Scala analysis: kesred12_5
Kees reduced 5-limit 12-note scale = Hahn reduced
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 16/15 111.731285 minor diatonic semitone 2: 9/8 203.910002 major whole tone 3: 6/5 315.641287 minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 25/18 568.717426 classic augmented fourth 7: 3/2 701.955001 perfect fifth 8: 8/5 813.686286 minor sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 9/5 1017.596288 just minor seventh, BP seventh 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: 16/15 111.7313 minor diatonic semitone 2: 135/128 92.1787 major chroma, major limma 3: 16/15 111.7313 minor diatonic semitone 4: 25/24 70.6724 classic chromatic semitone, minor chroma 5: 16/15 111.7313 minor diatonic semitone 6: 25/24 70.6724 classic chromatic semitone, minor chroma 7: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 8: 16/15 111.7313 minor diatonic semitone 9: 25/24 70.6724 classic chromatic semitone, minor chroma 10: 27/25 133.2376 large limma, BP small semitone, Zarlino semitone 11: 25/24 70.6724 classic chromatic semitone, minor chroma 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 1 135/128 92.179 cents major chroma, major limma 1: 5 16/15 111.731 cents minor diatonic semitone 1: 2 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 2: 5 10/9 182.404 cents minor whole tone 2: 5 9/8 203.910 cents major whole tone 2: 1 256/225 223.463 cents Neapolitan diminished third 2: 1 144/125 244.969 cents classic diminished third 3: 1 125/108 253.076 cents semi-augmented whole tone 3: 2 75/64 274.582 cents classic augmented second 3: 2 32/27 294.135 cents Pythagorean minor third 3: 7 6/5 315.641 cents minor third 4: 1 100/81 364.807 cents grave major third 4: 7 5/4 386.314 cents major third 4: 3 32/25 427.373 cents classic diminished fourth 4: 1 162/125 448.879 cents 5: 1 125/96 456.986 cents classic augmented third 5: 8 4/3 498.045 cents perfect fourth 5: 3 27/20 519.551 cents acute fourth 6: 3 25/18 568.717 cents classic augmented fourth 6: 3 45/32 590.224 cents diatonic tritone 6: 3 64/45 609.776 cents 2nd tritone 6: 3 36/25 631.283 cents classic diminished fifth 7: 3 40/27 680.449 cents grave fifth 7: 8 3/2 701.955 cents perfect fifth 7: 1 192/125 743.014 cents classic diminished sixth 8: 1 125/81 751.121 cents 8: 3 25/16 772.627 cents classic augmented fifth 8: 7 8/5 813.686 cents minor sixth 8: 1 81/50 835.193 cents acute minor sixth 9: 7 5/3 884.359 cents major sixth, BP sixth 9: 2 27/16 905.865 cents Pythagorean major sixth 9: 2 128/75 925.418 cents diminished seventh 9: 1 216/125 946.924 cents semi-augmented sixth 10: 1 125/72 955.031 cents classic augmented sixth 10: 1 225/128 976.537 cents augmented sixth 10: 5 16/9 996.090 cents Pythagorean minor seventh 10: 5 9/5 1017.596 cents just minor seventh, BP seventh 11: 2 50/27 1066.762 cents grave major seventh 11: 5 15/8 1088.269 cents classic major seventh 11: 1 256/135 1107.821 cents octave - major chroma 11: 4 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: 3.81818 = 42/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 111.7 203.9 315.6 386.3 498.0 568.7 702.0 813.7 884.4 1017.6 1088.3 1200.0 111.7 : 92.2 203.9 274.6 386.3 457.0 590.2 702.0 772.6 905.9 976.5 1088.3 1200.0 203.9 : 111.7 182.4 294.1 364.8 498.0 609.8 680.4 813.7 884.4 996.1 1107.8 1200.0 315.6 : 70.7 182.4 253.1 