2328:). For higher primes, additional signs have been designed. To facilitate quick estimation of pitches, cents indications may be added (downward deviations below and upward deviations above the respective accidental). The convention used is that the cents written refer to the tempered pitch implied by the flat, natural, or sharp sign and the note name. One of the great advantages of any such a notation is that it allows the natural harmonic series to be precisely notated. A complete legend and fonts for the notation (see samples) are open source and available from Plainsound Music Edition. Thus a Pythagorean scale is
2349:
2343:
2337:
321:
276:
47:
401:
27:
7207:
138:
2385:, where they describe distinctions between musical intervals that are eliminated by that tuning system. A comma can be viewed as the distance between two musical intervals. When a given comma is tempered out in a tuning system, the ability to distinguish between those two intervals in that tuning is eliminated. For example, the difference between the
2270:
meantone), the comma is the opposite of a diminished second, and therefore the opposite of the above-listed differences. More exactly, in these tuning systems the diminished second is a descending interval, while the comma is its ascending opposite. For instance, the
Pythagorean comma (531441:524288,
2323:
and by Joe Monzo, who refers to it by the acronym HEWM (Helmholtz-Ellis-Wolf-Monzo). In the Sabat-Schweinitz design, syntonic commas are marked by arrows attached to the flat, natural or sharp sign, septimal commas using
Giuseppe Tartini's symbol, and undecimal quartertones using the common practice
2496:
The following table lists the number of steps used that correspond various just intervals in various tuning systems. Zeros indicate that the interval is a comma (i.e. is tempered out) in that particular equal temperament. All of the frequency ratios in the first column are linked to their wikipedia
5727:
The comma can also be considered to be the fractional interval that remains after a "full circle" of some repeated chosen interval; the repeated intervals are all the same size, in relative pitch, and all the tones produced are reduced or raised by whole octaves back to the octave surrounding the
1822:
The syntonic comma has a crucial role in the history of music. It is the amount by which some of the notes produced in
Pythagorean tuning were flattened or sharpened to produce just minor and major thirds. In Pythagorean tuning, the only highly consonant intervals were the
129:
are considered the same note because the 12-note
Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in
2090:
1972:
312:
5835:, etc. An interesting property of temperaments is that this difference remains whatever the tuning of the intervals forming the circle. In this sense, commas and similar minute intervals can never be completely tempered out, whatever the tuning.
58:
38:
313:
59:
39:
2271:
or about 23.5 cents) can be computed as the difference between a chromatic and a diatonic semitone, which is the opposite of a
Pythagorean diminished second (524288:531441, or about −23.5 cents).
5886:
There are also several intervals called commas, which are not technically commas because they are not rational fractions like those above, but are irrational approximations of them. These include the
1289:
5315:
2310:
pitch notation. Sabat and
Schweinitz take the "conventional" flats, naturals and sharps as a Pythagorean series of perfect fifths. Thus, a series of perfect fifths beginning with F proceeds
4944:
1377:
5538:
5390:
2361:
uses a "−" as an accidental to indicate a note is lowered a syntonic comma, or a "+" to indicate a note is raised a syntonic comma; however, Johnston's "basic scale" (the plain nominals
807:
663:
5686:
5612:
5240:
5092:
1983:
1865:
987:
36:
5464:
5166:
4648:
4426:
1747:
1571:
732:
871:
5018:
4870:
4796:
4722:
4574:
4500:
4352:
4278:
4204:
4130:
4056:
3316:
3242:
1810:
1637:
1453:
1071:
3982:
3168:
2724:
3908:
3834:
3760:
3686:
3612:
3538:
3464:
3390:
3094:
3020:
2946:
2872:
2798:
2650:
310:
56:
2319:
and so on. The advantage for musicians is that conventional reading of the basic fourths and fifths remains familiar. Such an approach has also been advocated by
497:. A schisma is not audible in many contexts, as its size is narrower than the smallest audible difference between tones (which is around six cents, also known as
1855:. For instance, if you decrease the frequency of E by a syntonic comma (81:80), C–E (a major third) and E–G (a minor third) become just: C–E is flattened by a
2115:. Since then, other tuning systems were developed, and the syntonic comma was used as a reference value to temper the perfect fifths throughout the family of
493:) are basic intervals that can be used as yardsticks to define some of the other commas. For instance, the difference between them is a small comma called
311:
268:
characterized by two different kinds of semitones (diatonic and chromatic), and hence by a comma of unique size. The same is true for
Pythagorean tuning.
359:, more than two kinds of semitones may be produced. Thus, a single tuning system may be characterized by several different commas. For instance, a
57:
2306:
worked together in Berlin to develop a method to exactly indicate pitches in staff notation. This method was called the extended
Helmholtz-Ellis
37:
2251:
In
Pythagorean tuning, and any kind of meantone temperament tuning system that tempers the fifth to a size larger than 700 cents (such as
6928:
5940:
2135:, and any kind of meantone temperament tuning system that tempers the fifth to a size smaller than 700 cents, the comma is a
1238:
6175:
6271:
214:
6430:
6196:
6127:
6080:
6050:
6018:
5992:
5283:
2274:
In each of the above-mentioned tuning systems, the above-listed differences have all the same size. For instance, in
6114:
see article "The
Extended Helmholtz-Ellis JI Pitch Notation: eine Notationsmetode für dienatürlichen Intervalle" in
165:
Translated in this context, "comma" means "a hair" as in "off by just a hair". The word "comma" came via Latin from
4912:
1345:
2085:{\displaystyle {\frac {32}{\ 27\ }}\cdot {\frac {81}{\ 80\ }}={\frac {2\cdot 3}{\ 1\cdot 5\ }}={\frac {6}{\ 5\ }}}
1967:{\displaystyle {\frac {\ 81\ }{64}}\cdot {\frac {\ 80\ }{81}}={\frac {\ 1\cdot 5\ }{4\cdot 1}}={\frac {\ 5\ }{4}}}
5506:
5358:
2283:
775:
631:
5945:
5654:
5580:
5208:
5060:
955:
7234:
5432:
5134:
4616:
4394:
1715:
1539:
700:
179:
839:
6720:
6700:
4986:
4838:
4764:
4690:
4542:
4468:
4320:
4246:
4172:
4098:
4024:
3284:
3210:
2400:
the diesis, and thus does not distinguish between the two different types of semitones. On the other hand,
1778:
1605:
1421:
1039:
3950:
3136:
2692:
6826:
4242:
3876:
3802:
3728:
3654:
3580:
3506:
3432:
3358:
3062:
2988:
2914:
2840:
2766:
2618:
2275:
2365:) is tuned to just-intonation and thus already includes the syntonic comma. Thus a Pythagorean scale is
6832:
6540:
4094:
498:
409:
360:
6393:
6356:
6615:
6154:
2401:
1852:
1832:
213:) may have a slightly different frequency, and the interval between them is a comma. For example, in
198:
6071:
Rasch, Rudolph (1988). "Farey systems of musical intonation". In Benitez, J.M.; et al. (eds.).
1851:
recognized that by slightly tempering the pitch of some notes, the Pythagorean thirds could be made
1763:
7229:
7210:
6948:
6887:
6264:
6159:
5915:
4316:
4020:
2358:
2303:
460:
90:, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the
7008:
6485:
5728:
starting pitch. The Pythagorean comma, for instance, is the difference obtained, say, between A
4168:
2132:
2100:
1477:
1411:
284:
119:
6302:
5972:
7003:
6814:
6042:
6037:
Rasch, Rudolph (2000). "A word or two on the tunings of Harry Partch". In Dunn, David (ed.).
4538:
7190:
6943:
6850:
6844:
5892:
3724:
3502:
2602:
2592:
2582:
2572:
2562:
2552:
2542:
2532:
2467:
2446:
2425:
2412:
2120:
2116:
1221:
1203:
1185:
1150:
1115:
1089:
594:
261:
131:
8:
6973:
6838:
6820:
5871:
5844:
4464:
3798:
3354:
2522:
2512:
2382:
1314:
7086:
6523:
6257:
5963:
5130:
4834:
3132:
2390:
2224:
1326:
328:
320:
178:: "the result or effect of cutting". A more complete etymology is given in the article
109:
7142:
364:
7028:
6968:
6529:
6505:
6493:
6474:
6288:
6236:
6192:
6171:
6123:
6076:
6046:
6014:
5988:
4908:
2492:, but does not temper out the diesis, septimal diesis, or septimal comma of Archytas.
2404:
does not temper out this comma, and thus it distinguishes between the two semitones.
2394:
2386:
2320:
2279:
2244:
2174:
2136:
1590:
1466:
1318:
1295:
472:
324:
246:
218:
91:
50:
7098:
6923:
6868:
6862:
6856:
6760:
6660:
6633:
6517:
6499:
6444:
6438:
6280:
5968:
5910:
5740:
after a circle of twelve just fifths. A circle of three just major thirds, such as
5428:
3280:
3206:
2836:
2348:
2342:
2336:
2204:
2194:
2190:
2164:
2154:
669:
550:
464:
275:
154:
75:
7185:
7066:
6790:
6578:
6468:
6462:
6450:
6119:
4686:
2435:
2307:
2234:
2214:
2184:
2104:
1856:
1844:
468:
356:
265:
6152:. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", p. 109,
5866:‑th odd prime (prime 2 being ignored because it represents the octave) and
347:). The interval from C to D is wider than in quarter-comma meantone (see above).
7046:
7040:
7034:
6806:
6333:
5887:
4982:
4390:
3428:
2456:
2439:
2180:
1828:
1768:
1643:
1595:
1383:
993:
921:
823:
690:
543:
483:
363:
of five-limit tuning produces a 12-tone scale with four kinds of semitones and
307:). The interval from C to D is narrower than in Pythagorean tuning (see below).
260:
Commas are often defined as the difference in size between two semitones. Each
166:
126:
99:
87:
79:
30:
86:
two different ways. Strictly speaking, there are only two kinds of comma, the
7223:
7170:
6938:
6879:
6740:
6680:
6566:
6554:
6419:
6382:
6339:
6149:
6058:
5851:
4612:
3058:
2762:
2688:
2240:
2230:
2200:
2096:
1840:
1836:
1824:
1406:
759:
122:
7137:
6998:
6644:
6401:
6364:
6213:
6187:
Rasch, Rudolf (2002). "Tuning and temperament". In Christensen, Th. (ed.).
