Scala analysis: xen15-gilson-aristoxenus-diatonic-syntonon
Aristoxenus' Diatonic Syntonon
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 20/19 88.800698 small undevicesimal semitone 2: 20/17 281.358304 septendecimal augmented second 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 30/19 790.755699 smaller undevicesimal minor sixth 6: 30/17 983.313305 septendecimal minor seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 17/15 216.6867 septendecimal whole tone 1: 20/19 88.8007 small undevicesimal semitone 2: 19/17 192.5576 quasi-meantone 3: 17/15 216.6867 septendecimal whole tone 4: 9/8 203.9100 major whole tone 5: 20/19 88.8007 small undevicesimal semitone 6: 19/17 192.5576 quasi-meantone 7: 17/15 216.6867 septendecimal whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 20/19 88.801 cents small undevicesimal semitone 1: 2 19/17 192.558 cents quasi-meantone 1: 1 9/8 203.910 cents major whole tone 1: 2 17/15 216.687 cents septendecimal whole tone 2: 2 20/17 281.358 cents septendecimal augmented second 2: 1 45/38 292.711 cents 2: 1 68/57 305.487 cents 2: 2 19/15 409.244 cents undevicesimal ditone 2: 1 51/40 420.597 cents septendecimal diminished fourth 3: 1 45/34 485.268 cents 3: 4 4/3 498.045 cents perfect fourth 3: 1 51/38 509.397 cents 3: 1 57/40 613.154 cents 4: 1 80/57 586.846 cents 4: 1 76/51 690.603 cents 4: 4 3/2 701.955 cents perfect fifth 4: 1 68/45 714.732 cents 5: 1 80/51 779.403 cents 5: 2 30/19 790.756 cents smaller undevicesimal minor sixth 5: 1 57/34 894.513 cents 5: 1 76/45 907.289 cents 5: 2 17/10 918.642 cents septendecimal diminished seventh 6: 2 30/17 983.313 cents septendecimal minor seventh 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 34/19 1007.442 cents quasi-mean seventh 6: 2 19/10 1111.199 cents undevicesimal major seventh 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 : 88.8 281.4 498.0 702.0 790.8 983.3 1200.0 88.8 : 192.6 409.2 613.2 702.0 894.5 1111.2 1200.0 281.4 : 216.7 420.6 509.4 702.0 918.6 1007.4 1200.0 498.0 : 203.9 292.7 485.3 702.0 790.8 983.3 1200.0 702.0 : 88.8 281.4 498.0 586.8 779.4 996.1 1200.0 790.8 : 192.6 409.2 498.0 690.6 907.3 1111.2 1200.0 983.3 : 216.7 305.5 498.0 714.7 918.6 1007.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 88.8007 .. 216.6867 cents 323/300, 127.8860 cents 20/19, 88.8007 cents 2: 281.3583 .. 420.5967 cents 867/800, 139.2384 cents 300/289, 64.6716 cents 3: 485.2683 .. 613.1543 cents 323/300, 127.8860 cents 300/289, 64.6716 cents 4: 586.8457 .. 714.7317 cents 323/300, 127.8860 cents 3200/3249,-26.3086 cents 5: 779.4033 .. 918.6417 cents 867/800, 139.2384 cents 300/289, 64.6716 cents 6: 983.3133 .. 