Scala analysis: xen15-gilson-generalized-pythagorean-15-8-10
Generalized Pythagorean Scale, 15/8 stacked 10=9+1 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 194.418 cents 194.418433 2: 306.150 cents 306.149718 3: 170859375/134217728 417.881003 4: 11390625/8388608 529.612288 5: 759375/524288 641.343574 6: 50625/32768 753.074859 7: 3375/2048 864.806144 double augmented fifth 8: 225/128 976.537429 augmented sixth 9: 15/8 1088.268715 classic major seventh 10: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 194.4184 cents 194.4184 2: 111.7313 cents 111.7313 3: 111.7313 cents 111.7313 4: 16/15 111.7313 minor diatonic semitone 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
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 9 111.73129 cents 1: 1 194.41843 cents 2: 8 223.46257 cents 2: 2 306.14972 cents 3: 7 335.19386 cents 3: 3 170859375/134217728 417.881 cents 4: 6 446.92514 cents 4: 4 11390625/8388608 529.612 cents 5: 5 558.65643 cents 5: 5 759375/524288 641.344 cents 6: 4 670.38771 cents 6: 6 50625/32768 753.075 cents 7: 3 782.11900 cents 7: 7 3375/2048 864.806 cents double augmented fifth 8: 2 893.85028 cents 8: 8 225/128 976.537 cents augmented sixth 9: 1 1005.58157 cents 9: 9 15/8 1088.269 cents classic 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 0.0 : 194.4 306.1 417.9 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 194.4 : 111.7 223.5 335.2 446.9 558.7 670.4 782.1 893.9 1005.6 1200.0 306.1 : 111.7 223.5 335.2 446.9 558.7 670.4 782.1 893.9 1088.3 1200.0 417.9 : 111.7 223.5 335.2 446.9 558.7 670.4 782.1 976.5 1088.3 1200.0 529.6 : 111.7 223.5 335.2 446.9 558.7 670.4 864.8 976.5 1088.3 1200.0 641.3 : 111.7 223.5 335.2 446.9 558.7 753.1 864.8 976.5 1088.3 1200.0 753.1 : 111.7 223.5 335.2 446.9 641.3 753.1 864.8 976.5 1088.3 1200.0 864.8 : 111.7 223.5 335.2 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 976.5 : 111.7 223.5 417.9 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 1088.3: 111.7 306.1 417.9 529.6 641.3 753.1 864.8 976.5 1088.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 111.7313 .. 194.4184 cents 82.68715 cents 111.73129 cents 2: 223.4626 .. 306.1497 cents 82.68715 cents 29.04414 cents 3: 335.1939 .. 417.8810 cents 82.68715 cents 29.04414 cents 4: 446.9251 .. 529.6123 cents 82.68715 cents 29.04414 cents 5: 558.6564 .. 641.3436 cents 82.68715 cents 29.04414 cents 6: 670.3877 .. 753.0749 cents 82.68715 cents 29.04414 cents 7: 782.1190 .. 864.8061 cents 82.68715 cents 29.04414 cents 8: 893.8503 .. 976.5374 cents 82.68715 cents 29.04414 cents 9: 1005.5816 .. 1088.2687 cents 82.68715 cents 29.04414 cents
SHOW DATA
Number of notes : 10 -- Interval properties -- Smallest interval : 111.73129 cents, class 1 Average step (divided formal octave): 120.0000 cents Largest one step interval : 194.41843 cents Average / Smallest step : 1.074005 Largest / Average step : 1.620154 Largest / Smallest step : 1.740054 Median interval of one step : 111.73129 cents, amount: 9 Most common interval of one step : 111.73129 cents, amount: 9 Least squares average step : 123.54373 cents, oct.: 1235.43735 cents Scale is strictly proper Scale has Myhill's property chromas: 82.68715 cents and 29.04414 cents, ratio: 2.846948 generators: 111.73129 cents and 9 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: 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: ABBBBBBBBB Step pattern size order : LSSSSSSSSS Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 29.04414 cents Scale diversity : 1.188710 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.379846 