73:
540:
20:
258:
1691:
1228:
1654:
1589:
1522:
1477:
1470:
1150:
993:
1382:
1735:
1431:
1424:
1336:
1267:
954:
1698:
1596:
535:{\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}}}
679:
185:
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
1098:
1126:
1282:
1165:
1400:
1312:
1243:
1204:
1006:
1911:. 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.
969:
1495:
1354:
1099:
869:
1606:
1127:
841:
1711:
1630:
1669:
930:
897:
1446:
1565:
1535:
1283:
813:
1166:
576:
1034:
74:
1401:
1313:
1244:
1062:
1205:
1007:
1753:
970:
1496:
1355:
870:
1607:
842:
1710:
1629:
1670:
931:
898:
1447:
1566:
1536:
814:
1035:
1063:
242:
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
1754:
167:
1709:
1628:
839:
895:
1563:
811:
1124:
1096:
763:
represent a superparticular ratio. Many individual superparticular ratios have their own names, either in historical mathematics or in music theory. These include the following:
1280:
186:
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.
1163:
1398:
1310:
1241:
1202:
1004:
1493:
1352:
967:
1604:
867:
1667:
1444:
928:
1533:
1097:
1125:
674:{\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 }
1032:
1060:
716:, 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
1281:
1751:
1164:
1399:
1311:
1242:
1203:
1005:
968:
1494:
1353:
868:
1605:
840:
1972:
1712:
1631:
230:
observed, the superparticular numbers (including also the multiply superparticular ratios, numbers formed by adding an integer other than one to a
1668:
929:
896:
1445:
1564:
1534:
812:
1033:
2027:
1061:
1752:
2199:
759:
Every pair of adjacent positive integers represent a superparticular ratio, and similarly every pair of adjacent harmonics in the
1832:
2121:, 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.
2108:) aspect ratio as another common choice for digital photography, but unlike 4:3 and 3:2 this ratio is not superparticular.
2265:
2172:
118:
2134:
2098:
2067:
1998:
1905:
2192:
1920:
1850:
2018:
1258:
1141:
202:
689:
2260:
2185:
1922:
559:
1219:
760:
721:
555:
215:
210:, the areas of study that most frequently refer to the superparticular ratios by this name are
2088:
1988:
2166:
2124:
2057:
984:
781:
688:, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise via the
1879:
945:
744:
736:
243:
8:
2117:
The 7:6 medium format aspect ratio is one of several ratios possible using medium-format
1461:
1327:
740:
1690:
2074:
The paramount principle in
Ptolemy's tunings was the use of superparticular proportion.
1955:
1921:
Leonhard Euler; translated into
English by Myra F. Wyman and Bostwick F. Wyman (1985),
1867:
1580:
713:
239:
1653:
1588:
1521:
1476:
1469:
1149:
992:
2130:
2094:
2063:
1994:
1966:
1959:
1901:
1828:
546:
2041:
2022:
1848:
Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music".
2156:
2036:
1945:
1937:
1859:
1645:
728:; that is, all ratios of this type in which both the numerator and denominator are
709:
207:
174:
2229:
2208:
1875:
1697:
1595:
1415:
1180:
235:
1893:
1550:
1510:
857:
250:
227:
106:
19:
2254:
2152:
1381:
1227:
829:
729:
725:
563:
231:
2014:
2234:
2224:
1734:
1430:
1423:
1335:
1266:
1022:
953:
748:
685:
567:
211:
1370:
913:
885:
693:
86:
1825:
Isidore of
Seville's Etymologies: Complete English Translation, Volume 1
2160:
2105:
1941:
1871:
197:
2129:, How to Photograph Series, vol. 9, Stackpole Books, p. 43,
1950:
570:
as its numerator and the nearest multiple of four as its denominator:
724:
can be used to list all possible superparticular numbers for a given
1863:
2177:
2118:
1184:
2219:
1726:
717:
712:
can be expressed as a superparticular ratio (for example, due to
705:
554:
in several ways as a product of superparticular ratios and their
801:
102:
1847:
720:'s formulation of musical harmony. In this application,
549:
206:. Although these numbers have applications in modern
1990:
The Legacy of
Leonhard Euler: A Tricentennial Tribute
579:
261:
121:
754:
735:These ratios are also important in visual harmony.
