Scala analysis: xen15-gilson-generalized-just-3
Scale based on product (21/20)**3 * (16/15)**2 * (15/14)**3 * (10/9)**2 = 2
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 15/14 119.442808 major diatonic semitone 2: 8/7 231.174094 septimal whole tone 3: 6/5 315.641287 minor third 4: 9/7 435.084095 septimal major third, BP third 5: 10/7 617.487807 Euler's tritone 6: 3/2 701.955001 perfect fifth 7: 8/5 813.686286 minor sixth 8: 12/7 933.129094 septimal major sixth 9: 9/5 1017.596288 just minor seventh, BP seventh 10: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 10/9 182.4037 minor whole tone 1: 15/14 119.4428 major diatonic semitone 2: 16/15 111.7313 minor diatonic semitone 3: 21/20 84.4672 minor semitone 4: 15/14 119.4428 major diatonic semitone 5: 10/9 182.4037 minor whole tone 6: 21/20 84.4672 minor semitone 7: 16/15 111.7313 minor diatonic semitone 8: 15/14 119.4428 major diatonic semitone 9: 21/20 84.4672 minor semitone 10: 10/9 182.4037 minor whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 21/20 84.467 cents minor semitone 1: 2 16/15 111.731 cents minor diatonic semitone 1: 3 15/14 119.443 cents major diatonic semitone 1: 2 10/9 182.404 cents minor whole tone 2: 2 28/25 196.198 cents middle second 2: 2 9/8 203.910 cents major whole tone 2: 2 8/7 231.174 cents septimal whole tone 2: 2 7/6 266.871 cents septimal minor third 2: 2 25/21 301.847 cents BP second, quasi-equal minor third 3: 4 6/5 315.641 cents minor third 3: 1 56/45 378.602 cents septimal narrow major third 3: 4 5/4 386.314 cents major third 3: 1 80/63 413.578 cents wide major third 4: 1 63/50 400.108 cents quasi-equal major third 4: 1 9/7 435.084 cents septimal major third, BP third 4: 1 21/16 470.781 cents narrow fourth 4: 6 4/3 498.045 cents perfect fourth 4: 1 75/56 505.757 cents marvelous fourth 5: 5 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 5: 5 10/7 617.488 cents Euler's tritone 6: 1 112/75 694.243 cents marvelous fifth 6: 6 3/2 701.955 cents perfect fifth 6: 1 32/21 729.219 cents wide fifth 6: 1 14/9 764.916 cents septimal minor sixth 6: 1 100/63 799.892 cents quasi-equal minor sixth 7: 1 63/40 786.422 cents narrow minor sixth 7: 4 8/5 813.686 cents minor sixth 7: 1 45/28 821.398 cents septimale wide minor sixth 7: 4 5/3 884.359 cents major sixth, BP sixth 8: 2 42/25 898.153 cents quasi-equal major sixth 8: 2 12/7 933.129 cents septimal major sixth 8: 2 7/4 968.826 cents harmonic seventh 8: 2 16/9 996.090 cents Pythagorean minor seventh 8: 2 25/14 1003.802 cents middle minor seventh 9: 2 9/5 1017.596 cents just minor seventh, BP seventh 9: 3 28/15 1080.557 cents grave major seventh 9: 2 15/8 1088.269 cents classic major seventh 9: 3 40/21 1115.533 cents acute major seventh Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 4.22222 = 38/9
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 0.0 : 119.4 231.2 315.6 435.1 617.5 702.0 813.7 933.1 1017.6 1200.0 119.4 : 111.7 196.2 315.6 498.0 582.5 694.2 813.7 898.2 1080.6 1200.0 231.2 : 84.5 203.9 386.3 470.8 582.5 702.0 786.4 968.8 1088.3 1200.0 315.6 : 119.4 301.8 386.3 498.0 617.5 702.0 884.4 1003.8 1115.5 1200.0 435.1 : 182.4 266.9 378.6 498.0 582.5 764.9 884.4 996.1 1080.6 1200.0 617.5 : 84.5 196.2 315.6 400.1 582.5 702.0 813.7 898.2 1017.6 1200.0 702.0 : 111.7 231.2 315.6 498.0 617.5 729.2 813.7 933.1 1115.5 1200.0 813.7 : 119.4 203.9 386.3 505.8 617.5 702.0 821.4 1003.8 1088.3 1200.0 933.1 : 84.5 266.9 386.3 498.0 582.5 702.0 884.4 968.8 1080.6 1200.0 1017.6: 182.4 301.8 413.6 498.0 617.5 799.9 884.4 996.1 1115.