Scala analysis: Horowitz
Leonard G.Horowitz's DMD,MA,MPH,DNM(hon.) scale completed by Sparschuh
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2227/2112 91.790068 2: 9/8 203.910002 major whole tone 3: 313/264 294.749673 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 45/32 590.223716 diatonic tritone 7: 3/2 701.955001 perfect fifth 8: 835/528 793.485922 9: 37/22 900.026096 10: 313/176 996.704674 11: 15/8 1088.268715 classic major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 2227/2112 91.7901 2: 2376/2227 112.1199 3: 313/297 90.8397 4: 330/313 91.5640 5: 16/15 111.7313 minor diatonic semitone 6: 135/128 92.1787 major chroma, major limma 7: 16/15 111.7313 minor diatonic semitone 8: 835/792 91.5309 9: 888/835 106.5402 10: 313/296 96.6786 11: 330/313 91.5640 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 313/297 90.840 cents 1: 1 835/792 91.531 cents 1: 2 330/313 91.564 cents 1: 1 2227/2112 91.790 cents 1: 1 135/128 92.179 cents major chroma, major limma 1: 1 313/296 96.679 cents 1: 1 888/835 106.540 cents 1: 3 16/15 111.731 cents minor diatonic semitone 1: 1 2376/2227 112.120 cents 2: 1 10/9 182.404 cents minor whole tone 2: 1 165/148 188.243 cents 2: 1 37/33 198.071 cents 2: 1 2504/2227 202.960 cents 2: 1 939/835 203.219 cents 2: 1 334/297 203.262 cents 2: 2 352/313 203.295 cents 2: 1 2227/1980 203.521 cents 2: 3 9/8 203.910 cents major whole tone 3: 1 32/27 294.135 cents Pythagorean minor third 3: 1 2640/2227 294.524 cents 3: 2 313/264 294.750 cents 3: 1 198/167 294.783 cents 3: 1 2227/1878 295.085 cents 3: 1 835/704 295.441 cents 3: 1 1485/1252 295.474 cents 3: 1 44/37 299.974 cents 3: 1 592/495 309.802 cents 3: 2 6/5 315.641 cents minor third 4: 3 5/4 386.314 cents major third 4: 1 2227/1776 391.764 cents 4: 1 111/88 401.981 cents 4: 1 2816/2227 406.255 cents 4: 2 626/495 406.481 cents 4: 1 1056/835 406.514 cents 4: 1 167/132 407.172 cents 4: 2 396/313 407.205 cents 5: 6 4/3 498.045 cents perfect fourth 5: 1 2227/1670 498.304 cents 5: 1 2970/2227 498.434 cents 5: 1 939/704 498.660 cents 5: 1 835/626 498.736 cents 5: 1 99/74 503.884 cents 5: 1 74/55 513.712 cents 6: 1 835/594 589.576 cents 6: 1 440/313 589.609 cents 6: 1 2227/1584 589.835 cents 6: 2 45/32 590.224 cents diatonic tritone 6: 1 313/222 594.724 cents 6: 1 444/313 605.276 cents 6: 2 64/45 609.776 cents 2nd tritone 6: 1 3168/2227 610.165 cents 6: 1 313/220 610.391 cents 6: 1 1188/835 610.424 cents 7: 1 55/37 686.288 cents 7: 1 148/99 696.116 cents 7: 1 1252/835 701.264 cents 7: 1 1408/939 701.340 cents 7: 1 2227/1485 701.566 cents 7: 1 3340/2227 701.696 cents 7: 6 3/2 701.955 cents perfect fifth 8: 2 313/198 792.795 cents 8: 1 264/167 792.828 cents 8: 1 835/528 793.486 cents 8: 2 495/313 793.519 cents 8: 1 2227/1408 793.745 cents 8: 1 176/111 798.019 cents 8: 1 3552/2227 808.236 cents 8: 3 8/5 813.686 cents minor sixth 9: 2 5/3 884.359 cents major sixth, BP sixth 9: 1 495/296 890.198 cents 9: 1 37/22 900.026 cents 9: 1 2504/1485 904.526 cents 9: 1 1408/835 904.559 cents 9: 1 3756/2227 904.915 cents 9: 1 167/99 905.217 cents 9: 2 528/313 905.250 cents 9: 1 2227/1320 905.476 cents 9: 1 27/16 905.865 cents Pythagorean major sixth 10: 3 16/9 996.090 cents Pythagorean minor seventh 10: 1 