2562:
4159:
4617:
3122:
2241:
3810:
3662:
3048:
4507:
4085:
3386:
2208:
4579:
4469:
2526:
2311:
2276:
4767:
4731:
4431:
2344:
3700:
3548:
3472:
2170:
4267:
4009:
3510:
3198:
2940:
2674:
37:
2134:
399:
197:
2905:
2450:
3971:
3736:
3586:
3236:
2782:
2096:
4305:
3933:
3274:
2870:
3348:
2488:
2412:
2377:
4655:
3845:
4693:
4393:
4229:
4194:
4047:
3160:
3010:
2600:
4835:
4802:
3315:
4548:
4360:
3777:
2975:
3624:
2641:
2063:
4126:
3900:
3089:
2837:
2744:
2709:
3441:
2008:
1975:
420:
31:
1425:
974:
5784:. Meantone is present in the design of Western instruments. For example, the keys of a piano and its predecessors are structured so that the white keys form a seven-note meantone scale and the black keys form a five-note meantone scale. Another example is that guitars and other string instruments with at least five strings are typically tuned so that their open strings form a five-note meantone scale.
542:(c. 1605), published posthumously nearly three centuries later in 1884. However, due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values. As a result, the frequency ratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is claimed by Gene Cho as incorrect.
1415:
150:
using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations." Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications." Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament and that neither of the two should be treated as inventors.
1906:, which is about 6.5 cents flatter than a semitone. However, because 17 and 19 are rather large for consonant ratios and most people are unfamiliar with 17 limit and 19 limit intervals, 17 limit and 19 limit intervals are not useful for most purposes, so they can likely not be judged as playing a part in any consonances of 12 ET.
411:, curator of the Conservatoire museum in Brussels, duplicated a set of pitch pipes according to Zhu Zaiyu's specification. He said that the Chinese theory of tones knew more about the length of pitch pipes than its Western counterpart, and that the set of pipes duplicated according to the Zaiyu data proved the accuracy of this theory.
1229:
823:
accuracy, was already done in the first decades of the 19th century. Using beat rates, first proposed in 1749, became common after their diffusion by
Helmholtz and Ellis in the second half of the 19th century. The ultimate precision was available with 2 decimal tables published by White in 1917.
145:
in 1585. According to Fritz A. Kuttner, a critic of the theory, it is known that "Chu-Tsaiyu presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that "Simon Stevin offered a mathematical definition of equal temperament plus
1444:
12 ET is very accurate in the 3 limit, but as one increases prime limits to 11, it gradually gets worse by about a sixth of a semitone each time. Its eleventh and thirteenth harmonics are extremely inaccurate. 12 ET's seventeenth and nineteenth harmonics are almost as accurate as its
822:
A precise equal temperament is possible using the 17th century
Sabbatini method of splitting the octave first into three tempered major thirds. This was also proposed by several writers during the Classical era. Tuning without beat rates but employing several checks, achieving virtually modern
814:
The progress of equal temperament from the mid-18th century on is described with detail in quite a few modern scholarly publications: It was already the temperament of choice during the
Classical era (second half of the 18th century), and it became standard during the Early Romantic era (first
5934:
It is probably not an accident that as tuning in
European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the
5749:
with the goal of dividing up the commas in order to reduce the dissonance of the worst-affected intervals. Werckmeister and
Kirnberger were each dissatisfied with his first temperament and therefore created multiple temperaments, the latter temperaments more closely approximating equal temperament
406:
Zhu Zaiyu illustrated his equal temperament theory by the construction of a set of 36 bamboo tuning pipes ranging in 3 octaves, with instructions of the type of bamboo, color of paint, and detailed specification on their length and inner and outer diameters. He also constructed a 12-string tuning
149:
Kenneth
Robinson attributes the invention of equal temperament to Zhu Zaiyu and provides textual quotations as evidence. Zhu Zaiyu is quoted as saying that, in a text dating from 1584, "I have founded a new system. I establish one foot as the number from which the others are to be extracted, and
5700:
Historically, multiple tuning systems have been used that can be seen as slight variations of 12-TEDO, with twelve notes per octave but with some variation among interval sizes so that the notes are not quite equally-spaced. One example of this a three-limit scale where equally-tempered perfect
1797:
or microtonality will not be familiar with the eleventh and thirteenth harmonics. Similarly, while the error of the eleventh or thirteenth harmonic could be mostly canceled out by the error of the seventh harmonic, most
Western musicians would not find the resulting fractions consonant since
790:
Twelve-tone equal temperament took hold for a variety of reasons. It was a convenient fit for the existing keyboard design, and permitted total harmonic freedom with the burden of moderate impurity in every interval, particularly imperfect consonances. This allowed greater expression through
1544:
is simply two perfect fourths combined, they, for the most part, retain the accuracy of their predecessors; the error is doubled, but it remains small – so small, in fact, that humans cannot perceive it. One can continue to use fractions with higher powers of three, the next two being
5715:
at least a century before He
Chengtian created the sequence of 12-TEDO. Likewise, Pythagorean tuning, which was developed by ancient Greeks, was the predominant system in Europe until during the Renaissance, when Europeans realized that dissonant intervals such as
2563:
4160:
756:
From 1450 to about 1800, plucked instrument players (lutenists and guitarists) generally favored equal temperament, and the
Brossard lute manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to
4618:
3123:
2242:
3811:
3663:
3049:
4508:
4086:
3387:
2209:
815:
decade of the 19th century), except for organs that switched to it more gradually, completing only in the second decade of the 19th century. (In
England, some cathedral organists and choirmasters held out against it even after that date;
382:
in China recorded this work in his personal journal and may have transmitted the work back to Europe. (Standard resources on the topic make no mention of any such transfer.) In 1620, Zhu's work was referenced by a European mathematician.
4580:
4470:
2527:
2312:
2277:
4768:
4732:
4432:
2345:
3701:
3549:
3473:
2171:
4268:
4010:
3511:
3199:
2941:
2675:
2135:
1410:{\displaystyle {\begin{alignedat}{3}P_{40}&=440\left({\sqrt{2}}\right)^{(40-49)}&&\approx 261.626\ \mathrm {Hz} \\P_{46}&=440\left({\sqrt{2}}\right)^{(46-49)}&&\approx 369.994\ \mathrm {Hz} \end{alignedat}}}
2906:
2451:
5771:
The most popular type of scale in 12-TEDO is meantone. Meantone refers to any scale where all of its notes are consecutive on the circle of fifths. Meantone scales of different sizes exist, and some meantone scales used include
1503:
especially for the division of the octave into a relatively small number of tones. Specifically, a just perfect fifth is only one fifty-first of a semitone sharper than the equally-tempered approximation. Because the major tone
3972:
3737:
3587:
3237:
2783:
2097:
811:. Twelve-tone equal temperament did have some disadvantages, such as imperfect thirds, but as Europe switched to equal temperament, it changed the music that it wrote in order to accommodate the system and minimize dissonance.
1874:, which is only one cent away from 12 ET's major seventh. The nineteenth harmonic is only about 2.5 cents flatter than three of 12 ET's semitones, so it can likewise be combined with the third harmonic to yield
4306:
3934:
3275:
2871:
458:) was one of the first practical advocates of twelve-tone equal temperament. He composed a set of dance suites on each of the 12 notes of the chromatic scale in all the "transposition keys", and published also, in his 1584 "
3349:
2489:
2413:
2378:
4656:
3846:
4694:
4394:
4230:
4195:
4048:
3161:
3011:
2601:
38:
1643:, therefore, sounds in tune as its frequency ratio is approximately 4:5:6, further, merged with its first inversion, and two sub-octave tonics, it is 1:2:3:4:5:6, all six lowest natural harmonics of the bass tone.
4836:
4803:
3316:
4549:
4361:
3778:
2976:
3625:
2642:
2064:
4127:
3901:
3090:
2838:
2745:
2710:
5740:
that slightly modified the interval sizes but could still be viewed as an approximate of 12-TEDO. Due to meantone temperaments' tendency to concentrate error onto one enharmonic perfect fifth, making it
1046:
1158:
3442:
2009:
1976:
387:
said, "The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyü's brilliant solution remains an enigma." The 19th-century German physicist
370:
188:
who lived from 370 to 447. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history: 900 849 802 758 715 677 638 601 570 536 509.5 479 450.
1196:
refer to numbers assigned to the desired pitch and the reference pitch, respectively. These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A
226:
published in 1580. This was followed by the publication of a detailed account of the new theory of the equal temperament with a precise numerical specification for 12-ET in his 5,000-page work
303:
1607:) is approximately one-seventh of a semitone off. Because intervals that are less than a quarter of a scale step off still sound in tune, other five-limit intervals in 12 ET, such as
1673:) is about one-third of a semitone off. Because the error is greater than a quarter of a semitone, seven-limit intervals in 12 ET tend to sound out of tune. In the tritone fractions
4996:, and it is 12-ET's only 3-limit comma. However, as one increases the prime limit and includes more intervals, the number of commas increases. 12-ET's most important five-limit comma is
1793:) are accurately approximated (specifically, by three semitones), since the errors of the eleventh and thirteenth harmonics mostly cancel out, most people who are not familiar with
1705:, the errors of the fifth and seventh harmonics partially cancel each other out so that the just fractions are within a quarter of a semitone of their equally-tempered equivalents.
1918:. Differences of less than six cents cannot be noticed by most people, and intervals that are more than a quarter of a step; which in this case is 25 cents, off sound out of tune.
7051:
6039:
Goodman, Howard L.; Lien, Y. Edmund (April 2009). "A Third Century AD Chinese System of Di-Flute Temperament: Matching Ancient Pitch-Standards and Confronting Modal Practice".
7025:
a foundational work on acoustics and the perception of sound by Hermann von Helmholtz. Especially Appendix XX: Additions by the Translator, pages 430–556, (pdf pages 451–577)]
1729:), at 551.32 cents, falls almost exactly halfway between the nearest two equally-tempered intervals in 12 ET and therefore is not approximated by either. In fact,
5750:
than the former temperaments. Likewise, Europe as a whole gradually transitioned from meantone and well temperaments to 12-TEDO, the system that it still uses today.
499:
1822:) is only about 5 cents sharper than one semitone in 12 ET. It can be combined with 12 ET's approximation of the third harmonic in order to yield
239:) in 1584. An extended account is also given by Joseph Needham. Zhu obtained his result mathematically by dividing the length of string and pipe successively by
748:
published a 100-cent monochord table, which contained several errors due to his use of logarithmic tables. He did not explain how he obtained his results.
5768:, use all twelve notes of 12-TEDO, most music only uses notes from a particular subset of 12-TEDO known as a scale. Many different types of scales exist.
7018:
217:
court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his
783:, were opposed to the adoption of equal temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music, although
395:
that a Chinese prince (see below) introduced a scale of seven notes, and that the division of the octave into twelve semitones was discovered in China.
473:
100:, since the 18th century, and Europe almost exclusively used approximations of it for millennia before that. It has also been used in other cultures.
96:
Twelve-tone equal temperament is the most widespread system in music today. It has been the predominant tuning system of Western music, starting with
6714:
5773:
5745:, European music theorists, such as Andreas Werckmeister, Johann Philipp Kirnberger, Francesco Antonio Vallotti, and Thomas Young, created various
998:
374:
Zhu Zaiyu has been credited as the first person to solve the equal temperament problem mathematically. At least one researcher has proposed that
7046:
5701:
fifths of 700 cents are replaced with justly-intoned perfect fifths of 701.955 cents. Because the two intervals differ by less than 2 cents, or
1084:
1914:
In the following table the sizes of various just intervals are compared against their equal-tempered counterparts, given as a ratio as well as
874:
174:
6221:
1234:
6986:
Khramov, Mykhaylo. "Approximation of 5-limit just intonation. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave",
312:
170:
in the Chinese Bronze Age), covers five full 7-note octaves in the key of C Major, including 12 note semi-tones in the middle of the range.
7219:
6550:
163:
A complete set of bronze chime bells, among many musical instruments found in the tomb of the Marquis Yi of Zeng (early Warring States,
146:
a somewhat less precise computation of the corresponding numerical values in 1585 or later." The developments occurred independently.
6672:
17:
258:
7151:
7284:
7001:
6866:
6813:
6760:
6634:
6584:
6539:
6518:
6497:
6347:
7011:
5032:
and is the factor between Pythagorean thirds and sixths and their just counterparts. 12-ET's other 5-limit commas include:
1761:), at two-fifths of a semitone sharper than a minor sixth, is almost as inaccurate. Although this means that the fraction
7294:
123:. The standard pitch has not always been 440 Hz. It has varied and generally risen over the past few hundred years.
476:
had written music based on equal temperament by 1567. Gorzanis was not the only lutenist to explore all modes or keys:
7090:
6981:
6885:
6845:
6703:
498:
drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by
131:
The two figures frequently credited with the achievement of exact calculation of twelve-tone equal temperament are
7064:
7257:
7084:
4960:) are each mapped to the tritone, so they are treated as nominally the same interval; therefore, their quotient,
185:
179:
6657:
4140:
2469:
6674:
Quantifying Ritual: Political Cosmology, Courtly Music, and Precision Mathematics in Seventeenth-Century China
6000:, p. vii: Chu-Tsaiyu the first formulator of the mathematics of "equal temperament" anywhere in the world
7442:
5788:
4598:
3103:
2257:
1745:
is almost as far from any equally-tempered approximation as possible in 12 ET. The thirteenth harmonic (
741:
presented several equal tempered chord lengths obtained by Jean Beaugrand, Ismael Bouillaud, and Jean Galle.
