Scala analysis: triharmon
The triharmonic scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 99.534 cents 99.533780 2: 257.542 cents 257.542480 3: 357.076 cents 357.076250 4: 456.610 cents 456.610030 5: 614.619 cents 614.618730 6: 714.153 cents 714.152510 7: 8/5 813.686286 minor sixth 8: 971.695 cents 971.694980 9: 1071.229 cents 1071.228760 10: 1170.763 cents 1170.762540 11: 1328.771 cents 1328.771240 12: 1428.305 cents 1428.305020 13: 5/2 1586.313714 major 10th 14: 1685.847 cents 1685.847490 15: 1785.381 cents 1785.381270 16: 1943.390 cents 1943.389970 17: 2042.924 cents 2042.923750 18: 2142.458 cents 2142.457520 19: 2300.466 cents 2300.466220 20: 4/1 2400.000000 2 octaves
SHOW/INTERVAL
0: 99.5338 cents 99.5338 1: 99.5338 cents 99.5338 2: 158.0087 cents 158.0087 3: 99.5338 cents 99.5338 4: 99.5338 cents 99.5338 5: 158.0087 cents 158.0087 6: 99.5338 cents 99.5338 7: 99.5338 cents 99.5338 8: 158.0087 cents 158.0087 9: 99.5338 cents 99.5338 10: 99.5338 cents 99.5338 11: 158.0087 cents 158.0087 12: 99.5338 cents 99.5338 13: 158.0087 cents 158.0087 14: 99.5338 cents 99.5338 15: 99.5338 cents 99.5338 16: 158.0087 cents 158.0087 17: 99.5338 cents 99.5338 18: 99.5338 cents 99.5338 19: 158.0087 cents 158.0087 20: 99.5338 cents 99.5338
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 13 99.53378 cents 1: 7 158.00870 cents 2: 6 199.06755 cents 2: 14 257.54248 cents 3: 19 357.07625 cents 3: 1 415.55117 cents 4: 12 456.61003 cents 4: 8 515.08495 cents 5: 5 556.14381 cents 5: 15 614.61873 cents 6: 18 714.15251 cents 6: 2 25/16 772.627 cents classic augmented fifth 7: 11 8/5 813.686 cents minor sixth 7: 9 872.16120 cents 8: 4 913.22006 cents 8: 16 971.69498 cents 9: 17 1071.22876 cents 9: 3 1129.70368 cents 10: 10 1170.76254 cents 10: 10 1229.23746 cents 11: 3 1270.29632 cents 11: 17 1328.77124 cents 12: 16 1428.30502 cents 12: 4 1486.77993 cents 13: 9 1527.83879 cents 13: 11 5/2 1586.314 cents major 10th 14: 2 64/25 1627.373 cents classic diminished 11th 14: 18 1685.84749 cents 15: 15 1785.38127 cents 15: 5 1843.85619 cents 16: 8 1884.91504 cents 16: 12 1943.38997 cents 17: 1 1984.44883 cents 17: 19 2042.92375 cents 18: 14 2142.45752 cents 18: 6 2200.93244 cents 19: 7 2241.99130 cents 19: 13 2300.46622 cents Highest number of different intervals for one interval class: 2 Average number of different intervals per interval class: 2.00000 = 2
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.0 : 99.5 257.5 357.1 456.6 614.6 714.2 813.7 971.7 1071.2 1170.8 1328.8 1428.3 1586.3 1685.8 1785.4 1943.4 2042.9 2142.5 2300.5 2400.0 99.5 : 158.0 257.5 357.1 515.1 614.6 714.2 872.2 971.7 1071.2 1229.2 1328.8 1486.8 1586.3 1685.8 1843.9 1943.4 2042.9 2200.9 2300.5 2400.0 257.5 : 99.5 199.1 357.1 456.6 556.1 714.2 813.7 913.2 1071.2 1170.8 1328.8 1428.3 1527.8 1685.8 1785.4 1884.9 2042.9 2142.5 2242.0 2400.0 357.1 : 99.5 257.5 357.1 456.6 614.6 714.2 813.7 971.7 1071.2 1229.2 1328.8 1428.3 1586.3 1685.8 1785.4 1943.4 2042.9 2142.5 2300.5 2400.0 456.6 : 158.0 257.5 357.1 515.1 614.6 714.2 872.2 971.7 1129.7 1229.2 1328.8 1486.8 1586.3 1685.8 1843.9 1943.4 2042.9 2200.9 2300.5 2400.0 614.6 : 99.5 199.1 357.1 456.6 556.1 714.2 813.7 971.7 1071.2 