147:
48:
109:
86:
133:
74:
121:
3179:
311:, which were composed into scales spanning an octave. A distinction can be made between extended Pythagorean tuning and a 12-tone Pythagorean temperament. Extended Pythagorean tuning corresponds 1-on-1 with western music notation and there is no limit to the number of fifths. In 12-tone Pythagorean temperament however one is limited by 12-tones per octave and one cannot play most music according to the Pythagorean system corresponding to the enharmonic notation. Instead one finds that for instance the diminished sixth becomes a "wolf fifth".
33:
103:
20:
66:
4071:. Where a performer has an unaccompanied passage based on scales, they will tend towards using Pythagorean intonation as that will make the scale sound best in tune, then reverting to other temperaments for other passages (just intonation for chordal or arpeggiated figures, and equal temperament when accompanied with piano or orchestra). Such changes are never explicitly notated and are scarcely noticeable to the audience, just sounding 'in tune'.
296:. The so-called "Pythagorean tuning" was used by musicians up to the beginning of the 16th century. "The Pythagorean system would appear to be ideal because of the purity of the fifths, but some consider other intervals, particularly the major third, to be so badly out of tune that major chords a dissonance."
359:
has only 12 keys). This dates to antiquity: in
Ancient Mesopotamia, rather than stacking fifths, tuning was based on alternating ascending fifths and descending fourths (equal to an ascending fifth followed by a descending octave), resulting in the notes of a pentatonic or heptatonic scale falling
4055:
However, meantone presented its own harmonic challenges. Its wolf intervals proved to be even worse than those of the
Pythagorean tuning (so much so that it often required 19 keys to the octave as opposed to the 12 in Pythagorean tuning). As a consequence, meantone was not suitable for all music.
3083:
are thought of as being exactly the same note—however, as the above table indicates, in
Pythagorean tuning they have different ratios with respect to D, which means they are at a different frequency. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a
4024:
Because most fifths in 12-tone
Pythagorean temperament are in the simple ratio of 3:2, they sound very "smooth" and consonant. The thirds, by contrast, most of which are in the relatively complex ratios of 81:64 (for major thirds) and 32:27 (for minor thirds), sound less smooth depending on the
3591:
Four of the above-mentioned intervals take a specific name in
Pythagorean tuning. In the following table, these specific names are provided, together with alternative names used generically for some other intervals. The Pythagorean comma does not coincide with the diminished second, as its size
2818:
In the formulas, the ratios 3:2 or 2:3 represent an ascending or descending perfect fifth (i.e. an increase or decrease in frequency by a perfect fifth, while 2:1 or 1:2 represent a rising or lowering octave). The formulas can also be expressed in terms of powers of the third and the second
3197:
As explained above, one of the twelve fifths (the wolf fifth) has a different size with respect to the other eleven. For a similar reason, each interval type except unisons and octaves has two different sizes. The table on the right shows their frequency ratios, with deviations of a
49:
347:), it is customary to divide or multiply the frequencies of some of these notes by 2 or by a power of 2. The purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the
3354:
110:
2689:
1762:
1201:
87:
134:
3608:
are used generically for all tuning systems. Despite its name, a semiditone (3 semitones, or about 300 cents) can hardly be viewed as half of a ditone (4 semitones, or about 400 cents). All the intervals with prefix
4020:
adventurous, the wolf interval is unlikely to be a problem, as not all the possible fifths will be heard in such pieces. In extended
Pythagorean tuning there is no wolf interval, all perfect fifths are exactly 3:2.
75:
2297:
809:
3263:
122:
4001:(who lived around 500 BCE) by modern authors of music theory; Ancient Greeks borrowed much of their music theory from Mesopotamia, including the diatonic scale, Pythagorean tuning, and modes. The Chinese
2483:
2111:
1556:
995:
623:
3424:
3186:
The tables above only show the frequency ratios of each note with respect to the base note. However, intervals can start from any note and so twelve intervals can be defined for each
4404:(about 21.5 cents) from the corresponding justly intonated interval. Intervals made up of 1, 2, 6, 10, or 11 semitones (e.g. major and minor seconds or sevenths, tritones, and their
1642:
3440: ≈ 1.955 cents). Since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of 700 − 11
2775:
2569:
2383:
2197:
2006:
1848:
1451:
1365:
1287:
1081:
895:
709:
518:
4396:
Wolf intervals are operationally defined herein as intervals composed of 3, 4, 5, 7, 8, or 9 semitones (i.e. major and minor thirds or sixths, perfect fourths or fifths, and their
319:
12-tone
Pythagorean temperament is based on a sequence of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1 (the octave). Starting from D for example (
1670:
1600:
3123:. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a
2803:
2733:
2527:
2341:
2155:
1964:
1876:
1806:
1409:
1245:
1039:
853:
737:
667:
3281:
2225:
476:
2411:
1315:
1109:
2597:
2034:
1920:
1479:
923:
546:
435:
3117:) is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as this one is known as a
3093:
To get around this problem, Pythagorean tuning constructs only twelve notes as above, with eleven fifths between them. For example, one may use only the 12 notes from E
3142:
need to be sounded together, the position of the wolf fifth can be changed. For example, a C-based
Pythagorean tuning would produce a stack of fifths running from D
235:
205:
4012:
when using a 12-tone
Pythagorean temperament, this tuning is rarely used today, although it is thought to have been widespread. In music which does not change
3105:. This, as shown above, implies that only eleven just fifths are used to build the entire chromatic scale. The remaining interval (the diminished sixth from G
150:
Comparison of equal-tempered (black) and
Pythagorean (green) intervals showing the relationship between frequency ratio and the intervals' values, in cents.
2624:
1697:
1136:
4056:
From around the 18th century, as the desire grew for instruments to change key, and therefore to avoid a wolf interval, this led to the widespread use of
4412:
even when they are justly tuned, thus they are not marked as wolf intervals even when they deviate from just intonation by more than one syntonic comma.
3217:
4067:
Pythagorean temperament can still be heard in some parts of modern classical music from singers and from instruments with no fixed tuning such as the
245:
put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the
4465:. Tadema Press, LondonThe book title is of second edition. The first edition was entitled 'The Musicology and Organology of the Ancient Near East'.
3538:
In short, similar differences in width are observed for all interval types, except for unisons and octaves, and they are all multiples of
4616:
2247:
759:
4548:
4681:
2433:
2061:
1506:
945:
573:
343:, it encompasses 77 keys). Since notes differing in frequency by a factor of 2 are perceived as similar and given the same name (
246:
4350:
323:
tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down:
4293:
The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy
5053:
4430:
4280:
3376:
3166:
the wolf interval. However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all
4691:
4382:
146:
4495:
4336:
4300:
3993:
The system dates to Ancient Mesopotamia, and consisted of alternating ascending fifths and descending fourths; see
3549:(≈ 23.460) cents narrower or wider than its enharmonic equivalent. For instance, the d6 (or wolf fifth) is 12
4654:
4456:
4476:
3646:
307:
which can be constructed from only pure perfect fifths (3:2) and octaves (2:1). In Greek music it was used to
292:
by only two intervals, called "semitonium" and "tonus" in Latin (256:243 × 9:8 × 9:8), to
4839:
1606:
2739:
2533:
2347:
2161:
1970:
1812:
1415:
1337:
1251:
1045:
859:
673:
482:
4541:
3444:
cents, which is about 678.495 cents (the wolf fifth). As shown in the table, the latter interval, although
4045:
3429:
and intervals of any given type have the same size, but none are justly tuned except unisons and octaves.
4777:
1648:
1562:
3994:
2781:
2695:
2489:
2303:
2117:
1926:
1854:
1768:
1371:
1207:
1001:
815:
715:
629:
269:
174:
4621:
4152:
4112:
2203:
441:
2389:
1293:
1087:
4606:
4409:
3445:
2575:
238:
4221:
2012:
1898:
1457:
901:
524:
413:
5048:
4890:
5026:
4860:
4686:
4611:
4534:
4162:
2820:
273:
4882:
4845:
4835:
4644:
4483:
4033:
3664:
3622:
4442:
However, 3/2 is described as "almost exactly a just major third." Sethares (2005), p. 60.
4005:
uses the same intervals as the Pythagorean scale and was invented between 600 BCE and 240 CE.
