Scala analysis: zarlin16
Zarlino's 16-note JI scale implemented on an instrument with split keys
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: 10/9 182.403712 minor whole tone 3: 9/8 203.910002 major whole tone 4: 32/27 294.134997 Pythagorean minor third 5: 6/5 315.641287 minor third 6: 5/4 386.313714 major third 7: 4/3 498.044999 perfect fourth 8: 25/18 568.717426 classic augmented fourth 9: 45/32 590.223716 diatonic tritone 10: 3/2 701.955001 perfect fifth 11: 25/16 772.627428 classic augmented fifth 12: 5/3 884.358713 major sixth, BP sixth 13: 16/9 996.089998 Pythagorean minor seventh 14: 9/5 1017.596288 just minor seventh, BP seventh 15: 15/8 1088.268715 classic major seventh 16: 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: 16/15 111.7313 minor diatonic semitone 3: 81/80 21.5063 syntonic comma, Didymus comma 4: 256/243 90.2250 limma, Pythagorean minor second 5: 81/80 21.5063 syntonic comma, Didymus comma 6: 25/24 70.6724 classic chromatic semitone, minor chroma 7: 16/15 111.7313 minor diatonic semitone 8: 25/24 70.6724 classic chromatic semitone, minor chroma 9: 81/80 21.5063 syntonic comma, Didymus comma 10: 16/15 111.7313 minor diatonic semitone 11: 25/24 70.6724 classic chromatic semitone, minor chroma 12: 16/15 111.7313 minor diatonic semitone 13: 16/15 111.7313 minor diatonic semitone 14: 81/80 21.5063 syntonic comma, Didymus comma 15: 25/24 70.6724 classic chromatic semitone, minor chroma 16: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 81/80 21.506 cents syntonic comma, Didymus comma 1: 5 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 1 256/243 90.225 cents limma, Pythagorean minor second 1: 6 16/15 111.731 cents minor diatonic semitone 2: 3 135/128 92.179 cents major chroma, major limma 2: 2 16/15 111.731 cents minor diatonic semitone 2: 3 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 2: 7 10/9 182.404 cents minor whole tone 2: 1 256/225 223.463 cents Neapolitan diminished third 3: 1 27/25 133.238 cents large limma, BP small semitone, Zarlino semitone 3: 1 10/9 182.404 cents minor whole tone 3: 7 9/8 203.910 cents major whole tone 3: 1 256/225 223.463 cents Neapolitan diminished third 3: 1 144/125 244.969 cents classic diminished third 3: 2 125/108 253.076 cents semi-augmented whole tone 3: 3 32/27 294.135 cents Pythagorean minor third 4: 1 9/8 203.910 cents major whole tone 4: 1 144/125 244.969 cents classic diminished third 4: 4 75/64 274.582 cents classic augmented second 4: 2 32/27 294.135 cents Pythagorean minor third 4: 6 6/5 315.641 cents minor third 4: 1 100/81 364.807 cents grave major third 4: 1 512/405 405.866 cents narrow diminished fourth 5: 1 1215/1024 296.089 cents wide augmented second 5: 3 6/5 315.641 cents minor third 5: 1 100/81 364.807 cents grave major third 5: 7 5/4 386.314 cents major third 5: 1 512/405 405.866 cents narrow diminished fourth 5: 3 32/25 427.373 cents classic diminished fourth 6: 3 5/4 386.314 cents major third 6: 2 81/64 407.820 cents Pythagorean major third 6: 2 32/25 427.373 cents classic diminished