1170.8 1328.8 1428.3 1527.8 1685.8 1785.4 1884.9 2042.9 2142.5 2242.0 2400.0 714.2 : 99.5 257.5 357.1 456.6 614.6 714.2 872.2 971.7 1071.2 1229.2 1328.8 1428.3 1586.3 1685.8 1785.4 1943.4 2042.9 2142.5 2300.5 2400.0 813.7 : 158.0 257.5 357.1 515.1 614.6 772.6 872.2 971.7 1129.7 1229.2 1328.8 1486.8 1586.3 1685.8 1843.9 1943.4 2042.9 2200.9 2300.5 2400.0 971.7 : 99.5 199.1 357.1 456.6 614.6 714.2 813.7 971.7 1071.2 1170.8 1328.8 1428.3 1527.8 1685.8 1785.4 1884.9 2042.9 2142.5 2242.0 2400.0 1071.2: 99.5 257.5 357.1 515.1 614.6 714.2 872.2 971.7 1071.2 1229.2 1328.8 1428.3 1586.3 1685.8 1785.4 1943.4 2042.9 2142.5 2300.5 2400.0 1170.8: 158.0 257.5 415.6 515.1 614.6 772.6 872.2 971.7 1129.7 1229.2 1328.8 1486.8 1586.3 1685.8 1843.9 1943.4 2042.9 2200.9 2300.5 2400.0 1328.8: 99.5 257.5 357.1 456.6 614.6 714.2 813.7 971.7 1071.2 1170.8 1328.8 1428.3 1527.8 1685.8 1785.4 1884.9 2042.9 2142.5 2242.0 2400.0 1428.3: 158.0 257.5 357.1 515.1 614.6 714.2 872.2 971.7 1071.2 1229.2 1328.8 1428.3 1586.3 1685.8 1785.4 1943.4 2042.9 2142.5 2300.5 2400.0 1586.3: 99.5 199.1 357.1 456.6 556.1 714.2 813.7 913.2 1071.2 1170.8 1270.3 1428.3 1527.8 1627.4 1785.4 1884.9 1984.4 2142.5 2242.0 2400.0 1685.8: 99.5 257.5 357.1 456.6 614.6 714.2 813.7 971.7 1071.2 1170.8 1328.8 1428.3 1527.8 1685.8 1785.4 1884.9 2042.9 2142.5 2300.5 2400.0 1785.4: 158.0 257.5 357.1 515.1 614.6 714.2 872.2 971.7 1071.2 1229.2 1328.8 1428.3 1586.3 1685.8 1785.4 1943.4 2042.9 2200.9 2300.5 2400.0 1943.4: 99.5 199.1 357.1 456.6 556.1 714.2 813.7 913.2 1071.2 1170.8 1270.3 1428.3 1527.8 1627.4 1785.4 1884.9 2042.9 2142.5 2242.0 2400.0 2042.9: 99.5 257.5 357.1 456.6 614.6 714.2 813.7 971.7 1071.2 1170.8 1328.8 1428.3 1527.8 1685.8 1785.4 1943.4 2042.9 2142.5 2300.5 2400.0 2142.5: 158.0 257.5 357.1 515.1 614.6 714.2 872.2 971.7 1071.2 1229.2 1328.8 1428.3 1586.3 1685.8 1843.9 1943.4 2042.9 2200.9 2300.5 2400.0 2300.5: 99.5 199.1 357.1 456.6 556.1 714.2 813.7 913.2 1071.2 1170.8 1270.3 1428.3 1527.8 1685.8 1785.4 1884.9 2042.9 2142.5 2242.0 2400.0 2400.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 99.5338 .. 158.0087 cents 58.47492 cents 99.53378 cents 2: 199.0675 .. 257.5425 cents 58.47493 cents 41.05885 cents 3: 357.0762 .. 415.5512 cents 58.47492 cents 99.53377 cents 4: 456.6100 .. 515.0850 cents 58.47492 cents 41.05886 cents 5: 556.1438 .. 614.6187 cents 58.47492 cents 41.05886 cents 6: 714.1525 .. 772.6274 cents 58.47492 cents 99.53378 cents 7: 813.6863 .. 872.1612 cents 58.47491 cents 128/125, 41.0589 cents 8: 913.2201 .. 971.6950 cents 58.47492 cents 41.05886 cents 9: 1071.2288 .. 1129.7037 cents 58.47492 cents 99.53378 cents 10: 1170.7625 .. 1229.2375 cents 58.47492 cents 41.05886 cents 11: 1270.2963 .. 1328.7712 cents 58.47492 cents 41.05886 cents 12: 1428.3050 .. 1486.7799 cents 58.47491 cents 99.53378 cents 13: 1527.8388 .. 1586.3137 cents 58.47492 cents 41.05886 cents 14: 1627.3726 .. 1685.8475 cents 58.47492 cents 128/125, 41.0589 cents 15: 1785.3813 .. 1843.8562 cents 58.47492 cents 99.53378 cents 16: 1884.9150 .. 1943.3900 cents 58.47493 cents 41.05885 cents 17: 1984.4488 .. 2042.9237 cents 58.47492 cents 41.05886 cents 18: 2142.4575 .. 2200.9324 cents 58.47492 cents 99.53377 cents 19: 2241.9913 .. 2300.4662 cents 58.47492 cents 41.05886 cents