351:
D and the D above it (a note with twice its frequency). This interval is typically called the
4878:
4397:
3592:(524288:531441) is the reciprocal of the Pythagorean diminished second (531441:524288). Also
4886:
4831:
4813:
4808:
4803:
4798:
4793:
4788:
4783:
4768:
4763:
4758:
4753:
4748:
4743:
4141:
4040:, became the most popular system for tuning keyboards. At the same time, syntonic-diatonic
4029:
3582:
308:
3349:{\displaystyle S_{2}={3^{7} \over 2^{11}}={2187 \over 2048}\approx 113.685{\text{ cents}}}
8:
5058:
4850:
4823:
4721:
4177:
4167:
3778:
214:
184:
4915:
4701:
4508:
4470:
4243:
4182:
344:
4083:
is a duo giving historically informed performances of mediaeval Welsh music using the
3178:
4894:
4855:
4730:
4669:
4491:
4426:
4332:
4296:
4276:
4061:
3750:
3712:
3693:
3566:
3562:
3500:
3367:
3199:
3086:
2684:{\displaystyle \left({\frac {3}{2}}\right)^{5}\times \left({\frac {1}{2}}\right)^{2}}
1757:{\displaystyle \left({\frac {3}{2}}\right)^{6}\times \left({\frac {1}{2}}\right)^{3}}
1196:{\displaystyle \left({\frac {3}{2}}\right)^{4}\times \left({\frac {1}{2}}\right)^{2}}
250:
4920:
4247:
4172:
4002:
4874:
4738:
4706:
4626:
4591:
4233:
4217:
4188:
4157:
4147:
4102:
4057:
3923:
3902:
3765:
3586:
3473:
3449:
1690:
1499:
166:
4514:
32:
4996:
4636:
4357:
4041:
3614:
162:
4674:
4649:
4596:
4557:
4401:
3877:
3659:
3618:
1330:
340:
170:
158:
102:
39:
4520:
4238:
3545:
As an obvious consequence, each augmented or diminished interval is exactly 12
5042:
4969:
4938:
4711:
4664:
4571:
4094:
4068:
4037:
4009:
3955:
3943:
3889:
3866:
3837:
3808:
3688:
3436:
in the table) have a size of approximately 701.955 cents (700+ε cents, where
3119:
2617:
2426:
1891:
304:
258:
254:
178:
19:
4933:
4773:
4576:
4423:
Science and Civilization in China, Vol. IV: Physics and Physical Technology
4116:
4108:
3793:
3730:
752:
566:
293:
262:
5021:
5016:
5006:
4601:
4581:
4013:
3850:
3821:
3488:
3461:
3202:
coloured. The deviations arise because the notes determine two different
3167:
2827:
2240:
2054:
1129:
938:
42:
5011:
4980:
4405:
3998:
3997:. Within Ancient Greek music, the system had been mainly attributed to
3831:
3068:
289:
277:
23:
The syntonic tuning continuum, showing Pythagorean tuning at 702 cents.
4275:, seventh edition, 2 vols. (Boston: McGraw-Hill). Vol. I: p. 56.
3542:, the difference between the Pythagorean fifth and the average fifth.
65:
5001:
4955:
4028:
From about 1510 onward, as thirds came to be treated as consonances,
336:
335:
This succession of eleven 3:2 intervals spans across a wide range of
2292:{\displaystyle \left({\frac {3}{2}}\right)^{3}\times {\frac {1}{2}}}
804:{\displaystyle \left({\frac {3}{2}}\right)^{2}\times {\frac {1}{2}}}
3804:
3515:
3203:
3191:
3124:
285:
208:
4526:
4460:
3258:{\displaystyle S_{1}={256 \over 243}\approx 90.225{\text{ cents}}}
4975:
4925:
4049:
4017:
281:
242:
4215:
4943:
4659:
4586:
4080:
3969:
3860:
3701:
406:
356:
4960:
4948:
4084:
211:
of a vibrating string, after the octave (which is the ratio
4965:
4088:
3573:
can be also defined as one twelfth of a Pythagorean comma.
2478:{\displaystyle \left({\frac {2}{3}}\right)^{2}\times 2^{2}}
2106:{\displaystyle \left({\frac {2}{3}}\right)^{4}\times 2^{3}}
1551:{\displaystyle \left({\frac {2}{3}}\right)^{6}\times 2^{4}}
990:{\displaystyle \left({\frac {2}{3}}\right)^{3}\times 2^{2}}
618:{\displaystyle \left({\frac {2}{3}}\right)^{5}\times 2^{3}}
4311:
4309:
3432:
By definition, in Pythagorean tuning 11 perfect fifths (
4036:, which tunes thirds to the relatively simple ratio of
3419:{\displaystyle S_{E}={\sqrt{2}}=100.000{\text{ cents}}}
3625:(or epimoric ratio). The same is true for the octave.
280:(sixth century BC) by modern authors of music theory.
272:.) It is named, and has been widely misattributed, to
4306:
3379:
3284:
3220:
2784:
2742:
2698:
2627:
2578:
2536:
2492:
2436:
2392:
2350:
2306:
2250:
2206:
2164:
2120:
2064:
2015:
1973:
1929:
1901:
1857:
1815:
1771:
1700:
1651:
1609:
1565:
1509:
1460:
1418:
1374:
1340:
1296:
1254:
1210:
1139:
1090:
1048:
1004:
948:
904:
862:
818:
762:
718:
676:
632:
576:
527:
485:
444:
416:
241:"pure" interval, and the easiest to tune by ear. As
217:
187:
38:
A series of fifths generated can give seven notes: a
268:
The system dates back to Ancient Mesopotamia;. (See
3507:) are ≈ 384.360 cents (400 − 8
3468:) are ≈ 294.135 cents (300 − 3
331:–A–E–B–F♯–C♯–G♯
4115:and the Cal Arts Percussion Ensemble conducted by
3557:cents wider than each m3. This interval of size 12
3522:) are ≈ 90.225 cents (100 − 5
3418:
3348:
3257:
2797:
2769:
2727:
2683:
2591:
2563:
2521:
2477:
2405:
2377:
2335:
2291:
2219:
2191:
2149:
2105:
2028:
2000:
1958:
1914:
1870:
1842:
1800:
1756:
1664:
1636:
1594:
1550:
1473:
1445:
1403:
1359:
1309:
1281:
1239:
1195:
1103:
1075:
1033:
989:
917:
889:
847:
803:
731:
703:
661:
617:
540:
512:
470:
429:
229:
199:
4291:Kenneth Sylvan Guthrie, David R. Fideler (1987).
5040:
4222:"Invariant Fingerings Across a Tuning Continuum"
3569:(≈ −23.460 cents). This implies that
3182:The 144 intervals in C-based Pythagorean tuning.
4271:Bruce Benward and Marilyn Nadine Saker (2003).
4521:Creating a Pythagorean Tuning in a Spreadsheet
4462:The Archaeomusicology of the Ancient Near East
4400:) the size of which deviates by more than one
3553:cents narrower than each P5, and each A2 is 12
4542:
4488:Companion to Medieval & Renaissance Music
4185:, in which Plato discusses Pythagorean tuning
3530:) are ≈ 113.685 cents (100 + 7
3495:) are ≈ 407.820 cents (400 + 4
3480:) are ≈ 317.595 cents (300 + 9
3278:), also called chromatic semitone, with size
4617:List of intervals in 5-limit just intonation
4509:"A Pythagorean tuning of the diatonic scale"
4486:(1997), "The good, the bad and the boring",
4383:Transactions of the Asiatic Society of Japan
3214:), also called diatonic semitone, with size
4515:"Pythagorean Tuning and Medieval Polyphony"
4351:"The Development of Musical Tuning Systems"
4211:
4209:
3600:are specific for Pythagorean tuning, while
4549:
4535:
2830:based on C, obtained from this tuning is:
327:E♭–B♭–F–C–G–
169:are determined by choosing a sequence of
4267:
4265:
4263:
4237:
4455:
4315:
4206:
4101:(Hyperion, CDA66336, 1989), directed by
3995:Music of Mesopotamia § Music theory
3576:
3177:
270:Music of Mesopotamia § Music theory
207:. This is chosen because it is the next
145:
101:
18:
4387:, p. 82. Asiatic Society of Japan.