1111.1993 cents 323/300, 127.8860 cents 300/289, 64.6716 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 20/19, 88.8007 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 17/15, 216.6867 cents Average / Smallest step : 1.930487 Largest / Average step : 1.264006 Largest / Smallest step : 2.440146 Median interval of one step : 19/17, 192.5576 cents, amount: 2 Least squares average step : 165.76522 cents, oct.: 1160.35653 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 : 1.423336 Lumma stability : 0.363573 Lumma impropriety factor : 0.021924 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 : 153/152, 11.3524 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 4 Number of recognisable fifths : 6, average 702.1924 cents Number of appreciable fifths : 5, average 699.6845 cents Scale is Dorian octave species of abstract tetrachord Scale contains two identical disjunct tetrachords Best major thirds form a closed circle Scale contains a complete diamond : 1 3 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 classes 3 and 4 combined Limited inverse transpositions with margin 139.2384 cents: 5 -- Rational properties -- Prime limit : 19 Odd number limit : 57 (O: 57 U: 57) Highest odd numerator or denominator: 19 Scale harmonicity : 0.012665 Average absolute harmonicity : 0.226289 Specific harmonicity : 0.066270 Fundamental : 1/1938, -10.9204 octaves, 0.1350 Hz. Guide tone : 60, 5.9069 octaves, 15697.534 Hz. Exponens Consonantiae : 1.162800E+05, 16.82724 octaves Euler's gradus suavitatis : 46 Sum of Mann's harmonic distance : 93.5, average 13.35714 Mersenne's string divisions : 16616145 Sum of van Prooijen's expressibility: 5.97265, average 0.85324 Sum of Tenney's harmonic distance : 12.73307, average 1.81901 Vogel's harmonic complexity : 34.57143 Wille's k value : 1453 Wilson's harmonic complexity : 47 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 57.97294, average 2.07046 Lattice compactness (without 2's) : 41.53320, average 1.48333 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -1 .. 2 1.14286 10 1 3: -1 .. 1 0.28571 4 2 5: 0 .. 1 0.57143 4 17: -1 .. 0 -0.28571 2 2 19: -1 .. 0 -0.28571 2 2 Average exponent except 2's : 2 / 7 = 0.28571 Average absolute exponent except 2's: 12 / 7 = 1.71429 Scale is weakly epimorphic with val: <7 10 15 30 31| Scale is weakly epimorphic with val: <7 10 16 31 32| Scale is weakly epimorphic with val: <7 10 17 26 32| Scale is weakly epimorphic with val: <7 10 17 32 33| Scale is weakly epimorphic with val: <7 10 18 26 27| Scale is weakly epimorphic with val: <7 10 18 27 33| Scale is JI-epimorphic with val: <7 11 15 27 28| Scale is JI-epimorphic with val: <7 11 16 28 29| Scale is JI-epimorphic with val: <7 11 17 29 30| Scale is JI-epimorphic with val: <7 11 18 30 31| Scale is weakly epimorphic with val: <7 12 15 26 30| Scale is weakly epimorphic with val: <7 12 15 30 33| Scale is weakly epimorphic with val: <7 12 16 27 31| Scale is weakly epimorphic with val: <7 12 17 28 32| Scale is weakly epimorphic with val: <7 12 18 26 29| Scale is weakly epimorphic with val: <7 12 18 29 33|
