Scala analysis: canton
A 2.3.11/7.13/7 subgroup scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 14/13 128.298245 2/3-tone 2: 9/8 203.910002 major whole tone 3: 13/11 289.209719 tridecimal minor third 4: 14/11 417.507964 undecimal diminished fourth or major third 5: 4/3 498.044999 perfect fourth 6: 39/28 573.656756 7: 3/2 701.955001 perfect fifth 8: 11/7 782.492036 undecimal augmented fifth 9: 22/13 910.790281 tridecimal major sixth 10: 16/9 996.089998 Pythagorean minor seventh 11: 13/7 1071.701755 16/3-tone 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 14/13 128.2982 2/3-tone 1: 14/13 128.2982 2/3-tone 2: 117/112 75.6118 3: 104/99 85.2997 4: 14/13 128.2982 2/3-tone 5: 22/21 80.5370 undecimal minor semitone 6: 117/112 75.6118 7: 14/13 128.2982 2/3-tone 8: 22/21 80.5370 undecimal minor semitone 9: 14/13 128.2982 2/3-tone 10: 104/99 85.2997 11: 117/112 75.6118 12: 14/13 128.2982 2/3-tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 117/112 75.612 cents 1: 2 22/21 80.537 cents undecimal minor semitone 1: 2 104/99 85.300 cents 1: 5 14/13 128.298 cents 2/3-tone 2: 1 429/392 156.149 cents 2: 2 169/154 160.911 cents 2: 3 9/8 203.910 cents major whole tone 2: 3 44/39 208.835 cents 2: 2 112/99 213.598 cents 2: 1 196/169 256.596 cents two (2/3-tone) 3: 3 33/28 284.447 cents undecimal minor third 3: 4 13/11 289.210 cents tridecimal minor third 3: 2 32/27 294.135 cents Pythagorean minor third 3: 2 63/52 332.208 cents 3: 1 616/507 337.134 cents 4: 1 121/98 364.984 cents 4: 3 26/21 369.747 cents 4: 1 81/64 407.820 cents Pythagorean major third 4: 2 33/26 412.745 cents tridecimal major third 4: 4 14/11 417.508 cents undecimal diminished fourth or major third 4: 1 448/351 422.433 cents 5: 1 507/392 445.359 cents 5: 2 117/88 493.120 cents tridecimal gentle fourth 5: 2 121/91 493.282 cents 5: 5 4/3 498.045 cents perfect fourth 5: 2 196/143 545.806 cents 6: 3 39/28 573.657 cents 6: 1 169/121 578.419 cents two (tridecimal minor third) 6: 2 88/63 578.582 cents 6: 2 63/44 621.418 cents 6: 1 242/169 621.581 cents 6: 3 56/39 626.343 cents 7: 2 143/98 654.194 cents 7: 5 3/2 701.955 cents perfect fifth 7: 2 182/121 706.718 cents 7: 2 176/117 706.880 cents 7: 1 784/507 754.641 cents 8: 1 351/224 777.567 cents 8: 4 11/7 782.492 cents undecimal augmented fifth 8: 2 52/33 787.255 cents tridecimal minor sixth 8: 1 128/81 792.180 cents Pythagorean minor sixth 8: 3 21/13 830.253 cents 8: 1 196/121 835.016 cents two (undecimal diminished fourth or major third) 9: 1 507/308 862.866 cents 9: 2 104/63 867.792 cents 9: 2 27/16 905.865 cents Pythagorean major sixth 9: 4 22/13 910.790 cents tridecimal major sixth 9: 3 56/33 915.553 cents 10: 1 169/98 943.404 cents 10: 2 99/56 986.402 cents 10: 3 39/22 991.165 cents 10: 3 16/9 996.090 cents Pythagorean minor seventh 10: 2 308/169 1039.089 cents 10: 1 784/429 1043.851 cents 11: 5 13/7 1071.702 cents 16/3-tone 11: 2 99/52 1114.700 cents 11: 2 21/11 1119.463 cents undecimal major seventh 11: 3 224/117 1124.388 cents Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 5.27273 = 58/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 128.3 203.9 289.2 417.5 498.0 573.7 702.0 782.5 910.8 996.1 1071.7 1200.0 128.3 : 75.6 160.9 289.2 369.7 445.4 573.7 654.2 782.5 867.8 943.4 1071.7 1200.0 203.9 : 85.3 213.6 294.1 369.7 498.0 578.6 706.9 792.2 867.8 996.1 1124.4 1200.0 289.2 : 128.3 