842:
36:
28:
173:
20:
860:
The scale "contains a network of harmonic relationships with the property to match harmonic interval cycles of 833 cents." Steps 2 and 5 were presumably chosen to fill in the gaps between what become steps 1 and 3 and 4 and 6 (267.64 cents). The value of step 2 (235.77) was chosen to create a perfect
392:
Hz produce combination tones at 220 Hz and 660 Hz. 660 Hz is a perfect fifth (3:2) above 440 Hz, and produce combination tones at 220 Hz and 1,100 Hz. 1,100 Hz is a major sixth (5:3) above 660 Hz, and produce combination tones at 440 Hz and 1,760 Hz.
861:
twelfth (compound perfect fifth) between steps 16 (235.77+833.09+833.09) and step 0, and once chosen determined the value of step 5 due to the symmetry of the scale. Step 10 and 0 form an octave. All notes 7 steps apart form the golden ratio with each other, for example 16 & 9 and 10 & 3.
864:
The repetition of frequencies and the coincidence of higher steps with consonances such as the perfect fifth and octave may be seen (the step number of intervals that coincide with the stack of golden ratios are in bold, while the ratios of repeated intervals are in bold):
830:
2315:
802:
774:
278:
829:
2237:
2156:
2071:
1988:
1879:
215:
1714:
1637:
1556:
1476:
1397:
1288:
1207:
1124:
1047:
257:
236:
184:
to a value close to 833 cents. That means nothing else than that for instance for an interval of 144:89 (833.11 cents) both the summation and the difference tone appear...again 833 cents distance from this interval".
812:
591:
134:
1796:
801:
773:
140:
572:
there are three infinite stacks of possible golden ratios which never return exactly back to the unison or octave. Scale step 5 is 597.32 cents and scale step -5 is 602.68 cents (5.37 cents apart).
964:
393:
1100 Hz and 1760 Hz are a minor sixth (8:5), and so on. "It is by the way unimportant which interval we choose as a starting point for the above exercise; the result is always 833 cent."
784:
756:
180:
Starting with any interval, take the interval produced by the highest original tone and the closest combination tone. Then do the same for that interval. These intervals "
2484:
2300:
2277:
2222:
2199:
2137:
2114:
2052:
2030:
1969:
1945:
1918:
1860:
1837:
1777:
1755:
1699:
1677:
1622:
1599:
1541:
1519:
1461:
1439:
1378:
1354:
1327:
1269:
1246:
1188:
1166:
1109:
1087:
1028:
1005:
943:
921:
719:
160:
interval (notes 833.09 cents apart are not "the same" in the 833 cents scale the way notes 1200 cents apart are in traditional tunings). Other music theorists such as
2595:
831:
164:, in his 1993 "The Tonality of the Golden Section", and Lorne Temes in 1970, appear to have also created this scale prior to Bohlen's discovery of it.
3027:
2527:
2660:
803:
775:
82:
2670:
388:
and 220 Hz (unison) produce combination tones at 0 and 440 Hz. 440 Hz is an octave above 220 Hz. 220 Hz and
2467:
2633:
841:
2818:
2520:
2756:
2600:
2337:
The scale contains .83333 × 12 steps per octave (≈10). While ideally untempered, the scale may be approximated by
2585:
157:
2869:
35:
3005:
176:
Lattice with golden ratios on the unison (center), on the perfect fourth (left), and the perfect fifth (right)
564:
Two stacks are also produced on 3:2 and its inverse 4:3 to provide steps 2 and 5, creating a two dimensional
2839:
2665:
2590:
2513:
2623:
27:
2861:
2824:
2814:
161:
153:
3032:
2857:
389:
172:
587:
scale, with the pitches of steps 0, 1, 3, 4, & 6 derived from the stack of golden ratio intervals.
31:
Stack of golden ratio (φ) intervals, measured in Hz ((11.09 + 6.854) ÷ 11.09 = 11.09 ÷ 6.854 = 1.618).
