Scala analysis: xen15-gilson-aristoxenus-chromatic-hemiolon
Aristoxenus' Chromatic Hemiolon
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 80/77 66.169865 2: 40/37 134.969675 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 120/77 768.124866 6: 60/37 836.924676 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 37/30 363.0753 1: 80/77 66.1699 2: 77/74 68.7998 3: 37/30 363.0753 4: 9/8 203.9100 major whole tone 5: 80/77 66.1699 6: 77/74 68.7998 7: 37/30 363.0753
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 80/77 66.170 cents 1: 2 77/74 68.800 cents 1: 1 9/8 203.910 cents major whole tone 1: 2 37/30 363.075 cents 2: 2 40/37 134.970 cents 2: 1 90/77 270.080 cents 2: 1 296/231 429.245 cents 2: 2 77/60 431.875 cents 2: 1 111/80 566.985 cents 3: 1 45/37 338.880 cents 3: 4 4/3 498.045 cents perfect fourth 3: 1 111/77 633.155 cents 3: 1 231/160 635.785 cents 4: 1 320/231 564.215 cents 4: 1 154/111 566.845 cents 4: 4 3/2 701.955 cents perfect fifth 4: 1 74/45 861.120 cents 5: 1 160/111 633.015 cents 5: 2 120/77 768.125 cents 5: 1 231/148 770.755 cents 5: 1 77/45 929.920 cents 5: 2 37/20 1065.030 cents 6: 2 60/37 836.925 cents 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 148/77 1131.200 cents 6: 2 77/40 1133.830 cents Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 4.33333 = 13/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 66.2 135.0 498.0 702.0 768.1 836.9 1200.0 66.2 : 68.8 431.9 635.8 702.0 770.8 1133.8 1200.0 135.0 : 363.1 567.0 633.2 702.0 1065.0 1131.2 1200.0 498.0 : 203.9 270.1 338.9 702.0 768.1 836.9 1200.0 702.0 : 66.2 135.0 498.0 564.2 633.0 996.1 1200.0 768.1 : 68.8 431.9 498.0 566.8 929.9 1133.8 1200.0 836.9 : 363.1 429.2 498.0 861.1 1065.0 1131.2 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 66.1699 .. 363.0753 cents 2849/2400, 296.9055 cents 80/77, 66.1699 cents 2: 134.9697 .. 566.9853 cents 4107/3200, 432.0157 cents 1200/1369,-228.1056 cents 3: 338.8797 .. 635.7851 cents 2849/2400, 296.9055 cents 1200/1369,-228.1056 cents 4: 564.2149 .. 861.1203 cents 2849/2400, 296.9055 cents 51200/53361,-71.5703 cents 5: 633.0147 .. 1065.0303 cents 4107/3200, 432.0157 cents 1200/1369,-228.1056 cents 6: 836.9247 .. 1133.8301 cents 2849/2400, 296.9055 cents 1200/1369,-228.1056 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 80/77, 66.1699 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 37/30, 363.0753 cents Average / Smallest step : 2.590735 Largest / Average step : 2.117939 Largest / Smallest step : 5.487019 Median interval of one step : 77/74, 68.7998 cents, amount: 2 Least squares average step : 156.43026 cents, oct.: 1095.01183 cents Scale is not proper Step pattern alph. order: ABCDABC Step pattern size order : ABDCABD Scale is a Constant Structure, by a margin of 0.14052 cents Scale diversity : 1.504425 Lumma stability : 0.110283 Lumma impropriety factor : 0.775051 Rothenberg