3370:chromatic scale, all semitones measure
1637:{\displaystyle {\frac {2^{10}}{3^{6}}}}
5041:
4415:
4348:
4260:
3448:to a fifth, is more properly called a
2770:{\displaystyle {\frac {3^{5}}{2^{7}}}}
2564:{\displaystyle {\frac {2^{4}}{3^{2}}}}
2378:{\displaystyle {\frac {3^{3}}{2^{4}}}}
2192:{\displaystyle {\frac {2^{7}}{3^{4}}}}
2001:{\displaystyle {\frac {3^{1}}{2^{1}}}}
1843:{\displaystyle {\frac {3^{6}}{2^{9}}}}
1446:{\displaystyle {\frac {2^{2}}{3^{1}}}}
1360:{\displaystyle {\frac {2}{3}}\times 2}
1282:{\displaystyle {\frac {3^{4}}{2^{6}}}}
1076:{\displaystyle {\frac {2^{5}}{3^{3}}}}
890:{\displaystyle {\frac {3^{2}}{2^{3}}}}
704:{\displaystyle {\frac {2^{8}}{3^{5}}}}
513:{\displaystyle {\frac {3^{0}}{2^{0}}}}
4530:
3909:
3650:
3565:, exactly equal to the opposite of a
3173:
106:Pythagorean (tonic) major chord on C
16:Method of tuning a musical instrument
3988:
3194:, twelve 2-semitone intervals, etc.
4556:
4390:
1665:{\displaystyle {\frac {1024}{729}}}
1595:{\displaystyle 3^{-6}\times 2^{10}}
13:
4408:equivalents) are considered to be
4123:(Etceter Records, KTC1071, 1990):
4052:as the normal tuning for singers.
3534:), and their average is 100 cents.
3511:), and their average is 400 cents;
3484:), and their average is 300 cents;
2798:{\displaystyle {\frac {243}{128}}}
2728:{\displaystyle 3^{5}\times 2^{-7}}
2522:{\displaystyle 3^{-2}\times 2^{4}}
2336:{\displaystyle 3^{3}\times 2^{-4}}
2150:{\displaystyle 3^{-4}\times 2^{7}}
1959:{\displaystyle 3^{1}\times 2^{-1}}
1871:{\displaystyle {\frac {729}{512}}}
1801:{\displaystyle 3^{6}\times 2^{-9}}
1404:{\displaystyle 3^{-1}\times 2^{2}}
1240:{\displaystyle 3^{4}\times 2^{-6}}
1034:{\displaystyle 3^{-3}\times 2^{5}}
848:{\displaystyle 3^{2}\times 2^{-3}}
732:{\displaystyle {\frac {256}{243}}}
662:{\displaystyle 3^{-5}\times 2^{8}}
14:
5070:
4502:
4380:Asiatic Society of Japan (1879).
4220:; Plamondon, J. (December 2007).
4016:very often, or which is not very
2220:{\displaystyle {\frac {128}{81}}}
471:{\displaystyle 3^{0}\times 2^{0}}
4692:Ptolemy's intense diatonic scale
4295:, p. 24. Red Wheel/Weiser.
2406:{\displaystyle {\frac {27}{16}}}
1310:{\displaystyle {\frac {81}{64}}}
1104:{\displaystyle {\frac {32}{27}}}
64:
31:
4436:
4349:Frazer, Peter A. (April 2001).
4329:Tuning, Timbre, Spectrum, Scale
4099:Music for the Lion-Hearted King
3067:In equal temperament, pairs of
2592:{\displaystyle {\frac {16}{9}}}
288:, ascribed the division of the
4374:
4342:
4321:
4285:
4127:for guitar and percussion and
4074:
2029:{\displaystyle {\frac {3}{2}}}
1915:{\displaystyle {\frac {3}{2}}}
1474:{\displaystyle {\frac {4}{3}}}
918:{\displaystyle {\frac {9}{8}}}
541:{\displaystyle {\frac {1}{1}}}
430:{\displaystyle {\frac {1}{1}}}
237:), and hence is the next most
1:
4421:Needham, Joseph (1962/2004).
4327:Sethares, William A. (2005).
4273:Music: In Theory and Practice
4194:
3801:tone, whole tone, whole step
5054:3-limit tuning and intervals
4655:Harry Partch's 43-tone scale
4199:
3931:tritone (τρίτονον) (729:512)
7:
4523:, video with audio samples.
4490:. Oxford University Press.
4144:, a near-Pythagorean tuning
4135:
3621:, shown in the table, is a
45:on C in Pythagorean tuning
10:
5075:
4622:List of meantone intervals
4475:: CS1 maint: postscript (
4449:
4153:List of meantone intervals
3885:diatessaron (διατεσσάρων)
3580:
3526:), 5 chromatic semitones (
83:12-tone equal tempered and
4989:
4906:
4869:
4822:
4729:
4720:
4635:
4612:List of musical intervals
4607:Consonance and dissonance
4564:
4239:10.1162/comj.2007.31.4.15
4158:List of musical intervals
3979:
3976:
3953:
3950:
3887:
3884:
3865:
3857:
3828:
3803:
3800:
3737:
3719:
3700:
3698:
3687:
3663:
3658:
3653:
3645:
3640:
3637:
3632:
3446:enharmonically equivalent
3190:– twelve unisons, twelve
3058:
3047:
3036:
3025:
3014:
3003:
2992:
2981:
2978:
2962:
2951:
2940:
2929:
2918:
2907:
2896:
2885:
2875:
2870:
2865:
2860:
2855:
2850:
2845:
2840:
355:(on a piano keyboard, an
314:
4091:using Pythagorean tuning
3651:Other naming conventions
4163:List of pitch intervals
4129:Plaint & Variations
4121:Guitar & Percussion
261:), which is ≈ 702
4484:Daniel Leech-Wilkinson
4398:enharmonic equivalents
4226:Computer Music Journal
4131:on "Song of Palestine"
4034:quarter-comma meantone
3954:hemiolion (ημιόλιον),
3888:epitrite (επίτριτος),
3623:superparticular number
3420:
3350:
3274:The augmented unison (
3259:
3183:
2799:
2771:
2729:
2685:
2593:
2565:
2523:
2479:
2407:
2379:
2337:
2293:
2221:
2193:
2151:
2107:
2030:
2002:
1960:
1916:
1872:
1844:
1802:
1758:
1666:
1638:
1596:
1552:
1475:
1447:
1405:
1361:
1311:
1283:
1241:
1197:
1105:
1077:
1035:
991:
919:
891:
849:
805:
733:
705:
663:
619:
542:
514:
472:
431:
257:(i.e., the untempered
231:
201:
151:
143:
24:
4879:Temperament ordinaire
4511:, with audio samples.
4044:was posited first by
3781:(αποτομή) (2187:2048)
3696: (524288:531441)
3577:Pythagorean intervals
3421:
3351:
3265:(e.g. between D and E
3260:
3181:
2800:
2772:
2730:
2686:
2594:
2566:
2524:
2480:
2408:
2380:
2338:
2294:
2222:
2194:
2152:
2108:
2031:
2003:
1961:
1917:
1873:
1845:
1803:
1759:
1667:
1639:
1597:
1553:
1476:
1448:
1406:
1362:
1312:
1284:
1242:
1198:
1106:
1078:
1036:
992:
920:
892:
850:
806:
734:
706:
664:
620:
543:
515:
473:
432:
232:
202:
149:
105:
22:
4682:List of compositions
4517:, by Margo Schulter.
4457:Dumbrill, Richard J.
4425:, pp. 170–171.
4142:53 equal temperament
4030:meantone temperament
3977:diapason (διαπασών)
3951:diapente (διαπέντε)
3583:Pythagorean interval
3377:
3282:
3218:
2782:
2740:
2696:
2625:
2576:
2534:
2490:
2434:
2390:
2348:
2304:
2248:
2204:
2162:
2118:
2062:
2013:
1971:
1927:
1899:
1855:
1813:
1769:
1698:
1649:
1607:
1563:
1507:
1458:
1416:
1372:
1338:
1294:
1252:
1208:
1137:
1088:
1046:
1002:
946:
902:
860:
816:
760:
716:
674:
630:
574:
525:
483:
442:
414:
247:syntonic temperament
215:
185:
71:Diatonic scale on C
4178:Musical temperament
4168:Regular temperament
4032:, and particularly
3773:chromatic semitone,
3656:(pitch ratio names)
3654:Pythagorean tuning
3366:By contrast, in an
230:{\displaystyle 2:1}
200:{\displaystyle 3:2}
130:equal tempered and
4916:Chinese musicology
4702:Scale of harmonics
4697:Pythagorean tuning
4645:Euler–Fokker genus
4183:Timaeus (dialogue)
3753:(λείμμα) (256:243)
3745:diatonic semitone,
3647:Quality and number
3501:diminished fourths
3416:
3346:
3255:
3210:The minor second (
3184:
3174:Sizes of intervals
2795:
2767:
2725:
2681:
2589:
2561:
2519:
2475:
2403:
2375:
2333:
2289:
2217:
2189:
2147:
2103:
2026:
1998:
1956:
1912:
1868:
1840:
1798:
1754:
1662:
1634:
1592:
1548:
1471:
1443:
1401:
1357:
1307:
1279:
1237:
1193:
1101:
1073:
1031:
987:
915:
887:
845:
801:
729:
701:
659:
615:
538:
510:
468:
427:
360:within an octave.
