Scala analysis: xen15-gilson-ptolemy-diatonic-malakon
Ptolemy's Diatonic Malakon
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 21/20 84.467193 minor semitone 2: 7/6 266.870906 septimal minor third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 63/40 786.422194 narrow minor sixth 6: 7/4 968.825906 harmonic seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 8/7 231.1741 septimal whole tone 1: 21/20 84.4672 minor semitone 2: 10/9 182.4037 minor whole tone 3: 8/7 231.1741 septimal whole tone 4: 9/8 203.9100 major whole tone 5: 21/20 84.4672 minor semitone 6: 10/9 182.4037 minor whole tone 7: 8/7 231.1741 septimal whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 21/20 84.467 cents minor semitone 1: 2 10/9 182.404 cents minor whole tone 1: 1 9/8 203.910 cents major whole tone 1: 2 8/7 231.174 cents septimal whole tone 2: 2 7/6 266.871 cents septimal minor third 2: 1 189/160 288.377 cents 2: 1 6/5 315.641 cents minor third 2: 2 80/63 413.578 cents wide major third 2: 1 9/7 435.084 cents septimal major third, BP third 3: 1 21/16 470.781 cents narrow fourth 3: 4 4/3 498.045 cents perfect fourth 3: 1 27/20 519.551 cents acute fourth 3: 1 10/7 617.488 cents Euler's tritone 4: 1 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 4: 1 40/27 680.449 cents grave fifth 4: 4 3/2 701.955 cents perfect fifth 4: 1 32/21 729.219 cents wide fifth 5: 1 14/9 764.916 cents septimal minor sixth 5: 2 63/40 786.422 cents narrow minor sixth 5: 1 5/3 884.359 cents major sixth, BP sixth 5: 1 320/189 911.623 cents 5: 2 12/7 933.129 cents septimal major sixth 6: 2 7/4 968.826 cents harmonic seventh 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 9/5 1017.596 cents just minor seventh, BP seventh 6: 2 40/21 1115.533 cents acute 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 : 84.5 266.9 498.0 702.0 786.4 968.8 1200.0 84.5 : 182.4 413.6 617.5 702.0 884.4 1115.5 1200.0 266.9 : 231.2 435.1 519.6 702.0 933.1 1017.6 1200.0 498.0 : 203.9 288.4 470.8 702.0 786.4 968.8 1200.0 702.0 : 84.5 266.9 498.0 582.5 764.9 996.1 1200.0 786.4 : 182.4 413.6 498.0 680.4 911.6 1115.5 1200.0 968.8 : 231.2 315.6 498.0 729.2 933.1 1017.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 84.4672 .. 231.1741 cents 160/147, 146.7069 cents 21/20, 84.4672 cents 2: 266.8709 .. 435.0841 cents 54/49, 168.2132 cents 49/48, 35.6968 cents 3: 470.7809 .. 617.4878 cents 160/147, 146.7069 cents 49/48, 35.6968 cents 4: 582.5122 .. 729.2191 cents 160/147, 146.7069 cents 49/50,-34.9756 cents 5: 764.9159 .. 933.1291 cents 54/49, 168.2132 cents 49/48, 35.6968 cents 6: 968.8259 .. 1115.5328 cents 160/147, 146.7069 cents 49/48, 35.6968 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 21/20, 84.4672 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 8/7, 231.1741 cents Average / Smallest step : 2.029528 Largest / Average step : 1.348516 Largest / Smallest step : 2.736851 Median interval of one step : 10/9, 182.4037 cents, amount: 2 Least squares average step : 164.75165 cents, oct.: 1153.26152 cents Scale is not proper Step pattern alph. order: ABCDABC Step pattern size order : ABDCABD Scale is a Constant Structure, by a margin of 34.97561 cents Scale diversity : 1.518220 Lumma stability : 0.259768 Lumma impropriety factor : 0.029146 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 : 81/80, 21.5063 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 : 2 at degree 2 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 : 7 Odd number limit : 189 (O: 189 U: 189) Highest odd numerator or denominator: 63 Scale harmonicity : 0.033686 Average absolute harmonicity : 0.246697 Specific harmonicity : 0.164932 Fundamental : 1/120, -6.9069 