191:
171:
122:
25:
1503:
148:
509:, in which the main intervals are typically perceived as consonant, but none is justly tuned and as consonant as a just interval, except for the unison and octave. Although the size of equally tuned intervals is typically similar to that of just intervals, in most cases it cannot be expressed by small-integer ratios. For instance, an
290:
Ratios, rather than direct frequency measurements, allow musicians to work with relative pitch measurements applicable to many instruments in an intuitive manner, whereas one rarely has the frequencies of fixed pitched instruments memorized and rarely has the capabilities to measure the changes of
201:
389:
178:
408:
334:
248:
318:
264:
366:
350:
158:
135:
299:
relationship to string length, for example stopping a string at two-thirds (2:3) its length produces a pitch one and one-half (3:2) that of the open string (not to be confused with
513:
perfect fifth has a frequency ratio of about 1.4983:1 (or 14983:10000). For a comparison between the size of intervals in different tuning systems, see section
431:
Frequency ratios are used to describe intervals in both
Western and non-Western music. They are most often used to describe intervals between notes tuned with
1605:
1224:
1537:
1670:
1680:
567:
381:, wherein the ratios whose limit, which includes its integer multiples, is lower are generally more consonant. For example, the 3-limit
423:, a logarithmic measurement. For example, the just perfect fifth is 701.955 cents while the equal tempered perfect fifth is 700 cents.
108:
1643:
42:
89:
46:
61:
1828:
1530:
1016:
996:
1766:
1122:
68:
310:. As such ratios with lower integers are generally more consonant than intervals with higher integers. For example,
2037:
1610:
1128:
35:
1595:
911:
506:
499:
307:
75:
1879:
2015:
1849:
1675:
1600:
1523:
1506:
1244:
1183:
560:
1633:
190:
57:
1871:
1834:
1824:
1304:
296:
535:
1867:
1299:
1110:
505:
Most commonly, however, musical instruments are nowadays tuned using a different tuning system, called
1875:
1820:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1757:
1752:
1747:
1742:
1737:
1732:
1486:
1239:
1146:
1140:
726:
444:
401:
416:). Despite having larger integers 128:81 is less dissonant than 14:9, as according to limit theory.
1839:
1812:
1710:
1269:
1134:
1116:
1904:
1690:
1685:
1382:
819:
553:
455:
436:
125:
1909:
1883:
1844:
1719:
1658:
1324:
1264:
825:
801:
789:
770:
584:
510:
300:
257:
1863:
1727:
1695:
1615:
1580:
1394:
1219:
1164:
1158:
1152:
1056:
956:
929:
836:
813:
795:
740:
734:
576:
514:
359:
292:
284:
277:
234:
170:
82:
2042:
1985:
1625:
1362:
1086:
874:
764:
758:
746:
689:
652:
459:
440:
343:
238:
1663:
1638:
1546:
1342:
1336:
1330:
1102:
629:
479:
463:
448:
327:
195:
2031:
1958:
1927:
1700:
1653:
1560:
1466:
1315:
1234:
1229:
1175:
1036:
976:
862:
850:
715:
678:
635:
475:
432:
378:
241:
230:
1922:
1762:
1565:
1481:
1438:
1433:
1294:
940:
697:
660:
420:
152:
129:
2010:
2005:
1995:
1590:
1570:
1443:
1274:
709:
703:
672:
666:
487:
483:
382:
121:
24:
2000:
1969:
1476:
1370:
868:
781:
147:
287:, at (440*1.5=) 660 Hz, while the equal tempered E5 is 659.255 Hz.
1990:
1944:
1376:
880:
226:
1408:
856:
604:
273:
462:
tuning system, the size of the main intervals can be expressed by small-
1964:
1914:
1414:
1402:
1388:
807:
752:
398:
1932:
1648:
1575:
1471:
1448:
1356:
1350:
641:
623:
471:
467:
311:
1949:
1937:
1515:
844:
280:
222:
214:
189:
169:
146:
120:
1954:
1519:
549:
18:
545:
419:
For ease of comparison intervals may also be measured in
490:). Intervals with small-integer ratios are often called
377:
Consonance and dissonance may more subtly be defined by
1978:
1895:
1858:
1811:
1718:
1709:
1624:
1553:
1457:
1424:
1313:
1283:
1253:
1210:
1201:
1174:
1101:
1084:
926:
909:
898:
835:
780:
725:
688:
651:
616:
598:
583:
49:. Unsourced material may be challenged and removed.
447:, the size of which can be expressed by small-
1531:
561:
8:
1606:List of intervals in 5-limit just intonation
539:. Trans. J.A. Shedlock. Augener, 1900. 782.
