Scala analysis: xen15-gilson-generalized-pythagorean-3-2-12
Generalized Pythagorean Scale, 3/2 stacked 12=7+5 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2187/2048 113.685006 apotome 2: 9/8 203.910002 major whole tone 3: 19683/16384 317.595008 Pythagorean augmented second 4: 81/64 407.820003 Pythagorean major third 5: 177147/131072 521.505010 Pythagorean augmented third 6: 729/512 611.730005 Pythagorean tritone 7: 3/2 701.955001 perfect fifth 8: 6561/4096 815.640007 Pythagorean augmented fifth 9: 27/16 905.865003 Pythagorean major sixth 10: 59049/32768 1019.550009 Pythagorean augmented sixth 11: 243/128 1109.775004 Pythagorean major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 256/243 90.2250 limma, Pythagorean minor second 1: 2187/2048 113.6850 apotome 2: 256/243 90.2250 limma, Pythagorean minor second 3: 2187/2048 113.6850 apotome 4: 256/243 90.2250 limma, Pythagorean minor second 5: 2187/2048 113.6850 apotome 6: 256/243 90.2250 limma, Pythagorean minor second 7: 256/243 90.2250 limma, Pythagorean minor second 8: 2187/2048 113.6850 apotome 9: 256/243 90.2250 limma, Pythagorean minor second 10: 2187/2048 113.6850 apotome 11: 256/243 90.2250 limma, Pythagorean minor second 12: 256/243 90.2250 limma, Pythagorean minor second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 7 256/243 90.225 cents limma, Pythagorean minor second 1: 5 2187/2048 113.685 cents apotome 2: 2 65536/59049 180.450 cents Pythagorean diminished third 2: 10 9/8 203.910 cents major whole tone 3: 9 32/27 294.135 cents Pythagorean minor third 3: 3 19683/16384 317.595 cents Pythagorean augmented second 4: 4 8192/6561 384.360 cents Pythagorean diminished fourth 4: 8 81/64 407.820 cents Pythagorean major third 5: 11 4/3 498.045 cents perfect fourth 5: 1 177147/131072 521.505 cents Pythagorean augmented third 6: 6 1024/729 588.270 cents Pythagorean diminished fifth 6: 6 729/512 611.730 cents Pythagorean tritone 7: 1 262144/177147 678.495 cents Pythagorean diminished sixth 7: 11 3/2 701.955 cents perfect fifth 8: 8 128/81 792.180 cents Pythagorean minor sixth 8: 4 6561/4096 815.640 cents Pythagorean augmented fifth 9: 3 32768/19683 882.405 cents Pythagorean diminished seventh 9: 9 27/16 905.865 cents Pythagorean major sixth 10: 10 16/9 996.090 cents Pythagorean minor seventh 10: 2 59049/32768 1019.550 cents Pythagorean augmented sixth 11: 5 4096/2187 1086.315 cents Pythagorean diminished octave 11: 7 243/128 1109.775 cents Pythagorean 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 : 113.7 203.9 317.6 407.8 521.5 611.7 702.0 815.6 905.9 1019.6 1109.8 1200.0 113.7 : 90.2 203.9 294.1 407.8 498.0 588.3 702.0 792.2 905.9 996.1 1086.3 1200.0 203.9 : 113.7 203.9 317.6 407.8 498.0 611.7 702.0 815.6 905.9 996.1 1109.8 1200.0 317.6 : 90.2 203.9 294.1 384.4 498.0 588.3 702.0 792.2 882.4 996.1 1086.3 1200.0 407.8 : 113.7 203.9 294.1 407.8 498.0 611.7 702.0 792.2 905.9 996.1 1109.8 1200.0 521.5 : 90.2 180.4 294.1 384.4 498.0 588.3 678.5 792.2 882.4 996.1 1086.3 1200.0 611.7 : 90.2 203.9 294.1 407.8 498.0 588.3 702.0 792.2 905.9 996.1 1109.8 1200.0 702.0 : 113.7 203.9 317.6 407.8 498.0 611.7 702.0 815.6 905.9 1019.6 1109.8 1200.0 815.6 : 90.2 203.9 294.1 384.4 498.0 588.3 702.0 792.2 905.9 996.1 1086.3 1200.0 905.9 : 113.7 203.9 294.1 407.8 498.0 611.7 702.0 815.6 905.9 996.1 1109.8 1200.0 1019.6: 90.2 180.4 294.1 384.4 498.0 588.3 702.0 792.2 882.4 996.1 1086.3 1200.0 1109.8: 90.2 203.9 294.1 407.8 498.0 611.7 702.0 792.2 905.9 996.1 1109.