386.3 498.0 568.7 702.0 772.6 884.4 996.1 1088.3 1200.0 386.3 : 111.7 182.4 315.6 427.4 498.0 631.3 702.0 813.7 925.4 1017.6 1129.3 1200.0 498.0 : 70.7 203.9 315.6 386.3 519.6 590.2 702.0 813.7 905.9 1017.6 1088.3 1200.0 568.7 : 133.2 245.0 315.6 448.9 519.6 631.3 743.0 835.2 946.9 1017.6 1129.3 1200.0 702.0 : 111.7 182.4 315.6 386.3 498.0 609.8 702.0 813.7 884.4 996.1 1066.8 1200.0 813.7 : 70.7 203.9 274.6 386.3 498.0 590.2 702.0 772.6 884.4 955.0 1088.3 1200.0 884.4 : 133.2 203.9 315.6 427.4 519.6 631.3 702.0 813.7 884.4 1017.6 1129.3 1200.0 1017.6: 70.7 182.4 294.1 386.3 498.0 568.7 680.4 751.1 884.4 996.1 1066.8 1200.0 1088.3: 111.7 223.5 315.6 427.4 498.0 609.8 680.4 813.7 925.4 996.1 1129.3 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 .. 244.9689 cents 648/625, 62.5651 cents 250/243, 49.1661 cents 3: 253.0761 .. 315.6413 cents 648/625, 62.5651 cents 15625/15552, 8.1073 cents 4: 364.8074 .. 448.8789 cents 6561/6250, 84.0714 cents 250/243, 49.1661 cents 5: 456.9861 .. 519.5513 cents 648/625, 62.5651 cents 15625/15552, 8.1073 cents 6: 568.7174 .. 631.2826 cents 648/625, 62.5651 cents 250/243, 49.1661 cents 7: 680.4487 .. 743.0139 cents 648/625, 62.5651 cents 250/243, 49.1661 cents 8: 751.1211 .. 835.1926 cents 6561/6250, 84.0714 cents 15625/15552, 8.1073 cents 9: 884.3587 .. 946.9239 cents 648/625, 62.5651 cents 250/243, 49.1661 cents 10: 955.0311 .. 1017.5963 cents 648/625, 62.5651 cents 15625/15552, 8.1073 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: 5 Most common interval of one step : 16/15, 111.7313 cents, amount: 5 Least squares average step : 99.75936 cents, oct.: 1197.11228 cents Scale is strictly proper Step pattern alph. order: ABACACDACDCA Step pattern size order : CBCACADCADAC Scale is a Constant Structure, by a margin of 8.10728 cents Scale diversity : 1.409045 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.390642 Rothenberg efficiency : 0.361072 redundancy: 0.638928 Efficiency x scale size : 4.332864 Number of different interval sizes : 42 = 3.81818 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.81818 = 42/11 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 15625/15552, 8.1073 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 8 Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 8, average 701.9550 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 5 15 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 9 = 75.0000% Scale is n-1 differentially coherent in interval classes 1 and 2 combined Scale is 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 8 and 9 combined -- Rational properties -- Prime limit : 5 Odd number limit : 225 (O: 225 U: 225) Highest odd numerator or denominator: 25 Scale harmonicity : 0.027125 Average absolute harmonicity : 0.194914 Specific harmonicity : 0.164567 Fundamental : 1/360, -8.4919 octaves, 0.7267 Hz. Guide tone : 3600, 11.8138 octaves, 941852.035 Hz. Exponens Consonantiae : 1.296000E+06, 20.30563 octaves Euler's gradus suavitatis : 28 Sum of Mann's harmonic distance : 92.0, average 7.66667 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 9.40873, average 0.78406 Sum of Tenney's harmonic distance : 18.33782, average 1.52815 Vogel's harmonic complexity : 15.00000 Wille's k value : 337 Wilson's harmonic complexity : 27 Rectangular lattice diameter : 7 Triangular lattice diameter : 4 Lattice compactness : 265.11246, average 3.39888 Lattice compactness (without 2's) : 153.65339, average 1.96992 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -3 .. 