Scala analysis: xen15-gilson-generalized-pythagorean-13-8-10
Generalized Pythagorean Scale, 13/8 stacked 10=7+3 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2197/2048 121.582985 2: 4826809/4194304 243.165971 3: 364.749 cents 364.748956 4: 169/128 481.055324 5: 371293/262144 602.638309 6: 815730721/536870912 724.221294 7: 13/8 840.527662 tridecimal neutral sixth 8: 28561/16384 962.110647 9: 62748517/33554432 1083.693632 10: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 67108864/62748517 116.3064 1: 2197/2048 121.5830 2: 2197/2048 121.5830 3: 121.5830 cents 121.5830 4: 116.3064 cents 116.3064 5: 2197/2048 121.5830 6: 2197/2048 121.5830 7: 67108864/62748517 116.3064 8: 2197/2048 121.5830 9: 2197/2048 121.5830 10: 67108864/62748517 116.3064
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 116.30637 cents 1: 7 2197/2048 121.583 cents 2: 6 32768/28561 237.889 cents 2: 4 4826809/4194304 243.166 cents 3: 9 16/13 359.472 cents tridecimal neutral third 3: 1 364.74896 cents 4: 2 475.77871 cents 4: 8 169/128 481.055 cents 5: 5 524288/371293 597.362 cents 5: 5 371293/262144 602.638 cents 6: 8 256/169 718.945 cents two (tridecimal neutral third) 6: 2 815730721/536870912 724.221 cents 7: 1 835.25104 cents 7: 9 13/8 840.528 cents tridecimal neutral sixth 8: 4 8388608/4826809 956.834 cents 8: 6 28561/16384 962.111 cents 9: 7 4096/2197 1078.417 cents 9: 3 62748517/33554432 1083.694 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 0.0 : 121.6 243.2 364.7 481.1 602.6 724.2 840.5 962.1 1083.7 1200.0 121.6 : 121.6 243.2 359.5 481.1 602.6 718.9 840.5 962.1 1078.4 1200.0 243.2 : 121.6 237.9 359.5 481.1 597.4 718.9 840.5 956.8 1078.4 1200.0 364.7 : 116.3 237.9 359.5 475.8 597.4 718.9 835.3 956.8 1078.4 1200.0 481.1 : 121.6 243.2 359.5 481.1 602.6 718.9 840.5 962.1 1083.7 1200.0 602.6 : 121.6 237.9 359.5 481.1 597.4 718.9 840.5 962.1 1078.4 1200.0 724.2 : 116.3 237.9 359.5 475.8 597.4 718.9 840.5 956.8 1078.4 1200.0 840.5 : 121.6 243.2 359.5 481.1 602.6 724.2 840.5 962.1 1083.7 1200.0 962.1 : 121.6 237.9 359.5 481.1 602.6 718.9 840.5 962.1 1078.4 1200.0 1083.7: 116.3 237.9 359.5 481.1 597.4 718.9 840.5 956.8 1078.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 116.3064 .. 121.5830 cents 5.27662 cents 116.30637 cents 2: 237.8894 .. 243.1660 cents 5.27662 cents 67108864/62748517, 116.3064 cents 3: 359.4723 .. 364.7490 cents 5.27662 cents 67108864/62748517, 116.3064 cents 4: 475.7787 .. 481.0553 cents 5.27662 cents 111.02975 cents 5: 597.3617 .. 602.6383 cents 5.27662 cents 67108864/62748517, 116.3064 cents 6: 718.9447 .. 724.2213 cents 5.27662 cents 67108864/62748517, 116.3064 cents 7: 835.2510 .. 840.5277 cents 5.27662 cents 111.02975 cents 8: 956.8340 .. 962.1106 cents 5.27662 cents 67108864/62748517, 116.3064 cents 9: 1078.4170 .. 1083.6936 cents 5.27662 cents 67108864/62748517, 116.3064 cents
