Scala analysis: xen15-gilson-ptolemy-chromatic-malakon
Ptolemy's Chromatic Malakon
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: 10/9 182.403712 minor whole tone 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 14/9 764.915905 septimal minor sixth 6: 5/3 884.358713 major sixth, BP sixth 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 6/5 315.6413 minor third 1: 28/27 62.9609 Archytas' 1/3-tone 2: 15/14 119.4428 major diatonic semitone 3: 6/5 315.6413 minor third 4: 9/8 203.9100 major whole tone 5: 28/27 62.9609 Archytas' 1/3-tone 6: 15/14 119.4428 major diatonic semitone 7: 6/5 315.6413 minor third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 28/27 62.961 cents Archytas' 1/3-tone 1: 2 15/14 119.443 cents major diatonic semitone 1: 1 9/8 203.910 cents major whole tone 1: 2 6/5 315.641 cents minor third 2: 2 10/9 182.404 cents minor whole tone 2: 1 7/6 266.871 cents septimal minor third 2: 1 56/45 378.602 cents septimal narrow major third 2: 2 9/7 435.084 cents septimal major third, BP third 2: 1 27/20 519.551 cents acute fourth 3: 1 5/4 386.314 cents major third 3: 4 4/3 498.045 cents perfect fourth 3: 1 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 3: 1 81/56 638.994 cents 4: 1 112/81 561.006 cents 4: 1 10/7 617.488 cents Euler's tritone 4: 4 3/2 701.955 cents perfect fifth 4: 1 8/5 813.686 cents minor sixth 5: 1 40/27 680.449 cents grave fifth 5: 2 14/9 764.916 cents septimal minor sixth 5: 1 45/28 821.398 cents septimale wide minor sixth 5: 1 12/7 933.129 cents septimal major sixth 5: 2 9/5 1017.596 cents just minor seventh, BP seventh 6: 2 5/3 884.359 cents major sixth, BP sixth 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 28/15 1080.557 cents grave major seventh 6: 2 27/14 1137.039 cents septimal 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 : 63.0 182.4 498.0 702.0 764.9 884.4 1200.0 63.0 : 119.4 435.1 639.0 702.0 821.4 1137.0 1200.0 182.4 : 315.6 519.6 582.5 702.0 1017.6 1080.6 1200.0 498.0 : 203.9 266.9 386.3 702.0 764.9 884.4 1200.0 702.0 : 63.0 182.4 498.0 561.0 680.4 996.1 1200.0 764.9 : 119.4 435.1 498.0 617.5 933.1 1137.0 1200.0 884.4 : 315.6 378.6 498.0 813.7 1017.6 1080.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 62.9609 .. 315.6413 cents 81/70, 252.6804 cents 28/27, 62.9609 cents 2: 182.4037 .. 519.5513 cents 243/200, 337.1476 cents 25/27,-133.2376 cents 3: 386.3137 .. 638.9941 cents 81/70, 252.6804 cents 25/27,-133.2376 cents 4: 561.0059 .. 813.6863 cents 81/70, 252.6804 cents 6272/6561,-77.9882 cents 5: 680.4487 .. 1017.5963 cents 243/200, 337.1476 cents 25/27,-133.2376 cents 6: 884.3587 .. 1137.0391 cents 81/70, 252.6804 cents 25/27,-133.2376 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 28/27, 62.9609 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 6/5, 315.6413 cents Average / Smallest step : 2.722778 Largest / Average step : 1.841241 Largest / Smallest step : 5.013290 Median interval of one step : 15/14, 119.4428 cents, amount: 2 Least squares average step : 159.00325 cents, oct.: 1113.02276 cents Scale is not proper Step pattern alph. order: ABCDABC Step pattern size order : ABDCABD Scale is a Constant Structure, by a margin of 7.71152 cents Scale diversity : 1.607392 Lumma stability : 0.104935 Lumma impropriety factor : 0.509115 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 : 225/224, 7.7115 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 n-1 differentially coherent in interval classes 2 and 3 combined Scale is differentially coherent in interval classes 3 and 4 combined Scale is n-1 differentially coherent in interval class 4 -- Rational properties -- Prime limit : 7 Odd number limit : 81 (O: 81 U: 81) Highest odd numerator or denominator: 27 Scale harmonicity : 0.032946 Average absolute harmonicity : 0.255044 Specific harmonicity : 0.182044 Fundamental : 1/54, -5.7549 octaves, 4.8449 Hz. Guide tone : 420, 8.7142 octaves, 109882.737 Hz. Exponens Consonantiae : 2.268000E+04, 14.46913 octaves Euler's gradus suavitatis : 22 Sum of Mann's harmonic distance : 60.0, average 8.57143 Mersenne's string divisions : 4037880 Sum of van Prooijen's expressibility: 4.99306, average 0.71329 Sum of Tenney's harmonic distance : 10.26759, average 1.46680 Vogel's harmonic complexity : 10.85714 Wille's k value : 40 Wilson's harmonic complexity : 24 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 65.85476, average 2.35196 Lattice compactness (without 2's) : 52.17328, average 1.86333 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 2 0.85714 8 1 3: -3 .. 