272:) which is 2 (about 1.498). If the A above
1715:
1538:
1524:
1516:
1502:
1207:
1098:
923:
906:
613:
595:
568:
554:
546:
109:Learn how and when to remove this message
526:
498:. To most people, just intervals sound
283:, the perfect fifth above it would be
256:), 1.5, and may be approximated by an
7:
306:Intervals may be ranked by relative
47:adding citations to reliable sources
16:In music, ratio of pitch frequencies
14:
194:11-limit 11:10 greater undecimal
1681:Ptolemy's intense diatonic scale
1501:
515:Size in different tuning systems
502:, i.e. pleasant and well-tuned.
174:7-limit 8:7 septimal whole tone
23:
34:needs additional citations for
291:adjustable pitch instruments (
1:
244:(for example C to G) is 3:2 (
1644:Harry Partch's 43-tone scale
1123:septimal chromatic semitone
2059:
1611:List of meantone intervals
1129:septimal diatonic semitone
917:(Numbers in brackets refer
1601:List of musical intervals
1596:Consonance and dissonance
1499:
919:to fractional semitones.)
912:24-tone equal temperament
507:12-tone equal temperament
308:consonance and dissonance
1507:List of pitch intervals
1245:Subminor and supermajor
1184:minor diatonic semitone
1094:refer to pitch ratios.)
1305:Undecimal quarter tone
210:
187:
167:
144:
1868:Temperament ordinaire
1300:Septimal quarter tone
1111:septimal quarter tone
466:ratios, such as 1:1 (
193:
173:
150:
124:
1671:List of compositions
1487:Incomposite interval
1240:Pythagorean interval
1092:(Numbers in brackets
600:(Numbers in brackets
445:meantone temperament
43:improve this article
1270:Pythagorean apotome
1117:septimal third tone
536:Dictionary of Music
1905:Chinese musicology
1691:Scale of harmonics
1686:Pythagorean tuning
1634:Euler–Fokker genus
1383:Septimal semicomma
456:musical instrument
437:Pythagorean tuning
295:). Ratios have an
211:
188:
168:
145:
2038:Intervals (music)
2025:
2024:
1891:
1890:
1513:
1512:
1495:
1494:
1325:Pythagorean comma
1265:Pythagorean limma
1197:
1196:
1193:
1192:
1159:supermajor fourth
1135:supermajor second
1080:
1079:
894:
893:
890:
889:
602:are the number of
458:is tuned using a
237:. For example, a
119:
118:
111:
93:
2050:
1864:Well temperament
1850:Regular diatonic
1716:
1696:Tonality diamond
1540:
1533:
1526:
1517:
1505:
1504:
1395:Septimal kleisma
1208:
1165:subminor seventh
1147:supermajor third
1099:
1087:Just intonations
1072:
1071:
1067:
1064:
1052:
1051:
1047:
1044:
1032:
1031:
1027:
1024:
1012:
1011:
1007:
1004:
992:
991:
987:
984:
972:
971:
967:
964:
952:
951:
947:
924:
907:
614:
596:
570:
563:
556:
547:
540:
531:
415:
414:
413:
411:
396:
395:
394:
392:
373:
372:
371:
369:
357:
356:
355:
353:
341:
340:
339:
337:
325:
324:
323:
321:
293:electronic tuner
271:
270:
269:
267:
255:
254:
253:
251:
235:musical interval
208:
207:
206:
204:
185:
184:
183:
181:
165:
164:
163:
161:
142:
141:
140:
138:
114:
107:
103:
100:
94:
92:
58:"Interval ratio"
51:
27:
19:
2058:
2057:
2053:
2052:
2051:
2049:
2048:
2047:
2028:
2027:
2026:
2021:
2018:(Bohlen–Pierce)
1986:833 cents scale
1974:
1897:
1887:
1854:
1807:
1705:
1626:Just intonation
1620:
1549:
1547:Musical tunings
1544:
1514:
1509:
1491:
1453:
1420:
1363:Septimal diesis
1309:
1279:
1249:
1203:
1189:
1170:
1093:
1090:
1076:
1069:
1065:
1062:
1060:
1049:
1045:
1042:
1040:
1029:
1025:
1022:
1020:
1009:
1005:
1002:
1000:
989:
985:
982:
980:
969:
965:
962:
960:
949:
945:
944:
934:
933:
932:
928:
918:
915:
902:
900:
886:
831:
776:
721:
684:
647:
608:
603:
601:
591:
589:
586:
579:
574:
544:
543:
533:Riemann, Hugo.
