Scala analysis: xen03-wilson-positive-17
Positive, linear-mapped intonational system, 17 notes
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 2: 10/9 182.403712 minor whole tone 3: 9/8 203.910002 major whole tone 4: 32/27 294.134997 Pythagorean minor third 5: 5/4 386.313714 major third 6: 81/64 407.820003 Pythagorean major third 7: 4/3 498.044999 perfect fourth 8: 1024/729 588.269995 Pythagorean diminished fifth 9: 40/27 680.448711 grave fifth 10: 3/2 701.955001 perfect fifth 11: 128/81 792.179997 Pythagorean minor sixth 12: 5/3 884.358713 major sixth, BP sixth 13: 27/16 905.865003 Pythagorean major sixth 14: 16/9 996.089998 Pythagorean minor seventh 15: 15/8 1088.268715 classic major seventh 16: 243/128 1109.775004 Pythagorean major seventh 17: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 256/243 90.2250 limma, Pythagorean minor second 1: 256/243 90.2250 limma, Pythagorean minor second 2: 135/128 92.1787 major chroma, major limma 3: 81/80 21.5063 syntonic comma, Didymus comma 4: 256/243 90.2250 limma, Pythagorean minor second 5: 135/128 92.1787 major chroma, major limma 6: 81/80 21.5063 syntonic comma, Didymus comma 7: 256/243 90.2250 limma, Pythagorean minor second 8: 256/243 90.2250 limma, Pythagorean minor second 9: 135/128 92.1787 major chroma, major limma 10: 81/80 21.5063 syntonic comma, Didymus comma 11: 256/243 90.2250 limma, Pythagorean minor second 12: 135/128 92.1787 major chroma, major limma 13: 81/80 21.5063 syntonic comma, Didymus comma 14: 256/243 90.2250 limma, Pythagorean minor second 15: 135/128 92.1787 major chroma, major limma 16: 81/80 21.5063 syntonic comma, Didymus comma 17: 256/243 90.2250 limma, Pythagorean minor second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 5 81/80 21.506 cents syntonic comma, Didymus comma 1: 7 256/243 90.225 cents limma, Pythagorean minor second 1: 5 135/128 92.179 cents major chroma, major limma 2: 5 16/15 111.731 cents minor diatonic semitone 2: 5 2187/2048 113.685 cents apotome 2: 2 65536/59049 180.450 cents Pythagorean diminished third 2: 5 10/9 182.404 cents minor whole tone 3: 2 4096/3645 201.956 cents narrow diminished third 3: 13 9/8 203.910 cents major whole tone 3: 2 2560/2187 272.629 cents grave minor third 4: 3 729/640 225.416 cents acute major second 4: 11 32/27 294.135 cents Pythagorean minor third 4: 3 1215/1024 296.089 cents wide augmented second 5: 5 6/5 315.641 cents minor third 5: 3 19683/16384 317.595 cents Pythagorean augmented second 5: 4 8192/6561 384.360 cents Pythagorean diminished fourth 5: 5 5/4 386.314 cents major third 6: 4 512/405 405.866 cents narrow diminished fourth 6: 9 81/64 407.820 cents Pythagorean major third 6: 4 320/243 476.539 cents grave fourth 7: 1 6561/5120 429.326 cents acute major third 7: 15 4/3 498.045 cents perfect fourth 7: 1 10935/8192 499.999 cents fourth + schisma, 5-limit approximation to ET fourth 8: 5 27/20 519.551 cents acute fourth 8: 1 177147/131072 521.505 cents Pythagorean augmented third 8: 6 1024/729 588.270 cents Pythagorean diminished fifth 8: 5 45/32 590.224 cents diatonic tritone 9: 5 64/45 609.776 cents 2nd tritone 9: 6 729/512 611.730 cents Pythagorean tritone 