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 50.1320 c. M:-82.6279 c. Lebeng 9: 1 1 2 1 1 1 2 P M ME S SD: 32.2335 c. M: 49.9800 c. Pseudo Pelog, Pelogic-7, Antiionian 10: 1 1 2 2 1 1 2 P D3 S SD: 30.0087 c. M:-49.2443 c. 12: 1 2 2 2 1 2 2 P M ME S SD: 10.9837 c. M:-18.6417 c. G.Dorian, M.Phrygian, G.M.Hypoaeolian, Bhairavi That, Mela Hanumatodi, Raga Asavari (Asaveri), Bilashkhani Todi, Ghanta, Makam Kurd, Gregorian nr.3, In, Zokuso: Japan, Ousak: Greece, Major inverse 17: 1 3 3 3 1 3 3 N M DE S SD: 9.1949 c. M: 18.2125 c. 26: 2 4 5 4 2 4 5 P T3 SD: 8.0546 c. M: 14.0825 c. 29: 2 5 5 5 2 5 5 N M DE S SD: 5.6864 c. M:-9.7901 c. 38: 3 6 7 6 3 6 7 P T3 SD: 4.9031 c. M:-7.2182 c. 43: 3 7 8 7 3 7 8 N T3 SD: 5.3246 c. M: 9.3603 c. 50: 4 8 9 8 4 8 9 P T3 SD: 4.9119 c. M:-7.1993 c. 51: 4 8 9 9 4 8 9 N SD: 4.9250 c. M:-9.2443 c. 55: 4 9 10 9 4 9 10 N T3 SD: 3.0784 c. M: 5.3012 c. 56: 4 9 10 10 4 9 10 N SD: 3.4080 c. M:-5.1879 c. 65: 5 10 12 11 5 10 12 N SD: 3.0568 c. M: 4.8518 c. 67: 5 11 12 11 5 11 12 N T3 SD: 2.9722 c. M:-5.2089 c. 68: 5 11 12 12 5 11 12 N SD: 3.1096 c. M:-4.9220 c. 77: 6 12 14 13 6 12 14 N SD: 2.4595 c. M:-4.7058 c. 82: 6 13 15 14 6 13 15 N SD: 1.7179 c. M: 3.3095 c. 94: 7 15 17 16 7 15 17 N SD: 0.4279 c. M:-0.7337 c. 243: 18 39 44 41 18 39 44 N SD: 0.5095 c. M:-0.7204 c. 299: 22 48 54 51 22 48 54 N SD: 0.3270 c. M: 0.5064 c. 311: 23 50 56 53 23 50 56 N SD: 0.3176 c. M:-0.6095 c. 393: 29 63 71 67 29 63 71 N SD: 0.2677 c. M: 0.4423 c. 405: 30 65 73 69 30 65 73 N SD: 0.2520 c. M:-0.3904 c. 460: 34 74 83 78 34 74 83 N SD: 0.2328 c. M:-0.3808 c. 499: 37 80 90 85 37 80 90 N SD: 0.2397 c. M:-0.4267 c. 542: 40 87 98 92 40 87 98 N SD: 0.2132 c. M: 0.3498 c. 554: 41 89 100 94 41 89 100 N SD: 0.1334 c. M:-0.2301 c. 648: 48 104 117 110 48 104 117 N SD: 0.0800 c. M:-0.1232 c. 742: 55 119 134 126 55 119 134 N SD: 0.0757 c. M:-0.1481 c. 947: 70 152 171 161 70 152 171 N SD: 0.0533 c. M: 0.0995 c. 1041: 77 167 188 177 77 167 188 N SD: 0.0518 c. M: 0.0903 c. 1501: 111 241 271 255 111 241 271 N SD: 0.0469 c. M: 0.0828 c. 1595: 118 256 288 271 118 256 288 N SD: 0.0190 c. M: 0.0347 c. 1689: 125 271 305 287 125 271 305 N SD: 0.0067 c. M: 0.0096 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 5 x 6 S SD: 31.2010 cents 4 x 5 x 6 x 7:8 S SD: 26.4876 cents 5 x 6:7 x 8:9:10 S SD: 20.0112 cents 6 x 7:8:9 x 11:12 SD: 13.5240 cents 7 x 8:9 x 11:12:14 SD: 19.0465 cents 8 x 9:11:12:13:14:16 SD: 17.8912 cents 9:10:11:12 x 14:16:18 S SD: 19.6886 cents 10:11:12:13:15:16:18:20 SD: 14.7112 cents 11:12:13:15 x 17:19:22 SD: 15.0504 cents 12:13:14:16:18:19:21:24 SD: 7.7197 cents 16:17:19:21:24:25:28:32 SD: 6.0673 cents 18:19:21:24:27:28:32:36 SD: 4.6596 cents 22:23:26:29:33:35:39:44 SD: 4.1052 cents 24:25:28:32:36:38:42:48 SD: 3.9679 cents 33:35:39:44:50:52:58:66 SD: 3.4890 cents 34:36:40:45:51:54:60:68 SD: 2.7459 cents 38:40:45:51:57:60:67:76 SD: 2.3038 cents 52:55:61:69:78:82:92:104 SD: 1.9596 cents 56:59:66:75:84:88:99:112 SD: 1.7451 cents 60:63:71:80:90:95:106:120 SD: 1.7317 cents 62:65:73:83:93:98:109:124 SD: 1.7231 cents 72:76:85:96:108:114:127:144 SD: 1.2997 cents 74:78:87:99:111:117:131:148 SD: 1.2462 cents 78:82:92:104:117:123:138:156 SD: 1.0018 cents 90:95:106:120:135:142:159:180 SD: 0.8094 cents 114:120:134:152:171:180:201:228 SD: 0.3069 cents 192:202:226:256:288:303:339:384 SD: 0.2576 cents 306:322:360:408:459:483:540:612 SD: 0.1143 cents 510:537:600:680:765:805:900:1020 SD: 0.1087 cents