551:
31:
269:
1702:
1239:
1665:
1600:
1533:
1488:
1481:
1161:
1004:
1393:
1746:
1442:
1435:
1347:
1278:
965:
1709:
1607:
546:{\displaystyle \prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdots ={\frac {4}{3}}\cdot {\frac {16}{15}}\cdot {\frac {36}{35}}\cdots =2\cdot {\frac {8}{9}}\cdot {\frac {24}{25}}\cdot {\frac {48}{49}}\cdots ={\frac {\pi }{2}}}
690:
196:
A superparticular number is when a great number contains a lesser number, to which it is compared, and at the same time one part of it. For example, when 3 and 2 are compared, they contain 2, plus the 3 has another 1, which is half of two. When 3 and 4 are compared, they each contain a 3, and the 4
1922:. On pp. 123–124 the book discusses the classification of ratios into various types including the superparticular ratios, and the tradition by which this classification was handed down from Nichomachus to Boethius, Campanus, Oresme, and Clavius.
587:
85:
1109:
1137:
1293:
1176:
1411:
1323:
1254:
1215:
253:
terminates after two terms. The numbers whose continued fraction terminates in one term are the integers, while the remaining numbers, with three or more terms in their continued fractions, are
1017:
178:
980:
1506:
1365:
1110:
880:
1617:
1138:
852:
1722:
1641:
1680:
941:
908:
1457:
1576:
1546:
1294:
824:
774:
represent a superparticular ratio. Many individual superparticular ratios have their own names, either in historical mathematics or in music theory. These include the following:
1177:
197:
has another 1, which is a third part of 3. Again, when 5, and 4 are compared, they contain the number 4, and the 5 has another 1, which is the fourth part of the number 4, etc.
1045:
1412:
1324:
1255:
1073:
1216:
1983:
1018:
1764:
981:
1507:
1366:
881:
685:{\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {7}{8}}\cdot {\frac {11}{12}}\cdot {\frac {13}{12}}\cdot {\frac {17}{16}}\cdots }
2038:
1618:
853:
1721:
1640:
727:, the ninth harmonic, 9/1, may be expressed as a superparticular ratio, 9/8). Indeed, whether a ratio was superparticular was the most important criterion in
1681:
942:
909:
1458:
2210:
1577:
1547:
825:
1046:
241:
observed, the superparticular numbers (including also the multiply superparticular ratios, numbers formed by adding an integer other than one to a
1074:
1843:
1765:
2276:
2183:
2145:
2109:
2078:
2009:
1916:
1108:
1136:
770:
Every pair of adjacent positive integers represent a superparticular ratio, and similarly every pair of adjacent harmonics in the
1292:
2203:
1175:
1410:
2132:, and the 5:4 ratio is achieved by two common sizes for large format film, 4×5 inches and 8×10 inches. See e.g.
2119:) aspect ratio as another common choice for digital photography, but unlike 4:3 and 3:2 this ratio is not superparticular.
1861:
1322:
1253:
1214:
2029:
1016:
979:
129:
1505:
1364:
1269:
879:
1616:
851:
1723:
1642:
700:
1679:
1152:
940:
907:
213:
1456:
2271:
2196:
1575:
1545:
823:
1044:
570:
1072:
1230:
771:
732:
1763:
566:
226:
995:
792:
956:
755:
747:
254:
1701:
1472:
1338:
751:
1933:
1966:
1878:
1664:
1599:
1591:
1532:
1487:
1480:
1160:
1003:
724:
250:
221:, the areas of study that most frequently refer to the superparticular ratios by this name are
2141:
2105:
2099:
2074:
2005:
1999:
1977:
1912:
1839:
557:
30:
2177:
2135:
2068:
2167:
2047:
1956:
1948:
1870:
1656:
720:
699:, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise via the
218:
185:
1890:
1708:
1606:
2240:
2219:
1886:
1426:
1191:
246:
2128:
The 7:6 medium format aspect ratio is one of several ratios possible using medium-format
1238:
1392:
2085:
The paramount principle in
Ptolemy's tunings was the use of superparticular proportion.
