Scala analysis: Sp_refi_Ki_3
Sparschuh's [2010] refined epimoric Kirnberger III variant
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: 161/144 193.190252 3: 32/27 294.134997 Pythagorean minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 1025/729 589.959828 7: 323/216 696.603423 8: 128/81 792.179997 Pythagorean minor sixth 9: 107/64 889.760384 10: 16/9 996.089998 Pythagorean minor seventh 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: 256/243 90.2250 limma, Pythagorean minor second 2: 4347/4096 102.9653 3: 512/483 100.9447 4: 135/128 92.1787 major chroma, major limma 5: 16/15 111.7313 minor diatonic semitone 6: 1025/972 91.9148 7: 8721/8200 106.6436 8: 1024/969 95.5766 9: 8667/8192 97.5804 10: 1024/963 106.3296 11: 135/128 92.1787 major chroma, major limma 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 256/243 90.225 cents limma, Pythagorean minor second 1: 1 1025/972 91.915 cents 1: 2 135/128 92.179 cents major chroma, major limma 1: 1 1024/969 95.577 cents 1: 1 8667/8192 97.580 cents 1: 1 512/483 100.945 cents 1: 1 4347/4096 102.965 cents 1: 1 1024/963 106.330 cents 1: 1 8721/8200 106.644 cents 1: 2 16/15 111.731 cents minor diatonic semitone 2: 1 180/161 193.123 cents 2: 1 2889/2584 193.157 cents 2: 1 161/144 193.190 cents 2: 1 120/107 198.508 cents 2: 1 323/288 198.558 cents 2: 1 4096/3645 201.956 cents narrow diminished third 2: 1 1152/1025 202.220 cents 2: 1 820/729 203.646 cents 2: 4 9/8 203.910 cents major whole tone 3: 3 32/27 294.135 cents Pythagorean minor third 3: 1 1025/864 295.825 cents 3: 2 1215/1024 296.089 cents wide augmented second 3: 1 384/323 299.487 cents 3: 1 78003/65600 299.801 cents 3: 1 192/161 304.855 cents 3: 1 161/135 304.922 cents 3: 1 128/107 310.240 cents 3: 1 323/270 310.290 cents 4: 1 5/4 386.314 cents major third 4: 1 405/323 391.665 cents 4: 1 321/256 391.715 cents 4: 1 16400/13041 396.770 cents 4: 1 161/128 397.100 cents 4: 1 32768/26001 400.465 cents 4: 1 323/256 402.468 cents 4: 2 512/405 405.866 cents narrow diminished fourth 4: 1 1296/1025 406.130 cents 4: 2 81/64 407.820 cents Pythagorean major third 5: 6 4/3 498.045 cents perfect fourth 5: 1 2187/1640 498.309 cents 5: 1 1025/768 499.735 cents 5: 1 432/323 503.397 cents 5: 1 646/483 503.413 cents 5: 1 1288/963 503.430 cents 5: 1 107/80 503.447 cents 6: 1 1025/729 589.960 cents 6: 2 45/32 590.224 cents diatonic tritone 6: 1 4096/2907 593.622 cents 6: 1 2889/2048 595.625 cents 6: 1 2048/1449 598.990 cents 6: 1 1449/1024 601.010 cents 6: 1 4096/2889 604.375 cents 6: 1 2907/2048 606.378 cents 6: 2 64/45 609.776 cents 2nd tritone 6: 1 1458/1025 610.040 cents 7: 1 160/107 696.553 cents Milliet de Chales 1/4-comma fifth 7: 1 963/644 696.570 cents 7: 1 483/323 696.587 cents 7: 1 323/216 696.603 cents 7: 1 1536/1025 700.265 cents 7: 1 3280/2187 701.691 cents 7: 6 3/2 701.955 cents perfect fifth 8: 2 128/81 792.180 cents Pythagorean minor sixth 8: 1 1025/648 793.870 cents 8: 2 405/256 794.134 cents wide augmented fifth 8: 1 512/323 797.532 cents 8: 1 26001/16384 799.535 cents 8: 1 256/161 802.900 cents 8: 1 13041/8200 803.230 cents 8: 1 512/321 808.285 cents 8: 1 646/405 808.335 cents 8: 1 8/5 813.686 cents minor sixth 9: 1 540/323 889.710 cents 9: 1 107/64 889.760 cents 9: 1 270/161 895.078 cents Kirnberger's