Scala analysis: xen15-gilson-ptolemy-diatonic-hemiolon
Ptolemy's Diatonic Hemiolon
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 12/11 150.637059 undecimal neutral second, 3/4-tone 2: 6/5 315.641287 minor third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 18/11 852.592059 undecimal neutral sixth 6: 9/5 1017.596288 just minor seventh, BP seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 10/9 182.4037 minor whole tone 1: 12/11 150.6371 undecimal neutral second, 3/4-tone 2: 11/10 165.0042 Ptolemy's second, 4/5-tone 3: 10/9 182.4037 minor whole tone 4: 9/8 203.9100 major whole tone 5: 12/11 150.6371 undecimal neutral second, 3/4-tone 6: 11/10 165.0042 Ptolemy's second, 4/5-tone 7: 10/9 182.4037 minor whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 2 11/10 165.004 cents Ptolemy's second, 4/5-tone 1: 2 10/9 182.404 cents minor whole tone 1: 1 9/8 203.910 cents major whole tone 2: 2 6/5 315.641 cents minor third 2: 1 40/33 333.041 cents 2: 2 11/9 347.408 cents undecimal neutral third 2: 1 27/22 354.547 cents neutral third, Zalzal wosta of al-Farabi 2: 1 5/4 386.314 cents major third 3: 4 4/3 498.045 cents perfect fourth 3: 1 27/20 519.551 cents acute fourth 3: 1 15/11 536.951 cents undecimal augmented fourth 3: 1 11/8 551.318 cents undecimal semi-augmented fourth 4: 1 16/11 648.682 cents undecimal semi-diminished fifth 4: 1 22/15 663.049 cents undecimal diminished fifth 4: 1 40/27 680.449 cents grave fifth 4: 4 3/2 701.955 cents perfect fifth 5: 1 8/5 813.686 cents minor sixth 5: 1 44/27 845.453 cents neutral sixth 5: 2 18/11 852.592 cents undecimal neutral sixth 5: 1 33/20 866.959 cents 5: 2 5/3 884.359 cents major sixth, BP sixth 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 9/5 1017.596 cents just minor seventh, BP seventh 6: 2 20/11 1034.996 cents large minor seventh 6: 2 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 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 : 150.6 315.6 498.0 702.0 852.6 1017.6 1200.0 150.6 : 165.0 347.4 551.3 702.0 867.0 1049.4 1200.0 315.6 : 182.4 386.3 537.0 702.0 884.4 1035.0 1200.0 498.0 : 203.9 354.5 519.6 702.0 852.6 1017.6 1200.0 702.0 : 150.6 315.6 498.0 648.7 813.7 996.1 1200.0 852.6 : 165.0 347.4 498.0 663.0 845.5 1049.4 1200.0 1017.6: 182.4 333.0 498.0 680.4 884.4 1035.0 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 150.6371 .. 203.9100 cents 33/32, 53.2729 cents 12/11, 150.6371 cents 2: 315.6413 .. 386.3137 cents 25/24, 70.6724 cents 16/15, 111.7313 cents 3: 498.0450 .. 551.3179 cents 33/32, 53.2729 cents 16/15, 111.7313 cents 4: 648.6821 .. 701.9550 cents 33/32, 53.2729 cents 128/121, 97.3641 cents 5: 813.6863 .. 884.3587 cents 25/24, 70.6724 cents 16/15, 111.7313 cents 6: 996.0900 .. 1049.3629 cents 33/32, 53.2729 cents 16/15, 111.7313 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 12/11, 150.6371 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.138024 Largest / Average step : 1.189475 Largest / Smallest step : 1.353651 Median interval of one step : 11/10, 165.0042 cents, amount: 2 Least squares average step : 170.37438 cents, oct.: 1192.62063 cents Scale is strictly proper Step pattern alph. order: ABCDABC Step pattern size order : ABCDABC Scale is a Constant Structure, by a margin of 97.36412 cents Scale