1 -1.14286 10 2 5: 0 .. 1 0.28571 2 2 7: 0 .. 1 0.28571 2 2 Average exponent except 2's :-4 / 7 =-0.57143 Average absolute exponent except 2's: 14 / 7 = 2.00000 Scale is weakly epimorphic with val: <7 10 18 22| Scale is weakly epimorphic with val: <7 11 16 21| Scale is JI-epimorphic with val: <7 11 17 20| Scale is weakly epimorphic with val: <7 12 16 18|
FIT/MODE
7: 0 1 2 1 0 1 2 N T3 SD: 40.7530 c. M: 79.2016 c. 8: 0 1 2 2 0 1 2 N SD: 37.9809 c. M: 62.9609 c. 12: 1 1 3 2 1 1 3 N SD: 21.2627 c. M:-37.0391 c. Chromatic Dorian, Mela Kanakangi, Raga Kanakambari 14: 1 1 4 2 1 1 4 N SD: 16.6854 c. M: 27.2159 c. 17: 1 2 4 3 1 2 4 N SD: 17.6987 c. M:-33.2883 c. 19: 1 2 5 3 1 2 5 N SD: 5.3924 c. M:-7.2182 c. Ratio 1:2 Chromatic Dorian 39: 2 4 10 7 2 4 10 N SD: 4.6988 c. M:-7.9490 c. 41: 2 4 11 7 2 4 11 N SD: 4.1669 c. M: 6.7940 c. 53: 3 5 14 9 3 5 14 N SD: 2.7262 c. M:-4.9636 c. 58: 3 6 15 10 3 6 15 N SD: 2.6223 c. M:-5.2965 c. 60: 3 6 16 10 3 6 16 N SD: 3.0555 c. M: 4.9159 c. 72: 4 7 19 12 4 7 19 N SD: 1.9405 c. M:-3.7058 c. 80: 4 8 21 14 4 8 21 N SD: 2.1879 c. M:-3.0450 c. 91: 5 9 24 15 5 9 24 N SD: 2.1745 c. M:-3.0539 c. 94: 5 9 25 16 5 9 25 N SD: 1.9910 c. M: 3.6803 c. 99: 5 10 26 17 5 10 26 N SD: 1.1999 c. M: 2.3548 c. 118: 6 12 31 20 6 12 31 N SD: 1.1552 c. M: 2.2040 c. 130: 7 13 34 22 7 13 34 N SD: 1.3486 c. M:-2.2117 c. 133: 7 13 35 23 7 13 35 N SD: 1.4338 c. M:-2.0014 c. 152: 8 15 40 26 8 15 40 N SD: 0.5879 c. M:-0.8736 c. 171: 9 17 45 29 9 17 45 N SD: 0.1434 c. M:-0.2006 c. 612: 32 61 161 104 32 61 161 N SD: 0.1167 c. M: 0.2158 c. 665: 35 66 175 113 35 66 175 N SD: 0.1317 c. M:-0.1970 c. 764: 40 76 201 130 40 76 201 N SD: 0.1212 c. M: 0.2048 c. 783: 41 78 206 133 41 78 206 N SD: 0.0857 c. M: 0.1649 c. 935: 49 93 246 159 49 93 246 N SD: 0.0834 c. M: 0.1577 c. 954: 50 95 251 162 50 95 251 N SD: 0.0749 c. M: 0.1360 c. 1106: 58 110 291 188 58 110 291 N SD: 0.0625 c. M: 0.1252 c. 1125: 59 112 296 191 59 112 296 N SD: 0.0740 c. M: 0.1159 c. 1277: 67 127 336 217 67 127 336 N SD: 0.0535 c. M: 0.1014 c. 1448: 76 144 381 246 76 144 381 N SD: 0.0525 c. M: 0.1046 c. 1506: 79 150 396 256 79 150 396 N SD: 0.0516 c. M:-0.1034 c. 1677: 88 167 441 285 88 167 441 N SD: 0.0399 c. M:-0.0778 c. 1848: 97 184 486 314 97 184 486 N SD: 0.0347 c. M:-0.0638 c. 2019: 106 201 531 343 106 201 531 N SD: 0.0349 c. M:-0.0629 c. 2190: 115 218 576 372 115 218 576 N SD: 0.0383 c. M:-0.0620 c. 2289: 120 228 602 389 120 228 602 N SD: 0.0331 c. M: 0.0513 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: 0.0000 cents 4 x x 5:6 x 7:8 S SD: 35.0166 cents 5 x 6:7 x 8 x 10 S SD: 41.2807 cents 6 x 7:8:9 x 10:12 S SD: 16.8934 cents 7 x 8:9 x 11:12:14 SD: 19.0054 cents 8 x 9:11:12 x 13:16 SD: 14.4522 cents 9 x 10:12 x 14:15:18 S SD: 0.0000 cents 10 x 11:13:15:16:17:20 SD: 12.6686 cents 11 x 12:15 x 17:18:22 SD: 12.0981 cents 12 x 13:16:18:19:20:24 SD: 8.9133 cents 13 x 14:17 x 20:22:26 SD: 14.3138 cents 14:15:16:19:21:22:23:28 SD: 12.3198 cents 15:16:17:20 x 23:25:30 SD: 10.7683 cents 16:17:18:21:24:25:27:32 SD: 8.4411 cents 17:18:19:23 x 26:28:34 SD: 9.6007 cents 18:19:20:24:27:28:30:36 S SD: 4.3774 cents 24:25:27:32:36:37:40:48 SD: 3.9463 cents 27:28:30:36:41:42:45:54 SD: 3.0346 cents 36:37:40:48:54:56:60:72 SD: 2.2181 cents 48:50:53:64:72:75:80:96 SD: 2.1981 cents 52:54:58:69:78:81:87:104 SD: 1.8585 cents 54:56:60:72:81:84:90:108 S SD: 0.0000 cents