Scala analysis: xen15-gilson-generalized-pythagorean-15-8-11
Generalized Pythagorean Scale, 15/8 stacked 11=10+1 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 82.687 cents 82.687147 2: 194.418 cents 194.418433 3: 306.150 cents 306.149718 4: 170859375/134217728 417.881003 5: 11390625/8388608 529.612288 6: 759375/524288 641.343574 7: 50625/32768 753.074859 8: 3375/2048 864.806144 double augmented fifth 9: 225/128 976.537429 augmented sixth 10: 15/8 1088.268715 classic major seventh 11: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 82.6871 cents 82.6871 2: 111.7313 cents 111.7313 3: 111.7313 cents 111.7313 4: 111.7313 cents 111.7313 5: 16/15 111.7313 minor diatonic semitone 6: 16/15 111.7313 minor diatonic semitone 7: 16/15 111.7313 minor diatonic semitone 8: 16/15 111.7313 minor diatonic semitone 9: 16/15 111.7313 minor diatonic semitone 10: 16/15 111.7313 minor diatonic semitone 11: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 82.68715 cents 1: 10 111.73129 cents 2: 2 194.41843 cents 2: 9 223.46257 cents 3: 3 306.14972 cents 3: 8 335.19386 cents 4: 4 170859375/134217728 417.881 cents 4: 7 446.92514 cents 5: 5 11390625/8388608 529.612 cents 5: 6 558.65643 cents 6: 6 759375/524288 641.344 cents 6: 5 670.38771 cents 7: 7 50625/32768 753.075 cents 7: 4 782.11900 cents 8: 8 3375/2048 864.806 cents double augmented fifth 8: 3 893.85028 cents 9: 9 225/128 976.537 cents augmented sixth 9: 2 1005.58157 cents 10: 10 15/8 1088.269 cents classic major seventh 10: 1 1117.31285 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 0.0 : 82.7 194.4 306.1 417.9 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 82.7 : 111.7 223.5 335.2 446.9 558.7 670.4 782.1 893.9 1005.6 1117.3 1200.0 194.4 : 111.7 223.5 335.2 446.9 558.7 670.4 782.1 893.9 1005.6 1088.3 1200.0 306.1 : 111.7 223.5 335.2 446.9 558.7 670.4 782.1 893.9 976.5 1088.3 1200.0 417.9 : 111.7 223.5 335.2 446.9 558.7 670.4 782.1 864.8 976.5 1088.3 1200.0 529.6 : 111.7 223.5 335.2 446.9 558.7 670.4 753.1 864.8 976.5 1088.3 1200.0 641.3 : 111.7 223.5 335.2 446.9 558.7 641.3 753.1 864.8 976.5 1088.3 1200.0 753.1 : 111.7 223.5 335.2 446.9 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 864.8 : 111.7 223.5 335.2 417.9 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 976.5 : 111.7 223.5 306.1 417.9 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 1088.3: 111.7 194.4 306.1 417.9 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 82.6871 .. 111.7313 cents 29.04414 cents 82.68715 cents 2: 194.4184 .. 223.4626 cents 29.04414 cents 82.68715 cents 3: 306.1497 .. 335.1939 cents 29.04414 cents 82.68715 cents 4: 417.8810 .. 446.9251 cents 29.04414 cents 82.68715 cents 5: 529.6123 .. 558.6564 cents 29.04414 cents 82.68715 cents 6: 641.3436 .. 670.3877 cents 29.04414 cents 82.68715 cents 7: 753.0749 .. 782.1190 cents 29.04414 cents 82.68715 cents 8: 864.8061 .. 893.8503 cents 29.04414 cents 82.68715 cents 9: 976.5374 .. 1005.5816 cents 29.04414 cents 82.68715 cents 10: 1088.2687 .. 1117.3129 cents 29.04414 cents 82.68715 cents
