Scala analysis: GoldenRatio
Successive divisions of the octave by the Golden Section
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 108.204 cents 108.203930 2: 175.078 cents 175.077640 3: 283.282 cents 283.281570 4: 391.486 cents 391.485510 5: 458.359 cents 458.359210 6: 566.563 cents 566.563150 7: 633.437 cents 633.436850 8: 741.641 cents 741.640790 9: 849.845 cents 849.844720 10: 916.718 cents 916.718430 11: 1024.922 cents 1024.922360 12: 1133.126 cents 1133.126290 13: 1200.000 cents 1200.000000
SHOW/INTERVAL
0: 66.8737 cents 66.8737 1: 108.2039 cents 108.2039 2: 66.8737 cents 66.8737 3: 108.2039 cents 108.2039 4: 108.2039 cents 108.2039 5: 66.8737 cents 66.8737 6: 108.2039 cents 108.2039 7: 66.8737 cents 66.8737 8: 108.2039 cents 108.2039 9: 108.2039 cents 108.2039 10: 66.8737 cents 66.8737 11: 108.2039 cents 108.2039 12: 108.2039 cents 108.2039 13: 66.8737 cents 66.8737
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 5 66.87371 cents 1: 8 108.20393 cents 2: 10 175.07764 cents 2: 3 216.40787 cents 3: 2 241.95134 cents 3: 11 283.28157 cents 4: 7 350.15528 cents 4: 6 391.48551 cents 5: 12 458.35921 cents 5: 1 499.68944 cents 6: 4 525.23292 cents 6: 9 566.56315 cents 7: 9 633.43685 cents 7: 4 674.76708 cents 8: 1 700.31056 cents 8: 12 741.64079 cents 9: 6 808.51450 cents 9: 7 849.84472 cents 10: 11 916.71843 cents 10: 2 958.04865 cents 11: 3 983.59213 cents 11: 10 1024.92236 cents 12: 8 1091.79607 cents 12: 5 1133.12629 cents Highest number of different intervals for one interval class: 2 Average number of different intervals per interval class: 2.00000 = 2
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 0.0 : 108.2 175.1 283.3 391.5 458.4 566.6 633.4 741.6 849.8 916.7 1024.9 1133.1 1200.0 108.2 : 66.9 175.1 283.3 350.2 458.4 525.2 633.4 741.6 808.5 916.7 1024.9 1091.8 1200.0 175.1 : 108.2 216.4 283.3 391.5 458.4 566.6 674.8 741.6 849.8 958.0 1024.9 1133.1 1200.0 283.3 : 108.2 175.1 283.3 350.2 458.4 566.6 633.4 741.6 849.8 916.7 1024.9 1091.8 1200.0 391.5 : 66.9 175.1 242.0 350.2 458.4 525.2 633.4 741.6 808.5 916.7 983.6 1091.8 1200.0 458.4 : 108.2 175.1 283.3 391.5 458.4 566.6 674.8 741.6 849.8 916.7 1024.9 1133.1 1200.0 566.6 : 66.9 175.1 283.3 350.2 458.4 566.6 633.4 741.6 808.5 916.7 1024.9 1091.8 1200.0 633.4 : 108.2 216.4 283.3 391.5 499.7 566.6 674.8 741.6 849.8 958.0 1024.9 1133.1 1200.0 741.6 : 108.2 175.1 283.3 391.5 458.4 566.6 633.4 741.6 849.8 916.7 1024.9 1091.8 1200.0 849.8 : 66.9 175.1 283.3 350.2 458.4 525.2 633.4 741.6 808.5 916.7 983.6 1091.8 1200.0 916.7 : 108.2 216.4 283.3 391.5 458.4 566.6 674.8 741.6 849.8 916.7 1024.9 1133.1 1200.0 1024.9: 108.2 175.1 283.3 350.2 458.4 566.6 633.4 741.6 808.5 916.7 1024.9 1091.8 1200.0 1133.1: 66.9 175.1 242.0 350.2 458.4 525.2 633.4 700.3 808.5 916.7 983.6 1091.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 66.8737 .. 108.2039 cents 41.33022 cents 66.87371 cents 2: 175.0776 .. 216.4079 cents 41.33023 cents 66.87371 cents 3: 241.9513 .. 283.2816 cents 41.33023 cents 25.54347 cents 4: 350.1553 .. 391.4855 cents 41.33023 cents 66.87371 cents 5: 458.3592 .. 499.6894 cents 41.33023 cents 66.87370 cents 6: 525.2329 .. 566.5631 cents 41.33023 cents 25.54348 cents 7: 633.4369 .. 674.7671 cents 41.33023 cents 66.87370 cents 8: 700.3106 .. 741.6408 cents 41.33023 cents 25.54348 cents 9: 808.5145 .. 849.8447 cents 41.33022 cents 66.87371 cents 10: 916.7184 .. 958.0486 cents 41.33022 cents 66.87371 cents 11: 983.5921 .. 1024.9224 cents 41.33023 cents 25.54348 cents 12: 1091.7961 .. 1133.1263 cents 41.33022 cents 66.87371 cents
