Scala analysis: xen11-chalmers-tetrachordal-08-13
Euler's Genus Musicum, 8 tones, an Octony
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 28/27 62.960904 Archytas' 1/3-tone 2: 16/15 111.731285 minor diatonic semitone 3: 448/405 174.692189 4: 4/3 498.044999 perfect fourth 5: 112/81 561.005903 6: 64/45 609.776284 2nd tritone 7: 1792/1215 672.737188 8: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 1215/896 527.2628 1: 28/27 62.9609 Archytas' 1/3-tone 2: 36/35 48.7704 septimal diesis, 1/4-tone 3: 28/27 62.9609 Archytas' 1/3-tone 4: 135/112 323.3528 septimal wide minor third 5: 28/27 62.9609 Archytas' 1/3-tone 6: 36/35 48.7704 septimal diesis, 1/4-tone 7: 28/27 62.9609 Archytas' 1/3-tone 8: 1215/896 527.2628
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 36/35 48.770 cents septimal diesis, 1/4-tone 1: 4 28/27 62.961 cents Archytas' 1/3-tone 1: 1 135/112 323.353 cents septimal wide minor third 1: 1 1215/896 527.263 cents 2: 4 16/15 111.731 cents minor diatonic semitone 2: 2 5/4 386.314 cents major third 2: 2 45/32 590.224 cents diatonic tritone 3: 2 448/405 174.692 cents 3: 2 9/7 435.084 cents septimal major third, BP third 3: 1 35/27 449.275 cents septimal semi-diminished fourth, 9/4-tone 3: 2 81/56 638.994 cents 3: 1 35/24 653.185 cents septimal semi-diminished fifth 4: 4 4/3 498.045 cents perfect fourth 4: 4 3/2 701.955 cents perfect fifth 5: 1 48/35 546.815 cents septimal semi-augmented fourth 5: 2 112/81 561.006 cents 5: 1 54/35 750.725 cents septimal semi-augmented fifth 5: 2 14/9 764.916 cents septimal minor sixth 5: 2 405/224 1025.308 cents 6: 2 64/45 609.776 cents 2nd tritone 6: 2 8/5 813.686 cents minor sixth 6: 4 15/8 1088.269 cents classic major seventh 7: 1 1792/1215 672.737 cents 7: 1 224/135 876.647 cents narrow septimal major sixth 7: 4 27/14 1137.039 cents septimal major seventh 7: 2 35/18 1151.230 cents septimal semi-diminished octave Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 3.71429 = 26/7
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 0.0 : 63.0 111.7 174.7 498.0 561.0 609.8 672.7 1200.0 63.0 : 48.8 111.7 435.1 498.0 546.8 609.8 1137.0 1200.0 111.7 : 63.0 386.3 449.3 498.0 561.0 1088.3 1151.2 1200.0 174.7 : 323.4 386.3 435.1 498.0 1025.3 1088.3 1137.0 1200.0 498.0 : 63.0 111.7 174.7 702.0 764.9 813.7 876.6 1200.0 561.0 : 48.8 111.7 639.0 702.0 750.7 813.7 1137.0 1200.0 609.8 : 63.0 590.2 653.2 702.0 764.9 1088.3 1151.2 1200.0 672.7 : 527.3 590.2 639.0 702.0 1025.3 1088.3 1137.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 48.7704 .. 527.2628 cents 675/512, 478.4924 cents 36/35, 48.7704 cents 2: 111.7313 .. 590.2237 cents 675/512, 478.4924 cents 14336/18225,-415.5315 cents 3: 174.6922 .. 653.1846 cents 675/512, 478.4924 cents 14336/18225,-415.5315 cents 4: 498.0450 .. 701.9550 cents 9/8, 203.9100 cents 32/35,-155.1396 cents 5: 546.8154 .. 1025.3078 cents 675/512, 478.4924 cents 32/35,-155.1396 cents 6: 609.7763 .. 1088.2687 cents 675/512, 478.4924 cents 14336/18225,-415.5315 cents 7: 672.7372 .. 1151.2296 cents 675/512, 478.4924 cents 14336/18225,-415.5315 cents
SHOW DATA