SHOW DATA
Number of notes : 20 -- Interval properties -- Smallest interval : 99.53377 cents, class 1 Average step (divided formal octave): 120.0000 cents Largest one step interval : 158.00870 cents Average / Smallest step : 1.205621 Largest / Average step : 1.316739 Largest / Smallest step : 1.587488 Median interval of one step : 99.53378 cents, amount: 13 Most common interval of one step : 99.53378 cents, amount: 13 Least squares average step : 120.05094 cents, oct.: 2401.01873 cents Scale is strictly proper Scale has Myhill's property chromas: 58.47492 cents and 41.05886 cents, ratio: 1.424173 generators: 3 of 357.07625 cents and 17 of 2042.92375 cents ET pair of linear temperament : 20&27 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 1929-tone equal temperament with octave 4/1 degrees: 80 207 287 367 494 574 654 781 861 941 1068 1148 1275 1355 1435 1562 1642 1722 1849 1929 generators: 287 and 1642 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABAABAABAABABAABAABA Step pattern size order : SLSSLSSLSSLSLSSLSSLS Scale is a Constant Structure, by a margin of 41.05885 cents Scale diversity :-0.035286 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.537074 Rothenberg efficiency : 0.700000 redundancy: 0.300000 Efficiency x scale size : 14.000000 Number of different interval sizes : 38 = 2.00000 / class Number of one step interval sizes : 2 Highest interval variety : 2 Mean interval variety : 2.00000 = 2 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 41.05885 cents Most common intervals : 357.07625 cents & inv., amount: 19 Scale is a chain of 10 triads 0.0 813.686 2042.924 cents Most common triad is 0.0 357.076 2042.924 cents, amount: 18 Number of recognisable fifths : 18, average 714.1525 cents Number of recognisable fourths : 0 Scale contains two identical disjunct pentachords Best major thirds form a closed circle Scale contains a complete diamond : 1 5 Formal octave complements present : 19 = 95.0000% 2/1 octave complements present : 0 = 0.0000% Limited inverse transpositions : 3 Inversional symmetry on intervals : 1-2 11-12 -- Rational properties -- Prime limit : 5 (not all pitches rational)
FIT/MODE
20: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 16.9224 c. M:-29.2375 c. 27: 1 2 1 1 2 1 1 2 1 1 2 1 2 1 1 2 1 1 2 1 P M ME SD: 8.8017 c. M: 15.2070 c. 47: 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 SP M ME SD: 2.1447 c. M:-3.7055 c. 74: 3 5 3 3 5 3 3 5 3 3 5 3 5 3 3 5 3 3 5 3 SP M DE SD: 1.8492 c. M: 3.1950 c. 121: 5 8 5 5 8 5 5 8 5 5 8 5 8 5 5 8 5 5 8 5 SP M DE SD: 0.2978 c. M: 0.5146 c. 289: 12 19 12 12 19 12 12 19 12 12 19 12 19 12 12 19 12 12 19 12 SP M DE SD: 0.0994 c. M:-0.1717 c. 410: 17 27 17 17 27 17 17 27 17 17 27 17 27 17 17 27 17 17 27 17 SP M DE SD: 0.0178 c. M: 0.0308 c. 1109: 46 73 46 46 73 46 46 73 46 46 73 46 73 46 46 73 46 46 73 46 SP M DE SD: 0.0127 c. M:-0.0220 c. 1519: 63 100 63 63 100 63 63 100 63 63 100 63 100 63 63 100 63 63 100 63 SP M DE SD: 0.0045 c. M:-0.0077 c. 1929: 80 127 80 80 127 80 80 127 80 80 127 80 127 80 80 127 80 80 127 80 SP M DE SD: 0.0003 c. M: 0.0005 c.