SHOW DATA
Number of notes : 13 -- Interval properties -- Smallest interval : 66.87370 cents, class 1 Average step (divided formal octave): 92.3077 cents Largest one step interval : 108.20393 cents Average / Smallest step : 1.380329 Largest / Average step : 1.172209 Largest / Smallest step : 1.618034 Median interval of one step : 108.20393 cents, amount: 8 Most common interval of one step : 108.20393 cents, amount: 8 Least squares average step : 93.06466 cents, oct.: 1209.84053 cents Scale is strictly proper Scale has Myhill's property chromas: 41.33022 cents and 25.54349 cents, ratio: 1.618033 generators: 5 of 458.35921 cents and 8 of 741.64079 cents ET pair of linear temperament : 13&21 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 987-tone equal temperament with octave 1200.000 cents degrees: 89 144 233 322 377 466 521 610 699 754 843 932 987 generators: 377 and 610 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABAABABAABAAB Step pattern size order : LSLLSLSLLSLLS Scale is a Constant Structure, by a margin of 25.54347 cents Scale diversity : 1.217807 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.586698 Rothenberg efficiency : 0.717949 redundancy: 0.282051 Efficiency x scale size : 9.333333 Number of different interval sizes : 24 = 2.00000 / class Number of one step interval sizes : 2 Highest interval variety : 2 Mean interval variety : 2.00000 = 2 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 25.54347 cents Most common intervals : 458.35921 cents & inv., amount: 12 Scale is a chain of 6 triads 0.0 458.359 916.718 cents Most common triad is 0.0 458.359 741.641 cents, amount: 11 Number of recognisable fifths : 1, average 700.3106 cents Scale contains two identical disjunct hexachords Best major thirds form a closed circle Formal octave complements present : 9 = 69.2308% Limited inverse transpositions : 6 Inversional symmetry on degrees : 3 Inversional symmetry on intervals : 9-10 -- Rational properties --
FIT/MODE
13: 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 13.4884 c. M: 25.4340 c. 21: 2 1 2 2 1 2 1 2 2 1 2 2 1 P M ME S SD: 5.1606 c. M:-9.7309 c. 34: 3 2 3 3 2 3 2 3 3 2 3 3 2 SP M ME S SD: 1.9699 c. M: 3.7145 c. 55: 5 3 5 5 3 5 3 5 5 3 5 5 3 SP M DE S SD: 0.7526 c. M:-1.4192 c. 89: 8 5 8 8 5 8 5 8 8 5 8 8 5 SP M DE S SD: 0.2875 c. M: 0.5420 c. 144: 13 8 13 13 8 13 8 13 13 8 13 13 8 SP M DE S SD: 0.1098 c. M:-0.2070 c. 233: 21 13 21 21 13 21 13 21 21 13 21 21 13 SP M DE S SD: 0.0419 c. M: 0.0791 c. 377: 34 21 34 34 21 34 21 34 34 21 34 34 21 SP M DE S SD: 0.0160 c. M:-0.0302 c. 610: 55 34 55 55 34 55 34 55 55 34 55 55 34 SP M DE S SD: 0.0061 c. M: 0.0115 c. 987: 89 55 89 89 55 89 55 89 89 55 89 89 55 SP M DE S SD: 0.0023 c. M:-0.0044 c. 1597: 144 89 144 144 89 144 89 144 144 89 144 144 89 SP M DE S SD: 0.0009 c. M: 0.0017 c.