efficiency : 0.421769 redundancy: 0.578231 Efficiency x scale size : 2.952381 Number of different interval sizes : 26 = 4.33333 / class Number of one step interval sizes : 4 Highest interval variety : 5 Mean interval variety : 4.33333 = 13/3 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 12321/12320, 0.1405 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 4 Number of recognisable fifths : 4, average 701.9550 cents Scale is Dorian octave species of abstract tetrachord Scale contains two identical disjunct tetrachords Scale contains a complete diamond : 1 3 Formal octave complements present : 3 = 42.8571% Scale is differentially coherent in interval classes 3 and 4 combined -- Rational properties -- Prime limit : 37 Odd number limit : 231 (O: 231 U: 231) Highest odd numerator or denominator: 77 Scale harmonicity : 0.022123 Fundamental : 1/17094, -14.0612 octaves, 0.0153 Hz. Guide tone : 240, 7.9069 octaves, 62790.136 Hz. Exponens Consonantiae : 4.102560E+06, 21.96809 octaves Euler's gradus suavitatis : 66 Sum of Mann's harmonic distance : 271.5, average 38.78571 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 7.86363, average 1.12338 Sum of Tenney's harmonic distance : 16.43023, average 2.34718 Vogel's harmonic complexity : 49.00000 Wille's k value : 12820 Wilson's harmonic complexity : 66 Rectangular lattice diameter : 6 Triangular lattice diameter : 3 Lattice compactness : 73.18727, average 2.61383 Lattice compactness (without 2's) : 41.02735, average 1.46526 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: -1 .. 4 2.00000 16 1 3: -1 .. 1 0.28571 4 2 5: 0 .. 1 0.57143 4 7: -1 .. 0 -0.28571 2 11: -1 .. 0 -0.28571 2 2 37: -1 .. 0 -0.28571 2 2 Average exponent except 2's : 0 / 7 = 0.00000 Average absolute exponent except 2's: 14 / 7 = 2.00000 Scale is JI-epimorphic with val: <7 11 15 19 23 34| Scale is weakly epimorphic with val: <7 11 16 19 23 36| Scale is JI-epimorphic with val: <7 11 16 19 24 35| Scale is JI-epimorphic with val: <7 11 16 20 23 35| Scale is weakly epimorphic with val: <7 11 17 19 24 37| Scale is JI-epimorphic with val: <7 11 17 19 25 36| Scale is weakly epimorphic with val: <7 11 17 20 23 37| Scale is JI-epimorphic with val: <7 11 17 20 24 36| Scale is JI-epimorphic with val: <7 11 17 21 23 36| Scale is weakly epimorphic with val: <7 12 16 20 25 34| Scale is weakly epimorphic with val: <7 12 16 21 24 34| Scale is weakly epimorphic with val: <7 12 17 21 25 35|
FIT/MODE
7: 0 1 2 1 0 1 2 N T3 SD: 43.8105 c. M: 82.4106 c. 9: 0 1 3 1 1 0 3 N SD: 36.3537 c. M: 66.1699 c. 10: 1 0 3 2 0 1 3 N SD: 29.5163 c. M:-53.8301 c. 12: 1 0 4 2 1 0 4 N SD: 26.0616 c. M: 36.9247 c. 14: 1 1 4 2 1 1 4 N SD: 19.4879 c. M:-36.4589 c. 16: 1 1 5 2 1 1 5 N SD: 17.8395 c. M:-26.9550 c. 17: 1 1 5 3 1 1 5 N SD: 6.1091 c. M:-10.1341 c. 36: 2 2 11 6 2 2 11 N SD: 1.9121 c. M: 3.5913 c. Soft Chromatic Dorian 53: 3 3 16 9 3 3 16 N SD: 