Scala analysis: kleis
Kleismic detempered circle of minor thirds
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: 27/25 133.237575 large limma, BP small semitone, Zarlino semitone 3: 10/9 182.403712 minor whole tone 4: 125/108 253.076139 semi-augmented whole tone 5: 6/5 315.641287 minor third 6: 5/4 386.313714 major third 7: 125/96 456.986141 classic augmented third 8: 4/3 498.044999 perfect fourth 9: 25/18 568.717426 classic augmented fourth 10: 36/25 631.282574 classic diminished fifth 11: 3/2 701.955001 perfect fifth 12: 125/81 751.121138 13: 8/5 813.686286 minor sixth 14: 5/3 884.358713 major sixth, BP sixth 15: 125/72 955.031140 classic augmented sixth 16: 9/5 1017.596288 just minor seventh, BP seventh 17: 50/27 1066.762425 grave major seventh 18: 48/25 1129.327573 classic diminished octave 19: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 25/24 70.6724 classic chromatic semitone, minor chroma 1: 25/24 70.6724 classic chromatic semitone, minor chroma 2: 648/625 62.5651 major diesis 3: 250/243 49.1661 maximal diesis, Porcupine comma 4: 25/24 70.6724 classic chromatic semitone, minor chroma 5: 648/625 62.5651 major diesis 6: 25/24 70.6724 classic chromatic semitone, minor chroma 7: 25/24 70.6724 classic chromatic semitone, minor chroma 8: 128/125 41.0589 minor diesis, diesis 9: 25/24 70.6724 classic chromatic semitone, minor chroma 10: 648/625 62.5651 major diesis 11: 25/24 70.6724 classic chromatic semitone, minor chroma 12: 250/243 49.1661 maximal diesis, Porcupine comma 13: 648/625 62.5651 major diesis 14: 25/24 70.6724 classic chromatic semitone, minor chroma 15: 25/24 70.6724 classic chromatic semitone, minor chroma 16: 648/625 62.5651 major diesis 17: 250/243 49.1661 maximal diesis, Porcupine comma 18: 648/625 62.5651 major diesis 19: 25/24 70.6724 classic chromatic semitone, minor chroma
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 128/125 41.059 cents minor diesis, diesis 1: 3 250/243 49.166 cents maximal diesis, Porcupine comma 1: 6 648/625 62.565 cents major diesis 1: 9 25/24 70.672 cents classic chromatic semitone, minor chroma 2: 6 16/15 111.731 cents minor diatonic semitone 2: 2 3125/2916 119.839 cents quasi-equal 10-tone semitone 2: 8 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 2: 3 625/576 141.345 cents two (classic chromatic semitone, minor chroma) 3: 2 3456/3125 174.296 cents 3: 10 10/9 182.404 cents minor whole tone 3: 7 9/8 203.910 cents major whole tone 4: 7 144/125 244.969 cents classic diminished third 4: 9 125/108 253.076 cents semi-augmented whole tone 4: 2 729/625 266.475 cents two (large limma, BP small semitone, Zarlino semitone) 4: 1 75/64 274.582 cents classic augmented second 5: 1 32/27 294.135 cents Pythagorean minor third 5: 15 6/5 315.641 cents minor third 5: 3 3125/2592 323.749 cents 6: 1 768/625 356.700 cents 6: 4 100/81 364.807 cents grave major third 6: 3 3888/3125 378.206 cents 6: 11 5/4 386.314 cents major third 7: 5 32/25 427.373 cents classic diminished fourth 7: 4 625/486 435.480 cents 7: 6 162/125 448.879 