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235:, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, modulo octaves. Since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partch's theoretical prediction of the sensory dissonance of intervals (his "One-Footed Bride") are very similar to those of theorists including
1031:, the simple ratios of just intonation are mapped to nearby irrational approximations. This operation, if successful, does not change the relative harmonic complexity of the different intervals, but it can complicate the use of the harmonic limit concept. Since some chords (such as the
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666:. Unlike Partch, who often took scales directly from the harmonic series, the composers of the American Gamelan movement tended to draw scales from the just intonation lattice, in a manner like that used to construct
87:
924:
662:, musicians in California and elsewhere began to build their own gamelan instruments, often tuning them in just intonation. The central figure of this movement was the American composer
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277:
below and including the (odd) limit in a tuning. For example, the identities included in 5-limit tuning are 1, 3, and 5. Each odd number represents a new pitch in the
166:
are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5. For example, the
1845:
222:, the n-odd-limit contains all rational numbers such that the largest odd number that divides either the numerator or denominator is not greater than
458:
1285:
1973:
670:. Such scales often contain ratios with very large numbers, that are nevertheless related by simple intervals to other notes in the scale.
1831:
1329:
338:, is nevertheless an identity in music, simply because it is an odd number." Partch defines "identity" as "one of the correlatives, '
1905:
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1818:
677:
based on the limit. For example, 7-limit tuning and intervals are called septimal, 11-limit is called undecimal, and so on.
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2011:
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In the late 1970s, a new genre of music began to take shape on the West coast of the United States, known as the
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1978:
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1963:
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399:, the n-prime-limit contains all rational numbers that can be factored using primes no greater than
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372:, respectively. According to music software producer Tonalsoft: "An udentity is an identity of an
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1264:. Mathematical World 28. (Providence, R.I.: American Mathematical Society, 2009), p. 137.
8:
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First 32 harmonics, with the harmonics unique to each limit sharing the same color.
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1181:
Genesis Of A Music: An
Account Of A Creative Work, Its Roots, And Its Fulfillments
1133:
Genesis of a Music: An
Account of a Creative Work, Its Roots, and Its Fulfillments
31:
The first 16 harmonics, with frequencies and log frequencies (not drawn to scale).
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403:. In other words, it is the set of rationals with numerator and denominator both
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Prime-limit tuning and intervals are often referred to using the term for the
546:{\displaystyle x=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}...p_{r}^{\alpha _{r}}}
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to suggest that music has been slowly evolving to employ higher and higher
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Wolf, Daniel James (2003), "Alternative
Tunings, Alternative Tonalities",
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of these triads invoke relationships among the first five harmonics.
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quickly became the fundamental building blocks of
Western music. The
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1135:, second edition, enlarged (New York: Da Capo Press, 1974), p. 73.
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131:) were considered consonant. In the West, triadic harmony arose (
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of music, or the harmonies that can be made using a particular
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if all interval ratios between pitches lie in this subgroup.
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1290:
Glen
Peterson's Musical Instruments and Tuning Systems
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approximates 4:5:6:7 (albeit very poorly), while the
646:). ... We say that a scale or system of tuning uses
1846:
Twelve
Microtonal Etudes for Electronic Music Media
1211:
Harry Partch: An
Anthology of Critical Perspectives
638:
603:
545:
443:
334:According to Partch: "The number 9, though not a
2397:
190:, intervals between pitches are drawn from the
162:. In conventional music theory pedagogy, these
127:(involving relationships among the first three
75:The harmonic series and the evolution of music
71:on the complexity of harmony; hence the name.
1899:
1323:
1974:List of intervals in 5-limit just intonation
206:do not include the same intervals even when
181:
1906:
1892:
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158:debuted as fundamental building blocks in
103:, and Ralph David Hill are among the many
1022:
620:
431:
383:
281:and may thus be considered an identity:
78:
26:
1170:(2001), pp. 1–3 (Accessed 29 May 2010).
639:{\displaystyle \mathbb {Q} ^{+},\cdot }
455:whose prime factorization has the form
83:Overtone series, partials 1-5 numbered
14:
2398:
1887:
1311:
1286:"Limits: Consonance Theory Explained"
1154:
604:{\displaystyle p_{1},...,p_{r}\leq p}
451:consisting of those rational numbers
154:Around the turn of the 20th century,
1337:
1106:(1/2), Abingdon, UK: Routledge: 13,
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1913:
24:
1167:Some Music Theory from Paul Erlich
25:
2427:
1279:
2049:Ptolemy's intense diatonic scale
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1501:
1162:The Forms of Tonality: A Preview
444:{\displaystyle \mathbb {Q} ^{+}}
47:are a way of characterizing the
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1203:
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113:emancipation of the dissonance
13:
1:
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2012:Harry Partch's 43-tone scale
1709:Harry Partch's 43-tone scale
1049:3-limit (Pythagorean) tuning
213:
202:. Odd limit and prime limit
7:
1832:Sonata for Microtonal Piano
1112:10.1080/0749446032000134715
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257:
10:
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1979:List of meantone intervals
1839:Suite for Microtonal Piano
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218:For a positive odd number
2406:Just tuning and intervals
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1992:
1969:List of musical intervals
1964:Consonance and dissonance
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1791:Huygens-Fokker Foundation
1779:Boston Microtonal Society
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1209:Dunn, David, ed. (2000).