566:of superparticular ratios in which each term has a
673:
534:
161:
2126:How to Photograph the Outdoors in Black and White
2013:
1898:The Oxford Handbook of the History of Mathematics
1763:The root of some of these terms comes from Latin
162:{\displaystyle {\frac {n+1}{n}}=1+{\frac {1}{n}}}
2252:
1800:
1798:
1796:
1794:
1792:
1790:
1788:
743:, and aspect ratios of 7:6 and 5:4 are used in
2193:
2028:Bulletin of the American Mathematical Society
1892:
196:Superparticular ratios were written about by
112:More particularly, the ratio takes the form:
1971:: CS1 maint: multiple names: authors list (
1785:
2059:Tuning and Temperament: A Historical Survey
2200:
2186:
2062:, Courier Dover Publications, p. 23,
221:
2040:
1949:
18:
2055:
1986:
2253:
2122:
2181:
2168:De Institutione Arithmetica, liber II
699:
558:. It is also possible to convert the
2207:
1843:
1841:
2086:
2023:"On the structure of linear graphs"
13:
2173:Anicius Manlius Severinus Boethius
278:
14:
2277:
2146:
1993:, World Scientific, p. 214,
1923:"An essay on continued fractions"
1838:
1775:"and") describing the ratio 3:2.
755:Ratio names and related intervals
16:Ratio of two consecutive integers
1977:. See in particular p. 304.
1749:
1733:
1707:
1696:
1689:
1665:
1652:
1626:
1602:
1594:
1587:
1561:
1531:
1520:
1491:
1475:
1468:
1442:
1429:
1422:
1396:
1380:
1350:
1334:
1308:
1278:
1265:
1239:
1226:
1200:
1161:
1148:
1122:
1094:
1077:greater undecimal neutral second
1058:
1030:
1002:
991:
965:
952:
926:
893:
865:
837:
809:
2042:10.1090/S0002-9904-1946-08715-7
1113:lesser undecimal neutral second
2111:
2090:Digital Photography Essentials
2080:
2056:Barbour, James Murray (2004),
2049:
2007:
1980:
1914:
1896:; Stedall, Jacqueline (2008),
1886:
1817:
692:as the possible values of the
1:
1851:American Mathematical Monthly
739:of 4:3 and 3:2 are common in
23:Just diatonic semitone on C:
1811:
7:
2104:. Ang also notes the 16:9 (
1930:Mathematical Systems Theory
1900:, Oxford University Press,
1259:septimal chromatic semitone
10:
2282:
1987:Debnath, Lokenath (2010),
1823:Throop, Priscilla (2006).
1142:septimal diatonic semitone
751:photography respectively.
203:Introduction to Arithmetic
2266:Superparticular intervals
2215:
2093:, Penguin, p. 107,
1778:
2153:Superparticular numbers
2123:Schaub, George (1999),
1767:"one and a half" (from
1297:just chromatic semitone
1220:minor diatonic semitone
761:harmonic series (music)
222:Mathematical properties
1049:sesquinona: minor tone
778:Name/musical interval
696:of an infinite graph.
675:
536:
282:
216:history of mathematics
194:
163:
95:superparticular number
82:
2240:Superparticular ratio
2155:applied to construct
1827:, p. III.6.12, n. 7.
1375:inferior quarter tone
985:septimal major second
676:
560:Leibniz formula for π
537:
262:
183:
164:
91:superparticular ratio
22:
946:septimal minor third
577:
259:
119:
1462:septimal sixth-tone
1328:septimal third-tone
768:
741:digital photography
690:Erdős–Stone theorem
105:of two consecutive
1942:10.1007/bf01699475
1581:septimal semicomma
766:
714:octave equivalency
700:Other applications
671:
532:
240:continued fraction
234:) are exactly the
159:
83:
2248:
2247:
2157:pentatonic scales
2087:Ang, Tom (2011),
2035:(12): 1087–1091.
1858:(10): 1096–1100.