fourth 6: 1 162/125 448.879 cents 6: 1 125/96 456.986 cents classic augmented third 6: 1 320/243 476.539 cents grave fourth 6: 6 4/3 498.045 cents perfect fourth 7: 1 81/64 407.820 cents Pythagorean major third 7: 1 675/512 478.492 cents wide augmented third 7: 6 4/3 498.045 cents perfect fourth 7: 3 27/20 519.551 cents acute fourth 7: 2 25/18 568.717 cents classic augmented fourth 7: 3 64/45 609.776 cents 2nd tritone 8: 3 27/20 519.551 cents acute fourth 8: 3 25/18 568.717 cents classic augmented fourth 8: 2 45/32 590.224 cents diatonic tritone 8: 2 64/45 609.776 cents 2nd tritone 8: 3 36/25 631.283 cents classic diminished fifth 8: 3 40/27 680.449 cents grave fifth 9: 3 45/32 590.224 cents diatonic tritone 9: 2 36/25 631.283 cents classic diminished fifth 9: 3 40/27 680.449 cents grave fifth 9: 6 3/2 701.955 cents perfect fifth 9: 1 1024/675 721.508 cents narrow diminished sixth 9: 1 128/81 792.180 cents Pythagorean minor sixth 10: 6 3/2 701.955 cents perfect fifth 10: 1 243/160 723.461 cents acute fifth 10: 1 192/125 743.014 cents classic diminished sixth 10: 1 125/81 751.121 cents 10: 2 25/16 772.627 cents classic augmented fifth 10: 2 128/81 792.180 cents Pythagorean minor sixth 10: 3 8/5 813.686 cents minor sixth 11: 3 25/16 772.627 cents classic augmented fifth 11: 1 405/256 794.134 cents wide augmented fifth 11: 7 8/5 813.686 cents minor sixth 11: 1 81/50 835.193 cents acute minor sixth 11: 3 5/3 884.359 cents major sixth, BP sixth 11: 1 2048/1215 903.911 cents narrow diminished seventh 12: 1 405/256 794.134 cents wide augmented fifth 12: 1 81/50 835.193 cents acute minor sixth 12: 6 5/3 884.359 cents major sixth, BP sixth 12: 2 27/16 905.865 cents Pythagorean major sixth 12: 4 128/75 925.418 cents diminished seventh 12: 1 125/72 955.031 cents classic augmented sixth 12: 1 16/9 996.090 cents Pythagorean minor seventh 13: 3 27/16 905.865 cents Pythagorean major sixth 13: 2 216/125 946.924 cents semi-augmented sixth 13: 1 125/72 955.031 cents classic augmented sixth 13: 1 225/128 976.537 cents augmented sixth 13: 7 16/9 996.090 cents Pythagorean minor seventh 13: 1 9/5 1017.596 cents just minor seventh, BP seventh 13: 1 50/27 1066.762 cents grave major seventh 14: 1 225/128 976.537 cents augmented sixth 14: 7 9/5 1017.596 cents just minor seventh, BP seventh 14: 3 50/27 1066.762 cents grave major seventh 14: 2 15/8 1088.269 cents classic major seventh 14: 3 256/135 1107.821 cents octave - major chroma 15: 6 15/8 1088.269 cents classic major seventh 15: 1 243/128 1109.775 cents Pythagorean major seventh 15: 5 48/25 1129.328 cents classic diminished octave 15: 4 160/81 1178.494 cents octave - syntonic comma Highest number of different intervals for one interval class: 7 Average number of different intervals per interval class: 6.00000 = 6