octaves, 2.1802 Hz. Guide tone : 252, 7.9773 octaves, 65929.642 Hz. Exponens Consonantiae : 3.024000E+04, 14.88417 octaves Euler's gradus suavitatis : 22 Sum of Mann's harmonic distance : 91.5, average 13.07143 Mersenne's string divisions : 63408345 Sum of van Prooijen's expressibility: 5.76600, average 0.82371 Sum of Tenney's harmonic distance : 11.25342, average 1.60763 Vogel's harmonic complexity : 13.57143 Wille's k value : 94 Wilson's harmonic complexity : 21 Rectangular lattice diameter : 5 Triangular lattice diameter : 4 Lattice compactness : 75.48379, average 2.69585 Lattice compactness (without 2's) : 45.21701, average 1.61489 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -0.85714 12 1 3: -1 .. 2 0.28571 6 2 5: -1 .. 0 -0.28571 2 7: 0 .. 1 0.57143 4 4 Average exponent except 2's : 4 / 7 = 0.57143 Average absolute exponent except 2's: 12 / 7 = 1.71429 Scale is JI-epimorphic with val: <7 11 16 20| = patent
FIT/MODE
7: 0 1 1 2 0 1 2 N T3 SD: 95.0234 c. M: 155.1879 c. 8: 1 1 1 2 0 1 2 N D3 SD: 47.9171 c. M: 68.8259 c. 9: 1 1 2 1 1 1 2 P M ME S SD: 30.0537 c. M:-48.8661 c. Pseudo Pelog, Pelogic-7, Antiionian 10: 1 1 2 2 1 1 2 P D3 S SD: 28.2447 c. M:-53.5778 c. 11: 1 1 3 1 1 2 2 N D3 S SD: 34.1431 c. M: 48.6891 c. 12: 1 2 2 2 1 2 2 P M ME S SD: 18.9082 c. M:-33.1291 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 14: 1 2 3 2 1 2 3 P T3 SD: 14.7498 c. M: 25.9688 c. Superpelog-Pelog 17: 1 3 3 3 1 3 3 N M DE S SD: 11.5815 c. M:-19.4094 c. 24: 2 3 5 4 2 3 5 P SD: 12.3769 c. M: 18.8259 c. Soft Diatonic Dorian 26: 2 4 5 4 2 4 5 P T3 SD: 7.0921 c. M:-10.0522 c. 31: 2 5 6 5 2 5 6 N T3 SD: 6.2204 c. M: 12.2286 c. Soft Diatonic Dorian 41: 3 6 8 7 3 6 8 N SD: 2.5910 c. M:-3.8217 c. 58: 4 9 11 10 4 9 11 N SD: 1.8779 c. M:-3.5879 c. 72: 5 11 14 12 5 11 14 N SD: 1.8195 c. M: 3.0889 c. 99: 7 15 19 17 7 15 19 N SD: 0.8767 c. M:-1.4566 c. 171: 12 26 33 29 12 26 33 N SD: 0.2831 c. M: 0.4573 c. 270: 19 41 52 46 19 41 52 N SD: 0.1885 c. M:-0.2672 c. 441: 31 67 85 75 31 67 85 N SD: 0.1096 c. M: 0.2042 c. 737: 52 112 142 125 52 112 142 N SD: 0.1410 c. M:-0.2004 c. 836: 59 127 161 142 59 127 161 N SD: 0.1218 c. M:-0.2218 c. 908: 64 138 175 154 64 138 175 N SD: 0.1267 c. M:-0.1929 c. 966: 68 147 186 164 68 147 186 N SD: 0.1085 c. M:-0.2098 c. 1007: 71 153 194 171 71 153 194 N SD: 0.0737 c. M:-0.1406 c. 1178: 83 179 227 200 83 179 227 N SD: 0.0620 c. M:-0.0874 c. 1236: 87 188 238 210 87 188 238 N SD: 0.0610 c. M:-0.1194 c. 1277: 90 194 246 217 90 194 246 N SD: 0.0575 c. M:-0.1087 c. 1407: 99 214 271 239 99 214 271 N SD: 0.0488 c. M:-0.0801 c. 1448: 102 220 279 246 102 220 279 N SD: 0.0354 c. M:-0.0632 c. 1677: 118 255 323 285 118 255 323 N SD: 0.0267 c. M:-0.0471 c. 2118: 149 322 408 360 149 322 408 N SD: 0.0258 c. M: 0.0479 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: 28.1557 cents 4 x x 5:6 x 7:8 S SD: 27.9328 cents 5 x 6:7 x 8:9:10 S SD: 22.4812 cents 6 x 7:8:9 x 11:12 SD: 16.1074 cents 7 x 8:9 x 11:12:14 SD: 16.1593 cents 8 x 9:11:12:13:14:16 SD: 16.4397 cents 9 x 11:12 x 14:16:18 SD: 17.5409 cents 10:11:12:13:15:16:18:20 SD: 16.8473 cents 11:12:13:15 x 17:19:22 SD: 14.8867 cents 12:13:14:16:18:19:21:24 SD: 7.8388 cents 13:14:15:17 x 20:23:26 SD: 12.2775 cents 14:15:16:19:21:22:25:28 SD: 9.7680 cents 16:17:19:21:24:25:28:32 SD: 6.8398 cents 18:19:21:24:27:28:32:36 SD: 5.1296 cents 24:25:28:32:36:38:42:48 SD: 2.3637 cents 36:38:42:48:54:57:63:72 SD: 1.8457 cents 60:63:70:80:90:95:105:120 SD: 1.3051 cents 78:82:91:104:117:123:137:156 SD: 0.9999 cents 84:88:98:112:126:132:147:168 SD: 0.7940 cents 96:101:112:128:144:151:168:192 SD: 0.5895 cents 120:126:140:160:180:189:210:240 S SD: 0.0000 cents