Scala analysis: xen03-wilson-positive-12
Positive, linear-mapped intonational system, 12 notes
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 2: 9/8 203.910002 major whole tone 3: 32/27 294.134997 Pythagorean minor third 4: 81/64 407.820003 Pythagorean major third 5: 4/3 498.044999 perfect fourth 6: 1024/729 588.269995 Pythagorean diminished fifth 7: 3/2 701.955001 perfect fifth 8: 128/81 792.179997 Pythagorean minor sixth 9: 27/16 905.865003 Pythagorean major sixth 10: 16/9 996.089998 Pythagorean minor seventh 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: 256/243 90.2250 limma, Pythagorean minor second 2: 2187/2048 113.6850 apotome 3: 256/243 90.2250 limma, Pythagorean minor second 4: 2187/2048 113.6850 apotome 5: 256/243 90.2250 limma, Pythagorean minor second 6: 256/243 90.2250 limma, Pythagorean minor second 7: 2187/2048 113.6850 apotome 8: 256/243 90.2250 limma, Pythagorean minor second 9: 2187/2048 113.6850 apotome 10: 256/243 90.2250 limma, Pythagorean minor second 11: 2187/2048 113.6850 apotome 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 : 90.2 203.9 294.1 407.8 498.0 588.3 702.0 792.2 905.9 996.1 1109.8 1200.0 90.2 : 113.7 203.9 317.6 407.8 498.0 611.7 702.0 815.6 905.9 1019.6 1109.8 1200.0 203.9 : 90.2 203.9 294.1 384.4 498.0 588.3 702.0 792.2 905.9 996.1 1086.3 1200.0 294.1 : 113.7 203.9 294.1 407.8 498.0 611.7 702.0 815.6 905.9 996.1 1109.8 1200.0 407.8 : 90.2 180.4 294.1 384.4 498.0 588.3 702.0 792.2 882.4 996.1 1086.3 1200.0 498.0 : 90.2 203.9 294.1 407.8 498.0 611.7 702.0 792.2 905.9 996.1 1109.8 1200.0 588.3 : 113.7 203.9 317.6 407.8 521.5 611.7 702.0 815.6 905.9 1019.6 1109.8 1200.0 702.0 : 90.2 203.9 294.1 407.8 498.0 588.3 702.0 792.2 905.9 996.1 1086.3 1200.0 792.2 : 113.7 203.9 317.6 407.8 498.0 611.7 702.0 815.6 905.9 996.1 1109.8 1200.0 905.9 : 90.2 203.9 294.1 384.4 498.0 588.3 702.0 792.2 882.4 996.1 1086.3 1200.0 996.1 : 113.7 203.9 294.1 407.8 498.0 611.7 702.0 792.2 905.9 996.1 1109.8 1200.0 1109.8: 90.2 180.4 294.1 384.4 498.0 588.3 678.5 792.2 882.4 996.1 1086.3 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 : 100.00602 cents, oct.: 1200.07218 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: 5 of 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: 50 113 163 226 276 326 389 439 502 552 615 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABABAABABABA Step pattern size order : SLSLSSLSLSLS 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 90.225 701.955 cents Most common triad is 0.0 498.045 701.955 cents, amount: 10 Number of recognisable fifths : 11, average 701.9550 cents Scale contains two identical disjunct hexachords Best fifths form a closed circle Scale is a complete genus musicum : 2^m 3^11 Scale contains a complete diamond : 1 3 9 27 81 243 Formal octave complements present : 11 = 91.6667% 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: 729 Scale harmonicity : 0.021583 Average absolute harmonicity : 0.177127 Specific harmonicity : 0.124021 Fundamental : 1/93312, -16.5098 octaves, 0.0028 Hz. Guide tone : 248832, 17.9248 octaves, 65100812.665 Hz. Exponens Consonantiae : 2.321901E+10, 34.43459 octaves Euler's gradus suavitatis : 40 Sum of Mann's harmonic distance : 1568.0, average 130.66667 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 17.17637, average 1.43136 Sum of Tenney's harmonic distance : 34.63610, average 2.88634 Vogel's harmonic complexity : 24.83333 Wille's k value : 88573 Wilson's harmonic complexity : 33 Rectangular lattice diameter : 11 Triangular lattice diameter : 11 Lattice compactness : 598.75278, average 7.67632 Lattice compactness (without 2's) : 322.00000, average 4.12821 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -7 .. 