:
4022:
4018:
4013:
4011:
4007:
4003:
3995:
3990:
3987:
3984:
3983:
3978:
3974:
3970:
3966:
3963:
3960:
3959:
3954:
3951:
3948:
3947:
3943:
3933:
3929:
3928:Turkish music
3921:
3917:
3913:
3909:
3906:
3903:
3902:
3897:
3894:
3893:
3888:
3885:
3882:
3881:
3876:
3873:
3870:
3869:
3865:
3861:
3857:
3853:
3850:
3847:
3846:
3842:
3837:
3834:
3831:
3830:
3826:
3822:
3818:
3815:
3814:
3810:
3802:
3799:
3798:
3794:
3790:
3786:
3783:
3780:
3779:
3774:
3771:
3768:
3767:
3762:
3759:
3756:
3755:
3751:
3727:
3723:
3720:
3719:
3714:
3705:
3697:equivalents.
3696:
3692:
3688:
3678:
3675:
3667:
3657:
3653:
3647:
3646:
3641:This section
3639:
3635:
3630:
3629:
3621:
3610:
3609:Chinese music
3606:
3597:
3588:
3587:
3586:
3575:
3565:
3560:
3551:
3547:
3543:
3539:
3535:
3531:
3518:are tuned to
3517:
3513:
3510:According to
3489:
3476:
3475:
3474:
3472:
3471:Figure 1
3468:
3446:
3441:
3429:
3417:
3409:
3396:
3388:
3364:
3354:
3344:
3311:
3303:
3284:
3281:
3260:
3257:
3252:
3243:
3240:
3237:
3230:
3229:
3225:
3222:
3201:
3198:
3195:
3181:
3178:
3175:
3168:
3167:
3163:
3160:
3139:
3136:
3133:
3119:
3116:
3108:
3101:
3100:
3096:
3093:
3072:
3069:
3066:
3052:
3049:
3046:
3040:Major sixth (
3039:
3038:
3034:
3031:
3010:
3007:
3004:
2990:
2987:
2979:
2973:Minor sixth (
2972:
2971:
2967:
2964:
2943:
2940:
2937:
2923:
2920:
2917:
2910:
2909:
2905:
2902:
2881:
2878:
2875:
2861:
2858:
2850:
2843:
2842:
2838:
2835:
2814:
2811:
2808:
2794:
2791:
2788:
2781:
2780:
2776:
2773:
2752:
2749:
2746:
2732:
2729:
2726:
2720:Major third (
2719:
2718:
2714:
2711:
2690:
2687:
2684:
2670:
2667:
2659:
2653:Minor third (
2652:
2651:
2647:
2644:
2623:
2620:
2617:
2603:
2600:
2597:
2590:
2589:
2585:
2582:
2561:
2558:
2555:
2541:
2538:
2530:
2523:
2522:
2518:
2515:
2494:
2491:
2488:
2479:
2476:
2473:
2466:
2465:
2462:tuning error
2453:
2450:
2447:
2444:
2436:
2428:
2425:
2424:
2421:
2420:
2419:
2416:
2414:
2384:
2372:
2365:
2360:
2351:
2344:
2339:
2328:
2323:
2316:
2307:
2303:
2298:
2294:
2290:
2279:
2267:
2260:
2255:
2252:
2214:
2208:
2199:
2190:
2183:
2178:
2173:
2166:
2147:
2144:
2139:
2135:
2127:
2126:
2111:
2099:
2092:
2087:
2073:
2068:
2061:
2052:
2048:
2043:
2039:
2035:
2021:
2009:
2002:
1997:
1994:
1956:
1950:
1941:
1932:
1925:
1920:
1915:
1908:
1889:
1886:
1881:
1877:
1869:
1868:
1867:
1855:
1844:
1837:
1828:
1816:
1798:
1784:
1779:
1772:
1763:
1759:
1754:
1750:
1746:
1735:
1723:
1716:
1708:
1702:
1672:
1665:
1658:
1650:
1646:
1642:
1637:
1633:
1622:
1621:
1620:
1576:
1573:
1565:
1562:
1559:
1538:
1534:
1519:
1500:
1497:
1492:
1488:
1480:
1450:
1447:
1439:
1436:
1433:
1412:
1408:
1393:
1374:
1371:
1366:
1362:
1354:
1353:
1352:
1340:
1329:
1313:
1307:
1304:
1291:
1285:
1276:
1253:
1245:
1240:
1212:
1209:
1206:
1188:
1180:
1161:
1153:
1149:
1145:
1140:
1136:
1125:
1124:
1123:
1114:
1104:
1094:
1077:
1074:
1067:
1064:
1058:
1054:
1048:
1040:
1030:
1029:
1028:
1026:
1022:
1018:
1003:
999:
992:
990:
986:
982:
978:
974:
970:
965:
956:
952:
949:based on the
942:
938:
936:
932:
928:
918:
915:
886:
871:
867:
858:
850:
849:
848:
845:
842:
839:
829:
819:
815:
799:
796:February 2019
789:
783:
780:This section
778:
774:
769:
768:
760:
758:
750:
749:pitch classes
746:
742:
738:
714:
707:
698:
695:
685:
684:
683:
669:
665:
661:
656:
654:
652:
613:
605:
599:
596:
586:
569:
566:
561:
557:
546:
545:
544:
542:
538:
535:to different
534:
533:transposition
530:
526:
522:
518:
507:
504:
496:
486:
482:
478:
472:
471:
467:
462:This section
460:
456:
451:
450:
339:
333:
326:
324:
320:
316:
312:
308:
304:
300:
295:
294:can be used.
