Scala analysis: xen15-gilson-just-chromatic
Just Intonation Chromatic Scale (JICS)
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 25/24 70.672427 classic chromatic semitone, minor chroma 2: 9/8 203.910002 major whole tone 3: 75/64 274.582429 classic augmented second 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 45/32 590.223716 diatonic tritone 7: 3/2 701.955001 perfect fifth 8: 25/16 772.627428 classic augmented fifth 9: 5/3 884.358713 major sixth, BP sixth 10: 225/128 976.537429 augmented sixth 11: 15/8 1088.268715 classic major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 25/24 70.6724 classic chromatic semitone, minor chroma 2: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 3: 25/24 70.6724 classic chromatic semitone, minor chroma 4: 16/15 111.7313 minor diatonic semitone 5: 16/15 111.7313 minor diatonic semitone 6: 135/128 92.1787 major chroma, major limma 7: 16/15 111.7313 minor diatonic semitone 8: 25/24 70.6724 classic chromatic semitone, minor chroma 9: 16/15 111.7313 minor diatonic semitone 10: 135/128 92.1787 major chroma, major limma 11: 16/15 111.7313 minor diatonic semitone 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 2 135/128 92.179 cents major chroma, major limma 1: 6 16/15 111.731 cents minor diatonic semitone 1: 1 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 2: 4 10/9 182.404 cents minor whole tone 2: 6 9/8 203.910 cents major whole tone 2: 2 256/225 223.463 cents Neapolitan diminished third 3: 3 75/64 274.582 cents classic augmented second 3: 3 32/27 294.135 cents Pythagorean minor third 3: 6 6/5 315.641 cents minor third 4: 8 5/4 386.314 cents major third 4: 4 32/25 427.373 cents classic diminished fourth 5: 1 675/512 478.492 cents wide augmented third 5: 9 4/3 498.045 cents perfect fourth 5: 2 27/20 519.551 cents acute fourth 6: 2 25/18 568.717 cents classic augmented fourth 6: 4 45/32 590.224 cents diatonic tritone 6: 4 64/45 609.776 cents 2nd tritone 6: 2 36/25 631.283 cents classic diminished fifth 7: 2 40/27 680.449 cents grave fifth 7: 9 3/2 701.955 cents perfect fifth 7: 1 1024/675 721.508 cents narrow diminished sixth 8: 4 25/16 772.627 cents classic augmented fifth 8: 8 8/5 813.686 cents minor sixth 9: 6 5/3 884.359 cents major sixth, BP sixth 9: 3 27/16 905.865 cents Pythagorean major sixth 9: 3 128/75 925.418 cents diminished seventh 10: 2 225/128 976.537 cents augmented sixth 10: 6 16/9 996.090 cents Pythagorean minor seventh 10: 4 9/5 1017.596 cents just minor seventh, BP seventh 11: 1 50/27 1066.762 cents grave major seventh 11: 6 15/8 1088.269 cents classic major seventh 11: 2 256/135 1107.821 cents octave - major chroma 11: 3 48/25 1129.328 cents classic diminished octave Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.09091 = 34/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 70.7 203.9 274.6 386.3 498.0 590.2 702.0 772.6 884.4 976.5 1088.3 1200.0 70.7 : 133.2 203.9 315.6 427.4 519.6 631.3 702.0 813.7 905.9 1017.6 1129.3 1200.0 203.9 : 70.7 182.4 294.1 386.3 498.0 568.7 680.4 772.6 884.4 996.1 1066.8 1200.0 274.6 : 111.7 223.5 315.6 427.4 498.0 609.8 702.0 813.7 925.4 996.1 1129.3 1200.0 386.3 : 111.7 203.9 315.6 386.3 498.0 590.2 702.0 813.7 884.4 1017.6 1088.3 1200.0 498.0 : 92.2 203.9 274.6 386.3 478.5 590.2 702.0 772.6 905.9 976.5 1088.3 1200.0 590.2 : 111.7 182.4 294.1 386.3 498.0 609.8 680.4 813.7 884.4 996.1 1107.8 1200.0 702.0 : 70.7 182.4 274.6 386.3 498.0 568.7 702.0 772.6 884.4 996.1 1088.3 1200.0 772.6 : 111.7 203.9 315.6 427.4 498.0 631.3 702.0 813.7 925.4 1017.6 1129.3 1200.0 884.4 : 92.2 203.9 315.6 386.3 519.6 590.2 702.0 813.7 905.9 1017.6 1088.3 1200.0 976.5 : 111.7 223.5 294.1 427.4 498.0 609.8 721.5 813.7 925.4 996.1 1107.8 1200.0 1088.3: 111.7 182.4 315.6 386.3 498.0 609.8 702.0 813.7 884.4 996.1 1088.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 70.6724 .. 