4 0.08333 21 1 3: -2 .. 2 0.16667 12 3 5: -1 .. 2 0.08333 9 8 Average exponent except 2's : 3 / 12 = 0.25000 Average absolute exponent except 2's: 21 / 12 = 1.75000 Scale is JI-epimorphic with val: <12 19 28| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 14.2897 c. M:-31.2826 c. Twelve-tone Chromatic (1/11-comma) 19: 2 1 2 1 2 1 2 2 1 2 1 2 P M ME SD: 8.6442 c. M: 14.5845 c. Genus diatonico-chromaticum, Yasser's Supradiatonic inverse 31: 3 2 3 2 3 2 3 3 2 3 2 3 SP M ME SD: 6.6993 c. M:-11.9277 c. Genus diatonico-chromaticum 34: 3 3 3 2 3 2 4 3 2 4 2 3 P SD: 4.3559 c. M:-7.8547 c. 46: 4 4 4 3 4 3 5 4 3 5 3 4 SP SD: 4.4133 c. M:-7.3835 c. 53: 5 4 5 3 5 3 6 5 3 6 3 5 P SD: 1.3147 c. M: 2.6797 c. 99: 9 8 9 6 9 6 11 9 6 11 6 9 SP SD: 1.5155 c. M:-2.6404 c. 118: 11 9 11 7 11 7 13 11 7 13 7 11 SP SD: 0.3863 c. M:-0.7741 c. 171: 16 13 16 10 16 10 19 16 10 19 10 16 SP SD: 0.3192 c. M:-0.5494 c. 289: 27 22 27 17 27 17 32 27 17 32 17 27 SP SD: 0.2510 c. M: 0.4498 c. 323: 30 25 30 19 30 19 36 30 19 36 19 30 SP SD: 0.2402 c. M:-0.4244 c. 388: 36 30 36 23 36 23 43 36 23 43 23 36 SP SD: 0.2466 c. M:-0.3911 c. 441: 41 34 41 26 41 26 49 41 26 49 26 41 SP SD: 0.1005 c. M:-0.1716 c. 559: 52 43 52 33 52 33 62 52 33 62 33 52 SP SD: 0.0813 c. M:-0.1556 c. 612: 57 47 57 36 57 36 68 57 36 68 36 57 SP SD: 0.0411 c. M: 0.0900 c. 730: 68 56 68 43 68 43 81 68 43 81 43 68 SP SD: 0.0419 c. M: 0.0744 c. 1171: 109 90 109 69 109 69 130 109 69 130 69 109 SP SD: 0.0198 c. M:-0.0319 c. 1783: 166 137 166 105 166 105 198 166 105 198 105 166 SP SD: 0.0093 c. M:-0.0160 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: 12.1926 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 2.7590 cents 6 x x 7 x 8 x 9 x 10:11 x 12 S SD: 9.7006 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 13.6476 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 7.2906 cents 9 x 10:11 x 12 x x 14:15:16:17:18 S SD: 8.3628 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.2981 cents 11:12 x 13:14 x 15:17 x 18:20:21:22 SD: 10.6287 cents 12:13 x 14:15:16:17:18:19:20:22:23:24 S SD: 7.6385 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 8.4209 cents 14:15:16:17:18:19 x 21:22:23:25:26:28 SD: 7.3846 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 4.0711 cents 24:26:27:29:30:32:33:36:38:40:43:45:48 SD: 3.2913 cents 30:32:34:36:38:40:42:45:48:50:54:56:60 SD: 2.5533 cents 39:42:44:47:49:52:54:59:62:65:70:73:78 SD: 2.4071 cents 40:43:45:48:50:53:56:60:64:67:72:75:80 SD: 1.9794 cents 48:51:54:58:60:64:67:72:77:80:86:90:96 SD: 1.5564 cents 56:60:63:67:70:75:78:84:90:93:101:105:112 SD: 1.3934 cents 60:64:68:72:75:80:83:90:96:100:108:113:120 SD: 1.3701 cents 64:68:72:77:80:85:89:96:102:107:115:120:128 SD: 1.1812 cents 72:77:81:86:90:96:100:108:115:120:130:135:144 SD: 0.9217 cents 80:85:90:96:100:107:111:120:128:133:144:150:160 SD: 0.8202 cents 90:96:101:108:113:120:125:135:144:150:162:169:180 SD: 0.7630 cents 105:112:118:126:131:140:146:158:168:175:189:197:210 SD: 0.5862 cents 120:128:135:144:150:160:167:180:192:200:216:225:240 SD: 0.2883 cents 240:256:270:288:300:320:333:360:384:400:432:450:480 SD: 0.1443 cents 360:384:405:432:450:480:500:540:576:600:648:675:720 SD: 0.0000 cents