Number of notes : 8 -- Interval properties -- Smallest interval : 36/35, 48.7704 cents, class 1 Average step (divided formal octave): 150.0000 cents Largest one step interval : 1215/896, 527.2628 cents Average / Smallest step : 3.075637 Largest / Average step : 3.515085 Largest / Smallest step : 10.811128 Median interval of one step : 28/27, 62.9609 cents, amount: 4 Most common interval of one step : 28/27, 62.9609 cents, amount: 4 Least squares average step : 115.56631 cents, oct.: 924.53049 cents Scale is not proper Step pattern alph. order: ABACABAD Step pattern size order : BABCBABD Scale is a Constant Structure, by a margin of 19.55257 cents Scale diversity : 1.361644 Lumma stability : 0.081284 Lumma impropriety factor : 0.813781 Rothenberg efficiency : 0.421429 redundancy: 0.578571 Efficiency x scale size : 3.371429 Number of different interval sizes : 26 = 3.71429 / class Number of one step interval sizes : 4 Highest interval variety : 5 Mean interval variety : 3.71429 = 26/7 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 245/243, 14.1905 cents Number of recognisable fifths : 4, average 701.9550 cents Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 1 = 12.5000% Scale is n-1 differentially coherent in interval classes 1 and 2 combined Scale is differentially coherent in interval classes 3 and 4 combined Scale is differentially coherent in interval class 4 Limited inverse transpositions : 7 Inversional symmetry on intervals : 3-4 7-8 -- Rational properties -- Prime limit : 7 Odd number limit : 1215 (O: 1215 U: 1215) Highest odd numerator or denominator: 1215 Scale harmonicity : 0.026303 Average absolute harmonicity : 0.186608 Specific harmonicity : 0.122532 Fundamental : 1/1215, -10.2467 octaves, 0.2153 Hz. Guide tone : 1792, 10.8074 octaves, 468833.013 Hz. Exponens Consonantiae : 2.177280E+06, 21.05410 octaves Euler's gradus suavitatis : 29 Sum of Mann's harmonic distance : 2129.0, average 266.12500 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 12.33831, average 1.54229 Sum of Tenney's harmonic distance : 25.65269, average 3.20659 Vogel's harmonic complexity : 17.62500 Wille's k value : 607 Wilson's harmonic complexity : 27 Rectangular lattice diameter : 7 Triangular lattice diameter : 6 Lattice compactness : 152.45262, average 4.23479 Lattice compactness (without 2's) : 89.81662, average 2.49491 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: 1 .. 8 4.12500 33 1 3: -5 .. 0 -2.50000 20 1 5: -1 .. 0 -0.50000 4 2 7: 0 .. 1 0.50000 4 4 Average exponent except 2's :-20 / 8 =-2.50000 Average absolute exponent except 2's: 28 / 8 = 3.50000 Scale is weakly epimorphic with val: <8 12 17 22| Scale is JI-epimorphic with val: <8 12 18 21| Scale is weakly epimorphic with val: <8 12 18 23| Scale is weakly epimorphic with val: <8 12 19 22| Scale is weakly epimorphic with val: <8 12 21 22| Scale is weakly epimorphic with val: <8 14 17 22| Scale is weakly epimorphic with val: <8 14 19 22| Scale is weakly epimorphic with val: <8 14 21 22|
FIT/MODE
8: 0 0 1 2 0 1 0 4 N SD: 68.0758 c. M: 111.7313 c. 9: 0 1 0 3 0 1 0 4 N SD: 37.7917 c. M: 62.9609 c. 10: 1 0 1 2 1 0 1 4 N SD: 38.3446 c. M:-65.3078 c. 11: 0 1 0 3 1 0 1 5 N SD: 45.8111 c. M: 65.6013 c. 12: 1 0 1 3 1 0 1 5 N SD: 23.7518 c. M:-38.9941 c. 13: 1 0 1 3 1 1 0 6 N G SD: 24.3695 c. M: 36.5065 c. 14: 1 0 1 4 1 0 1 6 N SD: 20.1666 c. M:-38.9941 c. 17: 1 1 1 4 1 1 1 7 N D3 SD: 22.5789 c. M:-37.0725 c. 19: 1 1 1 5 1 1 1 8 N D3 SD: 13.6824 c. M:-21.9997 c. 20: 1 1 1 5 1 1 1 9 N D3 SD: 11.8848 c. M: 21.0059 c. 21: 1 1 1 6 1 1 1 9 N D3 SD: 10.8690 c. M:-18.7951 c. 22: 1 1 1 6 1 1 1 10 N D3 SD: 10.7197 c. M: 18.1917 c. 25: 1 1 2 6 2 1 1 11 N SD: 13.8029 c. M: 18.0450 c. 27: 1 2 1 7 2 1 1 12 N SD: 