293:
285:
280:
278:
277:pseudo-octave
274:
269:
258:
254:
243:
231:
228:
224:
221:
214:
210:
196:
194:
183:
179:
175:
148:
144:
143:12 equal
140:
129:
118:
114:
110:
105:
103:
99:
95:
91:
87:
86:tuning system
83:
79:
69:
58:
51:
43:
39:
38:just interval
35:
30:
24:
8824:Lambda scale
8731:Arabic maqam
8688:Werckmeister
8527:
8519:Temperaments
8296:Other topics
8284:
8277:
8270:
8263:
8246:
8240:Compositions
8223:
8212:publications
8167:Concepts and
8062:
8052:Lambda scale
7993:Harry Partch
7988:Yuri Landman
7983:Lou Harrison
7968:Wendy Carlos
7963:Glenn Branca
7863:Ben Johnston
7858:Charles Ives
7818:Franklin Cox
7808:Heinz Bohlen
7730:Compositions
7716:Tone cluster
7694:Polytonality
7681:
7657:Chromaticism
7650:and concepts
7584:Mystic chord
7573:
7501:ARDINALITIES
7471:
7453:
7412:
7408:
7400:– via
7374:
7335:
7300:. Retrieved
7296:the original
7291:
7285:As cited by
7280:
7261:
7239:
7220:
7207:
7198:
7194:
7152:(4): 15â32.
7149:
7143:
7129:
7118:. Retrieved
7113:
7103:
7092:. Retrieved
7089:WolframAlpha
7079:
7068:. Retrieved
7064:the original
7054:
7042:. Retrieved
7037:
7027:
7022:, p. 58
7015:
6996:
6990:
6979:. Retrieved
6975:the original
6970:
6959:
6951:the original
6939:
6928:
6909:
6903:
6892:
6881:
6870:. Retrieved
6865:
6855:
6830:
6822:
6813:
6809:
6800:
6781:
6775:
6763:. Retrieved
6759:the original
6753:
6743:
6731:. Retrieved
6727:the original
6717:
6708:
6704:
6698:
6687:. Retrieved
6683:the original
6678:
6669:
6656:
6652:
6643:
6634:
6624:
6610:
6604:
6592:
6581:. Retrieved
6577:the original
6572:
6563:
6554:
6547:
6538:
6530:the original
6517:
6507:
6495:
6486:
6479:
6458:
6441:
6433:
6427:
6418:
6412:
6406:
6351:(1): 47â55.