sixth 9: 1 161/96 895.145 cents 9: 1 131200/78003 900.199 cents 9: 1 323/192 900.513 cents 9: 2 2048/1215 903.911 cents narrow diminished seventh 9: 1 1728/1025 904.175 cents 9: 3 27/16 905.865 cents Pythagorean major sixth 10: 4 16/9 996.090 cents Pythagorean minor seventh 10: 1 729/410 996.354 cents 10: 1 1025/576 997.780 cents 10: 1 3645/2048 998.044 cents wide augmented sixth 10: 1 576/323 1001.442 cents 10: 1 107/60 1001.492 cents 10: 1 288/161 1006.810 cents 10: 1 5168/2889 1006.843 cents 10: 1 161/90 1006.877 cents 11: 2 15/8 1088.269 cents classic major seventh 11: 1 16400/8721 1093.356 cents 11: 1 963/512 1093.670 cents 11: 1 8192/4347 1097.035 cents 11: 1 483/256 1099.055 cents 11: 1 16384/8667 1102.420 cents 11: 1 969/512 1104.423 cents 11: 2 256/135 1107.821 cents octave - major chroma 11: 1 1944/1025 1108.085 cents 11: 1 243/128 1109.775 cents Pythagorean major seventh Highest number of different intervals for one interval class: 10 Average number of different intervals per interval class: 9.09091 = 100/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 193.2 294.1 386.3 498.0 590.0 696.6 792.2 889.8 996.1 1088.3 1200.0 90.2 : 103.0 203.9 296.1 407.8 499.7 606.4 702.0 799.5 905.9 998.0 1109.8 1200.0 193.2 : 100.9 193.1 304.9 396.8 503.4 599.0 696.6 802.9 895.1 1006.8 1097.0 1200.0 294.1 : 92.2 203.9 295.8 402.5 498.0 595.6 702.0 794.1 905.9 996.1 1099.1 1200.0 386.3 : 111.7 203.6 310.3 405.9 503.4 609.8 702.0 813.7 903.9 1006.9 1107.8 1200.0 498.0 : 91.9 198.6 294.1 391.7 498.0 590.2 702.0 792.2 895.1 996.1 1088.3 1200.0 590.0 : 106.6 202.2 299.8 406.1 498.3 610.0 700.3 803.2 904.2 996.4 1108.1 1200.0 696.6 : 95.6 193.2 299.5 391.7 503.4 593.6 696.6 797.5 889.7 1001.4 1093.4 1200.0 792.2 : 97.6 203.9 296.1 407.8 498.0 601.0 702.0 794.1 905.9 997.8 1104.4 1200.0 889.8 : 106.3 198.5 310.2 400.5 503.4 604.4 696.6 808.3 900.2 1006.8 1102.4 1200.0 996.1 : 92.2 203.9 294.1 397.1 498.0 590.2 702.0 793.9 900.5 996.1 1093.7 1200.0 1088.3: 111.7 202.0 304.9 405.9 498.0 609.8 701.7 808.3 903.9 1001.5 1107.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 111.7313 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 2: 193.1235 .. 203.9100 cents 161/160, 10.7865 cents 675/644, 81.3922 cents 3: 294.1350 .. 310.2897 cents 323/320, 16.1547 cents 256/243, 90.2250 cents 4: 386.3137 .. 407.8200 cents 81/80, 21.5063 cents 675/646, 76.0240 cents 5: 498.0450 .. 503.4467 cents 321/320, 5.4017 cents 256/243, 90.2250 cents 6: 589.9598 .. 610.0402 cents 1062882/1050625, 20.0803 cents 82000/78003, 86.5132 cents 7: 696.5533 .. 701.9550 cents 321/320, 5.4017 cents 82000/78003, 86.5132 cents 8: 792.1800 .. 813.6863 cents 81/80, 21.5063 cents 256/243, 90.2250 cents 9: 889.7103 .. 905.8650 cents 323/320, 16.1547 cents 675/646, 76.0240 cents 10: 996.0900 .. 1006.8765 cents 161/160, 10.7865 cents 256/243, 90.2250 cents 11: 1088.2687 .. 