218:
1157:
1144:
1128:
992:
31:
62:
1273:
1223:
1173:
822:
48:
838:. In one study, changes in tone quality reduced student musicians' ability to recognize, as out-of-tune, pitches that deviated from their appropriate values by ±12 cents. It has also been established that increased tonal context enables listeners to judge pitch more accurately. "While intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords."
4028:
1328:
940:, where the unit corresponds to a semitone in equal temperament. Alexander John Ellis in 1880 describes a large number of pitch standards that he noted or calculated, indicating in pronys with two decimals, i.e. with a precision to the 1/100 of a semitone, the interval that separated them from a theoretical pitch of 370 Hz, taken as point of reference.
890:
manageable units, he suggests to take 7/301 to obtain units of 1/43 octave. The octave therefore is divided in 43 parts, named "merides", themselves divided in 7 parts, the "heptamerides". Sauveur also imagined the possibility to further divide each heptameride in 10, but does not really make use of such microscopic units.
898:
Félix Savart (1791-1841) took over
Sauveur's system, without limiting the number of decimals of the logarithm of 2, so that the value of his unit varies according to sources. With five decimals, the base-10 logarithm of 2 is 0.30103, giving 301.03 savarts in the octave. This value often is rounded to
208:
Ellis presents applications of the cent system in this paper on musical scales of various nations, which include: (I. Heptatonic scales) Ancient Greece and Modern Europe, Persia, Arabia, Syria and
Scottish Highlands, India, Singapore, Burmah and Siam,; (II. Pentatonic scales) South Pacific, Western
889:
of 1701, proposed the usage of base-10 logarithms, probably because tables were available. He made use of logarithms computed with three decimals. The base-10 logarithm of 2 is equal to approximately 0.301, which
Sauveur multiplies by 1000 to obtain 301 units in the octave. In order to work on more
184:
to be "the number of double or complete vibrations, backwards and forwards, made in each second by a particle of air while the note is heard". He later defined musical pitch to be "the pitch, or V of any named musical note which determines the pitch of all the other notes in a particular system of
873:
The representation of musical intervals by logarithms is almost as old as logarithms themselves. Logarithms had been invented by Lord Napier in 1614. As early as 1647, Juan
Caramuel y Lobkowitz (1606-1682) in a letter to Athanasius Kircher described the usage of base-2 logarithms in music. In this
209:
Africa, Java, China and Japan. And he reaches the conclusion that "the
Musical Scale is not one, not 'natural,' nor even founded necessarily on the laws of the constitution of musical sound, so beautifully worked out by Helmholtz, but very diverse, very artificial, and very capricious".
829:
It is difficult to establish how many cents are perceptible to humans; this precision varies greatly from person to person. One author stated that humans can distinguish a difference in pitch of about 5–6 cents. The threshold of what is perceptible, technically known as the
1859:
Acoustique et musique : Données physiques et technologiques, problèmes de l'audition des sons musicaux, principes de fonctionnement et signification acoustique des principaux archétypes d'instruments de musique, les musiques expérimentales, l'acoustique des
1290:
1240:
204:
or tuning, and is the system at present used throughout Europe. He further gives calculations to approximate the measure of a ratio in cents, adding that "it is, as a general rule, unnecessary to go beyond the nearest whole number of cents."
1190:
132:'s suggestion. Making extensive measurements of musical instruments from around the world, Ellis used cents to report and compare the scales employed, and further described and utilized the system in his 1875 edition of
1291:
1241:
1191:
467:
1111:
The following audio files play various intervals. In each case the first note played is middle C. The next note is sharper than C by the assigned value in cents. Finally, the two notes are played simultaneously.
1158:
1145:
993:
1119:
may be heard (for example if middle C and a note 10 cents higher are played). At any particular instant, the two waveforms reinforce or cancel each other more or less, depending on their instantaneous
663:
200:
Ellis noted that "the object of the tuner is to make the interval between any two notes answering to any two adjacent finger keys throughout the instrument precisely the same. The result is called
261:. Thus, raising a frequency by one cent corresponds to multiplying the original frequency by this constant value. Raising a frequency by 1200 cents doubles the frequency, resulting in its octave.
1129:
189:
of the instrument, and an interval between any two notes is measured by "the ratio of the smaller pitch number to the larger, or by the fraction formed by dividing the larger by the smaller".
1289:
1239:
1189:
159:
in 1885, officially introduced the cent system to be used in exploring, by comparing and contrasting, musical scales of various nations. The cent system had already been defined in his
813: × 50 ≅ 1.02973). This error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes.
677:
has a frequency ratio 5:4 or ~386 cents, but in equal temperament is 400 cents. This 14 cent difference is about a seventh of a half step and large enough to be audible.
987:
938:
598:
571:
544:
497:
390:
363:
316:
289:
795:
773:
751:
517:
336:
241:—two notes that have a frequency ratio of 2:1—spans twelve semitones and therefore 1200 cents. The ratio of frequencies one cent apart is precisely equal to
2267:
Geringer, J. M.; Worthy, M.D. (1999). "Effects of Tone-Quality
Changes on Intonation and Tone-Quality Ratings of High School and College Instrumentalists".
1115:
Note that the JND for pitch difference is 5–6 cents. Played separately, the notes may not show an audible difference, but when they are played together,
1292:
1242:
2038:
Barbieri, Patrizio (1987). "Juan
Caramuel Lobkowitz (1606–1682): über die musikalischen Logarithmen und das Problem der musikalischen Temperatur".
1192:
167:, 99 other notes were interposed, making exactly equal intervals with each other, we should divide the octave into 1200 equal hundrecths [
2471:
2078:
1032:(1.95 cents) are nearly the same (≈ 614 steps per octave) and both may be approximated by 600 steps per octave (2 cents). Yasser promoted the
398:
3749:
2648:
2305:
2544:
2580:
860:, however, have trouble recognizing differences of less than 100 cents and sometimes have trouble with these or larger intervals.
2713:
849:
found that vibrato span typically ranged between ±34 cents and ±123 cents with a mean of ±71 cents and noted higher variation in
606:
2433:
2412:
845:, there is evidence that humans perceive the mean frequency as the center of the pitch. One study of modern performances of
3092:
2723:
217:
129:
3251:
2452:
2257:
2062:
1282:
1232:
1182:
2686:
1123:
relationship. A piano tuner may verify tuning accuracy by timing the beats when two strings are sounded at once.
2319:
Peretz, I.; Hyde, K.L. (August 2003). "What is specific to music processing? Insights from congenital amusia".
2871:
87:
17:
2229:
3541:
3521:
2573:
4050:
3647:
2809:
1420:
856:
Normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with
2087:
4055:
3653:
3361:
2653:
2139:; Hipkins, Alfred J. (1884), "Tonometrical Observations on Some Existing Non-Harmonic Musical Scales",
831:
229:(green) intervals showing the relationship between frequency ratio and the intervals' values, in cents.