3960/2227 996.479 cents 10: 2 313/176 996.705 cents 10: 1 297/167 996.738 cents 10: 1 1670/939 996.781 cents 10: 1 2227/1252 997.040 cents 10: 1 66/37 1001.929 cents 10: 1 296/165 1011.757 cents 10: 1 9/5 1017.596 cents just minor seventh, BP seventh 11: 1 2227/1188 1087.880 cents 11: 3 15/8 1088.269 cents classic major seventh 11: 1 835/444 1093.460 cents 11: 1 592/313 1103.321 cents 11: 1 256/135 1107.821 cents octave - major chroma 11: 1 4224/2227 1108.210 cents 11: 2 313/165 1108.436 cents 11: 1 1584/835 1108.469 cents 11: 1 594/313 1109.160 cents Highest number of different intervals for one interval class: 10 Average number of different intervals per interval class: 8.72727 = 96/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 91.8 203.9 294.7 386.3 498.0 590.2 702.0 793.5 900.0 996.7 1088.3 1200.0 91.8 : 112.1 203.0 294.5 406.3 498.4 610.2 701.7 808.2 904.9 996.5 1108.2 1200.0 203.9 : 90.8 182.4 294.1 386.3 498.0 589.6 696.1 792.8 884.4 996.1 1087.9 1200.0 294.7 : 91.6 203.3 295.5 407.2 498.7 605.3 702.0 793.5 905.3 997.0 1109.2 1200.0 386.3 : 111.7 203.9 315.6 407.2 513.7 610.4 702.0 813.7 905.5 1017.6 1108.4 1200.0 498.0 : 92.2 203.9 295.4 402.0 498.7 590.2 702.0 793.7 905.9 996.7 1088.3 1200.0 590.2 : 111.7 203.3 309.8 406.5 498.0 609.8 701.6 813.7 904.5 996.1 1107.8 1200.0 702.0 : 91.5 198.1 294.7 386.3 498.0 589.8 702.0 792.8 884.4 996.1 1088.3 1200.0 793.5 : 106.5 203.2 294.8 406.5 498.3 610.4 701.3 792.8 904.6 996.7 1108.5 1200.0 900.0 : 96.7 188.2 300.0 391.8 503.9 594.7 686.3 798.0 890.2 1001.9 1093.5 1200.0 996.7 : 91.6 203.3 295.1 407.2 498.0 589.6 701.3 793.5 905.3 996.8 1103.3 1200.0 1088.3: 111.7 203.5 315.6 406.5 498.0 609.8 702.0 813.7 905.2 1011.8 1108.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.8397 .. 112.1199 cents 705672/697051, 21.2803 cents 313/297, 90.8397 cents 2: 182.4037 .. 203.9100 cents 81/80, 21.5063 cents 11135/10692, 70.2838 cents 3: 294.1350 .. 315.6413 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 4: 386.3137 .. 407.2053 cents 1584/1565, 20.8916 cents 25/24, 70.6724 cents 5: 498.0450 .. 513.7124 cents 111/110, 15.6674 cents 313/297, 90.8397 cents 6: 589.5759 .. 610.4241 cents 705672/697225, 20.8482 cents 4175/3996, 75.8635 cents 7: 686.2876 .. 701.9550 cents 111/110, 15.6674 cents 4175/3996, 75.8635 cents 8: 792.7947 .. 813.6863 cents 1584/1565, 20.8916 cents 313/297, 90.8397 cents 9: 884.3587 .. 905.8650 cents 81/80, 21.5063 cents 25/24, 70.6724 cents 10: 996.0900 .. 1017.5963 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 11: 1087.8801 .. 1109.1603 cents 705672/697051, 21.2803 cents 11135/10692, 70.2838 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 313/297, 90.8397 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 2376/2227, 112.1199 cents Average / Smallest step : 1.100841 Largest / Average step : 1.121199 Largest / Smallest step : 1.234262 Median interval of one step : 94.42865 cents, amount: 0 Most common interval of one step : 16/15, 111.7313 cents, amount: 3 Least squares average step : 99.47768 cents, oct.: 1193.73214 cents Scale is strictly proper Step pattern alph. order: ABCDEFEGHIDE Step pattern size order : DIACHEHBGFCH Scale is a Constant Structure, by a margin of 70.28378 