1932:
Leonhard Euler; translated into
English by Myra F. Wyman and Bostwick F. Wyman (1985),
1904:
1561:
1521:
868:
261:
238:
117:
1745:
1441:
1434:
1346:
1277:
964:
2265:
2025:
1970:
840:
740:
736:
574:
242:
2052:
2033:
1859:
Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music".
2245:
2235:
1033:
759:
739:; that is, all ratios of this type in which both the numerator and denominator are
696:
578:
222:
1762:
1720:
1678:
1639:
1615:
1574:
1544:
1504:
1455:
1409:
1363:
1321:
1291:
1252:
1213:
1174:
1135:
1107:
1071:
1043:
1015:
978:
939:
906:
878:
850:
822:
1381:
924:
896:
704:
97:
2116:
208:
2163:
17:
2129:
1195:
1836:
Isidore of
Seville's Etymologies: Complete English Translation, Volume 1
2230:
2171:
1952:
1882:
1737:
728:
716:
2140:, How to Photograph Series, vol. 9, Stackpole Books, p. 43,
1961:
581:
as its numerator and the nearest multiple of four as its denominator:
812:
735:
can be used to list all possible superparticular numbers for a given
1874:
2188:
113:
29:
723:
can be expressed as a superparticular ratio (for example, due to
565:
in several ways as a product of superparticular ratios and their
2192:
560:
731:'s formulation of musical harmony. In this application,
217:. Although these numbers have applications in modern
2001:
The Legacy of
Leonhard Euler: A Tricentennial Tribute
590:
272:
132:
746:These ratios are also important in visual harmony.
577:of superparticular ratios in which each term has a
684:
545:
172:
2137:How to Photograph the Outdoors in Black and White
1909:The Oxford Handbook of the History of Mathematics
1774:The root of some of these terms comes from Latin
173:{\displaystyle {\frac {n+1}{n}}=1+{\frac {1}{n}}}
1811:
1809:
1807:
1805:
1803:
1801:
1799:
754:, and aspect ratios of 7:6 and 5:4 are used in
194:
2204:
2039:Bulletin of the American Mathematical Society
207:Superparticular ratios were written about by
123:More particularly, the ratio takes the form:
8:
1982:: CS1 maint: multiple names: authors list (
2070:Tuning and Temperament: A Historical Survey
2211:
2197:
2189:
2073:, Courier Dover Publications, p. 23,
2051:
1960:
669:
656:
643:
630:
617:
604:
591:
589:
533:
517:
504:
491:
469:
456:
443:
427:
414:
401:
388:
375:
362:
328:
299:
288:
277:
271:
160:
133:
131:
776:
1827:
1795:
1975:
2179:De Institutione Arithmetica, liber II
569:. It is also possible to convert the
7:
1854:
1852:
2034:"On the structure of linear graphs"
2184:Anicius Manlius Severinus Boethius
289:
25:
2004:, World Scientific, p. 214,
1934:"An essay on continued fractions"
1786:"and") describing the ratio 3:2.
766:Ratio names and related intervals
27:Ratio of two consecutive integers
1988:. See in particular p. 304.
1760:
1744:
1718:
1707:
1700:
1676:
1663:
1637:
1613:
1605:
1598:
1572:
1542:
1531:
1502:
1486:
1479:
1453:
1440:
1433:
1407:
1391:
1361:
1345:
1319:
1289:
1276:
1250:
1237:
1211:
1172:
1159:
1133:
1105:
1088:greater undecimal neutral second
1069:
1041:
1013:
1002:
976:
963:
937:
904:
876:
848:
820:
2053:10.1090/S0002-9904-1946-08715-7
1124:lesser undecimal neutral second
2101:Digital Photography Essentials
2067:Barbour, James Murray (2004),
1907:; Stedall, Jacqueline (2008),
703:as the possible values of the
1:
1862:American Mathematical Monthly
750:of 4:3 and 3:2 are common in
34:Just diatonic semitone on C:
2115:. Ang also notes the 16:9 (
1941:Mathematical Systems Theory
1911:, Oxford University Press,
1270:septimal chromatic semitone
2293:
1998:Debnath, Lokenath (2010),
1834:Throop, Priscilla (2006).