9: 1 262144/177147 678.495 cents Pythagorean diminished sixth 9: 5 40/27 680.449 cents grave fifth 10: 1 16384/10935 700.001 cents fifth - schisma, 5-limit approximation to ET fifth 10: 15 3/2 701.955 cents perfect fifth 10: 1 10240/6561 770.674 cents grave minor sixth 11: 4 243/160 723.461 cents acute fifth 11: 9 128/81 792.180 cents Pythagorean minor sixth 11: 4 405/256 794.134 cents wide augmented fifth 12: 5 8/5 813.686 cents minor sixth 12: 4 6561/4096 815.640 cents Pythagorean augmented fifth 12: 3 32768/19683 882.405 cents Pythagorean diminished seventh 12: 5 5/3 884.359 cents major sixth, BP sixth 13: 3 2048/1215 903.911 cents narrow diminished seventh 13: 11 27/16 905.865 cents Pythagorean major sixth 13: 3 1280/729 974.584 cents grave minor seventh 14: 2 2187/1280 927.371 cents acute major sixth 14: 13 16/9 996.090 cents Pythagorean minor seventh 14: 2 3645/2048 998.044 cents wide augmented sixth 15: 5 9/5 1017.596 cents just minor seventh, BP seventh 15: 2 59049/32768 1019.550 cents Pythagorean augmented sixth 15: 5 4096/2187 1086.315 cents Pythagorean diminished octave 15: 5 15/8 1088.269 cents classic major seventh 16: 5 256/135 1107.821 cents octave - major chroma 16: 7 243/128 1109.775 cents Pythagorean major seventh 16: 5 160/81 1178.494 cents octave - syntonic comma Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.37500 = 27/8
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0.0 : 90.2 182.4 203.9 294.1 386.3 407.8 498.0 588.3 680.4 702.0 792.2 884.4 905.9 996.1 1088.3 1109.8 1200.0 90.2 : 92.2 113.7 203.9 296.1 317.6 407.8 498.0 590.2 611.7 702.0 794.1 815.6 905.9 998.0 1019.6 1109.8 1200.0 182.4 : 21.5 111.7 203.9 225.4 315.6 405.9 498.0 519.6 609.8 702.0 723.5 813.7 905.9 927.4 1017.6 1107.8 1200.0 203.9 : 90.2 182.4 203.9 294.1 384.4 476.5 498.0 588.3 680.4 702.0 792.2 884.4 905.9 996.1 1086.3 1178.5 1200.0 294.1 : 92.2 113.7 203.9 294.1 386.3 407.8 498.0 590.2 611.7 702.0 794.1 815.6 905.9 996.1 1088.3 1109.8 1200.0 386.3 : 21.5 111.7 202.0 294.1 315.6 405.9 498.0 519.6 609.8 702.0 723.5 813.7 903.9 996.1 1017.6 1107.8 1200.0 407.8 : 90.2 180.4 272.6 294.1 384.4 476.5 498.0 588.3 680.4 702.0 792.2 882.4 974.6 996.1 1086.3 1178.5 1200.0 498.0 : 90.2 182.4 203.9 294.1 386.3 407.8 498.0 590.2 611.7 702.0 792.2 884.4 905.9 996.1 1088.3 1109.8 1200.0 588.3 : 92.2 113.7 203.9 296.1 317.6 407.8 500.0 521.5 611.7 702.0 794.1 815.6 905.9 998.0 1019.6 1109.8 1200.0 680.4 : 21.5 111.7 203.9 225.4 315.6 407.8 429.3 519.6 609.8 702.0 723.5 813.7 905.9 927.4 1017.6 1107.8 1200.0 702.0 : 90.2 182.4 203.9 294.1 386.3 407.8 498.0 588.3 680.4 702.0 792.2 884.4 905.9 996.1 1086.3 1178.5 1200.0 792.2 : 92.2 113.7 203.9 296.1 317.6 407.8 498.0 590.2 611.7 702.0 794.1 815.6 905.9 996.1 1088.3 1109.8 1200.0 884.4 : 21.5 111.7 203.9 225.4 315.6 405.9 498.0 519.6 609.8 702.0 723.5 813.7 903.9 996.1 1017.6 1107.8 1200.0 905.9 : 90.2 182.4 203.9 294.1 384.4 476.5 498.0 588.3 680.4 702.0 792.2 882.4 974.6 996.1 1086.3 1178.5 1200.0 996.1 : 92.2 113.7 203.9 294.1 386.3 407.8 498.0 590.2 611.7 702.0 792.2 884.4 905.9 996.1 1088.3 1109.8 1200.0 1088.3: 21.5 111.7 202.0 294.1 315.6 405.9 498.0 519.6 609.8 700.0 792.2 813.7 903.9 996.1 1017.6 1107.8 1200.0 1109.8: 90.2 180.4 272.6 294.1 384.4 476.5 498.0 588.3 678.5 770.7 792.2 882.4 974.6 996.1 1086.3 1178.