Scala analysis: JoanAlbertBan18tone
Pure 18-tone JI tone-scale (in dutch: 'toonschaal')
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: 16/15 111.731285 minor diatonic semitone 3: 10/9 182.403712 minor whole tone 4: 9/8 203.910002 major whole tone 5: 75/64 274.582429 classic augmented second 6: 6/5 315.641287 minor third 7: 5/4 386.313714 major third 8: 4/3 498.044999 perfect fourth 9: 25/18 568.717426 classic augmented fourth 10: 45/32 590.223716 diatonic tritone 11: 3/2 701.955001 perfect fifth 12: 25/16 772.627428 classic augmented fifth 13: 8/5 813.686286 minor sixth 14: 5/3 884.358713 major sixth, BP sixth 15: 16/9 996.089998 Pythagorean minor seventh 16: 9/5 1017.596288 just minor seventh, BP seventh 17: 15/8 1088.268715 classic major seventh 18: 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: 128/125 41.0589 minor diesis, diesis 3: 25/24 70.6724 classic chromatic semitone, minor chroma 4: 81/80 21.5063 syntonic comma, Didymus comma 5: 25/24 70.6724 classic chromatic semitone, minor chroma 6: 128/125 41.0589 minor diesis, diesis 7: 25/24 70.6724 classic chromatic semitone, minor chroma 8: 16/15 111.7313 minor diatonic semitone 9: 25/24 70.6724 classic chromatic semitone, minor chroma 10: 81/80 21.5063 syntonic comma, Didymus comma 11: 16/15 111.7313 minor diatonic semitone 12: 25/24 70.6724 classic chromatic semitone, minor chroma 13: 128/125 41.0589 minor diesis, diesis 14: 25/24 70.6724 classic chromatic semitone, minor chroma 15: 16/15 111.7313 minor diatonic semitone 16: 81/80 21.5063 syntonic comma, Didymus comma 17: 25/24 70.6724 classic chromatic semitone, minor chroma 18: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 3 81/80 21.506 cents syntonic comma, Didymus comma 1: 3 128/125 41.059 cents minor diesis, diesis 1: 8 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 4 16/15 111.731 cents minor diatonic semitone 2: 4 135/128 92.179 cents major chroma, major limma 2: 6 16/15 111.731 cents minor diatonic semitone 2: 2 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 2: 6 10/9 182.404 cents minor whole tone 3: 2 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 3: 1 1125/1024 162.851 cents double augmented prime 3: 3 10/9 182.404 cents minor whole tone 3: 6 9/8 203.910 cents major whole tone 3: 4 256/225 223.463 cents Neapolitan diminished third 3: 2 125/108 253.076 cents semi-augmented whole tone 4: 4 9/8 203.910 cents major whole tone 4: 2 144/125 244.969 cents classic diminished third 4: 4 75/64 274.582 cents classic augmented second 4: 6 32/27 294.135 cents Pythagorean minor third 4: 2 6/5 315.641 cents minor third 5: 1 144/125 244.969 cents classic diminished third 5: 2 75/64 274.582 cents classic augmented second 5: 8 6/5 315.641 cents minor third 5: 2 100/81 364.807 cents grave major third 5: 4 5/4 386.314 cents major third 5: 1 512/405 405.866 cents narrow diminished fourth 6: 2 6/5 315.641 cents minor third 6: 8 5/4 386.314 cents major third 6: 6 32/25 427.373 cents classic diminished fourth 6: 2 125/96 456.986 cents classic augmented third 7: 1 5/4 386.314 cents major third 7: 2 81/64 407.820 cents Pythagorean major third 7: 2 32/25 427.373 cents classic diminished fourth 7: 1 162/125 448.879 cents 7: 2 125/96 456.986 cents classic augmented third 7: 10 4/3 498.045 cents perfect fourth 8: 2 675/512 478.492 cents wide augmented third 8: 4 4/3 498.045 cents perfect fourth 8: 4 27/20 519.551 