3214:
3177:
1040:(10, 100, and 1000 steps per whole tone = 60, 600, and 6000 steps per octave = 20, 2, and 0.2 cents).
3436:
2638:
718:
118:
2333:
721:. Thus, although cents represent a logarithmic scale, small intervals (under 100 cents) can be
2922:
163:, where Ellis writes: "If we supposed that, between each pair of adjacent notes, forming an equal
4065:
4031:
3769:
3708:
3085:
3058:
2892:
2718:
2643:
2566:
140:. It has become the standard method of representing and comparing musical pitches and intervals.
963:
914:
3829:
3306:
2914:
2877:
2867:
2676:
2328:
3123:
2423:
2311:
4060:
3824:
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2910:
846:
133:
2402:
1871:"Ordinary savart", 1/301 octave, and "modified savart", 1/300 octave. Herbert Arthur Klein,
4011:
3764:
3671:
3665:
2918:
2863:
2845:
2840:
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2830:
2825:
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2800:
2795:
2790:
2785:
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2541:
2379:
2310:(Master's). Department of Electrical and Computer Engineering, Georgia Tech. Archived from
2225:
2196:
2180:
2136:
2102:
576:
549:
522:
475:
368:
341:
294:
267:
148:
121:
in the 17th century. Ellis chose to base his measures on the hundredth part of a semitone,
110:
825:
The waveforms of a unison (blue) vis-à-vis a cent (red) are practically indistinguishable.
237:
semitone (the interval between two adjacent piano keys) spans 100 cents by definition. An
8:
3794:
3659:
3641:
2882:
2855:
2753:
2553:
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780:
758:
736:
502:
321:
226:
2459:
Interval proportions can be converted to the cent values which are in common use today
2342:
1415:
in the first half of the 19th century, divided the octave in 301 or 301,03 units. See
3849:
3789:
3350:
3326:
3314:
3295:
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1025:
201:
83:
79:
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2514:
1845:
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30:
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3689:
3683:
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3581:
3481:
3454:
3338:
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3259:
3101:
2906:
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2738:
2658:
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2492:
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2276:
2218:
2204:
2148:
2110:
1964:
1404:
1389:
1351:
1116:
957:
908:
234:
106:. For humans, a single cent is too small to be perceived between successive notes.
99:
1380:
Caramuel mentioned the possible use of binary logarithms for music in a letter to
4006:
3887:
3611:
3399:
3289:
3283:
3271:
3028:
2668:
2548:
2052:
1913:
The precision is the same as with cents, but Ellis had not yet devised this unit.
1902:
674:
1834:
Principes d'acoustique et de musique ou Système général des intervalles des sons
1412:
61:
3867:
3861:
3855:
3627:
3154:
2706:
2681:
2628:
2589:
1408:
1385:
1341:
882:
222:
194:
190:
821:
53:
Octaves increase exponentially when measured on a linear frequency scale (Hz).
4044:
3991:
3840:
3759:
3754:
3700:
3561:
3501:
3387:
3375:
3240:
3203:
3160:
3001:
2970:
2743:
2696:
2603:
2522:
1120:
1010:
186:
103:
98:
of 100 cents each. Typically, cents are used to express small intervals, to
3819:
3465:
3222:
3185:
2965:
2805:
2506:
2350:
2153:
1899:
Instruction élémentaire sur les moyens de calculer les intervalles musicaux
1393:
1333:
1018:
754:
instead of the true exponential relation 2. The rounded error is zero when
47:
2122:
1819:
Ramon Ceñal, "Juan
Caramuel, su epistolario con Athanasio Kircher, S.J.",
668:
185:
tunings." He notes that these notes, when sounded in succession, form the
3968:
3799:
3234:
3228:
3197:
3191:
3053:
3048:
3038:
2633:
2613:
2481:"Influence of tonal context and timbral variation on perception of pitch"
1048:
834:(JND), also varies as a function of the frequency, the amplitude and the
462:{\displaystyle c=1200\cdot \log _{2}\left({\frac {f_{2}}{f_{1}}}\right)}
67:
Octaves are equally spaced when measured on a logarithmic scale (cents).
4001:
3895:
3393:
3043:
3012:
2497:
2480:
2288:
1311:
1261:
1211:
2162:
1470:, p. 166:The system most often employed in the modern literature.
233:
A cent is a unit of measure for the ratio between two frequencies. An
3901:
3405:
3033:
2987:
2392:
2368:"Vibrato extent and intonation in professional Western lyric singing"
2367:
2209:
2114:
114:
35:
2280:
2171:
3933:
3381:
3129:
1299:
1249:
1199:
863:
164:
95:
2558:
3939:
3927:
3913:
3332:
3277:
3007:
2957:
1740:
1728:
1346:
1029:
874:
base, the octave is represented by 1, the semitone by 1/12, etc.
842:
1764:
3996:
3973:
3881:
3875:
3166:
3148:
2975:
2691:
2618:
2166:
1356:
1044:
1006:
857:
835:
238:
102:, or to compare the sizes of comparable intervals in different
91:
1043:
For example: Equal tempered perfect fifth = 700 cents = 175.6
3369:
2992:
2980:
1788:
1024:
Iring noticed that the Grad/Werckmeister (1.96 cents, 12 per
850:
3070:
989:) equal to two cents (2) proposed as a unit of measurement (
2997:
2307:
Instrument
Timbres and Pitch Estimation in Polyphonic Music
180:
Ellis defined the pitch of a musical note in his 1880 work
1945:
1451:
1449:
2445:
Intervals, Scales, Tones and the
Concert Pitch C = 128 Hz
1922:
Alexander John Ellis, "On the History of Musical Pitch,"
169:
1993:
1776:
1692:
1680:
1668:
1656:
1632:
1608:
1596:
1584:
1572:
658:{\displaystyle f_{2}=f_{1}\times 2^{\frac {c}{1200}}\,.}
1509:
1497:
1446:
1138:
776:
is 0 or 100, and is only about 0.72 cents high at
669:
Comparison of major third in just and equal temperament
2554:
Cent conversion: Online utility with several functions
1538:
1536:
1302:, middle C 24 cents sharp, then both at the same time.
1252:, middle C 12 cents sharp, then both at the same time.
2404:
The Harvard Concise Dictionary of Music and Musicians
1933:
1752:
1548:
966:
917:
783:
761:
739:
609:
579:
552:
525:
505:
478:
401:
371:
344:
324:
297:
270:
1981:
1716:
1704:
1644:
1620:
1560:
1473:
1461:
1323:
1202:, middle C 1 cent sharp, then both at the same time.