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.0 : 70.7 182.4 203.9 294.1 315.6 386.3 498.0 568.7 590.2 702.0 772.6 884.4 996.1 1017.6 1088.3 1200.0 70.7 : 111.7 133.2 223.5 245.0 315.6 427.4 498.0 519.6 631.3 702.0 813.7 925.4 946.9 1017.6 1129.3 1200.0 182.4 : 21.5 111.7 133.2 203.9 315.6 386.3 407.8 519.6 590.2 702.0 813.7 835.2 905.9 1017.6 1088.3 1200.0 203.9 : 90.2 111.7 182.4 294.1 364.8 386.3 498.0 568.7 680.4 792.2 813.7 884.4 996.1 1066.8 1178.5 1200.0 294.1 : 21.5 92.2 203.9 274.6 296.1 407.8 478.5 590.2 702.0 723.5 794.1 905.9 976.5 1088.3 1109.8 1200.0 315.6 : 70.7 182.4 253.1 274.6 386.3 457.0 568.7 680.4 702.0 772.6 884.4 955.0 1066.8 1088.3 1178.5 1200.0 386.3 : 111.7 182.4 203.9 315.6 386.3 498.0 609.8 631.3 702.0 813.7 884.4 996.1 1017.6 1107.8 1129.3 1200.0 498.0 : 70.7 92.2 203.9 274.6 386.3 498.0 519.6 590.2 702.0 772.6 884.4 905.9 996.1 1017.6 1088.3 1200.0 568.7 : 21.5 133.2 203.9 315.6 427.4 448.9 519.6 631.3 702.0 813.7 835.2 925.4 946.9 1017.6 1129.3 1200.0 590.2 : 111.7 182.4 294.1 405.9 427.4 498.0 609.8 680.4 792.2 813.7 903.9 925.4 996.1 1107.8 1178.5 1200.0 702.0 : 70.7 182.4 294.1 315.6 386.3 498.0 568.7 680.4 702.0 792.2 813.7 884.4 996.1 1066.8 1088.3 1200.0 772.6 : 111.7 223.5 245.0 315.6 427.4 498.0 609.8 631.3 721.5 743.0 813.7 925.4 996.1 1017.6 1129.3 1200.0 884.4 : 111.7 133.2 203.9 315.6 386.3 498.0 519.6 609.8 631.3 702.0 813.7 884.4 905.9 1017.6 1088.3 1200.0 996.1 : 21.5 92.2 203.9 274.6 386.3 407.8 498.0 519.6 590.2 702.0 772.6 794.1 905.9 976.5 1088.3 1200.0 1017.6: 70.7 182.4 253.1 364.8 386.3 476.5 498.0 568.7 680.4 751.1 772.6 884.4 955.0 1066.8 1178.5 1200.0 1088.3: 111.7 182.4 294.1 315.6 405.9 427.4 498.0 609.8 680.4 702.0 813.7 884.4 996.1 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 .. 223.4626 cents 32768/30375, 131.2839 cents 2025/2048,-19.5526 cents 3: 133.2376 .. 294.1350 cents 800/729, 160.8974 cents 243/256,-90.2250 cents 4: 203.9100 .. 405.8663 cents 4096/3645, 201.9563 cents 243/256,-90.2250 cents 5: 296.0887 .. 427.3726 cents 32768/30375, 131.2839 cents 492075/524288,-109.7776 cents 6: 386.3137 .. 498.0450 cents 16/15, 111.7313 cents 125/128,-41.0589 cents 7: 407.8200 .. 609.7763 cents 4096/3645, 201.9563 cents 243/256,-90.2250 cents 8: 519.5513 .. 680.4487 cents 800/729, 160.8974 cents 243/256,-90.2250 cents 9: 590.2237 .. 792.1800 cents 4096/3645, 201.9563 cents 243/256,-90.2250 cents 10: 701.9550 .. 813.6863 cents 16/15, 111.7313 cents 243/256,-90.2250 cents 11: 772.6274 .. 903.9113 cents 32768/30375, 131.2839 cents 125/128,-41.0589 cents 12: 794.1337 .. 996.0900 cents 4096/3645, 201.9563 cents 492075/524288,-109.7776 cents 13: 905.8650 .. 1066.7624 cents 800/729, 160.8974 cents 243/256,-90.2250 cents 14: 976.5374 .. 1107.8213 cents 32768/30375, 131.2839 cents 243/256,-90.2250 cents 15: 1088.2687 .. 1178.4937 cents 256/243, 90.2250 cents 2025/2048,-19.5526 cents
SHOW DATA