5949:
5875:
5848:
5776:
5204:
2325:
2150:
1581:
1197:
1143:
1077:
538:
405:
371:
137:
83:
67:
2476:
tempers out the syntonic comma, as well as the comma defined by the ratio
467: 1" and "Interval 2", all intervals are presumed to be tuned in
7147:
6978:
6413:
6407:
6376:
6370:
6163:
3650:
3576:
2984:
2910:
2220:
2210:
2170:
2160:
1848:
618:
228:
46:
2143:
minor second and augmented unison (also known as diatonic and chromatic
400:
7180:
7074:
6572:
6138:
5056:
3946:
3872:
2299:
1215:
1178:
1082:
877:
829:
292:
186:
26:
7080:
6584:
5920:
5279:
4760:
2112:
2108:
389:
1819:
Many other commas have been enumerated and named by microtonalists.
102:, which can be defined, for instance, as the difference between an F
7112:
6560:
6308:
5576:
2460:
2144:
1831:. The Pythagorean major third (81:64) and minor third (32:27) were
525:
417:
254:
94:, "the difference between twelve 5ths and seven octaves". The word
370:
The size of commas is commonly expressed and compared in terms of
7118:
7106:
7092:
6511:
6456:
6166:(2006). "A Notation System for Extended Just Intonation" (2003),
5650:
5502:
5354:
764:
738:
561:
494:
20:
2463:, but does not temper out the septimal diesis or syntonic comma.
7175:
7152:
7060:
7054:
6345:
6327:
6087:= Source for 32:33 as difference between 11:16 & 2:3 .
2614:
2287:
2139:, which can be equivalently defined as the difference between:
1459:
753:
684:
385:
280:
250:
153:
are both approximated by the same interval although they are a
6097:
335:), or interval between enharmonically equivalent notes (from D
19:
This article is about the interval. For the silent pause, see
6548:
2421:
tempers out the diesis, as well as a variety of other commas.
2103:, which permitted the full development of music with complex
253:, is an easily audible comma (its size is more than 40% of a
6249:
5890:
and Mercator's commas, and the pitch-to-pitch step size in
1977:
and at the same time E–G is sharpened to the just ratio of
408:. Equal-tempered semitone added for comparison. JND is the
6116:
Mikrotöne und mehr – Auf György Ligetis Hamburger Pfaden
1284:{\displaystyle {\bigl (}2{\bigr )}^{\tfrac {\ 1\ }{53}}}
16:
Very small interval arising from discrepancies in tuning
5874:. Subsequent commas are in prime limits, each the next
5659:
5585:
5511:
5437:
5363:
5288:
5213:
5139:
5065:
4991:
4917:
4843:
4769:
4695:
4621:
4547:
4473:
4399:
4325:
4251:
4177:
4103:
4029:
3955:
3881:
3807:
3733:
3659:
3585:
3511:
3437:
3363:
3289:
3215:
3141:
3067:
2993:
2919:
2845:
2771:
2697:
2623:
1783:
1720:
1610:
1544:
1426:
1350:
1261:
1044:
960:
844:
780:
705:
636:
6139:
Tonalsoft Encyclopaedia article about 'HEWM' notation
5657:
5583:
5509:
5435:
5361:
5286:
5211:
5137:
5063:
4989:
4915:
4841:
4767:
4693:
4619:
4545:
4471:
4397:
4323:
4249:
4175:
4101:
4027:
3953:
3879:
3805:
3731:
3657:
3583:
3509:
3435:
3361:
3287:
3213:
3139:
3065:
2991:
2917:
2843:
2769:
2695:
2621:
1986:
1868:
1843:, forcing them to write music with relatively simple
1781:
1718:
1608:
1542:
1424:
1348:
1241:
1042:
958:
842:
778:
703:
634:
6039:
Harry Partch: An anthology of critical perspectives
2393:is called the diesis. The widely used 12 tone
245:are not exactly the same note, as they would be in
5854:. The first comma of the comma sequence is in the
5847:through a unique sequence of commas at increasing
5680:
5606:
5532:
5458:
5384:
5309:
5234:
5160:
5086:
5012:
4938:
4864:
4790:
4716:
4642:
4568:
4494:
4420:
4346:
4272:
4198:
4124:
4050:
3976:
3902:
3828:
3754:
3680:
3606:
3532:
3458:
3384:
3310:
3236:
3162:
3088:
3014:
2940:
2866:
2792:
2718:
2644:
2084:
1966:
1804:
1741:
1631:
1565:
1447:
1371:
1283:
1065:
981:
865:
801:
726:
657:
431:is the augmented unison (chromatic semitone), and
6158:, vol. 29, no. 2 (Summer 1991), pp. 106–137. and
2381:Commas are frequently used in the description of
1835:, and this prevented musicians from freely using
7221:
5881:
416:In the column below labeled "Difference between
5310:{\displaystyle {\tfrac {\ 2109375\ }{2097152}}}
404:Comparison of the size of different commas, in
395:
5791:. A circle of four just minor thirds, such as
6265:
6189:The Cambridge History of Western Music Theory
6168:"Maximum clarity" and Other Writings on Music
4939:{\displaystyle {\tfrac {\ 531441\ }{524288}}}
1372:{\displaystyle {\tfrac {\ 531441\ }{524288}}}
1255:
1244:
6191:. Cambridge University Press. p. 201.
5533:{\displaystyle {\tfrac {\ 32805\ }{32768}}}
5385:{\displaystyle {\tfrac {\ 15625\ }{15552}}}
802:{\displaystyle {\tfrac {\ 15625\ }{15552}}}
658:{\displaystyle {\tfrac {\ 32805\ }{32768}}}
7206:
6272:
6258:
2126:
5941:Grove's Dictionary of Music and Musicians
5681:{\displaystyle {\tfrac {\ 4375\ }{4374}}}
5607:{\displaystyle {\tfrac {\ 2401\ }{2400}}}
5235:{\displaystyle {\tfrac {\ 1728\ }{1715}}}
5087:{\displaystyle {\tfrac {\ 2048\ }{2025}}}
982:{\displaystyle {\tfrac {\ 2048\ }{2025}}}
427:is the minor second (diatonic semitone),
98:used without qualification refers to the
6143:
2278:they are all equal to the opposite of a
1036:
1011:
399:
319:
274:
249:. The interval between those notes, the
45:
25:
6057:— Describes difference between 11
6032:
6030:
6007:The Mathematical Theory of Tone Systems
6004:
5973:10.1093/gmo/9781561592630.article.06186
5459:{\displaystyle {\tfrac {\ 225\ }{224}}}
5161:{\displaystyle {\tfrac {\ 126\ }{125}}}
4643:{\displaystyle {\tfrac {\ 128\ }{125}}}
4421:{\displaystyle {\tfrac {\ 648\ }{625}}}
1742:{\displaystyle {\tfrac {\ 648\ }{625}}}
1566:{\displaystyle {\tfrac {\ 128\ }{125}}}
1033:
727:{\displaystyle {\tfrac {\ 225\ }{224}}}
7222:
6630:
2376:
2324:quartertone signs (a single cross and
866:{\displaystyle {\tfrac {\ 100\ }{99}}}
6253:
6186:
6070:
6036:
5013:{\displaystyle {\tfrac {\ 81\ }{80}}}
4865:{\displaystyle {\tfrac {\ 64\ }{63}}}
4791:{\displaystyle {\tfrac {\ 50\ }{49}}}
4717:{\displaystyle {\tfrac {\ 49\ }{48}}}
4569:{\displaystyle {\tfrac {\ 36\ }{35}}}
4495:{\displaystyle {\tfrac {\ 28\ }{27}}}
4347:{\displaystyle {\tfrac {\ 25\ }{24}}}
4273:{\displaystyle {\tfrac {\ 21\ }{20}}}
4199:{\displaystyle {\tfrac {\ 16\ }{15}}}
4125:{\displaystyle {\tfrac {\ 15\ }{14}}}
4051:{\displaystyle {\tfrac {\ 27\ }{25}}}
3311:{\displaystyle {\tfrac {\ 45\ }{32}}}
3237:{\displaystyle {\tfrac {\ 64\ }{45}}}
2442:, but does not temper out the diesis.
1805:{\displaystyle {\tfrac {\ 27\ }{26}}}
1632:{\displaystyle {\tfrac {\ 33\ }{32}}}
1448:{\displaystyle {\tfrac {\ 64\ }{63}}}
1066:{\displaystyle {\tfrac {\ 81\ }{80}}}
6027:
3977:{\displaystyle {\tfrac {\ 10\ }{9}}}
3163:{\displaystyle {\tfrac {\ 10\ }{7}}}
2719:{\displaystyle {\tfrac {\ 15\ }{8}}}
6237:"Mercator-comma / Mercator's comma"
3903:{\displaystyle {\tfrac {\ 9\ }{8}}}
3829:{\displaystyle {\tfrac {\ 8\ }{7}}}
3755:{\displaystyle {\tfrac {\ 7\ }{6}}}
3681:{\displaystyle {\tfrac {\ 6\ }{5}}}
3607:{\displaystyle {\tfrac {\ 5\ }{4}}}
3533:{\displaystyle {\tfrac {\ 9\ }{7}}}
3459:{\displaystyle {\tfrac {\ 4\ }{3}}}
3385:{\displaystyle {\tfrac {\ 7\ }{5}}}
3089:{\displaystyle {\tfrac {\ 3\ }{2}}}
3015:{\displaystyle {\tfrac {\ 8\ }{5}}}
2941:{\displaystyle {\tfrac {\ 5\ }{3}}}
2867:{\displaystyle {\tfrac {\ 7\ }{4}}}
2793:{\displaystyle {\tfrac {\ 9\ }{5}}}
2645:{\displaystyle {\tfrac {\ 2\ }{1}}}
197:Within the same tuning system, two
125:. Intervals separated by the ratio
13:
6122:, von Bockel Verlag, Hamburg 2005
2095:This led to the creation of a new
14:
7246:
6234:
6211:
5838:
331:as difference between semitones (
287:as difference between semitones (
241:tuned as two major thirds above C
7205:
6098:"List of commas, by prime limit"
2347:
2341:
2335:
308:
136:
78:, the difference resulting from
54:
34:
6228:
6205:
6180:
5961:Clive Greated (2001). "Comma",
270:
6132:
6108:
6090:
6064:
5998:
5985:Music: A Mathematical Offering
5977:
5955:
5946:John Alexander Fuller Maitland
5932:
192:
1:
6279:
5926:
5882:Other intervals called commas
5878:in sequence above the last.