SHOW DATA
Number of notes : 10 -- Interval properties -- Smallest interval : 116.30637 cents, class 1 Average step (divided formal octave): 120.0000 cents Largest one step interval : 2197/2048, 121.5830 cents Average / Smallest step : 1.031758 Largest / Average step : 1.013192 Largest / Smallest step : 1.045368 Median interval of one step : 2197/2048, 121.5830 cents, amount: 7 Most common interval of one step : 2197/2048, 121.5830 cents, amount: 7 Least squares average step : 120.30837 cents, oct.: 1203.08374 cents Scale is strictly proper Scale has Myhill's property chromas: 5.27662 cents and 111.02975 cents, ratio: 21.041841 generators: 359.47234 cents and 7 of 840.52766 cents ET pair of linear temperament : 10&227 Scale is maximally even for L / S <= 2 Scale is distributional even Step pattern alph. order: AAABAABAAB Step pattern size order : LLLSLLSLLS Scale is a Constant Structure, by a margin of 111.02975 cents Scale diversity : 1.034109 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.960425 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 : 5.27662 cents Most common intervals : 16/13, 359.4723 cents & inv., amount: 9 Scale is a chain of 5 triads 0.0 602.638 840.528 cents Most common triad is 0.0 359.472 840.528 cents, amount: 8 Number of recognisable fifths : 8, average 718.9447 cents Number of appreciable fifths : 0 Best major thirds form a closed circle Best minor thirds form a closed circle Formal octave complements present : 1 = 10.0000% Limited transpositions with margin 5.2766 cents: 1 2 3 4 5 6 7 8 9 Limited inverse transpositions : 3 Inversional symmetry on intervals : 1-2 6-7 -- Rational properties -- Prime limit : 13 (not all pitches rational)
FIT/MODE
10: 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 2.8169 c. M: 4.7490 c. 114: 12 11 12 11 11 12 11 11 12 11 SP D3 SD: 2.8079 c. M:-4.7328 c. 115: 12 11 12 11 12 11 12 11 12 11 SP B ME SD: 2.7894 c. M:-4.6897 c. 117: 12 12 12 11 12 12 11 12 12 11 SP M ME SD: 2.6585 c. M:-4.4818 c. 118: 12 12 12 11 12 12 12 12 12 11 SP D3 SD: 2.6894 c. M:-4.4420 c. 122: 12 13 12 12 12 13 11 13 12 12 SP D3 SD: 2.6089 c. M: 4.4621 c. 124: 13 12 13 12 12 13 12 12 13 12 SP D3 SD: 2.5317 c. M:-4.2235 c. 127: 13 13 13 12 13 13 12 13 13 12 SP M ME SD: 2.2273 c. M:-3.7550 c. 137: 14 14 14 13 14 14 13 14 14 13 SP M ME SD: 1.8591 c. M:-3.1343 c. 147: 15 15 15 14 15 15 14 15 15 14 SP M ME SD: 1.5410 c. M:-2.5980 c. 157: 16 16 16 15 16 16 15 16 16 15 SP M ME SD: 1.2635 c. M:-2.1300 c. 167: 17 17 17 16 17 17 16 17 17 16 SP M ME SD: 1.0191 c. M:-1.7181 c. 177: 18 18 18 17 18 18 17 18 18 17 SP M ME SD: 0.8024 c. M:-1.3527 c. 187: 19 19 19 18 19 19 18 19 19 18 SP M ME SD: 0.6089 c. M:-1.0264 c. 197: 20 20 20 19 20 20 19 20 20 19 SP M ME SD: 0.4350 c. M:-0.7333 c. 207: 21 21 21 20 21 21 20 21 21 20 SP M ME SD: 0.2779 c. M:-0.4684 c. 217: 22 22 22 