6348:
6342:
6336:
6322:
6308:
6296:. Retrieved
6286:
6271:
6094:
6084:
6075:
6025:Piano tuning
5960:
5956:exactly, or
5934:
5922:
5917:
5913:
5909:
5902:
5898:
5894:
5884:
5880:
5876:
5872:
5868:
5858:
5854:
5850:
5846:
5837:
5833:
5822:
5818:
5808:
5804:
5754:
5750:
5746:
5726:
5717:
5698:
5643:
5639:
5635:
5615:
5606:
5587:
5535:
5531:
5527:
5488:
5396:
5338:
5297:
5277:
5253:
5184:greater tone
5166:
5141:
5128:
5115:
5107:
5099:
5091:
5080:
5067:
5059:
5037:
5029:
5021:
5013:
5005:
4997:
4981:
4961:
4959:
4952:
4948:
4944:
4928:
4920:
4912:
4891:greater tone
4882:
4859:
4857:
4843:
4750:
4721:
4621:
4617:
4607:
4603:
4597:
4589:
4568:
4547:
4512:
4475:
4439:
4419:
4412:
4408:
4396:
4383:
4375:
4367:
4359:
4351:
4343:
4329:
4325:
4323:
4308:
4299:
4288:Please help
4283:verification
4280:
4253:
4251:
4211:
4171:
4131:
4122:
4116:
4110:
4107:Wendy Carlos
4105:
4082:justly tuned
4051:
4033:
4017:denominators
4014:
3999:
3980:36 EDO.
3823:but not the
3793:Charles Ives
3670:
3661:
3650:Please help
3645:verification
3642:
3607:
3603:
3571:
3509:
3447:
3401:
3379:
3377:
3309:
3301:
3297:
3235:
3173:
3106:
3044:
2977:
2915:
2848:
2786:
2724:
2657:
2595:
2528:
2471:
2417:
2404:
1853:
1842:
1832:
1823:
1821:
1612:
1338:
1327:
1311:
1302:
1289:
1274:
1248:
1235:
1232:
1109:
1106:
1092:
1014:
998:
977:atonal music
973:polytonality
966:
954:
941:Simon Stevin
939:
924:
916:
887:
875:
855:
846:
843:
838:Simon Stevin
825:
817:
793:
781:
734:
657:
648:
634:
514:
499:
490:
475:Please help
463:
303:open strings
296:
287:
283:
281:
270:
255:, while the
232:
219:
197:
185:
181:
142:
131:
120:
116:
106:
77:
75:
56:
42:prime limits
8819:Delta scale
8814:Gamma scale
8804:Alpha scale
8706:non-Western
8704:Traditional
8399:Pitch class
8379:Millioctave
8362:Measurement
8259:just pieces
8047:Delta scale
8042:Gamma scale
8032:Alpha scale
8022:Non-octave-
8012:Tunings and
7973:Ivor Darreg
7798:BĂ©la BartĂłk
7380:Ellis, A.J.
7134:Milne, A.;
7044:26 February
7040:. Joe Monzo
6649:Galilei, V.
6328:Ellis, A.J.
6318:Ellis, A.J.
5760:21 + 14 + 8
5649:15 + 10 + 6
5284:minor tones
5173:lesser tone
5055:grave fifth
5044:permutation
4899:lesser tone
4302:August 2017
4021:convergents
4010:17 EDO
4006:15 EDO
4002:13 EDO
3989:96 EDO
3985:96 EDO
3965:72 EDO
3961:72 EDO
3949:58 EDO
3908:53 EDO
3904:53 EDO
3895:46 EDO
3883:41 EDO
3875:34 EDO
3871:34 EDO
3852:31 EDO
3848:31 EDO
3832:29 EDO
3816:27 EDO
3800:26 EDO
3785:24 EDO
3781:24 EDO
3773:23 EDO
3769:23 EDO
3761:22 EDO
3757:22 EDO
3726:19 EDO
3721:19 EDO
3532:(1966) and
1284:440 Hz
1019:, i.e. the
995:Mathematics
962: 1605
852:operations.
664:logarithmic
102:logarithmic
94:frequencies
8840:Categories
8809:Beta scale
8787:Non-octave
8778:Tetrachord
8680:Kirnberger
8643:Schismatic
8210:Groups and
8169:techniques
8037:Beta scale
7888:Joe Maneri
7848:Alois HĂĄba
7828:John Eaton
7711:Tone Clock
7662:Cyclic set
7648:Techniques
7603:Genres and
7565:Scales and
7472:Jim Kukula
7120:2016-09-01
7094:2014-06-18
7070:2014-06-18
6981:2007-06-25
6872:2010-06-02
6689:2012-03-20
6583:2012-03-20
6544:Zhu, Zaiyu
6476:Zhu, Zaiyu
6264:References
6244:(sequence
6231:(sequence
6218:(sequence
6205:(sequence
6192:(sequence
6179:(sequence
6166:(sequence
6153:(sequence
6140:(sequence
6127:(sequence
6114:(sequence
6101:(sequence
6015:Microtuner
4330:whole tone
3973:Joe Maneri
3776:territory.