1775:
1712:
1687:
1649:
1602:
1536:
1511:
1470:
1418:
1342:
1313:
1235:
1161:
1126:
952:
925:
884:
836:
772:
697:
628:
598:
568:
396:Commas in different contexts
160:
7:
6827:septimal chromatic semitone
6075:. Vol. 2. p. 40.
6061:and 3 limit intervals.
5938:Waldo Selden Pratt (1922).
5904:
5843:A comma sequence defines a
2293:
1772:
1709:
1599:
1533:
1415:
1339:
1232:
949:
833:
769:
694:
625:
10:
7251:
6833:septimal diatonic semitone
6621:(Numbers in brackets refer
2113:instrumental accompaniment
499:just-noticeable difference
463:. In the columns labeled "
410:just-noticeable difference
18:
7203:
7161:
7128:
7017:
6987:
6957:
6914:
6905:
6878:
6805:
6788:
6623:to fractional semitones.)
6616:24-tone equal temperament
6613:
6602:
6539:
6484:
6429:
6392:
6355:
6320:
6287:
6155:Perspectives of New Music
5911:List of musical intervals
2402:19 tone equal temperament
1023:
1008:
992:
542:
537:
534:
529:
522:
517:
514:
511:
508:
459:are semitones as defined
293:enharmonically equivalent
264:tuning system produces a
199:enharmonically equivalent
5807:produces an interval of
2369:, while a just scale is
2332:, while a just scale is
2298:In the years 2000–2004,
2286:they are all equal to a
118:tuned using the D-based
108:tuned using the D-based
7211:List of pitch intervals
6949:Subminor and supermajor
6888:minor diatonic semitone
6798:refer to pitch ratios.)
5916:List of pitch intervals
2304:Wolfgang von Schweinitz
2127:Alternative definitions
1827:and its inversion, the
945:4 perfect fifths +
615:8 perfect fifths +
291:), or interval between
7009:Undecimal quarter tone
6224:– via lumma.org.
5682:
5608:
5534:
5460:
5386:
5311:
5236:
5162:
5088:
5014:
4940:
4866:
4792:
4718:
4644:
4570:
4496:
4422:
4348:
4274:
4200:
4126:
4052:
3978:
3904:
3830:
3756:
3682:
3608:
3534:
3460:
3386:
3312:
3238:
3164:
3090:
3016:
2942:
2868:
2794:
2720:
2646:
2284:quarter comma meantone
2133:quarter-comma meantone
2101:quarter-comma meantone
2086:
1968:
1806:
1743:
1685:in 5 limit tuning
1633:
1567:
1509:in 5 limit tuning
1449:
1412:septimal minor seventh
1373:
1334:12 perfect fifths
1285:
1067:
1021:in 5 limit tuning
983:
867:
803:
728:
659:
413:
348:
317:
285:quarter-comma meantone
120:quarter-comma meantone
63:
43:
7004:Septimal quarter tone
6815:septimal quarter tone
6009:. CRC Press. p.
6005:Haluška, Ján (2003).
5983:Benson, Dave (2006).
5683:
5609:
5535:
5461:
5387:
5312:
5237:
5163:
5089:
5015:
4941:
4867:
4793:
4719:
4645:
4571:
4497:
4423:
4349:
4275:
4201:
4127:
4053:
3979:
3905:
3831:
3757:
3683:
3609:
3535:
3461:
3387:
3313:
3239:
3165:
3091:
3017:
2943:
2869:
2795:
2721:
2647:
2121:meantone temperaments
2117:syntonic temperaments
2087:
1969:
1807:
1756:tridecimal third-tone
1744:
1634:
1568:
1450:
1374:
1286:
1068:
1026:4 perfect fifths
984:
868:
813:small undecimal comma
804:
729:
660:
403:
361:commonly used version
323:
278:
112:system, and another F
49:
29:
7235:Musical temperaments
7191:Incomposite interval
6944:Pythagorean interval
6796:(Numbers in brackets
6304:(Numbers in brackets
5655:
5581:
5507:
5433:
5359:
5284:
5209:
5135:
5061:
4987:
4913:
4839:
4765:
4691:
4617:
4543:
4469:
4395:
4321:
4247:
4173:
4099:
4025:
3951:
3877:
3803:
3729:
3655:
3581:
3507:
3433:
3359:
3285:
3211:
3137:
3063:
2989:
2915:
2841:
2767:
2693:
2619:
2383:musical temperaments
1984:
1866:
1847:. Musicians in late
1779:
1716:
1606:
1540:
1422:
1346:
1239:
1040:
956:
840:
776:
701:
632:
262:meantone temperament
6974:Pythagorean apotome
6821:septimal third tone
5845:musical temperament
2377:Tempering of commas
2111:, or melodies with
1704:4 minor thirds
1531:3 major thirds
947:2 major thirds
680:3 major thirds
535:Difference between
7087:Septimal semicomma
5964:Grove Music Online
5944:, Vol. 1, p. 568.
5678:
5676:
5604:
5602:
5530:
5528:
5456:
5454:
5382:
5380:
5307:
5305:
5232:
5230:
5158:
5156:
5084:
5082:
5010:
5008:
4936:
4934:
4862:
4860:
4788:
4786:
4714:
4712:
4640:
4638:
4566:
4564:
4492:
4490:
4418:
4416:
4344:
4342:
4270:
4268:
4196:
4194:
4122:
4120:
4048:
4046:
3974:
3972:
3900:
3898:
3826:
3824:
3752:
3750:
3678:
3676:
3604:
3602:
3530:
3528:
3456:
3454:
3382:
3380:
3308:
3306:
3234:
3232:
3160:
3158:
3086:
3084:
3012:
3010:
2938:
2936:
2864:
2862:
2790:
2788:
2716:
2714:
2642:
2640:
2391:chromatic semitone
2367:C D E+ F G A+ B+ C
2276:Pythagorean tuning
2225:diminished seventh
2082:
1964:
1802:
1800:
1764:tridecimal tritone
1739:
1737:
1629:
1627:
1563:
1561:
1445:
1443:
1369:
1367:
1327:Pythagorean tuning
1281:
1278:
1063:
1061:
1031:1 major third
979:
977:
863:
861:
799:
797:
724:
722:
655:
653:
530:Difference between
523:Difference between
471:. Notice that the
414:
349:
329:Pythagorean tuning
318:
110:Pythagorean tuning
64:
44:
7217:
7216:
7199:
7198:
7029:Pythagorean comma
6969:Pythagorean limma
6901:
6900:
6897:
6896:
6863:supermajor fourth
6839:supermajor second
6784:
6783:
6598:
6597:
6594:
6593:
6306:are the number of
6214:"Comma sequences"
6176:978-0-252-03098-7
5952:, eds. Macmillan.
5870:is the number of
5723:
5722:
5675:
5670:
5664:
5601:
5596:
5590:
5527:
5522:
5516:
5453:
5448:
5442:
5379:
5374:
5368:
5304:
5299:
5293:
5229:
5224:
5218:
5155:
5150:
5144:
5081:
5076:
5070:
5007:
5002:
4996:
4933:
4928:
4922:
4859:
4854:
4848:
4785:
4780:
4774:
4711:
4706:
4700:
4637:
4632:
4626:
4563:
4558:
4552:
4489:
4484:
4478:
4415:
4410:
4404:
4341:
4336:
4330:
4267:
4262:
4256:
4193:
4188:
4182:
4119:
4114:
4108:
4045:
4040:
4034:
3971:
3966:
3960:
3897:
3892:
3886:
3823:
3818:
3812:
3749:
3744:
3738:
3675:
3670:
3664:
3601:
3596:
3590:
3527:
3522:
3516:
3453:
3448:
3442:
3379:
3374:
3368:
3305:
3300:
3294:
3231:
3226:
3220:
3157:
3152:
3146:
3083:
3078:
3072:
3009:
3004:
2998:
2935:
2930:
2924:
2861:
2856:
2850:
2787:
2782:
2776:
2713:
2708:
2702:
2639:
2634:
2628:
2508:(frequency ratio)
2395:equal temperament
2387:diatonic semitone
2321:Daniel James Wolf
2280:Pythagorean comma
2245:diminished octave
2175:diminished fourth
2137:diminished second
2080:
2078:
2072:
2059:
2057:
2045:
2024:
2022:
2016:
2003:
2001:
1995:
1962:
1957:
1951:
1941:
1928:
1916:
1906:
1901:
1895:
1885:
1880:
1874:
1815:
1814:
1799:
1794:
1788:
1736:
1731:
1725:
1626:
1621:
1615:
1591:undecimal tritone
1560:
1555:
1549:
1467:diminished second
1442:
1437:
1431:
1366:
1361:
1355:
1296:Pythagorean comma
1277:
1272:
1266:
1060:
1055:
1049:
976:
971:
965:
860:
855:
849:
796:
791:
785:
721:
716:
710:
652:
647:
641:
512:Alternative name
473:Pythagorean comma
353:
352:
325:Pythagorean comma
314:
247:equal temperament
219:five-limit tuning
92:Pythagorean comma
60:
51:Pythagorean comma
40:
7242:
7209:
7208:
7099:Septimal kleisma
6912:
6911:
6869:subminor seventh
6851:supermajor third
6803:
6802:
6791:Just intonations
6776:
6775:
6771:
6768:
6756:
6755:
6751:
6748:
6736:
6735:
6731:
6728:
6716:
6715:
6711:
6708:
6696:
6695:
6691:
6688:
6676:
6675:
6671:
6668:
6656:
6655:
6651:
6628:
6627:
6611:
6610:
6318:
6317:
6300:
6299:
6274:
6267:
6260:
6251:
6250:
6245:
6244:
6232:
6226:
6225:
6223:
6221:
6209:
6203:
6202:
6184:
6178:
6147:
6141:
6136:
6130:
6112:
6106:
6105:
6102:Xenharmonic wiki
6094:
6088:
6086:
6068:
6062:
6056:
6034:
6025:
6024:
6012:
6002:
5996:
5981:
5975:
5959:
5953:
5936:
5900:
5898:
5897:
5869:
5865:
5861:
5857:
5834:
5833:
5828:
5827:
5822:
5820:
5819:
5816:
5813:
5806:
5804:
5803:
5798:
5797:
5790:
5789:
5784:
5783:
5774:
5772:
5771:
5768:
5765:
5755:
5753:
5752:
5747:
5746:
5739:
5738:
5733:
5732:
5687:
5685:
5684:
5679:
5677:
5671:
5668:
5662:
5660:
5613:
5611:
5610:
5605:
5603:
5597:
5594:
5588:
5586:
5539:
5537:
5536:
5531:
5529:
5523:
5520:
5514:
5512:
5465:
5463:
5462:
5457:
5455:
5449:
5446:
5440:
5438:
5391:
5389:
5388:
5383:
5381:
5375:
5372:
5366:
5364:
5319:
5316:
5314:
5313:
5308:
5306:
5300:
5297:
5291:
5289:
5241:
5239:
5238:
5233:
5231:
5225:
5222:
5216:
5214:
5167:
5165:
5164:
5159:
5157:
5151:
5148:
5142:
5140:
5093:
5091:
5090:
5085:
5083:
5077:
5074:
5068:
5066:
5019:
5017:
5016:
5011:
5009:
5003:
5000:
4994:
4992:
4945:
4943:
4942:
4937:
4935:
4929:
4926:
4920:
4918:
4871:
4869:
4868:
4863:
4861:
4855:
4852:
4846:
4844:
4797:
4795:
4794:
4789:
4787:
4781:
4778:
4772:
4770:
4723:
4721:
4720:
4715:
4713:
4707:
4704:
4698:
4696:
4649:
4647:
4646:
4641:
4639:
4633:
4630:
4624:
4622:
4575:
4573:
4572:
4567:
4565:
4559:
4556:
4550:
4548:
4501:
4499:
4498:
4493:
4491:
4485:
4482:
4476:
4474:
4427:
4425:
4424:
4419:
4417:
4411:
4408:
4402:
4400:
4353:
4351:
4350:
4345:
4343:
4337:
4334:
4328:
4326:
4279:
4277:
4276:
4271:
4269:
4263:
4260:
4254:
4252:
4205:
4203:
4202:
4197:
4195:
4189:
4186:
4180:
4178:
4131:
4129:
4128:
4123:
4121:
4115:
4112:
4106:
4104:
4057:
4055:
4054:
4049:
4047:
4041:
4038:
4032:
4030:
3983:
3981:
3980:
3975:
3973:
3967:
3964:
3958:
3956:
3909:
3907:
3906:
3901:
3899:
3893:
3890:
3884:
3882:
3835:
3833:
3832:
3827:
3825:
3819:
3816:
3810:
3808:
3761:
3759:
3758:
3753:
3751:
3745:
3742:
3736:
3734:
3687:
3685:
3684:
3679:
3677:
3671:
3668:
3662:
3660:
3613:
3611:
3610:
3605:
3603:
3597:
3594:
3588:
3586:
3539:
3537:
3536:
3531:
3529:
3523:
3520:
3514:
3512:
3465:
3463:
3462:
3457:
3455:
3449:
3446:
3440:
3438:
3391:
3389:
3388:
3383:
3381:
3375:
3372:
3366:
3364:
3317:
3315:
3314:
3309:
3307:
3301:
3298:
3292:
3290:
3243:
3241:
3240:
3235:
3233:
3227:
3224:
3218:
3216:
3169:
3167:
3166:
3161:
3159:
3153:
3150:
3144:
3142:
3095:
3093:
3092:
3087:
3085:
3079:
3076:
3070:
3068:
3021:
3019:
3018:
3013:
3011:
3005:
3002:
2996:
2994:
2947:
2945:
2944:
2939:
2937:
2931:
2928:
2922:
2920:
2873:
2871:
2870:
2865:
2863:
2857:
2854:
2848:
2846:
2799:
2797:
2796:
2791:
2789:
2783:
2780:
2774:
2772:
2725:
2723:
2722:
2717:
2715:
2709:
2706:
2700:
2698:
2651:
2649:
2648:
2643:
2641:
2635:
2632:
2626:
2624:
2608:
2607:
2598:
2597:
2588:
2587:
2578:
2577:
2568:
2567:
2558:
2557:
2548:
2547:
2538:
2537:
2528:
2527:
2518:
2517:
2509:
2502:
2501:
2491:
2489:
2488:
2485:
2482:
2475:
2473:
2472:
2455:tempers out the
2454:
2452:
2451:
2434:tempers out the
2433:
2431:
2430:
2420:
2418:
2417:
2372:
2368:
2364:
2353:
2351:
2345:
2339:
2331:
2318:
2317:
2316:
2269:
2267:
2265:
2264:
2261:
2258:
2205:diminished sixth
2195:diminished fifth
2191:augmented fourth
2165:augmented second
2155:diminished third
2109:polyphonic music
2091:
2089:
2088:
2083:
2081:
2079:
2076:
2070:
2065:
2060:
2058:
2055:
2043:
2041:
2030:
2025:
2023:
2020:
2014:
2009:
2004:
2002:
1999:
1993:
1988:
1973:
1971:
1970:
1965:
1963:
1958:
1955:
1949:
1947:
1942:
1940:
1929:
1926:
1914:
1912:
1907:
1902:
1899:
1893:
1891:
1886:
1881:
1878:
1872:
1870:
1811:
1809:
1808:
1803:
1801:
1795:
1792:
1786:
1784:
1753:tridecimal comma
1748:
1746:
1745:
1740:
1738:
1732:
1729:
1723:
1721:
1698:
1691:
1673:
1671:
1669:
1668:
1665:
1662:
1652:
1638:
1636:
1635:
1630:
1628:
1622:
1619:
1613:
1611:
1572:
1570:
1569:
1564:
1562:
1556:
1553:
1547:
1545:
1522:
1515:
1495:
1493:
1491:
1490:
1487:
1484:
1473:
1454:
1452:
1451:
1446:
1444:
1438:
1435:
1429:
1427:
1394:
1378:
1376:
1375:
1370:
1368:
1362:
1359:
1353:
1351:
1321:
1303:
1290:
1288:
1287:
1282:
1280:
1279:
1273:
1270:
1264:
1262:
1259:
1258:
1248:
1247:
1227:
1226:
1209:
1208:
1191:
1190:
1181:
1177:
1175:
1174:
1171:
1168:
1156:
1155:
1146:
1142:
1140:
1139:
1136:
1133:
1121:
1120:
1103:
1095:
1094:
1072:
1070:
1069:
1064:
1062:
1056:
1053:
1047:
1045:
1029:2 octaves +
1001:
988:
986:
985:
980:
978:
972:
969:
963:
961:
936:
929:
908:
906:
904:
903:
900:
897:
887:
872:
870:
869:
864:
862:
856:
853:
847:
845:
808:
806:
805:
800:
798:
792:
789:
783:
781:
733:
731:
730:
725:
723:
717:
714:
708:
706:
670:septimal kleisma
664:
662:
661:
656:
654:
648:
645:
639:
637:
609:
602:
593:
591:
589:
588:
585:
582:
572:
556:Interval 2
506:
505:
489:
478:
424:
384:fractions of an
383:
382:
378:
346:
345:
340:
339:
334:
327:(PC) defined in
316:
315:
306:
305:
300:
299:
290:
271:
240:
239:
226:
225:
212:
211:
206:
205:
201:notes (such as G
182:
177:
174:, from earlier *
172:
155:septimal kleisma
152:
151:
146:
145:
141:
140:
117:
116:
107:
106:
74:is a very small
62:
61:
42:
41:
7250:
7249:
7245:
7244:
7243:
7241:
7240:
7239:
7220:
7219:
7218:
7213:
7195:
7157:
7124:
7067:Septimal diesis
7013:
6983:
6953:
6907:
6893:
6874:
6797:
6794:
6780:
6773:
6769:
6766:
6764:
6753:
6749:
6746:
6744:
6733:
6729:
6726:
6724:
6713:
6709:
6706:
6704:
6693:
6689:
6686:
6684:
6673:
6669:
6666:
6664:
6653:
6649:
6648:
6638:
6637:
6636:
6632:
6622:
6619:
6606:
6604:
6590:
6535:
6480:
6425:
6388:
6351:
6312:
6307:
6305:
6295:
6293:
6290:
6283:
6278:
6248:
6233:
6229:
6219:
6217:
6210:
6206:
6199:
6185:
6181:
6148:
6144:
6137:
6133:
6120:Manfred Stahnke
6113:
6109:
6096:
6095:
6091:
6083:
6069:
6065:
6053:
6035:
6028:
6021:
6010:
6003:
5999:
5987:, p. 171.