cents acute fourth 8: 2 512/375 539.104 cents double diminished fifth 8: 4 25/18 568.717 cents classic augmented fourth 8: 2 64/45 609.776 cents 2nd tritone 9: 2 27/20 519.551 cents acute fourth 9: 2 25/18 568.717 cents classic augmented fourth 9: 5 45/32 590.224 cents diatonic tritone 9: 5 64/45 609.776 cents 2nd tritone 9: 2 36/25 631.283 cents classic diminished fifth 9: 2 40/27 680.449 cents grave fifth 10: 2 45/32 590.224 cents diatonic tritone 10: 4 36/25 631.283 cents classic diminished fifth 10: 2 375/256 660.896 cents double augmented fourth 10: 4 40/27 680.449 cents grave fifth 10: 4 3/2 701.955 cents perfect fifth 10: 2 1024/675 721.508 cents narrow diminished sixth 11: 10 3/2 701.955 cents perfect fifth 11: 2 192/125 743.014 cents classic diminished sixth 11: 1 125/81 751.121 cents 11: 2 25/16 772.627 cents classic augmented fifth 11: 2 128/81 792.180 cents Pythagorean minor sixth 11: 1 8/5 813.686 cents minor sixth 12: 2 192/125 743.014 cents classic diminished sixth 12: 6 25/16 772.627 cents classic augmented fifth 12: 8 8/5 813.686 cents minor sixth 12: 2 5/3 884.359 cents major sixth, BP sixth 13: 1 405/256 794.134 cents wide augmented fifth 13: 4 8/5 813.686 cents minor sixth 13: 2 81/50 835.193 cents acute minor sixth 13: 8 5/3 884.359 cents major sixth, BP sixth 13: 2 128/75 925.418 cents diminished seventh 13: 1 125/72 955.031 cents classic augmented sixth 14: 2 5/3 884.359 cents major sixth, BP sixth 14: 6 27/16 905.865 cents Pythagorean major sixth 14: 4 128/75 925.418 cents diminished seventh 14: 2 125/72 955.031 cents classic augmented sixth 14: 4 16/9 996.090 cents Pythagorean minor seventh 15: 2 216/125 946.924 cents semi-augmented sixth 15: 4 225/128 976.537 cents augmented sixth 15: 6 16/9 996.090 cents Pythagorean minor seventh 15: 3 9/5 1017.596 cents just minor seventh, BP seventh 15: 1 2048/1125 1037.149 cents double diminished octave 15: 2 50/27 1066.762 cents grave major seventh 16: 6 9/5 1017.596 cents just minor seventh, BP seventh 16: 2 50/27 1066.762 cents grave major seventh 16: 6 15/8 1088.269 cents classic major seventh 16: 4 256/135 1107.821 cents octave - major chroma 17: 4 15/8 1088.269 cents classic major seventh 17: 8 48/25 1129.328 cents classic diminished octave 17: 3 125/64 1158.941 cents classic augmented seventh, octave - minor diesis 17: 3 160/81 1178.494 cents octave - syntonic comma Highest number of different intervals for one interval class: 6 Average number of different intervals per interval class: 5.17647 = 88/17
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0.0 : 70.7 111.7 182.4 203.9 274.6 315.6 386.3 498.0 568.7 590.2 702.0 772.6 813.7 884.4 996.1 1017.6 1088.3 1200.0 70.7 : 41.1 111.7 133.2 203.9 245.0 315.6 427.4 498.0 519.6 631.3 702.0 743.0 813.7 925.4 946.9 1017.6 1129.3 1200.0 111.7 : 70.7 92.2 162.9 203.9 274.6 386.3 457.0 478.5 590.2 660.9 702.0 772.6 884.4 905.9 976.5 1088.3 1158.9 1200.0 182.4 : 21.5 92.2 133.2 203.9 315.6 386.3 407.8 519.6 590.2 631.3 702.0 813.7 835.2 905.9 1017.6 1088.3 1129.3 1200.0 203.9 : 70.7 111.7 182.4 294.1 364.8 386.3 498.0 568.7 609.8 680.4 792.2 813.7 884.4 996.1 1066.8 1107.8 1178.5 1200.0 274.6 : 41.1 111.7 223.5 294.1 315.6 427.4 498.0 539.1 609.8 721.5 743.0 813.7 925.4 996.1 1037.1 1107.8 1129.3 1200.0 315.6 : 70.7 182.4 253.1 274.6 386.3 457.0 498.0 568.7 680.4 702.0 772.6 884.4 955.0 996.1 1066.8 1088.3 1158.9 1200.0 386.3 : 111.7 