21 22 22 21 22 22 21 SP M ME SD: 0.1352 c. M:-0.2280 c. 227: 23 23 23 22 23 23 22 23 23 22 SP M ME SD: 0.0052 c. M:-0.0088 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: 11.1331 cents 3 x x x 4 x x 5 x x 6 S SD: 15.6695 cents 4 x x 5 x x 6 x 7 x 8 S SD: 7.9291 cents 5 x x 6 x 7 x 8:9 x 10 S SD: 16.2674 cents 6 x 7 x 8 x 9:10 x 11:12 S SD: 11.1132 cents 7 x 8 x 9:10 x 11:12:13:14 S SD: 11.8059 cents 8 x 9:10 x 11:12:13:14:15:16 S SD: 9.0153 cents 9 x 10:11:12:13:14:15:16:17:18 S SD: 11.3686 cents 10:11 x 12:13:14:15:16:17:19:20 SD: 10.7064 cents 12:13:14:15:16:17:18:20:21:22:24 S SD: 7.2423 cents 13:14:15:16:17:18:20:21:23:24:26 SD: 6.0291 cents 16:17:18:20:21:23:24:26:28:30:32 SD: 6.0059 cents 19:20:22:23:25:27:29:31:33:36:38 SD: 5.5472 cents 21:23:24:26:28:30:32:34:37:39:42 SD: 5.0171 cents 22:24:25:27:29:31:33:36:38:41:44 SD: 4.9430 cents 24:26:28:30:32:34:36:39:42:45:48 S SD: 4.6543 cents 25:27:29:31:33:35:38:41:44:47:50 SD: 3.7449 cents 26:28:30:32:34:37:40:42:45:49:52 SD: 3.6638 cents 27:29:31:33:36:38:41:44:47:50:54 SD: 3.2384 cents 29:31:33:36:38:41:44:47:51:54:58 SD: 3.1692 cents 31:33:36:38:41:44:47:50:54:58:62 SD: 2.7952 cents 38:41:44:47:50:54:58:62:66:71:76 SD: 2.1072 cents 46:49:53:57:61:65:70:75:80:86:92 SD: 1.8045 cents 53:57:61:65:70:75:81:86:92:99:106 SD: 1.7839 cents 54:58:62:67:71:76:82:88:94:101:108 SD: 1.7336 cents 56:60:64:69:74:79:85:91:98:105:112 SD: 1.6700 cents 59:63:68:73:78:84:90:96:103:110:118 SD: 1.6036 cents 65:70:75:80:86:92:99:106:113:122:130 SD: 1.5040 cents 67:72:77:83:88:95:102:109:117:125:134 SD: 1.3295 cents 69:74:79:85:91:98:105:112:120:129:138 SD: 1.2106 cents 81:87:93:100:107:115:123:132:141:151:162 SD: 0.9960 cents 85:91:98:105:112:120:129:138:148:159:170 SD: 0.8871 cents 93:100:107:115:123:132:141:151:162:174:186 SD: 0.8080 cents 94:101:108:116:124:133:143:153:164:176:188 SD: 0.6376 cents 106:114:122:131:140:150:161:172:185:198:212 SD: 0.6341 cents 125:134:144:154:165:177:190:203:218:234:250 SD: 0.4806 cents 166:178:191:205:219:235:252:270:289:310:332 SD: 0.4390 cents 179:192:206:221:236:254:272:291:312:335:358 SD: 0.4341 cents 192:206:221:237:254:272:292:312:335:359:384 SD: 0.4134 cents 206:221:237:254:272:292:313:335:359:385:412 SD: 0.3098 cents 272:292:313:336:359:385:413:442:474:509:544 SD: 0.2737 cents 277:297:319:342:366:392:421:450:483:518:554 SD: 0.2628 cents 291:312:335:359:384:412:442:473:507:544:582 SD: 0.2423 cents 312:335:359:385:412:442:474:507:544:583:624 SD: 0.2299 cents 331:355:381:409:437:469:503:538:577:619:662 SD: 0.1863 cents 384:412:442:474:507:544:583:624:669:718:768 SD: 0.1825 cents 397:426:457:490:524:562:603:645:692:742:794 SD: 0.1780 cents 452:485:520:558:597:640:687:735:788:845:904 SD: 0.1780 cents 456:489:525:563:602:646:693:741:795:853:912 SD: 0.1273 cents 562:603:647:694:742:796:854:913:980:1051:1124 SD: 0.1160 cents