345:octave equivalence
227:
197:
155:Pythagorean tuning
152:
144:
25:
5036:
5035:
4902:
4901:
4431:978-0-521-05802-5
4281:978-0-07-294262-0
4105:(Leech-Wilkinson)
4087:and six-stringed
4062:equal temperament
4058:well temperaments
3989:History and usage
3984:
3983:
3863:(δίτονον) (81:64)
3713:diminished second
3694:Pythagorean comma
3617:tuned, and their
3567:diminished second
3563:Pythagorean comma
3474:augmented seconds
3414:
3403:
3344:
3333:
3320:
3253:
3242:
3200:Pythagorean comma
3087:Pythagorean comma
3063:
3062:
2814:
2813:
2793:
2765:
2669:
2641:
2587:
2559:
2450:
2401:
2373:
2287:
2264:
2215:
2187:
2078:
2024:
1996:
1910:
1866:
1838:
1742:
1714:
1660:
1632:
1523:
1469:
1441:
1349:
1305:
1277:
1181:
1153:
1099:
1071:
962:
913:
885:
799:
776:
727:
699:
590:
536:
508:
425:
301:Pythagorean scale
5066:
4875:Well temperament
4861:Regular diatonic
4727:
4726:
4707:Tonality diamond
4551:
4544:
4537:
4528:
4527:
4480:
4474:
4466:
4443:
4440:
4434:
4419:
4413:
4394:
4388:
4378:
4372:
4371:
4369:
4368:
4362:
4356:. Archived from
4355:
4346:
4340:
4325:
4319:
4313:
4304:
4289:
4283:
4269:
4258:
4257:
4255:
4254:
4241:
4213:
4189:Whole-tone scale
4148:Enharmonic scale
4103:Christopher Page
3924:augmented fourth
3903:diminished fifth
3766:augmented unison
3722:(531441:524288)
3630:
3629:
3587:Interval (music)
3450:diminished sixth
3425:
3423:
3422:
3417:
3415:
3412:
3404:
3402:
3394:
3389:
3388:
3368:equally tempered
3361:
3360:
3355:
3353:
3352:
3347:
3345:
3342:
3334:
3326:
3321:
3319:
3318:
3309:
3308:
3299:
3294:
3293:
3270:
3269:
3264:
3262:
3261:
3256:
3254:
3251:
3243:
3235:
3230:
3229:
3165:
3164:
3159:
3158:
3153:
3152:
3147:
3146:
3141:
3140:
3135:
3134:
3116:
3115:
3110:
3109:
3104:
3103:
3098:
3097:
3082:
3081:
3076:
3075:
3056:
3055:
3051:
3045:
3044:
3040:
3034:
3033:
3029:
3023:
3022:
3018:
3012:
3011:
3007:
3001:
3000:
2996:
2990:
2989:
2985:
2971:
2970:
2966:
2960:
2959:
2955:
2949:
2948:
2944:
2938:
2937:
2933:
2927:
2926:
2922:
2916:
2915:
2911:
2905:
2904:
2900:
2894:
2893:
2889:
2835:
2834:
2804:
2802:
2801:
2796:
2794:
2786:
2776:
2774:
2773:
2768:
2766:
2764:
2763:
2754:
2753:
2744:
2734:
2732:
2731:
2726:
2724:
2723:
2708:
2707:
2690:
2688:
2687:
2682:
2680:
2679:
2674:
2670:
2662:
2652:
2651:
2646:
2642:
2634:
2614:
2613:
2598:
2596:
2595:
2590:
2588:
2580:
2570:
2568:
2567:
2562:
2560:
2558:
2557:
2548:
2547:
2538:
2528:
2526:
2525:
2520:
2518:
2517:
2505:
2504:
2484:
2482:
2481:
2476:
2474:
2473:
2461:
2460:
2455:
2451:
2443:
2412:
2410:
2409:
2404:
2402:
2394:
2384:
2382:
2381:
2376:
2374:
2372:
2371:
2362:
2361:
2352:
2342:
2340:
2339:
2334:
2332:
2331:
2316:
2315:
2298:
2296:
2295:
2290:
2288:
2280:
2275:
2274:
2269:
2265:
2257:
2226:
2224:
2223:
2218:
2216:
2208:
2198:
2196:
2195:
2190:
2188:
2186:
2185:
2176:
2175:
2166:
2156:
2154:
2153:
2148:
2146:
2145:
2133:
2132:
2112:
2110:
2109:
2104:
2102:
2101:
2089:
2088:
2083:
2079:
2071:
2051:
2050:
2035:
2033:
2032:
2027:
2025:
2017:
2007:
2005:
2004:
1999:
1997:
1995:
1994:
1985:
1984:
1975:
1965:
1963:
1962:
1957:
1955:
1954:
1939:
1938:
1921:
1919:
1918:
1913:
1911:
1903:
1877:
1875:
1874:
1869:
1867:
1859:
1849:
1847:
1846:
1841:
1839:
1837:
1836:
1827:
1826:
1817:
1807:
1805:
1804:
1799:
1797:
1796:
1781:
1780:
1763:
1761:
1760:
1755:
1753:
1752:
1747:
1743:
1735:
1725:
1724:
1719:
1715:
1707:
1691:augmented fourth
1687:
1686:
1671:
1669:
1668:
1663:
1661:
1653:
1643:
1641:
1640:
1635:
1633:
1631:
1630:
1621:
1620:
1611:
1601:
1599:
1598:
1593:
1591:
1590:
1578:
1577:
1557:
1555:
1554:
1549:
1547:
1546:
1534:
1533:
1528:
1524:
1516:
1500:diminished fifth
1496:
1495:
1480:
1478:
1477:
1472:
1470:
1462:
1452:
1450:
1449:
1444:
1442:
1440:
1439:
1430:
1429:
1420:
1410:
1408:
1407:
1402:
1400:
1399:
1387:
1386:
1366:
1364:
1363:
1358:
1350:
1342:
1316:
1314:
1313:
1308:
1306:
1298:
1288:
1286:
1285:
1280:
1278:
1276:
1275:
1266:
1265:
1256:
1246:
1244:
1243:
1238:
1236:
1235:
1220:
1219:
1202:
1200:
1199:
1194:
1192:
1191:
1186:
1182:
1174:
1164:
1163:
1158:
1154:
1146:
1126:
1125:
1110:
1108:
1107:
1102:
1100:
1092:
1082:
1080:
1079:
1074:
1072:
1070:
1069:
1060:
1059:
1050:
1040:
1038:
1037:
1032:
1030:
1029:
1017:
1016:
996:
994:
993:
988:
986:
985:
973:
972:
967:
963:
955:
924:
922:
921:
916:
914:
906:
896:
894:
893:
888:
886:
884:
883:
874:
873:
864:
854:
852:
851:
846:
844:
843:
828:
827:
810:
808:
807:
802:
800:
792:
787:
786:
781:
777:
769:
738:
736:
735:
730:
728:
720:
710:
708:
707:
702:
700:
698:
697:
688:
687:
678:
668:
666:
665:
660:
658:
657:
645:
644:
624:
622:
621:
616:
614:
613:
601:
600:
595:
591:
583:
563:
562:
547:
545:
544:
539:
537:
529:
519:
517:
516:
511:
509:
507:
506:
497:
496:
487:
477:
475:
474:
469:
467:
466:
454:
453:
436:
434:
433:
428:
426:
418:
374:Interval from D
368:
367:
309:tune tetrachords
236:
234:
233:
228:
206:
204:
203:
198:
163:frequency ratios
141:
140:
139:
137:
129:
128:
127:
125:
117:
116:
115:
113:
95:just intonation.
94:
93:
92:
90:
82:
81:
80:
78:
68:
56:
55:
54:
52:
35:
5074:
5073:
5069:
5068:
5067:
5065:
5064:
5063:
5049:Music of Greece
5039:
5038:
5037:
5032:
5029:(Bohlen–Pierce)
4997:833 cents scale
4985:
4908:
4898:
4865:
4818:
4716:
4637:Just intonation
4631:
4560:
4558:Musical tunings
4555:
4505:
4468:
4467:
4452:
4447:
4446:
4441:
4437:
4420:
4416:
4395:
4391:
4379:
4375:
4366:
4364:
4360:
4353:
4347:
4343:
4331:, p. 163.