3695:enharmonic
3664:March 2020
3351:See also:
3157:1.77777...
3090:1.66666...
2899:1.42222...
2832:1.33333...
2579:1.06666...
1101:See also:
910:â 1.029302
897:â 1.059463
407:, yellow:
391:, indigo:
297:Unfretted
8799:A12 scale
8753:Octoechos
8718:ShĂ-Ăšr-lÇ
8667:Irregular
8484:Otonality
8424:Microtone
8191:Sonido 13
7956:Inventors
7908:Ezra Sims
7786:Composers
7634:Serialism
7554:Atonality
7432:Kyle Gann
7372:(2005) .
7292:telia.com
7178:1531-5169
7168:0148-9267
6526:1000-4270
6358:0906.0127
6052:Footnotes
5893: 3
5783:meantone.
5745: 3
5634: 3
5541:9 + 6 + 4
5526: 3
5441:t t t t t
5190:T = s c Îș
5182:into the
4994:in a rowâ
4877:(or some
4019:of first
3353:Sonido 13
3223:1088.270
3196:1.887749
3134:1.781797
3067:1.681793
3005:1.587401
2938:1.498307
2876:1.414214
2844:Tritone (
2747:1.259921
2685:1.189207
2618:1.122462
2556:1.059463
2445:Pitch in
2304:
2209:≈
2167:⋅
2049:
1951:≈
1909:⋅
1760:
1709:≡
1659:⋅
1574:≈
1563:−
1520:⋅
1448:≈
1437:−
1394:⋅
1210:−
1162:⋅
1075:≈
985:serialism
870:Zhu Zaiyu
828:Zhu Zaiyu
788:talk page
741:logarithm
493:June 2011
464:does not
375:, green:
323:trombones
180:the term
8684:Vallotti
8637:septimal
8629:Meantone
8389:Interval
8186:Semitone
7398:71425252
7260:(2005).
7201:: 42â47.
7172:Online:
7060:"665edo"
6808:(1707).
6765:20 March
6651:(1584).
6546:(1580).
6478:(1584).
6383:20827087
6298:11 March
5991:See also
5970:â 21.506
5963:= 22.642
5954: Âą
5762:=
5676:meantone
5651:=
5543:=
5148:♯
5135:♯
5122:♯
4988:key of C
4986:(in the
4586:, where
4390:♯
4326:semitone
4060:plus an
3937:♭
3516:gamelans
3282:1200.00
3231:Octave (
3113:♭
2984:♭
2855:♭
2809:1.33484
2664:♭
2535:♭
2467:Unison (
2454:12
1860:♯
1345:♯
1318:♯
1078:1.059463
1017:semitone
541:interval
517:interval
423:, cyan:
315:keyboard
174:semitone
8773:Slendro
8723:Dastgah
8648:Miracle
8611:96-tone
8606:72-tone
8601:58-tone
8596:53-tone
8591:41-tone
8586:34-tone
8581:31-tone
8571:24-tone
8566:23-tone
8561:22-tone
8556:19-tone
8551:17-tone
8546:15-tone
8541:12-tone
8472:7-limit
8467:5-limit
7605:schools
7186:Sources
6733:14 June
6363:Bibcode
6250:in the
6247:A061416
6237:in the
6234:A060529
6224:in the
6221:A061919
6211:in the
6208:A061918
6198:in the
6195:A061921
6185:in the
6182:A061920
6172:in the
6169:A060233
6159:in the
6156:A060527
6146:in the
6143:A060526
6133:in the
6130:A060525
6120:in the
6117:A054540
6111:) â 3:2
6107:in the
6104:A060528
5968:
5958:
5951:
5939:
5932:
5866:
5831:
5827:
5816:
5812:
5802:
5779:
5767:
5715:
5703:
5672:
5660:
5604:
5592:
5560:
5548:
5486:
5481:s = 2 c
5479:
5394:
5390:
5386:
5229:
5206:
5198:octave
5188:
5178:t = s c
5176:
5153:fourths
5075:T t t s
5049:T T t s
4933:
4917:
4883:regular
4829:
4817:
4813:
4801:
4791:
4779:
4775:
4763:
4717:
4705:
4701:
4689:
4679:
4667:
4663:
4651:
4529:
4517:
4492:
4480:
4456:
4444:
4232:
4220:
4216:√
4192:
4180:
4176:√
4152:
4140:
4136:√
4098:√
4066:tritave
4040:in the
4037:A060528
3956:sixths.