1100:Contemporary Music Review
996:Pythagorean major seventh
668:Fokker periodicity blocks
658:. Inspired by Indonesian
182:Odd-limit and prime-limit
178:approximates 8:10:12:15.
135:) around the time of the
67:, who used it to give an
1033:diminished seventh chord
883:greater septimal tritone
1741:Otonality and Utonality
1069:Otonality and Utonality
939:Pythagorean major sixth
656:American gamelan school
111:in its constructs (see
1438:Claus-Steffen Mahnkopf
1179:Partch, Harry (1979).
1023:Beyond just intonation
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168:dominant seventh chord
160:African-American music
119:, only chords made of
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2236:Temperament ordinaire
1614:(Bohlen–Pierce scale)
1558:Tui St. George Tucker
1262:Mathematics and Music
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611:forms a subgroup of (
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548:
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418:Given a prime number
413:
387:
237:Hermann von Helmholtz
82:
30:
2039:List of compositions
1868:Generalized keyboard
1363:Easley Blackwood Jr.
1200:Partch (1979), p.71.
1151:(pbk reprint, 1979).
615:
557:
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426:
262:For other uses, see
51:found in a piece or
1863:Enharmonic keyboard
1814:quarter tone pieces
1808:Beauty in the Beast
1488:Ivan Wyschnegradsky
1029:musical temperament
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511:
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176:major seventh chord
133:contenance angloise
2273:Chinese musicology
2059:Scale of harmonics
2054:Pythagorean tuning
2002:Euler–Fokker genus
1785:Genesis of a Music
1413:Christiaan Huygens
1242:on 29 October 2013
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232:Genesis of a Music
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63:was introduced by
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1873:Modernism (music)
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1623:Equal temperament
1393:Brian Ferneyhough
1054:Five-limit tuning
1020:
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210:is an odd prime.
16:(Redirected from
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2232:Well temperament
2218:Regular diatonic
2084:
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2064:Tonality diamond
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1761:Tonality diamond
1584:repeating scales
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1563:Nicola Vicentino
1510:
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1473:Nicola Vicentino
1398:Michael Finnissy
1339:Microtonal music
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1296:"Harmonic Limit"
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1238:. Archived from
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264:Identity (music)
241:William Sethares
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2354:833 cents scale
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1493:La Monte Young
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1280:External links
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1260:David Wright,
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1160:Paul Erlich, "
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1131:Harry Partch,
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1064:Numerary nexus
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1059:7-limit tuning
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738:perfect fourth
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648:p-limit tuning
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370:under-identity
364:are short for
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164:seventh chords
125:perfect fifths
117:medieval music
105:microtonalists
99:Harry Partch,
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2327:Carnatic raga
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2296:Turkish makam
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2194:
2193:quarter-comma
2190:
2187:
2186:
2184:
2182:
2178:
2172:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2136:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2093:
2091:
2089:
2085:
2082:
2080:
2076:
2070:
2069:Tonality flux
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2033:
2030:
2028:
2025:
2024:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1999:
1997:
1995:
1991:
1985:
1982:
1980:
1977:
1975:
1972:
1970:
1967:
1965:
1962:
1960:
1957:
1955:
1952:
1950:
1947:
1945:
1942:
1940:
1937:
1935:
1932:
1930:
1927:
1926:
1924:
1920:
1916:
1909:
1904:
1902:
1897:
1895:
1890:
1889:
1886:
1874:
1871:
1869:
1866:
1864:
1861:
1860:
1858:
1854:
1848:
1847:
1843:
1841:
1840:
1836:
1834:
1833:
1829:
1827:
1826:
1822:
1820:
1817:
1815:
1812:
1810:
1809:
1805:
1804:
1802:
1798:
1792:
1789:
1787:
1786:
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1780:
1777:
1776:
1774:
1768:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1744:
1742:
1739:
1737:
1734:
1733:
1731:
1725:
1715:
1712:
1710:
1707:
1706:
1704:
1702:
1698:
1692:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
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1626:
1624:
1620:
1613:
1610:
1608:
1605:
1603:
1600:
1598:
1595:
1593:
1590:
1589:
1587:
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1579:
1576:
1570:
1564:
1561:
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1556:
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1551:
1549:
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1539:
1536:
1534:
1531:
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1526:
1524:
1521:
1520:
1518:
1514:
1509:
1504:
1494:
1491:
1489:
1486:
1484:
1483:Elaine Walker
1481:
1479:
1478:Claude Vivier
1476:
1474:
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1453:Roger Redgate
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1433:Stu Mackenzie
1431:
1429:
1428:György Ligeti
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1350:
1348:
1344:
1340:
1333:
1328:
1326:
1321:
1319:
1314:
1313:
1310:
1306:
1301:
1297:
1294:
1291:
1287:
1284:
1283:
1271:
1270:0-8218-4873-9
1267:
1263:
1257:
1241:
1237:
1233:
1227:
1220:
1219:9789057550652
1216:
1212:
1206:
1197:
1190:
1189:0-306-80106-X
1186:
1182:
1176:
1169:
1168:
1163:
1157:
1150:
1149:0-306-80106-X
1146:
1142:
1141:0-306-71597-X
1138:
1134:
1128:
1121:
1117:
1113:
1109:
1105:
1101:
1094:
1090:
1080:
1079:Tonality flux
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1046:
1040:
1038:
1034:
1030:
1013:
1005:
1002:
999:
997:
994:
991:
990:
984:
976:
973:
970:
968:
965:
962:
961:
955:
947:
944:
941:
938:
935:
934:
928:
920:
917:
914:
912:
909:
906:
905:
899:
891:
888:
885:
882:
879:
878:
872:
864:
861:
858:
856:
852:
849:
848:
842:
834:
831:
828:
826:
823:
820:
819:
813:
805:
802:
799:
797:
794:
791:
790:
784:
776:
773:
770:
768:
765:
762:
761:
755:
747:
744:
741:
739:
736:
733:
732:
726:
718:
715:
712:
710:
709:perfect fifth
707:
704:
703:
699:
696:
693:
690:
687:
686:
678:
676:
671:
669:
665:
661:
657:
651:
649:
633:
630:
625:
598:
595:
590:
586:
582:
579:
576:
573:
570:
565:
561:
536:
532:
526:
522:
518:
515:
512:
505:
501:
495:
491:
483:
479:
473:
469:
465:
462:
454:
436:
421:
417:
412:
410:
406:
402:
398:
395:
386:
377:
375:
371:
367:
366:over-identity
363:
362:
357:
356:
351:
349:
345:
341:
337:
282:
280:
276:
272:
265:
255:
253:
248:
246:
242:
238:
234:
233:
227:
225:
221:
211:
209:
205:
201:
197:
193:
189:
179:
177:
173:
169:
165:
161:
157:
152:
150:
146:
142:
138:
134:
130:
126:
122:
118:
114:
110:
106:
102:
91:
81:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
29:
19:
2416:Harry Partch
2384:Lambda scale
2291:Arabic maqam
2248:Werckmeister
2079:Temperaments
2021:
1856:Other topics
1844:
1837:
1830:
1823:
1806:
1800:Compositions
1783:
1772:publications
1735:
1727:Concepts and
1612:Lambda scale
1553:Harry Partch
1548:Yuri Landman
1543:Lou Harrison
1528:Wendy Carlos
1523:Glenn Branca
1423:Ben Johnston
1418:Charles Ives
1378:Franklin Cox
1368:Heinz Bohlen
1305:
1299:
1289:
1261:
1256:
1244:. Retrieved
1240:the original
1235:
1226:
1210:
1205:
1196:
1180:
1175:
1165:
1156:
1132:
1127:
1103:
1099:
1093:
1026:
911:major second
672:
664:Lou Harrison
653:
647:
452:
419:
415:
414:
404:
400:
396:
394:prime number
391:
369:
365:
360:
359:
354:
353:
352:
333:
270:
268:
249:
230:
228:
223:
219:
217:
207:
203:
199:
195:
185:
153:
149:minor thirds
98:
65:Harry Partch
60:
44:
40:
37:music theory
34:
2379:Delta scale
2374:Gamma scale
2364:Alpha scale
2266:non-Western
2264:Traditional
1959:Pitch class
1939:Millioctave
1922:Measurement
1819:just pieces
1607:Delta scale
1602:Gamma scale
1592:Alpha scale
1582:Non-octave-
1572:Tunings and
1533:Ivor Darreg
1358:BĂ©la BartĂłk
1300:Xenharmonic
825:major sixth
796:major tenth
767:major third
697:prime-limit
380:Prime limit
275:odd numbers
245:Paul Erlich
200:prime limit
137:Renaissance
101:Ivor Darreg
69:upper bound
59:. The term
18:Prime limit
2400:Categories
2369:Beta scale
2347:Non-octave
2338:Tetrachord
2240:Kirnberger
2203:Schismatic
1770:Groups and
1729:techniques
1597:Beta scale
1448:Joe Maneri
1408:Alois Hába
1388:John Eaton
1246:23 October
1232:"Udentity"
1085:References
2359:A12 scale
2313:Octoechos
2278:ShĂ-èr-lÇś
2227:Irregular
2044:Otonality
1984:Microtone
1751:Sonido 13
1516:Inventors
1468:Ezra Sims
1346:Composers
1236:Tonalsoft
1120:191457676
694:odd-limit
634:⋅
596:≤
533:α
502:α
480:α
374:utonality
254:, below.