cents Scale diversity : 1.151885 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.814540 Rothenberg efficiency : 0.224928 redundancy: 0.775072 Efficiency x scale size : 2.699134 Number of different interval sizes : 96 = 8.72727 / class Number of one step interval sizes : 9 Highest interval variety : 10 Mean interval variety : 8.72727 = 96/11 Median interval variety : 7 Lowest interval variety : 7 Smallest interval difference : 52272/52271, 0.0331 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 6 Most common triad is 0.0 498.045 701.955 cents, amount: 3 Number of recognisable fifths : 12, average 700.0000 cents Number of appreciable fifths : 11, average 701.2466 cents Scale is a well-temperament. It has 7 different sizes of fifth Best fifths form a closed circle Scale contains a complete diamond : 1 3 Formal octave complements present : 3 = 25.0000% Limited transpositions with margin 21.5063 cents: 5 7 9 Limited inverse transpositions with margin 21.5063 cents: 1 6 9 11 -- Rational properties -- Prime limit : 313 Odd number limit : 2227 (O: 2227 U: 2227) Highest odd numerator or denominator: 2227 Scale harmonicity : 0.014144 Fundamental : 1/2112, -11.0444 octaves, 0.1239 Hz. Guide tone : too high to compute Exponens Consonantiae : 1.637379E+15, 50.54031 octaves Euler's gradus suavitatis : 689 Sum of Mann's harmonic distance : 3484.0, average 290.33333 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 18.26546, average 1.52212 Sum of Tenney's harmonic distance : 35.43983, average 2.95332 Vogel's harmonic complexity : 96.66667 Wille's k value : 1113 Wilson's harmonic complexity : 690 Rectangular lattice diameter : 10 Triangular lattice diameter : 5 Lattice compactness : 268.81440, average 3.44634 Lattice compactness (without 2's) : 131.91961, average 1.69128 Number of different primes : 9 Prime exponents' range, average, count, tones@limit: 2: -6 .. 2 -2.41667 35 1 3: -1 .. 2 0.16667 10 3 5: 0 .. 1 0.33333 4 3 11: -1 .. 0 -0.41667 5 17: 0 .. 1 0.08333 1 37: 0 .. 1 0.08333 1 1 131: 0 .. 1 0.08333 1 1 167: 0 .. 1 0.08333 1 1 313: 0 .. 1 0.16667 2 2 Average exponent except 2's : 7 / 12 = 0.58333 Average absolute exponent except 2's: 25 / 12 = 2.08333 Scale is JI-epimorphic with val: <12 19 28 41 48 62 85 88 99| Scale is JI-epimorphic with val: <12 19 28 41 49 62 84 88 99| Scale is JI-epimorphic with val: <12 19 28 41 50 62 83 88 99| Scale is weakly epimorphic with val: <12 19 28 42 49 62 85 90 100| Scale is JI-epimorphic with val: <12 19 28 42 49 63 85 89 100| Scale is weakly epimorphic with val: <12 19 28 42 50 62 84 90 100| Scale is JI-epimorphic with val: <12 19 28 42 50 63 84 89 100|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 7.0220 c. M:-13.6863 c. Twelve-tone Chromatic (1/11-comma) 40: 3 4 3 3 4 3 3 3 4 3 3 4 SP D3 SD: 7.9481 c. M: 13.4859 c. 41: 3 4 3 3 4 3 4 3 4 3 3 4 SP M ME SD: 3.8058 c. M:-7.2910 c. 53: 4 5 4 4 5 4 5 4 5 4 4 5 SP M ME SD: 1.8548 c. M:-5.6343 c. 65: 5 6 5 5 6 5 6 5 6 5 5 6 SP M ME SD: 1.4741 c. M:-4.5893 c. 77: 6 7 6 6 7 6 7 6 7 6 6 7 SP M ME SD: 1.9652 c. M:-3.8700 c. 106: 8 10 8 8 10 8 10 8 9 9 8 10 SP D3 S SD: 1.8680 c. M: 5.6865 c. 118: 9 11 9 9 11 9 11 9 11 9 9 11 SP M DE SD: 1.4821 c. M:-5.0586 c. 140: 11 13 10 11 13 