182.4 203.9 315.6 386.3 427.4 498.0 609.8 631.3 702.0 813.7 884.4 925.4 996.1 1017.6 1088.3 1129.3 1200.0 498.0 : 70.7 92.2 203.9 274.6 315.6 386.3 498.0 519.6 590.2 702.0 772.6 813.7 884.4 905.9 976.5 1017.6 1088.3 1200.0 568.7 : 21.5 133.2 203.9 245.0 315.6 427.4 448.9 519.6 631.3 702.0 743.0 813.7 835.2 905.9 946.9 1017.6 1129.3 1200.0 590.2 : 111.7 182.4 223.5 294.1 405.9 427.4 498.0 609.8 680.4 721.5 792.2 813.7 884.4 925.4 996.1 1107.8 1178.5 1200.0 702.0 : 70.7 111.7 182.4 294.1 315.6 386.3 498.0 568.7 609.8 680.4 702.0 772.6 813.7 884.4 996.1 1066.8 1088.3 1200.0 772.6 : 41.1 111.7 223.5 245.0 315.6 427.4 498.0 539.1 609.8 631.3 702.0 743.0 813.7 925.4 996.1 1017.6 1129.3 1200.0 813.7 : 70.7 182.4 203.9 274.6 386.3 457.0 498.0 568.7 590.2 660.9 702.0 772.6 884.4 955.0 976.5 1088.3 1158.9 1200.0 884.4 : 111.7 133.2 203.9 315.6 386.3 427.4 498.0 519.6 590.2 631.3 702.0 813.7 884.4 905.9 1017.6 1088.3 1129.3 1200.0 996.1 : 21.5 92.2 203.9 274.6 315.6 386.3 407.8 478.5 519.6 590.2 702.0 772.6 794.1 905.9 976.5 1017.6 1088.3 1200.0 1017.6: 70.7 182.4 253.1 294.1 364.8 386.3 457.0 498.0 568.7 680.4 751.1 772.6 884.4 955.0 996.1 1066.8 1178.5 1200.0 1088.3: 111.7 182.4 223.5 294.1 315.6 386.3 427.4 498.0 609.8 680.4 702.0 813.7 884.4 925.4 996.1 1107.8 1129.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 21.5063 .. 111.7313 cents 256/243, 90.2250 cents 81/80, 21.5063 cents 2: 92.1787 .. 182.4037 cents 256/243, 90.2250 cents 2025/2048,-19.5526 cents 3: 133.2376 .. 253.0761 cents 3125/2916, 119.8386 cents 243/250,-49.1661 cents 4: 203.9100 .. 315.6413 cents 16/15, 111.7313 cents 243/250,-49.1661 cents 5: 244.9689 .. 405.8663 cents 800/729, 160.8974 cents 24/25,-70.6724 cents 6: 315.6413 .. 456.9861 cents 625/576, 141.3449 cents 243/256,-90.2250 cents 7: 386.3137 .. 498.0450 cents 16/15, 111.7313 cents 24/25,-70.6724 cents 8: 478.4924 .. 609.7763 cents 32768/30375, 131.2839 cents 2025/2048,-19.5526 cents 9: 519.5513 .. 680.4487 cents 800/729, 160.8974 cents 243/256,-90.2250 cents 10: 590.2237 .. 721.5076 cents 32768/30375, 131.2839 cents 243/256,-90.2250 cents 11: 701.9550 .. 813.6863 cents 16/15, 111.7313 cents 2025/2048,-19.5526 cents 12: 743.0139 .. 884.3587 cents 625/576, 141.3449 cents 24/25,-70.6724 cents 13: 794.1337 .. 955.0311 cents 800/729, 160.8974 cents 243/256,-90.2250 cents 14: 884.3587 .. 996.0900 cents 16/15, 111.7313 cents 24/25,-70.6724 cents 15: 946.9239 .. 1066.7624 cents 3125/2916, 119.8386 cents 243/250,-49.1661 cents 16: 1017.5963 .. 1107.8213 cents 256/243, 90.2250 cents 243/250,-49.1661 cents 17: 1088.2687 .. 1178.4937 cents 256/243, 90.2250 cents 2025/2048,-19.5526 cents
SHOW DATA
Number of notes : 18 -- Interval properties -- Smallest interval : 81/80, 21.5063 cents, class 1 Average step (divided formal octave): 66.6667 cents Largest one step interval : 16/15, 111.7313 cents Average / Smallest step : 3.099868 Largest / Average step : 1.675969 Largest / Smallest step : 5.195284 Median interval of one step : 25/24, 70.6724 cents, amount: 8 Most common interval of one step : 25/24, 70.6724 cents, amount: 8 Least squares average step : 63.49380 cents, oct.: 1142.88847 cents Scale is not proper Step pattern alph. order: ABACABADACDABADCAD Step pattern size order : CBCACBCDCADCBCDACD Scale is not a Constant Structure Scale diversity : 1.354295 Lumma stability : 0.035844 Lumma impropriety factor : 0.702994 Number of