1109.7750 cents 81/80, 21.5063 cents 675/644, 81.3922 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 16/15, 111.7313 cents Average / Smallest step : 1.108340 Largest / Average step : 1.117313 Largest / Smallest step : 1.238363 Median interval of one step : 99.26257 cents, amount: 0 Least squares average step : 99.21171 cents, oct.: 1190.54054 cents Scale is strictly proper Step pattern alph. order: ABCDEFGHIJDE Step pattern size order : AGFCJBIDEHCJ Scale is a Constant Structure, by a margin of 76.02400 cents Scale diversity : 1.119178 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.857674 Rothenberg efficiency : 0.218194 redundancy: 0.781806 Efficiency x scale size : 2.618326 Number of different interval sizes : 100 = 9.09091 / class Number of one step interval sizes : 10 Highest interval variety : 10 Mean interval variety : 9.09091 = 100/11 Median interval variety : 7 Lowest interval variety : 7 Smallest interval difference : 104329/104328, 0.0166 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 6 Most common triad is 0.0 498.045 701.955 cents, amount: 4 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 7 different sizes of fifth Scale has monotonic third-sizes over circle of fifths Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 21.5063 cents: 2 3 5 7 9 10 Limited inverse transpositions with margin 21.5063 cents: 1 2 4 6 7 9 11 -- Rational properties -- Prime limit : 107 Odd number limit : 78003 (O: 78003 U: 78003) Highest odd numerator or denominator: 1025 Scale harmonicity : 0.006170 Fundamental : 1/46656, -15.5098 octaves, 0.0056 Hz. Guide tone : too high to compute Exponens Consonantiae : 2.043642E+17, 57.50392 octaves Euler's gradus suavitatis : 245 Sum of Mann's harmonic distance : 1801.5, average 150.12500 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 18.78802, average 1.56567 Sum of Tenney's harmonic distance : 37.58270, average 3.13189 Vogel's harmonic complexity : 38.00000 Wille's k value : 39001 Wilson's harmonic complexity : 245 Rectangular lattice diameter : 15 Triangular lattice diameter : 7 Lattice compactness : 486.20384, average 6.23338 Lattice compactness (without 2's) : 223.08779, average 2.86010 Number of different primes : 9 Prime exponents' range, average, count, tones@limit: 2: -6 .. 8 0.75000 45 1 3: -6 .. 1 -2.08333 27 5 5: 0 .. 2 0.33333 4 2 7: 0 .. 1 0.08333 1 17: 0 .. 1 0.08333 1 19: 0 .. 1 0.08333 1 1 23: 0 .. 1 0.08333 1 1 41: 0 .. 1 0.08333 1 1 107: 0 .. 1 0.08333 1 1 Average exponent except 2's :-15 / 12 =-1.25000 Average absolute exponent except 2's: 37 / 12 = 3.08333 Scale is weakly epimorphic with val: <12 19 28 33 48 51 55 65 81| Scale is JI-epimorphic with val: <12 19 28 33 48 52 55 64 81| Scale is weakly epimorphic with val: <12 19 28 33 49 50 55 65 81| Scale is JI-epimorphic with val: <12 19 28 33 49 51 55 64 81| Scale is JI-epimorphic with val: <12 19 28 33 50 50 55 64 81| Scale is weakly epimorphic with val: <12 19 28 34 48 51 54 65 81| Scale is JI-epimorphic with val: <12 19 28 34 48 52 54 64 81| Scale is weakly epimorphic with val: <12 19 28 34 49 50 54 65 81| Scale is JI-epimorphic with val: <12 19 28 34 49 51 54 64 81| = patent Scale is JI-epimorphic with val: <12 19 28 34 50 50 54 64 81| Scale is weakly epimorphic with val: <12 19 28 35 48 51 53 65 81| Scale is JI-epimorphic with val: <12 19 28 35 48 52 53 64 81| Scale is weakly epimorphic with val: <12 19 28 35 49 50 53 65 81| Scale is JI-epimorphic with val: <12 19 28 35 49 51 53 64 81| Scale is JI-epimorphic with val: <12 19 28 35 50 50 53 64 81|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 8.1584 c. M:-13.6863 c. Twelve-tone Chromatic (1/11-comma) 40: 3 3 4 3 4 3 3 3 4 3 