7656:
7144:
3791:
3643:
3529:
3029:
2655:
775:
in all keys, with connecting enharmonically modulating passages. Among the 17th-century keyboard composers
466:. He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).
1445:
third harmonic, but by this point, the prime limit has gotten too high to sound consonant to most people.
7651:
7380:
4488:
4066:
3367:
2189:
7022:
7224:
6041:
5842:
4896:
by 12-ET due to its mapping of different fractions to the same equally-tempered interval. For example,
2507:
804:
119:
and all other notes are defined as some multiple of semitones apart from it, either higher or lower in
6749:
Kwang-chih Chang; Pingfang Xu; Liancheng Lu (2005). "The eastern Zhou and the growth of regionalism".
808:
7209:
6790:
Moody, Richard (February 2003). "Early Equal Temperament, An Aural Perspective: Claude Montal 1836".
6715:"Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory"
4712:
4412:
2292:
768:
398:
196:
34:
12-tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps.
7493:
6858:
A Critical Study of Chu Tsai-yü's Contribution to the Theory of Equal Temperament in Chinese Music
6225:
510:
with Galilei initially opposed equal temperament but eventually conceded to it in relation to the
7629:
7463:
7289:
7214:
7137:
4674:
4248:
3990:
3179:
2921:
2581:
2115:
816:
800:
408:
7485:
7448:
7438:
7247:
6558:
5847:
4210:
2886:
2431:
1524:
is simply two perfect fifths minus an octave, and its inversion, the Pythagorean minor seventh
1480:
981:
In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a
491:
200:
Prince Zhu Zaiyu constructed 12 string equal temperament tuning instrument, front and back view
6930:
6803:
1858:, only about three cents away from the equally-tempered tritone (the square root of two), and
826:
It is in the environment of equal temperament that the new styles of symmetrical tonality and
7481:
7074:
6922:
6508:
6487:
5832:
5519:
4336:
4327:
3952:
3876:
3867:
3605:
3567:
3417:
3408:
3217:
2813:
2804:
2039:
2030:
1201:
1063:
835:
388:
108:
6833:
6551:"The Significance of the Discovery of the Musical Equal Temperament in the Cultural History"
1216:
are the 40th and 46th key respectively. These numbers can be used to find the frequency of C
7489:
7434:
7416:
7411:
7406:
7401:
7396:
7391:
7386:
7371:
7366:
7361:
7356:
7351:
5737:
4286:
3914:
3255:
2851:
2393:
784:
776:
6678:
8:
7453:
7426:
7324:
7012:
https://web.archive.org/web/20050127000731/http://web.telia.com/~u57011259/pelog_main.htm
6695:
English Song, 1600-1675: Facsimiles of Twenty-six Manuscripts and an Edition of the Texts
4450:
3491:
3329:
2956:
2763:
2077:
990:
796:
529:
495:
477:
69:
253:, such that after twelve divisions (an octave) the length was divided by a factor of 2:
7518:
7304:
7299:
7096:
6926:
6737:
6677:, Departments of History and Asian Studies, University of Texas, Austin, archived from
6531:
The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century
6050:
5430:
5360:
4636:
3826:
3681:
1577:, but as the terms of the fractions grow larger, they become less pleasing to the ear.
1454:
792:
407:
instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890,
787:
emphatically advocated equal temperament in his 1707 treatise published posthumously.
7497:
7458:
7333:
7272:
6997:
6987:
6977:
6881:
6862:
6841:
6834:
6809:
6756:
6699:
6630:
6611:
6580:
6535:
6514:
6493:
6343:
6020:
5826:
5816:
4993:
1586:
795:, which became extremely important in the 18th century in music of such composers as
384:
307:
Similarly, after 84 divisions (7 octaves) the length was divided by a factor of 128:
65:
61:
7523:
6755:. Xu Pingfang, Shao Wangping, Zhang Zhongpei, Wang Renxiang. Yale University Press.
5712:
7477:
7341:
7309:
7229:
7194:
6908:
6729:
6653:
6603:
5852:
5837:
5800:
5781:
5746:
5318:
4789:
4529:
4374:
4175:
4107:
4028:
3758:
3296:
3141:
3070:
2991:
2622:
1962:
954:
890:
780:
745:
451:
112:
1890:, which is about 4.5 cents flatter than an equally-tempered minor sixth, and
7599:
7239:
7101:
7068:
6856:
6750:
6693:
6663:
6624:
6574:
6529:
5821:
1433:
459:
455:
97:
7277:
7252:
7199:
7160:
6799:
5857:
5796:
5792:
5777:
5246:
5029:
1652:
986:
938:
738:
419:
104:
7041:
7008:
Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta
6941:
6607:
5711:
of an octave, the two scales are very similar. In fact, the Chinese developed
444:
One of the earliest discussions of equal temperament occurs in the writing of
7645:
7572:
7541:
7314:
7267:
7174:
6615:
6422:(5th enlarged ed.). Boston, Massachusetts: Tuners Supply Co. p. 68.
5922:
5759:
5176:
4861:
1200:(the reference pitch) is the 49th key from the left end of a piano (tuned to
6913:
6896:
7536:
7376:
7179:
1915:
1794:
1051:
922:
827:
758:
533:
375:
214:
142:
6932:
On the Sensations of Tone as a Physiological Basis for the Theory of Music
849:
490:
in all the Tones) as early as 1507. In the 17th century lutenist-composer
30:
7624:
7619:
7609:
7204:
7184:
1839:
1640:
772:
445:
6594:
Cohen, H. Floris (1987). "Simon Stevin's equal division of the octave".
6054:
7614:
7583:
7061:
6771:
6748:
6026:
5742:
5106:
2725:
2690:
6741:
6450:
5726:
could be made more consonant by tempering them to simpler ratios like
7604:
7558:
6375:
5765:
5605:
906:
839:
831:
494:
wrote a set of 30 preludes including 24 in all the major/minor keys.
204:
132:
120:
7017:
Stewart, P. J. (2006) "From Galaxy to Galaxy: Music of the Spheres"
6752:
The Formation of Chinese Civilization: An Archaeological Perspective
6010:
6008:
6006:
5949:
6897:"The Battle Between Impeccable Intonation and Maximized Modulation"
6733:
5804:
1209:
982:
487:
470:
463:
90:
7129:
7036:
7578:
7528:
6399:
6154:
6003:
5036:
507:
503:
7546:
7262:
7189:
4816:
1989:
1424:
973:
379:
57:
6967:
How Equal Temperament Ruined Harmony (and Why You Should Care)
1459:
12 ET has a very good approximation of the perfect fifth
7563:
7551:
7085:
All existing 18th century quotes on J.S. Bach and temperament
6993:
6132:
6130:
1432:
The intervals of 12-ET closely approximate some intervals in
1174:
116:
7006:
Surjodiningrat, W., Sudarjana, P.J., and Susanto, A. (1972)
1079:, of a note in 12-ET, the following definition may be used:
1041:{\displaystyle {\sqrt{2}}=2^{\frac {1}{12}}\approx 1.059463}
7568:
6942:"Thirty preludes in all (24) keys for lute [DP 49]"
6085:
843:
511:
7010:, Gadjah Mada University Press, Jogjakarta 1972. Cited on
6861:. Volume 9 of Sinologica Coloniensia. Wiesbaden: Steiner.
6462:
6308:
6127:
1153:{\displaystyle P_{n}=P_{a}\left({\sqrt{2}}\right)^{(n-a)}}
761:
written in all keys, including the last prelude, entitled
6320:
846:(at least its piano component) developed and flourished.
6426:
6272:
6178:
5981:
5979:
5966:
5964:
5317:
One of the 7-limit commas that 12-ET tempers out is the
819:, for instance, opposed it all along. He died in 1876.)
173:
An approximation for equal temperament was described by
7097:
Well Temperaments, based on the Werckmeister Definition
6989:
Proceedings of the International Conference SIGMAP-2008
6202:
6117:
6115:
6061:
850:
Comparison of historical approximations of the semitone
6629:. Intabolatura di liuto: I-III (in Italian). Minkoff.
5991:
528:
The first mention of equal temperament related to the
365:{\displaystyle \left({\sqrt{2}}\right)^{84}=2^{7}=128}
103:
In modern times, 12-ET is usually tuned relative to a
7042:
Huygens-Fokker Foundation Centre for Microtonal Music
6438:
6166:
6027:
Kwang-chih Chang, Pingfang Xu & Liancheng Lu 2005
5976:
5961:
1232:
1087:
1001:
779:
advocated equal temperament. Some theorists, such as
315:
261:
6190:
6142:
6112:
4864:, meaning that there are several fractions close to
1657:
12 ET's approximation of the seventh harmonic (
545:
The following were Simon Stevin's chord length from
7091:
Rosetta Revisited: Bach's Very Ordinary Temperament
6798:
6456:
6251:
6014:
1057:
989:of the interval between two adjacent notes, is the
767:, which enharmonically modulates through all keys.
6921:
6646:Unequal Temperaments: Theory, History and Practice
6405:
6296:
6284:
6239:
6160:
6102:
6100:
5955:
1591:12 ET's approximation of the fifth harmonic (
1409:
1152:
1040:
364:
297:
6808:. Volume 4 - Part 1. Cambridge University Press.
6387:
1798:12 ET does not approximate them accurately.
7643:
6827:. Volume 4 - Part 1. Cambridge University Press.
6073:
7037:Xenharmonic wiki on EDOs vs. Equal Temperaments
6097:
6269:"Scintille de musica", (Brescia, 1533), p. 132
6224:. Appalachian State University. Archived from
4992:) is mapped to/treated as unison. This is the
1188:refers to the frequency of a reference pitch.
1173:refers to the pitch, or frequency (usually in
298:{\displaystyle \left({\sqrt{2}}\right)^{12}=2}
79:≈ 1.05946). That resulting smallest interval,
7145:
7071:. (2008) Latina, Il Levante Libreria Editrice
6825:The Shorter Science and Civilisation in China
6576:The Cambridge History of Western Music Theory
5736:, resulting in Europe developing a series of
515:
481:
234:
221:
208:
136:
7220:List of intervals in 5-limit just intonation
6822:
6648:(2nd ed.). Bray, Ireland: Bray Baroque.
6506:
6468:
6368:
6214:
6091:
1428:5-Limit just intervals approximated in 12-ET
762:
7062:Enharmonic instruments and music, 1470–1900
6802:; Ling, Wang; Robinson, Kenneth G. (1962).
6622:
6572:
6489:Tuning and Temperament: A Historical Survey
6326:
6314:
6208:
6136:
6038:
546:
537:
424:
7152:
7138:
968:
7054:A supplement to Mr. Chambers's cyclopædia
6976:. Michigan State University Press, 1991.
6912:
6880:(2nd ed.). London: Springer-Verlag.
6823:Needham, Joseph; Ronan, Colin A. (1978).
6643:
6381:
6263:
5787:Other scales used in 12-TEDO include the
5695:
737:A generation later, French mathematician
402:Zhu Zaiyu's equal temperament pitch pipes
56:) is the musical system that divides the
6935:(2nd ed.). London: Longmans, Green.
6875:
6854:
6769:
6555:Journal of Xinghai Conservatory of Music
6121:
5997:
1423:
972:
418:
397:
195:
135:(also romanized as Chu-Tsaiyu. Chinese:
29:
6712:
6691:
6652:
6507:Benward, Bruce; Saker, Marilyn (2003).
6485:
6432:
6257:
6196:
6184:
6148:
6067:
5985:
5970:
14:
7644:
6939:
6831:
6444:
6245:
5356:12-ET's other 7-limit commas include:
228:Complete Compendium of Music and Pitch
7133:
6789:
6593:
6417:
6393:
6278:
6894:
6670:
6172:
6079:
5891:12-tone equal division of the octave
5875:
89:the width of an octave, is called a
7159:
6548:
6527:
6302:
6290:
6106:
5764:While some types of music, such as
1801:
1708:
1639:, have similarly sized errors. The
246:≈ 1.059463, and for pipe length by
24:
6969:. W.W. Norton & Company, 2007.
6958:
6602:(5). Informa UK Limited: 471–488.
1945:Just intonation interval fraction
1399:
1396:
1314:
1311:
1050:This interval is divided into 100
977:One octave of 12-ET on a monochord
439:
25:
7668:
7030:
6805:Science and Civilisation in China
6457:Needham, Ling & Robinson 1962
6222:"Spinacino 1507a: Thematic Index"
6015:Needham, Ling & Robinson 1962
1419:
27:Equal temperament system in music
7295:Ptolemy's intense diatonic scale
6692:Jorgens, Elise Bickford (1986).
6370:Musicalische Paradoxal-Discourse
2360:Septadecimal chromatic semitone
2224:Novendecimal chromatic semitone
1058:Calculating absolute frequencies
548:Van de Spiegheling der singconst
539:Van De Spiegheling der singconst
469:Galilei's countryman and fellow
426:Van de Spiegheling der singconst
158:
60:into 12 parts, all of which are
6878:Tuning, Timbre, Spectrum, Scale
6840:(2nd ed.). Da Capo Press.