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 21.5063 .. 92.1787 cents 25/24, 70.6724 cents 81/80, 21.5063 cents 2: 111.7313 .. 182.4037 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 3: 201.9563 .. 272.6287 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 4: 225.4163 .. 296.0887 cents 25/24, 70.6724 cents 1594323/1638400,-47.2124 cents 5: 315.6413 .. 386.3137 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 6: 405.8663 .. 476.5387 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 7: 429.3263 .. 499.9987 cents 25/24, 70.6724 cents 1594323/1638400,-47.2124 cents 8: 519.5513 .. 590.2237 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 9: 609.7763 .. 680.4487 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 10: 700.0013 .. 770.6737 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 11: 723.4613 .. 794.1337 cents 25/24, 70.6724 cents 1594323/1638400,-47.2124 cents 12: 813.6863 .. 884.3587 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 13: 903.9113 .. 974.5837 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 14: 927.3713 .. 998.0437 cents 25/24, 70.6724 cents 1594323/1638400,-47.2124 cents 15: 1017.5963 .. 1088.2687 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents 16: 1107.8213 .. 1178.4937 cents 25/24, 70.6724 cents 2048/2025, 19.5526 cents
SHOW DATA
Number of notes : 17 -- Interval properties -- Smallest interval : 81/80, 21.5063 cents, class 1 Average step (divided formal octave): 70.5882 cents Largest one step interval : 135/128, 92.1787 cents Average / Smallest step : 3.282214 Largest / Average step : 1.305865 Largest / Smallest step : 4.286128 Median interval of one step : 256/243, 90.2250 cents, amount: 7 Most common interval of one step : 256/243, 90.2250 cents, amount: 7 Least squares average step : 71.42077 cents, oct.: 1214.15309 cents Scale is not proper Step pattern alph. order: ABCABCAABCABCABCA Step pattern size order : MLSMLSMMLSMLSMLSM Scale is a Constant Structure, by a margin of 19.55257 cents Scale diversity : 1.158767 Lumma stability : 0.215076 Lumma impropriety factor : 0.157375 Number of different interval sizes : 54 = 3.37500 / class Number of one step interval sizes : 3 Highest interval variety : 4 Mean interval variety : 3.37500 = 27/8 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 32805/32768, 1.9537 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 15 Most common triad is 0.0 498.045 701.955 cents, amount: 13 Number of recognisable fifths : 16, average 701.8329 cents Scale contains two identical disjunct "tetrachords" of 8 notes Scale contains a complete diamond : 1 3 9 27 81 243 Formal octave complements present : 11 = 64.7059% Limited inverse transpositions with margin 70.6724 cents: 8 -- Rational properties -- Prime limit : 5 Odd number limit : 177147 (O: 177147 U: 177147) Highest odd numerator or denominator: 729 Scale harmonicity : 0.018963 Average absolute harmonicity : 0.151397 Specific harmonicity : 0.119985 Fundamental : 1/93312, -16.5098 octaves, 