1833:978-1-4116-6523-1
1761:
1760:
1755:
1713:
1684:255th subharmonic
1671:
1632:
1621:127th subharmonic
1608:
1567:
1537:
1497:
1448:
1402:
1356:
1314:
1284:
1245:
1206:
1167:
1128:
1100:
1064:
1036:
1008:
971:
932:
899:
871:
843:
815:
722:Størmer's theorem
666:
653:
640:
627:
614:
601:
588:
547:irrational number
530:
514:
501:
488:
466:
453:
440:
424:
411:
398:
385:
372:
359:
341:
312:
157:
138:
2273:
2261:Rational numbers
2209:Rational numbers
2202:
2195:
2188:
2179:
2178:
2141:
2139:
2115:
2109:
2103:
2084:
2078:
2076:
2053:
2047:
2046:
2044:
2011:
2005:
2003:
1984:
1978:
1976:
1970:
1962:
1953:
1927:
1918:
1912:
1910:
1890:
1884:
1883:
1845:
1836:
1821:
1805:
1802:
1757:
1756:
1746:
1743:
1742:
1738:
1737:
1715:
1714:
1704:
1701:
1700:
1694:
1693:
1673:
1672:
1662:
1661:
1657:
1656:
1646:septimal kleisma
1634:
1633:
1610:
1609:
1599:
1598:
1592:
1591:
1569:
1568:
1558:
1539:
1538:
1528:
1525:
1524:
1515:63rd subharmonic
1499:
1498:
1488:
1485:
1484:
1480:
1479:
1473:
1472:
1450:
1449:
1439:
1438:
1434:
1433:
1427:
1426:
1404:
1403:
1393:
1390:
1389:
1385:
1384:
1358:
1357:
1347:
1344:
1343:
1339:
1338:
1316:
1315:
1305:
1304:
1286:
1285:
1275:
1274:
1270:
1269:
1247:
1246:
1236:
1235:
1231:
1230:
1208:
1207:
1197:
1194:
1193:
1169:
1168:
1158:
1157:
1153:
1152:
1130:
1129:
1119:
1102:
1101:
1091:
1088:
1087:
1083:
1066:
1065:
1055:
1038:
1037:
1010:
1009:
999:
996:
995:
973:
972:
962:
961:
957:
956:
934:
933:
923:
922:
901:
900:
873:
872:
845:
844:
817:
816:
769:
765:
704:In the study of
680:
678:
677:
672:
667:
659:
654:
646:
641:
633:
628:
620:
615:
607:
602:
594:
589:
581:
552:
541:
539:
538:
533:
531:
523:
515:
507:
502:
494:
489:
481:
467:
459:
454:
446:
441:
433:
425:
417:
412:
404:
399:
391:
386:
378:
373:
365:
360:
352:
347:
343:
342:
340:
326:
318:
313:
311:
297:
289:
281:
276:
236:rational numbers
208:pure mathematics
200:in his treatise
192:
175:positive integer
172:
168:
166:
165:
160:
158:
150:
139:
134:
123:
93:, also called a
81:
80:
79:
77:
70:
68:
67:
64:
61:
54:
52:
51:
48:
45:
38:
36:
35:
32:
29:
2281:
2280:
2276:
2275:
2274:
2272:
2271:
2270:
2251:
2250:
2249:
2244:
2230:Dyadic rational
2211:
2206:
2149:
2144:
2137:
2116:
2112:
2101:
2085:
2081:
2070:
2054:
2050:
2012:
2008:
2001:
1985:
1981:
1964:
1963:
1925:
1919:
1915:
1908:
1894:Robson, Eleanor
1891:
1887:
1864:10.2307/2317424
1846:
1839:
1822:
1818:
1814:
1809:
1808:
1803:
1786:
1781:
1750:
1744:
1740:
1739:
1732:
1708:
1702:
1695:
1688:
1666:
1659:
1658:
1651:
1627:
1603:
1593:
1586:
1562:
1556:
1532:
1526:
1519:
1514:
1492:
1486:
1482:
1481:
1474:
1467:
1443:
1436:
1435:
1428:
1421:
1416:septimal diesis
1397:
1391:
1387:
1386:
1379:
1374:
1351:
1345:
1341:
1340:
1333:
1309:
1302:
1301:
1279:
1272:
1271:
1264:
1240:
1233:
1232:
1225:
1201:
1195:
1191:
1190:
1162:
1155:
1154:
1147:
1123:
1117:
1095:
1089:
1085:
1084:
1081:
1059:
1053:
1031:
1021:sesquioctavum:
1003:
997:
990:
966:
959:
958:
951:
927:
920:
919:
912:sesquiquintum:
894:
884:sesquiquartum:
866:
856:sesquitertium:
838:
828:sesquialterum:
810:
783:
757:
708:, many musical
702:
658:
645:
632:
619:
606:
593:
580:
578:
575:
574:
550:
545:represents the
522:
506:
493:
480:
458:
445:
432:
416:
403:
390:
377:
364:
351:
327:
319:
317:
298:
290:
288:
287:
283:
277:
266:
260:
257:
256:
224:
193:
191:Throop (2006),
190:
170:
149:
124:
122:
120:
117:
116:
107:integer numbers
75:
72:
71:
65:
62:
59:
58:
56:
49:
46:
43:
42:
40:
33:
30:
27:
26:
24:
17:
12:
11:
5:
2279:
2269:
2268:
2263:
2246:
2245:
2243:
2242:
2237:
2232:
2227:
2222:
2216:
2213:
2212:
2205:
2204:
2197:
2190:
2182:
2176:
2175:
2164:
2161:David Canright
2148:
2147:External links
2145:
2143:
2142:
2135:
2110:
2099:
2079:
2068:
2048:
2006:
1999:
1979:
1913:
1906:
1885:
1837:
1815:
1813:
1810:
1807:
1806:
1783:
1782:
1780:
1777:
1759:
1758:
1747:
1729:
1724:
1721:
1717:
1716:
1705:
1685:
1682:
1679:
1675:
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1663:
1648:
1643:
1640:
1636:
1635:
1624:
1622:
1619:
1616:
1612:
1611:
1600:
1583:
1578:
1575:
1571:
1570:
1559:
1553:
1551:syntonic comma
1548:
1545:
1541:
1540:
1529:
1516:
1511:septimal comma
1508:
1505:
1501:
1500:
1489:
1464:
1459:
1456:
1452:
1451:
1440:
1418:
1413:
1410:
1406:
1405:
1394:
1376:
1367:
1364:
1360:
1359:
1348:
1330:
1325:
1322:
1318:
1317:
1306:
1298:
1295:
1292:
1288:
1287:
1276:
1261:
1256:
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1249:
1248:
1237:
1222:
1217:
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1209:
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1159:
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1136:
1132:
1131:
1120:
1114:
1111:
1108:
1104:
1103:
1092:
1078:
1075:
1072:
1068:
1067:
1056:
1050:
1047:
1044:
1040:
1039:
1028:
1025:
1019:
1016:
1012:
1011:
1000:
987:
982:
979:
975:
974:
963:
948:
943:
940:
936:
935:
924:
916:
910:
907:
903:
902:
891:
888:
882:
879:
875:
874:
863:
860:
858:perfect fourth
854:
851:
847:
846:
835:
832:
826:
823:
819:
818:
807:
804:
798:
795:
791:
790:
787:
779:
776:
773:
756:
753:
730:smooth numbers
701:
698:
682:
681:
670:
665:
662:
657:
652:
649:
644:
639:
636:
631:
626:
623:
618:
613:
610:
605:
600:
597:
592:
587:
584:
543:
542:
529:
526:
521:
518:
513:
510:
505:
500:
497:
492:
487:
484:
479:
476:
473:
470:
465:
462:
457:
452:
449:
444:
439:
436:
431:
428:
423:
420:
415:
410:
407:
402:
397:
394:
389:
384:
381:
376:
371:
368:
363:
358:
355:
350:
346:
339:
336:
333:
330:
325:
322:
316:
310:
307:
304:
301:
296:
293:
286:
280:
275:
272:
269:
265:
251:Wallis product
228:Leonhard Euler
223:
220:
188:
179:
178:
156:
153:
148:
145:
142:
137:
133:
130:
127:
99:epimoric ratio
15:
9:
6:
4:
3:
2:
2278:
2267:
2264:
2262:
2259:
2258:
2256:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2217:
2214:
2210:
2203:
2198:
2196:
2191:
2189:
2184:
2183:
2180:
2174:
2170:
2169:
2165:
2162:
2158:
2154:
2151:
2150:
2138:
2136:9780811724500
2132:
2128:
2127:
2120:
2114:
2107:
2102:
2100:9780756685263
2096:
2092:
2091:
2083:
2075:
2071:
2069:9780486434063
2065:
2061:
2060:
2052:
2043:
2038:
2034:
2030:
2029:
2024:
2020:
2016:
2010:
2002:
2000:9781848165267
1996:
1992:
1991:
1983:
1974:
1968:
1961:
1957:
1952:
1947:
1943:
1939:
1935:
1931:
1924:
1917:
1909:
1907:9780191607448
1903:
1899:
1895:
1889:
1881:
1877:
1873:
1869:
1865:
1861:
1857:
1853:
1852:
1844:
1842:
1834:
1830:
1826:
1820:
1816:
1801:
1799:
1797:
1795:
1793:
1791:
1789:
1784:
1776:
1774:
1771:"a half" and
1770:
1766:
1748:
1736:
1730:
1728:
1725:
1722:
1719:
1718:
1706:
1699:
1692:
1686:
1683:
1680:
1677:
1676:
1664:
1655:
1649:
1647:
1644:
1641:
1638:
1637:
1625:
1623:
1620:
1617:
1614:
1613:
1601:
1597:
1590:
1584:
1582:
1579:
1576:
1573:
1572:
1560:
1554:
1552:
1549:
1546:
1543:
1542:
1530:
1523:
1517:
1512:
1509:
1506:
1503:
1502:
1490:
1478:
1471:
1465:
1463:
1460:
1457:
1454:
1453:
1441:
1432:
1425:
1419:
1417:
1414:
1411:
1408:
1407:
1395:
1383:
1377:
1372:
1368:
1365:
1362:
1361:
1349:
1337:
1331:
1329:
1326:
1323:
1320:
1319:
1307:
1299:
1296:
1293:
1290:
1289:
1277:
1268:
1262:
1260:
1257:
1254:
1251:
1250:
1238:
1229:
1223:
1221:
1218:
1215:
1212:
1211:
1199:
1188:
1186:
1182:
1179:
1176:
1173:
1172:
1160:
1151:
1145:
1143:
1140:
1137:
1134:
1133:
1121:
1115:
1112:
1109:
1106:
1105:
1093:
1079:
1076:
1073:
1070:
1069:
1057:
1051:
1048:
1045:
1042:
1041:
1029:
1026:
1024:
1020:
1017:
1014:
1013:
1001:
994:
988:
986:
983:
980:
977:
976:
964:
955:
949:
947:
944:
941:
938:
937:
925:
917:
915:
911:
908:
905:
904:
892:
889:
887:
883:
880:
877:
876:
864:
861:
859:
855:
852:
849:
848:
836:
833:
831:
830:perfect fifth
827:
824:
821:
820:
808:
805:
803:
799:
796:
793:
792:
788:
785:
780:
777:
774:
771:
770:
764:
762:
752:
750:
746:
745:medium format
742:
738:
737:Aspect ratios
733:
731:
727:
723:
719:
715:
711:
707:
697:
695:
694:upper density
691:
687:
668:
663:
660:
655:
650:
647:
642:
637:
634:
629:
624:
621:
616:
611:
608:
603:
598:
595:
590:
585:
582:
573:
572:
571:
569:
565:
564:Euler product
561:
557:
553:
548:
527:
524:
519:
516:
511:
508:
503:
498:
495:
490:
485:
482:
477:
474:
471:
468:
463:
460:
455:
450:
447:
442:
437:
434:
429:
426:
421:
418:
413:
408:
405:
400:
395:
392:
387:
382:
379:
374:
369:
366:
361:
356:
353:
348:
344:
337:
334:
331:
328:
323:
320:
314:
308:
305:
302:
299:
294:
291:
284:
273:
270:
267:
263:
255:
254:
253:
252:
247:
245:
244:superpartient
241:
237:
233:
232:unit fraction
229:
219:
217:
213:
209:
205:
204:
199:
187:
182:
176:
154:
151:
146:
143:
140:
135:
131:
128:
125:
115:
114:
113:
110:
108:
104:
100:
96:
92:
88:
78:
21:
2239:
2235:Half-integer
2225:Dedekind cut
2167:
2125:
2113:
2089:
2082:
2073:
2058:
2051:
2032:
2026:
2019:Stone, A. H.
2009:
1989:
1982:
1933:
1929:
1916:
1897:
1888:
1855:
1849:
1824:
1819:
1804:Ancient name
1772:
1768:
1764:
1762:
1023:major second
782:Ben Johnston
758:
749:large format
734:
703:
686:graph theory
683:
568:prime number
544:
248:
225:
212:music theory
201:
195:
184:
180:
111:
98:
94:
90:
84:
1936:: 295–328,
1371:subharmonic
914:minor third
886:major third
87:mathematics
2255:Categories
2106:widescreen
1951:1811/32133
198:Nicomachus
2015:Erdős, P.