6411:
6361:
6352:
6332:
5928:
5907:12 equal division of the octave
2327:Novendecimal diatonic semitone
523:
186:Southern and Northern Dynasties
6713:Kuttner, Fritz A. (May 1975).
6579:. Cambridge University Press.
6486:Barbour, James Murray (2004).
6418:White, William Braid (1946) .
6406:von Helmholtz & Ellis 1885
6161:von Helmholtz & Ellis 1885
6032:
5956:von Helmholtz & Ellis 1885
4141:Pythagorean diminished seventh
2470:Pythagorean chromatic semitone
1942:Just intonation interval name
1376:
1364:
1291:
1279:
1145:
1133:
834:such as that written with the
751:
560:Chord 10000 from Simon Stevin
502:in his "Scintille de musica".
13:
1:
6876:Sethares, William A. (2005).
6367:Andreas Werckmeister (1707),
5913:); informally abbreviated to
5883:twelve-tone equal temperament
5863:
5789:ascending melodic minor scale
4599:Pythagorean diminished octave
3104:Pythagorean diminished fourth
2258:Pythagorean diatonic semitone
429:
164:
40:Play ascending and descending
7258:Harry Partch's 43-tone scale
7047:A.Orlandini: Music Acoustics
6776:The Engines of Our Ingenuity
6626:Intabolatura di liuto: I-III
6573:Christensen, Thomas (2002).
6513:. Vol. 1. McGraw-Hill.
6510:Music in Theory and Practice
6420:Piano Tuning and Allied Arts
6384:, pp. 140, 142 and 256.
5942:
5868:
3792:Pythagorean diminished sixth
3644:Pythagorean augmented fourth
3530:Pythagorean diminished fifth
3030:Pythagorean augmented second
2656:Pythagorean diminished third
483:"Recercare de tutti li Toni"
219:Fusion of Music and Calendar
191:
68:, with a ratio equal to the
7:
6644:Di Veroli, Claudio (2009).
5810:
4562:Tridecimal neutral seventh
4489:Pythagorean augmented sixth
4067:Pythagorean augmented fifth
3368:Pythagorean augmented third
2545:Lesser tridecimal 2/3-tone
2190:Septimal chromatic semitone
1439:
1177:), you are trying to find.
10:
7673:
7225:List of meantone intervals
6855:Robinson, Kenneth (1980).
6770:Lienhard, John H. (1997).
6478:
6340:Lutes, Viols, Temperaments
6042:The Galpin Society Journal
5843:List of meantone intervals
5757:
5753:
4860:12-ET tempers out several
2508:Septimal diatonic semitone
2073:
2060:
2057:
2048:
2029:
2026:
1806:The seventeenth harmonic (
1650:
1646:
1584:
1580:
1452:
1448:
1061:
805:Carl Philipp Emmanuel Bach
126:
7592:
7509:
7472:
7425:
7332:
7323:
7238:
7215:List of musical intervals
7210:Consonance and dissonance
7167:
7014:. Retrieved May 19, 2006.
6792:Piano Technicians Journal
6608:10.1080/00033798700200311
5921:without qualification in
5028:), which is known as the
4855:
4713:Pythagorean major seventh
4558:
4545:
4542:
4533:
4528:
4525:
4413:Pythagorean minor seventh
4370:
4357:
4354:
4345:
4326:
4323:
4136:
4123:
4120:
4111:
4106:
4103:
3910:
3897:
3894:
3885:
3866:
3863:
3787:
3774:
3771:
3762:
3757:
3754:
3719:Classic diminished fifth
3455:Classic augmented fourth
3451:
3438:
3435:
3426:
3407:
3404:
3325:
3312:
3309:
3300:
3295:
3292:
3099:
3086:
3083:
3074:
3069:
3066:
2847:
2834:
2831:
2822:
2803:
2800:
2651:
2638:
2635:
2626:
2621:
2618:
2293:Larger chromatic semitone
1951:Cents in just intonation
769:Angelo Michele Bartolotti
764:Prélude sur tous les tons
482:
414:
393:On the Sensations of Tone
235:
222:
209:
184:, a mathematician of the
137:
18:12-tone equal temperament
7075:Fractal Microtonal Music
6623:de Gorzanis, G. (1981).
6469:Benward & Saker 2003
6092:Needham & Ronan 1978
5899:12 equal division of 2/1
1909:
1838:, which is, as the next
1777:and also its inversion (
532:in the West appeared in
153:
6914:10.15385/jmo.2018.9.2.2
6778:. University of Houston
6492:. Courier Corporation.
5713:3-limit just intonation
4750:Septimal major seventh
4675:Seventeenth subharmonic
4249:Pythagorean major sixth
3991:Pythagorean minor sixth
3180:Pythagorean major third
2922:Pythagorean minor third
2582:Major diatonic semitone
2116:Just chromatic semitone
1936:Equally-tempered audio
1933:Decimal value in 12-ET
1713:The eleventh harmonic (
1068:To find the frequency,
969:Mathematical properties
817:Samuel Sebastian Wesley
809:Johann Gottfried Müthel
801:Wilhelm Friedemann Bach
500:Giovani Maria Lanfranco
448:in the 4th century BC.
409:Victor-Charles Mahillon
107:of 440 Hz, called
6923:von Helmholtz, Hermann
6895:True, Timothy (2018).
6832:Partch, Harry (1979).
6369:
5848:Diatonic and chromatic
5829:(the physics of music)
5696:Similar tuning systems
4211:Nineteenth subharmonic
2887:Tridecimal minor third
2432:Just diatonic semitone
1429:
1411:
1154:
1042:
978:
771:published a series of
763:
547:
538:
516:
436:
425:
403:
366:
299:
201:
168: 5th century BCE
64:(equally spaced) on a
46:
7482:Temperament ordinaire
6940:Wilson, John (1997).
6549:Cho, Gene J. (2010).
6528:Cho, Gene J. (2003).
6049:. Galpin Society: 7.
5833:Music and mathematics
5738:meantone temperaments
5520:Septimal quarter tone
3953:Undecimal minor sixth
3606:Just diminished fifth
3568:Just augmented fourth
3218:Undecimal major third
1948:Justly-intoned audio
1930:Exact value in 12-ET
1927:Note going up from C
1802:17 and 19 limits
1709:11 and 13 limits
1483:, the perfect fourth
1427:
1412:
1155:
1064:Piano key frequencies
1043:
976:
836:twelve tone technique
793:enharmonic modulation
710:five-tone and a half
682:four-tone and a half
422:
401:
389:Hermann von Helmholtz
367:
300:
199:
33:
7285:List of compositions
7089:Dominic Eckersley: "
7060:Barbieri, Patrizio.
6996:, pp. 181–184,
6944:. The Diapason Press
6671:Hart, Roger (1998),
5321:, which is equal to
4880:that are treated as
4287:Septimal major sixth
3915:Septimal minor sixth
3256:Septimal major third
2852:Septimal minor third
2394:Seventeenth harmonic
1230:
1085:
999:
785:Andreas Werckmeister
777:Girolamo Frescobaldi
517:Sopplimenti musicali
313:
259:
111:, meaning one note,
50:12 equal temperament
7657:Chinese discoveries
7052:"Temperament" from
6992:, 26–29 July 2008,
6927:Ellis, Alexander J.
6772:"Equal Temperament"
6534:. E. Mellen Press.
6459:, pp. 170–171.
6281:, pp. 471–488.
5958:, pp. 493–511.
5778:seven-note meantone
4451:Large minor seventh
3330:Just perfect fourth
2957:Nineteenth harmonic
2764:Septimal whole tone
2153:Undecimal semitone
2078:Septimal third tone
991:twelfth root of two
797:Francesco Geminiani
654:tritone and a half
530:twelfth root of two
496:Henricus Grammateus
478:Francesco Spinacino
213:), a prince of the
7652:Equal temperaments
7519:Chinese musicology
7305:Scale of harmonics
7300:Pythagorean tuning
7248:Euler–Fokker genus
7067:2009-02-15 at the
7023:Sensations of Tone
6836:Genesis of a Music
5917:or referred to as
5782:nine-note meantone
5774:five-note meantone
5361:Septimal semicomma
4637:Just major seventh
3827:Just perfect fifth
1840:Pell approximation
1455:Pythagorean tuning
1430:
1407:
1405:
1150:
1038:
979:
626:ditone and a half
437:
404:
362:
295:
202:
115:, is tuned to 440
47:
7639:
7638:
7505:
7504:
7126:
7122:
7117:
7113:
7109:
7105:
7002:978-989-8111-60-9
6972:Jorgensen, Owen.
6901:Musical Offerings
6868:978-3-515-02732-8
6815:978-0-521-05802-5
6762:978-0-300-09382-7
6654:Galilei, Vincenzo
6636:978-2-8266-0721-2
6596:Annals of Science
6586:978-0-521-62371-1
6541:978-0-7734-6941-9
6520:978-0-07-294261-3
6499:978-0-486-43406-3
6435:, pp. 55–78.
6348:978-0-521-28883-5
6187:, pp. 80–89.
6175:, pp. 61–74.
6070:, pp. 55–56.
5923:Western countries
5919:equal temperament
5827:Musical acoustics
5817:Equal temperament
5747:well temperaments
4994:Pythagorean comma
4853:
4852:
1587:Five-limit tuning
1394:
1357:
1309:
1272:
1126:
1029:
1012:
966:
965:
735:
734:
331:
277:
66:logarithmic scale
16:(Redirected from
7664:
7478:Well temperament
7464:Regular diatonic
7330:
7329:
7310:Tonality diamond
7154:
7147:
7140:
7131:
7130:
7124:
7120:
7115:
7111:
7107:
7103:
6965:Duffin, Ross W.