214:Odd limit
196:odd limit
129:harmonics
109:harmonics
2244:Vallotti
2197:septimal
2189:Meantone
1949:Interval
1746:Semitone
1213:, p.28.
1183:, p.93.
1043:See also
691:interval
681:Examples
361:udentity
355:Odentity
348:tonality
346:', in a
271:identity
258:Identity
2411:Harmony
2333:Slendro
2283:Dastgah
2208:Miracle
2171:96-tone
2166:72-tone
2161:58-tone
2156:53-tone
2151:41-tone
2146:34-tone
2141:31-tone
2131:24-tone
2126:23-tone
2121:22-tone
2116:19-tone
2111:17-tone
2106:15-tone
2101:12-tone
2032:7-limit
2027:5-limit
992:243/128
853:lesser
660:gamelan
331:12 ...
307:G ...
156:tetrads
121:octaves
49:harmony
2301:Mugham
2287:Maqam
2181:Linear
2135:pieces
2096:6-tone
2017:Hexany
1944:Savart
1825:Mother
1574:scales
1463:Sevish
1268:
1217:
1187:
1147:
1139:
1118:
967:ditone
700:audio
409:smooth
392:For a
342:' or '
243:, and
141:triads
139:, and
115:). In
41:limits
2318:Pelog
2306:Muqam
2252:Young
2213:Magic
2088:Equal
2022:Limit
1929:Pitch
1736:Limit
1116:S2CID
1037:12-ET
963:81/64
936:27/16
688:ratio
553:with
344:minor
340:major
336:prime
172:12-ET
145:major
61:limit
57:scale
53:genre
2323:Raga
1934:Cent
1266:ISBN
1248:2013
1215:ISBN
1185:ISBN
1145:ISBN
1137:ISBN
1011:Play
982:Play
953:Play
926:Play
897:Play
880:10/7
870:Play
840:Play
811:Play
782:Play
753:Play
724:Play
368:and
358:and
250:See
198:and
147:and
123:and
89:Play
1164:".
1108:doi
1035:in
1027:In
1000:243
907:9/8
850:7/5
821:5/3
792:5/2
763:5/4
734:4/3
705:3/2
376:".
327:10
323:8
319:6
315:4
311:2
303:E
299:C
295:G
291:C
287:C
269:An
229:In
186:In
170:in
43:or
35:In
2402::
2250:,
2246:,
2242:,
2195:,
1691:96
1686:72
1681:58
1676:53
1671:41
1666:34
1661:31
1656:24
1651:23
1646:22
1641:19
1636:17
1631:15
1298:,
1288:,
1234:.
1143:;
1114:,
1104:22
1102:,
971:81
942:27
411:.
329:11
247:.
239:,
226:.
39:,
2329:)
2325:(
2254:)
2238:(
2234:/
2199:)
2191:(
2137:)
2133:(
1907:e
1900:t
1893:v
1331:e
1324:t
1317:v
1302:.
1292:.
1272:.
1250:.
1221:.
1191:.
1110::
1003:3
974:3
945:3
918:3
915:9
889:7
886:7
862:7
859:7
832:5
829:5
803:5
800:5
774:5
771:5
745:3
742:3
716:3
713:3
631:,
626:+
621:Q
599:p
591:r
587:p
583:,
580:.
577:.
574:.
571:,
566:1
562:p
537:r
527:r
523:p
519:.
516:.
513:.
506:2
496:2
492:p
484:1
474:1
470:p
466:=
463:x
453:x
437:+
432:Q
420:p
407:-
405:n
401:n
397:n
325:9
321:7
317:5
313:3
309:1
305:F
301:D
297:B
293:E
289:G
285:C
266:.
224:n
220:n
208:n
204:n
95:.
20:)
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