cents 7: 4 125/96 456.986 cents classic augmented third 8: 13 4/3 498.045 cents perfect fourth 8: 5 27/20 519.551 cents acute fourth 8: 1 3125/2304 527.659 cents 9: 5 864/625 560.610 cents 9: 12 25/18 568.717 cents classic augmented fourth 9: 1 4374/3125 582.116 cents 9: 1 45/32 590.224 cents diatonic tritone 10: 1 64/45 609.776 cents 2nd tritone 10: 1 3125/2187 617.884 cents 10: 12 36/25 631.283 cents classic diminished fifth 10: 5 625/432 639.390 cents 11: 1 4608/3125 672.341 cents 11: 5 40/27 680.449 cents grave fifth 11: 13 3/2 701.955 cents perfect fifth 12: 4 192/125 743.014 cents classic diminished sixth 12: 6 125/81 751.121 cents 12: 4 972/625 764.520 cents 12: 5 25/16 772.627 cents classic augmented fifth 13: 11 8/5 813.686 cents minor sixth 13: 3 3125/1944 821.794 cents 13: 4 81/50 835.193 cents acute minor sixth 13: 1 625/384 843.300 cents 14: 3 5184/3125 876.251 cents 14: 15 5/3 884.359 cents major sixth, BP sixth 14: 1 27/16 905.865 cents Pythagorean major sixth 15: 1 128/75 925.418 cents diminished seventh 15: 2 1250/729 933.525 cents 15: 9 216/125 946.924 cents semi-augmented sixth 15: 7 125/72 955.031 cents classic augmented sixth 16: 7 16/9 996.090 cents Pythagorean minor seventh 16: 10 9/5 1017.596 cents just minor seventh, BP seventh 16: 2 3125/1728 1025.704 cents 17: 3 1152/625 1058.655 cents 17: 8 50/27 1066.762 cents grave major seventh 17: 2 5832/3125 1080.161 cents 17: 6 15/8 1088.269 cents classic major seventh 18: 9 48/25 1129.328 cents classic diminished octave 18: 6 625/324 1137.435 cents octave - major diesis 18: 3 243/125 1150.834 cents octave - maximal diesis 18: 1 125/64 1158.941 cents classic augmented seventh, octave - minor diesis Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.66667 = 11/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0.0 : 70.7 133.2 182.4 253.1 315.6 386.3 457.0 498.0 568.7 631.3 702.0 751.1 813.7 884.4 955.0 1017.6 1066.8 1129.3 1200.0 70.7 : 62.6 111.7 182.4 245.0 315.6 386.3 427.4 498.0 560.6 631.3 680.4 743.0 813.7 884.4 946.9 996.1 1058.7 1129.3 1200.0 133.2 : 49.2 119.8 182.4 253.1 323.7 364.8 435.5 498.0 568.7 617.9 680.4 751.1 821.8 884.4 933.5 996.1 1066.8 1137.4 1200.0 182.4 : 70.7 133.2 203.9 274.6 315.6 386.3 448.9 519.6 568.7 631.3 702.0 772.6 835.2 884.4 946.9 1017.6 1088.3 1150.8 1200.0 253.1 : 62.6 133.2 203.9 245.0 315.6 378.2 448.9 498.0 560.6 631.3 702.0 764.5 813.7 876.3 946.9 1017.6 1080.2 1129.3 1200.0 315.6 : 70.7 141.3 182.4 253.1 315.6 386.3 435.5 498.0 568.7 639.4 702.0 751.1 813.7 884.4 955.0 1017.6 1066.8 1137.4 1200.0 386.3 : 70.7 111.7 182.4 245.0 315.6 364.8 427.4 498.0 568.7 631.3 680.4 743.0 813.7 884.4 946.9 996.1 1066.8 1129.3 1200.0 457.0 : 41.1 111.7 174.3 245.0 294.1 356.7 427.4 498.0 560.6 609.8 672.3 743.0 813.7 876.3 925.4 996.1 1058.7 1129.3 1200.0 498.0 : 70.7 133.2 203.9 253.1 315.6 386.3 457.0 519.6 568.7 631.3 702.0 772.6 835.2 884.4 955.0 1017.6 1088.3 1158.9 1200.0 568.7 : 62.6 133.2 182.4 245.0 315.6 386.3 448.9 498.0 560.6 631.3 702.0 764.5 813.7 884.4 946.9 1017.6 