6953:
6951:
6949:
6936:
6918:
6916:
6891:
6872:
6851:
6839:
6828:
6819:
6795:
6786:
6784:
6783:
6766:
6745:
6719:
6709:
6688:
6687:
6686:
6667:
6649:
6640:
6619:
6590:
6569:
6567:
6566:
6557:. Archived from
6545:
6524:
6503:
6472:
6466:
6460:
6454:
6448:
6442:
6436:
6430:
6424:
6423:
6415:
6409:
6403:
6397:
6391:
6385:
6379:
6373:
6372:
6365:
6359:
6356:
6350:
6336:
6330:
6327:Christensen 2002
6324:
6318:
6315:Christensen 2002
6312:
6306:
6300:
6294:
6288:
6282:
6276:
6270:
6267:
6261:
6255:
6249:
6243:
6237:
6236:
6234:
6233:
6218:
6212:
6209:de Gorzanis 1981
6206:
6200:
6194:
6188:
6182:
6176:
6170:
6164:
6158:
6152:
6146:
6140:
6137:Christensen 2002
6134:
6125:
6119:
6110:
6104:
6095:
6089:
6083:
6077:
6071:
6065:
6059:
6058:
6036:
6030:
6024:
6018:
6012:
6001:
5995:
5989:
5983:
5974:
5968:
5959:
5953:
5936:
5932:
5926:
5879:
5853:Electronic tuner
5838:Microtonal music
5801:diminished scale
5735:
5734:
5730:
5725:
5724:
5720:
5710:
5709:
5705:
5691:
5689:
5687:
5686:
5683:
5680:
5671:
5669:
5668:
5665:
5662:
5655:
5653:
5652:
5649:
5646:
5639:
5637:
5636:
5633:
5630:
5623:
5621:
5620:
5617:
5614:
5601:
5599:
5598:
5595:
5592:
5585:
5583:
5582:
5579:
5576:
5569:
5567:
5566:
5563:
5560:
5553:
5551:
5550:
5547:
5544:
5537:
5535:
5534:
5531:
5528:
5516:
5514:
5512:
5511:
5508:
5505:
5498:
5496:
5495:
5492:
5489:
5482:
5480:
5479:
5476:
5473:
5464:
5462:
5461:
5458:
5455:
5448:
5446:
5445:
5442:
5439:
5426:
5424:
5423:
5420:
5417:
5410:
5408:
5407:
5404:
5401:
5394:
5392:
5391:
5388:
5385:
5378:
5376:
5375:
5372:
5369:
5355:
5353:
5351:
5350:
5347:
5344:
5336:
5334:
5333:
5330:
5327:
5319:septimal kleisma
5312:
5310:
5309:
5306:
5303:
5296:
5294:
5293:
5290:
5287:
5280:
5278:
5277:
5274:
5271:
5264:
5262:
5261:
5258:
5255:
5242:
5240:
5239:
5236:
5233:
5226:
5224:
5223:
5220:
5217:
5210:
5208:
5207:
5204:
5201:
5194:
5192:
5191:
5188:
5185:
5172:
5170:
5169:
5166:
5163:
5156:
5154:
5153:
5150:
5147:
5140:
5138:
5137:
5134:
5131:
5124:
5122:
5121:
5118:
5115:
5102:
5100:
5099:
5096:
5093:
5086:
5084:
5083:
5080:
5077:
5070:
5068:
5067:
5064:
5061:
5054:
5052:
5051:
5048:
5045:
5027:
5025:
5024:
5021:
5018:
5011:
5009:
5008:
5005:
5002:
4991:
4989:
4988:
4985:
4982:
4975:
4973:
4972:
4969:
4966:
4959:
4957:
4956:
4953:
4950:
4943:
4941:
4940:
4937:
4934:
4927:
4925:
4924:
4921:
4918:
4911:
4909:
4908:
4905:
4902:
4895:
4893:
4892:
4889:
4886:
4879:
4877:
4876:
4873:
4870:
4843:
4842:
4841:
4839:
4829:
4828:
4824:
4810:
4809:
4808:
4806:
4775:
4774:
4773:
4771:
4761:
4760:
4756:
4739:
4738:
4737:
4735:
4725:
4724:
4720:
4701:
4700:
4699:
4697:
4687:
4686:
4682:
4663:
4662:
4661:
4659:
4649:
4648:
4644:
4625:
4624:
4623:
4621:
4611:
4610:
4606:
4587:
4586:
4585:
4583:
4573:
4572:
4568:
4556:
4555:
4554:
4552:
4540:
4539:
4515:
4514:
4513:
4511:
4501:
4500:
4496:
4477:
4476:
4475:
4473:
4463:
4462:
4458:
4439:
4438:
4437:
4435:
4425:
4424:
4420:
4401:
4400:
4399:
4397:
4387:
4386:
4382:
4375:Harmonic seventh
4368:
4367:
4366:
4364:
4352:
4351:
4342:
4341:
4333:
4332:
4313:
4312:
4311:
4309:
4299:
4298:
4294:
4275:
4274:
4273:
4271:
4261:
4260:
4256:
4237:
4236:
4235:
4233:
4223:
4222:
4218:
4202:
4201:
4200:
4198:
4188:
4187:
4183:
4176:Just major sixth
4167:
4166:
4165:
4163:
4153:
4152:
4148:
4134:
4133:
4132:
4130:
4118:
4117:
4093:
4092:
4091:
4089:
4079:
4078:
4074:
4055:
4054:
4053:
4051:
4041:
4040:
4036:
4029:Just minor sixth
4017:
4016:
4015:
4013:
4003:
4002:
3998:
3979:
3978:
3977:
3975:
3965:
3964:
3960:
3941:
3940:
3939:
3937:
3927:
3926:
3922:
3908:
3907:
3906:
3904:
3892:
3891:
3882:
3881:
3873:
3872:
3853:
3852:
3851:
3849:
3839:
3838:
3834:
3818:
3817:
3816:
3814:
3804:
3803:
3799:
3785:
3784:
3783:
3781:
3769:
3768:
3744:
3743:
3742:
3740:
3730:
3729:
3725:
3708:
3707:
3706:
3704:
3694:
3693:
3689:
3670:
3669:
3668:
3666:
3656:
3655:
3651:
3632:
3631:
3630:
3628:
3618:
3617:
3613:
3594:
3593:
3592:
3590:
3580:
3579:
3575:
3556:
3555:
3554:
3552:
3542:
3541:
3537:
3518:
3517:
3516:
3514:
3504:
3503:
3499:
3492:Huygens' tritone
3480:
3479:
3478:
3476:
3466:
3465:
3461:
3449:
3448:
3447:
3445:
3433:
3432:
3423:
3422:
3414:
3413:
3394:
3393:
3392:
3390:
3380:
3379:
3375:
3356:
3355:
3354:
3352:
3342:
3341:
3337:
3323:
3322:
3321:
3319:
3307:
3306:
3282:
3281:
3280:
3278:
3268:
3267:
3263:
3244:
3243:
3242:
3240:
3230:
3229:
3225:
3206:
3205:
3204:
3202:
3192:
3191:
3187:
3168:
3167:
3166:
3164:
3154:
3153:
3149:
3142:Just major third
3130:
3129:
3128:
3126:
3116:
3115:
3111:
3097:
3096:
3095:
3093:
3081:
3080:
3056:
3055:
3054:
3052:
3042:
3041:
3037:
3018:
3017:
3016:
3014:
3004:
3003:
2999:
2992:Just minor third
2983:
2982:
2981:
2979:
2969:
2968:
2964:
2948:
2947:
2946:
2944:
2934:
2933:
2929:
2913:
2912:
2911:
2909:
2899:
2898:
2894:
2878:
2877:
2876:
2874:
2864:
2863:
2859:
2845:
2844:
2843:
2841:
2829:
2828:
2819:
2818:
2810:
2809:
2790:
2789:
2788:
2786:
2776:
2775:
2771:
2752:
2751:
2750:
2748:
2738:
2737:
2733:
2717:
2716:
2715:
2713:
2703:
2702:
2698:
2682:
2681:
2680:
2678:
2668:
2667:
2663:
2649:
2648:
2647:
2645:
2633:
2632:
2608:
2607:
2606:
2604:
2594:
2593:
2589:
2570:
2569:
2568:
2566:
2556:
2555:
2551:
2534:
2533:
2532:
2530:
2520:
2519:
2515:
2496:
2495:
2494:
2492:
2482:
2481:
2477:
2458:
2457:
2456:
2454:
2444:
2443:
2439:
2420:
2419:
2418:
2416:
2406:
2405:
2401:
2385:
2384:
2383:
2381:
2371:
2370:
2366:
2352:
2351:
2350:
2348:
2338:
2337:
2333:
2319:
2318:
2317:
2315:
2305:
2304:
2300:
2284:
2283:
2282:
2280:
2270:
2269:
2265:
2249:
2248:
2247:
2245:
2235:
2234:
2230:
2216:
2215:
2214:
2212:
2202:
2201:
2197:
2178:
2177:
2176:
2174:
2164:
2163:
2159:
2142:
2141:
2140:
2138:
2128:
2127:
2123:
2104:
2103:
2102:
2100:
2090:
2089:
2085:
2071:
2070:
2069:
2067:
2055:
2054:
2045:
2044:
2036:
2035:
2016:
2015:
2014:
2012:
2002:
2001:
1997:
1983:
1982:
1981:
1979:
1924:Number of steps
1921:
1920:
1905:
1903:
1902:
1899:
1896:
1889:
1887:
1886:
1883:
1880:
1873:
1871:
1870:
1867:
1864:
1857:
1855:
1854:
1851:
1848:
1837:
1835:
1834:
1831:
1828:
1821:
1819:
1818:
1815:
1812:
1792:
1790:
1789:
1786:
1783:
1776:
1774:
1773:
1770:
1767:
1760:
1758:
1757:
1754:
1751:
1744:
1742:
1741:
1738:
1735:
1728:
1726:
1725:
1722:
1719:
1704:
1702:
1701:
1698:
1695:
1688:
1686:
1685:
1682:
1679:
1672:
1670:
1669:
1666:
1663:
1638:
1636:
1635:
1632:
1629:
1622:
1620:
1619:
1616:
1613:
1606:
1604:
1603:
1600:
1597:
1576:
1574:
1573:
1570:
1567:
1560:
1558:
1557:
1554:
1551:
1543:
1541:
1539:
1538:
1535:
1532:
1523:
1521:
1519:
1518:
1515:
1512:
1502:
1500:
1498:
1497:
1494:
1491:
1478:
1476:
1474:
1473:
1470:
1467:
1416:
1414:
1413:
1408:
1406:
1402:
1392:
1382:
1380:
1379:
1362:
1358:
1356:
1348:
1331:
1330:
1317:
1307:
1297:
1295:
1294:
1277:
1273:
1271:
1263:
1246:
1245:
1195:
1191:
1187:
1172:
1162:In this formula
1159:
1157:
1156:
1151:
1149:
1148:
1131:
1127:
1125:
1117:
1110:
1109:
1097:
1096:
1078:
1047:
1045:
1044:
1039:
1031:
1030:
1022:
1013:
1011:
1003:
955:Johann Faulhaber
891:Vincenzo Galilei
854:
853:
781:Giuseppe Tartini
766:
746:Johann Faulhaber
598:tone and a half
566:Corrected chord
554:
553:
550:
541:
519:
485:
484:
474:Giacomo Gorzanis
452:Vincenzo Galilei
434:
431:
428:
371:
369:
368:
363:
355:
354:
342:
341:
336:
332:
330:
322:
304:
302:
301:
296:
288:
287:
282:
278:
276:
268:
252:
251:
245:
244:
238:
237:
225:
224:
212:
211:
183:
169:
166:
140:
139:
88:
87:
83:
78:
77:
62:equally tempered
45:
44:
43:
41:
21:
7672:
7671:
7667:
7666:
7665:
7663:
7662:
7661:
7642:
7641:
7640:
7635:
7632:(Bohlen–Pierce)
7600:833 cents scale
7588:
7511:
7501:
7468:
7421:
7319:
7240:Just intonation
7234:
7163:
7161:Musical tunings
7158:
7069:Wayback Machine
7033:
7028:
6961:
6959:Further reading
6956:
6947:
6945:
6888:
6869:
6848:
6816:
6800:Needham, Joseph
6781:
6779:
6763:
6722:Ethnomusicology
6717:
6706:
6684:
6682:
6664:Girolamo Scotto
6637:
6587:
6564:
6562:
6542:
6521:
6500:
6481:
6476:
6475:
6467:
6463:
6455:
6451:
6443:
6439:
6431:
6427:
6416:
6412:
6404:
6400:
6392:
6388:
6380:
6376:
6366:
6362:
6357:
6353:
6338:Lindley, Mark.
