Scala analysis: kirkwood
Scale based on Kirkwood gaps of the asteroid belt
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 9/8 203.910002 major whole tone 2: 7/6 266.870906 septimal minor third 3: 5/4 386.313714 major third 4: 4/3 498.044999 perfect fourth 5: 3/2 701.955001 perfect fifth 6: 5/3 884.358713 major sixth, BP sixth 7: 7/4 968.825906 harmonic seventh 8: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 8/7 231.1741 septimal whole tone 1: 9/8 203.9100 major whole tone 2: 28/27 62.9609 Archytas' 1/3-tone 3: 15/14 119.4428 major diatonic semitone 4: 16/15 111.7313 minor diatonic semitone 5: 9/8 203.9100 major whole tone 6: 10/9 182.4037 minor whole tone 7: 21/20 84.4672 minor semitone 8: 8/7 231.1741 septimal whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 28/27 62.961 cents Archytas' 1/3-tone 1: 1 21/20 84.467 cents minor semitone 1: 1 16/15 111.731 cents minor diatonic semitone 1: 1 15/14 119.443 cents major diatonic semitone 1: 1 10/9 182.404 cents minor whole tone 1: 2 9/8 203.910 cents major whole tone 1: 1 8/7 231.174 cents septimal whole tone 2: 1 10/9 182.404 cents minor whole tone 2: 1 8/7 231.174 cents septimal whole tone 2: 2 7/6 266.871 cents septimal minor third 2: 2 6/5 315.641 cents minor third 2: 1 5/4 386.314 cents major third 2: 1 9/7 435.084 cents septimal major third, BP third 3: 1 32/27 294.135 cents Pythagorean minor third 3: 1 5/4 386.314 cents major third 3: 1 9/7 435.084 cents septimal major third, BP third 3: 1 21/16 470.781 cents narrow fourth 3: 3 4/3 498.045 cents perfect fourth 3: 1 27/20 519.551 cents acute fourth 4: 2 4/3 498.045 cents perfect fourth 4: 2 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 4: 2 10/7 617.488 cents Euler's tritone 4: 2 3/2 701.955 cents perfect fifth 5: 1 40/27 680.449 cents grave fifth 5: 3 3/2 701.955 cents perfect fifth 5: 1 32/21 729.219 cents wide fifth 5: 1 14/9 764.916 cents septimal minor sixth 5: 1 8/5 813.686 cents minor sixth 5: 1 27/16 905.865 cents Pythagorean major sixth 6: 1 14/9 764.916 cents septimal minor sixth 6: 1 8/5 813.686 cents minor sixth 6: 2 5/3 884.359 cents major sixth, BP sixth 6: 2 12/7 933.129 cents septimal major sixth 6: 1 7/4 968.826 cents harmonic seventh 6: 1 9/5 1017.596 cents just minor seventh, BP seventh 7: 1 7/4 968.826 cents harmonic seventh 7: 2 16/9 996.090 cents Pythagorean minor seventh 7: 1 9/5 1017.596 cents just minor seventh, BP seventh 7: 1 28/15 1080.557 cents grave major seventh 7: 1 15/8 1088.269 cents classic major seventh 7: 1 40/21 1115.533 cents acute major seventh 7: 1 27/14 1137.039 cents septimal major seventh 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 0.0 : 203.9 266.9 386.3 498.0 702.0 884.4 968.8 1200.0 203.9 : 63.0 182.4 294.1 498.0 680.4 764.9 996.1 1200.0 266.9 : 119.4 231.2 435.1 617.5 702.0 933.1 1137.0 1200.0 386.3 : 111.7 315.6 498.0 582.5 813.7 1017.6 1080.6 1200.0 498.0 : 203.9 386.3 470.8 702.0 905.9 968.8 1088.3 1200.0 702.0 : 182.4 266.9 498.0 702.0 764.9 884.4 996.1 1200.0 884.4 : 84.5 315.6 519.6 582.5 702.0 813.7 1017.6 1200.0 968.8 : 231.2 435.1 498.0 617.5 729.2 933.1 1115.