2088:"Pitch Center of Stringed Instrument Vibrato Tones"
2005:
1533:
1521:
1485:
981:
932:
789:
767:
745:
657:
592:
565:
538:
511:
491:
461:
384:
357:
330:
310:
283:
1396:described musical logarithms using the semitone (
680:
4042:
2372:The Journal of the Acoustical Society of America
864:Other representations of intervals by logarithms
1873:The Science of Measurement. A Historical Survey
251:, the 1200th root of 2, which is approximately
113:, follow a tradition of measuring intervals by
34:One cent compared to a semitone on a truncated
2479:Warrier, C.M.; Zatorre, R.J. (February 2002).
2478:
2266:
1957:
1746:
1734:
3086:
2574:
2447:, translated by Bevis Stevens, Temple Lodge,
2649:List of intervals in 5-limit just intonation
2135:
2095:Journal of the Acoustical Society of America
2086:Brown, J.C.; Vaughn, K.V. (September 1996).
1836:, Minkoff Reprint, Geneva, 1973; see online
2542:Cent conversion: Whole number ratio to cent
2085:
1808:John Napier and the invention of logarithms
1770:
1426:
1384:in 1647; this usage often is attributed to
4027:
3093:
3079:
2581:
2567:
2470:: CS1 maint: location missing publisher (
2428:(4th ed.). Harvard University Press.
2318:
2247:
2230:"On the Musical Scales of Various Nations"
2141:Proceedings of the Royal Society of London
2077:: CS1 maint: location missing publisher (
1999:
1838:Mémoires de l'Académie royale des sciences
1794:
2496:
2443:Renold, Maria (2004) , Anna Meuss (ed.),
2391:
2332:
2208:
2170:
2152:
799:50 (whose correct value of 2 ≅
651:
197:were also defined based on these ratios.
2303:
2037:
1930:, Frits Knuf, Amsterdam, 1968, p. 11-62.
1722:
1416:
820:
216:
153:On the Musical Scales of Various Nations
29:
1928:Studies in the History of Musical Pitch
1810:, 1614, Cambridge, The University Press
1374:
1017:(1932) as 100 steps per equal tempered
725:with the linear relation 1 +
14:
4043:
3451:
2521:
2442:
2421:
2400:
2269:Journal of Research in Music Education
2050:
1987:
1951:
1939:
1758:
1479:
1467:
1432:301 can be divided only by 7 or by 43.
3074:
2562:
2365:
2224:
2179:
1782:
1710:
1698:
1686:
1674:
1662:
1650:
1638:
1626:
1614:
1602:
1590:
1578:
1566:
1554:
1542:
1527:
1515:
1503:
1491:
1455:
1147:Play middle C & 10.06 cents above
2028:
2011:
911:proposed a logarithmic unit of base
887:Principes d'acoustique et de musique
2588:
1862:, Masson, 1989, 4th edition, p. 16.
816:
24:
2248:Farnsworth, Paul Randolph (1969).
1271:
1221:
1171:
1160:Play middle C & 25 cents above
318:of two notes, the number of cents
130:Robert Holford Macdowell Bosanquet
25:
4077:
2535:
2529:. American Library of Musicology.
806:is approximated by 1 +
4026:
2724:Ptolemy's intense diatonic scale
1888:, London, 1944, ²2007, p. 53-54.
1326:
1310:Problems playing this file? See
1287:
1260:Problems playing this file? See
1237:
1210:Problems playing this file? See
1187:
1166:, beat frequency = 3.81 Hz
1131:Play middle C & 1 cent above
177:as they may be briefly called."
60:
46:
3963:
2425:The Harvard Dictionary of Music
2252:. Iowa State University Press.
1965:"Logarithmic Interval Measures"
1916:
1907:
1891:
1878:
1865:
1851:
1826:
1823:XII/44, Madrid 1954, p. 134 ss.
1813:
1800:
1153:, beat frequency = 1.53 Hz
1003:Die reine Stimmung in der Musik
877:
841:When listening to pitches with
173:] of an equal semitone, or
2485:Perception & Psychophysics
2250:The Social Psychology of Music
2234:Journal of the Society of Arts
2189:Journal of the Society of Arts
2054:Music: A Mathematical Offering
1924:Journal of the Society of Arts
1806:Ernest William Hobson (1914),
1106:
719:piecewise linear approximation
681:Piecewise linear approximation
157:Journal of the Society of Arts
13:
1:
3100:
2527:A Theory of Evolving Tonality
2343:10.1016/S1364-6613(03)00150-5
2304:Loeffler, D.B. (April 2006).
1362:
1015:A Theory of Evolving Tonality
943:
264:If one knows the frequencies
88:Twelve-tone equal temperament
2687:Harry Partch's 43-tone scale
2422:Randel, Don Michael (2003).
2407:. Harvard University Press.
2401:Randel, Don Michael (1999).
2321:Trends in Cognitive Sciences
1439:
1419:, pp. 145–168 and also
1367:
338:measuring the interval from
7:
3648:septimal chromatic semitone
2031:Harvard Dictionary of Music
1319:
907:Early in the 19th century,
699:, the function 2 increases
10:
4082:
3654:septimal diatonic semitone
3442:(Numbers in brackets refer
2654:List of meantone intervals
2185:"History of Musical Pitch"
2021:
1747:Warrier & Zatorre 2002
1735:Geringer & Worthy 1999
982:{\displaystyle {\sqrt{2}}}
933:{\displaystyle {\sqrt{2}}}
899:1/301 or to 1/300 octave.
832:just noticeable difference
143:
134:Hermann von Helmholtz
4024:
3982:
3949:
3838:
3808:
3778:
3735:
3726:
3699:
3626:
3609:
3444:to fractional semitones.)
3437:24-tone equal temperament
3434:
3423:
3360:
3305:
3250:
3213:
3176:
3141:
3108:
3021:
2938:
2901:
2854:
2761:
2752:
2667:
2644:List of musical intervals
2639:Consonance and dissonance
2596:
1283:Twenty-four Cent Interval
1096:
1088:
1080:
1072:
1064:
1059:
1056:
893:
868:
138:On the Sensations of Tone
119:Juan Caramuel y Lobkowitz
82:unit of measure used for
1875:, New York, 1974, p. 605
1421:Stigler's law of eponymy
1137:, beat frequency = 0.16
1005:(1898) as 600 steps per
902:
499:and the number of cents
182:History of Musical Pitch
161:History of Musical Pitch
4032:List of pitch intervals
3770:Subminor and supermajor
3709:minor diatonic semitone
3619:refer to pitch ratios.)
2366:Prame, E. (July 1997).
2033:. Taylor & Francis.
1901:, Paris, 1832. Online:
1771:Brown & Vaughn 1996
1001:) by Widogast Iring in
472:Likewise, if one knows
109:Cents, as described by
3830:Undecimal quarter tone
2154:10.1098/rspl.1884.0041
1795:Peretz & Hyde 2003
1407:did the same in 1832.
1276:
1226:
1176:
983:
934:
826:
791:
769:
747:
659:
594:
567:
540:
513:
493:
463:
386:
359:
332:
312:
285:
230:
212:
39:
3825:Septimal quarter tone
3636:septimal quarter tone
2911:Temperament ordinaire
2051:Benson, Dave (2007).