208.8 284.4 412.7 493.3 621.6 706.9 782.5 910.8 1039.1 1114.7 1200.0 417.5 : 80.5 156.1 284.4 365.0 493.3 578.6 654.2 782.5 910.8 986.4 1071.7 1200.0 498.0 : 75.6 203.9 284.4 412.7 498.0 573.7 702.0 830.3 905.9 991.2 1119.5 1200.0 573.7 : 128.3 208.8 337.1 422.4 498.0 626.3 754.6 830.3 915.6 1043.9 1124.4 1200.0 702.0 : 80.5 208.8 294.1 369.7 498.0 626.3 702.0 787.3 915.6 996.1 1071.7 1200.0 782.5 : 128.3 213.6 289.2 417.5 545.8 621.4 706.7 835.0 915.6 991.2 1119.5 1200.0 910.8 : 85.3 160.9 289.2 417.5 493.1 578.4 706.7 787.3 862.9 991.2 1071.7 1200.0 996.1 : 75.6 203.9 332.2 407.8 493.1 621.4 702.0 777.6 905.9 986.4 1114.7 1200.0 1071.7: 128.3 256.6 332.2 417.5 545.8 626.3 702.0 830.3 910.8 1039.1 1124.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 75.6118 .. 128.2982 cents 1568/1521, 52.6865 cents 117/112, 75.6118 cents 2: 156.1488 .. 256.5965 cents 76832/72501, 100.4477 cents 5577/5488, 27.8505 cents 3: 284.4470 .. 337.1335 cents 1568/1521, 52.6865 cents 5577/5488, 27.8505 cents 4: 364.9841 .. 422.4332 cents 43904/42471, 57.4492 cents 5577/5488, 27.8505 cents 5: 445.3585 .. 545.8062 cents 76832/72501, 100.4477 cents 177957/175616, 22.9253 cents 6: 573.6568 .. 626.3432 cents 1568/1521, 52.6865 cents 5577/5488, 27.8505 cents 7: 654.1938 .. 754.6415 cents 76832/72501, 100.4477 cents 5577/5488, 27.8505 cents 8: 777.5668 .. 835.0159 cents 43904/42471, 57.4492 cents 177957/175616, 22.9253 cents 9: 862.8665 .. 915.5530 cents 1568/1521, 52.6865 cents 5577/5488, 27.8505 cents 10: 943.4035 .. 1043.8512 cents 76832/72501, 100.4477 cents 5577/5488, 27.8505 cents 11: 1071.7018 .. 1124.3882 cents 1568/1521, 52.6865 cents 5577/5488, 27.8505 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 117/112, 75.6118 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 14/13, 128.2982 cents Average / Smallest step : 1.322546 Largest / Average step : 1.282982 Largest / Smallest step : 1.696803 Median interval of one step : 104/99, 85.2997 cents, amount: 2 Most common interval of one step : 14/13, 128.2982 cents, amount: 5 Least squares average step : 99.27123 cents, oct.: 1191.25473 cents Scale is strictly proper Step pattern alph. order: ABCADBADACBA Step pattern size order : DACDBADBDCAD Scale is a Constant Structure, by a margin of 22.92527 cents Scale diversity : 1.375542 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.349899 Rothenberg efficiency : 0.266709 redundancy: 0.733291 Efficiency x scale size : 3.200505 Number of different interval sizes : 58 = 5.27273 / class Number of one step interval sizes : 4 Highest interval variety : 6 Mean interval variety : 5.27273 = 58/11 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 10648/10647, 0.1626 cents Most common triad is 0.0 498.045 701.955 cents, amount: 3 Number of recognisable fifths : 9, average 704.1079 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 9 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 11 = 91.6667% -- Rational properties -- Prime limit : 13 Odd number limit : 507 (O: 507 U: 507) Highest odd numerator or denominator: 39 Scale harmonicity : 0.008410 Average absolute harmonicity : 0.159990 Specific harmonicity : 0.059475 Fundamental : 1/72072, -16.1372 octaves, 0.0036 Hz. Guide tone : 144144, 17.1372 octaves, 37711755.485 Hz. Exponens Consonantiae : 1.038875E+10, 33.27430 octaves Euler's gradus suavitatis : 72 Sum of Mann's harmonic