4326:
4322:
4314:
4307:
4290:
4286:
4270:
4261:
4252:
4250:
4216:Milne, Andrew;
4214:
4207:
4202:
4197:
4138:
4077:
4060:and eventually
4042:just intonation
4008:Because of the
4003:Shí-èr-lǜ scale
3991:
3774:
3746:
3741:
3739:
3666:
3655:
3641:Specific names
3634:
3619:frequency ratio
3589:
3581:Main articles:
3579:
3411:
3398:
3393:
3384:
3380:
3378:
3375:
3374:
3358:
3357:
3356:(e.g. between E
3341:
3325:
3314:
3310:
3304:
3300:
3298:
3289:
3285:
3283:
3280:
3279:
3267:
3266:
3250:
3234:
3225:
3221:
3219:
3216:
3215:
3176:
3162:
3161:
3156:
3155:
3150:
3149:
3144:
3143:
3138:
3137:
3132:
3131:
3113:
3112:
3107:
3106:
3101:
3100:
3095:
3094:
3079:
3078:
3073:
3072:
3071:notes such as A
3053:
3049:
3048:
3042:
3038:
3037:
3031:
3027:
3026:
3020:
3016:
3015:
3009:
3005:
3004:
2998:
2994:
2993:
2987:
2983:
2982:
2968:
2964:
2963:
2957:
2953:
2952:
2946:
2942:
2941:
2935:
2931:
2930:
2924:
2920:
2919:
2913:
2909:
2908:
2902:
2898:
2897:
2891:
2887:
2886:
2785:
2783:
2780:
2779:
2759:
2755:
2749:
2745:
2743:
2741:
2738:
2737:
2716:
2712:
2703:
2699:
2697:
2694:
2693:
2675:
2661:
2657:
2656:
2647:
2633:
2629:
2628:
2626:
2623:
2622:
2611:
2610:
2579:
2577:
2574:
2573:
2553:
2549:
2543:
2539:
2537:
2535:
2532:
2531:
2513:
2509:
2497:
2493:
2491:
2488:
2487:
2469:
2465:
2456:
2442:
2438:
2437:
2435:
2432:
2431:
2393:
2391:
2388:
2387:
2367:
2363:
2357:
2353:
2351:
2349:
2346:
2345:
2324:
2320:
2311:
2307:
2305:
2302:
2301:
2279:
2270:
2256:
2252:
2251:
2249:
2246:
2245:
2207:
2205:
2202:
2201:
2181:
2177:
2171:
2167:
2165:
2163:
2160:
2159:
2141:
2137:
2125:
2121:
2119:
2116:
2115:
2097:
2093:
2084:
2070:
2066:
2065:
2063:
2060:
2059:
2048:
2047:
2016:
2014:
2011:
2010:
1990:
1986:
1980:
1976:
1974:
1972:
1969:
1968:
1947:
1943:
1934:
1930:
1928:
1925:
1924:
1902:
1900:
1897:
1896:
1858:
1856:
1853:
1852:
1832:
1828:
1822:
1818:
1816:
1814:
1811:
1810:
1789:
1785:
1776:
1772:
1770:
1767:
1766:
1748:
1734:
1730:
1729:
1720:
1706:
1702:
1701:
1699:
1696:
1695:
1684:
1683:
1652:
1650:
1647:
1646:
1626:
1622:
1616:
1612:
1610:
1608:
1605:
1604:
1586:
1582:
1570:
1566:
1564:
1561:
1560:
1542:
1538:
1529:
1515:
1511:
1510:
1508:
1505:
1504:
1493:
1492:
1461:
1459:
1456:
1455:
1435:
1431:
1425:
1421:
1419:
1417:
1414:
1413:
1395:
1391:
1379:
1375:
1373:
1370:
1369:
1341:
1339:
1336:
1335:
1297:
1295:
1292:
1291:
1271:
1267:
1261:
1257:
1255:
1253:
1250:
1249:
1228:
1224:
1215:
1211:
1209:
1206:
1205:
1187:
1173:
1169:
1168:
1159:
1145:
1141:
1140:
1138:
1135:
1134:
1123:
1122:
1091:
1089:
1086:
1085:
1065:
1061:
1055:
1051:
1049:
1047:
1044:
1043:
1025:
1021:
1009:
1005:
1003:
1000:
999:
981:
977:
968:
954:
950:
949:
947:
944:
943:
905:
903:
900:
899:
879:
875:
869:
865:
863:
861:
858:
857:
836:
832:
823:
819:
817:
814:
813:
791:
782:
768:
764:
763:
761:
758:
757:
719:
717:
714:
713:
693:
689:
683:
679:
677:
675:
672:
671:
653:
649:
637:
633:
631:
628:
627:
609:
605:
596:
582:
578:
577:
575:
572:
571:
560:
559:
528:
526:
523:
522:
502:
498:
492:
488:
486:
484:
481:
480:
462:
458:
449:
445:
443:
440:
439:
417:
415:
412:
411:
397:
392:
387:
363:
317:
216:
213:
212:
186:
183:
182:
157:is a system of
135:
132:
131:
123:
120:
119:
111:
108:
107:
100:
99:
98:
97:
96:
88:
85:
84:
76:
73:
72:
69:
60:
59:
58:
50:
47:
46:
36:
17:
12:
11:
5:
5072:
5062:
5061:
5056:
5051:
5034:
5033:
5031:
5030:
5024:
5019:
5014:
5009:
5004:
4999:
4993:
4991:
4987:
4986:
4984:
4983:
4978:
4973:
4963:
4958:
4953:
4952:
4951:
4946:
4941:
4936:
4928:
4923:
4918:
4912:
4910:
4904:
4903:
4900:
4899:
4873:
4871:
4867:
4866:
4864:
4863:
4858:
4853:
4848:
4843:
4828:
4826:
4820:
4819:
4817:
4816:
4811:
4806:
4801:
4796:
4791:
4786:
4781:
4771:
4766:
4761:
4756:
4751:
4746:
4741:
4735:
4733:
4724:
4718:
4717:
4715:
4714:
4709:
4704:
4699:
4694:
4689:
4684:
4679:
4678:
4677:
4672:
4662:
4657:
4652:
4650:Harmonic scale
4647:
4641:
4639:
4633:
4632:
4630:
4629:
4624:
4619:
4614:
4609:
4604:
4599:
4597:Interval ratio
4594:
4589:
4584:
4579:
4574:
4568:
4566:
4562:
4561:
4554:
4553:
4546:
4539:
4531:
4525:
4524:
4518:
4512:
4504:
4503:External links
4501:
4500:
4499:
4481:
4451:
4448:
4445:
4444:
4435:
4414:
4402:syntonic comma
4389:
4373:
4341:
4320:
4305:
4284:
4259:
4218:Sethares, W.A.