3 4 SP D3 SD: 8.5050 c. M: 13.1903 c. 41: 3 3 4 3 4 3 4 3 3 4 3 4 SP M ME SD: 6.9328 c. M: 17.5805 c. 43: 3 4 4 3 4 3 4 3 4 4 3 4 SP M ME SD: 6.2325 c. M:-12.8417 c. Meantone Chromatic (1/5-comma) 50: 4 4 4 4 5 4 4 4 4 5 3 5 SP D3 SD: 5.9704 c. M:-11.9100 c. 53: 4 5 4 4 5 4 5 4 4 5 4 5 SP M ME SD: 3.9940 c. M:-10.5833 c. Schismic-12 57: 4 5 5 4 6 4 5 5 4 5 5 5 SP D3 SD: 5.3072 c. M:-7.8200 c. 62: 5 5 5 5 6 4 6 5 5 5 5 6 SP D3 SD: 4.7802 c. M: 9.3147 c. 65: 5 5 6 5 6 5 6 5 5 6 5 6 SP M ME SD: 3.2134 c. M: 8.5749 c. 74: 6 6 6 6 7 5 7 6 6 6 6 7 SP D3 SD: 3.9452 c. M:-7.0723 c. 77: 6 6 7 6 7 6 7 6 6 7 6 7 SP M ME SD: 3.0123 c. M: 6.1773 c. 89: 7 7 8 7 8 7 8 7 7 8 7 8 SP M ME SD: 3.2397 c. M:-4.6975 c. 93: 7 8 8 7 9 7 8 7 8 8 7 9 SP SD: 2.8784 c. M:-5.1808 c. 105: 8 9 9 8 10 8 9 8 9 9 8 10 SP SD: 2.6542 c. M:-4.8121 c. 106: 8 9 9 8 10 8 10 8 9 9 8 10 SP D3 SD: 2.1503 c. M:-5.2834 c. 118: 9 10 10 9 11 9 10 10 9 11 9 11 SP D3 SD: 2.1351 c. M: 5.0780 c. 120: 9 10 10 10 11 9 11 9 10 11 9 11 SP T3 SD: 2.5718 c. M: 4.1350 c. 124: 9 11 10 10 11 10 11 10 10 11 9 12 SP SD: 2.3706 c. M: 4.4966 c. 130: 10 11 11 10 12 10 11 11 10 12 10 12 SP D3 SD: 1.9347 c. M: 4.2957 c. 133: 10 11 12 10 12 10 12 11 11 11 11 12 SP SD: 2.7210 c. M: 3.7166 c. 135: 10 12 11 10 13 10 12 11 11 12 10 13 SP SD: 2.2947 c. M: 4.0915 c. 136: 10 12 11 11 12 11 12 11 11 12 10 13 SP SD: 2.0397 c. M: 3.9274 c. 142: 11 12 12 11 13 11 13 11 11 13 11 13 SP D3 S SD: 2.2121 c. M:-4.8050 c. 143: 11 12 12 11 13 11 13 11 12 13 11 13 SP SD: 1.9252 c. M: 3.3688 c. 147: 11 13 12 11 14 11 13 12 12 13 11 14 SP SD: 1.6742 c. M:-2.7281 c. 159: 12 14 13 12 15 12 14 13 13 14 12 15 SP SD: 1.3244 c. M:-3.0362 c. 171: 13 15 14 13 16 13 15 14 14 15 13 16 SP SD: 1.2717 c. M:-3.3010 c. 183: 14 16 15 14 17 14 16 15 15 16 14 17 SP SD: 1.4396 c. M:-3.5311 c. 190: 14 17 16 14 18 14 17 15 16 17 14 18 SP SD: 1.9244 c. M:-2.7071 c. 191: 14 17 16 14 18 15 17 15 16 17 14 18 SP SD: 1.7868 c. M: 3.0676 c. 193: 14 17 16 15 18 15 17 15 16 17 15 18 SP SD: 1.4173 c. M: 3.1784 c. 200: 15 17 17 15 19 15 18 16 16 18 15 19 SP S SD: 1.2693 c. M: 2.3137 c. 205: 15 18 17 16 19 16 18 16 17 18 16 19 SP SD: 1.1081 c. M: 2.4201 c. 212: 16 18 18 16 20 16 19 17 17 19 16 20 SP S SD: 0.8120 c. M: 1.4763 c. 224: 17 19 19 17 21 17 20 18 18 20 17 21 SP S SD: 0.5282 c. M:-0.8464 c. 236: 18 20 20 18 22 18 21 19 19 21 18 22 SP S SD: 0.5610 c. M:-1.3004 c. 417: 31 36 35 32 39 32 37 33 34 37 32 39 SP SD: 0.5407 c. M: 1.0164 c. 429: 32 37 36 33 40 33 38 34 35 38 33 40 SP SD: 0.3423 c. M: 0.7145 c. 441: 33 38 37 34 41 34 39 35 36 39 34 41 SP SD: 0.2428 c. M:-0.5164 c. 653: 49 56 55 50 61 50 58 52 53 58 50 61 SP SD: 0.2140 c. M: 0.4025 c. 665: 50 57 56 51 62 51 59 53 54 59 51 62 SP SD: 0.0872 c. M: 0.1484 c. 1342: 101 115 113 103 125 103 119 107 109 119 103 125 SP SD: 0.0742 c. M:-0.2041 c. 2007: 151 172 169 154 187 154 178 160 163 178 154 187 SP SD: 0.0708 c. M:-0.1747 c. 2224: 167 191 187 171 207 170 198 177 181 197 171 207 SP SD: 0.0807 c. M: 0.2116 c. 2231: 168 191 188 171 208 171 198 178 181 198 171 208 SP SD: 0.0959 c. M: 0.1468 c. 2236: 168 192 188 172 208 171 199 178 182 198 172 208 SP SD: 0.0673 c. M: 0.1566 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: 2.6758 cents 3 x x x x 4 x x x 5 x x 6 S SD: 1.8006 