Scala analysis: xen15-gilson-aristoxenus-enharmonic
Aristoxenus' Enharmonic
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 40/39 43.831051 tridecimal minor diesis 2: 20/19 88.800698 small undevicesimal semitone 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 20/13 745.786052 tridecimal semi-augmented fifth 6: 30/19 790.755699 smaller undevicesimal minor sixth 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 19/15 409.2443 undevicesimal ditone 1: 40/39 43.8311 tridecimal minor diesis 2: 39/38 44.9696 3: 19/15 409.2443 undevicesimal ditone 4: 9/8 203.9100 major whole tone 5: 40/39 43.8311 tridecimal minor diesis 6: 39/38 44.9696 7: 19/15 409.2443 undevicesimal ditone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 40/39 43.831 cents tridecimal minor diesis 1: 2 39/38 44.970 cents 1: 1 9/8 203.910 cents major whole tone 1: 2 19/15 409.244 cents undevicesimal ditone 2: 2 20/19 88.801 cents small undevicesimal semitone 2: 1 15/13 247.741 cents tridecimal 5/4-tone 2: 1 152/117 453.075 cents 2: 2 13/10 454.214 cents tridecimal semi-diminished fourth 2: 1 57/40 613.154 cents 3: 1 45/38 292.711 cents 3: 4 4/3 498.045 cents perfect fourth 3: 1 19/13 656.985 cents 3: 1 117/80 658.124 cents 4: 1 160/117 541.876 cents 4: 1 26/19 543.015 cents 4: 4 3/2 701.955 cents perfect fifth 4: 1 76/45 907.289 cents 5: 1 80/57 586.846 cents 5: 2 20/13 745.786 cents tridecimal semi-augmented fifth 5: 1 117/76 746.925 cents 5: 1 26/15 952.259 cents tridecimal semi-augmented sixth 5: 2 19/10 1111.199 cents undevicesimal major seventh 6: 2 30/19 790.756 cents smaller undevicesimal minor sixth 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 76/39 1155.030 cents 6: 2 39/20 1156.169 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 : 43.8 88.8 498.0 702.0 745.8 790.8 1200.0 43.8 : 45.0 454.2 658.1 702.0 746.9 1156.2 1200.0 88.8 : 409.2 613.2 657.0 702.0 1111.2 1155.0 1200.0 498.0 : 203.9 247.7 292.7 702.0 745.8 790.8 1200.0 702.0 : 43.8 88.8 498.0 541.9 586.8 996.1 1200.0 745.8 : 45.0 454.2 498.0 543.0 952.3 1156.2 1200.0 790.8 : 409.2 453.1 498.0 907.3 1111.2 1155.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 43.8311 .. 409.2443 cents 247/200, 365.4133 cents 40/39, 43.8311 cents 2: 88.8007 .. 613.1543 cents 1083/800, 524.3536 cents 300/361,-320.4436 cents 3: 292.7107 .. 658.1239 cents 247/200, 365.4133 cents 300/361,-320.4436 cents 4: 541.8761 .. 907.2893 cents 247/200, 365.4133 cents 12800/13689,-116.2479 cents 5: 586.8457 .. 1111.1993 cents 1083/800, 524.3536 cents 300/361,-320.4436 cents 6: 790.7557 .. 1156.1689 cents 247/200, 365.4133 cents 300/361,-320.4436 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 40/39, 43.8311 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 19/15, 409.2443 cents Average / Smallest step : 3.911122 Largest / Average step : 2.387258 Largest / Smallest step : 9.336858 Median interval of one step : 39/38, 44.9696 cents, amount: 2 Least squares average step : 152.83466 cents, oct.: 1069.84259 cents Scale is not proper Step pattern alph. order: ABCDABC Step pattern size order : ABDCABD Scale is a Constant Structure, by a margin of 26.30861 cents Scale diversity : 1.452385 Lumma stability : 0.073052 Lumma impropriety factor : 0.851999 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 : 1521/1520, 1.1386 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 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 2 at degree 0 Formal octave complements present : 3 = 