4204:
4203:
4201:
4198:
4196:
4193:
4192:
4191:
4186:
4180:
4175:
4170:
4165:
4160:
4155:
4150:
4145:
4137:
4134:
4133:
4132:
4113:John Schneider
4106:
4092:
4076:
4073:
3990:
3987:
3986:
3985:
3982:
3981:
3978:
3975:
3972:
3966:
3962:
3961:
3959:
3952:
3949:
3946:
3941:
3937:
3936:
3934:
3932:
3929:
3926:
3921:
3917:
3916:
3914:
3912:
3910:
3908:
3905:
3900:
3896:
3895:
3893:
3886:
3883:
3880:
3878:perfect fourth
3875:
3871:
3870:
3864:
3858:
3856:
3853:
3848:
3844:
3843:
3841:
3835:
3829:
3827:
3824:
3819:
3815:
3814:
3812:
3802:
3799:
3796:
3791:
3787:
3786:
3784:
3782:
3776:
3775:major semitone
3771:
3768:
3763:
3759:
3758:
3756:
3754:
3748:
3747:minor semitone
3743:
3736:
3733:
3728:
3724:
3723:
3720:
3718:
3715:
3710:
3706:
3705:
3699:
3697:
3691:
3686:
3684:
3682:
3678:
3677:
3674:
3670:
3669:
3662:
3660:5-limit tuning
3657:
3652:
3649:
3643:
3642:
3639:
3636:
3578:
3575:
3561:is known as a
3536:
3535:
3512:
3485:
3456:). Similarly,
3427:
3426:
3410:
3407:
3401:
3397:
3392:
3387:
3383:
3364:
3363:
3340:
3337:
3332:
3329:
3324:
3317:
3313:
3307:
3303:
3297:
3292:
3288:
3272:
3249:
3246:
3241:
3238:
3233:
3228:
3224:
3175:
3172:
3130:If the notes G
3065:
3064:
3061:
3060:
3057:
3046:
3035:
3024:
3013:
3002:
2991:
2980:
2977:
2973:
2972:
2961:
2950:
2939:
2928:
2917:
2906:
2895:
2884:
2880:
2879:
2874:
2869:
2864:
2859:
2854:
2849:
2844:
2839:
2816:
2815:
2812:
2811:
2808:
2805:
2792:
2789:
2777:
2762:
2758:
2752:
2748:
2735:
2722:
2719:
2715:
2711:
2706:
2702:
2691:
2678:
2673:
2668:
2665:
2660:
2655:
2650:
2645:
2640:
2637:
2632:
2620:
2615:
2606:
2605:
2602:
2599:
2586:
2583:
2571:
2556:
2552:
2546:
2542:
2529:
2516:
2512:
2508:
2503:
2500:
2496:
2485:
2472:
2468:
2464:
2459:
2454:
2449:
2446:
2441:
2429:
2424:
2420:
2419:
2416:
2413:
2400:
2397:
2385:
2370:
2366:
2360:
2356:
2343:
2330:
2327:
2323:
2319:
2314:
2310:
2299:
2286:
2283:
2278:
2273:
2268:
2263:
2260:
2255:
2243:
2238:
2234:
2233:
2230:
2227:
2214:
2211:
2199:
2184:
2180:
2174:
2170:
2157:
2144:
2140:
2136:
2131:
2128:
2124:
2113:
2100:
2096:
2092:
2087:
2082:
2077:
2074:
2069:
2057:
2052:
2043:
2042:
2039:
2036:
2023:
2020:
2008:
1993:
1989:
1983:
1979:
1966:
1953:
1950:
1946:
1942:
1937:
1933:
1922:
1909:
1906:
1894:
1889:
1885:
1884:
1881:
1878:
1865:
1862:
1850:
1835:
1831:
1825:
1821:
1808:
1795:
1792:
1788:
1784:
1779:
1775:
1764:
1751:
1746:
1741:
1738:
1733:
1728:
1723:
1718:
1713:
1710:
1705:
1693:
1688:
1679:
1678:
1675:
1672:
1659:
1656:
1644:
1629:
1625:
1619:
1615:
1602:
1589:
1585:
1581:
1576:
1573:
1569:
1558:
1545:
1541:
1537:
1532:
1527:
1522:
1519:
1514:
1502:
1497:
1488:
1487:
1484:
1481:
1468:
1465:
1453:
1438:
1434:
1428:
1424:
1411:
1398:
1394:
1390:
1385:
1382:
1378:
1367:
1356:
1353:
1348:
1345:
1333:
1331:perfect fourth
1328:
1324:
1323:
1320:
1317:
1304:
1301:
1289:
1274:
1270:
1264:
1260:
1247:
1234:
1231:
1227:
1223:
1218:
1214:
1203:
1190:
1185:
1180:
1177:
1172:
1167:
1162:
1157:
1152:
1149:
1144:
1132:
1127:
1118:
1117:
1114:
1111:
1098:
1095:
1083:
1068:
1064:
1058:
1054:
1041:
1028:
1024:
1020:
1015:
1012:
1008:
997:
984:
980:
976:
971:
966:
961:
958:
953:
941:
936:
932:
931:
928:
925:
912:
909:
897:
882:
878:
872:
868:
855:
842:
839:
835:
831:
826:
822:
811:
798:
795:
790:
785:
780:
775:
772:
767:
755:
750:
746:
745:
742:
739:
726:
723:
711:
696:
692:
686:
682:
669:
656:
652:
648:
643:
640:
636:
625:
612:
608:
604:
599:
594:
589:
586:
581:
569:
564:
555:
554:
551:
548:
535:
532:
520:
505:
501:
495:
491:
478:
465:
461:
457:
452:
448:
437:
424:
421:
409:
404:
400:
399:
394:
389:
384:
381:
378:
375:
372:
341:piano keyboard
333:
332:
316:
313:
274:Ancient Greeks
226:
223:
220:
196:
193:
190:
159:musical tuning
70:
63:
62:
61:
37:
30:
29:
28:
27:
26:
15:
9:
6:
4:
3:
2:
5071:
5060:
5057:
5055:
5052:
5050:
5047:
5046:
5044:
5028:
5025:
5023:
5020:
5018:
5015:
5013:
5010:
5008:
5005:
5003:
5000:
4998:
4995:
4994:
4992:
4988:
4982:
4979:
4977:
4974:
4971:
4970:Carnatic raga
4967:
4964:
4962:
4959:
4957:
4954:
4950:
4947:
4945:
4942:
4940:
4939:Turkish makam
4937:
4935:
4932:
4931:
4929:
4927:
4924:
4922:
4919:
4917:
4914:
4913:
4911:
4905:
4896:
4892:
4888:
4884:
4880:
4876:
4872:
4868:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4841:
4837:
4836:quarter-comma
4833:
4830:
4829:
4827:
4825:
4821:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4795:
4792:
4790:
4787:
4785:
4782:
4779:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4736:
4734:
4732:
4728:
4725:
4723:
4719:
4713:
4712:Tonality flux
4710:
4708:
4705:
4703:
4700:
4698:
4695:
4693:
4690:
4688:
4685:
4683:
4680:
4676:
4673:
4671:
4668:
4667:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4642:
4640:
4638:
4634:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4603:
4600:
4598:
4595:
4593:
4590:
4588:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4569:
4567:
4563:
4559:
4552:
4547:
4545:
4540:
4538:
4533:
4532:
4529:
4522:
4519:
4516:
4513:
4510:
4507:
4506:
4497:
4496:0-19-816540-4
4493:
4489:
4485:
4482:
4478:
4472:
4464:
4463:
4458:
4454:
4453:
4439:
4432:
4428:
4424:
4418:
4411:
4407:
4403:
4399:
4393:
4386:
4384:
4377:
4363:on 2006-05-06
4359:
4352:
4345:
4338:
4337:1-85233-797-4
4334:
4330:
4324:
4318:, p. 18.