3942:kleisma
3743:
3731:
3546:slendro
3333:
3321:
3317:√
3310:C G D A
3302:G D A E
3274:
3262:
3226:+11.73
3215:
3203:
3187:√
3161:996.09
3153:
3141:
3125:√
3097:+15.64
3094:884.36
3086:
3074:
3058:√
3035:-13.69
3032:813.69
3024:
3012:
2996:√
2965:701.96
2957:
2945:
2929:√
2903:609.78
2895:
2883:
2867:√
2836:498.04
2828:
2816:
2800:√
2777:+13.69
2774:386.31
2766:
2754:
2738:√
2715:-15.64
2712:315.64
2704:
2692:
2676:√
2645:203.91
2637:
2625:
2609:√
2586:-11.73
2583:111.73
2575:
2563:
2547:√
2508:
2496:
2215:554.365
1957:659.255
1577:369.994
1451:261.626
1308:), and
1299:middle
1286:), and
905:√
892:√
822:History
682:parts:
485:removed
470:sources
437:
425:
421:
409:
405:
393:
389:
377:
373:
361:
319:fretted
170:
158:
151:√
8741:Mugham
8727:Maqam
8621:Linear
8575:pieces
8536:6-tone
8457:Hexany
8384:Savart
8265:Mother
8014:scales
7903:Sevish
7567:tuning
7497:AVORED
7456:(1753)
7396:
7386:
7350:
7302:19 May
7268:
7246:
7240:Tuning
7227:
7176:
7166:
7003:
6916:
6843:
6788:
6524:
6381:
5890:Hence
5262:, and
5226:commas
5224:, and
5212:(or a
5026:, and
4938:or as
4897:, and
4625:steps.
4565:where
4380:, and
4121:, and
4096:), or
4062:octave
4023:of log
3841:58 EDO
3787:, the
3538:Tenzer
3534:McPhee
3501:each).
3164:+3.91
2968:-1.96
2906:-9.78
2839:+1.96
2648:-3.91
2382:
2376:
2370:
2355:
2349:
2320:
2314:
2277:
2271:
2265:
2231:
2218:
2212:
2206:
2194:
2188:
2170:
2164:
2151:
2109:
2103:
2097:
2078:
2065:
2059:
2033:
2019:
2013:
2007:
1973:
1960:
1954:
1948:
1936:
1930:
1912:
1906:
1893:
1796:
1776:
1770:
1733:
1727:
1721:
1712:
1706:
1700:
1678:
1670:
1662:
1656:
1630:
1593:
1580:
1545:
1523:
1517:
1504:
1467:
1454:
1419:
1397:
1391:
1378:
1218:
1195:
1184:
1175:
1165:
1159:
1133:
1044:
987:, and
933:, and
921:Europe
720:
711:
705:
693:
653:below.