42.8571% Scale is differentially coherent in interval classes 3 and 4 combined -- Rational properties -- Prime limit : 19 Odd number limit : 117 (O: 117 U: 117) Highest odd numerator or denominator: 39 Scale harmonicity : 0.013890 Average absolute harmonicity : 0.227875 Specific harmonicity : 0.072438 Fundamental : 1/1482, -10.5333 octaves, 0.1765 Hz. Guide tone : 120, 6.9069 octaves, 31395.068 Hz. Exponens Consonantiae : 1.778400E+05, 17.44022 octaves Euler's gradus suavitatis : 43 Sum of Mann's harmonic distance : 107.5, average 15.35714 Mersenne's string divisions : 8303295 Sum of van Prooijen's expressibility: 6.21676, average 0.88811 Sum of Tenney's harmonic distance : 13.10212, average 1.87173 Vogel's harmonic complexity : 31.14286 Wille's k value : 1111 Wilson's harmonic complexity : 43 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 63.42603, average 2.26522 Lattice compactness (without 2's) : 42.56847, average 1.52030 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -1 .. 3 1.42857 12 1 3: -1 .. 1 0.00000 4 2 5: 0 .. 1 0.57143 4 13: -1 .. 0 -0.28571 2 2 19: -1 .. 0 -0.28571 2 2 Average exponent except 2's : 0 / 7 = 0.00000 Average absolute exponent except 2's: 12 / 7 = 1.71429 Scale is weakly epimorphic with val: <7 10 15 27 31| Scale is weakly epimorphic with val: <7 10 15 28 30| Scale is weakly epimorphic with val: <7 10 16 28 32| Scale is weakly epimorphic with val: <7 10 16 29 31| Scale is weakly epimorphic with val: <7 10 17 29 33| Scale is weakly epimorphic with val: <7 10 18 24 33| Scale is JI-epimorphic with val: <7 11 15 24 27| Scale is weakly epimorphic with val: <7 11 16 24 29| Scale is JI-epimorphic with val: <7 11 16 25 28| Scale is weakly epimorphic with val: <7 11 17 25 30| Scale is JI-epimorphic with val: <7 11 17 26 29| Scale is weakly epimorphic with val: <7 11 18 26 31| Scale is JI-epimorphic with val: <7 11 18 27 30| Scale is weakly epimorphic with val: <7 12 15 25 33| Scale is weakly epimorphic with val: <7 12 15 28 30| Scale is weakly epimorphic with val: <7 12 16 26 27| Scale is weakly epimorphic with val: <7 12 16 29 31| Scale is weakly epimorphic with val: <7 12 17 27 28| Scale is weakly epimorphic with val: <7 12 18 24 33| Scale is weakly epimorphic with val: <7 12 18 28 29|
FIT/MODE
7: 0 0 3 1 0 0 3 N SD: 59.7334 c. M: 105.0414 c. 8: 0 1 2 2 0 0 3 N SD: 41.3382 c. M:-61.1993 c. 9: 0 1 3 1 1 0 3 N SD: 36.6829 c. M:-54.2139 c. 10: 0 1 3 2 0 1 3 N SD: 30.7890 c. M:-49.2443 c. 11: 0 1 4 1 1 0 4 N SD: 33.5565 c. M:-47.4095 c. 12: 0 1 4 2 0 1 4 N SD: 24.5998 c. M: 45.7861 c. 13: 0 1 4 3 0 1 4 N SD: 29.8894 c. M: 43.8311 c. 14: 1 0 5 2 1 0 5 N SD: 21.7857 c. M:-41.8832 c. 15: 1 0 5 3 0 1 5 N SD: 19.9545 c. M:-36.1689 c. 17: 1 0 6 3 1 0 6 N SD: 17.8249 c. M:-30.6845 c. 19: 1 0 7 3 1 1 6 N SD: 17.7273 c. M:-30.2969 c. 20: 1 0 7 4 0 1 7 N SD: 18.9841 c. M: 28.8007 c. 21: 1 1 7 3 1 1 7 N SD: 14.3838 c. M:-25.4850 c. 22: 1 1 7 4 1 1 7 N SD: 15.5800 c. M:-27.4261 c. 24: 1 1 8 4 1 1 8 N SD: 6.2603 c. M:-11.1993 c. Enharmonic Dorian 26: 1 1 9 4 1 1 9 N SD: 6.4938 c. M:-9.6473 c. 29: 1 1 10 5 1 1 10 N SD: 3.1302 c. M: 6.0421 c. 53: 2 2 18 9 2 2 18 N SD: 1.1970 c. M:-1.7653 c. 82: 3 3 28 14 3 3 28 N SD: 0.5393 c. M: 0.9958 c. 135: 5 5 46 23 5 5 46 N SD: 0.4518 c. M:-0.8806 c. 217: 8 8 74 37 8 8 74 N SD: 0.3944 c. M:-0.7577 c. 243: 9 9 83 41 9 9 83 N SD: 0.5118 c. M:-0.7204 c. 296: 11 11 101 50 11 11 101 N SD: 0.4682 c. M:-0.7635 c. 299: 11 11 102 51 11 11 102 N SD: 0.4075 c. M:-0.7022 