2865:
2810:
2792:
2787:
2782:
2777:
2772:
2767:
2762:
2747:
2742:
2737:
2732:
2727:
2722:
2338:
854:
2829:
2802:
2700:
181:
2894:
2680:
2675:
2899:
2873:
2834:
2709:
2648:
2463:
2285:
2262:
2207:
2184:
2122:
2099:
2037:
2015:
1954:
1930:
1903:
1845:
1822:
1762:
1740:
1684:
1662:
1607:
1584:
1526:
1504:
1446:
1424:
1363:
1339:
1312:
1254:
1231:
1173:
1151:
1094:
1072:
1013:
990:
928:
906:
704:
569:
2853:
2717:
2685:
2605:
2570:
850:
584:
72:
64:
60:
2615:
2350:
581:
565:
2653:
2628:
2575:
2536:
2406:
48:
3021:
2948:
2917:
2690:
2643:
2550:
2355:
52:
2380:
2912:
2752:
2555:
76:
68:
56:
845:
Three stacks of 13 golden ratios on the unison, perfect fourth, and perfect fifth
828:
800:
772:
3000:
2995:
2985:
2580:
2560:
19:
2990:
2959:
396:
Once the interval of 833.09 cents is determined, a stack of them is produced:
2980:
2934:
148:). In the 833 cents scale this interval is taken as an alternative to the
129:{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .}
39:
833 cents scale in 36-tet notation, with slurs indicating a golden ratio
2954:
2904:
2440:
2922:
2638:
2565:
2495:
149:
2423:
2939:
2927:
2505:
840:
385:
171:
18:
2944:
23:
Line segments in the golden ratio ((A + B) / A = A / B = 1.618).
2509:
2341:, one advantage being that 36-TET includes traditional 12-TET.
849:
This is comparable to the derivation of the major scale from a
2485:
Fun with
Emulator X: Bohlen 833 cents scale and harmonics
2288:
2265:
2210:
2187:
2125:
2102:
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2018:
1957:
1933:
1906:
1848:
1825:
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85:
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713:
128:
2454:
2452:
156:, however the golden ratio is not regarded as an
136:, which as a musical interval is 833.09 cents (
2521:
8:
2596:List of intervals in 5-limit just intonation
2381:An 833 Cents Scale: An experiment on harmony
2375:
2373:
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2705:
2528:
2514:
2506:
2287:
2264:
2209:
2186:
2124:
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1956:
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101:
92:
84:
867:
600:
398:
187:
34:
26:
2367:
2401:
2399:
2397:
2395:
2393:
7:
2379:Bohlen, Heinz (last updated 2012). "
815:
787:
759:
568:. Given that the golden ratio is an
2424:The Tonality of the Golden Section
14:
2671:Ptolemy's intense diatonic scale
2443:", University of Toronto, 1970.
826:
798:
770:
2407:833 Cent Golden Scale (Bohlen)
1:
813:Tetrachord, steps: 0, 3, 4, 7
2634:Harry Partch's 43-tone scale
3028:Non–octave-repeating scales
2422:O'Connell, Walter (1993). "
71:, and, coincidentally, the
3049:
2601:List of meantone intervals
2460:On Musical Self-Similarity
853:(FCGDAEB = CDEFGAB). See:
2591:List of musical intervals
2586:Consonance and dissonance
2458:Pareyon, Gabriel (2011).