620:
609:
594:
573:
554:
317:, and
34:octave
8758:Pelog
8746:Muqam
8692:Young
8653:Magic
8528:Equal
8462:Limit
8369:Pitch
8176:Limit
7509:CALES
7344:Porto
6812:[
6707:[
6655:[
6633:[
6553:[
6485:[
6379:S2CID
6353:arXiv
5921:= 53
5916:+ 10
5912:+ 16
5781:comma
5723:with
5674:comma
5612:with
5562:comma
5388:s = c
5280:major
5086:comma
5081:grave
4940:cents
4611:steps
4509:sets
4472:sets
4436:sets
4212:gamma
4132:alpha
4124:gamma
4112:alpha
3745:comma
3550:pelog
3512:Kunst
3258:1200
3219:1.875
3199:1100
3137:1000
2641:1.125
2460:cents
2280:round
2022:round
1736:round
1244:hertz
861:China
660:cents
521:ratio
286:, or
259:uses
227:hertz
213:A 440
98:pitch
80:is a
8763:Raga
8374:Cent
7622:List
7556:and
7514:ETER
7512:by P
7394:OCLC
7384:ISBN
7348:ISBN
7304:2006
7266:ISBN
7244:ISBN
7225:ISBN
7174:ISSN
7164:ISSN
7046:2019
7001:ISBN
6914:ISBN
6841:ISBN
6786:ISBN
6767:2012
6735:2012
6551:ćŸæŠèé
6522:ISSN
6483:æšćŸć
šæž
6300:2017
6252:OEIS
6239:OEIS
6226:OEIS
6213:OEIS
6200:OEIS
6187:OEIS
6174:OEIS
6161:OEIS
6148:OEIS
6135:OEIS
6122:OEIS
6109:OEIS
5974:the
5942:1300
5926:for
5901:+ 2
5897:+ 2
5883:= 9
5857:= 8
5836:= 5
5807:= 3
5753:+ 2
5749:+ 2
5642:+ 2
5638:+ 2
5534:+ 2
5530:+ 2
5434:and
5417:and
5392:and
5371:and
5354:and
5309:and
5282:and
5247:and
5196:just
4863:(12
4815:and
4606:â 2
4497:and
4328:and
4243:Play
4203:Play
4172:beta
4163:Play
4118:beta
4090:play
4074:play
4042:OEIS
4029:just
4008:and
3912:just
3858:and
3703:Play
3595:Play
3574:Thai
3558:Play
3548:and
3530:Hood
3456:and
3439:Play
3427:Play
3415:Play
3402:{{7
3400:and
3394:Play
3190:2048
3070:900
3008:800
2941:700
2879:600
2812:500
2770:1.25
2750:400
2688:300
2621:200
2559:100
1850:and
1335:and
1259:and
989:jazz
757:MIDI
649:See
537:keys
468:any
466:cite
350:and
311:wind
244:and
7617:Row
7518:UCH
7430:by
7154:doi
6837:134
6438:vii
6371:doi
5972:Âą ,
5908:27
5793:TET
5790:53
5738:TET
5735:43
5729:= 0
5688:TET
5685:43
5627:TET
5624:31
5618:= 0
5577:TET
5574:31
5519:=
5501:TET
5498:19
5491:= 0
5470:TET
5467:19
5423:TET
5412:TET
5399:= 0
5380:TET
5377:12
5363:TET
5345:TET
5319:TET
5316:12
5300:= 0
4973:TET
4970:12
4968:in
4962:not
4866:TET
4796:EDO
4758:EDO
4744:EDO
4738:EDO
4730:EDO
4724:12
4719:).
4684:EDO
4646:EDO
4640:EDO
4634:EDO
4592:= 1
4582:EDO
4571:= 0
4560:EDO
4541:EDO
4505:EDO
4499:31
4468:EDO
4462:19
4432:EDO
4409:q t
4292:by
3654:by
3616:TET
3581:TET
3564:).
3523:TET
3495:TET
3490:In
3482:TET
3477:In
3461:TET
3452:TET
3405:TET
3384:TET
3371:TET
3028:1.6
2961:1.5
2932:128
2708:1.2
2457:TET
2440:TET
2432:TET
2408:TET
2324:440
2317:550
2295:log
2148:440
2140:550
2069:440
2062:660
2040:log
1890:440
1882:660
1751:log
1616:TET
1501:440
1375:440
1119:TET
1008:TET
983:or
946:TET
881:TET
878:12
833:æ±èŒć
479:by
355:TET
352:60
346:TET
343:10
290:EDO
264:TET
261:24
250:TET
247:31
239:TET
236:19
203:TET
200:12
190:TET
187:12
133:12
130:or
125:TET
122:12
107:In
84:or
76:An
8842::
8690:,
8686:,
8682:,
8635:,
8131:96
8126:72
8121:58
8116:53
8111:41
8106:34
8101:31
8096:24
8091:23
8086:22
8081:19
8076:17
8071:15
7470:,
7392:.
7342:.
7290:.
7199:13
7197:.
7162:.
7150:31
7148:.
7142:.
7112:.
7087:.
7036:.
6864:.
6839:.
6784:.
6677:.
6571:.
6520:.
6516:.
6467:^
6449:^
6440:.
6419:19
6417:.
6391:^
6377:.
6369:.
6361:.
6349:78
6347:.
6316:;
6278:,
6060:^
5948:53
5937:=
5906:=
5879:+
5875:+
5871:=
5853:+
5849:=
5821:=
5776:5
5770:1
5758:=
5712:4
5706:3
5701:=
5669:4
5663:1
5647:=
5601:3
5595:2
5590:=
5557:3
5551:1
5539:=
5420:7
5409:5
5360:5
5342:7
5336:.