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 62.9609 .. 231.1741 cents 54/49, 168.2132 cents 28/27, 62.9609 cents 2: 182.4037 .. 435.0841 cents 81/70, 252.6804 cents 35/36,-48.7704 cents 3: 294.1350 .. 519.5513 cents 729/640, 225.4163 cents 224/243,-140.9491 cents 4: 498.0450 .. 701.9550 cents 9/8, 203.9100 cents 80/81,-21.5063 cents 5: 680.4487 .. 905.8650 cents 729/640, 225.4163 cents 80/81,-21.5063 cents 6: 764.9159 .. 1017.5963 cents 81/70, 252.6804 cents 224/243,-140.9491 cents 7: 968.8259 .. 1137.0391 cents 54/49, 168.2132 cents 35/36,-48.7704 cents
SHOW DATA
Number of notes : 8 -- Interval properties -- Smallest interval : 28/27, 62.9609 cents, class 1 Average step (divided formal octave): 150.0000 cents Largest one step interval : 8/7, 231.1741 cents Average / Smallest step : 2.382431 Largest / Average step : 1.541161 Largest / Smallest step : 3.671709 Median interval of one step : 150.92326 cents, amount: 0 Most common interval of one step : 9/8, 203.9100 cents, amount: 2 Least squares average step : 142.58079 cents, oct.: 1140.64634 cents Scale is not proper Step pattern alph. order: ABCDAEFG Step pattern size order : FADCFEBG Scale is not a Constant Structure Scale diversity : 1.573527 Lumma stability : 0.104935 Lumma impropriety factor : 0.352043 Rothenberg efficiency : 0.410714 redundancy: 0.589286 Efficiency x scale size : 3.285714 Number of different interval sizes : 32 = 4.57143 / class Number of one step interval sizes : 7 Highest interval variety : 7 Mean interval variety : 6.00000 = 6 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 225/224, 7.7115 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 5 Most common triad is 0.0 498.045 701.955 cents, amount: 2 Number of recognisable fifths : 5, average 701.9550 cents Scale contains a complete diamond : 1 3 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 2 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 37.5000% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval class 2 Scale is differentially coherent in interval class 3 Scale is differentially coherent in interval class 4 Scale is differentially coherent in interval classes 5 and 6 combined Scale is differentially coherent in interval class 6 -- Rational properties -- Prime limit : 7 Odd number limit : 27 (O: 27 U: 27) Highest odd numerator or denominator: 9 Scale harmonicity : 0.033686 Average absolute harmonicity : 0.248677 Specific harmonicity : 0.198399 Fundamental : 1/24, -4.5850 octaves, 10.9011 Hz. Guide tone : 1260, 10.2992 octaves, 329648.212 Hz. Exponens Consonantiae : 3.024000E+04, 14.88417 octaves Euler's gradus suavitatis : 22 Sum of Mann's harmonic distance : 36.5, average 4.56250 Mersenne's string divisions : 6126120 Sum of van Prooijen's expressibility: 4.99662, average 0.62458 Sum of Tenney's harmonic distance : 9.56322, average 1.19540 Vogel's harmonic complexity : 8.25000 Wille's k value : 13 Wilson's harmonic complexity : 21 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 92.65044, average 2.57362 Lattice compactness (without 2's) : 54.87506, average 1.52431 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -0.75000 12 1 3: -1 .. 