5982:
5978:
5960:
5956:
5937:
5933:
5929:
5907:
5895:
5894:
5891:
5884:
5867:
5863:
5859:
5855:
5841:
5831:
5830:
5825:
5824:
5817:
5814:
5812: 648
5811:
5810:
5808:
5801:
5800:
5795:
5794:
5792:
5787:
5786:
5781:
5780:
5769:
5766:
5763:
5762:
5760:
5750:
5749:
5744:
5743:
5741:
5736:
5735:
5730:
5729:
5661:
5658:
5656:
5653:
5652:
5587:
5584:
5582:
5579:
5578:
5513:
5510:
5508:
5505:
5504:
5439:
5436:
5434:
5431:
5430:
5365:
5362:
5360:
5357:
5356:
5290:
5287:
5285:
5282:
5281:
5278:
5215:
5212:
5210:
5207:
5206:
5141:
5138:
5136:
5133:
5132:
5067:
5064:
5062:
5059:
5058:
4993:
4990:
4988:
4985:
4984:
4919:
4916:
4914:
4911:
4910:
4845:
4842:
4840:
4837:
4836:
4771:
4768:
4766:
4763:
4762:
4697:
4694:
4692:
4689:
4688:
4623:
4620:
4618:
4615:
4614:
4549:
4546:
4544:
4541:
4540:
4475:
4472:
4470:
4467:
4466:
4401:
4398:
4396:
4393:
4392:
4327:
4324:
4322:
4319:
4318:
4253:
4250:
4248:
4245:
4244:
4179:
4176:
4174:
4171:
4170:
4105:
4102:
4100:
4097:
4096:
4031:
4028:
4026:
4023:
4022:
3957:
3954:
3952:
3949:
3948:
3883:
3880:
3878:
3875:
3874:
3809:
3806:
3804:
3801:
3800:
3735:
3732:
3730:
3727:
3726:
3661:
3658:
3656:
3653:
3652:
3587:
3584:
3582:
3579:
3578:
3513:
3510:
3508:
3505:
3504:
3439:
3436:
3434:
3431:
3430:
3365:
3362:
3360:
3357:
3356:
3291:
3288:
3286:
3283:
3282:
3217:
3214:
3212:
3209:
3208:
3143:
3140:
3138:
3135:
3134:
3069:
3066:
3064:
3061:
3060:
2995:
2992:
2990:
2987:
2986:
2921:
2918:
2916:
2913:
2912:
2847:
2844:
2842:
2839:
2838:
2773:
2770:
2768:
2765:
2764:
2699:
2696:
2694:
2691:
2690:
2625:
2622:
2620:
2617:
2616:
2605:
2604:
2595:
2594:
2585:
2584:
2575:
2574:
2565:
2564:
2555:
2554:
2545:
2544:
2535:
2534:
2525:
2524:
2515:
2514:
2507:
2506:
2486:
2483:
2480:
2479:
2477:
2470:
2469:
2466:
2449:
2448:
2445:
2436:septimal diesis
2428:
2427:
2424:
2415:
2414:
2411:
2379:
2370:
2366:
2362:
2333:
2330:C D E F G A B C
2329:
2314:
2313:
2311:
2296:
2262:
2259:
2256:
2255:
2253:
2252:
2235:augmented sixth
2215:augmented fifth
2185:augmented third
2129:
2069:
2064:
2042:
2031:
2029:
2013:
2008:
1992:
1987:
1985:
1982:
1981:
1948:
1946:
1930:
1913:
1911:
1892:
1890:
1871:
1869:
1867:
1864:
1863:
1785:
1782:
1780:
1777:
1776:
1722:
1719:
1717:
1714:
1713:
1701:
1696:
1694:
1689:
1684:
1683:
1679:
1675:
1666:
1663:
1660:
1659:
1657:
1656:
1654:
1650:
1612:
1609:
1607:
1604:
1603:
1577:undecimal comma
1546:
1543:
1541:
1538:
1537:
1525:
1520:
1518:
1513:
1508:
1507:
1503:
1499:
1488:
1485:
1482:
1481:
1479:
1478:
1475:
1471:
1465:
1428:
1425:
1423:
1420:
1419:
1399:
1397:
1392:
1389:
1388:Archytas' comma
1352:
1349:
1347:
1344:
1343:
1324:
1319:
1308:
1306:
1301:
1298:
1263:
1260:
1254:
1253:
1252:
1243:
1242:
1240:
1237:
1236:
1230:
1224:
1223:
1218:
1212:
1206:
1205:
1200:
1194:
1188:
1187:
1182:
1172:
1169:
1166:
1165:
1163:
1162:
1159:
1153:
1152:
1147:
1137:
1134:
1131:
1130:
1128:
1127:
1124:
1118:
1117:
1112:
1108:
1106:
1101:
1098:
1092:
1091:
1046:
1043:
1041:
1038:
1037:
1030:
1020:
1019:
1015:
1006:
1004:
999:
996:
962:
959:
957:
954:
953:
946:
939:
934:
932:
927:
919:
918:
914:
910:
901:
898:
895:
894:
892:
891:
889:
885:
846:
843:
841:
838:
837:
782:
779:
777:
774:
773:
762:
757:
707:
704:
702:
699:
698:
688:
638:
635:
633:
630:
629:
616:
612:
607:
605:
600:
586:
583:
580:
579:
577:
576:
574:
570:
531:
524:
492:
487:
481:
476:
469:just intonation
457:
450:
443:
436:
422:
398:
380:
376:
375:
357:just intonation
343:
342:
337:
336:
332:
309:
303:
302:
297:
296:
288:
244:
237:
236:
234:
223:
222:
215:extended scales
209:
208:
203:
202:
195:
183:(Ancient Greek)
180:
175:
170:
163:
149:
148:
143:
142:
135:
114:
113:
104:
103:
55:
35:
24:
17:
12:
11:
5:
7248:
7238:
7237:
7232:
7230:Commas (music)
7215:
7214:
7204:
7201:
7200:
7197:
7196:
7194:
7193:
7188:
7183:
7178:
7173:
7167:
7165:
7159:
7158:
7156:
7155:
7150:
7145:
7140:
7134:
7132:
7126:
7125:
7123:
7122:
7116:
7110:
7103:
7102:
7096:
7090:
7084:
7078:
7071:
7070:
7064:
7061:Greater diesis
7058:
7051:
7050:
7047:Septimal comma
7044:
7041:Holdrian comma
7038:
7035:Syntonic comma
7032:
7025:
7023:
7015:
7014:
7012:
7011:
7006:
7001:
6995:
6993:
6985:
6984:
6982:
6981:
6976:
6971:
6965:
6963:
6955:
6954:
6952:
6951:
6946:
6941:
6936:
6931:
6926:
6920:
6918:
6909:
6903:
6902:
6899:
6898:
6895:
6894:
6892:
6891:
6884:
6882:
6876:
6875:
6873:
6872:
6866:
6860:
6857:subminor fifth
6854:
6848:
6845:subminor third
6842:
6836:
6830:
6824:
6818:
6811:
6809:
6800:
6786:
6785:
6782:
6781:
6779:
6778:
6758:
6738:
6718:
6698:
6678:
6658:
6641:
6639:
6631:
6625:
6608:
6600:
6599:
6596:
6595:
6592:
6591:
6589:
6588:
6582:
6576:
6570:
6564:
6558:
6552:
6545:
6543:
6537:
6536:
6534:
6533:
6527:
6521:
6515:
6509:
6503:
6497:
6490:
6488:
6482:
6481:
6479:
6478:
6472:
6466:
6460:
6454:
6448:
6442:
6435:
6433:
6427:
6426:
6424:
6423:
6417:
6411:
6405:
6398:
6396:
6390:
6389:
6387:
6386:
6380:
6374:
6368:
6361:
6359:
6353:
6352:
6350:
6349:
6343:
6337:
6331:
6324:
6322:
6315:
6297:
6285:
6284:
6277:
6276:
6269:
6262:
6254:
6247:
6246:
6227:
6204:
6197:
6179:
6142:
6131:
6107:
6089:
6081:
6063:
6051:
6026:
6019:
5997:
5976:
5954:
5930:
5928:
5925:
5924:
5923:
5918:
5913:
5906:
5903:
5883:
5880:
5858:-limit, where
5840:
5839:Comma sequence
5837:
5725:
5724:
5721:
5720:
5717:
5714:
5711:
5708:
5705:
5702:
5699:
5696:
5693:
5690:
5674:
5667:
5647:
5646:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5616:
5600:
5593:
5573:
5572:
5569:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5545:
5542:
5526:
5519:
5499:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5452:
5445:
5425:
5424:
5421:
5418:
5415:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5378:
5371:
5351:
5350:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5323:
5320:
5303:
5296:
5275:
5274:
5271:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5247:
5244:
5228:
5221:
5201:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5176:
5173:
5170:
5154:
5147:
5127:
5126:
5123:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5080:
5073:
5053:
5052:
5049:
5046:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5022:
5006:
4999:
4979:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4957:
4954:
4951:
4948:
4932:
4925:
4905:
4904:
4901:
4898:
4895:
4892:
4889:
4886:
4883:
4880:
4877:
4874:
4858:
4851:
4831:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4784:
4777:
4757:
4756:
4753:
4750:
4747:
4744:
4741:
4738:
4735:
4732:
4729:
4726:
4710:
4703:
4683:
4682:
4679:
4676:
4673:
4670:
4667:
4664:
4661:
4658:
4655:
4652:
4636:
4629:
4609:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4562:
4555:
4535:
4534:
4531:
4528:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4488:
4481:
4461:
4460:
4457:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4433:
4430:
4414:
4407:
4387:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4340:
4333:
4313:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4266:
4259:
4239:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4208:
4192:
4185:
4165:
4164:
4161:
4158:
4155:
4152:
4149:
4146:
4143:
4140:
4137:
4134:
4118:
4111:
4091:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4044:
4037:
4017:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3970:
3963:
3943:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3896:
3889:
3869:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3822:
3815:
3795:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3748:
3741:
3721:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3674:
3667:
3647:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3600:
3593:
3573:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3526:
3519:
3499:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3452:
3445:
3425:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3378:
3371:
3351:
3350:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3304:
3297:
3277:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3230:
3223:
3203:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3156:
3149:
3129:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3082:
3075:
3055:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3008:
3001:
2981:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2934:
2927:
2907:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2860:
2853:
2833:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2786:
2779:
2759:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2712:
2705:
2685:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2638:
2631:
2611:
2610:
2600:
2590:
2580:
2570:
2560:
2550:
2540:
2530:
2520:
2510:
2494:
2493:
2464:
2457:septimal comma
2443:
2440:syntonic comma
2422:
2378:
2375:
2326:backwards flat
2295:
2292:
2249:
2248:
2238:
2228:
2218:
2208:
2198:
2188:
2181:perfect fourth
2178:
2168:
2158:
2148:
2128:
2125:
2093:
2092:
2075:
2068:
2063:
2054:
2051:
2048:
2040:
2037:
2034:
2028:
2019:
2012:
2007:
1998:
1991:
1975:
1974:
1961:
1954:
1945:
1939:
1936:
1933:
1925:
1922:
1919:
1910:
1905:
1898:
1889:
1884:
1877:
1829:perfect fourth
1817:
1816:
1813:
1812:
1798:
1791:
1774:
1771:
1769:perfect fourth
1766:
1761:
1759:
1757:
1754:
1750:
1749:
1735:
1728:
1711:
1708:
1705:
1702:
1699:
1692:
1686:
1681:
1677:
1648:
1646:
1644:greater diesis
1640:
1639:
1625:
1618:
1601:
1598:
1596:perfect fourth
1593:
1588:
1586:
1584:
1578:
1574:
1573:
1559:
1552:
1535:
1532:
1529:
1526:
1523:
1516:
1510:
1505:
1501:
1469:
1462:
1456:
1455:
1441:
1434:
1417:
1414:
1409:
1404:
1402:
1400:
1395:
1390:
1386:
1384:septimal comma
1380:
1379:
1365:
1358:
1341:
1338:
1337:7 octaves
1335:
1332:
1330:
1312:
1309:
1304:
1299:
1292:
1291:
1276:
1269:
1257:
1251:
1246:
1234:
1231:
1219:
1213:
1201:
1195:
1183:
1160:
1148:
1125:
1113:
1109:
1104:
1099:
1086:
1085:
1080:
1074:
1073:
1059:
1052:
1035:
1032:
1027:
1024:
1022:
1017:
1013:
1010:
1009:Didymus' comma
1007:
1002:
997:
994:syntonic comma
990:
989:
975:
968:
951:
948:
943:
942:3 octaves
940:
937:
930:
924:
922:5-limit tuning
916:
912:
883:
880:
874:
873:
859:
852:
835:
832:
826:
824:neutral second
820:
818:
816:
814:
810:
809:
795:
788:
771:
768:
750:
749:6 minor thirds
747:
745:
743:
741:
735:
734:
720:
713:
696:
693:
691:septimal comma
681:
678:
676:
674:
672:
666:
665:
651:
644:
627:
624:
623:5 octaves
621:
613:
610:
603:
597:
567:
564:
558:
557:
554:
547:
546:
541:
536:
533:
528:
520:
519:
516:
513:
510:
509:Name of comma
490:
484:syntonic comma
479:
455:
448:
441:
434:
412:between tones.
