Scala analysis: xen15-gilson-eratosthenes-enharmonic
Eratosthenes' Enharmonic
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 46/45 38.050632 23rd-partial chroma 2: 16/15 111.731285 minor diatonic semitone 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 23/15 740.005633 6: 8/5 813.686286 minor sixth 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 5/4 386.3137 major third 1: 46/45 38.0506 23rd-partial chroma 2: 24/23 73.6807 vicesimotertial minor semitone 3: 5/4 386.3137 major third 4: 9/8 203.9100 major whole tone 5: 46/45 38.0506 23rd-partial chroma 6: 24/23 73.6807 vicesimotertial minor semitone 7: 5/4 386.3137 major third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 46/45 38.051 cents 23rd-partial chroma 1: 2 24/23 73.681 cents vicesimotertial minor semitone 1: 1 9/8 203.910 cents major whole tone 1: 2 5/4 386.314 cents major third 2: 2 16/15 111.731 cents minor diatonic semitone 2: 1 23/20 241.961 cents 2: 1 23/18 424.364 cents vicesimotertial major third 2: 2 30/23 459.994 cents 2: 1 45/32 590.224 cents diatonic tritone 3: 1 6/5 315.641 cents minor third 3: 4 4/3 498.045 cents perfect fourth 3: 1 23/16 628.274 cents 23rd harmonic 3: 1 135/92 663.904 cents 4: 1 184/135 536.096 cents 4: 1 32/23 571.726 cents 23rd subharmonic 4: 4 3/2 701.955 cents perfect fifth 4: 1 5/3 884.359 cents major sixth, BP sixth 5: 1 64/45 609.776 cents 2nd tritone 5: 2 23/15 740.006 cents 5: 1 36/23 775.636 cents 5: 1 40/23 958.039 cents 5: 2 15/8 1088.269 cents classic major seventh 6: 2 8/5 813.686 cents minor sixth 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 23/12 1126.319 cents vicesimotertial major seventh 6: 2 45/23 1161.949 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 : 38.1 111.7 498.0 702.0 740.0 813.7 1200.0 38.1 : 73.7 460.0 663.9 702.0 775.6 1161.9 1200.0 111.7 : 386.3 590.2 628.3 702.0 1088.3 1126.3 1200.0 498.0 : 203.9 242.0 315.6 702.0 740.0 813.7 1200.0 702.0 : 38.1 111.7 498.0 536.1 609.8 996.1 1200.0 740.0 : 73.7 460.0 498.0 571.7 958.0 1161.9 1200.0 813.7 : 386.3 424.4 498.0 884.4 1088.3 1126.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 38.0506 .. 386.3137 cents 225/184, 348.2631 cents 46/45, 38.0506 cents 2: 111.7313 .. 590.2237 cents 675/512, 478.4924 cents 64/75,-274.5824 cents 3: 315.6413 .. 663.9044 cents 225/184, 348.2631 cents 64/75,-274.5824 cents 4: 536.0956 .. 884.3587 cents 225/184, 348.2631 cents 16928/18225,-127.8087 cents 5: 609.7763 .. 1088.2687 cents 675/512, 478.4924 cents 64/75,-274.5824 cents 6: 813.6863 .. 1161.9494 cents 225/184, 348.2631 cents 64/75,-274.5824 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 46/45, 38.0506 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 5/4, 386.3137 cents Average / Smallest step : 4.505275 Largest / Average step : 2.253497 Largest / Smallest step : 10.152623 Median interval of one step : 24/23, 73.6807 cents, amount: 2 Least squares average step : 153.89724 cents, oct.: 1077.28070 cents Scale is not proper Step pattern alph. order: ABCDABC Step pattern size order : ABDCABD Scale is a Constant Structure, by a margin of 18.49806 cents Scale diversity : 1.600846 Lumma stability : 0.063418 Lumma impropriety factor : 0.813781 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 : 1035/1024, 18.4981 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 All steps are superparticular (epimoric) Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 42.8571% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval classes 3 and 4 combined -- Rational properties -- Prime limit : 23 Odd number limit : 135 (O: 135 U: 135) Highest odd numerator or denominator: 45 Scale harmonicity : 0.016264 Average absolute harmonicity : 0.243959 Specific harmonicity : 0.086085 Fundamental : 1/90, -6.4919 octaves, 2.9070 Hz. Guide tone : 1104, 10.1085 octaves, 288834.624 Hz. Exponens Consonantiae : 9.936000E+04, 16.60038 octaves Euler's gradus suavitatis : 38 Sum of Mann's harmonic distance : 94.0, average 13.42857 Mersenne's string divisions : 1607970 Sum of van Prooijen's expressibility: 5.84424, average 0.83489 Sum of Tenney's harmonic distance : 11.99442, average 1.71349 Vogel's harmonic complexity : 15.42857 Wille's k value : 67 Wilson's harmonic complexity : 37 Rectangular lattice diameter : 5 Triangular lattice diameter : 4 Lattice compactness : 74.93201, average 2.67614 Lattice compactness (without 2's) : 42.23288, average 1.50832 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 4 1.42857 12 1 3: -2 .. 