1926:, 1880, reprinted in
1773:, pp. 1728–1735.
1275:
1225:
1175:
984:
935:
824:
792:
770:
748:
660:
595:
593:{\displaystyle f_{2}}
568:
566:{\displaystyle f_{2}}
541:
539:{\displaystyle f_{1}}
519:in the interval from
514:
494:
492:{\displaystyle f_{1}}
464:
387:
385:{\displaystyle f_{2}}
360:
358:{\displaystyle f_{1}}
333:
313:
311:{\displaystyle f_{2}}
286:
284:{\displaystyle f_{1}}
220:
33:
27:Musical interval unit
4012:Incomposite interval
3765:Pythagorean interval
3617:(Numbers in brackets
3125:(Numbers in brackets
2714:List of compositions
2147:(232–234): 368–385,
2029:Apel, Willi (1970).
1954:, pp. 154, 416.
1886:The Physics of Music
1821:Revista de Filosofia
1233:Twelve Cent Interval
964:
915:
847:Schubert's Ave Maria
781:
759:
737:
723:loosely approximated
689:increases from 0 to
607:
577:
550:
523:
503:
476:
399:
369:
342:
322:
295:
268:
149:Alexander John Ellis
111:Alexander John Ellis
3795:Pythagorean apotome
3642:septimal third tone
2384:1997ASAJ..102..616P
2226:Ellis, Alexander J.
2201:1880Natur..21..550E
2181:Ellis, Alexander J.
2137:Ellis, Alexander J.
2107:1996ASAJ..100.1728B
1797:, pp. 362–367.
1785:, pp. 616–621.
1749:, pp. 198–207.
1737:, pp. 135–149.
1403:) as base in 1665;
1298:Sine wave plays at
1248:Sine wave plays at
1198:Sine wave plays at
1100:200 per whole tone
1097:100 per whole tone
673:The major third in
155:, published by the
4051:Equal temperaments
3908:Septimal semicomma
2948:Chinese musicology
2734:Scale of harmonics
2729:Pythagorean tuning
2677:Euler–Fokker genus
2547:2017-04-22 at the
2498:10.3758/BF03195786
1969:Huygens-Fokker.org
1897:Gaspard de Prony,
1701:, p. 520-525.
1689:, p. 514-520.
1677:, p. 508-514.
1665:, p. 507-508.
1641:, p. 506-507.
1617:, p. 505-506.
1605:, p. 500-505.
1593:, p. 492-500.
1581:, p. 491-492.
1518:, p. 293-294.
1506:, p. 293-336.
1458:, p. 485-527.
1382:Athanasius Kircher
1277:
1227:
1177:
1051:= 350 centitones.
979:
930:
827:
787:
765:
743:
655:
590:
563:
536:
509:
489:
459:
382:
355:
328:
308:
281:
231:
40:
4056:Intervals (music)
4038:
4037:
4020:
4019:
3850:Pythagorean comma
3790:Pythagorean limma
3722:
3721:
3718:
3717:
3684:supermajor fourth
3660:supermajor second
3605:
3604:
3419:
3418:
3415:
3414:
3127:are the number of
3068:
3067:
2934:
2933:
2435:978-0-674-01163-2
2414:978-0-674-00084-1
1293:
1243:
1193:
1183:One Cent Interval
1104:
1103:
1092:100 per semitone
1036:, centitone, and
1026:Pythagorean comma
977:
928:
790:{\displaystyle c}
768:{\displaystyle c}
746:{\displaystyle c}
717:, allowing for a
648:
512:{\displaystyle c}
453:
331:{\displaystyle c}
202:equal temperament
84:musical intervals
16:(Redirected from
4073:
4030:
4029:
3920:Septimal kleisma
3733:
3732:
3690:subminor seventh
3672:supermajor third
3624:
3623:
3612:Just intonations
3597:
3596:
3592:
3589:
3577:
3576:
3572:
3569:
3557:
3556:
3552:
3549:
3537:
3536:
3532:
3529:
3517:
3516:
3512:
3509:
3497:
3496:
3492:
3489:
3477:
3476:
3472:
3449:
3448:
3432:
3431:
3139:
3138:
3121:
3120:
3095:
3088:
3081:
3072:
3071:
2907:Well temperament
2893:Regular diatonic
2759:
2758:
2739:Tonality diamond
2583:
2576:
2569:
2560:
2559:
2530:
2518:
2500:
2475:
2469:
2461:
2439:
2418:
2397:
2395:
2393:10.1121/1.419735
2362:
2336:
2315:
2300:
2263:
2244:
2243:
2241:
2221:
2212:
2210:10.1038/021550a0
2195:(545): 293–337,
2176:
2174:
2156:
2132:
2130:
2129:
2115:10.1121/1.416070
2101:(3): 1728–1735.
2092:
2082:
2076:
2068:
2047:
2034:
2015:
2009:
2003:
1997:
1991:
1985:
1979:
1978:
1976:
1975:
1961:
1955:
1949:
1943:
1937:
1931:
1920:
1914:
1911:
1905:
1895:
1889:
1884:Alexander Wood,
1882:
1876:
1869:
1863:
1855:
1849:
1832:Joseph Sauveur,
1830:
1824:
1817:
1811:
1804:
1798:
1792:
1786:
1780:
1774:
1768:
1762:
1756:
1750:
1744:
1738:
1732:
1726:
1720:
1714:
1708:
1702:
1696:
1690:
1684:
1678:
1672:
1666:
1660:
1654:
1648:
1642:
1636:
1630:
1624:
1618:
1612:
1606:
1600:
1594:
1588:
1582:
1576:
1570:
1564:
1558:
1552:
1546:
1540:
1531:
1525:
1519:
1513:
1507:
1501:
1495:
1489:
1483:
1477:
1471:
1465:
1459:
1453:
1433:
1430:
1424:
1405:Gaspard de Prony
1402:
1401:
1390:Binary logarithm
1378:
1352:Microtonal music
1336:
1331:
1330:
1329:
1295:
1294:
1274:
1245:
1244:
1224:
1195:
1194:
1174:
1165:
1164:
1163:
1161:
1152:
1151:
1150:
1148:
1136:
1135:
1134:
1132:
1089:50 per semitone
1054:
1053:
1000:
999:
998:
996:
988:
986:
985:
980:
978:
976:
968:
958:musical interval
939:
937:
936:
931:
929:
927:
919:
909:Gaspard de Prony
853:'s opera arias.