397:
394:
351:
350:
242:
232:
217:produced with
194:
191:
162:
159:
100:syntonic comma
88:syntonic comma
31:Syntonic comma
15:
9:
6:
4:
3:
2:
7247:
7236:
7233:
7231:
7228:
7227:
7225:
7212:
7202:
7192:
7189:
7187:
7184:
7182:
7179:
7177:
7174:
7172:
7169:
7168:
7166:
7164:
7160:
7154:
7151:
7149:
7146:
7144:
7141:
7139:
7136:
7135:
7133:
7131:
7127:
7120:
7117:
7114:
7111:
7108:
7105:
7104:
7100:
7097:
7094:
7091:
7088:
7085:
7082:
7079:
7076:
7073:
7072:
7068:
7065:
7062:
7059:
7056:
7055:Lesser diesis
7053:
7052:
7048:
7045:
7042:
7039:
7036:
7033:
7030:
7027:
7026:
7024:
7022:
7021:
7016:
7010:
7007:
7005:
7002:
7000:
6997:
6996:
6994:
6992:
6991:
6990:Quarter tones
6986:
6980:
6977:
6975:
6972:
6970:
6967:
6966:
6964:
6962:
6961:
6956:
6950:
6947:
6945:
6942:
6940:
6939:Pseudo-octave
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6921:
6919:
6917:
6913:
6910:
6904:
6889:
6886:
6885:
6883:
6881:
6877:
6870:
6867:
6864:
6861:
6858:
6855:
6852:
6849:
6846:
6843:
6840:
6837:
6834:
6831:
6828:
6825:
6822:
6819:
6816:
6813:
6812:
6810:
6808:
6804:
6801:
6799:
6793:
6792:
6787:
6762:
6759:
6742:
6739:
6722:
6719:
6702:
6699:
6682:
6679:
6662:
6659:
6646:
6643:
6642:
6640:
6635:
6629:
6626:
6624:
6618:
6617:
6612:
6609:
6601:
6586:
6583:
6580:
6577:
6574:
6571:
6568:
6565:
6562:
6559:
6556:
6553:
6550:
6547:
6546:
6544:
6542:
6538:
6531:
6528:
6525:
6522:
6519:
6516:
6513:
6510:
6507:
6504:
6501:
6498:
6495:
6492:
6491:
6489:
6487:
6483:
6476:
6473:
6470:
6467:
6464:
6461:
6458:
6455:
6452:
6449:
6446:
6443:
6440:
6437:
6436:
6434:
6432:
6428:
6421:
6418:
6415:
6412:
6409:
6406:
6403:
6400:
6399:
6397:
6395:
6391:
6384:
6381:
6378:
6375:
6372:
6369:
6366:
6363:
6362:
6360:
6358:
6354:
6347:
6344:
6341:
6338:
6335:
6332:
6329:
6326:
6325:
6323:
6319:
6316:
6314:
6310:
6301:
6298:
6292:
6286:
6282:
6275:
6270:
6268:
6263:
6261:
6256:
6255:
6252:
6242:
6241:tonalsoft.com
6238:
6231:
6215:
6208:
6200:
6198:0-521-62371-5
6194:
6190:
6183:
6177:
6173:
6169:
6165:
6161:
6160:Johnston, Ben
6157:
6156:
6151:
6150:John Fonville
6146:
6140:
6135:
6129:
6128:3-932696-62-X
6125:
6121:
6117:
6111:
6103:
6099:
6093:
6084:
6082:3-7186-4846-6
6078:
6074:
6067:
6060:
6054:
6052:90-5755-065-2
6048:
6044:
6040:
6033:
6031:
6022:
6020:0-8247-4714-3
6016:
6008:
6001:
5994:
5993:0-521-85387-7
5990:
5986:
5980:
5974:
5970:
5966:
5965:
5958:
5951:
5947:
5943:
5942:
5935:
5931:
5922:
5919:
5917:
5914:
5912:
5909:
5908:
5902:
5899:
5889:
5879:
5877:
5873:
5853:
5850:
5846:
5836:
5778:
5759:
5756:produces the
5718:
5715:
5712:
5709:
5706:
5703:
5700:
5697:
5694:
5691:
5689:
5672:
5665:
5649:
5648:
5644:
5641:
5638:
5635:
5632:
5629:
5626:
5623:
5620:
5617:
5615:
5598:
5591:
5575:
5574:
5570:
5567:
5564:
5561:
5558:
5555:
5552:
5549:
5546:
5543:
5541:
5524:
5517:
5501:
5500:
5496:
5493:
5490:
5487:
5484:
5481:
5478:
5475:
5472:
5469:
5467:
5450:
5443:
5427:
5426:
5422:
5419:
5416:
5413:
5410:
5407:
5404:
5401:
5398:
5395:
5393:
5376:
5369:
5353:
5352:
5348:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5324:
5321:
5318:
5301:
5294:
5277:
5276:
5272:
5269:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5243:
5226:
5219:
5203:
5202:
5198:
5195:
5192:
5189:
5186:
5183:
5180:
5177:
5174:
5171:
5169:
5152:
5145:
5129:
5128:
5124:
5121:
5118:
5115:
5112:
5109:
5106:
5103:
5100:
5097:
5095:
5078:
5071:
5055:
5054:
5050:
5047:
5044:
5041:
5038:
5035:
5032:
5029:
5026:
5023:
5021:
5004:
4997:
4981:
4980:
4976:
4973:
4970:
4967:
4964:
4961:
4958:
4955:
4952:
4949:
4947:
4930:
4923:
4907:
4906:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4873:
4856:
4849:
4833:
4832:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4799:
4782:
4775:
4759:
4758:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4725:
4708:
4701:
4685:
4684:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4651:
4634:
4627:
4611:
4610:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4577:
4560:
4553:
4537:
4536:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4505:
4503:
4486:
4479:
4463:
4462:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4429:
4412:
4405:
4389:
4388:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4355:
4338:
4331:
4315:
4314:
4310:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4283:
4281:
4264:
4257:
4241:
4240:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4207:
4190:
4183:
4167:
4166:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4133:
4116:
4109:
4093:
4092:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4059:
4042:
4035:
4019:
4018:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3985:
3968:
3961:
3945:
3944:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3911:
3894:
3887:
3871:
3870:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3837:
3820:
3813:
3797:
3796:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3763:
3746:
3739:
3723:
3722:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3689:
3672:
3665:
3649:
3648:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3615:
3598:
3591:
3575:
3574:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3541:
3524:
3517:
3501:
3500:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3467:
3450:
3443:
3427:
3426:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3393:
3376:
3369:
3353:
3352:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3319:
3302:
3295:
3279:
3278:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3245:
3228:
3221:
3205:
3204:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3171:
3154:
3147:
3131:
3130:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3097:
3080:
3073:
3057:
3056:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3023:
3006:
2999:
2983:
2982:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2949:
2932:
2925:
2909:
2908:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2875:
2858:
2851:
2835:
2834:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2801:
2784:
2777:
2761:
2760:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2727:
2710:
2703:
2687:
2686:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2653:
2636:
2629:
2613:
2612:
2609:
2601:
2599:
2591:
2589:
2581:
2579:
2571:
2569:
2561:
2559:
2551:
2549:
2541:
2539:
2531:
2529:
2521:
2519:
2511:
2504:
2503:
2500:
2499:
2498:
2474:
2465:
2462:
2458:
2453:
2444:
2441:
2437:
2432:
2423:
2419:
2410:
2409:
2408:
2405:
2403:
2399:
2396:
2392:
2388:
2384:
2374:
2371:C D E F G A B
2363:A B C D E F G
2360:
2355:
2350:
2344:
2338:
2327:
2322:
2312:C G D A E B F
2309:
2305:
2301:
2291:
2289:
2285:
2281:
2277:
2272:
2246:
2242:
2241:major seventh
2239:
2236:
2232:
2231:minor seventh
2229:
2226:
2222:
2219:
2216:
2212:
2209:
2206:
2202:
2201:perfect fifth
2199:
2196:
2192:
2189:
2186:
2182:
2179:
2176:
2172:
2169:
2166:
2162:
2159:
2156:
2152:
2149:
2146:
2142:
2141:
2140:
2138:
2134:
2124:
2122:
2118:
2114:
2110:
2106:
2102:
2098:
2097:tuning system
2073:
2066:
2061:
2052:
2049:
2046:
2038:
2035:
2032:
2026:
2017:
2010:
2005:
1996:
1989:
1980:
1979:
1978:
1959:
1952:
1943:
1937:
1934:
1931:
1923:
1920:
1917:
1908:
1903:
1896:
1887:
1882:
1875:
1862:
1861:
1860:
1858:
1854:
1850:
1846:
1842:
1838:
1834:
1830:
1826:
1825:perfect fifth
1820:
1796:
1789:
1770:
1767:
1765:
1762:
1760:
1758:
1755:
1752:
1751:
1733:
1726:
1706:
1703:
1647:
1645:
1642:
1641:
1623:
1616:
1597:
1594:
1592:
1589:
1587:
1585:
1583:
1579:
1576:
1575:
1557:
1550:
1530:
1527:
1497:
1468:
1464:lesser diesis
1463:
1461:
1458:
1457:
1439:
1432:
1413:
1410:
1408:
1407:minor seventh
1405:
1403:
1401:
1387:
1385:
1382:
1381:
1363:
1356:
1336:
1333:
1331:
1328:
1323:
1316:
1311:ditonic comma
1310:
1297:
1294:
1293:
1274:
1267:
1249:
1228:
1217:
1214:
1210:
1199:
1196:
1192:
1180:
1157:
1145:
1122:
1110:
1096:
1088:
1087:
1084:
1081:
1079:
1076:
1075:
1057:
1050:
1028:
1025:
995:
991:
973:
966:
944:
941:
923:
881:
879:
876:
875:
857:
850:
831:
827:
825:
821:
819:
817:
815:
812:
811:
793:
786:
766:
761:
760:perfect fifth
755:
751:
748:
746:
744:
742:
740:
737:
736:
718:
711:
692:
686:
682:
679:
677:
675:
673:
671:
668:
667:
649:
642:
622:
620:
614:
596:
565:
563:
560:
559:
555:
552:
549:
548:
545:
540:
527:
521:
507:
504:
503:
502:
500:
496:
485:
474:
470:
466:
462:
458:
451:
444:
437:
430:
426:
419:
411:
407:
402:
393:
391:
387:
373:
368:
366:
362:
358:
330:
326:
322:
295:notes (from C
294:
286:
282:
277:
273:
272:
269:
267:
266:12-tone scale
263:
258:
256:
252:
248:
230:
220:
216:
200:
190:
188:
184:
173:
168:
158:
156:
139:
133:
128:
124:
123:tuning system
121:
111:
101:
97:
93:
89:
85:
81:
77:
73:
69:
52:
48:
32:
28:
22:
7162:
7129:
7115:(0.72 cents)
7109:(1.95 cents)
7089:(13.8 cents)
7083:(10.1 cents)
7077:(19.5 cents)
7069:(35.7 cents)
7063:(62.6 cents)
7057:(41.1 cents)
7049:(27.3 cents)
7043:(22.6 cents)
7037:(21.5 cents)
7031:(23.5 cents)
7019:
7018:
6999:Quarter tone
6989:
6988:
6959:
6958:
6933:
6915:
6880:Higher-limit
6795:
6789:
6701:major fourth
6645:quarter tone
6620:
6614:
6303:
6240:
6235:Monzo, Joe.