11 13 11 12 11 11 13 SP SD: 1.8989 c. M:-3.6569 c. 142: 11 13 11 11 13 11 13 11 13 11 11 13 SP M DE SD: 1.6822 c. M:-4.1993 c. 147: 11 14 11 11 14 11 14 11 13 12 11 14 SP SD: 1.6325 c. M: 2.6402 c. 159: 12 15 12 12 15 12 15 12 14 13 12 15 SP SD: 1.0487 c. M: 1.9129 c. 171: 13 16 13 13 16 13 16 13 15 14 13 16 SP SD: 0.6463 c. M: 1.7805 c. 183: 14 17 14 14 17 14 17 14 16 15 14 17 SP SD: 0.5686 c. M: 1.6654 c. 195: 15 18 15 15 18 15 18 15 17 16 15 18 SP SD: 0.7885 c. M: 1.5646 c. 212: 16 20 16 16 20 16 20 16 19 17 16 20 SP SD: 0.8916 c. M: 1.5445 c. 224: 17 21 17 17 21 17 21 17 20 18 17 21 SP SD: 0.5002 c. M: 0.9380 c. 236: 18 22 18 18 22 18 22 18 21 19 18 22 SP SD: 0.2536 c. M: 0.5202 c. 643: 49 60 49 49 60 49 60 49 57 52 49 60 SP SD: 0.3102 c. M: 0.4927 c. 655: 50 61 50 50 61 50 61 50 58 53 50 61 SP SD: 0.2842 c. M: 0.5512 c. 720: 55 67 55 55 67 55 67 55 64 58 55 67 SP SD: 0.2551 c. M: 0.5767 c. 795: 61 74 60 61 74 61 74 61 70 64 61 74 SP SD: 0.2752 c. M: 0.4783 c. 836: 64 78 63 64 78 64 78 64 74 67 64 78 SP SD: 0.2567 c. M: 0.5324 c. 848: 65 79 64 65 79 65 79 65 75 68 65 79 SP SD: 0.2288 c. M: 0.4783 c. 860: 66 80 65 66 80 66 80 66 76 69 66 80 SP SD: 0.2425 c. M:-0.4676 c. 901: 69 84 68 69 84 69 84 69 80 72 69 84 SP SD: 0.2401 c. M: 0.4783 c. 913: 70 85 69 70 85 70 85 70 81 73 70 85 SP SD: 0.2324 c. M: 0.4287 c. 969: 74 91 73 74 90 75 90 74 86 78 74 90 SP SD: 0.2638 c. M:-0.4884 c. 976: 75 91 74 74 91 75 91 74 87 79 74 91 SP SD: 0.2613 c. M: 0.4531 c. 988: 76 92 75 75 92 76 92 75 88 80 75 92 SP SD: 0.2597 c. M:-0.5176 c. 1007: 77 94 76 77 94 77 94 77 89 81 77 94 SP SD: 0.2595 c. M: 0.4783 c. 1019: 78 95 77 78 95 78 95 78 90 82 78 95 SP SD: 0.2226 c. M: 0.4338 c. 1031: 79 96 78 79 96 79 96 79 91 83 79 96 SP SD: 0.2142 c. M: 0.3904 c. 1043: 80 97 79 80 97 80 97 80 92 84 80 97 SP SD: 0.2354 c. M:-0.3779 c. 1063: 81 100 80 81 99 82 99 81 94 86 81 99 SP SD: 0.2244 c. M:-0.4174 c. 1072: 82 100 81 82 100 82 100 82 95 86 82 100 SP SD: 0.2112 c. M: 0.4360 c. 1075: 82 101 81 82 100 83 100 82 95 87 82 100 SP SD: 0.2079 c. M:-0.3691 c. 1084: 83 101 82 83 101 83 101 83 96 87 83 101 SP SD: 0.1868 c. M: 0.3947 c. 1096: 84 102 83 84 102 84 102 84 97 88 84 102 SP SD: 0.1934 c. M: 0.3543 c. 1128: 86 106 85 86 105 87 105 86 100 91 86 105 SP SD: 0.1744 c. M:-0.3453 c. 1140: 87 107 86 87 106 88 106 87 101 92 87 106 SP SD: 0.1717 c. M:-0.3026 c. 1193: 91 112 90 91 111 92 111 91 106 96 91 111 SP SD: 0.1652 c. M:-0.2811 c. 1205: 92 113 91 92 112 93 112 92 107 97 92 112 SP SD: 0.1762 c. M:-0.3157 c. 1299: 99 122 98 99 121 100 121 99 115 105 99 121 SP SD: 0.1625 c. M: 0.3351 c. 1311: 100 123 99 100 122 101 122 100 116 106 100 122 SP SD: 0.1442 c. M: 0.2569 c. 1323: 101 124 100 101 123 102 123 101 117 107 101 123 SP SD: 0.1532 c. M: 0.2529 c. 1364: 104 128 103 104 127 105 127 104 121 110 104 127 SP SD: 0.1202 c. M: 0.2945 c. 1376: 105 129 104 105 128 106 128 105 122 111 105 128 SP SD: 0.1141 c. M: 0.2203 c. 1441: 110 135 109 110 134 111 134 110 128 116 110 134 SP SD: 0.1241 c. M:-0.1996 c. 1547: 118 145 117 118 144 119 144 118 137 125 118 144 SP SD: 0.1094 c. M: 0.2581 c. 1600: 122 150 121 122 149 123 149 122 142 129 122 149 SP SD: 0.0932 c. M: 0.2901 c. 1612: 123 151 122 123 150 124 150 123 143 