2330:
2311:
2308:
2305:
2256:
2253:
2248:
2233:
2230:
2227:
2179:
2174:
2167:
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2142:
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1999:
1984:
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1978:
1898:
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1710:
1707:
1704:
1656:
1653:
1648:
1633:
1630:
1627:
1579:
1574:
1567:
1552:
1549:
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1499:
1494:
1487:
1472:
1469:
1466:
1418:
1413:
1408:
1393:
1390:
1387:
1307:
1304:
1299:
1284:
1279:
1274:
1226:
1223:
1218:
1203:
1198:
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1145:
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1135:
1120:
1117:
1114:
1066:
1063:
1058:
1043:
1038:
1033:
985:
982:
975:
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953:
948:
901:
896:
745:
742:
739:
737:
734:
731:
729:
726:
723:
693:
690:
687:
684:
681:
678:
675:
672:
662:
659:
656:
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650:
647:
644:
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638:
630:
627:
624:
621:
618:
615:
612:
609:
606:
377:
2295:{\displaystyle \varphi }
2272:{\displaystyle \varphi }
2217:{\displaystyle \varphi }
2194:{\displaystyle \varphi }
2132:{\displaystyle \varphi }
2109:{\displaystyle \varphi }
2047:{\displaystyle \varphi }
2025:{\displaystyle \varphi }
1964:{\displaystyle \varphi }
1940:{\displaystyle \varphi }
1913:{\displaystyle \varphi }
1855:{\displaystyle \varphi }
1832:{\displaystyle \varphi }
1772:{\displaystyle \varphi }
1750:{\displaystyle \varphi }
1694:{\displaystyle \varphi }
1672:{\displaystyle \varphi }
1617:{\displaystyle \varphi }
1594:{\displaystyle \varphi }
1536:{\displaystyle \varphi }
1514:{\displaystyle \varphi }
1456:{\displaystyle \varphi }
1434:{\displaystyle \varphi }
1373:{\displaystyle \varphi }
1349:{\displaystyle \varphi }
1322:{\displaystyle \varphi }
1264:{\displaystyle \varphi }
1241:{\displaystyle \varphi }
1183:{\displaystyle \varphi }
1161:{\displaystyle \varphi }
1104:{\displaystyle \varphi }
1082:{\displaystyle \varphi }
1023:{\displaystyle \varphi }
1000:{\displaystyle \varphi }
938:{\displaystyle \varphi }
916:{\displaystyle \varphi }
785:Trichord, steps: 0, 4, 7
757:Trichord, steps: 0, 3, 7
714:{\displaystyle \varphi }
16:Musical tuning and scale
851:stack of perfect fifths
2296:
2273:
2218:
2195:
2133:
2110:
2048:
2026:
1965:
1941:
1914:
1856:
1833:
1773:
1751:
1695:
1673:
1618:
1595:
1537:
1515:
1457:
1435:
1374:
1350:
1323:
1265:
1242:
1184:
1162:
1105:
1083:
1024:
1001:
939:
917:
846:
715:
177:
130:
40:
32:
24:
2858:Temperament ordinaire
2297:
2274:
2219:
2196:
2134:
2111:
2049:
2027:
1966:
1942:
1915:
1857:
1834:
1774:
1752:
1696:
1674:
1619:
1596:
1538:
1516:
1458:
1436:
1375:
1351:
1324:
1266:
1243:
1185:
1163:
1106:
1084:
1025:
1002:
940:
918:
844:
716:
175:
131:
38:
30:
22:
2661:List of compositions
2339:36 equal temperament
2286:
2263:
2208:
2185:
2123:
2100:
2038:
2016:
1955:
1931:
1904:
1846:
1823:
1763:
1741:
1685:
1663:
1608:
1585:
1527:
1505:
1447:
1425:
1364:
1340:
1313:
1255:
1232:
1174:
1152:
1095:
1073:
1014:
991:
929:
907:
855:Generated collection
705:
83:
2356:Zipf's distribution
580:Bohlen describes a
199:Closest combination
194:Closest combination
152:as the interval of
2895:Chinese musicology
2681:Scale of harmonics
2676:Pythagorean tuning
2624:Euler–Fokker genus
2385:Huygens-Fokker.org
2292:
2269:
2214:
2191:
2129:
2106:
2044:
2022:
1961:
1937:
1910:
1852:
1829:
1769:
1747:
1691:
1669:
1614:
1591:
1533:
1511:
1453:
1431:
1370:
1346:
1319:
1261:
1238:
1180:
1158:
1101:
1079:
1020:
997:
935:
913:
847:
711:
178:
126:
73:Fibonacci sequence
41:
33:
25:
3015:
3014:
2881:
2880:
2335:
2334:
832:
804:
776:
750:
749:
570:irrational number
562:
561:
384:For example, 220
382:
381:
112:
106:
61:combination tones
3040:
2854:Well temperament
2840:Regular diatonic
2706:
2686:Tonality diamond
2530:
2523:
2516:
2507:
2500:Xenharmonic Wiki
2471:
2456:
2447:
2437:
2431:
2420:
2414:
2411:Xenharmonic Wiki
2403:
2388:
2377:
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2321:
2320:
2318:
2303:
2302:
2301:
2299:
2298:
2293:
2279:
2278:
2276:
2275:
2270:
2244:
2243:
2242:
2240:
2225:
2224:
2223:
2221:
2220:
2215:
2201:
2200:
2198:
2197:
2192:
2170:
2163:
2162:
2161:
2159:
2140:
2139:
2138:
2136:
2135:
2130:
2116:
2115:
2113:
2112:
2107:
2085:
2078:
2077:
2076:
2074:
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2054:
2053:
2051:
2050:
2045:
2032:
2031:
2029:
2028:
2023:
1995:
1994:
1993:
1991:
1976:
1975:
1974:
1972:
1971:
1970:
1968:
1967:
1962:
1950:
1947:
1946:
1944:
1943:
1938:
1920:
1919:
1917:
1916:
1911:
1886:
1885:
1884:
1882:
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1862:
1861:
1859:
1858:
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1839:
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1597:
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1559:
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1517:
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1490:
1483:
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1479:
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1463:
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1460:
1459:
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1437:
1432:
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1376:
1371:
1359:
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1329:
1328:
1326:
1325:
1320:
1295:
1294:
1293:
1291:
1272:
1271:
1270:
1268:
1267:
1262:
1248:
1247:
1245:
1244:
1239:
1214:
1213:
1212:
1210:
1191:
1190:
1189:
1187:
1186:
1181:
1168:
1167:
1165:
1164:
1159:
1131:
1130:
1129:
1127:
1112:
1111:
1110:
1108:
1107:
1102:
1089:
1088:
1086:
1085:
1080:
1054:
1053:
1052:
1050:
1031:
1030:
1029:
1027:
1026:
1021:
1007:
1006:
1004:
1003:
998:
978:
971:
970:
969:
967:
946:
945:
944:
942:
941:
936:
923:
922:
920:
919:
914:
868:
834:
833:
822:
818:
806:
805:
794:
790:
778:
777:
766:
762:
720:
718:
717:
712:
601:
598:
597:
596:
594:
399:
285:
284:
283:
281:
264:
263:
262:
260:
243:
242:
241:
239:
222:
221:
220:
218:
188:
162:Walter O'Connell
147:
146:
145:
143:
135:
133:
132:
127:
113:
108:
107:
102:
93:
3048:
3047:
3043:
3042:
3041:
3039:
3038:
3037:
3018:
3017:
3016:
3011:
3008:(Bohlen–Pierce)
2976:833 cents scale
2964:
2887:
2877:
2844:
2797:
2695:
2616:Just intonation
2610:
2539:
2537:Musical tunings
2534:
2480:
2475:
2474:
2457:
2450:
2439:Temes, Lorne, "
2438:
2434:
2421:
2417:
2404:
2391:
2378:
2369:
2364:
2351:Kepler triangle
2347:
2316:
2314:
2313:
2312:
2284:
2283:
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2260:
2258:
2257:
2238:
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2206:
2205:
2203:
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2182:
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2180:
2168:
2157:
2155:
2154:
2153:
2121:
2120:
2118:
2098:
2097:
2096:
2095:
2083:
2072:
2070:
2069:
2068:
2036:
2035:
2034:
2014:
2013:
2011:
2010:
1989:
1987:
1986:
1985:
1953:
1952:
1951:
1948:
1929:
1928:
1926:
1925:
1923:
1922:
1902:
1901:
1900:
1899:
1880:
1878:
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1876:
1844:
1843:
1841:
1821:
1820:
1819:
1818:
1797:
1795:
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1793:
1761:
1760:
1759:
1739:
1738:
1737:
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1713:
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1711:
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1681:
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1638:
1636:
1635:
1634:
1606:
1605:
1603:
1583:
1582:
1581:
1580:
1568:
1557:
1555:
1554:
1553:
1525:
1524:
1523:
1503:
1502:
1501:
1500:
1488:
1477:
1475:
1474:
1473:
1445:
1444:
1443:
1423:
1422:
1420:
1419:
1398:
1396:
1395:
1394:
1362:
1361:
1360:
1357:
1338:
1337:
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1334:
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1331:
1311:
1310:
1309:
1308:
1289:
1287:
1286:
1285:
1253:
1252:
1250:
1230:
1229:
1228:
1227:
1208:
1206:
1205:
1204:
1172:
1171:
1170:
1150:
1149:
1147:
1146:
1125:
1123:
1122:
1121:
1093:
1092:
1091:
1071:
1070:
1068:
1067:
1048:
1046:
1045:
1044:
1012:
1011:
1009:
989:
988:
987:
986:
976:
965:
963:
962:
961:
927:
926:
925:
905:
904:
903:
902:
888:
883:
878:
839:
838:
837:
836:
835:
827:
824:
820:
816:
809:
808:
807:
799:
796:
792:
788:
781:
780:
779:
771:
768:
764:
760:
703:
702:
592:
590:
589:
588:
578:
279:
277:
276:
275:
258:
256:
255:
254:
237:
235:
234:
233:
216:
214:
213:
212:
200:
195:
170:
141:
139:
138:
137:
94:
81:
80:
45:833 cents scale
17:
12:
11:
5:
3046:
3044:
3036:
3035:
3030:
3020:
3019:
3013:
3012:
3010:
3009:
3003:
2998:
2993:
2988:
2983:
2978:
2972:
2970:
2966:
2965:
2963:
2962:
2957:
2952:
2942:
2937:
2932:
2931:
2930:
2925:
2920:
2915:
2907:
2902:
2897:
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2889:
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2882:
2879:
2878:
2852:
2850:
2846:
2845:
2843:
2842:
2837:
2832:
2827:
2822:
2807:
2805:
2799:
2798:
2796:
2795:
2790:
2785:
2780:
2775:
2770:
2765:
2760:
2750:
2745:
2740:
2735:
2730:
2725:
2720:
2714:
2712:
2703:
2697:
2696:
2694:
2693:
2688:
2683:
2678:
2673:
2668:
2663:
2658:
2657:
2656:
2651:
2641:
2636:
2631:
2629:Harmonic scale
2626:
2620:
2618:
2612:
2611:
2609:
2608:
2603:
2598:
2593:
2588:
2583:
2578:
2576:Interval ratio
2573:
2568:
2563:
2558:
2553:
2547:
2545:
2541:
2540:
2535:
2533:
2532:
2525:
2518:
2510:
2504:
2503:
2492:
2479:
2478:External links
2476:
2473:
2472:
2448:
2432:
2415:
2389:
2366:
2365:
2363:
2360:
2359:
2358:
2353:
2346:
2343:
2333:
2332:
2328:
2327:
2324:
2323:
2310:
2307:
2304:
2291:
2268:
2255:
2251:
2250:
2246:
2245:
2232:
2229:
2226:
2213:
2190:
2178:
2172:
2171:
2165:
2164:
2151:
2146:
2141:
2128:
2105:
2093:
2087:
2086:
2080:
2079:
2066:
2061:
2056:
2043:
2021:
2008:
2002:
2001:
1997:
1996:
1983:
1980:
1977:
1960:
1936:
1909:
1897:
1893:
1892:
1888:
1887:
1874:
1869:
1864:
1851:
1828:
1816:
1810:
1809:
1805:
1804:
1791:
1786:
1781:
1768:
1746:
1734:
1728:
1727:
1723:
1722:
1709:
1706:
1703:
1690:
1668:
1655:
1651:
1650:
1646:
1645:
1632:
1629:
1626:
1613:
1590:
1578:
1572:
1571:
1565:
1564:
1551:
1548:
1545:
1532:
1510:
1498:
1492:
1491:
1485:
1484:
1471:
1468:
1465:
1452:
1430:
1417:
1411:
1410:
1406:
1405:
1392:
1389:
1386:
1369:
1345:
1318:
1306:
1302:
1301:
1297:
1296:
1283:
1278:
1273:
1260:
1237:
1225:
1221:
1220:
1216:
1215:
1202:
1197:
1192:
1179:
1157:
1144:
1138:
1137:
1133:
1132:
1119:
1116:
1113:
1100:
1078:
1065:
1061:
1060:
1056:
1055:
1042:
1037:
1032:
1019:
996:
984:
980:
979:
973:
972:
959:
957:
952:
947:
934:
912:
900:
894:
893:
890:
885:
880:
875:
872:
825:
811:
810:
797:
783:
782:
769:
755:
754:
753:
752:
751:
748:
747:
744:
741:
738:
736:
733:
730:
728:
725:
722:
710:
698:
697:
695:
692:
689:
686:
683:
680:
677:
674:
671:
669:
665:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
633:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
577:
574:
560:
559:
556:
553:
550:
547:
544:
541:
538:
535:
533:
531:
529:
525:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
487:
486:
483:
480:
475:
470:
465:
460:
455:
450:
447:
444:
441:
437:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
380:
379:
375:
374:
371:
368:
364:
363:
360:
357:
353:
352:
349:
346:
342:
341:
338:
335:
331:
330:
327:
324:
320:
319:
316:
313:
309:
308:
305:
302:
298:
297:
294:
291:
287:
286:
273:
270:
266:
265:
252:
249:
245:
244:
231:
228:
224:
223:
210:
207:
203:
202:
197:
192:
191:Base interval
169:
166:
125:
122:
119:
116:
111:
105:
100:
97:
91:
88:
49:musical tuning
15:
13:
10:
9:
6:
4:
3:
2:
3045:
3034:
3031:
3029:
3026:
3025:
3023:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2973:
2971:
2967:
2961:
2958:
2956:
2953:
2950:
2949:Carnatic raga
2946:
2943:
2941:
2938:
2936:
2933:
2929:
2926:
2924:
2921:
2919:
2918:Turkish makam
2916:
2914:
2911:
2910:
2908:
2906:
2903:
2901:
2898:
2896:
2893:
2892:
2890:
2884:
2875:
2871:
2867:
2863:
2859:
2855:
2851:
2847:
2841:
2838:
2836:
2833:
2831:
2828:
2826:
2823:
2820:
2816:
2815:quarter-comma
2812:
2809:
2808:
2806:
2804:
2800:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2758:
2754:
2751:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2715:
2713:
2711:
2707:
2704:
2702:
2698:
2692:
2691:Tonality flux
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2655:
2652:
2650:
2647:
2646:
2645:
2642:
2640:
2637:
2635:
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2630:
2627:
2625:
2622:
2621:
2619:
2617:
2613:
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2597:
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2579:
2577:
2574:
2572:
2569:
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2564:
2562:
2559:
2557:
2554:
2552:
2549:
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2538:
2531:
2526:
2524:
2519:
2517:
2512:
2511:
2508:
2501:
2497:
2493:
2490:
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2482:
2481:
2477:
2469:
2468:9789525431322
2465:
2461:
2455:
2453:
2449:
2446:
2445:Anaphoria.com
2442:
2436:
2433:
2429:
2428:Anaphoria.com
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2408:
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2400:
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2342:
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2329:
2326:
2325:
2319:
2289:
2266:
2252:
2247:
2241:
2211:
2188:
2177:
2173:
2166:
2160:
2150:
2145:
2126:
2103:
2092:
2088:
2081:
2075:
2065:
2060:
2041:
2019:
2007:
2003:
1998:
1992:
1958:
1934:
1907:
1894:
1889:
1883:
1873:
1868:
1849:
1826:
1815:
1811:
1806:
1800:
1790:
1785:
1766:
1744:
1733:
1729:
1724:
1718:
1688:
1666:
1652:
1647:
1641:
1611:
1588:
1577:
1573:
1566:
1560:
1530:
1508:
1497:
1493:
1486:
1480:
1450:
1428:
1416:
1412:
1407:
1401:
1367:
1343:
1316:
1303:
1298:
1292:
1282:
1277:
1258:
1235:
1222:
1217:
1211:
1201:
1196:
1177:
1155:
1143:
1139:
1134:
1128:
1098:
1076:
1062:
1057:
1051:
1041:
1036:
1017:
994:
981:
974:
968:
958:
956:
951:
932:
910:
899:
895:
891:
886:
881:
876:
873:
870:
869:
866:
862:
858:
856:
852:
843:
823:
814:
795:
786:
767:
758:
708:
700:
699:
696:
670:
667:
666:
635:
634:
603:
602:
599:
595:
586:
583:
575:
573:
571:
567:
557:
554:
551:
548:
545:
542:
539:
536:
534:
532:
530:
527:
526:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
488:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
448:
445:
442:
439:
438:
434:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
400:
397:
394:
391:
387:
376:
372:
369:
366:
365:
361:
358:
355:
354:
350:
347:
344:
343:
339:
336:
333:
332:
328:
325:
322:
321:
317:
314:
311:
310:
306:
303:
300:
299:
295:
292:
289:
288:
282:
274:
271:
268:
267:
261:
253:
250:
247:
246:
240:
232:
229:
226:
225:
219:
211:
208:
205:
204:
201:tone (cents)
198:
196:tone (ratio)
193:
190:
189:
186:
183:
174:
167:
165:
163:
159:
155:
151:
144:
123:
120:
117:
114:
109:
103:
98:
95:
89:
86:
78:
74:
70:
66:
62:
58:
54:
50:
46:
37:
29:
21:
3033:Golden ratio
3006:Lambda scale
2975:
2913:Arabic maqam
2870:Werckmeister
2701:Temperaments
2499:
2496:Golden Ratio
2488:
2459:
2444:
2441:Golden Tones
2435:
2427:
2418:
2410:
2384:
2336:
2175:
2148:
2143:
2090:
2063:
2058:
2005:
1871:
1866:
1813:
1788:
1783:
1731:
1575:
1495:
1414:
1280:
1275:
1199:
1194:
1141:
1039:
1034:
954:
949:
897:
863:
859:
848:
579:
563:
477:
472:
467:
462:
457:
452:
395:
383:
179:
118:1.6180339887
77:golden ratio
57:Heinz Bohlen
55:proposed by
44:
42:
3001:Delta scale
2996:Gamma scale
2986:Alpha scale
2888:non-Western
2886:Traditional
2581:Pitch class
2561:Millioctave
2544:Measurement
668:Step width
604:Scale step
582:symmetrical
528:Scale step
3022:Categories
2991:Beta scale
2969:Non-octave
2960:Tetrachord
2862:Kirnberger
2825:Schismatic
585:seven tone
168:Derivation
158:equivalent
154:repetition
67:of 833.09
2981:A12 scale
2935:Octoechos
2900:Shí-èr-lǜ
2849:Irregular
2666:Otonality
2606:Microtone
2462:, p.398.
2290:φ
2267:φ
2212:φ
2189:φ
2127:φ
2104:φ
2042:φ
2020:φ
1959:φ
1935:φ
1908:φ
1850:φ
1827:φ
1767:φ
1745:φ
1708:1,068.86
1689:φ
1667:φ
1612:φ
1589:φ
1531:φ
1509:φ
1451:φ
1429:φ
1368:φ
1344:φ
1317:φ
1259:φ
1236:φ
1178:φ
1156:φ
1099:φ
1077:φ
1018:φ
995:φ
933:φ
911:φ
709:φ
121:…
87:φ
59:based on
2866:Vallotti
2819:septimal
2811:Meantone
2571:Interval
2489:CatSynth
889:(cents)
884:(cents)
701:Stacked
449:1100.73
373:833.094
370:610:377
367:377:233
362:833.081
359:377:233
356:233:144
351:833.113
348:233:144
340:833.030
329:833.248
318:832.676
307:834.175
296:830.253
182:converge
65:interval
2955:Slendro
2905:Dastgah
2830:Miracle
2793:96-tone
2788:72-tone
2783:58-tone
2778:53-tone
2773:41-tone
2768:34-tone
2763:31-tone
2753:24-tone
2748:23-tone
2743:22-tone
2738:19-tone
2733:17-tone
2728:15-tone
2723:12-tone
2654:7-limit
2649:5-limit
2362:Sources
2309:701.96
2280:⁄
2254:16 (2)
2249:136.50
2231:565.45
2228:1.3863
2202:⁄
2117:⁄
2033:⁄
2000:136.50
1982:230.41
1979:1.1424
1973:
1924:
1921:⁄
1896:12 (5)
1891:131.14
1840:⁄
1758:⁄
1726:131.14
1705:1.8541
1680:⁄
1649:136.50
1631:932.36
1628:1.7135
1602:⁄
1550:833.09
1547:1.6180
1522:⁄
1470:733.82
1467:1.5279
1442:⁄
1409:136.50
1391:597.32
1388:1.4120
1382:
1333:
1330:⁄
1300:131.14
1249:⁄
1169:⁄
1136:131.14
1118:235.77
1115:1.1459
1090:⁄
1059:136.50
1008:⁄
924:⁄
879:(dec.)