817:Human perception
812:
811:
805:
804:
796:
794:
793:
788:
774:
772:
771:
766:
752:
750:
749:
744:
731:
730:
716:
715:
709:
708:
698:
697:
693:
664:
662:
661:
656:
650:
649:
641:
632:
631:
619:
618:
599:
597:
596:
591:
589:
588:
572:
570:
569:
564:
562:
561:
545:
543:
542:
537:
535:
534:
518:
516:
515:
510:
498:
496:
495:
490:
488:
487:
468:
466:
465:
460:
458:
454:
452:
451:
442:
441:
432:
423:
422:
391:
389:
388:
383:
381:
380:
364:
362:
361:
356:
354:
353:
337:
335:
334:
329:
317:
315:
314:
309:
307:
306:
290:
288:
287:
282:
280:
279:
260:
259:
256:
250:
249:
248:
235:equally tempered
195:relative pitches
127:
126:
117:that began with
100:check intonation
64:
50:
21:
4081:
4080:
4076:
4075:
4074:
4072:
4071:
4070:
4041:
4040:
4039:
4034:
4016:
3978:
3945:
3888:Septimal diesis
3834:
3804:
3774:
3728:
3714:
3695:
3618:
3615:
3601:
3594:
3590:
3587:
3585:
3574:
3570:
3567:
3565:
3554:
3550:
3547:
3545:
3534:
3530:
3527:
3525:
3514:
3510:
3507:
3505:
3494:
3490:
3487:
3485:
3474:
3470:
3469:
3459:
3458:
3457:
3453:
3443:
3440:
3427:
3425:
3411:
3356:
3301:
3246:
3209:
3172:
3133:
3128:
3126:
3116:
3114:
3111:
3104:
3099:
3069:
3064:
3061:(Bohlen–Pierce)
3029:833 cents scale
3017:
2940:
2930:
2897:
2850:
2748:
2669:Just intonation
2663:
2592:
2590:Musical tunings
2587:
2549:Wayback Machine
2538:
2533:
2463:
2462:
2455:
2436:
2415:
2334:10.1.1.585.2171
2281:10.2307/3345719
2260:
2239:
2237:
2127:
2125:
2090:
2070:
2069:
2065:
2024:
2019:
2018:
2010:
2006:
2000:Farnsworth 1969
1998:
1994:
1986:
1982:
1973:
1971:
1963:
1962:
1958:
1950:
1946:
1938:
1934:
1921:
1917:
1912:
1908:
1896:
1892:
1883:
1879:
1870:
1866:
1856:
1852:
1831:
1827:
1818:
1814:
1805:
1801:
1793:
1789:
1781:
1777:
1769:
1765:
1757:
1753:
1745:
1741:
1733:
1729:
1721:
1717:
1709:
1705:
1697:
1693:
1685:
1681:
1673:
1669:
1661:
1657:
1649:
1645:
1637:
1633:
1625:
1621:
1613:
1609:
1601:
1597:
1589:
1585:
1577:
1573:
1565:
1561:
1557:, p. 491-.
1553:
1549:
1541:
1534:
1526:
1522:
1514:
1510:
1502:
1498:
1490:
1486:
1478:
1474:
1466:
1462:
1454:
1447:
1442:
1437:
1436:
1431:
1427:
1399:
1397:
1379:
1375:
1370:
1365:
1332:
1327:
1325:
1322:
1317:
1316:
1308:
1306:
1305:
1304:
1303:
1296:
1288:
1285:
1278:
1272:
1267:
1266:
1258:
1256:
1255:
1254:
1253:
1246:
1238:
1235:
1228:
1222:
1217:
1216:
1208:
1206:
1205:
1204:
1203:
1196:
1188:
1185:
1178:
1172:
1159:
1156:
1155:
1154:
1146:
1143:
1142:
1141:
1130:
1127:
1126:
1109:
994:
991:
990:
972:
967:
965:
962:
961:
946:
923:
918:
916:
913:
912:
905:
896:
880:
871:
866:
819:
809:
807:
802:
800:
782:
779:
778:
760:
757:
756:
738:
735:
734:
728:
726:
713:
711:
706:
704:
701:almost linearly
695:
691:
690:
683:
675:just intonation
671:
640:
636:
627:
623:
614:
610:
608:
605:
604:
584:
580:
578:
575:
574:
557:
553:
551:
548:
547:
530:
526:
524:
521:
520:
504:
501:
500:
483:
479:
477:
474:
473:
447:
443:
437:
433:
431:
427:
418:
414:
400:
397:
396:
376:
372:
370:
367:
366:
349:
345:
343:
340:
339:
323:
320:
319:
302:
298:
296:
293:
292:
275:
271:
269:
266:
265:
257:
254:
252:
246:
244:
242:
215:
146:
124:
122:
72:
71:
70:
69:
68:
65:
56:
55:
54:
51:
28:
23:
22:
15:
12:
11:
5:
4079:
4069:
4068:
4066:Units of level
4063:
4058:
4053:
4036:
4035:
4025:
4022:
4021:
4018:
4017:
4015:
4014:
4009:
4004:
3999:
3994:
3988:
3986:
3980:
3979:
3977:
3976:
3971:
3966:
3961:
3955:
3953:
3947:
3946:
3944:
3943:
3937:
3931:
3924:
3923:
3917:
3911:
3905:
3899:
3892:
3891:
3885:
3882:Greater diesis
3879:
3872:
3871:
3868:Septimal comma
3865:
3862:Holdrian comma
3859:
3856:Syntonic comma
3853:
3846:
3844:
3836:
3835:
3833:
3832:
3827:
3822:
3816:
3814:
3806:
3805:
3803:
3802:
3797:
3792:
3786:
3784:
3776:
3775:
3773:
3772:
3767:
3762:
3757:
3752:
3747:
3741:
3739:
3730:
3724:
3723:
3720:
3719:
3716:
3715:
3713:
3712:
3705:
3703:
3697:
3696:
3694:
3693:
3687:
3681:
3678:subminor fifth
3675:
3669:
3666:subminor third
3663:
3657:
3651:
3645:
3639:
3632:
3630:
3621:
3607:
3606:
3603:
3602:
3600:
3599:
3579:
3559:
3539:
3519:
3499:
3479:
3462:
3460:
3452:
3446:
3429:
3421:
3420:
3417:
3416:
3413:
3412:
3410:
3409:
3403:
3397:
3391:
3385:
3379:
3373:
3366:
3364:
3358:
3357:
3355:
3354:
3348:
3342:
3336:
3330:
3324:
3318:
3311:
3309:
3303:
3302:
3300:
3299:
3293:
3287:
3281:
3275:
3269:
3263:
3256:
3254:
3248:
3247:
3245:
3244:
3238:
3232:
3226:
3219:
3217:
3211:
3210:
3208:
3207:
3201:
3195:
3189:
3182:
3180:
3174:
3173:
3171:
3170:
3164:
3158:
3152:
3145:
3143:
3136:
3118:
3106:
3105:
3098:
3097:
3090:
3083:
3075:
3066:
3065:
3063:
3062:
3056:
3051:
3046:
3041:
3036:
3031:
3025:
3023:
3019:
3018:
3016:
3015:
3010:
3005:
2995:
2990:
2985:
2984:
2983:
2978:
2973:
2968:
2960:
2955:
2950:
2944:
2942:
2936:
2935:
2932:
2931:
2905:
2903:
2899:
2898:
2896:
2895:
2890:
2885:
2880:
2875:
2860:
2858:
2852:
2851:
2849:
2848:
2843:
2838:
2833:
2828:
2823:
2818:
2813:
2803:
2798:
2793:
2788:
2783:
2778:
2773:
2767:
2765:
2756:
2750:
2749:
2747:
2746:
2741:
2736:
2731:
2726:
2721:
2716:
2711:
2710:
2709:
2704:
2694:
2689:
2684:
2682:Harmonic scale
2679:
2673:
2671:
2665:
2664:
2662:
2661:
2656:
2651:
2646:
2641:
2636:
2631:
2629:Interval ratio
2626:
2621:
2616:
2611:
2606:
2600:
2598:
2594:
2593:
2586:
2585:
2578:
2571:
2563:
2557:
2556:
2551:
2537:
2536:External links
2534:
2532:
2531:
2523:Yasser, Joseph
2519:
2491:(2): 198–207.