1.0256 c. M:-1.7547 c. 89: 5 5 27 15 5 5 27 N SD: 0.7621 c. M:-1.2459 c. 142: 8 8 43 24 8 8 43 N SD: 0.7172 c. M:-1.4358 c. 195: 11 11 59 33 11 11 59 N SD: 0.7616 c. M:-1.5224 c. 222: 12 13 67 38 12 13 67 N SD: 0.7553 c. M: 1.3050 c. 231: 13 13 70 39 13 13 70 N SD: 0.7112 c. M:-1.3626 c. 241: 13 14 73 41 13 14 73 N SD: 0.7830 c. M: 1.4396 c. 258: 14 15 78 44 14 15 78 N SD: 0.5264 c. M: 1.0536 c. 275: 15 16 83 47 15 16 83 N SD: 0.5496 c. M:-0.8935 c. 294: 16 17 89 50 16 17 89 N SD: 0.4595 c. M: 0.8637 c. 311: 17 18 94 53 17 18 94 N SD: 0.3229 c. M: 0.5750 c. 347: 19 20 105 59 19 20 105 N SD: 0.2377 c. M: 0.4638 c. 364: 20 21 110 62 20 21 110 N SD: 0.2401 c. M:-0.4380 c. 400: 22 23 121 68 22 23 121 N SD: 0.0887 c. M: 0.1699 c. 453: 25 26 137 77 25 26 137 N SD: 0.0891 c. M:-0.1614 c. 489: 27 28 148 83 27 28 148 N SD: 0.0827 c. M: 0.1149 c. 853: 47 49 258 145 47 49 258 N SD: 0.0622 c. M:-0.1210 c. 889: 49 51 269 151 49 51 269 N SD: 0.0387 c. M: 0.0709 c. 1342: 74 77 406 228 74 77 406 N SD: 0.0265 c. M:-0.0527 c. 1742: 96 100 527 296 96 100 527 N SD: 0.0328 c. M:-0.0475 c. 1831: 101 105 554 311 101 105 554 N SD: 0.0299 c. M:-0.0435 c. 2231: 123 128 675 379 123 128 675 N SD: 0.0257 c. M: 0.0388 c.
FIT/HARMONIC
1 x x x x x x 2 S SD: 0.0000 cents 2 x x x 3 x x 4 S SD: 0.0000 cents 3 x x 4 x x 5:6 S SD: 15.8113 cents 4 x x 5:6 x 7:8 S SD: 43.2159 cents 5 x 6:7 x x 8:10 S SD: 50.1977 cents 6 x 7:8:9 x 10:12 S SD: 28.0342 cents 7 x 8:9 x 11 x 14 SD: 28.9674 cents 8 x 9:11:12 x 13:16 SD: 17.4399 cents 9 x 10:12 x 14:15:18 S SD: 13.4317 cents 10 x 11:13:15 x 16:20 SD: 11.5987 cents 11 x 12:15 x 17:18:22 SD: 9.4116 cents 12 x 13:16:18:19 x 24 SD: 5.5337 cents 13 x 14:17 x 20:21:26 SD: 8.2901 cents 14 x 15:19:21:22:23:28 SD: 7.2529 cents 15 x 16:20 x 23:24:30 SD: 8.6504 cents 16 x 17:21:24:25:26:32 SD: 6.8261 cents 17:18 x 23 x 26:28:34 SD: 11.8311 cents 18:19 x 24:27:28:29:36 SD: 4.9710 cents 22:23:24:29:33:34:36:44 SD: 4.9665 cents 24:25:26:32:36:37:39:48 SD: 2.8474 cents 46:48:50:61:69:72:75:92 SD: 2.7772 cents 47:49:51:63:71:73:76:94 SD: 2.7459 cents 48:50:52:64:72:75:78:96 SD: 1.1650 cents 72:75:78:96:108:112:117:144 SD: 1.0741 cents 84:87:91:112:126:131:136:168 SD: 1.0237 cents 98:102:106:131:147:153:159:196 SD: 0.9046 cents 100:104:108:133:150:156:162:200 SD: 0.7924 cents 122:127:132:163:183:190:198:244 SD: 0.7735 cents 124:129:134:165:186:193:201:248 SD: 0.6883 cents 132:137:143:176:198:206:214:264 SD: 0.6731 cents 150:156:162:200:225:234:243:300 SD: 0.4946 cents 174:181:188:232:261:271:282:348 SD: 0.3940 cents 222:231:240:296:333:346:360:444 SD: 0.3762 cents 256:266:277:341:384:399:415:512 SD: 0.3365 cents 258:268:279:344:387:402:418:516 SD: 0.2446 cents 306:318:331:408:459:477:496:612 SD: 0.1974 cents 360:374:389:480:540:561:584:720 SD: 0.1531 cents 492:511:532:656:738:767:798:984 SD: 0.1344 cents 516:536:558:688:774:804:837:1032 SD: 0.1222 cents