Scala analysis: xen15-gilson-generalized-pythagorean-18-17-12
Generalized Pythagorean Scale, 18/17 stacked 12=1+11 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 18/17 98.954592 Arabic lute index finger 2: 324/289 197.909184 two (Arabic lute index finger) 3: 5832/4913 296.863777 4: 104976/83521 395.818369 5: 1889568/1419857 494.772961 6: 34012224/24137569 593.727553 7: 612220032/410338673 692.682146 8: 791.637 cents 791.636738 9: 890.591 cents 890.591330 10: 989.546 cents 989.545922 11: 1088.501 cents 1088.500515 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 111.4995 cents 111.4995 1: 18/17 98.9546 Arabic lute index finger 2: 18/17 98.9546 Arabic lute index finger 3: 18/17 98.9546 Arabic lute index finger 4: 18/17 98.9546 Arabic lute index finger 5: 18/17 98.9546 Arabic lute index finger 6: 18/17 98.9546 Arabic lute index finger 7: 18/17 98.9546 Arabic lute index finger 8: 98.9546 cents 98.9546 9: 98.9546 cents 98.9546 10: 98.9546 cents 98.9546 11: 98.9546 cents 98.9546 12: 111.4995 cents 111.4995
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 11 18/17 98.955 cents Arabic lute index finger 1: 1 111.49949 cents 2: 10 324/289 197.909 cents two (Arabic lute index finger) 2: 2 210.45408 cents 3: 9 5832/4913 296.864 cents 3: 3 309.40867 cents 4: 8 104976/83521 395.818 cents 4: 4 408.36326 cents 5: 7 1889568/1419857 494.773 cents 5: 5 410338673/306110016 507.318 cents 6: 6 34012224/24137569 593.728 cents 6: 6 24137569/17006112 606.272 cents 7: 5 612220032/410338673 692.682 cents 7: 7 1419857/944784 705.227 cents 8: 4 791.63674 cents 8: 8 83521/52488 804.182 cents 9: 3 890.59133 cents 9: 9 4913/2916 903.136 cents 10: 2 989.54592 cents 10: 10 289/162 1002.091 cents 11: 1 1088.50051 cents 11: 11 17/9 1101.045 cents septendecimal major seventh 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 0.0 : 99.0 197.9 296.9 395.8 494.8 593.7 692.7 791.6 890.6 989.5 1088.5 1200.0 99.0 : 99.0 197.9 296.9 395.8 494.8 593.7 692.7 791.6 890.6 989.5 1101.0 1200.0 197.9 : 99.0 197.9 296.9 395.8 494.8 593.7 692.7 791.6 890.6 1002.1 1101.0 1200.0 296.9 : 99.0 197.9 296.9 395.8 494.8 593.7 692.7 791.6 903.1 1002.1 1101.0 1200.0 395.8 : 99.0 197.9 296.9 395.8 494.8 593.7 692.7 804.2 903.1 1002.1 1101.0 1200.0 494.8 : 99.0 197.9 296.9 395.8 494.8 593.7 705.2 804.2 903.1 1002.1 1101.0 1200.0 593.7 : 99.0 197.9 296.9 395.8 494.8 606.3 705.2 804.2 903.1 1002.1 1101.0 1200.0 692.7 : 99.0 197.9 296.9 395.8 507.3 606.3 705.2 804.2 903.1 1002.1 1101.0 1200.0 791.6 : 99.0 197.9 296.9 408.4 507.3 606.3 705.2 804.2 903.1 1002.1 1101.0 1200.0 890.6 : 99.0 197.9 309.4 408.4 507.3 606.3 705.2 804.2 903.1 1002.1 1101.0 1200.0 989.5 : 99.0 210.5 309.4 408.4 507.3 606.3 705.2 804.2 903.1 1002.1 1101.0 1200.0 1088.5: 111.5 210.5 309.4 408.4 507.3 606.3 705.2 804.2 903.1 1002.1 1101.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 98.9546 .. 111.4995 cents 12.54489 cents 18/17, 98.9546 cents 2: 197.9092 .. 210.4541 cents 12.54489 cents 86.40970 cents 3: 296.8638 .. 309.4087 cents 12.54489 cents 86.40970 cents 4: 395.8184 .. 408.3633 cents 12.54489 cents 86.40970 cents 5: 494.7730 .. 507.3179 cents 12.54489 cents 86.40970 cents 6: 593.7276 .. 