5332:=
5290:,
5258:,
5194:a
5186:,
5164:.
5072:=
5010:,
4976:).
4957:.
4951:â
4947:=
4915:=
4903:t
4901:,
4895:T
4893:,
4887:s
4826:3
4820:2
4810:3
4804:1
4788:9
4782:8
4772:7
4766:1
4714:5
4708:3
4698:5
4692:2
4676:3
4670:2
4660:3
4654:1
4620:â
4616:4
4602:7
4576:7
4554:5
4526:5
4520:2
4515:=
4478:=
4442:=
4426:12
4411:=
4372:,
4364:,
4356:,
4341:.
4258:.
4214::
4174::
4134::
4115:,
4103:.
4012:.
4004:,
3887:41
3836:29
3740:3
3734:1
3620:.
3613:7
3578:7
3572:A
3520:5
3492:7
3479:5
3473:.
3458:7
3449:5
3408:}}
3381:5
3368:7
3285:0
3276:=
3247:=
3241:)
3217:=
3206:15
3185:=
3179:)
3155:=
3144:16
3128:32
3123:=
3117:)
3088:=
3056:=
3050:)
3026:=
2994:=
2988:)
2959:=
2927:=
2921:)
2897:=
2892:45
2886:64
2865:=
2859:)
2830:=
2803:32
2798:=
2792:)
2768:=
2736:=
2730:)
2706:=
2674:=
2668:)
2639:=
2607:=
2601:)
2577:=
2572:15
2566:16
2545:=
2539:)
2519:0
2516:0
2510:=
2492:0
2489:1
2483:=
2477:)
2352:12
2291:12
2268:12
2100:12
2036:12
2010:12
1933:12
1747:12
1724:12
1566:49
1560:46
1539:12
1493:46
1440:49
1434:40
1413:12
1367:40
1351::
1189:12
1068:12
1049:12
1027::
975:,
959:c.
929:,
907:2
894:2
655:)
434:8
428:13
418:8
412:11
402:4
386:4
370:2
313:,
195:.
167:12
136:ET
119:,
8769:)
8765:(
8694:)
8678:(
8674:/
8639:)
8631:(
8577:)
8573:(
8347:e
8340:t
8333:v
7771:e
7764:t
7757:v
7546:e
7539:t
7532:v
7516:B
7507:S
7505:F
7503:O
7499:C
7495:F
7486:"
7474:.
7417:)
7356:.
7306:.
7274:.
7252:.
7233:.
7214:.
7170:.
7156::
7123:.
7097:.
7073:.
7048:.
7009:.
6984:.
6922:.
6875:.
6849:.
6794:.
6737:.
6692:.
6619:.
6586:.
6385:.
6373::
6365::
6355::
6302:.
5986:.
5965:Âą
5961:Îș
5945:/
5935:Îș
5923:Îș
5918:Îș
5914:Îș
5910:Îș
5903:s
5899:t
5895:T
5888:.
5885:Îș
5881:Îș
5877:c
5873:s
5869:T
5862:,
5859:Îș
5855:c
5851:s
5847:t
5841:,
5838:Îș
5834:s
5825:Âł
5823:Îș
5819:c
5809:Îș
5805:c
5773:/
5755:s
5751:t
5747:T
5731:,
5727:Îș
5721:,
5718:s
5709:/
5699:c
5678:.
5666:/
5644:s
5640:t
5636:T
5620:,
5616:Îș
5610:,
5607:s
5598:/
5588:c
5567:.
5554:/
5536:s
5532:t
5528:T
5521:t
5517:T
5513:s
5509:c
5505:,
5493:,
5489:Îș
5458:,
5451:Îș
5447:s
5443:,
5436:Îș
5432:s
5401:.
5397:Îș
5373:Îș
5369:s
5356:Îș
5352:c
5334:t
5330:T
5326:s
5311:c
5306:s
5298:Îș
5292:t
5288:T
5286:(
5264:Îș
5260:c
5256:s
5249:c
5245:s
5241:c
5237:s
5233:.
5231:Îș
5222:c
5218:s
5210:,
5192:,
5180:,
5169:Îș
5142:C
5138:â
5129:F
5116:F
5112:â
5108:B
5100:B
5096:â
5092:E
5078:(
5068:A
5064:â
5060:D
5051:.