1153:septimal diatonic semitone
762:photography respectively.
214:Introduction to Arithmetic
2277:Superparticular intervals
2226:
2104:, Penguin, p. 107,
2164:Superparticular numbers
2134:Schaub, George (1999),
1778:"one and a half" (from
1308:just chromatic semitone
1231:minor diatonic semitone
772:harmonic series (music)
233:Mathematical properties
1060:sesquinona: minor tone
789:Name/musical interval
707:of an infinite graph.
686:
547:
293:
227:history of mathematics
205:
174:
106:superparticular number
93:
2251:Superparticular ratio
2166:applied to construct
1838:, p. III.6.12, n. 7.
1386:inferior quarter tone
996:septimal major second
687:
571:Leibniz formula for π
548:
273:
175:
102:superparticular ratio
33:
957:septimal minor third
588:
270:
130:
1473:septimal sixth-tone
1339:septimal third-tone
779:
752:digital photography
701:Erdős–Stone theorem
116:of two consecutive
1953:10.1007/bf01699475
1592:septimal semicomma
777:
725:octave equivalency
711:Other applications
682:
543:
251:continued fraction
245:) are exactly the
170:
94:
2259:
2258:
2168:pentatonic scales
2098:Ang, Tom (2011),
2046:(12): 1087–1091.
1869:(10): 1096–1100.
1844:978-1-4116-6523-1
1772:
1771:
1766:
1724:
1695:255th subharmonic
1682:
1643:
1632:127th subharmonic
1619:
1578:
1548:
1508:
1459:
1413:
1367:
1325:
1295:
1256:
1217:
1178:
1139:
1111:
1075:
1047:
1019:
982:
943:
910:
882:
854:
826:
733:Størmer's theorem
677:
664:
651:
638:
625:
612:
599:
558:irrational number
541:
525:
512:
499:
477:
464:
451:
435:
422:
409:
396:
383:
370:
352:
323:
168:
149:
16:(Redirected from
2284:
2272:Rational numbers
2220:Rational numbers
2213:
2206:
2199:
2190:
2152:
2150:
2126:
2120:
2114:
2095:
2089:
2087:
2064:
2058:
2057:
2055:
2022:
2016:
2014:
1995:
1989:
1987:
1981:
1973:
1964:
1938:
1929:
1923:
1921:
1901:
1895:
1894:
1856:
1847:
1832:
1816:
1813:
1768:
1767:
1757:
1754:
1753:
1749:
1748:
1726:
1725:
1715:
1712:
1711:
1705:
1704:
1684:
1683:
1673:
1672:
1668:
1667:
1657:septimal kleisma
1645:
1644:
1621:
1620:
1610:
1609:
1603:
1602:
1580:
1579:
1569:
1550:
1549:
1539:
1536:
1535:
1526:63rd subharmonic
1510:
1509:
1499:
1496:
1495:
1491:
1490:
1484:
1483:
1461:
1460:
1450:
1449:
1445:
1444:
1438:
1437:
1415:
1414:
1404:
1401:
1400:
1396:
1395:
1369:
1368:
1358:
1355:
1354:
1350:
1349:
1327:
1326:
1316:
1315:
1297:
1296:
1286:
1285:
1281:
1280:
1258:
1257:
1247:
1246:
1242:
1241:
1219:
1218:
1208:
1205:
1204:
1180:
1179:
1169:
1168:
1164:
1163:
1141:
1140:
1130:
1113:
1112:
1102:
1099:
1098:
1094:
1077:
1076:
1066:
1049:
1048:
1021:
1020:
1010:
1007:
1006:
984:
983:
973:
972:
968:
967:
945:
944:
934:
933:
912:
911:
884:
883:
856:
855:
828:
827:
780:
715:In the study of
691:
689:
688:
683:
678:
670:
665:
657:
652:
644:
639:
631:
626:
618:
613:
605:
600:
592:
563:
552:
550:
549:
544:
542:
534:
526:
518:
513:
505:
500:
492:
478:
470:
465:
457:
452:
444:
436:
428:
423:
415:
410:
402:
397:
389:
384:
376:
371:
363:
358:
354:
353:
351:
337:
329:
324:
322:
308:
300:
292:
287:
247:rational numbers
219:pure mathematics
211:in his treatise
203:
186:positive integer
183:
179:
177:
176:
171:
169:
161:
150:
145:
134:
104:, also called a
92:
91:
90:
88:
81:
79:
78:
75:
72:
65:
63:
62:
59:
56:
49:
47:
46:
43:
40:
21:
2292:
2291:
2287:
2286:
2285:
2283:
2282:
2281:
2262:
2261:
2260:
2255:
2241:Dyadic rational
2222:
2217:
2160:
2155:
2148:
2133:
2127:
2123:
2112:
2097:
2096:
2092:
2081:
2066:
2065:
2061:
2024:
2023:
2019:
2012:
1997:
1996:
1992:
1974:
1936:
1931:
1930:
1926:
1919:
1905:Robson, Eleanor
1903:
1902:
1898:
1875:10.2307/2317424
1858:
1857:
1850:
1833:
1829:
1825:
1820:
1819:
1814:
1797:
1792:
1761:
1755:
1751:
1750:
1743:
1719:
1713:
1706:
1699:
1677:
1670:
1669:
1662:
1638:
1614:
1604:
1597:
1573:
1567:
1543:
1537:
1530:
1525:
1503:
1497:
1493:
1492:
1485:
1478:
1454:
1447:
1446:
1439:
1432:
1427:septimal diesis
1408:
1402:
1398:
1397:
1390:
1385:
1362:
1356:
1352:
1351:
1344:
1320:
1313:
1312:
1290:
1283:
1282:
1275:
1251:
1244:
1243:
1236:
1212:
1206:
1202:
1201:
1173:
1166:
1165:
1158:
1134:
1128:
1106:
1100:
1096:
1095:
1092:
1070:
1064:
1042:
1032:sesquioctavum:
1014:
1008:
1001:
977:
970:
969:
962:
938:
931:
930:
923:sesquiquintum:
905:
895:sesquiquartum:
877:
867:sesquitertium:
849:
839:sesquialterum:
821:
794:
768:
719:, many musical
713:
586:
585:
561:
556:represents the
338:
330:
309:
301:
298:
294:
268:
267:
235:
204:
202:Throop (2006),
201:
181:
135:
128:
127:
118:integer numbers
86:
84:
83:
82:
76:
73:
70:
69:
67:
60:
57:
54:
53:
51:
44:
41:
38:
37:
35:
28:
23:
22:
15:
12:
11:
5:
2290:
2288:
2280:
2279:
2274:
2264:
2263:
2257:
2256:
2254:
2253:
2248:
2243:
2238:
2233:
2227:
2224:
2223:
2218:
2216:
2215:
2208:
2201:
2193:
2187:
2186:
2175:
2172:David Canright
2159:
2158:External links
2156:
2154:
2153:
2146:
2121:
2110:
2090:
2079:
2059:
2017:
2010:
1990:
1924:
1917:
1896:
1848:
1826:
1824:
1821:
1818:
1817:
1794:
1793:
1791:
1788:
1770:
1769:
1758:
1740:
1735:
1732:
1728:
1727:
1716:
1696:
1693:
1690:
1686:
1685:
1674:
1659:
1654:
1651:
1647:
1646:
1635:
1633:
1630:
1627:
1623:
1622:
1611:
1594:
1589:
1586:
1582:
1581:
1570:
1564:
1562:syntonic comma
1559:
1556:
1552:
1551:
1540:
1527:
1522:septimal comma
1519:
1516:
1512:
1511:
1500:
1475:
1470:
1467:
1463:
1462:
1451:
1429:
1424:
1421:
1417:
1416:
1405:
1387:
1378:
1375:
1371:
1370:
1359:
1341:
1336:
1333:
1329:
1328:
1317:
1309:
1306:
1303:
1299:
1298:
1287:
1272:
1267:
1264:
1260:
1259:
1248:
1233:
1228:
1225:
1221:
1220:
1209:
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1155:
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1147:
1143:
1142:
1131:
1125:
1122:
1119:
1115:
1114:
1103:
1089:
1086:
1083:
1079:
1078:
1067:
1061:
1058:
1055:
1051:
1050:
1039:
1036:
1030:
1027:
1023:
1022:
1011:
998:
993:
990:
986:
985:
974:
959:
954:
951:
947:
946:
935:
927:
921:
918:
914:
913:
902:
899:
893:
890:
886:
885:
874:
871:
869:perfect fourth
865:
862:
858:
857:
846:
843:
837:
834:
830:
829:
818:
815:
809:
806:
802:
801:
798:
790:
787:
784:
767:
764:
741:smooth numbers
712:
709:
693:
692:
681:
676:
673:
668:
663:
660:
655:
650:
647:
642:
637:
634:
629:
624:
621:
616:
611:
608:
603:
598:
595:
554:
553:
540:
537:
532:
529:
524:
521:
516:
511:
508:
503:
498:
495:
490:
487:
484:
481:
476:
473:
468:
463:
460:
455:
450:
447:
442:
439:
434:
431:
426:
421:
418:
413:
408:
405:
400:
395:
392:
387:
382:
379:
374:
369:
366:
361:
357:
350:
347:
344:
341:
336:
333:
327:
321:
318:
315:
312:
307:
304:
297:
291:
286:
283:
280:
276:
262:Wallis product
239:Leonhard Euler
234:
231:
199:
190:
189:
167:
164:
159:
156:
153:
148:
144:
141:
138:
110:epimoric ratio
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2289:
2278:
2275:
2273:
2270:
2269:
2267:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2228:
2225:
2221:
2214:
2209:
2207:
2202:
2200:
2195:
2194:
2191:
2185:
2181:
2180:
2176:
2173:
2169:
2165:
2162:
2161:
2157:
2149:
2147:9780811724500
2143:
2139:
2138:
2131:
2125:
2122:
2118:
2113:
2111:9780756685263
2107:
2103:
2102:
2094:
2091:
2086:
2082:
2080:9780486434063
2076:
2072:
2071:
2063:
2060:
2054:
2049:
2045:
2041:
2040:
2035:
2031:
2027:
2021:
2018:
2013:
2011:9781848165267
2007:
2003:
2002:
1994:
1991:
1985:
1979:
1972:
1968:
1963:
1958:
1954:
1950:
1946:
1942:
1935:
1928:
1925:
1920:
1918:9780191607448
1914:
1910:
1906:
1900:
1897:
1892:
1888:
1884:
1880:
1876:
1872:
1868:
1864:
1863:
1855:
1853:
1849:
1845:
1841:
1837:
1831:
1828:
1822:
1812:
1810:
1808:
1806:
1804:
1802:
1800:
1796:
1789:
1787:
1785:
1782:"a half" and
1781:
1777:
1759:
1747:
1741:
1739:
1736:
1733:
1730:
1729:
1717:
1710:
1703:
1697:
1694:
1691:
1688:
1687:
1675:
1666:
1660:
1658:
1655:
1652:
1649:
1648:
1636:
1634:
1631:
1628:
1625:
1624:
1612:
1608:
1601:
1595:
1593:
1590:
1587:
1584:
1583:
1571:
1565:
1563:
1560:
1557:
1554:
1553:
1541:
1534:
1528:
1523:
1520:
1517:
1514:
1513:
1501:
1489:
1482:
1476:
1474:
1471:
1468:
1465:
1464:
1452:
1443:
1436:
1430:
1428:
1425:
1422:
1419:
1418:
1406:
1394:
1388:
1383:
1379:
1376:
1373:
1372:
1360:
1348:
1342:
1340:
1337:
1334:
1331:
1330:
1318:
1310:
1307:
1304:
1301:
1300:
1288:
1279:
1273:
1271:
1268:
1265:
1262:
1261:
1249:
1240:
1234:
1232:
1229:
1226:
1223:
1222:
1210:
1199:
1197:
1193:
1190:
1187:
1184:
1183:
1171:
1162:
1156:
1154:
1151:
1148:
1145:
1144:
1132:
1126:
1123:
1120:
1117:
1116:
1104:
1090:
1087:
1084:
1081:
1080:
1068:
1062:
1059:
1056:
1053:
1052:
1040:
1037:
1035:
1031:
1028:
1025:
1024:
1012:
1005:
999:
997:
994:
991:
988:
987:
975:
966:
960:
958:
955:
952:
949:
948:
936:
928:
926:
922:
919:
916:
915:
903:
900:
898:
894:
891:
888:
887:
875:
872:
870:
866:
863:
860:
859:
847:
844:
842:
841:perfect fifth
838:
835:
832:
831:
819:
816:
814:
810:
807:
804:
803:
799:
796:
791:
788:
785:
782:
781:
775:
773:
765:
763:
761:
757:
756:medium format
753:
749:
748:Aspect ratios
744:
742:
738:
734:
730:
726:
722:
718:
710:
708:
706:
705:upper density
702:
698:
679:
674:
671:
666:
661:
658:
653:
648:
645:
640:
635:
632:
627:
622:
619:
614:
609:
606:
601:
596:
593:
584:
583:
582:
580:
576:
575:Euler product
572:
568:
564:
559:
538:
535:
530:
527:
522:
519:
514:
509:
506:
501:
496:
493:
488:
485:
482:
479:
474:
471:
466:
461:
458:
453:
448:
445:
440:
437:
432:
429:
424:
419:
416:
411:
406:
403:
398:
393:
390:
385:
380:
377:
372:
367:
364:
359:
355:
348:
345:
342:
339:
334:
331:
325:
319:
316:
313:
310:
305:
302:
295:
284:
281:
278:
274:
266:
265:
264:
263:
258:
256:
255:superpartient
252:
248:
244:
243:unit fraction
240:
232:
230:
228:
224:
220:
216:
215:
210:
198:
193:
187:
165:
162:
157:
154:
151:
146:
142:
139:
136:
126:
125:
124:
121:
119:
115:
111:
107:
103:
99:
89:
32:
19:
18:Sesquiquartum
2250:
2246:Half-integer
2236:Dedekind cut
2178:
2136:
2124:
2100:
2093:
2084:
2069:
2062:
2043:
2037:
2030:Stone, A. H.
2020:
2000:
1993:
1944:
1940:
1927:
1908:
1899:
1866:
1860:
1835:
1830:
1815:Ancient name
1783:
1779:
1775:
1773:
1034:major second
793:Ben Johnston
769:
760:large format
745:
714:
697:graph theory
694:
579:prime number
555:
259:
236:
223:music theory
212:
206:
195:
191:
122:
109:
105:
101:
95:
1947:: 295–328,
1382:subharmonic
925:minor third
897:major third
98:mathematics
2266:Categories
2117:widescreen
1962:1811/32133
209:Nicomachus
2026:Erdős, P.
1971:126941824
1823:Citations
1731:4375:4374
1194:diatonic
778:Examples
721:intervals
680:⋯
667:⋅
654:⋅
641:⋅
628:⋅
615:⋅
594:π
536:π
528:⋯
515:⋅
502:⋅
489:⋅
480:⋯
467:⋅
454:⋅
438:⋯
425:⋅
412:⋅
399:⋅
386:⋅
373:⋅
326:⋅
317:−
290:∞
275:∏
112:, is the
2130:120 film
2032:(1946).