1088.3 1129.3 1200.0 631.3 : 70.7 119.8 182.4 253.1 323.7 386.3 435.5 498.0 568.7 639.4 702.0 751.1 821.8 884.4 955.0 1025.7 1066.8 1137.4 1200.0 702.0 : 49.2 111.7 182.4 253.1 315.6 364.8 427.4 498.0 568.7 631.3 680.4 751.1 813.7 884.4 955.0 996.1 1066.8 1129.3 1200.0 751.1 : 62.6 133.2 203.9 266.5 315.6 378.2 448.9 519.6 582.1 631.3 702.0 764.5 835.2 905.9 946.9 1017.6 1080.2 1150.8 1200.0 813.7 : 70.7 141.3 203.9 253.1 315.6 386.3 457.0 519.6 568.7 639.4 702.0 772.6 843.3 884.4 955.0 1017.6 1088.3 1137.4 1200.0 884.4 : 70.7 133.2 182.4 245.0 315.6 386.3 448.9 498.0 568.7 631.3 702.0 772.6 813.7 884.4 946.9 1017.6 1066.8 1129.3 1200.0 955.0 : 62.6 111.7 174.3 245.0 315.6 378.2 427.4 498.0 560.6 631.3 702.0 743.0 813.7 876.3 946.9 996.1 1058.7 1129.3 1200.0 1017.6: 49.2 111.7 182.4 253.1 315.6 364.8 435.5 498.0 568.7 639.4 680.4 751.1 813.7 884.4 933.5 996.1 1066.8 1137.4 1200.0 1066.8: 62.6 133.2 203.9 266.5 315.6 386.3 448.9 519.6 590.2 631.3 702.0 764.5 835.2 884.4 946.9 1017.6 1088.3 1150.8 1200.0 1129.3: 70.7 141.3 203.9 253.1 323.7 386.3 457.0 527.7 568.7 639.4 702.0 772.6 821.8 884.4 955.0 1025.7 1088.3 1137.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 41.0589 .. 70.6724 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents 2: 111.7313 .. 141.3449 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents 3: 174.2964 .. 203.9100 cents 3125/3072, 29.6136 cents 1990656/1953125, 32.9516 cents 4: 244.9689 .. 274.5824 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents 5: 294.1350 .. 323.7486 cents 3125/3072, 29.6136 cents 2048/2025, 19.5526 cents 6: 356.7001 .. 386.3137 cents 3125/3072, 29.6136 cents 1990656/1953125, 32.9516 cents 7: 427.3726 .. 456.9861 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents 8: 498.0450 .. 527.6586 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents 9: 560.6101 .. 590.2237 cents 3125/3072, 29.6136 cents 1990656/1953125, 32.9516 cents 10: 609.7763 .. 639.3899 cents 3125/3072, 29.6136 cents 2048/2025, 19.5526 cents 11: 672.3414 .. 701.9550 cents 3125/3072, 29.6136 cents 1990656/1953125, 32.9516 cents 12: 743.0139 .. 772.6274 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents 13: 813.6863 .. 843.2999 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents 14: 876.2514 .. 905.8650 cents 3125/3072, 29.6136 cents 1990656/1953125, 32.9516 cents 15: 925.4176 .. 955.0311 cents 3125/3072, 29.6136 cents 2048/2025, 19.5526 cents 16: 996.0900 .. 1025.7036 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents 17: 1058.6551 .. 1088.2687 cents 3125/3072, 29.6136 cents 1990656/1953125, 32.9516 cents 18: 1129.3276 .. 1158.9411 cents 3125/3072, 29.6136 cents 128/125, 41.0589 cents
SHOW DATA