13.1145 c. M:-21.6020 c. 30: 1 2 1 8 2 1 2 13 N SD: 12.6807 c. M: 22.9609 c. 32: 2 1 2 8 2 1 2 14 N SD: 8.0909 c. M:-12.8078 c. 34: 2 1 2 9 2 1 2 15 N SD: 5.3016 c. M: 9.7763 c. 41: 2 2 2 11 2 2 2 18 N D3 SD: 3.4833 c. M:-5.3419 c. 53: 3 2 3 14 3 2 3 23 N SD: 4.1586 c. M:-6.5081 c. 55: 3 2 3 15 3 2 3 24 N SD: 3.1850 c. M:-6.2668 c. 75: 4 3 4 20 4 3 4 33 N SD: 1.2122 c. M: 2.0450 c. 118: 6 5 6 32 6 5 6 52 N SD: 1.2519 c. M: 1.9440 c. 130: 7 5 7 35 7 5 7 57 N SD: 1.1254 c. M:-2.0710 c. 152: 8 6 8 41 8 6 8 67 N SD: 1.0939 c. M: 1.8815 c. 171: 9 7 9 46 9 7 9 75 N SD: 0.5652 c. M:-0.9470 c. 246: 13 10 13 66 13 10 13 108 N SD: 0.4591 c. M:-0.9176 c. 248: 13 10 13 67 13 10 13 109 N SD: 0.2913 c. M: 0.4986 c. 323: 17 13 17 87 17 13 17 142 N SD: 0.2475 c. M: 0.4884 c. 419: 22 17 22 113 22 17 22 184 N SD: 0.2066 c. M:-0.3306 c. 494: 26 20 26 133 26 20 26 217 N SD: 0.1251 c. M:-0.2066 c. 742: 39 30 39 200 39 30 39 326 N SD: 0.0982 c. M:-0.1801 c. 858: 45 35 45 231 45 35 45 377 N SD: 0.1069 c. M: 0.1667 c. 1031: 54 42 54 278 54 42 54 453 N SD: 0.0784 c. M:-0.1170 c. 1106: 58 45 58 298 58 45 58 486 N SD: 0.0336 c. M: 0.0656 c. 1525: 80 62 80 411 80 62 80 670 N SD: 0.0369 c. M:-0.0598 c. 1600: 84 65 84 431 84 65 84 703 N SD: 0.0319 c. M:-0.0578 c. 1848: 97 75 97 498 97 75 97 812 N SD: 0.0259 c. M: 0.0430 c.
FIT/HARMONIC
1 x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x x 3:4 S SD: 14.6089 cents 3 x x x 4 x x x 6 S SD: 0.0000 cents 4 x x x 5 x x 6:8 S SD: 38.4961 cents 5 x x 6 x 7 x x 10 SD: 47.5268 cents 6 x x 7:8 x x 9:12 S SD: 24.1746 cents 7 x x 8:9 x 10 x 14 SD: 21.2334 cents 8 x x 9 x 11 x 12:16 SD: 10.6102 cents 9 x x 10:12 x 13 x 18 SD: 6.9818 cents 10 x x 11:13:14 x 15:20 SD: 11.5433 cents 11 x x 12 x 15 x 16:22 SD: 10.4162 cents 12 x 13 x 16 x 17:18:24 SD: 8.0500 cents 13 x 14 x 17:18 x 19:26 SD: 8.1446 cents 14 x 15 x 19 x 20:21:28 SD: 8.7443 cents 15 x 16:17:20:21 x 22:30 SD: 8.0276 cents 16 x 17:18:21:22:23:24:32 S SD: 7.7363 cents 17 x 18:19:23 x 24:25:34 SD: 6.0325 cents 18 x 19:20:24:25:26:27:36 S SD: 6.4233 cents 19 x 20:21:25:26:27:28:38 SD: 5.3096 cents 20:21 x 22:27:28 x 30:40 SD: 8.0538 cents 21:22 x 23:28:29:30:31:42 SD: 3.7015 cents 22:23 x 24:29:30:31:32:44 SD: 7.2579 cents 23:24 x 25:31:32:33:34:46 SD: 5.9725 cents 24:25:26:27:32:33:34:35:48 SD: 5.8037 cents 26:27:28:29:35:36:37:38:52 SD: 3.9835 cents 27:28:29:30:36:37:38:40:54 SD: 3.6060 cents 28:29:30:31:37:39:40:41:56 SD: 3.2680 cents 29:30:31:32:39:40:41:43:58 SD: 2.7387 cents 45:47:48:50:60:62:64:66:90 SD: 2.3168 cents 47:49:50:52:63:65:67:69:94 SD: 2.0562 cents 55:57:59:61:73:76:78:81:110 SD: 1.8222 cents 57:59:61:63:76:79:81:84:114 SD: 1.0119 cents 76:79:81:84:101:105:108:112:152 SD: 0.9617 cents 102:106:109:113:136:141:145:150:204 SD: 0.9407 cents 104:108:111:115:139:144:148:153:208 SD: 0.8882 cents 105:109:112:116:140:145:149:155:210 SD: 0.6864 cents 133:138:142:147:177:184:189:196:266 SD: 0.5738 cents 162:168:173:179:216:224:230:239:324 SD: 0.5159 cents 196:203:209:217:261:271:279:289:392 SD: 0.4827 cents 225:233:240:249:300:311:320:332:450 SD: 0.3469 cents 270:280:288:299:360:373:384:398:540 SD: 0.3320 cents 272:282:290:301:363:376:387:401:544 SD: 0.2822 cents 301:312:321:333:401:416:428:444:602 SD: 0.2428 cents 303:314:323:335:404:419:431:447:606 SD: 0.2400 cents 348:361:371:385:464:481:495:513:696 SD: 0.1977 cents 405:420:432:448:540:560:576:597:810 SD: 0.1208 cents