1978:citation
1752:♯
1671:♯
1494:♯
1448:♭
1399:♭
1353:♭
1314:♯
1284:♭
1245:♯
1203:♭
1196:semitone
1167:♯
1097:♭
971:♭
932:♭
811:duplex:
797:above C
795:notation
573:into an
567:inverses
225:and the
200:—
2231:Integer
1891:0313189
1883:2317424
1776:sesqui-
1738:ragisma
1689:256:255
1650:225:224
1626:128:127
1585:126:125
729:Ptolemy
717:harmony
80:
68:
64:
52:
48:
36:
2144:
2108:
2077:
2008:
1969:
1915:
1889:
1881:
1842:
1227:104.96
1188:111.73
1149:119.44
1121:150.64
1085:165.00
1057:182.40
1029:203.91
992:231.17
953:266.87
920:315.64
892:386.31
864:498.04
836:701.96
813:octave
800:Audio
786:Cents
783:Ratio
249:whose
192:Thus:
180:where
66:= 1 +
55:15 + 1
1967:S2CID
1937:(PDF)
1879:JSTOR
1790:Notes
1780:semis
1629:13.58
1588:13.79
1558:21.51
1555:81:80
1518:27.26
1515:64:63
1469:34.98
1466:50:49
1423:35.70
1420:49:48
1380:31st
1377:54.96
1374:32:31
1335:62.96
1332:28:27
1305:70.67
1302:25:24
1266:84.47
1263:21:20
1224:17:16
1185:16:15
1146:15:14
1118:12:11
1082:11:10
737:limit
184:is a
114:ratio
2142:ISBN
2106:ISBN
2075:ISBN
2006:ISBN
1984:link
1913:ISBN
1840:ISBN
1784:-que
1734:0.40
1692:6.78
1653:7.71
1192:just
1054:10:9
808:1200
758:and
260:The
100:, a
87:Play
2182:by
2170:by
2048:doi
1957:hdl
1949:doi
1871:doi
1026:9:8
989:8:7
950:7:6
917:6:5
889:5:4
861:4:3
833:3:2
805:2:1
695:In
237:As
108:or
96:In
2268::
2083:,
2044:52
2042:.
2036:.
2028:;
1980:}}
1976:{{
1965:,
1955:,
1945:18
1943:,
1939:,
1887:MR
1885:.
1877:.
1867:79
1865:.
1851:^
1798:^
817:C'
743:.
675:16
672:17
662:12
659:13
649:12
646:11
523:49
520:48
510:25
507:24
475:35
472:36
462:15
459:16
257:.
229:.
120:.
77:15
61:15
50:=
45:15
39:16
2212:e
2205:t
2198:v
2174:.
2151:.
2088:.
2056:.
2050::
2015:.
1986:)
1959::
1951::
1893:.
1873::
1846:.
1756:-
1742:C
1714:-
1698:D
1661:B
1596:D
1568:+
1566:C
1538:-
1529:C
1524:,
1498:-
1477:B
1431:D
1403:-
1389:D
1384:,
1357:-
1343:D
1311:C
1274:D
1235:C
1207:-
1200:D
1157:C
1129:↓
1127:D
1101:-
1093:↑
1091:D
1065:-
1063:D
1038:D
1009:-
1000:D
961:E
929:E
901:E
873:F
845:G
636:8
633:7
623:4
620:5
610:4
607:3
602:=
597:4
562:π
539:2
531:=
497:9
494:8
486:2
483:=
449:3
446:4
441:=
433:7
430:6
420:5
417:6
407:5
404:4
394:3
391:4
381:3
378:2
368:1
365:2
360:=
356:)
349:1
346:+
343:n
340:2
335:n
332:2
320:1
314:n
311:2
306:n
303:2
296:(
285:1
282:=
279:n
188:.
182:n
166:n
163:1
158:+
155:1
152:=
147:n
143:1
140:+
137:n
74:/
71:1
58:/
42:/
20:)
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