532:
528:
523:
460:just intonation
441:just intonation
429:
409:
407:
406:
405:
390:
388:
387:
386:
367:
365:
364:
363:
351:
349:
348:
347:
335:
333:
332:
331:
319:
317:
316:
315:
265:
263:
262:
261:
260:perfect fifth (
249:
247:
246:
245:
202:
200:
199:
198:
179:
177:
176:
175:
159:
157:
156:
155:
136:
134:
133:
132:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
2056:
2054:
2046:
2045:
2040:
2030:
2029:
2023:
2022:
2020:
2019:
2013:
2008:
2003:
1998:
1993:
1988:
1982:
1980:
1976:
1975:
1973:
1972:
1967:
1962:
1952:
1947:
1942:
1941:
1940:
1935:
1930:
1925:
1917:
1912:
1907:
1901:
1899:
1893:
1892:
1889:
1888:
1862:
1860:
1856:
1855:
1853:
1852:
1847:
1842:
1837:
1832:
1817:
1815:
1809:
1808:
1806:
1805:
1800:
1795:
1790:
1785:
1780:
1775:
1770:
1760:
1755:
1750:
1745:
1740:
1735:
1730:
1724:
1722:
1713:
1707:
1706:
1704:
1703:
1698:
1693:
1688:
1683:
1678:
1673:
1668:
1667:
1666:
1661:
1651:
1646:
1641:
1639:Harmonic scale
1636:
1630:
1628:
1622:
1621:
1619:
1618:
1613:
1608:
1603:
1598:
1593:
1588:
1586:Interval ratio
1583:
1578:
1573:
1568:
1563:
1557:
1555:
1551:
1550:
1545:
1543:
1542:
1535:
1528:
1520:
1511:
1510:
1500:
1497:
1496:
1493:
1492:
1490:
1489:
1484:
1479:
1474:
1469:
1463:
1461:
1455:
1454:
1452:
1451:
1446:
1441:
1436:
1430:
1428:
1422:
1421:
1419:
1418:
1412:
1406:
1399:
1398:
1392:
1386:
1380:
1374:
1367:
1366:
1360:
1357:Greater diesis
1354:
1347:
1346:
1343:Septimal comma
1340:
1337:Holdrian comma
1334:
1331:Syntonic comma
1328:
1321:
1319:
1311:
1310:
1308:
1307:
1302:
1297:
1291:
1289:
1281:
1280:
1278:
1277:
1272:
1267:
1261:
1259:
1251:
1250:
1248:
1247:
1242:
1237:
1232:
1227:
1222:
1216:
1214:
1205:
1199:
1198:
1195:
1194:
1191:
1190:
1188:
1187:
1180:
1178:
1172:
1171:
1169:
1168:
1162:
1156:
1153:subminor fifth
1150:
1144:
1141:subminor third
1138:
1132:
1126:
1120:
1114:
1107:
1105:
1096:
1082:
1081:
1078:
1077:
1075:
1074:
1054:
1034:
1014:
994:
974:
954:
937:
935:
927:
921:
904:
896:
895:
892:
891:
888:
887:
885:
884:
878:
872:
866:
860:
854:
848:
841:
839:
833:
832:
830:
829:
823:
817:
811:
805:
799:
793:
786:
784:
778:
777:
775:
774:
768:
762:
756:
750:
744:
738:
731:
729:
723:
722:
720:
719:
713:
707:
701:
694:
692:
686:
685:
683:
682:
676:
670:
664:
657:
655:
649:
648:
646:
645:
639:
633:
627:
620:
618:
611:
593:
581:
580:
575:
573:
572:
565:
558:
550:
542:
541:
525:
524:
522:
519:
511:equal tempered
496:pure intervals
492:just intervals
480:perfect fourth
433:tuning systems
428:
425:
258:equal tempered
219:interval ratio
196:neutral second
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
2055:
2044:
2041:
2039:
2036:
2035:
2033:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1983:
1981:
1977:
1971:
1968:
1966:
1963:
1960:
1959:Carnatic raga
1956:
1953:
1951:
1948:
1946:
1943:
1939:
1936:
1934:
1931:
1929:
1928:Turkish makam
1926:
1924:
1921:
1920:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1902:
1900:
1894:
1885:
1881:
1877:
1873:
1869:
1865:
1861:
1857:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1830:
1826:
1825:quarter-comma
1822:
1819:
1818:
1816:
1814:
1810:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1779:
1776:
1774:
1771:
1768:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1725:
1723:
1721:
1717:
1714:
1712:
1708:
1702:
1701:Tonality flux
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1665:
1662:
1660:
1657:
1656:
1655:
1652:
1650:
1647:
1645:
1642:
1640:
1637:
1635:
1632:
1631:
1629:
1627:
1623:
1617:
1614:
1612:
1609:
1607:
1604:
1602:
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1558:
1556:
1552:
1548:
1541:
1536:
1534:
1529:
1527:
1522:
1521:
1518:
1508:
1498:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1465:
1464:
1462:
1460:
1456:
1450:
1447:
1445:
1442:
1440:
1437:
1435:
1432:
1431:
1429:
1427:
1423:
1416:
1413:
1410:
1407:
1404:
1401:
1400:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1368:
1364:
1361:
1358:
1355:
1352:
1351:Lesser diesis
1349:
1348:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1322:
1320:
1318:
1317:
1312:
1306:
1303:
1301:
1298:
1296:
1293:
1292:
1290:
1288:
1287:
1286:Quarter tones
1282:
1276:
1273:
1271:
1268:
1266:
1263:
1262:
1260:
1258:
1257:
1252:
1246:
1243:
1241:
1238:
1236:
1235:Pseudo-octave
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1217:
1215:
1213:
1209:
1206:
1200:
1185:
1182:
1181:
1179:
1177:
1173:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1108:
1106:
1104:
1100:
1097:
1095:
1089:
1088:
1083:
1058:
1055:
1038:
1035:
1018:
1015:
998:
995:
978:
975:
958:
955:
942:
939:
938:
936:
931:
925:
922:
920:
914:
913:
908:
905:
897:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
842:
840:
838:
834:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
787:
785:
783:
779:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
732:
730:
728:
724:
717:
714:
711:
708:
705:
702:
699:
696:
695:
693:
691:
687:
680:
677:
674:
671:
668:
665:
662:
659:
658:
656:
654:
650:
643:
640:
637:
634:
631:
628:
625:
622:
621:
619:
615:
612:
610:
606:
597:
594:
588:
582:
578:
571:
566:
564:
559:
557:
552:
551:
548:
538:
537:
530:
527:
520:
518:
516:
512:
508:
503:
501:
497:
493:
489:
485:
481:
477:
476:perfect fifth
473:
469:
465:
461:
457:
452:
450:
446:
442:
438:
434:
426:
424:
422:
417:
412:
403:
400:
393:
384:
380:
375:
370:
361:
354:
345:
338:
329:
322:
313:
309:
304:
302:
298:
294:
288:
286:
282:
279:
275:
268:
259:
252:
243:
242:perfect fifth
240:
236:
232:
228:
224:
220:
216:
205:
197:
192:
182:
172:
162:
154:
151:5-limit 10:9
149:
139:
131:
127:
123:
113:
110:
102:
99:February 2024
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
2016:Lambda scale
1923:Arabic maqam
1880:Werckmeister
1711:Temperaments
1585:
1458:
1425:
1411:(0.72 cents)
1405:(1.95 cents)
1385:(13.8 cents)
1379:(10.1 cents)
1373:(19.5 cents)
1365:(35.7 cents)
1359:(62.6 cents)
1353:(41.1 cents)
1345:(27.3 cents)
1339:(22.6 cents)
1333:(21.5 cents)
1327:(23.5 cents)
1314:
1295:Quarter tone
1285:
1284:
1255:
1254:
1211:
1176:Higher-limit
1091:
1085:
997:major fourth
941:quarter tone
916:
910:
599:
534:
529:
504:
495:
491:
453:
430:
418:
376:
305:
289:
218:
212:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2011:Delta scale
2006:Gamma scale
1996:Alpha scale
1898:non-Western
1896:Traditional
1591:Pitch class
1571:Millioctave
1554:Measurement
1444:Millioctave
1426:Measurement
1417:(0.4 cents)
1397:(7.7 cents)
1391:(8.1 cents)
1275:Major limma
1017:minor fifth
488:minor third
484:major third
360:65536:59049
227:frequencies
2032:Categories
2001:Beta scale
1979:Non-octave
1970:Tetrachord
1872:Kirnberger
1835:Schismatic
1477:Semiditone
1371:Diaschisma
1186:(17-limit)
877:(22 or 23)
875:fourteenth
871:(20 or 21)
869:thirteenth
865:(18 or 19)
859:(17 or 18)
853:(15 or 16)
847:(13 or 14)
782:Diminished
609:interval.)