6337:
6333:
6325:
6321:
6313:
6309:
6301:
6297:
6289:
6285:
6277:
6273:
6268:
6264:
6256:
6252:
6244:
6240:
6231:
6229:
6220:
6219:
6215:
6207:
6203:
6195:
6191:
6183:
6179:
6171:
6167:
6159:
6155:
6147:
6143:
6135:
6128:
6120:
6113:
6105:
6098:
6090:
6086:
6078:
6074:
6066:
6062:
6037:
6033:
6025:
6021:
6013:
6004:
5996:
5992:
5984:
5977:
5969:
5962:
5954:
5950:
5945:
5940:
5939:
5933:
5929:
5880:
5876:
5871:
5866:
5822:Just intonation
5813:
5762:
5756:
5732:
5728:
5727:
5722:
5718:
5717:
5707:
5703:
5702:
5698:
5684:
5681:
5678:
5677:
5675:
5673:
5666:
5663:
5660:
5659:
5657:
5650:
5647:
5644:
5643:
5641:
5634:
5631:
5628:
5627:
5625:
5618:
5615:
5612:
5611:
5609:
5596:
5593:
5590:
5589:
5587:
5580:
5577:
5574:
5573:
5571:
5564:
5561:
5558:
5557:
5555:
5548:
5545:
5542:
5541:
5539:
5532:
5529:
5526:
5525:
5523:
5509:
5506:
5503:
5502:
5500:
5493:
5490:
5487:
5486:
5484:
5477:
5474:
5471:
5470:
5468:
5466:
5459:
5456:
5453:
5452:
5450:
5443:
5440:
5437:
5436:
5434:
5431:Archytas' comma
5421:
5418:
5415:
5414:
5412:
5405:
5402:
5399:
5398:
5396:
5389:
5386:
5383:
5382:
5380:
5373:
5370:
5367:
5366:
5364:
5348:
5345:
5342:
5341:
5339:
5338:
5331:
5328:
5325:
5324:
5322:
5307:
5304:
5301:
5300:
5298:
5291:
5288:
5285:
5284:
5282:
5275:
5272:
5269:
5268:
5266:
5259:
5256:
5253:
5252:
5250:
5237:
5234:
5231:
5230:
5228:
5221:
5218:
5215:
5214:
5212:
5205:
5202:
5199:
5198:
5196:
5189:
5186:
5183:
5182:
5180:
5167:
5164:
5161:
5160:
5158:
5151:
5148:
5145:
5144:
5142:
5135:
5132:
5129:
5128:
5126:
5119:
5116:
5113:
5112:
5110:
5097:
5094:
5091:
5090:
5088:
5081:
5078:
5075:
5074:
5072:
5065:
5062:
5059:
5058:
5056:
5049:
5046:
5043:
5042:
5040:
5022:
5019:
5016:
5015:
5013:
5006:
5003:
5000:
4999:
4997:
4986:
4983:
4980:
4979:
4977:
4970:
4967:
4964:
4963:
4961:
4954:
4951:
4948:
4947:
4945:
4938:
4935:
4932:
4931:
4929:
4922:
4919:
4916:
4915:
4913:
4906:
4903:
4900:
4899:
4897:
4890:
4887:
4884:
4883:
4881:
4874:
4871:
4868:
4867:
4865:
4858:
4837:
4834:
4833:
4826:
4822:
4821:
4804:
4801:
4800:
4769:
4766:
4765:
4758:
4754:
4753:
4733:
4730:
4729:
4722:
4718:
4717:
4695:
4692:
4691:
4684:
4680:
4679:
4657:
4654:
4653:
4646:
4642:
4641:
4619:
4616:
4615:
4608:
4604:
4603:
4581:
4578:
4577:
4570:
4566:
4565:
4550:
4547:
4546:
4537:
4535:
4509:
4506:
4505:
4498:
4494:
4493:
4471:
4468:
4467:
4460:
4456:
4455:
4433:
4430:
4429:
4422:
4418:
4417:
4395:
4392:
4391:
4384:
4380:
4379:
4362:
4359:
4358:
4349:
4347:
4339:
4338:
4330:
4329:
4307:
4304:
4303:
4296:
4292:
4291:
4269:
4266:
4265:
4258:
4254:
4253:
4231:
4228:
4227:
4220:
4216:
4215:
4196:
4193:
4192:
4185:
4181:
4180:
4161:
4158:
4157:
4150:
4146:
4145:
4128:
4125:
4124:
4115:
4113:
4087:
4084:
4083:
4076:
4072:
4071:
4049:
4046:
4045:
4038:
4034:
4033:
4011:
4008:
4007:
4000:
3996:
3995:
3973:
3970:
3969:
3962:
3958:
3957:
3935:
3932:
3931:
3924:
3920:
3919:
3902:
3899:
3898:
3889:
3887:
3879:
3878:
3870:
3869:
3847:
3844:
3843:
3836:
3832:
3831:
3812:
3809:
3808:
3801:
3797:
3796:
3779:
3776:
3775:
3766:
3764:
3738:
3735:
3734:
3727:
3723:
3722:
3702:
3699:
3698:
3691:
3687:
3686:
3682:Euler's tritone
3664:
3661:
3660:
3653:
3649:
3648:
3626:
3623:
3622:
3615:
3611:
3610:
3588:
3585:
3584:
3577:
3573:
3572:
3550:
3547:
3546:
3539:
3535:
3534:
3512:
3509:
3508:
3501:
3497:
3496:
3474:
3471:
3470:
3463:
3459:
3458:
3443:
3440:
3439:
3430:
3428:
3420:
3419:
3411:
3410:
3388:
3385:
3384:
3377:
3373:
3372:
3350:
3347:
3346:
3339:
3335:
3334:
3317:
3314:
3313:
3304:
3302:
3276:
3273:
3272:
3265:
3261:
3260:
3238:
3235:
3234:
3227:
3223:
3222:
3200:
3197:
3196:
3189:
3185:
3184:
3162:
3159:
3158:
3151:
3147:
3146:
3124:
3121:
3120:
3113:
3109:
3108:
3091:
3088:
3087:
3078:
3076:
3050:
3047:
3046:
3039:
3035:
3034:
3012:
3009:
3008:
3001:
2997:
2996:
2977:
2974:
2973:
2966:
2962:
2961:
2942:
2939:
2938:
2931:
2927:
2926:
2907:
2904:
2903:
2896:
2892:
2891:
2872:
2869:
2868:
2861:
2857:
2856:
2839:
2836:
2835:
2826:
2824:
2816:
2815:
2807:
2806:
2784:
2781:
2780:
2773:
2769:
2768:
2746:
2743:
2742:
2735:
2731:
2730:
2711:
2708:
2707:
2700:
2696:
2695:
2676:
2673:
2672:
2665:
2661:
2660:
2643:
2640:
2639:
2630:
2628:
2602:
2599:
2598:
2591:
2587:
2586:
2564:
2561:
2560:
2553:
2549:
2548:
2528:
2525:
2524:
2517:
2513:
2512:
2490:
2487:
2486:
2479:
2475:
2474:
2452:
2449:
2448:
2441:
2437:
2436:
2414:
2411:
2410:
2403:
2399:
2398:
2379:
2376:
2375:
2368:
2364:
2363:
2346:
2343:
2342:
2335:
2331:
2330:
2313:
2310:
2309:
2302:
2298:
2297:
2278:
2275:
2274:
2267:
2263:
2262:
2243:
2240:
2239:
2232:
2228:
2227:
2210:
2207:
2206:
2199:
2195:
2194:
2172:
2169:
2168:
2161:
2157:
2156:
2136:
2133:
2132:
2125:
2121:
2120:
2098:
2095:
2094:
2087:
2083:
2082:
2065:
2062:
2061:
2052:
2050:
2042:
2041:
2033:
2032:
2010:
2007:
2006:
1999:
1995:
1994:
1977:
1974:
1973:
1912:
1900:
1897:
1894:
1893:
1891:
1884:
1881:
1878:
1877:
1875:
1868:
1865:
1862:
1861:
1859:
1852:
1849:
1846:
1845:
1843:
1832:
1829:
1826:
1825:
1823:
1816:
1813:
1810:
1809:
1807:
1804:
1787:
1784:
1781:
1780:
1778:
1771:
1768:
1765:
1764:
1762:
1755:
1752:
1749:
1748:
1746:
1739:
1736:
1733:
1732:
1730:
1723:
1720:
1717:
1716:
1714:
1711:
1699:
1696:
1693:
1692:
1690:
1683:
1680:
1677:
1676:
1674:
1667:
1664:
1661:
1660:
1658:
1655:
1649:
1633:
1630:
1627:
1626:
1624:
1617:
1614:
1611:
1610:
1608:
1601:
1598:
1595:
1594:
1592:
1589:
1583:
1571:
1568:
1565:
1564:
1562:
1555:
1552:
1549:
1548:
1546:
1536:
1533:
1530:
1529:
1527:
1525:
1516:
1513:
1510:
1509:
1507:
1505:
1495:
1492:
1489:
1488:
1486:
1484:
1471:
1468:
1465:
1464:
1462:
1460:
1457:
1451:
1442:
1434:just intonation
1422:
1420:Just intervals
1404:
1403:
1395:
1381:
1363:
1352:
1347:
1343:
1342:
1332:
1326:
1322:
1319:
1318:
1310:
1296:
1278:
1267:
1262:
1258:
1257:
1247:
1241:
1237:
1233:
1231:
1228:
1227:
1223:
1219:
1215:
1207:
1199:
1193:
1189:
1186:
1185:
1181:
1178:
1171:
1170:
1166:
1163:
1132:
1121:
1116:
1112:
1111:
1105:
1101:
1092:
1088:
1086:
1083:
1082:
1077:
1076:
1072:
1069:
1066:
1060:
1021:
1017:
1007:
1002:
1000:
997:
996:
987:frequency ratio
971:
852:
754:
526:
456:Galileo Galilei
442:
432:
423:Simon Stevin's
417:
350:
346:
337:
326:
321:
317:
316:
314:
311:
310:
283:
272:
267:
263:
262:
260:
257:
256:
249:
247:
242:
240:
194:
177:
167:
161:
156:
129:
98:classical music
85:
81:
80:
75:
73:
39:
36:
35:
28:
23:
22:
15:
12:
11:
5:
7670:
7660:
7659:
7654:
7637:
7636:
7634:
7633:
7627:
7622:
7617:
7612:
7607:
7602:
7596:
7594:
7590:
7589:
7587:
7586:
7581:
7576:
7566:
7561:
7556:
7555:
7554:
7549:
7544:
7539:
7531:
7526:
7521:
7515:
7513:
7507:
7506:
7503:
7502:
7476:
7474:
7470:
7469:
7467:
7466:
7461:
7456:
7451:
7446:
7431:
7429:
7423:
7422:
7420:
7419:
7414:
7409:
7404:
7399:
7394:
7389:
7384:
7374:
7369:
7364:
7359:
7354:
7349:
7344:
7338:
7336:
7327:
7321:
7320:
7318:
7317:
7312:
7307:
7302:
7297:
7292:
7287:
7282:
7281:
7280:
7275:
7265:
7260:
7255:
7253:Harmonic scale
7250:
7244:
7242:
7236:
7235:
7233:
7232:
7227:
7222:
7217:
7212:
7207:
7202:
7200:Interval ratio
7197:
7192:
7187:
7182:
7177:
7171:
7169:
7165:
7164:
7157:
7156:
7149:
7142:
7134:
7128:
7127:
7099:
7094:
7087:
7082:
7072:
7058:
7049:
7044:
7039:
7032:
7031:External links
7029:
7027:
7026:
7020:
7015:
7004:
6984:
6970:
6962:
6960:
6957:
6955:
6954:
6937:
6919:
6892:
6886:
6873:
6867:
6852:
6846:
6829:
6820:
6814:
6796:
6794:. Kansas City.
6787:
6767:
6761:
6746:
6734:10.2307/850355
6728:(2): 163–206.
6710:
6704:
6689:
6668:
6650:
6641:
6635:
6620:
6591:
6585:
6570:
6546:
6540:
6525:
6519:
6504:
6498:
6482:
6480:
6477:
6474:
6473:
6461:
6449:
6447:, p. 134.
6437:
6425:
6410:
6408:, p. 548.
6398:
6386:
6382:Di Veroli 2009
6374:
6360:
6351:
6331:
6319:
6317:, p. 207.
6307:
6305:, p. 222.
6295:
6293:, p. 223.
6283:
6271:
6262:
6250:
6238:
6213:
6201:
6189:
6177:
6165:
6163:, p. 258.
6153:
6141:
6139:, p. 205.
6126:
6111:
6096:
6094:, p. 385.
6084:
6072:
6060:
6031:
6029:, p. 140.
6019:
6017:, p. 221.
6002:
5990:
5988:, p. 200.
5975:
5973:, p. 163.
5960:
5947:
5946:
5944:
5941:
5938:
5937:
5927:
5881:Also known as
5873:
5872:
5870:
5867:
5865:
5862:
5861:
5860:
5858:Musical tuning
5855:
5850:
5845:
5840:
5835:
5830:
5824:
5819:
5812:
5809:
5797:harmonic major
5793:harmonic minor
5755:
5752:
5743:very dissonant
5697:
5694:
5693:
5692:
5603:
5517:
5428:
5315:
5314:
5247:Greater diesis
5244:
5174:
5104:
5030:syntonic comma
4857:
4854:
4851:
4850:
4847:
4844:
4831:
4819:
4814:
4811:
4798:
4795:
4792:
4787:
4783:
4782:
4779:
4776:
4763:
4751:
4747:
4746:
4743:
4740:
4727:
4715:
4709:
4708:
4705:
4702:
4689:
4677:
4671:
4670:
4667:
4664:
4651:
4639:
4633:
4632:
4629:
4626:
4613:
4601:
4595:
4594:
4591:
4588:
4575:
4563:
4560:
4557:
4544:
4541:
4532:
4527:
4523:
4522:
4519:
4516:
4503:
4491:
4485:
4484:
4481:
4478:
4465:
4453:
4447:
4446:
4443:
4440:
4427:
4415:
4409:
4408:
4405:
4402:
4389:
4377:
4372:
4369:
4356:
4353:
4344:
4325:
4321:
4320:
4317:
4314:
4301:
4289:
4283:
4282:
4279:
4276:
4263:
4251:
4245:
4244:
4241:
4238:
4225:
4213:
4207:
4206:
4203:
4190:
4178:
4172:
4171:
4168:
4155:
4143:
4138:
4135:
4122:
4119:
4110:
4105:
4101:
4100:
4097:
4094:
4081:
4069:
4063:
4062:
4059:
4056:
4043:
4031:
4025:
4024:
4021:
4018:
4005:
3993:
3987:
3986:
3983:
3980:
3967:
3955:
3949:
3948:
3945:
3942:
3929:
3917:
3912:
3909:
3896:
3893:
3884:
3865:
3861:
3860:
3857:
3854:
3841:
3829:
3823:
3822:
3819:
3806:
3794:
3789:
3786:
3773:
3770:
3761:
3756:
3752:
3751:
3748:
3745:
3732:
3720:
3716:
3715:
3712:
3709:
3696:
3684:
3678:
3677:
3674:
3671:
3658:
3646:
3640:
3639:
3636:
3633:
3620:
3608:
3602:
3601:
3598:
3595:
3582:
3570:
3564:
3563:
3560:
3557:
3544:
3532:
3526:
3525:
3522:
3519:
3506:
3494:
3488:
3487:
3484:
3481:
3468:
3456:
3453:
3450:
3437:
3434:
3425:
3406:
3402:
3401:
3398:
3395:
3382:
3370:
3364:
3363:
3360:
3357:
3344:
3332:
3327:
3324:
3311:
3308:
3299:
3294:
3290:
3289:
3286:
3283:
3270:
3258:
3252:
3251:
3248:
3245:
3232:
3220:
3214:
3213:
3210:
3207:
3194:
3182:
3176:
3175:
3172:
3169:
3156:
3144:
3138:
3137:
3134:
3131:
3118:
3106:
3101:
3098:
3085:
3082:
3073:
3068:
3064:
3063:
3060:
3057:
3044:
3032:
3026:
3025:
3022:
3019:
3006:
2994:
2988:
2987:
2984:
2971:
2959:
2953:
2952:
2949:
2936:
2924:
2918:
2917:
2914:
2901:
2889:
2883:
2882:
2879:
2866:
2854:
2849:
2846:
2833:
2830:
2821:
2802:
2798:
2797:
2794:
2791:
2778:
2766:
2760:
2759:
2756:
2753:
2740:
2728:
2722:
2721:
2718:
2705:
2693:
2687:
2686:
2683:
2670:
2658:
2653:
2650:
2637:
2634:
2625:
2620:
2616:
2615:
2612:
2609:
2596:
2584:
2578:
2577:
2574:
2571:
2558:
2546:
2542:
2541:
2538:
2535:
2522:
2510:
2504:
2503:
2500:
2497:
2484:
2472:
2466:
2465:
2462:
2459:
2446:
2434:
2428:
2427:
2424:
2421:
2408:
2396:
2390:
2389:
2386:
2373:
2361:
2357:
2356:
2353:
2340:
2328:
2324:
2323:
2320:
2307:
2295:
2289:
2288:
2285:
2272:
2260:
2254:
2253:
2250:
2237:
2225:
2221:
2220:
2217:
2204:
2192:
2186:
2185:
2182:
2179:
2166:
2154:
2150:
2149:
2146:
2143:
2130:
2118:
2112:
2111:
2108:
2105:
2092:
2080:
2075:
2072:
2059:
2056:
2047:
2028:
2024:
2023:
2020:
2017:
2004:
1992:
1987:
1984:
1971:
1968:
1965:
1960:
1956:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1911:
1908:
1803:
1800:
1710:
1707:
1653:7-limit tuning
1648:
1645:
1582:
1579:
1450:
1447:
1441:
1438:
1421:
1418:
1401:
1398:
1391:
1388:
1385:
1383:
1378:
1375:
1372:
1369:
1366:
1361:
1355:
1351:
1346:
1341:
1338:
1335:
1333:
1329:
1325:
1321:
1320:
1316:
1313:
1306:
1303:
1300:
1298:
1293:
1290:
1287:
1284:
1281:
1276:
1270:
1266:
1261:
1256:
1253:
1250:
1248:
1244:
1240:
1236:
1235:
1221:
1217:
1213:
1205:
1197:
1183:
1182:
1179:
1168:
1167:
1164:
1147:
1144:
1141:
1138:
1135:
1130:
1124:
1120:
1115:
1108:
1104:
1100:
1095:
1091:
1074:
1073:
1070:
1059:
1056:
1037:
1034:
1028:
1025:
1020:
1016:
1010:
1006:
970:
967:
964:
963:
960:
957:
952:
948:
947:
944:
941:
939:Marin Mersenne
936:
932:
931:
928:
925:
920:
916:
915:
912:
909:
904:
900:
899:
896:
893:
888:
884:
883:
880:
877:
872:
868:
867:
864:
861:
858:
851:
848:
753:
750:
739:Marin Mersenne
733:
732:
730:
727:
725:
721:
720:
717:
714:
711:
707:
706:
703:
700:
697:
693:
692:
689:
686:
683:
679:
678:
675:
672:
669:
665:
664:
661:
658:
655:
651:
650:
647:
644:
641:
637:
636:
633:
630:
627:
623:
622:
619:
616:
613:
609:
608:
605:
602:
599:
595:
594:
592:
589:
586:
582:
581:
578:
575:
572:
568:
567:
564:
561:
558:
536:'s manuscript
525:
522:
441:
438:
416:
413:
385:Murray Barbour
361:
358:
353:
349:
345:
340:
335:
329:
325:
320:
294:
291:
286:
281:
275:
271:
266:
232:Yuelü quan shu
193:
190:
160:
157:
155:
152:
141:) in 1584 and
128:
125:
105:standard pitch
93:or half step.