6230:
6218:. Retrieved
6216:. Xenharmony
6212:Smith, G.W.
6207:
6188:
6182:
6167:
6164:Gilmore, Bob
6153:
6145:
6134:
6115:
6110:
6101:
6092:
6072:
6066:
6038:
6006:
6000:
5984:
5979:
5962:
5957:
5950:George Grove
5939:
5934:
5885:
5842:
5758:small diesis
5757:
5726:
2495:
2406:
2397:
2380:
2359:Ben Johnston
2356:
2297:
2273:
2250:
2151:major second
2130:
2119:, including
2094:
1976:
1821:
1818:
1582:quarter-tone
515:Definitions
453:
446:
439:
432:
428:
421:
415:
369:
354:
259:
196:
169:
164:
95:
71:
68:music theory
65:
7148:Millioctave
7130:Measurement
7121:(0.4 cents)
7101:(7.7 cents)
7095:(8.1 cents)
6979:Major limma
6721:minor fifth
5779:) between G
5775:(41.1
2398:tempers out
2221:major sixth
2211:minor sixth
2171:major third
2161:minor third
2099:, known as
1849:Middle Ages
1111:1 step
619:major third
501:, or JND).
390:logarithmic
365:four commas
283:defined in
229:major third
227:tuned as a
193:Description
7224:Categories
7181:Semiditone
7075:Diaschisma
6890:(17-limit)
6581:(22 or 23)
6579:fourteenth
6575:(20 or 21)
6573:thirteenth
6569:(18 or 19)
6563:(17 or 18)
6557:(15 or 16)
6551:(13 or 14)
6486:Diminished
6313:interval.)
6294:(post-Bach
6041:. p.
5927:References
5872:generators
2407:Examples:
2300:Marc Sabat
2107:, such as
1580:Undecimal
1216:minor tone
1198:major tone
1179:minor tone
1144:major tone
1083:minor tone
1078:major tone
882:diaskhisma
878:diaschisma
830:minor tone
482:) and the
187:Wiktionary
7143:Centitone
7081:Semicomma
6960:Semitones
6924:Microtone
6908:intervals
6585:fifteenth
6431:Augmented
6309:semitones
6281:Intervals
6170:, p. 78.
6073:Listening
5921:Semicomma
5823:between A
2497:article.
2357:Composer
2282:, and in
2145:semitones
2050:⋅
2036:⋅
2006:⋅
1935:⋅
1921:⋅
1888:⋅
1859:ratio of
1853:consonant
1833:dissonant
1674:meantone,
909:meantone,
526:semitones
418:semitones
161:Etymology
7113:Breedsma
6561:eleventh
6541:Compound
6296:Western)
6291:semitone
5905:See also
5888:Holdrian
5832:♯
5826:♭
5802:♭
5796:♯
5793: G
5788:♭
5782:♯
5751:♯
5745:♭
5742: A
5737:♯
5731:♭
5317:
2505:Interval
2461:Archytas
2315:♯
2294:Notation
1496:meantone
1222:53
1204:53
1186:53
1151:53
1116:53
1090:53
595:meantone
551:Interval
465:Interval
344:♯
338:♭
304:♭
298:♯
255:semitone
238:♯
224:♭
210:♭
204:♯
150:♯
144:♭
115:♯
105:♯
76:interval
7119:Ragisma
7107:Schisma
7093:Kleisma
6929:5-limit
6835:(15:14)
6829:(21:20)
6823:(28:27)
6817:(36:35)
6807:7-limit
6772:⁄
6761:seventh
6752:⁄
6732:⁄
6712:⁄
6692:⁄
6672:⁄
6652:⁄
6634:Neutral
6607:systems
6567:twelfth
6524:seventh
6475:seventh
6420:seventh
6383:seventh
6321:Perfect
6289:Twelve-
6220:26 July
5862:is the
5821:
5809:
5799:B D F A
5773:
5761:
5688:
5614:
5540:
5466:
5392:
5302:2097152
5295:2109375
5280:
5242:
5168:
5094:
5020:
4946:
4872:
4798:
4724:
4650:
4576:
4502:
4428:
4354:
4280:
4206:
4132:
4058:
3984:
3910:
3836:
3762:
3688:
3614:
3540:
3466:
3392:
3318:
3244:
3170:
3096:
3022:
2948:
2874:
2800:
2726:
2652:
2490:
2478:
2266:
2254:
2105:texture
1845:texture
1670:
1658:
1492:
1480:
1176:
1164:
1141:
1129:
905:
893:
828:1
822:1
765:tritave
758:1
752:1
739:kleisma
689:1
687:−
683:1
617:1
590:
578:
566:skhisma
562:schisma
553: 1
532:commas
495:schisma
392:scale.
379:⁄
333:A1 − m2
289:m2 − A1
279:Lesser
235:and a G
231:below C
185:in the
157:apart.
21:Caesura
7176:Ditone
7163:Others
7153:Savart
7020:Commas
6916:Groups
6865:(10:7)
6661:second
6605:tuning
6530:octave
6506:fourth
6494:second
6457:fourth
6445:second
6439:unison
6402:second
6365:second
6346:octave
6334:fourth
6328:unison
6311:in the
6195:
6174:
6126:
6118:, ed.
6079:
6049:
6017:
5991:
5948:, Sir
5852:limits
5669:
5663:
5651:
5595:
5589:
5577:
5521:
5515:
5503:
5447:
5441:
5429:
5373:
5367:
5355:
5298:
5292:
5223:
5217:
5205:
5149:
5143:
5131:
5075:
5069:
5057:
5001:
4995:
4983:
4931:524288
4927:
4924:531441
4921:
4909:
4853:
4847:
4835:
4779:
4773:
4761:
4705:
4699:
4687:
4631:
4625:
4613:
4557:
4551:
4539:
4483:
4477:
4465:
4409:
4403:
4391:
4335:
4329:
4317:
4261:
4255:
4243:
4187:
4181:
4169:
4113:
4107:
4095:
4039:
4033:
4021:
3965:
3959:
3947:
3891:
3885:
3873:
3817:
3811:
3799:
3743:
3737:
3725:
3669:
3663:
3651:
3595:
3589:
3577:
3521:
3515:
3503:
3447:
3441:
3429:
3373:
3367:
3355:
3299:
3293:
3281:
3225:
3219:
3207:
3151:
3145:
3133:
3077:
3071:
3059:
3003:
2997:
2985:
2929:
2923:
2911:
2855:
2849:
2837:
2781:
2775:
2763:
2707:
2701:
2689:
2633:
2627:
2615:
2340:F G A
2288:diesis
2077:
2071:
2056:
2044:
2021:
2015:
2000:
1994:
1956:
1950:
1927:
1915:
1900:
1894:
1879:
1873:
1841:chords
1837:triads
1793:
1787:
1730:
1724:
1707:octave
1620:
1614:
1554:
1548:
1528:octave
1460:diesis
1436:
1430:
1364:524288
1360:
1357:531441
1354:
1271:
1265:
1054:
1048:
970:
964:
854:
848:
790:
784:
754:octave
715:
709:
685:octave
646:
640:
386:octave
281:diesis
251:diesis
176:κοπ-μα
80:tuning
7186:Secor
6934:Comma
6906:Other
6871:(7:4)
6859:(7:5)
6853:(9:7)
6847:(7:6)
6841:(8:7)
6741:sixth
6681:third
6603:Other
6555:tenth
6549:ninth
6518:sixth
6512:fifth
6500:third
6469:sixth
6463:fifth
6451:third
6414:sixth
6408:third
6394:Minor
6377:sixth
6371:third
6357:Major
6340:fifth
6059:limit
5876:prime
5849:prime
5829:and G
5785:and A
5748:C E G
5734:and G
5525:32768
5518:32805
5377:15552
5370:15625
2606:T EDO
2596:T EDO
2586:T EDO
2576:T EDO
2566:T EDO
2556:T EDO
2546:T EDO
2536:T EDO
2526:T EDO
2516:T EDO
2334:C D E
2268:comma
2147:), or
1773:65.34
1710:62.57
1672:comma
1653:2 − 1
1600:53.27
1534:41.06
1494:comma
1474:2 − 1
1416:27.26
1340:23.46
1233:22.64
1097:comma
1034:21.51
950:19.55
907:comma
888:2 − 1
834:17.40
794:15552
787:15625
650:32768
643:32805
592:comma
544:Ratio
539:Cents
518:Size
406:cents
388:on a
372:cents
207:and A
181:κόμμα
171:κόμμα
167:Greek
147:and A
132:53TET
127:81:80
96:comma
72:comma
53:on C
7171:Wolf
7138:Cent
6587:(24)
6532:(11)
6477:(12)
6471:(10)
6422:(10)
6385:(11)
6348:(12)
6222:2012
6193:ISBN
6172:ISBN
6162:and
6124:ISBN
6077:ISBN
6047:ISBN
6015:ISBN
6011:xxvi
5989:ISBN
5818:625
5777:cent
5770:125
5764:128
5673:4374
5666:4375
5599:2400
5592:2401
5227:1715
5220:1728
5079:2025
5072:2048
2438:and
2389:and
2302:and
2243:and
2237:, or
2233:and
2227:, or
2223:and
2217:, or
2213:and
2207:, or
2203:and
2197:, or
2193:and
2187:, or
2183:and
2177:, or
2173:and
2167:, or
2163:and
2157:, or
2153:and
1857:just
1839:and
1695:− 1
1519:− 1
1325:(in
1220:(in
1202:(in
1184:(in
1149:(in
1114:(in
974:2025
967:2048
933:− 1
770:8.11
695:7.71
626:1.95
606:− 1
569:1 −
461:here
381:1200
341:to C
301:to D
221:an A
84:note
82:one
70:, a
33:on C
6526:(9)
6520:(7)
6514:(6)
6508:(4)
6502:(2)
6496:(0)
6465:(8)
6459:(6)
6453:(5)
6447:(3)
6441:(1)
6416:(8)
6410:(3)
6404:(1)
6379:(9)
6373:(4)
6367:(2)
6342:(7)
6336:(5)
6330:(0)
5969:doi
5896:TET
5893:53
5571:−1
5451:224
5444:225
5349:−1
5153:125
5146:126
4635:125
4628:128
4413:625
4406:648
4015:11
3941:12
3867:14
3793:16
3719:19
3645:23
3571:26
3497:30
3423:35
3349:35
3275:37
3201:37
3127:42
3053:49
2979:53
2905:58
2831:61
2757:65
2683:72
2603:72
2593:53
2583:41
2573:34
2563:31
2553:22
2543:19
2533:12
2487:98
2481:99
2471:TET
2468:31
2459:of
2450:TET
2447:22
2429:TET
2426:19
2416:TET
2413:12
2263:12
2131:In
1734:625
1727:648
1680:− S
1655:in
1651:min
1558:125
1551:128
1504:− S
1476:in
1472:min
1320:min
1225:TET
1207:TET
1189:TET
1154:TET
1119:TET
1093:TET
1016:− S
920:in
915:− S
890:in
886:min
851:100
719:224
712:225
587:12
575:in
571:min
423:min
420:",
355:In
257:).