0.0028 Hz. Guide tone : 1244160, 20.2467 octaves, 325504063.325 Hz. Exponens Consonantiae : 1.160951E+11, 36.75652 octaves Euler's gradus suavitatis : 44 Sum of Mann's harmonic distance : 1631.0, average 95.94118 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 22.13600, average 1.30212 Sum of Tenney's harmonic distance : 44.18007, average 2.59883 Vogel's harmonic complexity : 25.41176 Wille's k value : 88573 Wilson's harmonic complexity : 38 Rectangular lattice diameter : 12 Triangular lattice diameter : 11 Lattice compactness : 1024.73453, average 6.69761 Lattice compactness (without 2's) : 556.84273, average 3.63949 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -7 .. 10 0.88235 67 1 3: -6 .. 5 -0.64706 43 11 5: 0 .. 1 0.29412 5 5 Average exponent except 2's :-6 / 17 =-0.35294 Average absolute exponent except 2's: 48 / 17 = 2.82353 Scale is weakly epimorphic with val: <17 25 38| Scale is JI-epimorphic with val: <17 27 39| = patent Scale is weakly epimorphic with val: <17 29 40|
FIT/MODE
17: 1 2 0 1 1 1 1 1 2 0 1 2 0 1 1 1 1 N T3 SD: 20.0730 c. M: 33.3725 c. 18: 1 2 0 1 2 0 1 2 1 1 1 1 1 1 1 1 1 N D3 SD: 19.3196 c. M: 31.3783 c. 19: 1 2 0 1 2 0 2 1 2 0 1 2 0 2 1 0 2 N T3 SD: 21.2435 c. M: 41.5034 c. 21: 1 2 0 2 2 0 1 2 2 0 2 1 1 1 2 0 2 N SD: 20.1521 c. M: 40.9021 c. 22: 2 1 1 1 2 0 2 2 1 1 1 2 0 2 2 0 2 N SD: 18.1360 c. M: 33.1377 c. 23: 1 2 1 1 2 1 1 2 2 0 2 2 0 2 2 0 2 N T3 SD: 18.9428 c. M: 38.0511 c. 24: 2 2 0 2 2 0 2 2 2 0 2 2 0 2 2 0 2 N M DE S SD: 10.3455 c. M:-19.5513 c. 26: 2 2 0 2 2 1 2 2 2 0 2 2 1 2 2 0 2 N SD: 12.4943 c. M:-19.4236 c. 27: 2 2 1 2 2 0 2 2 2 1 2 2 0 2 2 1 2 N SD: 12.2150 c. M: 21.6020 c. 28: 2 2 1 2 2 1 2 2 2 0 2 3 0 2 2 1 2 N SD: 12.1944 c. M:-20.7514 c. 29: 2 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2 P M ME S SD: 9.4949 c. M: 18.3797 c. 30: 2 3 0 2 3 0 2 3 2 1 2 2 1 2 2 1 2 N SD: 11.2064 c. M:-18.0450 c. 32: 2 3 0 3 2 1 2 3 2 1 2 3 0 3 2 1 2 N SD: 10.5821 c. M:-16.4100 c. 33: 2 3 1 2 3 0 3 2 3 0 3 2 1 2 3 1 2 N SD: 10.5379 c. M: 17.4977 c. 34: 3 2 1 2 3 1 2 3 2 1 2 3 1 2 3 0 3 N SD: 10.0811 c. M:-15.7094 c. 37: 3 3 0 3 3 1 2 3 3 1 2 3 1 3 3 0 3 N SD: 9.0520 c. M:-14.4340 c. 39: 3 3 1 3 3 0 3 3 3 1 3 3 0 3 3 1 3 N SD: 8.5325 c. M:-13.6863 c. 41: 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 1 3 N M DE S SD: 3.7229 c. M: 7.2780 c. 53: 4 4 1 4 4 1 4 4 4 1 4 4 1 4 4 1 4 N M DE S SD: 0.7554 c. M: 1.4763 c. 171: 13 13 3 13 13 3 13 13 13 3 13 13 3 13 13 3 13 N M DE S SD: 0.6130 c. M:-1.2037 c. 224: 17 17 4 17 17 4 17 17 17 4 17 17 4 17 17 4 17 N M DE S SD: 0.5634 c. M:-1.0157 c. 277: 21 21 5 21 21 5 21 21 21 5 21 21 5 21 21 5 21 N M DE S SD: 0.5601 c. M: 0.9041 c. 335: 25 26 6 25 26 6 25 25 26 6 25 26 6 25 26 6 25 N SD: 0.4662 c. M: 0.8073 c. 388: 29 30 7 29 30 7 29 29 30 7 29 30 7 29 30 7 29 N SD: 0.3377 c. M: 0.6411 c. 400: 30 31 7 30 31 7 30 30 31 7 30 31 7 30 31 7 30 N SD: 0.3735 c. M:-0.7313 c. 441: 33 34 8 33 34 8 33 33 34 8 33 34 8 33 34 8 33 N SD: 0.2600 c. M: 0.5149 c. 453: 34 35 8 34 35 8 34 34 35 8 34 35 8 34 35 8 34 N SD: 0.2420 c. M:-0.4730 c. 494: 37 38 9 37 38 9 37 37 38 9 37 38 9 37 38 9 37 N SD: 0.2248 c. M: 0.4157 c. 506: 38 39 