2476:
2453:
2440:
2434:
2419:
2413:
2398:
2378:(1): 616–621.
2363:
2327:(8): 362–367.
2316:
2314:on 2007-12-18.
2301:
2275:(2): 135–149.
2264:
2258:
2245:
2222:
2177:
2133:
2083:
2063:
2048:
2035:
2025:
2023:
2020:
2017:
2016:
2014:, p. 363.
2004:
1992:
1980:
1956:
1944:
1942:, p. 123.
1932:
1915:
1906:
1890:
1877:
1864:
1850:
1825:
1812:
1799:
1787:
1775:
1763:
1761:, p. 368.
1751:
1739:
1727:
1715:
1713:, p. 526.
1703:
1691:
1679:
1667:
1655:
1653:, p. 507.
1643:
1631:
1629:, p. 506.
1619:
1607:
1595:
1583:
1571:
1569:, p. 488.
1559:
1547:
1545:, p. 487.
1532:
1530:, p. 294.
1520:
1508:
1496:
1494:, p. 295.
1484:
1482:, p. 138.
1472:
1460:
1444:
1443:
1441:
1438:
1435:
1434:
1425:
1409:Joseph Sauveur
1386:Leonhard Euler
1372:
1371:
1369:
1366:
1364:
1361:
1360:
1359:
1354:
1349:
1344:
1338:
1337:
1321:
1318:
1307:
1297:
1286:
1281:
1280:
1279:
1270:
1269:
1268:
1257:
1247:
1236:
1231:
1230:
1229:
1220:
1219:
1218:
1207:
1197:
1186:
1181:
1180:
1179:
1170:
1169:
1168:
1108:
1105:
1102:
1101:
1098:
1094:
1093:
1090:
1086:
1085:
1082:
1078:
1077:
1074:
1073:0.5 centitone
1070:
1069:
1066:
1062:
1061:
1058:
975:
971:
945:
942:
926:
922:
904:
901:
895:
892:
883:Joseph Sauveur
879:
876:
870:
867:
865:
862:
818:
815:
786:
764:
742:
682:
679:
670:
667:
666:
665:
654:
647:
644:
639:
635:
630:
626:
622:
617:
613:
587:
583:
560:
556:
533:
529:
508:
486:
482:
470:
469:
457:
450:
446:
440:
436:
430:
426:
421:
417:
413:
410:
407:
404:
379:
375:
352:
348:
327:
305:
301:
278:
274:
223:equal-tempered
221:Comparison of
214:
211:
145:
142:
104:tuning systems
66:
59:
58:
57:
52:
45:
44:
43:
42:
41:
26:
9:
6:
4:
3:
2:
4078:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4049:
4048:
4046:
4033:
4023:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3989:
3987:
3985:
3981:
3975:
3972:
3970:
3967:
3965:
3962:
3960:
3957:
3956:
3954:
3952:
3948:
3941:
3938:
3935:
3932:
3929:
3926:
3925:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3893:
3889:
3886:
3883:
3880:
3877:
3876:Lesser diesis
3874:
3873:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3847:
3845:
3843:
3842:
3837:
3831:
3828:
3826:
3823:
3821:
3818:
3817:
3815:
3813:
3812:
3811:Quarter tones
3807:
3801:
3798:
3796:
3793:
3791:
3788:
3787:
3785:
3783:
3782:
3777:
3771:
3768:
3766:
3763:
3761:
3760:Pseudo-octave
3758:
3756:
3753:
3751:
3748:
3746:
3743:
3742:
3740:
3738:
3734:
3731:
3725:
3710:
3707:
3706:
3704:
3702:
3698:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3633:
3631:
3629:
3625:
3622:
3620:
3614:
3613:
3608:
3583:
3580:
3563:
3560:
3543:
3540:
3523:
3520:
3503:
3500:
3483:
3480:
3467:
3464:
3463:
3461:
3456:
3450:
3447:
3445:
3439:
3438:
3433:
3430:
3422:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3367:
3365:
3363:
3359:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3312:
3310:
3308:
3304:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3257:
3255:
3253:
3249:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3220:
3218:
3216:
3212:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3183:
3181:
3179:
3175:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3146:
3144:
3140:
3137:
3135:
3131:
3122:
3119:
3113:
3107:
3103:
3096:
3091:
3089:
3084:
3082:
3077:
3076:
3073:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3026:
3024:
3020:
3014:
3011:
3009:
3006:
3003:
3002:Carnatic raga
2999:
2996:
2994:
2991:
2989:
2986:
2982:
2979:
2977:
2974:
2972:
2971:Turkish makam
2969:
2967:
2964:
2963:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2945:
2943:
2937:
2928:
2924:
2920:
2916:
2912:
2908:
2904:
2900:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2873:
2869:
2868:quarter-comma
2865:
2862:
2861:
2859:
2857:
2853:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2811:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2768:
2766:
2764:
2760:
2757:
2755:
2751:
2745:
2744:Tonality flux
2742:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2715:
2712:
2708:
2705:
2703:
2700:
2699:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2674:
2672:
2670:
2666:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2601:
2599:
2595:
2591:
2584:
2579:
2577:
2572:
2570:
2565:
2564:
2561:
2555:
2552:
2550:
2546:
2543:
2540:
2539:
2528:
2524:
2520:
2516:
2512:
2508:
2504:
2499:
2494:
2490:
2486:
2482:
2477:
2473:
2467:
2460:
2456:
2454:9781902636467
2450:
2446:
2441:
2437:
2431:
2427:
2426:
2420:
2416:
2410:
2406:
2405:
2399:
2394:
2389:
2385:
2381:
2377:
2373:
2369:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2335:
2330:
2326:
2322:
2317:
2313:
2309:
2308:
2302:
2298:
2294:
2290:
2286:
2282:
2278:
2274:
2270:
2265:
2261:
2259:9780813815473
2255:
2251:
2246:
2235:
2231:
2227:
2223:
2220:
2216:
2211:
2206:
2202:
2198:
2194:
2190:
2186:
2182:
2178:
2173:
2168:
2164:
2160:
2155:
2150:
2146:
2142:
2138:
2134:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2089:
2084:
2080:
2074:
2066:
2064:9780521853873
2060:
2057:. Cambridge.