Number of notes : 16 -- Interval properties -- Smallest interval : 81/80, 21.5063 cents, class 1 Average step (divided formal octave): 75.0000 cents Largest one step interval : 16/15, 111.7313 cents Average / Smallest step : 3.487352 Largest / Average step : 1.489750 Largest / Smallest step : 5.195284 Median interval of one step : 25/24, 70.6724 cents, amount: 5 Most common interval of one step : 16/15, 111.7313 cents, amount: 6 Least squares average step : 72.40522 cents, oct.: 1158.48354 cents Scale is not proper Step pattern alph. order: ABCDCABACBABBCAB Step pattern size order : BDACABDBADBDDABD Scale is not a Constant Structure Scale diversity : 1.338372 Lumma stability : 0.035844 Lumma impropriety factor : 0.771425 Number of different interval sizes : 56 = 3.73333 / class Number of one step interval sizes : 4 Highest interval variety : 7 Mean interval variety : 6.00000 = 6 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: 12 Most common triad is 0.0 498.045 701.955 cents, amount: 8 Number of recognisable fifths : 12, average 701.9550 cents Scale contains a complete diamond : 1 3 9 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 9 = 56.2500% Scale is n-1 differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval classes 2 and 3 combined -- Rational properties -- Prime limit : 5 Odd number limit : 1215 (O: 1215 U: 1215) Highest odd numerator or denominator: 45 Scale harmonicity : 0.023511 Average absolute harmonicity : 0.162005 Specific harmonicity : 0.144096 Fundamental : 1/4320, -12.0768 octaves, 0.0606 Hz. Guide tone : 7200, 12.8138 octaves, 1883704.070 Hz. Exponens Consonantiae : 3.110400E+07, 24.89060 octaves Euler's gradus suavitatis : 33 Sum of Mann's harmonic distance : 205.0, average 12.81250 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 15.32261, average 0.95766 Sum of Tenney's harmonic distance : 29.94324, average 1.87145 Vogel's harmonic complexity : 20.75000 Wille's k value : 3037 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 8 Triangular lattice diameter : 6 Lattice compactness : 560.75070, average 4.12317 Lattice compactness (without 2's) : 315.27732, average 2.31822 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -5 .. 5 -0.50000 36 1 3: -3 .. 2 -0.18750 21 5 5: -1 .. 2 0.56250 13 10 Average exponent except 2's : 6 / 16 = 0.37500 Average absolute exponent except 2's: 34 / 16 = 2.12500
FIT/MODE
16: 0 2 0 1 1 1 1 1 0 2 1 1 2 0 1 2 N SD: 44.0615 c. M: 70.6724 c. 17: 1 1 1 1 0 1 2 1 0 2 1 1 2 0 1 2 N SD: 22.3844 c. M: 41.2272 c. 19: 1 2 0 2 0 1 2 1 0 2 1 2 2 0 1 2 N SD: 11.4725 c. M: 21.8027 c. 28: 2 2 1 2 0 2 3 1 1 2 2 3 2 1 1 3 N SD: 11.8549 c. M: 16.8401 c. 29: 2 2 1 2 1 1 3 2 0 3 2 2 3 1 1 3 N G SD: 11.1677 c. M: 16.8865 c. 31: 2 3 0 3 0 2 3 2 0 3 2 3 3 0 2 3 N SD: 8.4093 c. M:-15.5424 c. 33: 2 3 1 2 1 2 3 2 0 3 2 3 3 1 2 3 N SD: 9.5351 c. M:-14.2718 c. 34: 2 3 1 2 1 2 3 2 1 3 2 3 3 1 2 3 N SD: 5.7871 c. M: 11.7821 c. 41: 2 4 1 3 1 2 4 2 1 4 2 4 4 1 2 4 N SD: 6.6369 c. M: 12.6199 c. 46: 3 4 1 3 1 3 4 3 1 4 3 4 4 1 3 4 N SD: 5.4862 c. M:-9.9813 c. 53: 3 5 1 4 1 3 5 3 1 5 3 5 5 1 3 5 N SD: 1.5025 c. M: 2.8161 c. 118: 7 11 2 9 2 7 11 7 2 11 7 11 11 2 7 11 N SD: 0.4703 c. M:-0.7803 c. 171: 10 16 3 13 3 10 16 10 3 16 10 16 16 3 10 16 N SD: 0.4048 c. M: 0.7500 c. 289: 17 27 5 22 5 17 27 17 5 27 17 27 27 5 17 27 N SD: 0.3355 c. M:-0.6747 c. 323: 19 30 6 24 6 19 30 19 6 30 19 30 30 6 19 30 N SD: 0.3152 c. M: 0.6365 c. 376: 22 35 7 28 7 