606.2724 cents 12.54489 cents 86.40970 cents 7: 692.6821 .. 705.2270 cents 12.54489 cents 86.40970 cents 8: 791.6367 .. 804.1816 cents 12.54489 cents 86.40970 cents 9: 890.5913 .. 903.1362 cents 12.54489 cents 86.40970 cents 10: 989.5459 .. 1002.0908 cents 12.54489 cents 86.40970 cents 11: 1088.5005 .. 1101.0454 cents 12.54489 cents 86.40970 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 98.95459 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 111.49949 cents Average / Smallest step : 1.010565 Largest / Average step : 1.114995 Largest / Smallest step : 1.126774 Median interval of one step : 18/17, 98.9546 cents, amount: 11 Most common interval of one step : 18/17, 98.9546 cents, amount: 11 Least squares average step : 99.18619 cents, oct.: 1190.23428 cents Scale is strictly proper Scale has Myhill's property chromas: 12.54489 cents and 86.40970 cents, ratio: 6.888038 generators: 1 of 98.95459 cents and 1101.04541 cents ET pair of linear temperament : 12&85 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 861-tone equal temperament with octave 2/1 degrees: 71 142 213 284 355 426 497 568 639 710 781 861 generators: 71 and 790 Number of contiguous 1-step segments: 0 Step pattern alph. order: AAAAAAAAAAAB Step pattern size order : SSSSSSSSSSSL Scale is a Constant Structure, by a margin of 86.40970 cents Scale diversity : 1.089877 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.885005 Rothenberg efficiency : 0.722222 redundancy: 0.277778 Efficiency x scale size : 8.666667 Number of different interval sizes : 22 = 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 : 12.54489 cents Most common intervals : 18/17, 98.9546 cents & inv., amount: 11 Scale is a chain of 6 triads 0.0 494.773 1101.045 cents Most common triad is 0.0 98.955 197.909 cents, amount: 10 Number of recognisable fifths : 12, average 700.0000 cents Scale contains two identical disjunct hexachords Best fifths form a closed circle Steps are increasing Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 12.5449 cents: 1 2 3 4 5 6 7 8 9 10 11 Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 17 (not all pitches rational)
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 6.7884 c. M:-11.4995 c. Twelve-tone Chromatic (1/11-comma) 49: 4 4 4 4 4 4 4 4 4 4 4 5 SP M ME SD: 6.4638 c. M: 10.9495 c. 52: 4 5 4 4 4 5 4 4 5 4 4 5 SP D3 SD: 6.2267 c. M: 10.1576 c. 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 6.1721 c. M: 10.9127 c. Schismic-12 55: 5 4 5 4 5 4 5 4 5 4 5 5 SP D3 SD: 5.8398 c. M:-10.1363 c. 56: 5 4 5 4 5 5 4 5 5 4 5 5 SP D3 SD: 5.9142 c. M: 10.1041 c. 57: 5 4 5 5 5 4 5 5 4 5 5 5 SP D3 SD: 5.8799 c. M:-10.4902 c. 61: 5 5 5 5 5 5 5 5 5 5 5 6 SP M ME SD: 3.8568 c. M: 6.5333 c. 73: 6 6 6 6 6 6 6 6 6 6 6 7 SP M ME SD: 2.1069 c. M: 3.5690 c. 85: 7 7 7 7 7 7 7 7 7 7 7 8 SP M ME SD: 0.8511 c. M: 1.4417 c. 97: 8 8 8 8 8 8 8 8 8 8 8 9 SP M ME SD: 0.0940 c. M:-0.1593 c. 473: 39 39 39 39 39 39 39 39 39 39 39 44 SP M DE SD: 0.0758 c. M: 0.1284 c. 570: 47 47 47 47 47 47 47 47 47 47 47 53 SP M DE SD: 0.0469 c. M: 0.0795 c. 667: 55 55 55 55 55 55 55 55 55 55 55 62 SP M DE SD: 0.0264 c. M: 0.0447 c. 764: 63 63 63 63 63 63 63 63 63 63 63 71 SP M DE SD: 0.0111 c. M: 