FIT/HARMONIC
1 x x x x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x x x 3 x x x x 4 S SD: 19.8429 cents 3 x x x x 4 x x x x 5 x x 6 S SD: 17.0689 cents 4 x x x 5 x x x 6 x 7 x x 8 S SD: 16.4258 cents 5 x x 6 x x 7 x x 8 x 9 x 10 S SD: 10.3201 cents 6 x x 7 x 8 x x 9 x 10:11 x 12 S SD: 11.8600 cents 7 x x 8 x 9 x 10:11 x 12:13 x 14 S SD: 12.4849 cents 8 x 9 x 10 x 11 x 12:13:14 x 15:16 S SD: 10.8106 cents 9 x 10 x 11:12 x 13:14 x 15:16:17:18 S SD: 9.3168 cents 10 x 11:12 x 13:14 x 15:16:17:18:19:20 S SD: 6.9575 cents 11 x 12:13:14 x 15:16:17:18:19:20:21:22 S SD: 5.5503 cents 12:13 x 14:15:16 x 17:18 x 20:22:23:24 S SD: 8.3579 cents 13:14 x 15:16:17:18:19:20:21:22:24:25:26 S SD: 5.9296 cents 14:15 x 16 x 18:19:20:21:23:24:25:27:28 SD: 7.8573 cents 15:16:17:18:19:20:21:22:23 x 25:27:29:30 SD: 6.9637 cents 16:17:18:19:20:21:22:23:25:26:27:29:31:32 SD: 4.0797 cents 18:19:20:21:23 x 25:26:28:29:31:33:35:36 SD: 5.4512 cents 19:20:21:22:24:25:26:27:29:31:32:34:37:38 SD: 4.8130 cents 20:21:22:24:25:26:28:29:31:33:34:36:38:40 SD: 4.3716 cents 26:28:29:31:33:34:36:37:40:42:44:47:50:52 SD: 3.8441 cents 27:29:30:32:34:35:37:39:41:44:46:49:52:54 SD: 2.9274 cents 30:32:33:35:38:39:42:43:46:49:51:54:58:60 SD: 2.6660 cents 36:38:40:42:45:47:50:52:55:59:61:65:69:72 SD: 2.0989 cents 43:46:48:51:54:56:60:62:66:70:73:78:83:86 SD: 1.9963 cents 46:49:51:54:58:60:64:66:71:75:78:83:89:92 SD: 1.6455 cents 47:50:52:55:59:61:65:68:72:77:80:85:90:94 SD: 1.4680 cents 56:60:62:66:70:73:78:81:86:91:95:101:108:112 SD: 1.4567 cents 62:66:69:73:78:81:86:89:95:101:105:112:119:124 SD: 1.3121 cents 63:67:70:74:79:82:87:91:97:103:107:114:121:126 SD: 1.0840 cents 75:80:83:88:94:98:104:108:115:123:127:136:144:150 SD: 1.0797 cents 79:84:87:93:99:103:110:114:121:129:134:143:152:158 SD: 0.9072 cents 90:96:100:106:113:117:125:130:138:147:153:163:173:180 SD: 0.8608 cents 93:99:103:110:117:121:129:134:143:152:158:168:179:186 SD: 0.8224 cents 95:101:105:112:119:124:132:137:146:155:161:172:183:190 SD: 0.6142 cents 106:113:117:125:133:138:147:153:163:173:180:192:204:212 SD: 0.6122 cents 122:130:135:144:153:159:169:176:187:199:207:221:235:244 SD: 0.5743 cents 129:137:143:152:162:168:179:186:198:211:219:233:248:258 SD: 0.5210 cents 142:151:157:167:178:185:197:205:218:232:241:257:273:284 SD: 0.3808 cents 197:210:218:232:247:257:273:284:302:322:335:356:379:394 SD: 0.3747 cents 215:229:238:253:270:280:298:310:330:351:365:389:414:430 SD: 0.3546 cents 224:238:248:264:281:292:311:323:344:366:380:405:431:448 SD: 0.3542 cents 235:250:260:277:295:306:326:339:361:384:399:425:452:470 SD: 0.2905 cents 264:281:292:311:331:344:366:381:405:431:448:477:508:528 SD: 0.2235 cents 281:299:311:331:352:366:390:405:431:459:477:508:541:562 SD: 0.2118 cents 339:361:375:399:425:442:470:489:520:554:576:613:652:678 SD: 0.2115 cents 355:378:393:418:445:463:492:512:545:580:603:642:683:710 SD: 0.2110 cents 383:408:424:451:480:499:531:552:588:626:650:692:737:766 SD: 0.2033 cents 385:410:426:453:483:502:534:555:591:629:654:696:741:770 SD: 0.1959 cents 387:412:428:456:485:504:537:558:594:632:657:700:745:774 SD: 0.1822 cents 405:431:448:477:508:528:562:584:622:662:688:732:779:810 SD: 0.1730 cents 421:448:466:496:528:549:584:607:646:688:715:761:810:842 SD: 0.1432 cents 462:492:511:544:579:602:641:666:709:755:785:835:889:924 SD: 0.1350 cents 478:509:529:563:599:623:663:689:734:781:812:864:920:956 SD: 0.1318 cents 527:561:583:621:661:687:731:760:809:861:895:953:1014:1054 SD: 0.1236 cents 557:593:616:656:698:726:773:803:855:910:946:1007:1072:1114 SD: 0.1206 cents 591:629:654:696:741:770:820:852:907:966:1004:1068:1137:1182 SD: 0.1097 cents