2056:
2055:
2049:
2046:(2): 145–168.
2045:
2041:
2036:
2032:
2027:
2026:
2013:
2008:
2002:, p. 24.
2001:
1996:
1990:, p. 14.
1989:
1984:
1970:
1966:
1960:
1953:
1948:
1941:
1936:
1929:
1925:
1919:
1910:
1903:
1900:
1894:
1887:
1881:
1874:
1868:
1861:
1857:Émile Leipp,
1854:
1847:
1843:
1839:
1835:
1829:
1822:
1816:
1809:
1803:
1796:
1791:
1784:
1779:
1772:
1767:
1760:
1755:
1748:
1743:
1736:
1731:
1724:
1723:Loeffler 2006
1719:
1712:
1707:
1700:
1695:
1688:
1683:
1676:
1671:
1664:
1659:
1652:
1647:
1640:
1635:
1628:
1623:
1616:
1611:
1604:
1599:
1592:
1587:
1580:
1575:
1568:
1563:
1556:
1551:
1544:
1539:
1537:
1529:
1524:
1517:
1512:
1505:
1500:
1493:
1488:
1481:
1476:
1469:
1464:
1457:
1452:
1450:
1445:
1429:
1422:
1418:
1417:Barbieri 1987
1414:
1411:in 1701, and
1410:
1406:
1395:
1391:
1388:in 1739 (see
1387:
1383:
1377:
1373:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1340:
1339:
1335:
1324:
1315:
1313:
1301:
1284:
1265:
1263:
1251:
1234:
1215:
1213:
1201:
1184:
1167:
1162:
1149:
1140:
1133:
1124:
1122:
1118:
1113:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1055:
1052:
1050:
1046:
1041:
1039:
1035:
1031:
1027:
1022:
1020:
1016:
1012:
1011:Joseph Yasser
1009:and later by
1008:
1004:
997:
973:
969:
959:
955:
951:
941:
924:
920:
910:
900:
891:
888:
884:
875:
861:
859:
854:
852:
848:
844:
839:
837:
833:
823:
814:
798:
784:
775:
762:
753:
740:
724:
720:
702:
688:
678:
676:
652:
645:
642:
637:
633:
628:
624:
620:
615:
611:
603:
602:
601:
585:
581:
558:
554:
531:
527:
506:
484:
480:
455:
448:
444:
438:
434:
428:
424:
419:
415:
411:
408:
405:
402:
395:
394:
393:
377:
373:
350:
346:
325:
303:
299:
276:
272:
262:
240:
236:
228:
224:
219:
210:
206:
203:
198:
196:
192:
188:
183:
178:
176:
172:
171:
166:
162:
158:
154:
150:
141:
139:
135:
131:
120:
116:
112:
107:
105:
101:
97:
93:
89:
85:
81:
77:
63:
49:
37:
32:
19:
18:Cents (music)
4061:100 (number)
3983:
3958:
3950:
3936:(0.72 cents)
3930:(1.95 cents)
3910:(13.8 cents)
3904:(10.1 cents)
3898:(19.5 cents)
3890:(35.7 cents)
3884:(62.6 cents)
3878:(41.1 cents)
3870:(27.3 cents)
3864:(22.6 cents)
3858:(21.5 cents)
3852:(23.5 cents)
3839:
3820:Quarter tone
3810:
3809:
3780:
3779:
3736:
3701:Higher-limit
3616:
3610:
3522:major fourth
3466:quarter tone
3441:
3435:
3124:
3059:Lambda scale
2966:Arabic maqam
2923:Werckmeister
2754:Temperaments
2608:
2526:
2488:
2484:
2458:
2444:
2424:
2403:
2375:
2371:
2324:
2320:
2312:the original
2306:
2272:
2268:
2249:
2238:, retrieved
2233:
2192:
2188:
2144:
2140:
2126:. Retrieved
2098:
2094:
2053:
2043:
2040:Musiktheorie
2039:
2030:
2007:
1995:
1983:
1972:. Retrieved
1968:
1959:
1947:
1935:
1927:
1923:
1918:
1909:
1898:
1893:
1885:
1880:
1872:
1867:
1858:
1853:
1837:
1833:
1828:
1820:
1815:
1807:
1802:
1790:
1778:
1766:
1754:
1742:
1730:
1718:
1706:
1694:
1682:
1670:
1658:
1646:
1634:
1622:
1610:
1598:
1586:
1574:
1562:
1550:
1523:
1511:
1499:
1487:
1475:
1463:
1428:
1413:Félix Savart
1394:Isaac Newton
1376:
1334:Music portal
1309:
1259:
1209:
1125:
1114:
1110:
1065:1 centitone
1049:millioctaves
1042:
1037:
1033:
1023:
1014:
1002:
953:
949:
947:
906:
897:
886:
881:
878:Heptamerides
872:
855:
840:
828:
777:
755:
733:
722:
700:
686:
684:
672:
471:
263:
232:
225:(black) and
207:
199:
181:
179:
174:
168:
160:
156:
152:
147:
137:
108:
90:divides the
75:
73:
3969:Millioctave
3951:Measurement
3942:(0.4 cents)
3922:(7.7 cents)
3916:(8.1 cents)
3800:Major limma
3542:minor fifth
3054:Delta scale
3049:Gamma scale
3039:Alpha scale
2941:non-Western
2939:Traditional
2634:Pitch class
2614:Millioctave
2597:Measurement
1988:Yasser 1932
1952:Randel 2003
1940:Randel 1999
1759:Benson 2007
1480:Renold 2004
1468:Benson 2007
1107:Sound files
1057:Centitones
227:Pythagorean
80:logarithmic
4045:Categories
4002:Semiditone
3896:Diaschisma
3711:(17-limit)
3402:(22 or 23)
3400:fourteenth
3396:(20 or 21)
3394:thirteenth
3390:(18 or 19)
3384:(17 or 18)
3378:(15 or 16)
3372:(13 or 14)
3307:Diminished
3134:interval.)