130 123 150 SP SD: 0.0836 c. M: 0.2268 c. 1624: 124 152 123 124 151 125 151 124 144 131 124 151 SP SD: 0.1035 c. M:-0.1704 c. 1665: 127 156 126 127 155 128 155 127 148 134 127 155 SP SD: 0.0920 c. M: 0.2585 c. 1836: 140 172 139 140 171 141 171 140 163 148 140 171 SP SD: 0.0858 c. M: 0.2868 c. 1848: 141 173 140 141 172 142 172 141 164 149 141 172 SP SD: 0.0742 c. M: 0.2316 c. 2383: 182 223 180 182 222 183 222 182 211 192 182 222 SP SD: 0.1011 c. M: 0.1630 c. 2385: 182 223 181 182 222 183 222 182 212 192 182 222 SP SD: 0.0948 c. M: 0.2177 c. 2395: 183 224 181 183 223 184 223 183 212 193 183 223 SP SD: 0.0895 c. M:-0.1675 c.
FIT/HARMONIC
1 x x x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x x 3 x x x x 4 S SD: 0.0000 cents 3 x x x x 4 x x x 5 x x 6 S SD: 5.2225 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.9697 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.3224 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.3937 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 10.4553 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.3888 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 8.1300 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.5151 cents 11 x 12:13:14:15 x x 17:19:20:21:22 SD: 11.9227 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.8054 cents 13:14:15 x 16:17:18:20:21:22:23:24:26 S SD: 9.0976 cents 14:15:16:17:18:19:20:21:22:24:25:26:28 S SD: 7.8274 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.3877 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 5.0129 cents 17:18:19:20:21:23:24:26:27:29:30:32:34 SD: 4.9155 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 3.8895 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.5370 cents 31:33:35:37:39:41:44:47:49:52:55:58:62 SD: 3.1355 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.0146 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 2.0126 cents 48:51:54:57:60:64:68:72:76:81:85:90:96 SD: 1.7425 cents 54:57:61:64:68:72:76:81:85:91:96:101:108 SD: 1.4802 cents 64:67:72:76:80:85:90:96:101:108:114:120:128 SD: 1.3548 cents 69:73:78:82:86:92:97:104:109:116:123:129:138 SD: 1.3456 cents 71:75:80:84:89:95:100:107:112:119:126:133:142 SD: 1.2762 cents 72:76:81:85:90:96:101:108:114:121:128:135:144 SD: 0.7573 cents 96:101:108:114:120:128:135:144:152:161:171:180:192 SD: 0.6375 cents 128:135:144:152:160:171:180:192:202:215:228:240:256 SD: 0.5580 cents 144:152:162:171:180:192:203:216:228:242:256:270:288 SD: 0.4966 cents 160:169:180:190:200:213:225:240:253:269:285:300:320 SD: 0.4692 cents 167:176:188:198:209:223:235:251:264:281:297:313:334 SD: 0.4412 cents 168:177:189:199:210:224:236:252:266:283:299:315:336 SD: 0.3883 cents 184:194:207:218:230:245:259:276:291:309:327:345:368 SD: 0.3510 cents 200:211:225:237:250:267:281:300:316:336:356:375:400 SD: 0.3439 cents 216:228:243:256:270:288:304:324:342:363:384:405:432 SD: 0.2893 cents 280:295:315:332:350:373:394:420:443:471:498:525:560 SD: 0.2115 cents 296:312:333:351:370:395:416:444:468:498:526:555:592 SD: 0.2058 cents 312:329:351:370:390:416:439:468:493:525:555:585:624 SD: 0.1710 cents 384:405:432:455:480:512:540:576:607:646:683:720:768 SD: 0.1214 cents 480:506:540:569:600:640:675:720:759:807:854:900:960 SD: 0.0922 cents