SHOW DATA
Number of notes : 11 -- Interval properties -- Smallest interval : 82.68715 cents, class 1 Average step (divided formal octave): 109.0909 cents Largest one step interval : 111.73129 cents Average / Smallest step : 1.319321 Largest / Average step : 1.024203 Largest / Smallest step : 1.351253 Median interval of one step : 111.73129 cents, amount: 10 Most common interval of one step : 111.73129 cents, amount: 10 Least squares average step : 107.94292 cents, oct.: 1187.37211 cents Scale is strictly proper Scale has Myhill's property chromas: 29.04414 cents and 53.64301 cents, ratio: 1.846948 generators: 111.73129 cents and 10 of 1088.26871 cents ET pair of linear temperament : 10&11 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 537-tone equal temperament with octave 2/1 degrees: 37 87 137 187 237 287 337 387 437 487 537 generators: 50 and 487 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABBBBBBBBBB Step pattern size order : SLLLLLLLLLL Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 82.68715 cents Scale diversity : 1.186801 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.757966 Rothenberg efficiency : 0.727273 redundancy: 0.272727 Efficiency x scale size : 8.000000 Number of different interval sizes : 20 = 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 : 29.04414 cents Most common intervals : 111.73129 cents & inv., amount: 10 Scale is a chain of 5 triads 0.0 529.612 1088.269 cents Most common triad is 0.0 111.731 1088.269 cents, amount: 9 Number of recognisable fifths : 0 Best minor thirds form a closed circle Steps are increasing Formal octave complements present : 1 = 9.0909% Scale is n-1 differentially coherent in interval class 1 Limited inverse transpositions : 1 Inversional symmetry on degrees : 6 Inversional symmetry on intervals : 0-1 -- Rational properties -- Prime limit : 5 (not all pitches rational)
FIT/MODE
11: 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 15.6207 c. M:-26.4038 c. 21: 1 2 2 2 2 2 2 2 2 2 2 SP M ME S SD: 15.1122 c. M: 25.5443 c. 24: 2 2 2 2 3 2 2 2 3 2 2 P D3 S SD: 13.8032 c. M:-23.4626 c. 25: 2 2 2 3 2 2 3 2 2 3 2 SP D3 S SD: 12.6477 c. M: 18.1497 c. 28: 2 3 2 3 2 3 3 2 3 2 3 SP M ME S SD: 11.8956 c. M:-19.8673 c. 29: 2 3 2 3 3 2 3 3 3 2 3 SP D3 SD: 11.4968 c. M: 20.6539 c. 30: 2 3 3 2 3 3 3 3 2 3 3 SP D3 S SD: 10.7230 c. M: 17.8810 c. 31: 2 3 3 3 3 3 2 3 3 3 3 SP M ME S SD: 10.0698 c. M: 17.5910 c. 32: 2 3 3 3 3 3 3 3 3 3 3 SP M ME S SD: 4.5478 c. M: 7.6871 c. 43: 3 4 4 4 4 4 4 4 4 4 4 SP M ME S SD: 0.6116 c. M:-1.0338 c. 161: 11 15 15 15 15 15 15 15 15 15 15 SP M DE S SD: 0.4139 c. M: 0.6996 c. 204: 14 19 19 19 19 19 19 19 19 19 19 SP M DE S SD: 0.1977 c. M: 0.3342 c. 247: 17 23 23 23 23 23 23 23 23 23 23 SP M DE S SD: 0.0568 c. M: 0.0961 c. 290: 20 27 27 27 27 27 27 27 27 27 27 SP M DE S SD: 0.0423 c. M:-0.0715 c. 537: 37 50 50 50 50 50 50 50 50 50 50 SP M DE S SD: 0.0033 c. M: 0.0056 c.