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 84.4672 .. 182.4037 cents 200/189, 97.9365 cents 21/20, 84.4672 cents 2: 196.1985 .. 301.8465 cents 625/588, 105.6480 cents 126/125, 13.7948 cents 3: 315.6413 .. 413.5778 cents 200/189, 97.9365 cents 126/125, 13.7948 cents 4: 400.1085 .. 505.7565 cents 625/588, 105.6480 cents 3969/4000,-13.4693 cents 5: 582.5122 .. 617.4878 cents 50/49, 34.9756 cents 392/375, 76.7557 cents 6: 694.2435 .. 799.8915 cents 625/588, 105.6480 cents 392/375, 76.7557 cents 7: 786.4222 .. 884.3587 cents 200/189, 97.9365 cents 3969/4000,-13.4693 cents 8: 898.1535 .. 1003.8015 cents 625/588, 105.6480 cents 126/125, 13.7948 cents 9: 1017.5963 .. 1115.5328 cents 200/189, 97.9365 cents 126/125, 13.7948 cents
SHOW DATA
Number of notes : 10 -- Interval properties -- Smallest interval : 21/20, 84.4672 cents, class 1 Average step (divided formal octave): 120.0000 cents Largest one step interval : 10/9, 182.4037 cents Average / Smallest step : 1.420670 Largest / Average step : 1.520031 Largest / Smallest step : 2.159462 Median interval of one step : 115.58705 cents, amount: 0 Least squares average step : 116.59071 cents, oct.: 1165.90711 cents Scale is not proper Step pattern alph. order: ABCADCBACD Step pattern size order : CBACDABCAD Scale is a Constant Structure, by a margin of 13.46933 cents Scale diversity : 1.394628 Lumma stability : 0.314687 Lumma impropriety factor : 0.022449 Rothenberg efficiency : 0.362698 redundancy: 0.637302 Efficiency x scale size : 3.626984 Number of different interval sizes : 38 = 4.22222 / class Number of one step interval sizes : 4 Highest interval variety : 5 Mean interval variety : 4.22222 = 38/9 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 225/224, 7.7115 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 6 Number of recognisable fifths : 7, average 700.8534 cents All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 0 Length of longest subharmonic part : 2 at degree 6 Formal octave complements present : 1 = 10.0000% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval classes 2 and 3 combined Scale is 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 5 and 6 combined Scale is differentially coherent in interval classes 6 and 7 combined Scale is n-1 differentially coherent in interval classes 7 and 8 combined -- Rational properties -- Prime limit : 7 Odd number limit : 75 (O: 75 U: 75) Highest odd numerator or denominator: 15 Scale harmonicity : 0.030846 Average absolute harmonicity : 0.187552 Specific harmonicity : 0.152130 Fundamental : 1/70, -6.1293 octaves, 3.7375 Hz. Guide tone : 360, 8.4919 octaves, 94185.204 Hz. Exponens Consonantiae : 2.520000E+04, 14.62114 octaves Euler's gradus suavitatis : 23 Sum of Mann's harmonic distance : 71.0, average 7.10000 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 7.49493, average 0.74949 Sum of Tenney's harmonic distance : 15.45070, average 1.54507 Vogel's harmonic complexity : 12.30000 Wille's k value : 52 Wilson's harmonic complexity : 23 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 134.19380, average 2.43989 Lattice compactness (without 2's) : 86.06591, average 1.56483 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 3 0.90000 13 1 3: 0 .. 2 0.80000 8 1 5: -1 .. 1 -0.10000 5 3 7: -1 .. 0 -0.50000 5 5 Average exponent except 2's : 2 / 10 = 0.20000 Average absolute exponent except 2's: 18 / 10 = 1.80000 Scale is JI-epimorphic with val: <10 16 23 28| = patent Scale is weakly epimorphic with val: <10 17 21 26| Scale is weakly epimorphic with val: <10 18 24 25| Scale is weakly epimorphic with val: <10 18 24 27| Scale is weakly epimorphic with val: <10 18 24 29| Scale is weakly epimorphic with val: <10 18 24 31|