70:12th root of 2
26:
9:
6:
4:
3:
2:
7669:
7658:
7655:
7653:
7650:
7649:
7647:
7631:
7628:
7626:
7623:
7621:
7618:
7616:
7613:
7611:
7608:
7606:
7603:
7601:
7598:
7597:
7595:
7591:
7585:
7582:
7580:
7577:
7574:
7573:Carnatic raga
7570:
7567:
7565:
7562:
7560:
7557:
7553:
7550:
7548:
7545:
7543:
7542:Turkish makam
7540:
7538:
7535:
7534:
7532:
7530:
7527:
7525:
7522:
7520:
7517:
7516:
7514:
7508:
7499:
7495:
7491:
7487:
7483:
7479:
7475:
7471:
7465:
7462:
7460:
7457:
7455:
7452:
7450:
7447:
7444:
7440:
7439:quarter-comma
7436:
7433:
7432:
7430:
7428:
7424:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7398:
7395:
7393:
7390:
7388:
7385:
7382:
7378:
7375:
7373:
7370:
7368:
7365:
7363:
7360:
7358:
7355:
7353:
7350:
7348:
7345:
7343:
7340:
7339:
7337:
7335:
7331:
7328:
7326:
7322:
7316:
7315:Tonality flux
7313:
7311:
7308:
7306:
7303:
7301:
7298:
7296:
7293:
7291:
7288:
7286:
7283:
7279:
7276:
7274:
7271:
7270:
7269:
7266:
7264:
7261:
7259:
7256:
7254:
7251:
7249:
7246:
7245:
7243:
7241:
7237:
7231:
7228:
7226:
7223:
7221:
7218:
7216:
7213:
7211:
7208:
7206:
7203:
7201:
7198:
7196:
7193:
7191:
7188:
7186:
7183:
7181:
7178:
7176:
7173:
7172:
7170:
7166:
7162:
7155:
7150:
7148:
7143:
7141:
7136:
7135:
7132:
7118:
7100:
7098:
7095:
7092:
7088:
7086:
7083:
7080:
7076:
7073:
7070:
7066:
7063:
7059:
7057:
7055:
7050:
7048:
7045:
7043:
7040:
7038:
7035:
7034:
7024:
7021:
7019:
7016:
7013:
7009:
7005:
7003:
6999:
6995:
6991:
6990:
6985:
6983:
6982:0-87013-290-3
6979:
6975:
6971:
6968:
6964:
6963:
6943:
6938:
6934:
6933:
6928:
6924:
6920:
6915:
6910:
6906:
6902:
6898:
6893:
6889:
6887:1-85233-797-4
6883:
6879:
6874:
6870:
6864:
6860:
6859:
6853:
6849:
6847:0-306-80106-X
6843:
6838:
6837:
6830:
6826:
6821:
6817:
6811:
6807:
6806:
6801:
6797:
6793:
6788:
6777:
6773:
6768:
6764:
6758:
6754:
6753:
6747:
6743:
6739:
6735:
6731:
6727:
6723:
6716:
6711:
6707:
6705:9780824082314
6701:
6697:
6696:
6690:
6681:on 2012-03-05
6680:
6676:
6675:
6669:
6665:
6661:
6660:
6655:
6651:
6647:
6642:
6638:
6632:
6628:
6627:
6621:
6617:
6613:
6609:
6605:
6601:
6597:
6592:
6588:
6582:
6578:
6577:
6571:
6561:on 2012-03-15
6560:
6556:
6552:
6547:
6543:
6537:
6533:
6532:
6526:
6522:
6516:
6512:
6511:
6505:
6501:
6495:
6491:
6490:
6484:
6483:
6471:, p. 56.
6470:
6465:
6458:
6453:
6446:
6441:
6434:
6429:
6421:
6414:
6407:
6402:
6395:
6390:
6383:
6378:
6371:
6364:
6355:
6349:
6345:
6341:
6335:
6329:, p. 78.
6328:
6323:
6316:
6311:
6304:
6299:
6292:
6287:
6280:
6275:
6266:
6259:
6254:
6247:
6242:
6228:on 2011-07-25
6227:
6223:
6217:
6210:
6205:
6198:
6193:
6186:
6181:
6174:
6169:
6162:
6157:
6150:
6145:
6138:
6133:
6131:
6123:
6122:Lienhard 1997
6118:
6116:
6108:
6103:
6101:
6093:
6088:
6081:
6076:
6069:
6064:
6056:
6052:
6048:
6044:
6043:
6035:
6028:
6023:
6016:
6011:
6009:
6007:
5999:
5998:Robinson 1980
5994:
5987:
5982:
5980:
5972:
5967:
5965:
5957:
5952:
5948:
5931:
5924:
5920:
5916:
5912:
5908:
5904:
5900:
5896:
5892:
5888:
5884:
5878:
5874:
5859:
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5839:
5836:
5834:
5831:
5828:
5825:
5823:
5820:
5818:
5815:
5814:
5808:
5806:
5802:
5798:
5794:
5790:
5785:
5783:
5779:
5775:
5769:
5767:
5761:
5760:Scale (music)
5751:
5748:
5744:
5739:
5714:
5607:
5604:
5521:
5518:
5432:
5429:
5362:
5359:
5358:
5357:
5320:
5248:
5245:
5178:
5177:Lesser diesis
5175:
5108:
5105:
5038:
5035:
5034:
5033:
5031:
4995:
4863:
4848:
4845:
4840:
4832:
4820:
4818:
4815:
4812:
4807:
4799:
4796:
4793:
4791:
4788:
4785:
4784:
4780:
4777:
4772:
4764:
4762:= 1.92857...
4752:
4749:
4748:
4744:
4741:
4736:
4728:
4726:= 1.89843...
4716:
4714:
4711:
4710:
4706:
4703:
4698:
4690:
4688:= 1.88235...
4678:
4676:
4673:
4672:
4668:
4665:
4660:
4652:
4640:
4638:
4635:
4634:
4630:
4627:
4622:
4614:
4612:= 1.87288...
4602:
4600:
4597:
4596:
4592:
4589:
4584:
4576:
4574:= 1.85714...
4564:
4561:
4553:
4531:
4524:
4520:
4517:
4512:
4504:
4502:= 1.80203...
4492:
4490:
4487:
4486:
4482:
4479:
4474:
4466:
4454:
4452:
4449:
4448:
4444:
4441:
4436:
4428:
4426:= 1.77777...
4416:
4414:
4411:
4410:
4406:
4403:
4398:
4390:
4378:
4376:
4373:
4365:
4343:
4334:
4322:
4318:
4315:
4310:
4302:
4300:= 1.71428...
4290:
4288:
4285:
4284:
4280:
4277:
4272:
4264:
4252:
4250:
4247:
4246:
4242:
4239:
4234:
4226:
4224:= 1.68421...
4214:
4212:
4209:
4208:
4204:
4199:
4191:
4189:= 1.66666...
4179:
4177:
4174:
4173:
4169:
4164:
4156:
4154:= 1.66478...
4144:
4142:
4139:
4131:
4109:
4102:
4098:
4095:
4090:
4082:
4080:= 1.60180...
4070:
4068:
4065:
4064:
4060:
4057:
4052:
4044:
4032:
4030:
4027:
4026:
4022:
4019:
4014:
4006:
4004:= 1.58024...
3994:
3992:
3989:
3988:
3984:
3981:
3976:
3968:
3966:= 1.57142...
3956:
3954:
3951:
3950:
3946:
3943:
3938:
3930:
3928:= 1.55555...
3918:
3916:
3913:
3905:
3883:
3874:
3862:
3858:
3855:
3850:
3842:
3830:
3828:
3825:
3824:
3820:
3815:
3807:
3805:= 1.47981...
3795:
3793:
3790:
3782:
3760:
3753:
3749:
3746:
3741:
3733:
3721:
3718:
3717:
3713:
3710:
3705:
3697:
3695:= 1.42857...
3685:
3683:
3680:
3679:
3675:
3672:
3667:
3659:
3657:= 1.42382...
3647:
3645:
3642:
3641:
3637:
3634:
3629:
3621:
3619:= 1.42222...
3609:
3607:
3604:
3603:
3599:
3596:
3591:
3583:
3571:
3569:
3566:
3565:
3561:
3558:
3553:
3545:
3543:= 1.40466...
3533:
3531:
3528:
3527:
3523:
3520:
3515:
3507:
3495:
3493:
3490:
3489:
3485:
3482:
3477:
3469:
3467:= 1.38888...
3457:
3454:
3446:
3424:
3415:
3403:
3399:
3396:
3391:
3383:
3381:= 1.35152...
3371:
3369:
3366:
3365:
3361:
3358:
3353:
3345:
3343:= 1.33333...
3333:
3331:
3328:
3320:
3298:
3291:
3287:
3284:
3279:
3271:
3269:= 1.28571...
3259:
3257:
3254:
3253:
3249:
3246:
3241:
3233:
3231:= 1.27272...
3221:
3219:
3216:
3215:
3211:
3208:
3203:
3195:
3183:
3181:
3178:
3177:
3173:
3170:
3165:
3157:
3145:
3143:
3140:
3139:
3135:
3132:
3127:
3119:
3117:= 1.24859...
3107:
3105:
3102:
3094:
3072:
3065:
3061:
3058:
3053:
3045:
3043:= 1.20135...
3033:
3031:
3028:
3027:
3023:
3020:
3015:
3007:
2995:
2993:
2990:
2989:
2985:
2980:
2972:
2960:
2958:
2955:
2954:
2950:
2945:
2937:
2935:= 1.18518...
2925:
2923:
2920:
2919:
2915:
2910:
2902:
2900:= 1.18181...
2890:
2888:
2885:
2884:
2880:
2875:
2867:
2865:= 1.16666...
2855:
2853:
2850:
2842:
2820:
2811:
2799:
2795:
2792:
2787:
2779:
2777:= 1.14285...
2767:
2765:
2762:
2761:
2757:
2754:
2749:
2741:
2729:
2727:
2724:
2723:
2719:
2714:
2706:
2704:= 1.11111...
2694:
2692:
2689:
2688:
2684:
2679:
2671:
2669:= 1.10985...
2659:
2657:
2654:
2646:
2624:
2617:
2613:
2610:
2605:
2597:
2585:
2583:
2580:
2579:
2575:
2572:
2567:
2559:
2557:= 1.07692...