Number of notes : 19 -- Interval properties -- Smallest interval : 128/125, 41.0589 cents, class 1 Average step (divided formal octave): 63.1579 cents Largest one step interval : 25/24, 70.6724 cents Average / Smallest step : 1.538228 Largest / Average step : 1.118980 Largest / Smallest step : 1.721247 Median interval of one step : 648/625, 62.5651 cents, amount: 6 Most common interval of one step : 25/24, 70.6724 cents, amount: 9 Least squares average step : 63.14618 cents, oct.: 1199.77742 cents Scale is strictly proper Step pattern alph. order: ABCABAADABACBAABCBA Step pattern size order : DCBDCDDADCDBCDDCBCD Scale is a Constant Structure, by a margin of 19.55257 cents Scale diversity : 1.299028 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.555796 Number of different interval sizes : 66 = 3.66667 / class Number of one step interval sizes : 4 Highest interval variety : 4 Mean interval variety : 3.66667 = 11/3 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 15625/15552, 8.1073 cents Most common intervals : 6/5, 315.6413 cents & inv., amount: 15 Most common triad is 0.0 315.641 884.359 cents, amount: 12 Number of recognisable fifths : 13, average 701.9550 cents Best fifths form a closed circle Best major thirds form a closed circle Best minor thirds form a closed circle Scale is fully connected 5-limit major-minor triadic Scale contains a complete diamond : 3 5 9 25 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 15 = 78.9474% Scale is differentially coherent in interval classes 2 and 3 combined -- Rational properties -- Prime limit : 5 Odd number limit : 3125 (O: 3125 U: 3125) Highest odd numerator or denominator: 125 Scale harmonicity : 0.016760 Average absolute harmonicity : 0.133079 Specific harmonicity : 0.098668 Fundamental : 1/64800, -15.9837 octaves, 0.0040 Hz. Guide tone : 54000, 15.7207 octaves, 14127780.526 Hz. Exponens Consonantiae : 3.499200E+09, 31.70438 octaves Euler's gradus suavitatis : 44 Sum of Mann's harmonic distance : 650.5, average 34.23684 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 22.50074, average 1.18425 Sum of Tenney's harmonic distance : 44.91570, average 2.36398 Vogel's harmonic complexity : 28.47368 Wille's k value : 42187 Wilson's harmonic complexity : 46 Rectangular lattice diameter : 12 Triangular lattice diameter : 7 Lattice compactness : 814.44456, average 4.28655 Lattice compactness (without 2's) : 593.56491, average 3.12403 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -5 .. 4 -0.10526 32 1 3: -4 .. 3 -0.52632 30 2 5: -2 .. 3 0.63158 30 16 Average exponent except 2's : 2 / 19 = 0.10526 Average absolute exponent except 2's: 60 / 19 = 3.15789 Scale is JI-epimorphic with val: <19 30 44| = patent
FIT/MODE
19: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 6.6803 c. M: 14.8809 c. 34: 2 2 1 2 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 P M ME I SD: 4.6892 c. M: 9.9447 c. 51: 3 3 2 3 2 3 3 2 3 3 3 2 3 3 3 2 2 3 3 SP D3 SD: 6.7662 c. M: 9.9273 c. 52: 3 3 2 3 3 3 3 2 3 2 3 3 2 3 3 3 2 3 3 P D3 SD: 6.4089 c. M:-10.4173 c. 53: 3 3 2 3 3 3 3 2 3 3 3 2 3 3 3 3 2 3 3 SP M ME I SD: 2.5095 c. M: 4.1560 c. 99: 6 5 4 6 5 6 6 3 6 5 6 4 5 6 6 5 4 5 6 SP SD: 1.4627 c. M:-3.6199 c. 118: 7 6 5 7 6 7 7 4 7 6 7 5 6 7 7 6 5 6 7 SP SD: 0.7168 c. M:-1.4212 c. 152: 9 8 6 9 8 9 9 5 9 8 9 6 8 9 9 8 6 8 9 SP SD: 0.6312 c. M: 1.1211 c. 171: 10 9 7 10 9 10 