3115:(post-Bach
3044:Beta scale
3022:Non-octave
3013:Tetrachord
2915:Kirnberger
2878:Schismatic
2128:2008-09-28
1974:2021-06-25
1846:Acoustique
1842:Acoustique
1783:Prame 1997
1711:Ellis 1885
1699:Ellis 1885
1687:Ellis 1885
1675:Ellis 1885
1663:Ellis 1885
1651:Ellis 1885
1639:Ellis 1885
1627:Ellis 1885
1615:Ellis 1885
1603:Ellis 1885
1591:Ellis 1885
1579:Ellis 1885
1567:Ellis 1885
1555:Ellis 1885
1543:Ellis 1885
1528:Ellis 1880
1516:Ellis 1880
1504:Ellis 1880
1492:Ellis 1880
1456:Ellis 1885
1363:References
1312:media help
1262:media help
1212:media help
1028:) and the
1019:whole tone
944:Centitones
115:logarithms
3964:Centitone
3902:Semicomma
3781:Semitones
3745:Microtone
3729:intervals
3406:fifteenth
3252:Augmented
3130:semitones
3102:Intervals
3034:A12 scale
2988:Octoechos
2953:Shí-èr-lǜ
2902:Irregular
2719:Otonality
2659:Microtone
2329:CiteSeerX
2297:144918272
2240:1 January
2236:: 485–527
2073:cite book
2012:Apel 1970
1440:Citations
1368:Footnotes
1038:millitone
950:centitone
885:, in his
634:×
425:
412:⋅
96:semitones
36:monochord
3934:Breedsma
3382:eleventh
3362:Compound
3117:Western)
3112:semitone
2919:Vallotti
2872:septimal
2864:Meantone
2624:Interval
2545:Archived
2525:(1932).
2515:15094971
2507:12013375
2466:citation
2351:12907232
2228:(1885),
2183:(1880),
1840:, 1700,
1320:See also
1300:middle C
1250:middle C
1200:middle C
1068:2 cents
1047:= 583.3
1034:decitone
600:equals:
191:Absolute
165:semitone
151:' paper
94:into 12
3940:Ragisma
3928:Schisma
3914:Kleisma
3750:5-limit
3656:(15:14)
3650:(21:20)
3644:(28:27)
3638:(36:35)
3628:7-limit
3593:⁄
3582:seventh
3573:⁄
3553:⁄
3533:⁄
3513:⁄
3493:⁄
3473:⁄
3455:Neutral
3428:systems
3388:twelfth
3345:seventh
3296:seventh
3241:seventh
3204:seventh
3142:Perfect
3110:Twelve-
3008:Slendro
2958:Dastgah
2883:Miracle
2846:96-tone
2841:72-tone
2836:58-tone
2831:53-tone
2826:41-tone
2821:34-tone
2816:31-tone
2806:24-tone
2801:23-tone
2796:22-tone
2791:19-tone
2786:17-tone
2781:15-tone
2776:12-tone
2707:7-limit
2702:5-limit
2380:Bibcode
2359:3224978
2289:3345719
2219:4107831
2197:Bibcode
2172:1432077
2169::
2123:8817899
2103:Bibcode
2022:Sources
1844:; 1701
1398:√
1347:Gradian
1117:beating
1076:1 cent
1045:savarts
1030:schisma
956:) is a
843:vibrato
694:⁄
573:, then
245:√
144:History
123:√
3997:Ditone
3984:Others
3974:Savart
3841:Commas
3737:Groups
3686:(10:7)
3482:second
3426:tuning
3351:octave
3327:fourth
3315:second
3278:fourth
3266:second
3260:unison
3223:second
3186:second
3167:octave
3155:fourth
3149:unison
3132:in the
2976:Mugham
2962:Maqam
2856:Linear
2810:pieces
2771:6-tone
2692:Hexany
2619:Savart
2513:
2505:
2451:
2432:
2411:
2357:
2349:
2331:
2295:
2287:
2256:
2217:
2167:Zenodo
2163:114325
2161:
2121:
2061:
1860:salles
1357:Radian
1342:Degree
1060:Cents
1007:octave
952:(also
894:Savart
869:Octave
858:amusia
836:timbre
732:
239:octave
92:octave
4007:Secor
3755:Comma
3727:Other
3692:(7:4)
3680:(7:5)
3674:(9:7)
3668:(7:6)
3662:(8:7)
3562:sixth
3502:third
3424:Other
3376:tenth
3370:ninth
3339:sixth
3333:fifth
3321:third
3290:sixth
3284:fifth
3272:third
3235:sixth
3229:third
3215:Minor
3198:sixth
3192:third
3178:Major
3161:fifth
2993:Pelog
2981:Muqam
2927:Young
2888:Magic
2763:Equal
2697:Limit
2604:Pitch
2511:S2CID
2355:S2CID
2293:S2CID
2285:JSTOR
2215:S2CID
2159:JSTOR
2091:(PDF)
1121:phase
954:Iring
903:Prony
851:Verdi
808:0.000
801:1.029
727:0.000
712:1.059
705:1.000
703:from
253:1.000
187:scale
175:cents
128:, at
78:is a
3992:Wolf
3959:Cent
3408:(24)
3353:(11)
3298:(12)
3292:(10)
3243:(10)
3206:(11)
3169:(12)
2998:Raga
2609:Cent
2503:PMID
2472:link
2449:ISBN
2430:ISBN
2409:ISBN
2347:PMID
2254:ISBN
2242:2020
2119:PMID
2079:link
2059:ISBN
995:Play
960:(2,
810:5946
729:5946
646:1200
409:1200
392:is:
291:and
258:7895
243:2 =
193:and
76:cent
74:The
3347:(9)
3341:(7)
3335:(6)
3329:(4)
3323:(2)
3317:(0)
3286:(8)
3280:(6)
3274:(5)
3268:(3)
3262:(1)
3237:(8)
3231:(3)
3225:(1)
3200:(9)
3194:(4)
3188:(2)
3163:(7)
3157:(5)
3151:(0)
2493:doi
2388:doi
2376:102
2339:doi
2277:doi
2205:doi
2149:doi
2111:doi
2099:100
1392:).
1013:in
974:600
710:to
685:As
546:to
416:log
365:to
255:577
213:Use
170:sic
136:'s
4047::
3586:10
2925:,
2921:,
2917:,
2870:,
2509:.
2501:.
2489:64
2487:.
2483:.
2468:}}
2464:{{
2457:,
2386:.
2374:.
2370:.
2353:.
2345:.
2337:.
2323:.
2291:.
2283:.
2273:47
2271:.
2232:,
2213:,
2203:,
2193:21
2191:,
2187:,
2165:,
2157:,
2145:37
2143:,
2117:.
2109:.
2097:.
2093:.
2075:}}
2071:{{
2042:.
1967:.
1535:^
1448:^
1139:Hz
1084:2
1081:2
1021:.
948:A
925:12
803:30
714:46
707:00
696:12
86:.
3598:)
3595:2
3591:1
3588:+
3584:(
3578:)
3575:2
3571:1
3568:+
3566:8
3564:(
3558:)
3555:2
3551:1
3548:+
3546:6
3544:(
3538:)
3535:2
3531:1
3528:+
3526:5
3524:(
3518:)
3515:2
3511:1
3508:+
3506:3
3504:(
3498:)
3495:2
3491:1
3488:+
3486:1
3484:(
3478:)
3475:2
3471:1
3468:(
3094:e
3087:t
3080:v
3004:)
3000:(
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