Rothenberg efficiency : 0.733333 redundancy: 0.266667 Efficiency x scale size : 7.333333 Number of different interval sizes : 18 = 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: 9 Scale is a chain of 5 triads 0.0 641.344 1088.269 cents Number of recognisable fifths : 0 Best major thirds form a closed circle Steps are decreasing Formal octave complements present : 1 = 10.0000% Scale is n-1 differentially coherent in interval class 1 Limited inverse transpositions : 1 Inversional symmetry on intervals : 0-1 5-6 -- Rational properties -- Prime limit : 5 (not all pitches rational)
FIT/MODE
10: 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 44.1429 c. M: 74.4184 c. 11: 2 1 1 1 1 1 1 1 1 1 P M ME SD: 14.0958 c. M:-23.7634 c. 21: 3 2 2 2 2 2 2 2 2 2 SP M ME SD: 13.6369 c. M: 22.9899 c. 23: 4 2 2 2 2 2 3 2 2 2 P SD: 13.4987 c. M: 22.6401 c. 24: 4 2 2 3 2 2 2 3 2 2 P T3 SD: 13.4018 c. M:-23.4626 c. 25: 4 2 3 2 2 3 2 2 3 2 P SD: 12.5793 c. M: 18.1497 c. 28: 5 2 3 2 3 3 2 3 2 3 N SD: 12.4394 c. M:-19.8673 c. 30: 5 3 2 3 3 3 3 2 3 3 P SD: 11.2142 c. M: 17.8810 c. 31: 5 3 3 3 3 2 3 3 3 3 P G D3 SD: 10.4291 c. M: 17.5910 c. 32: 5 3 3 3 3 3 3 3 3 3 SP M DE SD: 4.1038 c. M: 6.9184 c. 43: 7 4 4 4 4 4 4 4 4 4 SP M DE SD: 0.5519 c. M:-0.9304 c. 161: 26 15 15 15 15 15 15 15 15 15 SP M DE SD: 0.3735 c. M: 0.6296 c. 204: 33 19 19 19 19 19 19 19 19 19 SP M DE SD: 0.1784 c. M: 0.3008 c. 247: 40 23 23 23 23 23 23 23 23 23 SP M DE SD: 0.0513 c. M: 0.0864 c. 290: 47 27 27 27 27 27 27 27 27 27 SP M DE SD: 0.0382 c. M:-0.0643 c. 537: 87 50 50 50 50 50 50 50 50 50 SP M DE SD: 0.0030 c. M: 0.0050 c.
FIT/HARMONIC
1 x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x 3 x x x 4 S SD: 25.5599 cents 3 x x x 4 x x 5 x x 6 S SD: 12.3774 cents 4 x x 5 x x 6 x 7 x 8 S SD: 15.1435 cents 5 x 6 x 7 x x 8:9 x 10 S SD: 16.9559 cents 6 x 7 x 8 x 9:10 x 11:12 S SD: 13.9926 cents 7:8 x 9 x 10:11 x 12:13:14 S SD: 10.3413 cents 8:9 x 10:11 x 12:13:14:15:16 S SD: 8.6782 cents 9:10:11 x 12:13:14:15:16:17:18 S SD: 6.9686 cents 10:11:12:13:14 x 15:16:18:19:20 S SD: 12.4285 cents 11:12:13:14:15:16:17:18:19:21:22 SD: 6.5991 cents 15:17:18:19:20:22:23:25:26:28:30 SD: 5.7708 cents 16:18:19:20:22:23:25:26:28:30:32 SD: 5.3238 cents 17:19:20:22:23:25:26:28:30:32:34 SD: 5.0607 cents 22:25:26:28:30:32:34:36:39:41:44 SD: 3.9902 cents 25:28:30:32:34:36:39:41:44:47:50 SD: 2.5875 cents 33:37:39:42:45:48:51:54:58:62:66 SD: 2.3873 cents 40:45:48:51:54:58:62:66:70:75:80 SD: 1.9709 cents 42:47:50:53:57:61:65:69:74:79:84 SD: 1.8863 cents 51:57:61:65:69:74:79:84:90:96:102 SD: 1.3597 cents 62:69:74:79:84:90:96:102:109:116:124 SD: 1.2442 cents 73:82:87:93:99:106:113:120:128:137:146 SD: 1.1381 cents 83:93:99:106:113:120:128:137:146:156:166 SD: 1.0258 cents 92:103:110:117:125:133:142:152:162:173:184 SD: 0.9103 cents 103:115:123:131:140:149:159:170:181:193:206 SD: 0.5915 cents 125:140:149:159:170:181:193:206:220:234:250 SD: 0.5518 cents 136:152:162:173:185:197:210:224:239:255:272 SD: 0.5237 cents 145:162:173:185:197:210:224:239:255:272:290 SD: 0.4886 cents 154:172:184:196:209:223:238:254:271:289:308 SD: 0.4763 cents 165:185:197:210:224:239:255:272:290:309:330 SD: 0.4388 cents 176:197:210:224:239:255:272:290:309:330:352 SD: 0.2432 cents 279:312:333:355:379:404:431:460:490:523:558 SD: 0.2315 cents 310:347:370:395:421:449:479:511:545:581:620 SD: 0.2051 cents 330:369:394:420:448:478:510:544:580:619:660 SD: 0.1749 cents 352:394:420:448:478:510:544:580:619:660:704 SD: 0.1412 cents 455:509:543:579:618:659:703:750:800:853:910 SD: 0.1072 cents