10 6 10 9 10 7 9 10 10 9 7 9 10 SP SD: 0.3591 c. M: 0.8458 c. 270: 16 14 11 16 14 16 16 9 16 14 16 11 14 16 16 14 11 14 16 SP SD: 0.3164 c. M:-0.7916 c. 289: 17 15 12 17 15 17 17 10 17 15 17 12 15 17 17 15 12 15 17 SP SD: 0.2231 c. M:-0.4360 c. 441: 26 23 18 26 23 26 26 15 26 23 26 18 23 26 26 23 18 23 26 SP SD: 0.0776 c. M:-0.1567 c. 612: 36 32 25 36 32 36 36 21 36 32 36 25 32 36 36 32 25 32 36 SP SD: 0.0830 c. M: 0.1407 c. 730: 43 38 30 43 38 43 43 25 43 38 43 30 38 43 43 38 30 38 43 SP SD: 0.0522 c. M:-0.1117 c. 1171: 69 61 48 69 61 69 69 40 69 61 69 48 61 69 69 61 48 61 69 SP SD: 0.0282 c. M:-0.0591 c. 1342: 79 70 55 79 70 79 79 46 79 70 79 55 70 79 79 70 55 70 79 SP SD: 0.0226 c. M: 0.0562 c. 1783: 105 93 73 105 93 105 105 61 105 93 105 73 93 105 105 93 73 93 105 SP SD: 0.0129 c. M: 0.0275 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: 3.4487 cents 5 .. 10 match: 5 H SD: 2.7590 cents 6 .. 12 match: 6 H SD: 3.7007 cents 7 .. 14 match: 7 H SD: 7.3219 cents 8 .. 16 match: 8 H SD: 5.7805 cents 9 .. 18 match: 9 H SD: 5.5447 cents 10 .. 20 match: 10 H SD: 4.4774 cents 11 .. 22 match: 11 H SD: 6.2852 cents 12 .. 24 match: 12 H SD: 3.5837 cents 13 .. 26 match: 13 H SD: 4.5159 cents 14 .. 28 match: 14 H SD: 5.0740 cents 15 .. 30 match: 15 H SD: 4.1248 cents 16 .. 32 match: 16 H SD: 4.3219 cents 17 .. 34 match: 14 SD: 5.6084 cents 18 .. 36 match: 17 SD: 3.9853 cents 19 .. 38 match: 16 SD: 4.1531 cents 20 .. 40 match: 18 SD: 2.6661 cents 21 .. 42 match: 17 S SD: 3.7554 cents 22 .. 44 match: 17 SD: 4.2784 cents 23 .. 46 match: 18 SD: 3.6790 cents 24 .. 48 match: 19 SD: 2.7343 cents 25 .. 50 match: 18 SD: 2.6935 cents 26 .. 52 match: 19 SD: 2.5384 cents 36 .. 72 match: 19 SD: 2.0350 cents 39 .. 78 match: 19 SD: 1.8317 cents 44 .. 88 match: 19 SD: 1.7880 cents 45 .. 90 match: 19 SD: 1.7126 cents 46 .. 92 match: 19 SD: 1.5289 cents 50 .. 100 match: 19 SD: 1.4346 cents 52 .. 104 match: 19 SD: 1.3102 cents 60 .. 120 match: 19 SD: 1.3050 cents 63 .. 126 match: 19 SD: 1.2349 cents 64 .. 128 match: 19 SD: 1.0234 cents 72 .. 144 match: 19 SD: 0.8474 cents 75 .. 150 match: 19 SD: 0.7812 cents 90 .. 180 match: 19 SD: 0.6706 cents 96 .. 192 match: 19 SD: 0.6457 cents 100 .. 200 match: 19 SD: 0.5185 cents 120 .. 240 match: 19 SD: 0.5145 cents 136 .. 272 match: 19 SD: 0.4725 cents 152 .. 304 match: 19 SD: 0.4612 cents 160 .. 320 match: 19 SD: 0.4142 cents 164 .. 328 match: 19 SD: 0.4039 cents 186 .. 372 match: 19 SD: 0.3903 cents 188 .. 376 match: 19 SD: 0.3666 cents 189 .. 378 match: 19 SD: 0.3508 cents 215 .. 430 match: 19 SD: 0.3096 cents 216 .. 432 match: 19 SD: 0.2214 cents 285 .. 570 match: 19 SD: 0.2156 cents 288 .. 576 match: 19 SD: 0.2036 cents 350 .. 700 match: 19 SD: 0.1920 cents 360 .. 720 match: 19 SD: 0.1513 cents 414 .. 828 match: 19 SD: 0.1443 cents 440 .. 880 match: 19 SD: 0.1329 cents 450 .. 900 match: 19 SD: 0.1243 cents 504 .. 1008 match: 19 SD: 0.1170 cents 552 .. 1104 match: 19 SD: 0.1127 cents 560 .. 1120 match: 19 SD: 0.1102 cents 576 .. 1152 match: 19 SD: 0.0900 cents