9 38 39 9 38 38 39 9 38 39 9 38 39 9 38 N SD: 0.1386 c. M:-0.2688 c. 559: 42 43 10 42 43 10 42 42 43 10 42 43 10 42 43 10 42 N SD: 0.0573 c. M:-0.1034 c. 612: 46 47 11 46 47 11 46 46 47 11 46 47 11 46 47 11 46 N SD: 0.0300 c. M: 0.0566 c. 1171: 88 90 21 88 90 21 88 88 90 21 88 90 21 88 90 21 88 N SD: 0.0286 c. M: 0.0548 c. 1783: 134 137 32 134 137 32 134 134 137 32 134 137 32 134 137 32 134 N SD: 0.0244 c. M: 0.0479 c. 2395: 180 184 43 180 184 43 180 180 184 43 180 184 43 180 184 43 180 N SD: 0.0242 c. M: 0.0445 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: 7.5385 cents 6 .. 12 match: 6 H SD: 7.9180 cents 7 .. 14 match: 7 H SD: 8.3934 cents 8 .. 16 match: 8 H SD: 7.9352 cents 9 .. 18 match: 9 H SD: 7.2473 cents 10 .. 20 match: 10 H SD: 6.1724 cents 11 .. 22 match: 11 H SD: 8.8040 cents 12 .. 24 match: 11 SD: 5.6845 cents 13 .. 26 match: 13 H SD: 8.0951 cents 14 .. 28 match: 14 H SD: 6.5588 cents 15 .. 30 match: 13 SD: 4.7783 cents 16 .. 32 match: 12 SD: 4.9038 cents 17 .. 34 match: 15 S SD: 4.8594 cents 18 .. 36 match: 12 SD: 3.9355 cents 19 .. 38 match: 13 SD: 4.5581 cents 20 .. 40 match: 14 SD: 4.5931 cents 21 .. 42 match: 16 SD: 4.3359 cents 22 .. 44 match: 14 SD: 3.5179 cents 23 .. 46 match: 15 SD: 3.4728 cents 24 .. 48 match: 14 SD: 3.3960 cents 25 .. 50 match: 14 SD: 4.1074 cents 26 .. 52 match: 13 SD: 3.3903 cents 27 .. 54 match: 14 SD: 3.0419 cents 28 .. 56 match: 14 SD: 3.8173 cents 29 .. 58 match: 17 SD: 3.3068 cents 30 .. 60 match: 16 SD: 2.9940 cents 31 .. 62 match: 15 SD: 3.0902 cents 32 .. 64 match: 16 SD: 2.4085 cents 33 .. 66 match: 16 SD: 2.1769 cents 34 .. 68 match: 14 SD: 2.5575 cents 35 .. 70 match: 14 SD: 2.6496 cents 36 .. 72 match: 16 SD: 2.3309 cents 37 .. 74 match: 16 SD: 2.2357 cents 38 .. 76 match: 16 SD: 1.8094 cents 39 .. 78 match: 16 SD: 1.8582 cents 41 .. 82 match: 16 SD: 2.0167 cents 42 .. 84 match: 15 SD: 1.5318 cents 43 .. 86 match: 15 SD: 2.2533 cents 44 .. 88 match: 17 SD: 2.0562 cents 45 .. 90 match: 17 SD: 1.8432 cents 46 .. 92 match: 16 SD: 2.0828 cents 47 .. 94 match: 16 SD: 2.0396 cents 48 .. 96 match: 17 SD: 1.4243 cents 50 .. 100 match: 15 SD: 1.5440 cents 51 .. 102 match: 16 SD: 1.9922 cents 52 .. 104 match: 17 SD: 1.6278 cents 54 .. 108 match: 16 SD: 1.0571 cents 58 .. 116 match: 16 SD: 1.5243 cents 59 .. 118 match: 16 SD: 1.5057 cents 60 .. 120 match: 17 SD: 1.2311 cents 62 .. 124 match: 16 SD: 1.4369 cents 63 .. 126 match: 17 SD: 1.3964 cents 64 .. 128 match: 17 SD: 1.0114 cents 71 .. 142 match: 17 SD: 0.9646 cents 72 .. 144 match: 17 SD: 0.7785 cents 96 .. 192 match: 17 SD: 0.6864 cents 108 .. 216 match: 17 SD: 0.6560 cents 112 .. 224 match: 17 SD: 0.6087 cents 128 .. 256 match: 17 SD: 0.5038 cents 135 .. 270 match: 17 SD: 0.4613 cents 144 .. 288 match: 17 SD: 0.4490 cents 168 .. 336 match: 17 SD: 0.3877 cents 189 .. 378 match: 17 SD: 0.3859 cents 200 .. 400 match: 17 SD: 0.3434 cents 207 .. 414 match: 17 SD: 0.2992 cents 240 .. 480 match: 17 SD: 0.2974 cents 260 .. 520 match: 17 SD: 0.2866 cents 264 .. 528 match: 17 SD: 0.2073 cents 297 .. 594 match: 17 SD: 0.1775 cents 336 .. 672 match: 17 SD: 0.1447 cents 432 .. 864 match: 17 SD: 0.0804 cents