685:131.14
679:131.14
660:833.09
657:733.82
654:597.32
651:466.18
648:366.91
645:235.77
566:lattice
485:565.45
482:932.36
446:267.64
443:634.55
345:144:89
337:144:89
280:840.528
259:813.686
238:884.359
217:701.955
2923:Mugham
2909:Maqam
2803:Linear
2757:pieces
2718:6-tone
2639:Hexany
2566:Savart
2466:
2176:15 (1)
2149:466.18
2144:1.3090
2091:14 (0)
2064:366.91
2059:1.2361
2006:13 (6)
1867:1.0590
1814:11 (4)
1808:99.27
1784:1.0000
1732:10 (3)
1654:9 (2)
1281:466.18
1276:1.3090
1219:99.27
1200:366.91
1195:1.2361
1035:1.0590
950:1.0000
892:Audio
874:Ratio
691:99.27
688:136.5
682:99.27
676:136.5
673:99.27
642:99.27
639:00.00
636:Cents
523:1.386
520:1.713
517:1.059
514:1.309
511:1.618
508:1.000
505:1.236
502:1.528
499:1.889
496:1.167
493:1.443
490:Ratio
473:466.18
468:833.09
458:366.91
453:733.82
440:Cents
334:89:55
326:89:55
323:55:34
315:55:34
312:34:21
304:34:21
301:21:13
293:21:13
150:octave
75:. The
2940:Pelog
2928:Muqam
2874:Young
2835:Magic
2710:Equal
2644:Limit
2551:Pitch
2169:99.27
2084:99.27
1872:99.27
1576:8 (1)
1569:99.27
1496:7 (0)
1489:99.27
1040:99.27
977:99.27
887:Width
882:Ratio
877:Ratio
871:Step
821:]
817:[
793:]
789:[
765:]
761:[
721:tone
576:Scale
478:99.27
402:Tone
290:13:8
272:13:8
69:cents
63:, an
53:scale
47:is a
2945:Raga
2556:Cent
2464:ISBN
2331:...
2317:Play
2306:1.5
2239:Play
2158:Play
2073:Play
1990:Play
1881:Play
1798:Play
1716:Play
1639:Play
1558:Play
1478:Play
1399:Play
1290:Play
1209:Play
1126:Play
1049:Play
966:Play
746:...
694:...
663:...
631:...
593:Play
378:...
269:8:5
251:8:5
248:5:3
230:5:3
227:3:2
209:3:2
206:2:1
142:Play
51:and
43:The
2498:",
2487:",
2426:",
2409:",
2383:",
2345:See
740:−2
732:−1
558:15
417:−1
414:−2
411:−3
408:−4
405:−5
390:440
79:is
3024::
2872:,
2868:,
2864:,
2817:,
2451:^
2392:^
2370:^
1305:5
1224:4
1064:2
983:1
857:.
743:1
735:2
727:3
724:0
628:7
625:6
622:5
619:4
616:3
613:2
610:1
607:0
555:8
552:1
549:4
546:7
543:0
540:3
537:6
435:5
432:4
429:3
426:2
423:1
420:0
386:Hz
2951:)
2947:(
2876:)
2860:(
2856:/
2821:)
2813:(
2759:)
2755:(
2529:e
2522:t
2515:v
2502:.
2494:"
2491:.
2483:"
2470:.
2430:.
2413:.
2405:"
2387:.
2282:2
2259:3
2204:8
2119:2
2012:2
1949:/
1927:6
1842:4
1789:0
1659:3
1604:4
1421:4
1415:6
1358:/
1336:3
1251:2
1148:2
1142:3
1069:3
1010:4
955:0
898:0
819:2
791:2
763:2
463:0
124:.
115:=
110:2
104:5
99:+
96:1
90:=
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