2547:
2544:
2543:
2539:
2536:
2531:
2523:
2521:= 1.07142...
2511:
2509:
2506:
2505:
2501:
2498:
2493:
2485:
2483:= 1.06787...
2473:
2471:
2468:
2467:
2463:
2460:
2455:
2447:
2445:= 1.06666...
2435:
2433:
2430:
2429:
2425:
2422:
2417:
2409:
2397:
2395:
2392:
2391:
2387:
2382:
2374:
2372:= 1.05882...
2362:
2359:
2358:
2354:
2349:
2341:
2339:= 1.05555...
2329:
2326:
2325:
2321:
2316:
2308:
2306:= 1.05468...
2296:
2294:
2291:
2290:
2286:
2281:
2273:
2271:= 1.05349...
2261:
2259:
2256:
2255:
2251:
2246:
2238:
2236:= 1.05263...
2226:
2223:
2222:
2218:
2213:
2205:
2193:
2191:
2188:
2187:
2183:
2180:
2175:
2167:
2165:= 1.04761...
2155:
2152:
2151:
2147:
2144:
2139:
2131:
2129:= 1.04166...
2119:
2117:
2114:
2113:
2109:
2106:
2101:
2093:
2091:= 1.03703...
2081:
2079:
2076:
2068:
2046:
2037:
2025:
2021:
2018:
2013:
2005:
1993:
1991:
1988:
1985:
1980:
1972:
1969:
1966:
1964:
1961:
1958:
1957:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1922:
1919:
1917:
1907:
1841:
1799:
1796:
1795:quarter tones
1706:
1654:
1644:
1642:
1588:
1578:
1482:
1456:
1446:
1437:
1435:
1426:
1417:
1389:
1386:
1384:
1373:
1370:
1367:
1359:
1353:
1349:
1344:
1339:
1336:
1334:
1327:
1323:
1304:
1301:
1299:
1288:
1285:
1282:
1274:
1268:
1264:
1259:
1254:
1251:
1249:
1242:
1238:
1225:
1211:
1203:
1176:
1160:
1142:
1139:
1136:
1128:
1122:
1118:
1113:
1106:
1102:
1098:
1093:
1089:
1080:
1065:
1055:
1053:
1048:
1035:
1032:
1026:
1023:
1018:
1014:
1008:
1004:
994:
992:
988:
984:
975:
961:
958:
956:
953:
950:
949:
945:
942:
940:
937:
934:
933:
929:
926:
924:
921:
918:
917:
913:
910:
908:
905:
902:
901:
897:
894:
892:
889:
886:
885:
881:
878:
876:
873:
870:
869:
865:
862:
859:
856:
855:
847:
845:
841:
837:
833:
829:
824:
820:
818:
812:
810:
806:
802:
798:
794:
788:
786:
782:
778:
774:
770:
765:
760:
749:
747:
742:
740:
731:
728:
726:
723:
722:
718:
715:
712:
709:
708:
704:
701:
698:
695:
694:
690:
687:
684:
681:
680:
676:
673:
670:
667:
666:
662:
659:
656:
653:
652:
648:
645:
642:
639:
638:
634:
631:
628:
625:
624:
620:
617:
614:
611:
610:
606:
603:
600:
597:
596:
593:
590:
587:
584:
583:
579:
576:
573:
570:
569:
565:
562:
559:
556:
555:
552:
549:
543:
540:
535:
531:
521:
518:
513:
509:
505:
501:
497:
493:
489:
479:
475:
472:
467:
465:
461:
457:
453:
449:
447:
440:Early history
427:
421:
412:
410:
400:
396:
394:
390:
386:
381:
377:
372:
359:
356:
351:
347:
343:
338:
333:
327:
323:
318:
308:
305:
292:
289:
284:
279:
273:
269:
264:
254:
233:
229:
220:
216:
206:
198:
189:
187:
181:
176:
171:
159:Early history
151:
147:
144:
134:
124:
122:
118:
114:
110:
106:
101:
99:
94:
92:
71:
67:
63:
59:
55:
51:
42:
32:
19:
7630:Lambda scale
7537:Arabic maqam
7494:Werckmeister
7346:
7325:Temperaments
7108:ARDINALITIES
7078:
7053:
7007:
6988:
6973:
6966:
6946:. Retrieved
6931:
6907:(2): 61–74.
6904:
6900:
6877:
6857:
6835:
6824:
6804:
6791:
6780:. Retrieved
6775:
6751:
6725:
6721:
6694:
6683:, retrieved
6679:the original
6673:
6658:
6645:
6625:
6599:
6595:
6575:
6563:. Retrieved
6559:the original
6554:
6530:
6509:
6488:
6464:
6452:
6440:
6433:Barbour 2004
6428:
6419:
6413:
6401:
6389:
6377:
6363:
6354:
6339:
6334:
6322:
6310:
6298:
6286:
6274:
6265:
6258:Jorgens 1986
6253:
6241:
6230:. Retrieved
6226:the original
6216:
6204:
6199:, p. 8.
6197:Barbour 2004
6192:
6185:Galilei 1584
6180:
6168:
6156:
6151:, p. 7.
6149:Barbour 2004
6144:
6087:
6075:
6068:Barbour 2004
6063:
6046:
6040:
6034:
6022:
5993:
5986:Kuttner 1975
5971:Kuttner 1975
5951:
5930:
5918:
5915:twelve equal
5914:
5910:
5906:
5902:
5898:
5894:
5890:
5886:
5882:
5877:
5786:
5770:
5763:
5699:
5316:
4859:
2407:= 1.0625...
1913:
1805:
1712:
1656:
1647:7 limit
1590:
1581:5 limit
1458:
1449:3 limit
1443:
1431:
1226:
1161:
1081:
1067:
1049:
995:
980:
959:1.059490385
943:1.059322034
927:1.059546514
923:Simon Stevin
911:1.059463094
879:1.060070671
875:He Chengtian
832:atonal music
828:polytonality
825:
821:
813:
789:
773:passacaglias
755:
743:
736:
544:
534:Simon Stevin
527:
524:Simon Stevin
468:
450:
443:
405:
392:
376:Matteo Ricci
373:
309:
306:
255:
231:
227:
218:
203:
175:He Chengtian
172:
162:
148:
143:Simon Stevin
130:
102:
95:
53:
49:
48:
7625:Delta scale
7620:Gamma scale
7610:Alpha scale
7512:non-Western
7510:Traditional
7205:Pitch class
7185:Millioctave
7168:Measurement
6698:. Garland.
6445:Partch 1979
6246:Wilson 1997
5935:performers.
4543:1.88774...
4355:1.78179...
4121:1.68179...
3895:1.58740...
3772:1.49830...
3436:1.41421...
3310:1.33484...
3193:= 1.265625
3084:1.25992...
2832:1.18920...
2636:1.12246...
2058:1.05946...
1954:Difference
1641:major triad
985:, i.e. the
752:Baroque era
585:whole tone
492:John Wilson
454:(father of
446:Aristoxenus
433: 1605
178: [
7646:Categories
7615:Beta scale
7593:Non-octave
7584:Tetrachord
7486:Kirnberger
7449:Schismatic
7079:Jim Kukula
6948:27 October
6782:2014-10-05
6685:2012-03-20
6662:. Venice:
6659:Il Fronimo
6565:2020-04-06
6394:Moody 2003
6279:Cohen 1987
6232:2012-06-14
5864:References
5803:, and the
5758:See also:
5384:2 × 3 × 7
5107:Diaschisma
3581:= 1.40625
2726:Major tone
2691:Minor tone
1651:See also:
1585:See also:
1453:See also:
1062:See also:
729:1.0592000
724:full tone
716:1.0594788
702:1.0593477
696:five-tone
688:1.0595558
674:1.0597014
668:four-tone
660:1.0594845
646:1.0593975
632:1.0594046
618:1.0594758
604:1.0600904
591:1.0593781
577:1.0595465
462:", 24 + 1
7605:A12 scale
7559:Octoechos
7524:Shí-èr-lǜ
7473:Irregular
7290:Otonality
7230:Microtone
6616:0003-3790
6173:True 2018
6080:Hart 1998
5943:Citations
5869:Footnotes
5766:serialism
5606:Jubilisma
4262:= 1.6875
2970:= 1.1875
1481:inversion
1387:≈
1371:−
1302:≈
1286:−
1212:), and F#
1140:−
1033:≈
907:Zhu Zaiyu
840:serialism
571:semitone
520:in 1588.
464:ricercars
391:wrote in
205:Zhu Zaiyu
192:Zhu Zaiyu
133:Zhu Zaiyu
121:frequency
7490:Vallotti
7443:septimal
7435:Meantone
7195:Interval
7065:Archived
6929:(1885).
6656:(1584).
6358:Vm7 6214
6303:Cho 2003
6291:Cho 2003
6107:Cho 2010
6055:20753625
5811:See also
5805:in scale
4846:1200.00
4778:1137.04
4742:1109.78
4704:1095.04
4666:1088.27
4650:= 1.875
4628:1086.31
4590:1071.70
4518:1019.55
4480:1017.60
4340:♭
4331:♯
3880:♭
3871:♯
3421:♭
3412:♯
2817:♭
2808:♯
2739:= 1.125
2043:♭
2034:♯
1479:and its
1440:By limit
1210:middle C
1204:), and C
1036:1.059463
983:semitone
744:In 1630
640:tritone
488:Ricercar
480:wrote a
471:lutenist
91:semitone
7579:Slendro
7529:Dastgah
7454:Miracle
7417:96-tone
7412:72-tone
7407:58-tone
7402:53-tone
7397:41-tone
7392:34-tone
7387:31-tone
7377:24-tone
7372:23-tone
7367:22-tone
7362:19-tone
7357:17-tone
7352:15-tone
7347:12-tone
7278:7-limit
7273:5-limit
6479:Sources
5895:12-TEDO
5754:Subsets
5731:⁄
5721:⁄
5706:⁄
5688:
5676:
5670:
5658:
5654:
5651:524288
5642:
5638:
5626:
5622:
5610:
5600:
5588:
5584:
5572:
5568:
5565:524288
5556:
5552:
5540:
5536:
5524:
5513:
5501:
5497:
5485:
5481:
5478:524288
5469:
5463:
5451:
5447:
5435:
5425:
5413:
5409:
5397:
5393:
5381:
5377:
5365:
5352:
5340:
5335:
5323:
5311:
5299:
5295:
5292:524288
5283:
5279:
5267:
5263:
5251:
5241:
5229:
5225:
5222:524288
5213:
5209:
5197:
5193:
5181:
5171:
5159:
5155:
5152:524288
5143:
5139:
5127:
5123:
5111:
5101:
5089:
5085:
5082:524288
5073:
5069:
5057:
5053:
5041:
5037:Schisma
5026:
5014:
5010:
4998:
4990:
4978:
4974:
4971:524288
4962:
4958:
4946:
4942:
4930:
4926:
4914:
4910:
4898:
4894:
4882:
4878:
4866:
4825:⁄
4781:+37.04
4757:⁄
4721:⁄
4683:⁄
4669:-11.73
4645:⁄
4631:-13.69
4607:⁄
4593:-28.30
4569:⁄
4536:√
4521:+19.55
4497:⁄
4483:+17.60
4459:⁄
4442:996.09
4421:⁄
4407:-31.17
4404:968.83
4388:= 1.75
4383:⁄
4348:√
4319:+33.13
4316:933.13
4295:⁄
4278:905.87
4257:⁄
4240:902.49
4219:⁄
4205:884.36
4184:⁄
4170:882.40
4149:⁄
4114:√
4099:+15.64
4096:815.64
4075:⁄
4061:+13.69
4058:813.69
4037:⁄
4020:792.18
3999:⁄
3985:-17.51
3982:782.49
3961:⁄
3947:-35.08
3944:764.92
3923:⁄
3888:√
3856:701.96
3835:⁄
3821:678.49
3800:⁄
3765:√
3750:+31.28
3747:631.28
3731:= 1.44
3726:⁄
3714:+17.49
3711:617.49
3690:⁄
3676:+11.73
3673:611.73
3652:⁄
3635:609.78
3614:⁄
3597:590.22
3576:⁄
3562:-11.73
3559:588.27
3538:⁄
3524:-17.49
3521:582.51
3500:⁄
3486:-31.28
3483:568.72
3462:⁄
3429:√
3400:+21.51
3397:521.51
3376:⁄
3359:498.04
3338:⁄
3303:√
3288:+35.08
3285:435.08
3264:⁄
3250:+17.51
3247:417.51
3226:⁄
3209:407.82
3188:⁄
3174:-13.69
3171:386.31
3155:= 1.25
3150:⁄
3136:-15.64
3133:384.36
3112:⁄
3077:√
3062:+17.60
3059:317.60
3038:⁄
3024:+15.64
3021:315.64
3000:⁄
2986:297.51
2965:⁄
2951:294.13
2930:⁄
2916:289.21
2895:⁄
2881:266.87
2860:⁄
2825:√
2796:+31.17
2793:231.17
2772:⁄
2755:203.91
2734:⁄
2720:182.40
2699:⁄
2685:180.45
2664:⁄
2629:√
2614:+33.24
2611:133.24
2595:= 1.08
2590:⁄
2576:+28.30
2573:128.30
2552:⁄
2540:+19.44
2537:119.44
2516:⁄
2502:+13.69
2499:113.69
2478:⁄
