Scala analysis: xen15-gilson-ptolemy-chromatic-syntonon
Ptolemy's Chromatic Syntonon
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 22/21 80.537035 undecimal minor semitone 2: 8/7 231.174094 septimal whole tone 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 11/7 782.492036 undecimal augmented fifth 6: 12/7 933.129094 septimal major sixth 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 7/6 266.8709 septimal minor third 1: 22/21 80.5370 undecimal minor semitone 2: 12/11 150.6371 undecimal neutral second, 3/4-tone 3: 7/6 266.8709 septimal minor third 4: 9/8 203.9100 major whole tone 5: 22/21 80.5370 undecimal minor semitone 6: 12/11 150.6371 undecimal neutral second, 3/4-tone 7: 7/6 266.8709 septimal minor third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 22/21 80.537 cents undecimal minor semitone 1: 2 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 1 9/8 203.910 cents major whole tone 1: 2 7/6 266.871 cents septimal minor third 2: 2 8/7 231.174 cents septimal whole tone 2: 1 33/28 284.447 cents undecimal minor third 2: 1 11/9 347.408 cents undecimal neutral third 2: 2 14/11 417.508 cents undecimal diminished fourth or major third 2: 1 21/16 470.781 cents narrow fourth 3: 1 9/7 435.084 cents septimal major third, BP third 3: 4 4/3 498.045 cents perfect fourth 3: 1 11/8 551.318 cents undecimal semi-augmented fourth 3: 1 63/44 621.418 cents 4: 1 88/63 578.582 cents 4: 1 16/11 648.682 cents undecimal semi-diminished fifth 4: 4 3/2 701.955 cents perfect fifth 4: 1 14/9 764.916 cents septimal minor sixth 5: 1 32/21 729.219 cents wide fifth 5: 2 11/7 782.492 cents undecimal augmented fifth 5: 1 18/11 852.592 cents undecimal neutral sixth 5: 1 56/33 915.553 cents 5: 2 7/4 968.826 cents harmonic seventh 6: 2 12/7 933.129 cents septimal major sixth 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 6: 2 21/11 1119.463 cents undecimal 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 : 80.5 231.2 498.0 702.0 782.5 933.1 1200.0 80.5 : 150.6 417.5 621.4 702.0 852.6 1119.5 1200.0 231.2 : 266.9 470.8 551.3 702.0 968.8 1049.4 1200.0 498.0 : 203.9 284.4 435.1 702.0 782.5 933.1 1200.0 702.0 : 80.5 231.2 498.0 578.6 729.2 996.1 1200.0 782.5 : 150.6 417.5 498.0 648.7 915.6 1119.5 1200.0 933.1 : 266.9 347.4 498.0 764.9 968.8 1049.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 80.5370 .. 266.8709 cents 49/44, 186.3339 cents 22/21, 80.5370 cents 2: 231.1741 .. 470.7809 cents 147/128, 239.6068 cents 48/49,-35.6968 cents 3: 435.0841 .. 621.4180 cents 49/44, 186.3339 cents 48/49,-35.6968 cents 4: 578.5820 .. 764.9159 cents 49/44, 186.3339 cents 3872/3969,-42.8359 cents 5: 729.2191 .. 968.8259 cents 147/128, 239.6068 cents 48/49,-35.6968 cents 6: 933.1291 .. 1119.4630 cents 49/44, 186.3339 cents 48/49,-35.6968 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 22/21, 80.5370 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 7/6, 266.8709 cents Average / Smallest step : 2.128568 Largest / Average step : 1.556747 Largest / Smallest step : 3.313642 Median interval of one step : 12/11, 150.6371 cents, amount: 2 Least squares average step : 162.54339 cents, oct.: 1137.80375 cents Scale is not proper Step pattern alph. order: ABCDABC Step pattern size order : ABDCABD Scale is a Constant Structure, by a margin of 17.57613 cents Scale diversity : 1.595742 Lumma stability : 0.134228 Lumma impropriety factor : 0.154686 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 : 99/98, 17.5761 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 -- Rational properties -- Prime limit : 11 Odd number limit : 63 (O: 63 U: 63) Highest odd numerator or denominator: 21 Scale harmonicity : 0.026454 Average absolute harmonicity : 0.242201 Specific harmonicity : 0.130411 Fundamental : 1/42, -5.3923 octaves, 6.2292 Hz. Guide tone : 264, 8.0444 octaves, 69069.149 Hz. Exponens Consonantiae : 1.108800E+04, 13.43671 octaves Euler's gradus suavitatis : 25 Sum of Mann's harmonic distance : 55.0, average 7.85714 Mersenne's string divisions : 514710 Sum of van Prooijen's expressibility: 5.00805, average 0.71544 Sum of Tenney's harmonic distance : 10.38196, average 1.48314 Vogel's harmonic complexity : 11.28571 Wille's k value : 31 Wilson's harmonic complexity : 24 Rectangular lattice diameter : 4 Triangular lattice diameter : 3 Lattice compactness : 62.61661, average 2.23631 Lattice compactness (without 2's) : 38.57775, average 1.37778 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 3 1.14286 10 1 3: -1 .. 1 0.00000 4 2 7: -1 .. 0 -0.57143 4 2 11: 0 .. 1 0.28571 2 2 Average exponent except 2's :-2 / 7 =-0.28571 Average absolute exponent except 2's: 10 / 7 = 1.42857 Scale is weakly epimorphic with val: <7 10 22 23| Scale is JI-epimorphic with val: <7 11 19 24| Scale is weakly epimorphic with val: <7 11 20 26| Scale is weakly epimorphic with val: <7 12 18 22| Scale is weakly epimorphic with val: <7 12 22 23|
FIT/MODE
7: 0 1 2 1 0 1 2 N T3 SD: 60.6195 c. M: 96.7778 c. 8: 0 1 2 1 1 1 2 N D3 SD: 63.1686 c. M: 101.9550 c. 9: 1 1 2 1 1 1 2 P M ME S SD: 31.2691 c. M:-52.7963 c. Pseudo Pelog, Pelogic-7, Antiionian 10: 1 1 2 2 1 1 2 P D3 S SD: 30.0373 c. M:-57.5080 c. 12: 1 1 3 2 1 1 3 N SD: 19.8650 c. M: 33.1291 c. Chromatic Dorian, Mela Kanakangi, Raga Kanakambari 14: 1 2 3 2 1 2 3 P T3 SD: 14.3713 c. M:-25.9688 c. Superpelog-Pelog 17: 1 2 4 3 1 2 4 N SD: 10.5729 c. M: 19.4094 c. 26: 2 3 6 4 2 3 6 N SD: 7.8414 c. M:-11.7707 c. 31: 2 4 7 5 2 4 7 N SD: 4.6326 c. M: 8.2985 c. 41: 3 5 9 7 3 5 9 N SD: 4.3779 c. M:-7.7519 c. 46: 3 6 10 8 3 6 10 N SD: 3.0632 c. M:-6.0013 c. 58: 4 7 13 10 4 7 13 N SD: 2.4040 c. M:-3.7149 c. 63: 4 8 14 11 4 8 14 N SD: 2.5025 c. M: 4.3466 c. 72: 5 9 16 12 5 9 16 N SD: 1.7269 c. M:-2.7963 c. 89: 6 11 20 15 6 11 20 N SD: 1.3822 c. M: 2.7920 c. 104: 7 13 23 18 7 13 23 N SD: 1.4187 c. M:-2.1233 c. 118: 8 15 26 20 8 15 26 N SD: 1.4448 c. M:-2.7242 c. 130: 9 16 29 22 9 16 29 N SD: 1.3172 c. M:-2.5399 c. 135: 9 17 30 23 9 17 30 N SD: 0.2802 c. M: 0.5370 c. 342: 23 43 76 58 23 43 76 N SD: 0.2120 c. M:-0.4049 c. 477: 32 60 106 81 32 60 106 N SD: 0.1398 c. M:-0.2724 c. 566: 38 71 126 96 38 71 126 N SD: 0.1580 c. M: 0.2669 c. 612: 41 77 136 104 41 77 136 N SD: 0.1317 c. M:-0.2042 c. 670: 45 84 149 114 45 84 149 N SD: 0.1161 c. M:-0.1945 c. 701: 47 88 156 119 47 88 156 N SD: 0.1172 c. M: 0.1811 c. 805: 54 101 179 137 54 101 179 N SD: 0.1068 c. M:-0.1568 c. 836: 56 105 186 142 56 105 186 N SD: 0.1095 c. M: 0.1954 c. 877: 59 110 195 149 59 110 195 N SD: 0.1043 c. M:-0.1927 c. 1012: 68 127 225 172 68 127 225 N SD: 0.0671 c. M:-0.1167 c. 1147: 77 144 255 195 77 144 255 N SD: 0.0531 c. M:-0.0880 c. 1371: 92 172 305 233 92 172 305 N SD: 0.0515 c. M: 0.1019 c. 1506: 101 189 335 256 101 189 335 N SD: 0.0536 c. M: 0.0984 c. 1578: 106 198 351 268 106 198 351 N SD: 0.0441 c. M:-0.0713 c. 1713: 115 215 381 291 115 215 381 N SD: 0.0209 c. M: 0.0293 c. 1848: 124 232 411 314 124 232 411 N SD: 0.0130 c. M: 0.0245 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: 16.2568 cents 4 x x 5:6 x 7:8 S SD: 29.3238 cents 5 x 6:7 x 8:9:10 S SD: 29.9180 cents 6 x 7:8:9 x 10:12 S SD: 12.0877 cents 7 x 8:9 x 11:12:14 SD: 12.5922 cents 8 x 9:11:12:13:14:16 SD: 15.1141 cents 9 x 10:12 x 14:15:18 S SD: 14.2352 cents 10 x 11:13:15:16:17:20 SD: 14.4170 cents 11:12:13:15 x 17:19:22 SD: 17.3198 cents 12:13:14:16:18:19:21:24 SD: 11.1460 cents 13:14:15:17 x 20:22:26 SD: 12.3963 cents 14:15:16:19:21:22:24:28 SD: 7.0748 cents 18:19:21:24:27:28:31:36 SD: 6.0909 cents 20:21:23:27:30:31:34:40 SD: 5.2861 cents 21:22:24:28:32:33:36:42 S SD: 3.8949 cents 28:29:32:37:42:44:48:56 SD: 3.5930 cents 36:38:41:48:54:57:62:72 SD: 3.0022 cents 40:42:46:53:60:63:69:80 SD: 2.7899 cents 42:44:48:56:63:66:72:84 S SD: 0.0000 cents