134:, B
66:In
7226::
6765:10
6239:.
6100:.
6045:.
6043:34
6029:^
6013:.
5967:.
5901:.
5719:0
5707:−1
5698:−1
5692:−1
5645:0
5627:−1
5618:−1
5559:−1
5553:−1
5547:−1
5497:0
5473:−1
5423:0
5417:−1
5408:−1
5399:−1
5343:−1
5331:−1
5325:−2
5273:1
5264:−1
5252:−1
5249:−1
5199:1
5172:−1
5125:2
5098:−1
5051:1
5005:80
4998:81
4977:0
4965:−1
4959:−1
4953:−1
4903:2
4857:63
4850:64
4829:2
4805:−1
4783:49
4776:50
4755:2
4709:48
4702:49
4681:3
4654:−1
4607:3
4561:35
4554:36
4533:4
4487:27
4480:28
4459:4
4432:−1
4385:4
4339:24
4332:25
4311:5
4265:20
4258:21
4237:7
4191:15
4184:16
4163:7
4117:14
4110:15
4089:8
4043:25
4036:27
3962:10
3864:10
3790:12
3716:14
3713:11
3642:17
3639:13
3636:11
3633:10
3568:19
3565:15
3562:12
3559:11
3494:22
3491:17
3488:14
3485:13
3420:26
3417:20
3414:17
3411:15
3408:11
3346:26
3343:20
3340:17
3337:15
3334:11
3303:32
3296:45
3272:27
3269:21
3266:17
3263:16
3260:11
3257:10
3229:45
3222:64
3198:27
3195:21
3192:17
3189:16
3186:11
3183:10
3148:10
3124:31
3121:24
3118:20
3115:18
3112:13
3109:11
3050:36
3047:28
3044:23
3041:21
3038:15
3035:13
2976:39
2973:30
2970:25
2967:23
2964:16
2961:14
2902:43
2899:33
2896:28
2893:25
2890:18
2887:15
2884:10
2828:45
2825:35
2822:29
2819:26
2816:19
2813:16
2810:10
2754:48
2751:37
2748:31
2745:28
2742:20
2739:17
2736:11
2704:15
2680:53
2677:41
2674:34
2671:31
2668:22
2665:19
2662:12
2523:7
2513:5
2373:.
2354:.
2308:JI
2290:.
2257:1
2123:.
2018:80
2011:81
1997:27
1990:32
1904:81
1897:80
1883:64
1876:81
1797:26
1790:27
1700:𝜋
1688:4
1667:3
1661:1
1624:32
1617:33
1524:𝜋
1512:3
1489:4
1483:1
1440:63
1433:64
1317:−
1305:𝜋
1275:53
1173:8
1167:1
1138:9
1132:1
1105:53
1058:80
1051:81
938:𝜋
926:2
902:6
896:1
858:99
767:")
763:("
604:𝜋
599:1
581:1
480:𝜋
452:,
445:,
438:,
374:–
367:.
189:.
6777:)
6774:2
6770:1
6767:+
6763:(
6757:)
6754:2
6750:1
6747:+
6745:8
6743:(
6737:)
6734:2
6730:1
6727:+
6725:6
6723:(
6717:)
6714:2
6710:1
6707:+
6705:5
6703:(
6697:)
6694:2
6690:1
6687:+
6685:3
6683:(
6677:)
6674:2
6670:1
6667:+
6665:1
6663:(
6657:)
6654:2
6650:1
6647:(
6273:e
6266:t
6259:v
6243:.
6201:.
6104:.
6085:.
6055:.
6023:.
5995:.
5971::
5868:n
5864:n
5860:q
5856:q
5815:/
5805:,
5767:/
5754:,
5716:0
5713:1
5710:0
5704:1
5701:0
5695:0
5642:1
5639:0
5636:2
5633:0
5630:1
5624:1
5621:2
5568:0
5565:0
5562:1
5556:1
5550:0
5544:1
5494:0
5491:0
5488:0
5485:0
5482:0
5479:0
5476:0
5470:1
5420:0
5414:0
5411:1
5405:0
5402:1
5396:2
5346:0
5340:1
5337:0
5334:0
5328:1
5322:3
5270:0
5267:1
5261:0
5258:0
5255:1
5246:0
5196:1
5193:1
5190:1
5187:0
5184:1
5181:0
5178:0
5175:1
5122:1
5119:1
5116:0
5113:1
5110:0
5107:1
5104:0
5101:1
5048:1
5045:1
5042:1
5039:0
5036:1
5033:0
5030:0
5027:0
5024:0
4974:1
4971:1
4968:2
4962:2
4956:0
4950:1
4900:1
4897:1
4894:0
4891:1
4888:0
4885:1
4882:0
4879:0
4876:0
4826:1
4823:1
4820:0
4817:1
4814:0
4811:1
4808:0
4802:1
4752:2
4749:1
4746:2
4743:1
4740:1
4737:0
4734:1
4731:1
4728:0
4678:2
4675:2
4672:1
4669:1
4666:1
4663:1
4660:0
4657:1
4604:2
4601:2
4598:1
4595:1
4592:1
4589:1
4586:0
4583:0
4580:0
4530:3
4527:2
4524:2
4521:2
4518:1
4515:1
4512:1
4509:1
4506:0
4456:3
4453:3
4450:2
4447:1
4444:2
4441:1
4438:0
4435:1
4382:3
4379:2
4376:2
4373:2
4370:1
4367:1
4364:1
4361:0
4358:1
4308:4
4305:3
4302:3
4299:2
4296:2
4293:1
4290:1
4287:1
4284:0
4234:5
4231:4
4228:3
4225:3
4222:2
4219:2
4216:1
4213:1
4210:0
4160:5
4157:4
4154:3
4151:3
4148:2
4145:2
4142:1
4139:0
4136:1
4086:6
4083:5
4080:4
4077:3
4074:3
4071:2
4068:1
4065:1
4062:0
4012:8
4009:6
4006:5
4003:5
4000:3
3997:3
3994:2
3991:1
3988:1
3969:9
3938:9
3935:7
3932:6
3929:5
3926:4
3923:3
3920:2
3917:1
3914:1
3895:8
3888:9
3861:8
3858:6
3855:6
3852:4
3849:4
3846:2
3843:1
3840:1
3821:7
3814:8
3787:9
3784:8
3781:7
3778:5
3775:4
3772:3
3769:2
3766:1
3747:6
3740:7
3710:9
3707:8
3704:6
3701:5
3698:3
3695:2
3692:1
3673:5
3666:6
3630:7
3627:6
3624:4
3621:2
3618:2
3599:4
3592:5
3556:8
3553:7
3550:4
3547:2
3544:2
3525:7
3518:9
3482:9
3479:8
3476:5
3473:3
3470:2
3451:3
3444:4
3405:9
3402:6
3399:4
3396:2
3377:5
3370:7
3331:9
3328:6
3325:3
3322:3
3254:6
3251:4
3248:2
3180:6
3177:3
3174:3
3155:7
3106:7
3103:4
3100:3
3081:2
3074:3
3032:8
3029:5
3026:3
3007:5
3000:8
2958:9
2955:5
2952:4
2933:3
2926:5
2881:6
2878:4
2859:4
2852:7
2807:6
2804:4
2785:5
2778:9
2733:6
2730:5
2711:8
2659:7
2656:5
2637:1
2630:2
2484:/
2352:C
2346:B
2260:/
2247:.
2074:5
2067:6
2062:=
2053:5
2047:1
2039:3
2033:2
2027:=
1960:4
1953:5
1944:=
1938:1
1932:4
1924:5
1918:1
1909:=
1697:κ
1693:S
1690:κ
1682:1
1678:4
1676:S
1664:/
1521:κ
1517:S
1514:κ
1506:1
1502:3
1500:S
1498:,
1486:/
1398:)
1396:A
1393:κ
1391:(
1329:)
1322:2
1315:1
1307:)
1302:κ
1300:(
1268:1
1256:)
1250:2
1245:(
1229:)
1211:)
1193:)
1170:/
1158:)
1135:/
1123:)
1107:)
1102:κ
1100:(
1018:1
1014:2
1012:S
1005:)
1003:S
1000:κ
998:(
935:κ
931:S
928:κ
917:2
913:3
911:S
899:/
756:+
611:S
608:κ
601:κ
584:/
573:2
491:S
488:κ
486:(
477:κ
475:(
456:4
454:S
449:3
447:S
442:2
440:S
435:1
433:S
429:1
425:2
377:1
243:4
233:5
23:.
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