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 113.6850 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 2: 180.4500 .. 203.9100 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 3: 294.1350 .. 317.5950 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 4: 384.3600 .. 407.8200 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 5: 498.0450 .. 521.5050 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 7: 678.4950 .. 701.9550 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 8: 792.1800 .. 815.6400 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 9: 882.4050 .. 905.8650 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 10: 996.0900 .. 1019.5500 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 11: 1086.3150 .. 1109.7750 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 2187/2048, 113.6850 cents Average / Smallest step : 1.108340 Largest / Average step : 1.136850 Largest / Smallest step : 1.260017 Median interval of one step : 256/243, 90.2250 cents, amount: 7 Most common interval of one step : 256/243, 90.2250 cents, amount: 7 Least squares average step : 101.19706 cents, oct.: 1214.36474 cents Scale is strictly proper Scale has Myhill's property chromas: 531441/524288, 23.4600 cents and 134217728/129140163, 66.7650 cents, ratio: 2.845906 generators: 498.04500 cents and 7 of 701.95500 cents ET pair of linear temperament : 12&41 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 665-tone equal temperament with octave 2/1 degrees: 63 113 176 226 289 339 389 452 502 565 615 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABABABBABABB Step pattern size order : LSLSLSSLSLSS Scale is a Constant Structure, by a margin of 66.76499 cents Ratio Whole / Half tone : 2.260017 Scale diversity : 1.154991 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.784950 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 : 531441/524288, 23.4600 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 11 Scale is a chain of 6 triads 0.0 611.730 701.955 cents Most common triad is 0.0 498.045 701.955 cents, amount: 10 Number of recognisable fifths : 11, average 701.9550 cents Best fifths form a closed circle Scale is a complete genus musicum : 2^m 3^11 Formal octave complements present : 1 = 8.3333% Scale is n-1 differentially coherent in interval classes 4 and 5 combined Scale is n-1 differentially coherent in interval class 5 Scale is n-1 differentially coherent in interval classes 6 and 7 combined Scale is n-1 differentially coherent in interval class 7 Limited inverse transpositions : 5 Inversional symmetry on intervals : 2-3 8-9 -- Rational properties -- Prime limit : 3 Odd number limit : 177147 (O: 177147 U: 177147) Highest odd numerator or denominator: 177147 Scale harmonicity : 0.021127 Average absolute harmonicity : 0.146737 Specific harmonicity : 0.112917 Fundamental : 1/131072, -17.0000 octaves, 0.0020 Hz. Guide tone : 354294, 18.4346 octaves, 92692368.033 Hz. Exponens Consonantiae : 4.643802E+10, 35.43459 octaves Euler's gradus suavitatis : 41 Sum of Mann's harmonic distance : 226410.0, average 18867.50000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 31.49000, average 2.62417 Sum of Tenney's harmonic distance : 61.59300, average 5.13275 Vogel's harmonic complexity : 25.33333 Wille's k value : 88573 Wilson's harmonic complexity : 33 Rectangular lattice diameter : 11 Triangular lattice diameter : 11 Lattice compactness : 660.37746, average 8.46638 Lattice compactness (without 2's) : 352.00000, average 4.51282 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -17 .. 1 -8.16667 100 1 3: 0 .. 