2 0.00000 6 3 5: 0 .. 1 0.25000 2 2 7: 0 .. 1 0.25000 2 2 Average exponent except 2's : 4 / 8 = 0.50000 Average absolute exponent except 2's: 10 / 8 = 1.25000 Scale is weakly epimorphic with val: <8 13 20 22| Scale is weakly epimorphic with val: <8 14 17 21| Scale is weakly epimorphic with val: <8 14 19 23|
FIT/MODE
8: 1 1 0 1 1 2 0 2 N T3 SD: 60.3518 c. M: 101.9550 c. 9: 2 0 1 1 1 2 0 2 N SD: 35.8373 c. M:-62.7567 c. 10: 2 0 1 1 2 1 1 2 N SD: 26.0139 c. M: 44.3587 c. 12: 2 1 1 1 2 2 1 2 N D3 SD: 17.7632 c. M:-33.1291 c. Minor Bebop, Dorian Bebop, Raga Khokhar, Zilla, Mixolydian/Dorian mixed 19: 3 1 2 2 3 3 1 4 N SD: 11.3472 c. M: 21.4575 c. 22: 4 1 2 2 4 3 2 4 N SD: 9.1113 c. M:-14.2718 c. 27: 5 1 3 2 5 4 2 5 N SD: 9.9437 c. M:-18.3122 c. 31: 5 2 3 3 5 5 2 6 N SD: 5.1863 c. M: 10.3616 c. 41: 7 2 4 4 7 6 3 8 N SD: 3.4631 c. M: 6.3099 c. 46: 8 2 5 4 8 7 3 9 N SD: 3.7922 c. M: 6.0013 c. 53: 9 3 5 5 9 8 4 10 N SD: 2.4939 c. M:-4.8272 c. 72: 12 4 7 7 12 11 5 14 N SD: 2.1671 c. M: 3.9100 c. 94: 16 5 9 9 16 14 7 18 N SD: 1.8372 c. M: 3.5076 c. 99: 17 5 10 9 17 15 7 19 N SD: 1.1417 c. M:-2.1506 c. 130: 22 7 13 12 22 20 9 25 N SD: 0.9356 c. M:-1.7951 c. 171: 29 9 17 16 29 26 12 33 N SD: 0.2717 c. M: 0.4049 c. 441: 75 23 44 41 75 67 31 85 N SD: 0.1154 c. M: 0.2042 c. 612: 104 32 61 57 104 93 43 118 N SD: 0.1030 c. M: 0.2042 c. 665: 113 35 66 62 113 101 47 128 N SD: 0.1232 c. M:-0.1968 c. 836: 142 44 83 78 142 127 59 161 N SD: 0.1040 c. M: 0.1893 c. 935: 159 49 93 87 159 142 66 180 N SD: 0.0960 c. M:-0.1581 c. 1106: 188 58 110 103 188 168 78 213 N SD: 0.0558 c. M: 0.0911 c. 1277: 217 67 127 119 217 194 90 246 N SD: 0.0488 c. M: 0.0987 c. 1407: 239 74 140 131 239 214 99 271 N SD: 0.0543 c. M:-0.0801 c. 1547: 263 81 154 144 263 235 109 298 N SD: 0.0505 c. M:-0.0978 c. 1677: 285 88 167 156 285 255 118 323 N SD: 0.0483 c. M:-0.0906 c. 1718: 292 90 171 160 292 261 121 331 N SD: 0.0428 c. M: 0.0747 c. 1848: 314 97 184 172 314 281 130 356 N SD: 0.0278 c. M:-0.0569 c. 2019: 343 106 201 188 343 307 142 389 N SD: 0.0274 c. M: 0.0467 c.
FIT/HARMONIC
1 x x x x x x x 2 S SD: 0.0000 cents 2 x x x x 3 x x 4 S SD: 0.0000 cents 3 x x x 4 x 5 x 6 S SD: 0.0000 cents 4 x x 5 x 6 x 7:8 S SD: 0.0000 cents 5 x 6 x 7 x 8:9:10 S SD: 25.9895 cents 6 x 7 x 8:9:10:11:12 S SD: 13.4228 cents 7:8 x 9 x 11 x 12:14 SD: 20.8637 cents 8:9 x 10:11:12:13:14:16 S SD: 9.8552 cents 9:10 x 11:12:14:15:16:18 S SD: 11.6790 cents 10:11:12 x 13:15:17:18:20 SD: 13.8264 cents 11:12:13:14:15:17:18:19:22 SD: 12.5101 cents 12 x 14:15:16:18:20:21:24 S SD: 0.0000 cents 13 x 15:16:17:20:22:23:26 SD: 10.2959 cents 14:16 x 18:19:21:23:25:28 SD: 10.9774 cents 15:17:18:19:20:23:25:26:30 SD: 8.6512 cents 16:18:19:20:21:24:27:28:32 S SD: 5.7890 cents 20 x 23:25:27:30:33:35:40 SD: 5.3180 cents 23:26:27:29:31:35:38:40:46 SD: 5.1916 cents 24:27:28:30:32:36:40:42:48 S SD: 0.0000 cents