FIT/HARMONIC
1 .. 4 match: 3 H SD: 16.9039 cents 2 .. 8 match: 6 H SD: 9.7407 cents 3 .. 12 match: 9 H SD: 13.7201 cents 4 .. 16 match: 12 H SD: 6.5557 cents 5 .. 20 match: 13 SD: 8.1717 cents 6 .. 24 match: 16 SD: 7.6853 cents 7 .. 28 match: 17 S SD: 6.5868 cents 8 .. 32 match: 18 SD: 6.5770 cents 9 .. 36 match: 18 SD: 7.4576 cents 10 .. 40 match: 19 SD: 6.6055 cents 11 .. 44 match: 19 SD: 6.3147 cents 12 .. 48 match: 20 SD: 4.3911 cents 13 .. 52 match: 20 SD: 4.3354 cents 14 .. 56 match: 20 SD: 3.7523 cents 17 .. 68 match: 20 SD: 3.2257 cents 19 .. 76 match: 20 SD: 3.1132 cents 20 .. 80 match: 20 SD: 2.7640 cents 25 .. 100 match: 20 SD: 2.7047 cents 26 .. 104 match: 20 SD: 2.2601 cents 30 .. 120 match: 20 SD: 1.7491 cents 31 .. 124 match: 20 SD: 1.7053 cents 32 .. 128 match: 20 SD: 1.6898 cents 33 .. 132 match: 20 SD: 1.6716 cents 35 .. 140 match: 20 SD: 1.5971 cents 36 .. 144 match: 20 SD: 1.5034 cents 40 .. 160 match: 20 SD: 1.4338 cents 43 .. 172 match: 20 SD: 1.4081 cents 47 .. 188 match: 20 SD: 1.3322 cents 49 .. 196 match: 20 SD: 1.0473 cents 56 .. 224 match: 20 SD: 0.9012 cents 57 .. 228 match: 20 SD: 0.8911 cents 63 .. 252 match: 20 SD: 0.8755 cents 66 .. 264 match: 20 SD: 0.8713 cents 68 .. 272 match: 20 SD: 0.8586 cents 69 .. 276 match: 20 SD: 0.8218 cents 70 .. 280 match: 20 SD: 0.6085 cents 82 .. 328 match: 20 SD: 0.5108 cents 100 .. 400 match: 20 SD: 0.4908 cents 113 .. 452 match: 20 SD: 0.4445 cents 117 .. 468 match: 20 SD: 0.3993 cents 118 .. 472 match: 20 SD: 0.3867 cents 148 .. 592 match: 20 SD: 0.3651 cents 149 .. 596 match: 20 SD: 0.3567 cents 150 .. 600 match: 20 SD: 0.3428 cents 153 .. 612 match: 20 SD: 0.3277 cents 162 .. 648 match: 20 SD: 0.3141 cents 169 .. 676 match: 20 SD: 0.3062 cents 170 .. 680 match: 20 SD: 0.2974 cents 182 .. 728 match: 20 SD: 0.2757 cents 188 .. 752 match: 20 SD: 0.2151 cents 218 .. 872 match: 20 SD: 0.2143 cents 235 .. 940 match: 20 SD: 0.1836 cents 272 .. 1088 match: 20 SD: 0.1671 cents 288 .. 1152 match: 20 SD: 0.1526 cents 305 .. 1220 match: 20 SD: 0.1396 cents 354 .. 1416 match: 20 SD: 0.1386 cents 355 .. 1420 match: 20 SD: 0.1298 cents 368 .. 1472 match: 20 SD: 0.1225 cents 388 .. 1552 match: 20 SD: 0.1191 cents 425 .. 1700 match: 20 SD: 0.1097 cents 441 .. 1764 match: 20 SD: 0.1018 cents 474 .. 1896 match: 20 SD: 0.0978 cents 507 .. 2028 match: 20 SD: 0.0907 cents 511 .. 2044 match: 20 SD: 0.0906 cents 523 .. 2092 match: 20 SD: 0.0831 cents 537 .. 2148 match: 20 SD: 0.0747 cents