5038:D
5034:â
5030:G
5022:G
5018:â
5014:C
5006:C
5002:â
4998:F
4982:F
4953:t
4949:T
4945:Îș
4936:,
4929:t
4925:/
4921:T
4913:Îș
4907:Îș
4869:)
4823:/
4807:/
4799:(
4785:/
4769:/
4753:q
4733:,
4726:k
4711:/
4695:/
4673:/
4657:/
4629:n
4622:s
4618:t
4608:s
4604:t
4590:q
4578:k
4569:q
4563:,
4556:k
4548:C
4532:.
4523:/
4513:q
4501:k
4495:,
4489:3
4486:/
4483:1
4476:q
4464:k
4459:,
4453:2
4450:/
4447:1
4440:q
4428:k
4422:k
4413:s
4403:q
4397:C
4384:F
4376:F
4368:E
4360:D
4352:C
4339:t
4335:s
4315:)
4309:(
4304:)
4300:(
4286:.
4229:2
4226:/
4223:3
4189:2
4186:/
4183:3
4149:2
4146:/
4143:3
4100:3
4068:(
4025:2
3996:.
3924:2
3827:.
3811:.
3805:2
3737:/
3677:)
3671:(
3666:)
3662:(
3648:.
3526:,
3498:,
3485:,
3330:2
3327:/
3324:3
3278:2
3271:1
3268:/
3265:2
3254:2
3249:2
3245:2
3236:C
3212:8
3209:/
3183:2
3174:B
3150:9
3147:/
3121:2
3107:B
3083:3
3080:/
3077:5
3061:8
3054:2
3045:A
3021:5
3018:/
3015:8
2999:4
2992:2
2978:A
2954:2
2951:/
2948:3
2925:2
2916:G
2889:/
2870:2
2863:2
2849:G
2825:3
2822:/
2819:4
2796:2
2787:F
2763:4
2760:/
2757:5
2741:2
2734:2
2725:E
2701:5
2698:/
2695:6
2679:2
2672:2
2658:E
2634:8
2631:/
2628:9
2612:2
2605:2
2596:D
2569:/
2550:2
2543:2
2529:D
2512:1
2505:1
2502:/
2499:1
2485:1
2481:2
2472:C
2385:.
2373:3
2366:1
2361:=
2345:4
2340:=
2335:)
2329:)
2308:(
2299:2
2286:(
2261:1
2256:=
2253:x
2226:z
2223:H
2200:)
2191:3
2184:1
2179:(
2174:2
2159:z
2156:H
2145:=
2136:E
2112:.
2093:7
2088:=
2083:)
2074:)
2053:(
2044:2
2028:(
2003:1
1998:=
1995:x
1968:z
1965:H
1942:)
1926:7
1921:(
1916:2
1901:z
1898:H
1887:=
1878:E
1864:5
1854:C
1848:5
1843:E
1835:a
1833:E
1826:n
1824:E
1799:.
1791:)
1785:)
1780:a
1773:n
1764:(
1755:2
1742:(
1717:1
1703:x
1673:x
1666:2
1651:a
1647:E
1643:=
1638:n
1634:E
1588:z
1585:H
1569:)
1557:(
1551:)
1535:2
1528:(
1512:z
1509:H
1498:=
1489:P
1462:z
1459:H
1443:)
1431:(
1425:)
1409:2
1402:(
1386:z
1383:H
1372:=
1363:P
1349:4
1339:F
1333:4
1328:C
1322:4
1312:F
1303:C
1297:(
1295:4
1290:C
1280:4
1275:A
1269:a
1265:n
1261:a
1257:n
1251:a
1249:P
1238:n
1236:P
1213:a
1207:n
1201:)
1181:2
1170:(
1154:a
1150:P
1146:=
1141:n
1137:P
1112:n
1110:P
1065:1
1059:2
1055:=
1041:2
957:(
912:,
899:,
798:)
794:(
790:.
753:c
715:n
708:w
699:=
696:c
680:n
676:w
672:p
645:n
641:p
637:r
614:n
606:p
600:=
597:r
570:p
567:=
562:n
558:r
506:)
500:(
495:)
491:(
487:.
473:.
431:/
415:/
399:/
396:7
383:/
380:5
367:/
364:3
267:.
220:A
164:/
161:1
153:2
57:C
44:.
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