590:(post-Bach
521:References
397:) and the
153:minor tone
130:major tone
69:newspapers
1991:A12 scale
1945:Octoechos
1910:ShĂ-èr-lÇś
1859:Irregular
1676:Otonality
1616:Microtone
1439:Centitone
1377:Semicomma
1256:Semitones
1220:Microtone
1204:intervals
881:fifteenth
727:Augmented
605:semitones
577:Intervals
500:consonant
301:inversion
1876:Vallotti
1829:septimal
1821:Meantone
1581:Interval
1409:Breedsma
857:eleventh
837:Compound
592:Western)
587:semitone
486:), 6:5 (
482:), 5:4 (
478:), 4:3 (
474:), 3:2 (
470:), 2:1 (
451:ratios.
435:such as
374:), etc.
274:middle C
1965:Slendro
1915:Dastgah
1840:Miracle
1803:96-tone
1798:72-tone
1793:58-tone
1788:53-tone
1783:41-tone
1778:34-tone
1773:31-tone
1763:24-tone
1758:23-tone
1753:22-tone
1748:19-tone
1743:17-tone
1738:15-tone
1733:12-tone
1664:7-limit
1659:5-limit
1415:Ragisma
1403:Schisma
1389:Kleisma
1225:5-limit
1131:(15:14)
1125:(21:20)
1119:(28:27)
1113:(36:35)
1103:7-limit
1068:⁄
1057:seventh
1048:⁄
1028:⁄
1008:⁄
988:⁄
968:⁄
948:⁄
930:Neutral
903:systems
863:twelfth
820:seventh
771:seventh
716:seventh
679:seventh
617:Perfect
585:Twelve-
464:integer
454:When a
449:integer
399:7-limit
297:inverse
231:pitches
229:of the
225:of the
126:3-limit
83:scholar
2043:Ratios
1933:Mugham
1919:Maqam
1813:Linear
1767:pieces
1728:6-tone
1649:Hexany
1576:Savart
1472:Ditone
1459:Others
1449:Savart
1316:Commas
1212:Groups
1161:(10:7)
957:second
901:tuning
826:octave
802:fourth
790:second
753:fourth
741:second
735:unison
698:second
661:second
642:octave
630:fourth
624:unison
607:in the
472:octave
468:unison
443:, and
383:128:81
85:
78:
71:
64:
56:
1950:Pelog
1938:Muqam
1884:Young
1845:Magic
1720:Equal
1654:Limit
1561:Pitch
1482:Secor
1230:Comma
1202:Other
1167:(7:4)
1155:(7:5)
1149:(9:7)
1143:(7:6)
1137:(8:7)
1037:sixth
977:third
899:Other
851:tenth
845:ninth
814:sixth
808:fifth
796:third
765:sixth
759:fifth
747:third
710:sixth
704:third
690:Minor
673:sixth
667:third
653:Major
636:fifth
494:, or
427:Usage
421:cents
379:limit
233:in a
223:ratio
221:is a
217:, an
215:music
90:JSTOR
76:books
1955:Raga
1566:Cent
1467:Wolf
1434:Cent
883:(24)
828:(11)
773:(12)
767:(10)
718:(10)
681:(11)
644:(12)
410:Play
402:14:9
391:Play
368:Play
352:Play
336:Play
320:Play
266:Play
250:Play
239:just
203:Play
180:Play
160:Play
137:Play
128:9:8
62:news
822:(9)
816:(7)
810:(6)
804:(4)
798:(2)
792:(0)
761:(8)
755:(6)
749:(5)
743:(3)
737:(1)
712:(8)
706:(3)
700:(1)
675:(9)
669:(4)
663:(2)
638:(7)
632:(5)
626:(0)
358:),
344:9:8
342:),
328:4:3
326:),
312:2:1
303:).
278:440
276:is
213:In
45:by
2034::
1882:,
1878:,
1874:,
1827:,
1061:10
517:.
439:,
281:Hz
1961:)
1957:(
1886:)
1870:(
1866:/
1831:)
1823:(
1769:)
1765:(
1539:e
1532:t
1525:v
1073:)
1070:2
1066:1
1063:+
1059:(
1053:)
1050:2
1046:1
1043:+
1041:8
1039:(
1033:)
1030:2
1026:1
1023:+
1021:6
1019:(
1013:)
1010:2
1006:1
1003:+
1001:5
999:(
993:)
990:2
986:1
983:+
981:3
979:(
973:)
970:2
966:1
963:+
961:1
959:(
953:)
950:2
946:1
943:(
569:e
562:t
555:v
404:(
385:(
362:(
346:(
330:(
314:(
285:E
209:.
186:.
166:.
143:.
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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