133.2376 cents 648/625, 62.5651 cents 25/24, 70.6724 cents 2: 182.4037 .. 223.4626 cents 128/125, 41.0589 cents 250/243, 49.1661 cents 3: 274.5824 .. 315.6413 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 4: 386.3137 .. 427.3726 cents 128/125, 41.0589 cents 25/24, 70.6724 cents 5: 478.4924 .. 519.5513 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 6: 568.7174 .. 631.2826 cents 648/625, 62.5651 cents 250/243, 49.1661 cents 7: 680.4487 .. 721.5076 cents 128/125, 41.0589 cents 250/243, 49.1661 cents 8: 772.6274 .. 813.6863 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 9: 884.3587 .. 925.4176 cents 128/125, 41.0589 cents 25/24, 70.6724 cents 10: 976.5374 .. 1017.5963 cents 128/125, 41.0589 cents 16875/16384, 51.1199 cents 11: 1066.7624 .. 1129.3276 cents 648/625, 62.5651 cents 250/243, 49.1661 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 25/24, 70.6724 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 27/25, 133.2376 cents Average / Smallest step : 1.414979 Largest / Average step : 1.332376 Largest / Smallest step : 1.885284 Median interval of one step : 16/15, 111.7313 cents, amount: 6 Most common interval of one step : 16/15, 111.7313 cents, amount: 6 Least squares average step : 98.56819 cents, oct.: 1182.81831 cents Scale is strictly proper Step pattern alph. order: ABACCDCACDCC Step pattern size order : ADACCBCACBCC Scale is a Constant Structure, by a margin of 49.16614 cents Scale diversity : 1.332962 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.569861 Rothenberg efficiency : 0.439743 redundancy: 0.560257 Efficiency x scale size : 5.276912 Number of different interval sizes : 34 = 3.09091 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.09091 = 34/11 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 2048/2025, 19.5526 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 9 Most common triad is 0.0 498.045 701.955 cents, amount: 6 Number of recognisable fifths : 9, average 701.9550 cents Best fifths form a closed circle Scale is fully connected 5-limit major-minor triadic Scale is a complete genus musicum : 2^m 3^3 5^2 Scale contains a complete diamond : 1 3 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 25.0000% Scale is differentially coherent in interval classes 1 and 2 combined Scale is n-1 differentially coherent in interval classes 2 and 3 combined Scale is n-1 differentially coherent in interval classes 3 and 4 combined Scale is n-1 differentially coherent in interval classes 4 and 5 combined Scale is n-1 differentially coherent in interval classes 8 and 9 combined Limited inverse transpositions : 3 Inversional symmetry on intervals : 1-2 7-8 -- Rational properties -- Prime limit : 5 Odd number limit : 675 (O: 675 U: 675) Highest odd numerator or denominator: 225 Scale harmonicity : 0.033557 Average absolute harmonicity : 0.181586 Specific harmonicity : 0.167260 Fundamental : 1/384, -8.5850 octaves, 0.6813 Hz. Guide tone : 900, 9.8138 octaves, 235463.009 Hz. Exponens Consonantiae : 3.456000E+05, 18.39874 octaves Euler's gradus suavitatis : 24 Sum of Mann's harmonic distance : 365.5, average 30.45833 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 13.15885, average 1.09657 Sum of Tenney's harmonic distance : 25.25120, average 2.10427 Vogel's harmonic complexity : 13.91667 Wille's k value : 337 Wilson's harmonic complexity : 19 Rectangular lattice diameter : 5 Triangular lattice diameter : 5 Lattice compactness : 301.45509, average 3.86481 Lattice compactness (without 2's) : 144.81449, average 1.85660 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -7 .. 2 -2.58333 37 1 3: -1 .. 2 0.50000 12 3 5: 0 .. 