1 -0.57143 6 2 5: -1 .. 0 -0.57143 4 2 23: 0 .. 1 0.28571 2 2 Average exponent except 2's :-6 / 7 =-0.85714 Average absolute exponent except 2's: 12 / 7 = 1.71429 Scale is JI-epimorphic with val: <7 11 15 31| Scale is weakly epimorphic with val: <7 11 16 33| Scale is weakly epimorphic with val: <7 12 17 30|
FIT/MODE
7: 0 1 2 1 0 1 2 N T3 SD: 38.4989 c. M:-59.6973 c. 9: 0 1 3 1 1 0 3 N SD: 34.2088 c. M:-59.9944 c. 10: 0 1 3 2 0 1 3 N SD: 21.5808 c. M: 38.0506 c. 12: 0 1 4 2 0 1 4 N SD: 21.9769 c. M: 40.0056 c. 19: 1 1 6 3 1 1 6 N SD: 13.7403 c. M:-25.1073 c. 21: 1 1 7 3 1 1 7 N SD: 12.5015 c. M:-19.0922 c. 22: 1 1 7 4 1 1 7 N SD: 11.7078 c. M:-23.6307 c. 24: 1 1 8 4 1 1 8 N SD: 9.0653 c. M: 13.6863 c. Enharmonic Dorian 29: 1 2 9 5 1 2 9 N SD: 7.4251 c. M:-13.8999 c. 31: 1 2 10 5 1 2 10 N SD: 3.6745 c. M: 5.1808 c. Enharmonic Dorian 34: 1 2 11 6 1 2 11 N SD: 3.3321 c. M: 5.8489 c. 53: 2 3 17 9 2 3 17 N SD: 3.9244 c. M:-7.2324 c. 63: 2 4 20 11 2 4 20 N SD: 2.9071 c. M:-5.3613 c. 65: 2 4 21 11 2 4 21 N SD: 0.9877 c. M: 1.5441 c. 159: 5 10 51 27 5 10 51 N SD: 0.7944 c. M:-1.4763 c. 183: 6 11 59 31 6 11 59 N SD: 0.6784 c. M:-1.2936 c. 193: 6 12 62 33 6 12 62 N SD: 0.5457 c. M:-0.8215 c. 224: 7 14 72 38 7 14 72 N SD: 0.5112 c. M:-0.7687 c. 248: 8 15 80 42 8 15 80 N SD: 0.4746 c. M: 0.7831 c. 258: 8 16 83 44 8 16 83 N SD: 0.4286 c. M: 0.8413 c. 311: 10 19 100 53 10 19 100 N SD: 0.4458 c. M:-0.8304 c. 342: 11 21 110 58 11 21 110 N SD: 0.3628 c. M:-0.5494 c. 376: 12 23 121 64 12 23 121 N SD: 0.2133 c. M:-0.4199 c. 407: 13 25 131 69 13 25 131 N SD: 0.2036 c. M:-0.3080 c. 441: 14 27 142 75 14 27 142 N SD: 0.0986 c. M: 0.1667 c. 665: 21 41 214 113 21 41 214 N SD: 0.1151 c. M: 0.1560 c. 848: 27 52 273 144 27 52 273 N SD: 0.0807 c. M:-0.1569 c. 913: 29 56 294 155 29 56 294 N SD: 0.0691 c. M: 0.1047 c. 1041: 33 64 335 177 33 64 335 N SD: 0.0748 c. M:-0.1466 c. 1106: 35 68 356 188 35 68 356 N SD: 0.0441 c. M: 0.0759 c. 1547: 49 95 498 263 49 95 498 N SD: 0.0335 c. M:-0.0489 c. 1954: 62 120 629 332 62 120 629 N SD: 0.0223 c. M:-0.0394 c. 2019: 64 124 650 343 64 124 650 N SD: 0.0200 c. M: 0.0354 c. 2395: 76 147 771 407 76 147 771 N SD: 0.0182 c. M:-0.0361 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: 23.5575 cents 4 x x 5:6 x x 8 S SD: 37.2438 cents 5 x x 7 x x 8:10 SD: 28.1557 cents 6 x 7:8:9 x 10:12 S SD: 34.0957 cents 7 x 8:9 x x 11:14 SD: 34.6444 cents 8 x 9:11:12 x 13:16 SD: 21.9594 cents 9 x 10:12 x 14 x 18 SD: 18.7335 cents 10 x 11:13:15 x 16:20 SD: 13.7973 cents 11 x 12:15 x 17:18:22 SD: 13.7504 cents 12 x 13:16:18 x 19:24 SD: 6.4779 cents 13 x 14:17 x 20:21:26 SD: 8.2764 cents 14 x 15:19:21 x 22:28 SD: 8.8803 cents 15 x 16:20 x 23:24:30 SD: 0.0000 cents 16 x 17:21:24:25:26:32 SD: 8.4556 cents 17 x 18:23 x 26:27:34 SD: 6.2767 cents 18 x 19:24:27:28:29:36 SD: 5.5093 cents 19 x 20:25 x 29:30:38 SD: 8.1006 cents 20 x 21:27:30:31:32:40 SD: 6.5747 cents 21 x 22:28 x 32:34:42 SD: 7.7697 cents 22 x 23:29:33:34:35:44 SD: 7.2340 cents 23:24:25:31 x 35:37:46 SD: 9.0442 cents 24:25:26:32:36:37:38:48 SD: 6.7026 cents 25:26:27:33 x 38:40:50 SD: 7.2359 cents 26:27:28:35:39:40:42:52 SD: 5.7167 cents 27:28:29:36 x 41:43:54 SD: 5.5556 cents 28:29:30:37:42:43:45:56 SD: 4.2440 cents 29:30:31:39 x 44:46:58 SD: 5.8192 cents 30:31:32:40:45:46:48:60 S SD: 2.6737 cents 42:43:45:56:63:64:67:84 SD: 2.0685 cents 45:46:48:60:68:69:72:90 SD: 1.8252 cents 48:49:51:64:72:74:77:96 SD: 1.8057 cents 58:59:62:77:87:89:93:116 SD: 1.7894 cents 60:61:64:80:90:92:96:120 SD: 1.3478 cents 86:88:92:115:129:132:138:172 SD: 1.2927 cents 88:90:94:117:132:135:141:176 SD: 0.8783 cents 90:92:96:120:135:138:144:180 S SD: 0.0000 cents