c. 325: 12 12 111 55 12 12 111 N SD: 0.3731 c. M: 0.6019 c. 378: 14 14 129 64 14 14 129 N SD: 0.3366 c. M:-0.6134 c. 407: 15 15 139 69 15 15 139 N SD: 0.3283 c. M: 0.5837 c. 431: 16 16 147 73 16 16 147 N SD: 0.3721 c. M:-0.7165 c. 460: 17 17 157 78 17 17 157 N SD: 0.2841 c. M:-0.5168 c. 513: 19 19 175 87 19 19 175 N SD: 0.3041 c. M:-0.6134 c. 542: 20 20 185 92 20 20 185 N SD: 0.2729 c. M:-0.4494 c. 581: 21 22 198 99 21 22 198 N SD: 0.2636 c. M: 0.4576 c. 595: 22 22 203 101 22 22 203 N SD: 0.2756 c. M:-0.5387 c. 607: 22 23 207 103 22 23 207 N SD: 0.2430 c. M: 0.4813 c. 624: 23 23 213 106 23 23 213 N SD: 0.2803 c. M:-0.3997 c. 634: 23 24 216 108 23 24 216 N SD: 0.2430 c. M:-0.4115 c. 636: 23 24 217 108 23 24 217 N SD: 0.2678 c. M: 0.5030 c. 660: 24 25 225 112 24 25 225 N SD: 0.2046 c. M: 0.3315 c. 689: 25 26 235 117 25 26 235 N SD: 0.1788 c. M: 0.3579 c. 716: 26 27 244 122 26 27 244 N SD: 0.2126 c. M:-0.3058 c. 742: 27 28 253 126 27 28 253 N SD: 0.1306 c. M: 0.2335 c. 771: 28 29 263 131 28 29 263 N SD: 0.1451 c. M: 0.2608 c. 795: 29 30 271 135 29 30 271 N SD: 0.1358 c. M:-0.2559 c. 824: 30 31 281 140 30 31 281 N SD: 0.0811 c. M: 0.1550 c. 877: 32 33 299 149 32 33 299 N SD: 0.0762 c. M:-0.1389 c. 906: 33 34 309 154 33 34 309 N SD: 0.0649 c. M: 0.1224 c. 959: 35 36 327 163 35 36 327 N SD: 0.0361 c. M:-0.0681 c. 1041: 38 39 355 177 38 39 355 N SD: 0.0411 c. M: 0.0623 c. 1865: 68 70 636 317 68 70 636 N SD: 0.0389 c. M: 0.0777 c. 2000: 73 75 682 340 73 75 682 N SD: 0.0320 c. M:-0.0450 c. 2243: 82 84 765 381 82 84 765 N SD: 0.0275 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: 31.2010 cents 4 x x 5:6 x x 8 S SD: 37.2438 cents 5 x x 7 x x 8:10 SD: 29.1748 cents 6 x 7:8:9 x x 12 S SD: 44.5176 cents 7 x 8:9 x x 11:14 SD: 38.9732 cents 8 x 9:11:12 x 13:16 SD: 27.2510 cents 9 x 10:12 x 14 x 18 SD: 23.8845 cents 10 x 11:13:15 x 16:20 SD: 18.1703 cents 11 x 12:15 x 17 x 22 SD: 18.3695 cents 12 x 13:16:18 x 19:24 SD: 10.0006 cents 13 x 14:17 x 20:21:26 SD: 13.0388 cents 14 x 15:19:21 x 22:28 SD: 8.8231 cents 15 x 16:20 x 23:24:30 SD: 6.5880 cents 16 x 17:21:24 x 25:32 SD: 7.3019 cents 17 x 18:23 x 26:27:34 SD: 6.1621 cents 18 x 19:24:27:28 x 36 SD: 3.9447 cents 19 x 20:25 x 29:30:38 SD: 5.3445 cents 20 x 21:27:30:31:32:40 SD: 5.7117 cents 21 x 22:28 x 32:33:42 SD: 4.0548 cents 22 x 23:29:33:34:35:44 SD: 4.6071 cents 23 x 24:31 x 35:36:46 SD: 6.8280 cents 24 x 25:32:36:37:38:48 SD: 3.1828 cents 25 x 26:33 x 38:39:50 SD: 8.0330 cents 26:27 x 35:39:40:41:52 SD: 4.5397 cents 27:28 x 36:41:42:43:54 SD: 6.2474 cents 28:29 x 37:42:43:44:56 SD: 4.1003 cents 33:34:35:44:50:51:52:66 SD: 3.5254 cents 34:35:36:45:51:52:54:68 SD: 3.2392 cents 36:37:38:48:54:55:57:72 SD: 2.0437 cents 54:55:57:72:81:83:85:108 SD: 2.0194 cents 69:71:73:92:104:106:109:138 SD: 1.9436 cents 71:73:75:95:107:109:112:142 SD: 1.8846 cents 72:74:76:96:108:111:114:144 SD: 1.2129 cents 74:76:78:99:111:114:117:148 SD: 1.0686 cents 76:78:80:101:114:117:120:152 SD: 0.8467 cents 78:80:82:104:117:120:123:156 SD: 0.4487 cents 150:154:158:200:225:231:237:300 SD: 0.4202 cents 192:197:202:256:288:295:303:384 SD: 0.3826 cents 228:234:240:304:342:351:360:456 SD: 0.2300 cents 306:314:322:408:459:471:483:612 SD: 0.2060 cents 342:351:360:456:513:526:540:684 SD: 0.1780 cents 420:431:442:560:630:646:663:840 SD: 0.1672 cents 510:523:537:680:765:785:805:1020 SD: 0.1668 cents 516:529:543:688:774:794:815:1032 SD: 0.1596 cents 546:560:575:728:819:840:862:1092 SD: 0.1172 cents