4317:
4316:Dumbrill 1998
4312:
4310:
4302:
4301:9780933999510
4298:
4294:
4288:
4282:
4278:
4274:
4268:
4266:
4264:
4249:
4245:
4240:
4235:
4231:
4227:
4223:
4219:
4212:
4210:
4205:
4190:
4187:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4143:
4140:
4139:
4130:
4126:
4122:
4118:
4114:
4111:performed by
4110:
4107:
4104:
4100:
4096:
4095:Gothic Voices
4093:
4090:
4086:
4082:
4079:
4078:
4072:
4070:
4069:violin family
4065:
4063:
4059:
4053:
4051:
4047:
4043:
4039:
4035:
4031:
4026:
4022:
4019:
4015:
4011:
4010:wolf interval
4006:
4004:
4000:
3996:
3980:duplex (2:1)
3973:
3971:
3967:
3964:
3963:
3960:
3957:
3956:sesquialterum
3947:
3945:
3944:perfect fifth
3942:
3939:
3938:
3935:
3933:
3930:
3927:
3925:
3922:
3919:
3918:
3915:
3913:
3911:
3906:
3904:
3901:
3898:
3897:
3894:
3891:
3890:sesquitertium
3881:
3879:
3876:
3873:
3872:
3868:
3867:sesquiquartum
3862:
3859:
3854:
3852:
3849:
3846:
3845:
3842:
3839:
3838:sesquiquintum
3836:
3833:
3830:
3825:
3823:
3820:
3817:
3816:
3813:
3810:
3809:sesquioctavum
3806:
3797:
3795:
3792:
3789:
3788:
3785:
3783:
3780:
3777:
3772:
3769:
3767:
3764:
3761:
3760:
3757:
3755:
3752:
3749:
3744:
3734:
3732:
3729:
3726:
3725:
3721:
3716:
3714:
3711:
3708:
3707:
3703:
3695:
3692:
3690:
3685:
3683:
3680:
3679:
3675:
3672:
3671:
3668:
3661:
3648:
3644:
3638:Generic names
3631:
3628:
3627:
3626:
3624:
3620:
3616:
3612:
3607:
3603:
3599:
3595:
3588:
3584:
3574:
3572:
3568:
3564:
3560:
3556:
3552:
3548:
3543:
3541:
3533:
3529:
3525:
3521:
3517:
3513:
3510:
3506:
3502:
3498:
3494:
3490:
3486:
3483:
3479:
3475:
3471:
3467:
3463:
3459:
3458:
3457:
3455:
3451:
3447:
3443:
3439:
3435:
3430:
3408:
3405:
3399:
3395:
3390:
3385:
3381:
3373:
3372:
3371:
3369:
3338:
3335:
3330:
3327:
3322:
3315:
3311:
3305:
3301:
3295:
3290:
3286:
3277:
3273:
3247:
3244:
3239:
3236:
3231:
3226:
3222:
3213:
3209:
3208:
3207:
3205:
3201:
3195:
3193:
3189:
3188:interval type
3180:
3171:
3169:
3128:
3126:
3122:
3121:
3120:wolf interval
3091:
3089:
3088:
3070:
2975:
2974:
2882:
2881:
2878:
2873:
2868:
2863:
2858:
2853:
2848:
2843:
2837:
2836:
2833:
2832:
2831:
2829:
2824:
2822:
2809:
2806:
2790:
2787:
2778:
2760:
2756:
2750:
2746:
2736:
2720:
2717:
2713:
2709:
2704:
2700:
2692:
2676:
2671:
2666:
2663:
2658:
2653:
2648:
2643:
2638:
2635:
2630:
2621:
2619:
2618:major seventh
2616:
2608:
2607:
2603:
2600:
2584:
2581:
2572:
2554:
2550:
2544:
2540:
2530:
2514:
2510:
2506:
2501:
2498:
2494:
2486:
2470:
2466:
2462:
2457:
2452:
2447:
2444:
2439:
2430:
2428:
2427:minor seventh
2425:
2422:
2421:
2417:
2414:
2398:
2395:
2386:
2368:
2364:
2358:
2354:
2344:
2328:
2325:
2321:
2317:
2312:
2308:
2300:
2284:
2281:
2276:
2271:
2266:
2261:
2258:
2253:
2244:
2242:
2239:
2236:
2235:
2231:
2228:
2212:
2209:
2200:
2182:
2178:
2172:
2168:
2158:
2142:
2138:
2134:
2129:
2126:
2122:
2114:
2098:
2094:
2090:
2085:
2080:
2075:
2072:
2067:
2058:
2056:
2053:
2045:
2044:
2040:
2037:
2021:
2018:
2009:
1991:
1987:
1981:
1977:
1967:
1951:
1948:
1944:
1940:
1935:
1931:
1923:
1907:
1904:
1895:
1893:
1892:perfect fifth
1890:
1887:
1886:
1882:
1879:
1863:
1860:
1851:
1833:
1829:
1823:
1819:
1809:
1793:
1790:
1786:
1782:
1777:
1773:
1765:
1749:
1744:
1739:
1736:
1731:
1726:
1721:
1716:
1711:
1708:
1703:
1694:
1692:
1689:
1681:
1680:
1676:
1673:
1657:
1654:
1645:
1627:
1623:
1617:
1613:
1603:
1587:
1583:
1579:
1574:
1571:
1567:
1559:
1543:
1539:
1535:
1530:
1525:
1520:
1517:
1512:
1503:
1501:
1498:
1490:
1489:
1485:
1482:
1466:
1463:
1454:
1436:
1432:
1426:
1422:
1412:
1396:
1392:
1388:
1383:
1380:
1376:
1368:
1354:
1351:
1346:
1343:
1334:
1332:
1329:
1326:
1325:
1321:
1318:
1302:
1299:
1290:
1272:
1268:
1262:
1258:
1248:
1232:
1229:
1225:
1221:
1216:
1212:
1204:
1188:
1183:
1178:
1175:
1170:
1165:
1160:
1155:
1150:
1147:
1142:
1133:
1131:
1128:
1120:
1119:
1115:
1112:
1096:
1093:
1084:
1066:
1062:
1056:
1052:
1042:
1026:
1022:
1018:
1013:
1010:
1006:
998:
982:
978:
974:
969:
964:
959:
956:
951:
942:
940:
937:
934:
933:
929:
926:
910:
907:
898:
880:
876:
870:
866:
856:
840:
837:
833:
829:
824:
820:
812:
796:
793:
788:
783:
778:
773:
770:
765:
756:
754:
751:
748:
747:
743:
740:
724:
721:
712:
694:
690:
684:
680:
670:
654:
650:
646:
641:
638:
634:
626:
610:
606:
602:
597:
592:
587:
584:
579:
570:
568:
565:
557:
556:
552:
549:
533:
530:
521:
503:
499:
493:
489:
479:
463:
459:
455:
450:
446:
438:
422:
419:
410:
408:
405:
402:
401:
395:
390:
385:
382:
379:
376:
373:
370:
369:
366:
365:
364:
361:
358:
354:
350:
346:
342:
338:
330:
326:
325:
324:
322:
312:
310:
306:
302:
297:
295:
291:
287:
283:
279:
275:
271:
266:
264:
260:
259:perfect fifth
256:
253:is the ratio
252:
249:in which the
248:
244:
240:
224:
221:
218:
210:
194:
191:
188:
181:, with ratio
180:
176:
172:
168:
164:
161:in which the
160:
156:
148:
138:
126:
114:
104:
91:
79:
67:
53:
44:
41:
34:
21:
5027:Lambda scale
4934:Arabic maqam
4891:Werckmeister
4722:Temperaments
4696:
4487:
4461:
4438:
4422:
4417:
4392:
4381:
4376:
4365:. Retrieved
4358:the original
4344:
4328:
4323:
4292:
4287:
4272:
4251:. Retrieved
4232:(4): 15–32.
4229:
4225:
4128:
4124:
4120:
4117:John Bergamo
4109:Lou Harrison
4098:
4066:
4054:
4048:and then by
4027:
4025:instrument.
4023:
4018:harmonically
4007:
3992:
3807:(επόγδοον),
3794:major second
3731:minor second
3610:
3605:
3601:
3597:
3593:
3590:
3570:
3558:
3554:
3550:
3546:
3544:
3539:
3537:
3531:
3527:
3523:
3519:
3508:
3504:
3496:
3492:
3489:major thirds
3481:
3477:
3469:
3465:
3462:minor thirds
3453:
3441:
3437:
3433:
3431:
3428:
3365:
3275:
3211:
3196:
3187:
3185:
3129:
3118:
3092:
3085:
3066:
2876:
2871:
2866:
2861:
2856:
2851:
2846:
2841:
2825:
2817:
753:major second
567:minor second
362:
353:basic octave
352:
348:
334:
328:
320:
318:
300:
298:
294:Eratosthenes
284:, and later
267:
154:
153:
5022:Delta scale
5017:Gamma scale
5007:Alpha scale
4909:non-Western
4907:Traditional
4602:Pitch class
4582:Millioctave
4565:Measurement
4125:Suite No. 1
4075:Discography
3851:major third
3822:minor third
3514:7 diatonic
3413: cents
3343: cents
3252: cents
2828:major scale
2241:major sixth
2055:minor sixth
1130:major third
939:minor third
173:which are "
43:major scale
5059:Pythagoras
5043:Categories
5012:Beta scale
4990:Non-octave
4981:Tetrachord
4883:Kirnberger
4846:Schismatic
4406:enharmonic
4385:, Volume 7
4367:2014-02-02
4253:2013-07-11
4195:References
3999:Pythagoras
3968:(perfect)
3832:semiditone
3740:half tone,
3704:(128:125)
3598:semiditone
3154:, making F
3069:enharmonic
396:12-TET-dif
290:tetrachord
278:Pythagoras
276:, notably
5002:A12 scale
4956:Octoechos
4921:Shí-èr-lǜ
4870:Irregular
4687:Otonality
4627:Microtone
4471:cite book
4410:dissonant
4200:Citations
4173:Shí-èr-lǜ
3742:half step
3738:semitone,
3665:1/4-comma
3635:semitones
3633:Number of
3516:semitones
3336:≈
3245:≈
3204:semitones
3192:semitones
3170:in tune.
3127:flatter.