distance : 136.5, average 11.37500 Mersenne's string divisions : 43054200 Sum of van Prooijen's expressibility: 10.99235, average 0.91603 Sum of Tenney's harmonic distance : 22.11711, average 1.84309 Vogel's harmonic complexity : 41.08333 Wille's k value : 40540 Wilson's harmonic complexity : 74 Rectangular lattice diameter : 10 Triangular lattice diameter : 4 Lattice compactness : 225.15140, average 2.88656 Lattice compactness (without 2's) : 149.94864, average 1.92242 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -3 .. 4 0.33333 16 1 3: -2 .. 2 0.08333 7 4 7: -1 .. 1 -0.08333 5 11: -1 .. 1 0.00000 4 2 13: -1 .. 1 0.08333 5 5 Average exponent except 2's : 1 / 12 = 0.08333 Average absolute exponent except 2's: 21 / 12 = 1.75000 Scale is weakly epimorphic with val: <12 17 31 40 44| Scale is weakly epimorphic with val: <12 17 32 41 45| Scale is weakly epimorphic with val: <12 17 33 42 46| Scale is weakly epimorphic with val: <12 17 34 43 47| Scale is JI-epimorphic with val: <12 19 31 39 42| Scale is JI-epimorphic with val: <12 19 32 40 43| Scale is JI-epimorphic with val: <12 19 33 41 44| Scale is JI-epimorphic with val: <12 19 34 42 45| Scale is JI-epimorphic with val: <12 19 35 43 46| Scale is weakly epimorphic with val: <12 19 36 39 47| Scale is JI-epimorphic with val: <12 19 36 44 47|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 16.2780 c. M: 28.2982 c. Twelve-tone Chromatic (1/11-comma) 17: 2 1 1 2 1 1 2 1 2 1 1 2 P D3 SD: 7.8135 c. M: 12.8782 c. 29: 3 2 2 3 2 2 3 2 3 2 2 3 SP D3 SD: 3.1208 c. M:-5.6536 c. 46: 5 3 3 5 3 3 5 3 5 3 3 5 SP D3 SD: 2.5271 c. M:-4.7857 c. 75: 8 5 5 8 5 5 8 5 8 5 5 8 SP D3 SD: 2.1402 c. M:-4.0900 c. 84: 9 5 6 9 6 5 9 6 9 6 5 9 SP SD: 2.7163 c. M:-3.9100 c. 94: 10 6 7 10 6 6 10 6 10 7 6 10 SP SD: 2.3984 c. M: 4.4073 c. 104: 11 7 7 11 7 7 11 7 11 7 7 11 SP D3 SD: 2.2426 c. M:-3.7823 c. 111: 12 7 8 12 7 7 12 7 12 8 7 12 SP SD: 2.2058 c. M:-4.1137 c. 113: 12 7 8 12 8 7 12 8 12 8 7 12 SP SD: 1.9949 c. M:-3.3487 c. 123: 13 8 9 13 8 8 13 8 13 9 8 13 SP SD: 1.8849 c. M:-3.4732 c. 130: 14 8 9 14 9 8 14 9 14 9 8 14 SP SD: 1.6580 c. M: 3.0559 c. 140: 15 9 10 15 9 9 15 9 15 10 9 15 SP SD: 1.6060 c. M:-2.4920 c. 149: 16 9 11 16 10 9 16 10 16 11 9 16 SP SD: 1.4393 c. M:-2.5677 c. 159: 17 10 11 17 11 10 17 11 17 11 10 17 SP SD: 1.3961 c. M: 2.4173 c. 169: 18 11 12 18 11 11 18 11 18 12 11 18 SP SD: 1.4193 c. M:-2.0072 c. 178: 19 11 13 19 12 11 19 12 19 13 11 19 SP SD: 0.8538 c. M:-1.6628 c. 195: 21 12 14 21 13 12 21 13 21 14 12 21 SP SD: 0.7699 c. M: 1.3491 c. 207: 22 13 15 22 14 13 22 14 22 15 13 22 SP SD: 0.6221 c. M:-1.0115 c. 224: 24 14 16 24 15 14 24 15 24 16 14 24 SP SD: 0.2719 c. M: 0.4425 c. 253: 27 16 18 27 17 16 27 17 27 18 16 27 SP SD: 0.1402 c. M:-0.2563 c. 477: 51 30 34 51 32 30 51 32 51 34 30 51 SP SD: 0.0875 c. M:-0.1364 c. 730: 78 46 52 78 49 46 78 49 78 52 46 78 SP SD: 0.0656 c. M: 0.1053 c. 1000: 107 63 71 107 67 63 107 67 107 71 63 107 SP SD: 0.0715 c. M:-0.1018 c. 1207: 129 76 86 129 81 76 129 81 129 86 76 129 SP SD: 0.0684 c. M: 0.1026 c. 1253: 134 79 89 134 84 79 134 84 134 89 79 134 SP SD: 0.0447 c. M:-0.0804 c. 1506: 161 95 107 161 101 95 161 101 161 107 95 161 SP SD: 0.0403 c. M:-0.0741 c. 1730: 185 109 123 185 116 109 185 116 185 123 109 185 SP SD: 0.0340 c. M:-0.0643 c. 1983: 212 125 141 212 133 125 212 133 212 141 125 212 SP SD: 0.0290 c. M:-0.0490 c.