11 5.50000 66 11 Average exponent except 2's : 66 / 12 = 5.50000 Average absolute exponent except 2's: 66 / 12 = 5.50000 Scale is weakly epimorphic with val: <12 17| Scale is JI-epimorphic with val: <12 19| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 12.6950 c. M: 21.5050 c. Twelve-tone Chromatic (1/11-comma) 27: 3 2 2 2 3 2 2 2 2 3 2 2 SP D3 SD: 11.9029 c. M:-19.6483 c. 29: 3 2 3 2 3 2 2 3 2 3 2 2 SP M ME SD: 9.6967 c. M:-16.4260 c. 34: 3 3 3 3 3 2 3 3 3 3 2 3 SP D3 S SD: 9.1274 c. M:-15.7094 c. 39: 4 3 3 3 4 3 3 4 2 4 3 3 SP D3 SD: 8.4765 c. M:-15.1292 c. 41: 4 3 4 3 4 3 3 4 3 4 3 3 SP M ME SD: 3.1430 c. M:-5.3243 c. 53: 5 4 5 4 5 4 4 5 4 5 4 4 SP M ME SD: 0.4429 c. M: 0.7503 c. 200: 19 15 19 15 19 15 15 19 15 19 15 15 SP M DE SD: 0.2922 c. M:-0.4950 c. 253: 24 19 24 19 24 19 19 24 19 24 19 19 SP M DE SD: 0.1382 c. M:-0.2341 c. 306: 29 23 29 23 29 23 23 29 23 29 23 23 SP M DE SD: 0.0376 c. M:-0.0636 c. 359: 34 27 34 27 34 27 27 34 27 34 27 27 SP M DE SD: 0.0334 c. M: 0.0565 c. 665: 63 50 63 50 63 50 50 63 50 63 50 50 SP M DE SD: 0.0007 c. M: 0.0013 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: 10.6087 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 13.7737 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 5.8826 cents 6 x x 7 x 8 x 9 x 10:11 x 12 S SD: 11.1488 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 9.9093 cents 8 x 9 x 10:11 x 12:13 x 14:15:16 S SD: 8.8443 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 8.7370 cents 10 x 11:12:13 x 14:15:16:17:18:19:20 S SD: 6.8531 cents 11:12 x 13:14:15:16:17:18:19:20:21:22 S SD: 9.0291 cents 12:13 x 14:15:16:17:18:19:20:22:23:24 S SD: 7.2213 cents 13:14:15:16 x 18 x 20:21:22:23:25:26 SD: 9.6493 cents 14:15:16:17:18:19:20:21:22:24:25:27:28 SD: 5.7624 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.4954 cents 16:17:18:19:20:22:23:24:26:27:29:30:32 SD: 4.7777 cents 23:25:26:28:29:31:33:35:37:39:41:44:46 SD: 4.6034 cents 24:26:27:29:30:32:34:36:38:41:43:46:48 SD: 4.5924 cents 25:27:28:30:32:34:36:38:40:42:45:47:50 SD: 3.9138 cents 26:28:29:31:33:35:37:39:42:44:47:49:52 SD: 2.7709 cents 30:32:34:36:38:41:43:45:48:51:54:57:60 SD: 2.4304 cents 40:43:45:48:51:54:57:60:64:68:72:76:80 SD: 1.8229 cents 45:48:51:54:57:61:64:68:72:76:81:85:90 SD: 1.7725 cents 48:51:54:58:61:65:68:72:77:81:86:91:96 SD: 1.7217 cents 56:60:63:67:71:76:80:84:90:95:101:106:112 SD: 1.4967 cents 60:64:68:72:76:81:85:90:96:101:108:114:120 SD: 1.3967 cents 70:75:79:84:89:95:100:105:112:118:126:133:140 SD: 1.2571 cents 71:76:80:85:90:96:101:107:114:120:128:135:142 SD: 1.0836 cents 80:85:90:96:101:108:114:120:128:135:144:152:160 SD: 0.8892 cents 90:96:101:108:114:122:128:135:144:152:162:171:180 SD: 0.6976 cents 106:113:119:127:134:143:151:159:170:179:191:201:212 SD: 0.6912 cents 120:128:135:144:152:162:171:180:192:203:216:228:240 SD: 0.5490 cents 128:137:144:154:162:173:182:192:205:216:231:243:256 SD: 0.4884 cents 174:186:196:209:220:235:248:261:279:294:314:330:348 SD: 0.4769 cents 176:188:198:211:223:238:251:264:282:297:317:334:352 SD: 0.4341 cents 193:206:217:232:244:261:275:290:309:326:348:366:386 SD: 0.4247 cents 207:221:233:249:262:280:295:311:332:349:373:393:414 SD: 0.4221 cents 208:222:234:250:263:281:296:312:333:351:375:395:416 SD: 0.2262 cents 304:325:342:365:385:411:433:456:487:513:548:577:608 SD: 0.2256 cents 342:365:385:411:433:462:487:513:548:577:616:649:684 SD: 0.1932 cents 384:410:432:461:486:519:547:576:615:648:692:729:768 SD: 0.1241 cents 512:547:576:615:648:692:729:768:820:864:923:972:1024 SD: 0.0921 cents