1960:126941824
1812:Citations
1720:4375:4374
1183:diatonic
767:Examples
710:intervals
669:⋯
656:⋅
643:⋅
630:⋅
617:⋅
604:⋅
583:π
525:π
517:⋯
504:⋅
491:⋅
478:⋅
469:⋯
456:⋅
443:⋅
427:⋯
414:⋅
401:⋅
388:⋅
375:⋅
362:⋅
315:⋅
306:−
279:∞
264:∏
101:, is the
2119:120 film
2021:(1946).
1967:citation
1741:♯
1660:♯
1483:♯
1437:♭
1388:♭
1342:♭
1303:♯
1273:♭
1234:♯
1192:♭
1185:semitone
1156:♯
1086:♭
960:♭
921:♭
800:duplex:
786:above C
784:notation
562:into an
556:inverses
214:and the
189:—
2220:Integer
1880:0313189
1872:2317424
1765:sesqui-
1727:ragisma
1678:256:255
1639:225:224
1615:128:127
1574:126:125
718:Ptolemy
706:harmony
69:
57:
53:
41:
37:
25:
2133:
2097:
2066:
1997:
1958:
1904:
1878:
1870:
1831:
1216:104.96
1177:111.73
1138:119.44
1110:150.64
1074:165.00
1046:182.40
1018:203.91
981:231.17
942:266.87
909:315.64
881:386.31
853:498.04
825:701.96
802:octave
789:Audio
775:Cents
772:Ratio
238:whose
181:Thus:
169:where
55:= 1 +
44:15 + 1
1956:S2CID
1926:(PDF)
1868:JSTOR
1779:Notes
1769:semis
1618:13.58
1577:13.79
1547:21.51
1544:81:80
1507:27.26
1504:64:63
1458:34.98
1455:50:49
1412:35.70
1409:49:48
1369:31st
1366:54.96
1363:32:31
1324:62.96
1321:28:27
1294:70.67
1291:25:24
1255:84.47
1252:21:20
1213:17:16
1174:16:15
1135:15:14
1107:12:11
1071:11:10
726:limit
173:is a
103:ratio
2131:ISBN
2095:ISBN
2064:ISBN
1995:ISBN
1973:link
1902:ISBN
1829:ISBN
1773:-que
1723:0.40
1681:6.78
1642:7.71
1181:just
1043:10:9
797:1200
747:and
249:The
89:, a
76:Play
2171:by
2159:by
2037:doi
1946:hdl
1938:doi
1860:doi
1015:9:8
978:8:7
939:7:6
906:6:5
878:5:4
850:4:3
822:3:2
794:2:1
684:In
226:As
97:or
85:In
2257::
2072:,
2033:52
2031:.
2025:.
2017:;
1969:}}
1965:{{
1954:,
1944:,
1934:18
1932:,
1928:,
1876:MR
1874:.
1866:.
1856:79
1854:.
1840:^
1787:^
806:C'
732:.
664:16
661:17
651:12
648:13
638:12
635:11
512:49
509:48
499:25
496:24
464:35
461:36
451:15
448:16
246:.
218:.
109:.
66:15
50:15
39:=
34:15
28:16
2201:e
2194:t
2187:v
2163:.
2140:.
2077:.
2045:.
2039::
2004:.
1975:)
1948::
1940::
1882:.
1862::
1835:.
1745:-
1731:C
1703:-
1687:D
1650:B
1585:D
1557:+
1555:C
1527:-
1518:C
1513:,
1487:-
1466:B
1420:D
1392:-
1378:D
1373:,
1346:-
1332:D
1300:C
1263:D
1224:C
1196:-
1189:D
1146:C
1118:↓
1116:D
1090:-
1082:↑
1080:D
1054:-
1052:D
1027:D
998:-
989:D
950:E
918:E
890:E
862:F
834:G
625:8
622:7
612:4
609:5
599:4
596:3
591:=
586:4
551:π
528:2
520:=
486:9
483:8
475:2
472:=
438:3
435:4
430:=
422:7
419:6
409:5
406:6
396:5
393:4
383:3
380:4
370:3
367:2
357:1
354:2
349:=
345:)
338:1
335:+
332:n
329:2
324:n
321:2
309:1
303:n
300:2
295:n
292:2
285:(
274:1
271:=
268:n
177:.
171:n
155:n
152:1
147:+
144:1
141:=
136:n
132:1
129:+
126:n
63:/
60:1
47:/
31:/
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