FIT/HARMONIC
1 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 4 S SD: 25.5599 cents 3 x x x x 4 x x 5 x x 6 S SD: 12.3774 cents 4 x x x 5 x x 6 x 7 x 8 S SD: 15.1435 cents 5 x x 6 x 7 x x 8:9 x 10 S SD: 16.9559 cents 6 x x 7 x 8 x 9:10 x 11:12 S SD: 13.9926 cents 7 x 8 x 9 x 10:11 x 12:13:14 S SD: 10.3413 cents 8 x 9 x 10:11 x 12:13:14:15:16 S SD: 8.6782 cents 9 x 10:11 x 12:13:14:15:16:17:18 S SD: 6.9686 cents 10 x 11:12:13:14 x 15:16:18:19:20 S SD: 12.4285 cents 11 x 12:13:14:15:16:17:18:19:21:22 SD: 6.5991 cents 12 x 13:14:15:16:17:19:20:21:23:24 SD: 10.8716 cents 13:14:15:16:17:18:19:20:21:23:24:26 SD: 10.5281 cents 14:15:16:17:18:19:20:22:23:25:26:28 SD: 7.2605 cents 15:16:17:18:19:20:22:23:25:26:28:30 SD: 5.8732 cents 16:17:18:19:20:22:23:25:26:28:30:32 SD: 5.2462 cents 17:18:19:20:22:23:25:26:28:30:32:34 SD: 4.8325 cents 20:21:22:24:25:27:29:31:33:35:38:40 SD: 4.7077 cents 22:23:25:26:28:30:32:34:36:39:41:44 SD: 3.6647 cents 25:26:28:30:32:34:36:39:41:44:47:50 SD: 2.7093 cents 36:38:40:43:46:49:52:56:59:63:68:72 SD: 2.5730 cents 37:39:41:44:47:50:54:57:61:65:69:74 SD: 2.5053 cents 40:42:45:48:51:54:58:62:66:70:75:80 SD: 1.7990 cents 42:44:47:50:53:57:61:65:69:74:79:84 SD: 1.7259 cents 58:61:65:69:74:79:84:90:96:102:109:116 SD: 1.3921 cents 62:65:69:74:79:84:90:96:102:109:116:124 SD: 1.1339 cents 82:86:92:98:104:111:119:127:135:144:154:164 SD: 1.0849 cents 83:87:93:99:106:113:120:128:137:146:156:166 SD: 0.9389 cents 99:104:111:118:126:134:143:153:163:174:186:198 SD: 0.8773 cents 103:108:115:123:131:140:149:159:170:181:193:206 SD: 0.5407 cents 125:131:140:149:159:170:181:193:206:220:234:250 SD: 0.5203 cents 143:150:160:171:182:194:207:221:236:251:268:286 SD: 0.4901 cents 145:152:162:173:185:197:210:224:239:255:272:290 SD: 0.4546 cents 161:169:180:192:205:219:233:249:265:283:302:322 SD: 0.4504 cents 165:173:185:197:210:224:239:255:272:290:309:330 SD: 0.4043 cents 176:185:197:210:224:239:255:272:290:309:330:352 SD: 0.3989 cents 187:196:209:223:238:254:271:289:308:329:351:374 SD: 0.3552 cents 202:212:226:241:257:274:293:312:333:355:379:404 SD: 0.3382 cents 227:238:254:271:289:308:329:351:374:399:426:454 SD: 0.2669 cents 268:281:300:320:341:364:388:414:442:471:503:536 SD: 0.2633 cents 285:299:319:340:363:387:413:440:470:501:534:570 SD: 0.2432 cents 304:319:340:363:387:413:440:470:501:534:570:608 SD: 0.2413 cents 310:325:347:370:395:421:449:479:511:545:581:620 SD: 0.2030 cents 330:346:369:394:420:448:478:510:544:580:619:660 SD: 0.1720 cents 352:369:394:420:448:478:510:544:580:619:660:704 SD: 0.1590 cents 455:477:509:543:579:618:659:703:750:800:853:910 SD: 0.1296 cents 528:554:591:630:672:717:765:816:870:928:990:1056 SD: 0.1233 cents 553:580:619:660:704:751:801:854:911:972:1037:1106 SD: 0.1165 cents