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.9461 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.5974 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.0187 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 11.8048 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 8.5856 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.6519 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.4480 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.9417 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 6.7489 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.2756 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.1268 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.3993 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 5.2495 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.6701 cents 22:23:25:26:28:29:31:33:35:37:39:41:44 SD: 4.3893 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.3966 cents 27:28:30:32:34:36:38:40:43:45:48:51:54 SD: 3.3960 cents 31:33:35:37:39:41:44:46:49:52:55:58:62 SD: 3.3160 cents 32:34:36:38:40:43:45:48:51:54:57:60:64 SD: 2.6891 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.4000 cents 36:38:40:43:45:48:51:54:57:60:64:68:72 SD: 2.2233 cents 40:42:45:47:50:53:56:60:63:67:71:75:80 SD: 2.0820 cents 48:51:54:57:60:64:67:72:76:80:85:90:96 SD: 2.0706 cents 49:52:55:58:61:65:69:73:77:82:87:92:98 SD: 1.8545 cents 52:55:58:62:65:69:73:78:82:87:92:98:104 SD: 1.7489 cents 54:57:60:64:68:72:76:81:85:90:96:101:108 SD: 1.6995 cents 57:60:64:68:71:76:80:85:90:95:101:107:114 SD: 1.4964 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.2588 cents 69:73:77:82:86:92:97:103:109:115:123:129:138 SD: 1.1726 cents 76:80:85:90:95:101:107:114:120:127:135:143:152 SD: 0.8833 cents 91:96:102:108:114:121:128:136:144:152:162:171:182 SD: 0.8219 cents 93:98:104:110:116:124:131:139:147:155:165:174:186 SD: 0.8019 cents 109:115:122:129:136:145:153:163:172:182:194:204:218 SD: 0.7252 cents 113:119:126:134:141:151:159:169:179:189:201:212:226 SD: 0.6916 cents 128:135:143:152:160:171:180:191:202:214:228:240:256 SD: 0.6398 cents 129:136:144:153:161:172:181:193:204:216:229:242:258 SD: 0.5741 cents 135:142:151:160:169:180:190:202:213:226:240:253:270 SD: 0.4688 cents 152:160:170:180:190:203:214:227:240:254:270:285:304 SD: 0.4377 cents 167:176:187:198:209:223:235:250:264:279:297:313:334 SD: 0.4353 cents 168:177:188:199:210:224:236:251:265:281:299:315:336 SD: 0.3936 cents 204:215:228:242:255:272:287:305:322:341:363:383:408 SD: 0.3360 cents 243:256:272:288:304:324:342:363:384:406:432:456:486 SD: 0.3264 cents 244:257:273:289:305:325:343:365:386:408:434:458:488 SD: 0.3176 cents 264:278:295:313:330:352:371:395:417:441:469:495:528 SD: 0.2351 cents 339:357:379:402:424:452:477:507:536:567:603:636:678 SD: 0.2282 cents 355:374:397:421:444:473:499:531:561:594:631:666:710 SD: 0.2242 cents 357:376:399:423:446:476:502:534:564:597:635:669:714 SD: 0.1701 cents 372:392:416:441:465:496:523:556:588:622:661:698:744 SD: 0.1623 cents 432:455:483:512:540:576:607:646:683:722:768:810:864 SD: 0.1344 cents 448:472:501:531:560:597:630:670:708:749:796:840:896 SD: 0.1226 cents 567:597:634:672:709:756:797:848:896:948:1008:1063:1134 SD: 0.1081 cents 600:632:671:711:750:800:844:897:948:1003:1067:1125:1200 SD: 0.1031 cents