different interval sizes : 56 = 3.29412 / class Number of one step interval sizes : 4 Highest interval variety : 6 Mean interval variety : 5.17647 = 88/17 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 32805/32768, 1.9537 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 14 Most common triad is 0.0 498.045 701.955 cents, amount: 10 Number of recognisable fifths : 14, average 701.9550 cents Scale is fully connected 5-limit major-minor triadic Scale contains a complete diamond : 1 3 5 9 15 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 13 = 72.2222% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval classes 4 and 5 combined Scale is n-1 differentially coherent in interval classes 5 and 6 combined Scale is n-1 differentially coherent in interval classes 6 and 7 combined Scale is differentially coherent in interval classes 12 and 13 combined Limited inverse transpositions : 7 Inversional symmetry on intervals : 3-4 12-13 -- Rational properties -- Prime limit : 5 Odd number limit : 1125 (O: 1125 U: 1125) Highest odd numerator or denominator: 75 Scale harmonicity : 0.025084 Average absolute harmonicity : 0.152481 Specific harmonicity : 0.137471 Fundamental : 1/2880, -11.4919 octaves, 0.0908 Hz. Guide tone : 3600, 11.8138 octaves, 941852.035 Hz. Exponens Consonantiae : 1.036800E+07, 23.30563 octaves Euler's gradus suavitatis : 31 Sum of Mann's harmonic distance : 267.0, average 14.83333 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 17.64137, average 0.98008 Sum of Tenney's harmonic distance : 34.67023, average 1.92612 Vogel's harmonic complexity : 18.83333 Wille's k value : 1687 Wilson's harmonic complexity : 27 Rectangular lattice diameter : 7 Triangular lattice diameter : 5 Lattice compactness : 726.32830, average 4.24753 Lattice compactness (without 2's) : 379.71856, average 2.22058 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -6 .. 4 -0.66667 44 1 3: -2 .. 2 0.00000 20 4 5: -1 .. 2 0.50000 17 13 Average exponent except 2's : 9 / 18 = 0.50000 Average absolute exponent except 2's: 37 / 18 = 2.05556 Scale is weakly epimorphic with val: <18 28 43|
FIT/MODE
18: 1 1 1 0 1 1 1 2 1 0 2 1 0 1 2 0 1 2 N T3 SD: 19.3158 c. M:-35.2883 c. 19: 1 1 1 0 1 1 1 2 1 0 2 1 1 1 2 0 1 2 N SD: 11.5127 c. M: 21.9508 c. 28: 2 1 1 1 1 1 2 3 1 1 2 2 1 2 2 1 1 3 N SD: 12.4774 c. M: 17.4396 c. 30: 2 1 2 0 2 1 2 2 2 1 3 1 1 2 3 0 2 3 N SD: 11.2985 c. M:-18.0450 c. 31: 2 1 2 0 2 1 2 3 2 0 3 2 1 2 3 0 2 3 N SD: 7.1605 c. M:-11.9277 c. 34: 2 1 2 1 2 1 2 3 2 1 3 2 1 2 3 1 2 3 N SD: 5.2458 c. M:-9.7763 c. 53: 3 2 3 1 3 2 3 5 3 1 5 3 2 3 5 1 3 5 N SD: 1.6425 c. M: 2.8843 c. 118: 7 4 7 2 7 4 7 11 7 2 11 7 4 7 11 2 7 11 N SD: 0.4058 c. M:-0.7741 c. 289: 17 10 17 5 17 10 17 27 17 5 27 17 10 17 27 5 17 27 N SD: 0.3161 c. M: 0.6043 c. 323: 19 11 19 6 19 11 19 30 19 6 30 19 11 19 30 6 19 30 N SD: 0.2769 c. M:-0.4884 c. 441: 26 15 26 8 26 15 26 41 26 8 41 26 15 26 41 8 26 41 N SD: 0.1325 c. M:-0.2525 c. 559: 33 19 33 10 33 19 33 52 33 10 52 33 19 33 52 10 33 52 N SD: 0.1045 c. M:-0.1940 c. 612: 36 21 36 11 36 21 36 57 36 11 57 36 21 36 57 11 36 57 N SD: 0.0484 c. M: 0.0900 c. 730: 43 25 43 13 43 25 43 68 43 13 68 43 25 43 68 13 43 68 N SD: 0.0486 c. M: 0.0867 c. 1171: 69 40 69 21 69 40 69 109 69 21 109 69 40 69 109 21 69 109 N SD: 0.0270 c. M:-0.0546 c. 1783: 105 61 105 32 105 61 105 166 105 32 166 105 61 105 166 32 105 166 N SD: 0.0104 c. M:-0.0175 c.