22 35 22 7 35 22 35 35 7 22 35 N SD: 0.3535 c. M: 0.6323 c. 388: 23 36 7 29 7 23 36 23 7 36 23 36 36 7 23 36 N SD: 0.2993 c. M:-0.5684 c. 441: 26 41 8 33 8 26 41 26 8 41 26 41 41 8 26 41 N SD: 0.1338 c. M: 0.2574 c. 559: 33 52 10 42 10 33 52 33 10 52 33 52 52 10 33 52 N SD: 0.0940 c. M:-0.1812 c. 612: 36 57 11 46 11 36 57 36 11 57 36 57 57 11 36 57 N SD: 0.0465 c. M: 0.0900 c. 1171: 69 109 21 88 21 69 109 69 21 109 69 109 109 21 69 109 N SD: 0.0242 c. M:-0.0455 c. 1783: 105 166 32 134 32 105 166 105 32 166 105 166 166 32 105 166 N SD: 0.0120 c. M: 0.0239 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: 8.3554 cents 6 .. 12 match: 6 H SD: 6.9770 cents 7 .. 14 match: 7 H SD: 11.6194 cents 8 .. 16 match: 8 H SD: 6.8090 cents 9 .. 18 match: 9 H SD: 6.4638 cents 10 .. 20 match: 10 H SD: 7.5302 cents 11 .. 22 match: 9 SD: 8.3601 cents 12 .. 24 match: 12 H SD: 6.7463 cents 13 .. 26 match: 10 S SD: 9.4534 cents 14 .. 28 match: 12 SD: 7.7523 cents 15 .. 30 match: 11 SD: 5.4226 cents 16 .. 32 match: 14 SD: 5.0659 cents 17 .. 34 match: 13 SD: 5.5657 cents 18 .. 36 match: 13 SD: 4.8338 cents 19 .. 38 match: 13 SD: 5.1487 cents 20 .. 40 match: 13 SD: 4.5409 cents 21 .. 42 match: 15 SD: 4.2565 cents 22 .. 44 match: 14 SD: 4.4010 cents 23 .. 46 match: 13 SD: 3.8171 cents 24 .. 48 match: 14 SD: 3.1487 cents 25 .. 50 match: 13 SD: 3.1440 cents 26 .. 52 match: 14 SD: 3.3807 cents 27 .. 54 match: 13 SD: 2.5513 cents 28 .. 56 match: 14 SD: 3.3784 cents 29 .. 58 match: 15 SD: 3.3920 cents 30 .. 60 match: 14 SD: 2.7135 cents 31 .. 62 match: 15 SD: 3.1546 cents 32 .. 64 match: 14 SD: 2.3378 cents 33 .. 66 match: 13 SD: 2.9763 cents 34 .. 68 match: 15 SD: 2.8321 cents 35 .. 70 match: 14 SD: 3.0489 cents 36 .. 72 match: 15 SD: 2.5438 cents 37 .. 74 match: 15 SD: 2.7528 cents 38 .. 76 match: 14 SD: 2.8562 cents 39 .. 78 match: 16 SD: 2.0100 cents 40 .. 80 match: 15 SD: 2.2762 cents 41 .. 82 match: 14 SD: 2.0325 cents 42 .. 84 match: 14 SD: 2.1220 cents 43 .. 86 match: 14 SD: 2.3189 cents 44 .. 88 match: 16 SD: 1.7652 cents 45 .. 90 match: 15 SD: 1.7681 cents 46 .. 92 match: 15 SD: 1.7061 cents 47 .. 94 match: 15 SD: 1.9148 cents 48 .. 96 match: 16 SD: 1.5575 cents 50 .. 100 match: 15 SD: 1.7054 cents 51 .. 102 match: 15 SD: 1.7601 cents 52 .. 104 match: 15 SD: 1.8611 cents 54 .. 108 match: 16 SD: 1.2410 cents 57 .. 114 match: 15 SD: 1.5853 cents 59 .. 118 match: 15 SD: 1.5938 cents 63 .. 126 match: 16 SD: 1.5605 cents 64 .. 128 match: 16 SD: 0.9139 cents 80 .. 160 match: 16 SD: 0.8419 cents 96 .. 192 match: 16 SD: 0.5723 cents 120 .. 240 match: 16 SD: 0.5348 cents 135 .. 270 match: 16 SD: 0.5317 cents 136 .. 272 match: 16 SD: 0.5248 cents 144 .. 288 match: 16 SD: 0.3720 cents 190 .. 380 match: 16 SD: 0.3675 cents 215 .. 430 match: 16 SD: 0.3419 cents 216 .. 432 match: 16 SD: 0.2089 cents 288 .. 576 match: 16 SD: 0.1839 cents 360 .. 720 match: 16 SD: 0.1387 cents 384 .. 768 match: 16 SD: 0.1356 cents 432 .. 864 match: 16 SD: 0.1342 cents 480 .. 960 match: 16 SD: 0.0987 cents 576 .. 1152 match: 16 SD: 0.0649 cents