2464:+11.73
2461:111.73
2440:⁄
2423:104.96
2402:⁄
2367:⁄
2334:⁄
2301:⁄
2266:⁄
2231:⁄
2203:= 1.05
2198:⁄
2184:-19.46
2160:⁄
2148:-29.33
2124:⁄
2110:-37.04
2086:⁄
2051:√
1998:⁄
1904:
1892:
1888:
1876:
1872:
1860:
1856:
1844:
1836:
1824:
1820:
1808:
1791:
1779:
1775:
1763:
1759:
1747:
1743:
1731:
1727:
1715:
1703:
1691:
1687:
1675:
1671:
1659:
1637:
1625:
1621:
1609:
1605:
1593:
1575:
1563:
1559:
1547:
1540:
1528:
1520:
1508:
1499:
1487:
1475:
1463:
1390:369.994
1305:261.626
895:18:17
759:Bocquet
719:5297.2
705:5612.3
663:6674.2
649:7071.1
635:7491.5
612:ditone
580:9438.7
514:in his
508:polemic
506:in his
504:Zarlino
460:Fronimo
248:√
241:√
127:History
84:⁄
74:√
7547:Mugham
7533:Maqam
7427:Linear
7381:pieces
7342:6-tone
7263:Hexany
7190:Savart
7104:AVORED
7056:(1753)
7000:
6980:
6974:Tuning
6884:
6865:
6844:
6812:
6759:
6742:850355
6740:
6702:
6633:
6614:
6583:
6538:
6517:
6496:
6346:
6053:
5911:12-EDO
5903:12-ED2
5887:12-TET
5799:, the
5795:, the
5791:, the
5780:, and
5645:531441
5629:2 × 5
5559:531441
5543:2 × 3
5472:531441
5460:3 × 7
5286:531441
5270:2 × 3
5216:531441
5146:531441
5136:3 × 5
5076:531441
5060:3 × 5
5050:32768
5023:2 × 5
4965:531441
4928:) and
4862:commas
4856:Commas
4817:Octave
4794:2 = 2
4745:+9.78
4707:-4.96
4464:= 1.8
4445:-3.91
4281:+5.87
4243:+2.49
4042:= 1.6
4023:-7.82
3859:+1.96
3840:= 1.5
3802:177147
3798:262144
3638:+9.78
3600:-9.78
3505:= 1.4
3378:131072
3374:177147
3362:-1.96
3212:+7.82
3005:= 1.2
2758:+3.91
2426:+4.96
2388:98.95
2355:93.60
2322:92.18
2287:90.22
2252:88.80
2219:84.47
2181:80.54
2145:70.67
2107:62.96
1990:Unison
1967:2 = 1
1939:Cents
1842:after
1393:
1308:
1220:and F#
1202:440 Hz
962:100.0
930:100.1
914:100.0
882:101.0
866:Cents
863:Ratio
842:, and
807:, and
563:Ratio
415:Europe
380:Jesuit
58:octave
7564:Pelog
7552:Muqam
7498:Young
7459:Magic
7334:Equal
7268:Limit
7175:Pitch
7116:CALES
6994:Porto
6738:JSTOR
6718:(PDF)
6051:JSTOR
5656:) × (
5586:) × (
5570:) × (
5549:5 ×v7
5499:) × (
5483:) × (
5411:) × (
5337:, or
5297:) × (
5227:) × (
5157:) × (
5120:2025
5087:) × (
5044:32805
4933:1024
4813:1200
4559:1100
4499:32768
4495:59049
4371:1000
4151:19683
4147:32768
3040:16384
3036:19683
2666:59049
2662:65536
1916:cents
1910:Table
1175:hertz
1052:cents
951:1630
946:99.8
935:1630
919:1585
903:1581
898:99.0
887:1580
860:Name
857:Year
713:5296
699:5611
691:5946
685:5944
677:6299
671:6298
657:6674
643:7071
629:7491
621:7937
615:7936
607:8409
601:8404
588:8909
574:9438
557:Tone
182:]
154:China
117:hertz
54:12-ET
7569:Raga
7180:Cent
7121:ETER
7119:by P
6998:ISBN
6978:ISBN
6950:2020
6882:ISBN
6863:ISBN
6842:ISBN
6810:ISBN
6757:ISBN
6700:ISBN
6631:ISBN
6612:ISSN
6581:ISBN
6536:ISBN
6515:ISBN
6494:ISBN
6344:ISBN
5685:224
5672:) ×
5597:224
5510:224
5422:224
5374:125
5343:3×5
5332:224
5260:625
5190:125
5114:2048
4838:play
4830:= 2
4805:play
4770:play
4734:play
4696:play
4658:play
4620:play
4609:2187
4605:4096
4582:play
4551:play
4538:2048
4534:2 =
4510:play
4472:play
4434:play
4396:play
4363:play
4346:2 =
4308:Play
4270:play
4232:play
4197:play
4162:play
4137:900
4129:play
4112:2 =
4088:play
4077:4096
4073:6561
4050:play
4012:play
3974:play
3936:play
3911:800
3903:play
3886:2 =
3848:play
3813:play
3788:700
3780:play
3763:2 =
3739:play
3703:Play
3665:play
3627:play
3589:Play
3551:play
3536:1024
3513:play
3475:play
3452:600
3444:play
3427:2 =
3389:play
3351:play
3326:500
3318:play
3301:2 =
3277:play
3239:Play
3201:play
3163:play
3125:play
3114:6561
3110:8192
3100:400
3092:play
3075:2 =
3051:play
3013:play
2978:play
2943:play
2908:play
2873:play
2848:300
2840:play
2823:2 =
2785:play
2747:play
2712:play
2677:play
2652:200
2644:play
2627:2 =
2603:play
2565:play
2529:play
2491:play
2480:2048
2476:2187
2453:play
2415:play
2380:play
2347:play
2314:play
2279:play
2244:play
2211:play
2173:play
2137:Play
2099:play
2074:100
2066:play
2049:2 =
2011:play
2003:= 1
1978:play
1689:and
1623:and
1561:and
1192:and
871:400
844:jazz
512:lute
378:, a
236:樂律全書
223:律暦融通
215:Ming
109:A440
7125:UCH
6909:doi
6730:doi
6604:doi
5905:),
5897:),
5889:),
5708:600
5679:225
5667:80
5640:= (
5619:49
5591:225
5554:= (
5533:35
5504:225
5494:80
5444:63
5416:225
5406:80
5395:= (
5368:126
5349:2×7
5326:225
5308:80
5254:648
5238:80
5211:= (
5184:128
5168:80
5141:= (
5098:80
5071:= (
5007:80
4939:729
4907:512
4901:729
4786:12
4723:128
4719:243
4526:11
4335:or
4324:10
3997:128
3875:or
3767:128
3654:512
3650:729
3540:729
3416:or
2812:or
2303:128
2299:135
2268:243
2264:256
2038:or
1895:19
1879:19
1863:17
1827:17
1811:17
1782:22
1766:13
1750:13
1734:11
1718:11
1694:10
1566:32
1550:27
1531:16
1340:440
1255:440
838:or
360:128
210:朱載堉
138:朱載堉
7648::
7496:,
7492:,
7488:,
7441:,
7077:,
6925:;
6903:.
6899:.
6774:.
6736:.
6726:19
6724:.
6720:.
6610:.
6600:44
6598:.
6553:.
6342:.
6129:^
6114:^
6099:^
6047:62
6045:.
6005:^
5978:^
5963:^
5807:.
5776:,
5723:64
5719:81
5661:81
5624:=
5613:50
5608::
5581:80
5575:81
5538:=
5527:36
5522::
5488:81
5465:=
5449:=
5438:64
5433::
5400:81
5379:=
5363::
5302:81
5281:=(
5265:=
5249::
5232:81
5200:2
5195:=
5179::
5162:81
5125:=
5109::
5092:81
5055:=
5039::
5001:81
4981:3
4949:2
4917:3
4885:1
4869:1
4849:0
4797:2
4759:14
4755:27
4685:17
4681:32
4643:15
4567:13
4419:16
4350:32
4293:12
4259:16
4255:27
4221:19
4217:32
4104:9
4001:81
3959:11
3921:14
3864:8
3755:7
3728:25
3724:36
3688:10
3616:45
3612:64
3578:32
3574:45
3464:18
3460:25
3405:6
3305:32
3293:5
3228:11
3224:14
3190:64
3186:81
3067:4
2967:16
2963:19
2932:27
2928:32
2897:11
2893:13
2801:3
2697:10
2619:2
2592:25
2588:27
2554:13
2550:14
2518:14
2514:15
2442:15
2438:16
2404:16
2400:17
2369:17
2365:18
2336:18
2332:19
2233:19
2229:20
2200:20
2196:21
2162:21
2158:22
2126:24
2122:25
2088:27
2084:28
2027:1
2022:0
2019:0
1986:0
1970:1
1959:0
1901:18
1885:12
1847:7
1833:12
1817:16
1788:13
1772:11
1678:7
1662:7
1628:8
1612:5
1596:5
1572:27
1556:16
1542:),
1511:9
1501:),
1490:4
1466:3
1436:.
1374:49
1368:46
1354:12
1328:46
1289:49
1283:40
1269:12
1243:40
1224::
1123:12
1054:.
1027:12
1009:12
993::
830:,
803:,
799:,
551::
430:c.
339:84
328:12
285:12
274:12
180:zh
165:c.
86:12
7575:)
7571:(
7500:)
7484:(
7480:/
7445:)
7437:(
7383:)
7379:(
7153:e
7146:t
7139:v
7123:B
7114:S
7112:F
7110:O
7106:C
7102:F
7093:"
7081:.
6952:.
6917:.
6911::
6905:9
6890:.
6871:.
6850:.
6818:.
6785:.
6765:.
6744:.
6732::
6708:.
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6639:.
6618:.
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6568:.
6544:.
6523:.
6502:.
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6260:.
6248:.
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6211:.
6124:.
6109:.
6082:.
6057:.
5925:.
5909:(
5901:(
5893:(
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5729:5
5704:1
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5682:/
5674:(
5664:/
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5632:/
5616:/
5602:)
5594:/
5578:/
5562:/
5546:/
5530:/
5515:)
5507:/
5491:/
5475:/
5467:(
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5454:2
5441:/
5427:)
5419:/
5403:/
5390:5
5387:/
5371:/
5354:.
5346:/
5329:/
5313:)
5305:/
5289:/
5276:5
5273:/
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5243:)
5235:/
5219:/
5206:5
5203:/
5187:/
5173:)
5165:/
5149:/
5133:/
5130:2
5117:/
5103:)
5095:/
5079:/
5066:2
5063:/
5047:/
5020:/
5017:3
5012:(
5004:/
4987:2
4984:/
4976:(
4968:/
4955:3
4952:/
4944:(
4936:/
4923:2
4920:/
4912:(
4904:/
4891:1
4888:/
4875:1
4872:/
4827:1
4823:2
4790:C
4647:8
4571:7
4530:B
4461:5
4457:9
4423:9
4385:4
4381:7
4337:B
4328:A
4297:7
4186:3
4182:5
4116:8
4108:A
4039:5
4035:8
3963:7
3925:9
3890:4
3877:A
3868:G
3837:2
3833:3
3759:G
3692:7
3502:5
3498:7
3431:2
3418:G
3409:F
3340:3
3336:4
3297:F
3266:7
3262:9
3152:4
3148:5
3079:2
3071:E
3002:5
2998:6
2862:6
2858:7
2827:2
2814:E
2805:D
2774:7
2770:8
2736:8
2732:9
2701:9
2631:2
2623:D
2053:2
2040:D
2031:C
2000:1
1996:1
1963:C
1898:/
1882:/
1869:9
1866:/
1853:5
1850:/
1830:/
1814:/
1785:/
1769:/
1756:8
1753:/
1740:8
1737:/
1724:8
1721:/
1700:7
1697:/
1684:5
1681:/
1668:4
1665:/
1634:5
1631:/
1618:3
1615:/
1602:4
1599:/
1569:/
1553:/
1537:9
1534:/
1526:(
1522:)
1517:8
1514:/
1506:(
1496:3
1493:/
1485:(
1477:)
1472:2
1469:/
1461:(
1400:z
1397:H
1377:)
1365:(
1360:)
1350:2
1345:(
1337:=
1324:P
1315:z
1312:H
1292:)
1280:(
1275:)
1265:2
1260:(
1252:=
1239:P
1222:4
1218:4
1214:4
1208:(
1206:4
1198:4
1194:a
1190:n
1184:a
1180:P
1169:n
1165:P
1146:)
1143:a
1137:n
1134:(
1129:)
1119:2
1114:(
1107:a
1103:P
1099:=
1094:n
1090:P
1075:n
1071:P
1024:1
1019:2
1015:=
1005:2
486:(
435:.
357:=
352:7
348:2
344:=
334:)
324:2
319:(
293:2
290:=
280:)
270:2
265:(
250:2
243:2
230:(
207:(
113:A
82:1
76:2
72:(
52:(
20:)
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