FIT/HARMONIC
1 x x x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x x 3 x x x x 4 S SD: 0.0000 cents 3 x x x x 4 x x x 5 x x 6 S SD: 8.8105 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 10.3574 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 9.4078 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 6.8651 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 8.4182 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 9.5672 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 10.9331 cents 10:11 x 12:13 x 14:15:16:17:18:19:20 S SD: 8.0802 cents 11:12 x 13:14 x 15 x 17:19 x 20:22 SD: 9.0828 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 6.1324 cents 13:14 x 15 x 17:18 x 20:22:23:24:26 SD: 7.4352 cents 14:15:16:17:18:19:20:21:22:24:25:26:28 S SD: 6.8469 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.8592 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.2498 cents 18:19:20:21:23:24:25:27:28:30:32:33:36 SD: 5.0985 cents 22:24:25:26:28:29:31:33:35:37:39:41:44 SD: 3.9328 cents 23:25:26:27:29:31:32:35:36:39:41:43:46 SD: 3.6453 cents 26:28:29:31:33:35:36:39:41:44:46:48:52 SD: 2.7250 cents 33:36:37:39:42:44:46:50:52:56:59:61:66 SD: 2.6935 cents 38:41:43:45:48:51:53:57:60:64:68:71:76 SD: 2.3655 cents 39:42:44:46:50:52:54:59:61:66:69:72:78 SD: 2.2957 cents 40:43:45:47:51:53:56:60:63:68:71:74:80 SD: 1.7571 cents 48:52:54:57:61:64:67:72:75:81:85:89:96 SD: 1.6030 cents 54:58:61:64:69:72:75:81:85:91:96:100:108 SD: 1.3266 cents 66:71:74:78:84:88:92:99:104:112:117:123:132 SD: 1.0054 cents 88:95:99:104:112:117:123:132:138:149:156:163:176 SD: 0.9756 cents 94:101:106:111:120:125:131:141:148:159:167:175:188 SD: 0.8949 cents 104:112:117:123:132:139:145:156:163:176:185:193:208 SD: 0.6867 cents 114:123:128:135:145:152:159:171:179:193:203:212:228 SD: 0.6280 cents 120:129:135:142:153:160:167:180:189:203:213:223:240 SD: 0.5931 cents 142:153:160:168:181:189:198:213:223:240:252:264:284 SD: 0.5905 cents 143:154:161:169:182:191:199:215:225:242:254:266:286 SD: 0.5561 cents 144:155:162:170:183:192:201:216:226:244:256:267:288 SD: 0.5383 cents 154:166:173:182:196:205:215:231:242:261:274:286:308 SD: 0.5367 cents 158:170:178:187:201:211:220:237:248:267:281:293:316 SD: 0.5286 cents 166:179:187:196:211:221:231:249:261:281:295:308:332 SD: 0.4673 cents 168:181:189:199:214:224:234:252:264:284:299:312:336 SD: 0.4218 cents 194:209:218:229:247:259:270:291:305:328:345:360:388 SD: 0.3850 cents 198:213:223:234:252:264:276:297:311:335:352:368:396 SD: 0.2849 cents 224:241:252:265:285:299:312:336:352:379:398:416:448 SD: 0.2766 cents 264:284:297:312:336:352:368:396:415:447:469:490:528 SD: 0.2501 cents 312:336:351:369:397:416:435:468:490:528:555:579:624 SD: 0.2422 cents 336:362:378:397:428:448:468:504:528:569:597:624:672 SD: 0.1894 cents 352:379:396:416:448:469:490:528:553:596:626:654:704 SD: 0.1788 cents 390:420:439:461:496:520:543:585:613:660:693:724:780 SD: 0.1765 cents 392:422:441:463:499:523:546:588:616:663:697:728:784 SD: 0.1633 cents 440:474:495:520:560:587:613:660:691:745:782:817:880 SD: 0.1610 cents 456:491:513:539:580:608:635:684:717:772:811:847:912 SD: 0.1587 cents 494:532:556:584:629:659:688:741:776:836:878:917:988 SD: 0.1562 cents 504:543:567:596:641:672:702:756:792:853:896:936:1008 SD: 0.1488 cents 506:545:569:598:644:675:705:759:795:856:900:940:1012 SD: 0.1476 cents 534:575:601:631:680:712:744:801:839:904:949:992:1068 SD: 0.1393 cents 546:588:614:645:695:728:761:819:858:924:971:1014:1092 SD: 0.1377 cents 560:603:630:662:713:747:780:840:880:948:996:1040:1120 SD: 0.1243 cents 576:620:648:681:733:768:802:864:905:975:1024:1070:1152 SD: 0.1208 cents