FIT/HARMONIC
1 .. 2 match: 1 H SD: 0.0000 cents 2 .. 4 match: 2 H SD: 0.0000 cents 3 .. 6 match: 3 H SD: 0.0000 cents 4 .. 8 match: 4 H SD: 6.8160 cents 5 .. 10 match: 5 H SD: 1.5423 cents 6 .. 12 match: 6 H SD: 5.4482 cents 7 .. 14 match: 7 H SD: 11.6194 cents 8 .. 16 match: 8 H SD: 5.2538 cents 9 .. 18 match: 9 H SD: 6.4638 cents 10 .. 20 match: 10 H SD: 6.3123 cents 11 .. 22 match: 9 SD: 8.4990 cents 12 .. 24 match: 12 H SD: 5.5728 cents 13 .. 26 match: 11 S SD: 6.6541 cents 14 .. 28 match: 12 SD: 6.6328 cents 15 .. 30 match: 12 SD: 3.6057 cents 16 .. 32 match: 15 S SD: 4.6849 cents 17 .. 34 match: 13 SD: 4.7696 cents 18 .. 36 match: 14 SD: 4.0478 cents 19 .. 38 match: 14 SD: 5.0060 cents 20 .. 40 match: 14 SD: 3.9777 cents 21 .. 42 match: 16 SD: 4.2514 cents 22 .. 44 match: 15 SD: 4.2600 cents 23 .. 46 match: 15 SD: 3.8586 cents 24 .. 48 match: 15 SD: 2.8147 cents 25 .. 50 match: 16 SD: 3.0898 cents 26 .. 52 match: 17 SD: 3.4929 cents 27 .. 54 match: 15 SD: 2.7410 cents 28 .. 56 match: 16 SD: 3.0359 cents 29 .. 58 match: 17 SD: 2.9271 cents 30 .. 60 match: 17 SD: 2.2802 cents 32 .. 64 match: 16 SD: 2.4046 cents 33 .. 66 match: 15 SD: 2.8495 cents 34 .. 68 match: 17 SD: 2.6397 cents 35 .. 70 match: 16 SD: 2.5425 cents 36 .. 72 match: 17 SD: 2.2491 cents 37 .. 74 match: 17 SD: 2.7328 cents 38 .. 76 match: 16 SD: 3.0302 cents 39 .. 78 match: 18 SD: 2.1436 cents 40 .. 80 match: 17 SD: 1.9907 cents 41 .. 82 match: 17 SD: 1.8916 cents 42 .. 84 match: 17 SD: 2.0025 cents 43 .. 86 match: 16 SD: 2.2399 cents 44 .. 88 match: 18 SD: 1.8348 cents 45 .. 90 match: 17 SD: 1.5127 cents 48 .. 96 match: 18 SD: 1.5064 cents 50 .. 100 match: 17 SD: 1.7227 cents 51 .. 102 match: 17 SD: 1.6789 cents 54 .. 108 match: 18 SD: 1.4292 cents 57 .. 114 match: 18 SD: 1.4053 cents 59 .. 118 match: 17 SD: 1.4858 cents 60 .. 120 match: 18 SD: 1.4746 cents 62 .. 124 match: 18 SD: 1.4060 cents 63 .. 126 match: 18 SD: 1.3817 cents 64 .. 128 match: 18 SD: 0.9529 cents 72 .. 144 match: 18 SD: 0.8942 cents 80 .. 160 match: 18 SD: 0.8557 cents 90 .. 180 match: 18 SD: 0.8268 cents 96 .. 192 match: 18 SD: 0.7809 cents 100 .. 200 match: 18 SD: 0.7739 cents 105 .. 210 match: 18 SD: 0.6475 cents 120 .. 240 match: 18 SD: 0.5186 cents 135 .. 270 match: 18 SD: 0.4885 cents 144 .. 288 match: 18 SD: 0.4300 cents 168 .. 336 match: 18 SD: 0.4203 cents 180 .. 360 match: 18 SD: 0.3905 cents 190 .. 380 match: 18 SD: 0.3840 cents 192 .. 384 match: 18 SD: 0.3145 cents 216 .. 432 match: 18 SD: 0.2776 cents 240 .. 480 match: 18 SD: 0.2386 cents 280 .. 560 match: 18 SD: 0.2379 cents 288 .. 576 match: 18 SD: 0.2094 cents 345 .. 690 match: 18 SD: 0.2001 cents 360 .. 720 match: 18 SD: 0.1018 cents 576 .. 1152 match: 18 SD: 0.0824 cents