2 1.00000 12 8 Average exponent except 2's : 18 / 12 = 1.50000 Average absolute exponent except 2's: 24 / 12 = 2.00000 Scale is weakly epimorphic with val: <12 17 28| Scale is JI-epimorphic with val: <12 19 28| = patent Scale is weakly epimorphic with val: <12 21 25| Scale is weakly epimorphic with val: <12 21 26| Scale is weakly epimorphic with val: <12 21 28| Scale is weakly epimorphic with val: <12 21 29| Scale is weakly epimorphic with val: <12 21 31|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 17.0636 c. M:-29.3276 c. Twelve-tone Chromatic (1/11-comma) 19: 1 2 1 2 2 1 2 1 2 1 2 2 P M ME SD: 14.8917 c. M: 29.1690 c. Mandelbaum Twelve out of 19 20: 1 2 2 1 2 2 2 1 2 1 2 2 P D3 SD: 16.8615 c. M: 26.3137 c. 22: 1 3 1 2 2 2 2 1 2 2 2 2 P D3 SD: 8.8309 c. M: 16.1270 c. 31: 2 3 2 3 3 2 3 2 3 2 3 3 SP M ME SD: 6.1013 c. M: 10.3616 c. 34: 2 4 2 3 3 3 3 2 3 3 3 3 SP D3 SD: 6.0693 c. M:-11.6979 c. 53: 3 6 3 5 5 4 5 3 5 4 5 5 SP SD: 1.8459 c. M: 2.9525 c. 65: 4 7 4 6 6 5 6 4 6 5 6 6 SP SD: 1.6892 c. M:-3.1737 c. 118: 7 13 7 11 11 9 11 7 11 9 11 11 SP SD: 0.3087 c. M: 0.5202 c. 323: 19 36 19 30 30 25 30 19 30 25 30 30 SP SD: 0.2965 c. M:-0.5524 c. 376: 22 42 22 35 35 29 35 22 35 29 35 35 SP SD: 0.2329 c. M: 0.4597 c. 441: 26 49 26 41 41 34 41 26 41 34 41 41 SP SD: 0.1699 c. M:-0.3333 c. 494: 29 55 29 46 46 38 46 29 46 38 46 46 SP SD: 0.1100 c. M: 0.2271 c. 559: 33 62 33 52 52 43 52 33 52 43 52 52 SP SD: 0.1228 c. M:-0.2068 c. 612: 36 68 36 57 57 47 57 36 57 47 57 57 SP SD: 0.0488 c. M: 0.0842 c. 1171: 69 130 69 109 109 90 109 69 109 90 109 109 SP SD: 0.0346 c. M:-0.0638 c. 1665: 98 185 98 155 155 128 155 98 155 128 155 155 SP SD: 0.0314 c. M:-0.0540 c. 1783: 105 198 105 166 166 137 166 105 166 137 166 166 SP SD: 0.0105 c. M:-0.0189 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: 0.0000 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 1.9279 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 14.3066 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 6.6104 cents 7 x 8 x 9 x 10 x 11 x 12:13:14 S SD: 11.1769 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 7.3891 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.6446 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 10.1257 cents 11 x 12:13:14:15 x x 17:18:19:21:22 SD: 10.4244 cents 12 x x 14:15:16:17:18:19:20:21:23:24 SD: 4.7496 cents 13:14 x 15:16:17:18 x 20:22:23:24:26 S SD: 9.4220 cents 14:15:16 x 18:19:20:21:22:23:25:26:28 SD: 8.5852 cents 15:16:17:18:19:20:21 x 23:25:26:28:30 SD: 6.9433 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 5.5367 cents 18:19:20:21:23:24:25:27:28:30:32:34:36 SD: 4.9744 cents 20:21 x 23:25:27:28:30:31:33:35:38:40 SD: 4.8599 cents 22:23:25:26:28:29:31:33:34:37:39:41:44 SD: 4.4595 cents 23:24:26:27:29:31:32:35:36:38:40:43:46 SD: 3.9357 cents 24:25:27:28:30:32:34:36:38:40:42:45:48 SD: 2.3688 cents 40:42:45:47:50:53:56:60:63:67:70:75:80 SD: 2.2254 cents 42:44:47:49:53:56:59:63:66:70:74:79:84 SD: 2.1367 cents 46:48:52:54:58:61:65:69:72:77:81:86:92 SD: 1.9909 cents 47:49:53:55:59:63:66:71:73:78:83:88:94 SD: 1.9555 cents 48:50:54:56:60:64:68:72:75:80:84:90:96 SD: 1.3998 cents 64:67:72:75:80:85:90:96:100:107:113:120:128 SD: 1.2035 cents 71:74:80:83:89:95:100:107:111:118:125:133:142 SD: 1.1631 cents 72:75:81:84:90:96:101:108:113:120:127:135:144 SD: 1.0942 cents 87:91:98:102:109:116:122:131:136:145:153:163:174 SD: 0.9966 cents 94:98:106:110:118:125:132:141:147:157:165:176:188 SD: 0.9648 cents 96:100:108:113:120:128:135:144:150:160:169:180:192 SD: 0.6745 cents 120:125:135:141:150:160:169:180:188:200:211:225:240 SD: 0.5854 cents 128:133:144:150:160:171:180:192:200:213:225:240:256 SD: 0.5104 cents 144:150:162:169:180:192:203:216:225:240:253:270:288 SD: 0.4210 cents 168:175:189:197:210:224:236:252:263:280:295:315:336 SD: 0.3611 cents 192:200:216:225:240:256:270:288:300:320:338:360:384 SD: 0.2136 cents 336:350:378:394:420:448:473:504:525:560:591:630:672 SD: 0.2001 cents 360:375:405:422:450:480:506:540:563:600:633:675:720 SD: 0.1586 cents 384:400:432:450:480:512:540:576:600:640:675:720:768 SD: 0.0000 cents