0.0188 c. 861: 71 71 71 71 71 71 71 71 71 71 71 80 SP M DE SD: 0.0007 c. M:-0.0012 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: 4.6364 cents 3 x x x x 4 x x x 5 x x 6 S SD: 2.3464 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.1525 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 8.3699 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.4440 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 10.4863 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.6878 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 8.1554 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.6598 cents 11 x 12:13:14:15 x 16:17:18:19:21:22 SD: 11.0474 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.2032 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.2471 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 7.4300 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 4.8864 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 4.5961 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 3.9748 cents 27:29:30:32:34:36:38:40:43:45:48:51:54 SD: 3.1969 cents 31:33:35:37:39:41:44:46:49:52:55:58:62 SD: 2.2653 cents 39:41:44:46:49:52:55:58:62:65:69:73:78 SD: 2.0597 cents 47:50:53:56:59:63:66:70:74:79:83:88:94 SD: 1.9577 cents 48:51:54:57:60:64:68:72:76:80:85:90:96 S SD: 1.6662 cents 49:52:55:58:62:65:69:73:77:82:87:92:98 SD: 1.5009 cents 64:68:72:76:80:85:90:95:101:107:113:120:128 SD: 1.4388 cents 65:69:73:77:82:87:92:97:103:109:115:122:130 SD: 1.4280 cents 66:70:74:78:83:88:93:98:104:110:117:124:132 SD: 1.2331 cents 69:73:77:82:87:92:97:103:109:115:122:129:138 SD: 1.1863 cents 73:77:82:87:92:97:103:109:115:122:129:137:146 SD: 1.1196 cents 81:86:91:96:102:108:114:121:128:135:143:152:162 SD: 1.0014 cents 85:90:95:101:107:113:120:127:134:142:151:159:170 SD: 0.9238 cents 97:103:109:115:122:129:137:145:153:162:172:182:194 SD: 0.8110 cents 100:106:112:119:126:133:141:149:158:167:177:188:200 SD: 0.7414 cents 117:124:131:139:147:156:165:175:185:196:207:219:234 SD: 0.6693 cents 122:129:137:145:153:162:172:182:193:204:216:229:244 SD: 0.6340 cents 140:148:157:166:176:186:197:209:221:234:248:263:280 SD: 0.5267 cents 166:176:186:197:209:221:234:248:262:278:294:311:332 SD: 0.4638 cents 171:181:192:203:215:228:241:255:270:286:303:321:342 SD: 0.4071 cents 183:194:205:217:230:244:258:273:289:306:324:343:366 SD: 0.3994 cents 205:217:230:243:258:273:289:306:324:343:363:384:410 SD: 0.3777 cents 223:236:250:265:280:297:314:333:352:373:395:418:446 SD: 0.3282 cents 240:254:269:285:302:319:338:358:379:401:425:450:480 SD: 0.3231 cents 256:271:287:304:322:341:361:382:404:428:453:480:512 SD: 0.2962 cents 257:272:288:305:323:342:362:383:406:430:455:482:514 SD: 0.2108 cents 272:288:305:323:342:362:383:406:430:455:482:510:544 SD: 0.1977 cents 305:323:342:362:383:406:430:455:482:510:540:572:610 SD: 0.1837 cents 354:375:397:420:445:471:499:528:559:592:627:664:708 SD: 0.1563 cents 471:499:528:559:592:627:664:703:744:788:834:883:942 SD: 0.1460 cents 514:544:576:610:646:684:724:767:812:860:910:964:1028 SD: 0.1319 cents 529:560:593:628:665:704:745:789:836:885:937:992:1058 SD: 0.1197 cents 545:577:611:647:685:725:768:813:861:912:965:1022:1090 SD: 0.0974 cents