FIT/MODE
10: 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 31.6283 c. M:-62.4037 c. 12: 1 1 1 1 2 1 1 1 1 2 P B ME S SD: 21.7475 c. M: 35.0841 c. Messiaen mode 7 19: 2 2 1 2 3 1 2 2 1 3 N D3 SD: 10.5577 c. M:-21.4575 c. 22: 2 2 2 2 3 2 2 2 2 3 SP B ME S SD: 10.8460 c. M:-18.7673 c. Dynamic Symmetrical Minor 27: 3 2 2 3 4 2 2 3 2 4 P SD: 8.3440 c. M:-13.8905 c. 31: 3 3 2 3 5 2 3 3 2 5 N SD: 5.5293 c. M: 11.1447 c. 41: 4 4 3 4 6 3 4 4 3 6 P SD: 4.1257 c. M:-6.7940 c. 53: 5 5 4 5 8 4 5 5 4 8 P SD: 3.9016 c. M: 6.2353 c. 58: 6 5 4 6 9 4 5 6 4 9 N SD: 3.7519 c. M: 6.7897 c. 68: 7 6 5 7 10 5 6 7 5 10 P SD: 3.4638 c. M:-6.0924 c. 72: 7 7 5 7 11 5 7 7 5 11 N SD: 1.7542 c. M:-2.9804 c. 99: 10 9 7 10 15 7 9 10 7 15 N SD: 1.0118 c. M:-1.7693 c. 130: 13 12 9 13 20 9 12 13 9 20 N SD: 1.1755 c. M: 2.2117 c. 140: 14 13 10 14 21 10 13 14 10 21 N SD: 1.2360 c. M:-2.4037 c. 152: 15 14 11 15 23 11 14 15 11 23 N SD: 1.1615 c. M: 2.2267 c. 171: 17 16 12 17 26 12 16 17 12 26 N SD: 0.2040 c. M:-0.4049 c. 441: 44 41 31 44 67 31 41 44 31 67 N SD: 0.1682 c. M:-0.2901 c. 612: 61 57 43 61 93 43 57 61 43 93 N SD: 0.1356 c. M:-0.2100 c. 783: 78 73 55 78 119 55 73 78 55 119 N SD: 0.1322 c. M:-0.2435 c. 935: 93 87 66 93 142 66 87 93 66 142 N SD: 0.1011 c. M: 0.1616 c. 1106: 110 103 78 110 168 78 103 110 78 168 N SD: 0.0774 c. M: 0.1280 c. 1236: 123 115 87 123 188 87 115 123 87 188 N SD: 0.0889 c. M: 0.1326 c. 1277: 127 119 90 127 194 90 119 127 90 194 N SD: 0.0708 c. M: 0.1033 c. 1376: 137 128 97 137 209 97 128 137 97 209 N SD: 0.0671 c. M:-0.1363 c. 1407: 140 131 99 140 214 99 131 140 99 214 N SD: 0.0669 c. M: 0.1161 c. 1547: 154 144 109 154 235 109 144 154 109 235 N SD: 0.0545 c. M:-0.1154 c. 1677: 167 156 118 167 255 118 156 167 118 255 N SD: 0.0525 c. M: 0.0906 c. 1718: 171 160 121 171 261 121 160 171 121 261 N SD: 0.0524 c. M:-0.0987 c. 1848: 184 172 130 184 281 130 172 184 130 281 N SD: 0.0372 c. M: 0.0638 c. 2019: 201 188 142 201 307 142 188 201 142 307 N SD: 0.0318 c. M: 0.0629 c. 2190: 218 204 154 218 333 154 204 218 154 333 N SD: 0.0352 c. M: 0.0620 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: 0.0000 cents 3 x x x 4 x x x 5 x 6 S SD: 26.5469 cents 4 x x x 5 x 6 x 7 x 8 S SD: 15.1096 cents 5 x x 6 x 7 x 8 x 9:10 S SD: 6.9951 cents 6 x 7 x 8 x 9 x 10:11:12 S SD: 15.4794 cents 7 x 8 x 9:10 x 11:12:13:14 S SD: 8.9220 cents 8 x 9 x 10:11:12:13:14 x 16 S SD: 13.9204 cents 9 x 10:11:12:13 x 14:15:16:18 S SD: 14.2178 cents 10:11 x 12:13:14:15:16:17:18:20 S SD: 6.9166 cents 11:12 x 13:14:16:17:18:19:20:22 S SD: 9.6940 cents 12:13:14 x 15:17:18:19:21:22:24 SD: 9.1898 cents 13:14:15:16:17:19:20:21:22:23:26 SD: 9.0999 cents 14:15:16:17:18:20:21:22:24:25:28 SD: 3.9789 cents 28:30:32:34:36:40:42:45:48:50:56 SD: 2.5875 cents 40:43:46:48:51:57:60:64:69:72:80 SD: 2.2239 cents 42:45:48:50:54:60:63:67:72:76:84 SD: 1.7332 cents 55:59:63:66:71:79:83:88:94:99:110 SD: 1.7155 cents 56:60:64:67:72:80:84:90:96:101:112 SD: 0.9893 cents 70:75:80:84:90:100:105:112:120:126:140 S SD: 0.0000 cents