2821:harmonics
2718:−
2710:×
2654:×
2507:×
2499:−
2463:×
2326:−
2318:×
2277:×
2135:×
2127:−
2091:×
1949:−
1941:×
1791:−
1783:×
1727:×
1580:×
1572:−
1536:×
1389:×
1381:−
1352:×
1230:−
1222:×
1166:×
1019:×
1011:−
975:×
838:−
830:×
789:×
647:×
639:−
603:×
456:×
386:Frequency
349:base note
337:frequency
251:generator
239:consonant
167:intervals
4887:Vallotti
4840:septimal
4832:Meantone
4592:Interval
4459:(1998).
4248:27906745
4136:See also
3805:epogdoön
3667:meantone
3359:♭
3268:♭
3163:♭
3157:♯
3151:♯
3145:♭
3139:♭
3133:♯
3125:semitone
3114:♭
3108:♯
3102:♯
3096:♭
3080:♯
3074:♭
2807:1109.78
2612:♯
2049:♭
1685:♯
1494:♭
1124:♯
561:♭
398:(cents)
393:(cents)
377:Formula
286:Boethius
209:harmonic
118:(compare
40:diatonic
4976:Slendro
4926:Dastgah
4851:Miracle
4814:96-tone
4809:72-tone
4804:58-tone
4799:53-tone
4794:41-tone
4789:34-tone
4784:31-tone
4774:24-tone
4769:23-tone
4764:22-tone
4759:19-tone
4754:17-tone
4749:15-tone
4744:12-tone
4675:7-limit
4670:5-limit
4450:Sources
4050:Zarlino
3834:(32:27)
3779:apotome
3611:sesqui-
3606:tritone
3409:100.000
3339:113.685
3052:⁄
3041:⁄
3030:⁄
3019:⁄
3008:⁄
2997:⁄
2986:⁄
2967:⁄
2956:⁄
2945:⁄
2934:⁄
2923:⁄
2912:⁄
2901:⁄
2890:⁄
2601:996.09
2415:905.87
2229:792.18
2038:701.96
1880:611.73
1677:−11.73
1674:588.27
1483:498.04
1319:407.82
1113:294.13
927:203.91
321:D-based
303:is any
282:Ptolemy
243:Novalis
179:perfect
165:of all
4944:Mugham
4930:Maqam
4824:Linear
4778:pieces
4739:6-tone
4660:Hexany
4587:Savart
4494:
4429:
4335:
4299:
4279:
4246:
4081:Bragod
3970:octave
3869:(5:4)
3861:ditone
3702:diesis
3676:Short
3615:justly
3594:ditone
3362:and E)
3248:90.225
2883:Ratio
2604:−3.91
2232:−7.82
1883:11.73
1486:−1.96
1116:−5.87
744:−9.78
741:90.22
407:unison
388:ratio
357:octave
339:(on a
315:Method
265:wide.
171:fifths
142:just).
4961:Pelog
4949:Muqam
4895:Young
4856:Magic
4731:Equal
4665:Limit
4572:Pitch
4361:(PDF)
4354:(PDF)
4244:S2CID
4085:crwth
4046:Ramos
3958:(3:2)
3892:(4:3)
3840:(6:5)
3811:(9:8)
3751:limma
3689:comma
3499:), 4
3472:), 3
3136:and E
3077:and G
2976:Step
2838:Note
2810:9.78
2418:5.87
2041:1.96
1322:7.82
930:3.91
553:0.00
550:0.00
371:Note
305:scale
263:cents
177:" or
4966:Raga
4577:Cent
4492:ISBN
4477:link
4427:ISBN
4333:ISBN
4297:ISBN
4277:ISBN
4089:lyre
3673:Full
3613:are
3604:and
3602:tone
3596:and
3585:and
3331:2048
3328:2187
3168:keys
3148:to F
3111:to E
3099:to G
2826:The
1655:1024
391:Size
299:The
175:pure
136:Play
124:Play
112:Play
89:Play
77:Play
51:Play
4234:doi
4038:5:4
4014:key
3974:P8
3948:P5
3928:A4
3907:d5
3798:M2
3240:243
3237:256
3054:243
3050:256
3010:243
3006:256
2958:128
2954:243
2791:128
2788:243
2210:128
1864:512
1861:729
1658:729
725:243
722:256
255:3:2
5045::
4893:,
4889:,
4885:,
4838:,
4473:}}
4469:{{
4308:^
4262:^
4242:.
4230:31
4228:.
4224:.
4208:^
4119:-
4097:–
4064:.
3965:12
3882:P4
3855:M3
3826:m3
3770:A1
3735:m2
3717:d2
3528:A1
3520:m2
3505:d4
3493:M3
3487:8
3478:A2
3466:m3
3460:9
3454:d6
3434:P5
3400:12
3316:11
3276:A1
3212:m2
3206::
3160:-D
3090:.
3059:—
2979:—
2947:16
2943:27
2914:64
2910:81
2823:.
2582:16
2423:C
2399:16
2396:27
2237:B
2213:81
1888:A
1618:10
1588:10
1327:G
1303:64
1300:81
1097:27
1094:32
935:F
749:E
403:D
383:=
380:=
4972:)
4968:(
4897:)
4881:(
4877:/
4842:)
4834:(
4780:)
4776:(
4550:e
4543:t
4536:v
4498:.
4479:)
4433:.
4370:.
4339:.
4303:.
4256:.
4236::
3940:7
3920:6
3899:6
3874:5
3847:4
3818:3
3790:2
3762:1
3727:1
3709:0
3681:0
3571:ε
3559:ε
3555:ε
3551:ε
3547:ε
3540:ε
3532:ε
3524:ε
3518:(
3509:ε
3503:(
3497:ε
3491:(
3482:ε
3476:(
3470:ε
3464:(
3452:(
3442:ε
3438:ε
3406:=
3396:2
3391:=
3386:E
3382:S
3323:=
3312:2
3306:7
3302:3
3296:=
3291:2
3287:S
3271:)
3232:=
3227:1
3223:S
3043:8
3039:9
3032:8
3028:9
3021:8
3017:9
2999:8
2995:9
2988:8
2984:9
2969:1
2965:2
2936:2
2932:3
2925:3
2921:4
2903:8
2899:9
2892:1
2888:1
2877:C
2872:B
2867:A
2862:G
2857:F
2852:E
2847:D
2842:C
2761:7
2757:2
2751:5
2747:3
2721:7
2714:2
2705:5
2701:3
2677:2
2672:)
2667:2
2664:1
2659:(
2649:5
2644:)
2639:2
2636:3
2631:(
2609:C
2585:9
2555:2
2551:3
2545:4
2541:2
2515:4
2511:2
2502:2
2495:3
2471:2
2467:2
2458:2
2453:)
2448:3
2445:2
2440:(
2369:4
2365:2
2359:3
2355:3
2329:4
2322:2
2313:3
2309:3
2285:2
2282:1
2272:3
2267:)
2262:2
2259:3
2254:(
2183:4
2179:3
2173:7
2169:2
2143:7
2139:2
2130:4
2123:3
2099:3
2095:2
2086:4
2081:)
2076:3
2073:2
2068:(
2046:B
2022:2
2019:3
1992:1
1988:2
1982:1
1978:3
1952:1
1945:2
1936:1
1932:3
1908:2
1905:3
1834:9
1830:2
1824:6
1820:3
1794:9
1787:2
1778:6
1774:3
1750:3
1745:)
1740:2
1737:1
1732:(
1722:6
1717:)
1712:2
1709:3
1704:(
1682:G
1628:6
1624:3
1614:2
1584:2
1575:6
1568:3
1544:4
1540:2
1531:6
1526:)
1521:3
1518:2
1513:(
1491:A
1467:3
1464:4
1437:1
1433:3
1427:2
1423:2
1397:2
1393:2
1384:1
1377:3
1355:2
1347:3
1344:2
1273:6
1269:2
1263:4
1259:3
1233:6
1226:2
1217:4
1213:3
1189:2
1184:)
1179:2
1176:1
1171:(
1161:4
1156:)
1151:2
1148:3
1143:(
1121:F
1067:3
1063:3
1057:5
1053:2
1027:5
1023:2
1014:3
1007:3
983:2
979:2
970:3
965:)
960:3
957:2
952:(
911:8
908:9
881:3
877:2
871:2
867:3
841:3
834:2
825:2
821:3
797:2
794:1
784:2
779:)
774:2
771:3
766:(
695:5
691:3
685:8
681:2
655:8
651:2
642:5
635:3
611:3
607:2
598:5
593:)
588:3
585:2
580:(
558:E
534:1
531:1
504:0
500:2
494:0
490:3
464:0
460:2
451:0
447:3
423:1
420:1
329:D
225:1
222::
219:2
195:2
192::
189:3
57:.
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