10 1.33333 58 1 3: -6 .. 5 -0.50000 36 11 Average exponent except 2's :-6 / 12 =-0.50000 Average absolute exponent except 2's: 36 / 12 = 3.00000 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: 6.8192 c. M:-11.7300 c. Twelve-tone Chromatic (1/11-comma) 29: 2 3 2 3 2 2 3 2 3 2 3 2 SP M ME SD: 5.2087 c. M: 8.9596 c. 41: 3 4 3 4 3 3 4 3 4 3 4 3 SP M ME SD: 1.6883 c. M: 2.9041 c. 53: 4 5 4 5 4 4 5 4 5 4 5 4 SP M ME SD: 0.2379 c. M:-0.4093 c. 200: 15 19 15 19 15 15 19 15 19 15 19 15 SP M DE SD: 0.1570 c. M: 0.2700 c. 253: 19 24 19 24 19 19 24 19 24 19 24 19 SP M DE SD: 0.0742 c. M: 0.1277 c. 306: 23 29 23 29 23 23 29 23 29 23 29 23 SP M DE SD: 0.0202 c. M: 0.0347 c. 359: 27 34 27 34 27 27 34 27 34 27 34 27 SP M DE SD: 0.0179 c. M:-0.0308 c. 665: 50 63 50 63 50 50 63 50 63 50 63 50 SP M DE SD: 0.0004 c. M:-0.0007 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: 7.1688 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 8.6813 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.5385 cents 6 x x 7 x 8 x 9 x 10:11 x 12 S SD: 10.5986 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 9.7175 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 9.1612 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 9.2430 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.0858 cents 11 x 12:13:14:15 x x 17:19:20:21:22 SD: 10.6512 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.5878 cents 13:14:15 x 16:17:18:20:21:22:23:25:26 SD: 9.7153 cents 14:15:16:17:18:19:20:21:22:24:25:27:28 SD: 7.2344 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.7811 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.5201 cents 18:19:20:21:23:24:25:27:28:30:32:34:36 SD: 4.6810 cents 22:23:25:26:28:29:31:33:35:37:39:42:44 SD: 3.0113 cents 32:34:36:38:41:43:45:48:51:54:57:61:64 SD: 2.8185 cents 33:35:37:39:42:44:46:50:52:56:59:63:66 SD: 2.7319 cents 38:40:43:45:48:51:53:57:60:64:68:72:76 SD: 1.9556 cents 42:44:47:50:53:56:59:63:66:71:75:80:84 SD: 1.7862 cents 54:57:61:64:68:72:76:81:85:91:96:103:108 SD: 1.3635 cents 64:67:72:76:81:85:90:96:101:108:114:122:128 SD: 1.3116 cents 71:75:80:84:90:95:100:107:112:120:126:135:142 SD: 1.1887 cents 72:76:81:85:91:96:101:108:114:122:128:137:144 SD: 1.0068 cents 86:91:97:102:109:115:121:129:136:145:153:163:172 SD: 0.9586 cents 90:95:101:107:114:120:126:135:142:152:160:171:180 SD: 0.8558 cents 96:101:108:114:122:128:135:144:152:162:171:182:192 SD: 0.8319 cents 98:103:110:116:124:131:138:147:155:165:174:186:196 SD: 0.8248 cents 109:115:123:129:138:145:153:164:172:184:194:207:218 SD: 0.8206 cents 112:118:126:133:142:149:157:168:177:189:199:213:224 SD: 0.6377 cents 128:135:144:152:162:171:180:192:202:216:228:243:256 SD: 0.5730 cents 136:143:153:161:172:181:191:204:215:230:242:258:272 SD: 0.5633 cents 150:158:169:178:190:200:211:225:237:253:267:285:300 SD: 0.4325 cents 200:211:225:237:253:267:281:300:316:338:356:380:400 SD: 0.4174 cents 207:218:233:245:262:276:291:311:327:349:368:393:414 SD: 0.3654 cents 210:221:236:249:266:280:295:315:332:354:373:399:420 SD: 0.3511 cents 222:234:250:263:281:296:312:333:351:375:395:421:444 SD: 0.3193 cents 264:278:297:313:334:352:371:396:417:446:469:501:528 SD: 0.2401 cents 297:313:334:352:376:396:417:446:469:501:528:564:594 SD: 0.2297 cents 336:354:378:398:425:448:472:504:531:567:597:638:672 SD: 0.1458 cents 432:455:486:512:547:576:607:648:683:729:768:820:864 SD: 0.1139 cents 576:607:648:683:729:768:809:864:910:972:1024:1094:1152 SD: 0.1128 cents 598:630:673:709:757:797:840:897:945:1009:1063:1135:1196 SD: 0.1090 cents