diversity : 1.313255 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.704636 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 : 243/242, 7.1391 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 Best fifths form a closed circle Best major thirds form a closed circle Best minor thirds form a closed circle 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 : 4 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 classes 4 and 5 combined Limited inverse transpositions with margin 70.6724 cents: 6 -- Rational properties -- Prime limit : 11 Odd number limit : 33 (O: 33 U: 33) Highest odd numerator or denominator: 11 Scale harmonicity : 0.028104 Average absolute harmonicity : 0.250876 Specific harmonicity : 0.152508 Fundamental : 1/330, -8.3663 octaves, 0.7928 Hz. Guide tone : 36, 5.1699 octaves, 9418.520 Hz. Exponens Consonantiae : 1.188000E+04, 13.53625 octaves Euler's gradus suavitatis : 24 Sum of Mann's harmonic distance : 46.0, average 6.57143 Mersenne's string divisions : 1540770 Sum of van Prooijen's expressibility: 4.69024, average 0.67003 Sum of Tenney's harmonic distance : 9.70594, average 1.38656 Vogel's harmonic complexity : 18.85714 Wille's k value : 247 Wilson's harmonic complexity : 25 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 60.59909, average 2.16425 Lattice compactness (without 2's) : 44.51877, average 1.58996 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 2 0.85714 8 1 3: -1 .. 2 0.85714 8 2 5: -1 .. 0 -0.28571 2 2 11: -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 18 26| Scale is JI-epimorphic with val: <7 11 16 24| = patent Scale is weakly epimorphic with val: <7 11 17 23|
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 16.2202 c. M:-27.2159 c. Lebeng 17: 2 2 3 3 2 2 3 SP D3 S SD: 17.4075 c. M: 33.2883 c. 19: 2 3 3 3 2 3 3 SP M ME S SD: 15.7683 c. M: 31.5394 c. Nineteen-tone Phrygian 22: 3 3 3 4 3 3 3 SP M ME I SD: 12.8938 c. M:-20.1352 c. Minor quasi-equal Heptatonic, Miller's Porcupine-7 24: 3 3 4 4 3 3 4 SP D3 S SD: 9.0162 c. M: 17.5963 c. Maqam Hussaini, Ushaq, Neutral Diatonic Dorian 27: 3 4 4 5 3 4 4 SP T3 SD: 9.0664 c. M: 17.3037 c. 29: 4 4 4 5 4 4 4 SP M ME I SD: 12.0480 c. M:-16.8865 c. Miller's Porcupine-7 31: 4 4 5 5 4 4 5 SP D3 S SD: 5.7578 c. M: 11.1447 c. Neutral Diatonic Dorian 34: 4 5 5 6 4 5 5 SP T3 SD: 5.2125 c. M: 9.4606 c. 38: 5 5 6 6 5 5 6 SP D3 S SD: 5.4365 c. M:-7.2577 c. 41: 5 6 6 7 5 6 6 SP T3 SD: 4.1304 c. M:-6.7940 c. 46: 6 6 7 8 6 6 7 SP SD: 4.1643 c. M:-8.2775 c. 53: 7 7 8 9 7 7 8 SP SD: 4.2378 c. M:-7.8535 c. 58: 7 8 9 10 7 8 9 SP SD: 3.7675 c. M: 5.8095 c. 65: 8 9 10 11 8 9 10 SP SD: 2.0153 c. M: 3.3613 c. 72: 9 10 11 12 9 10 11 SP SD: 1.5438 c. M: 2.5921 c. Equable Diatonic Dorian 111: 14 15 17 19 14 15 17 SP SD: 1.2070 c. M: 2.1278 c. 118: 15 16 18 20 15 16 18 SP SD: 1.0029 c. M:-1.9053 c. 145: 18 20 22 25 18 20 22 SP SD: 1.1174 c. M: 1.6715 c. 152: 19 21 23 26 19 21 23 SP SD: 0.5379 c. M:-0.8248 c. 183: 23 25 28 31 23 25 28 SP SD: 0.5957 c. M: 1.2028 c. 224: 28 31 34 38 28 31 34 SP SD: 0.4418 c. M: 0.8063 c. 270: 34 37 41 46 34 37 41 SP SD: 0.3698 c. M:-0.7413 c. 335: 42 46 51 57 42 46 51 SP SD: 0.2169 c. M: 0.4174 c. 342: 43 47 52 58 43 47 52 SP SD: 0.1533 c. M:-0.2401 c. 494: 62 68 75 84 62 68 75 SP SD: 0.1078 c. M:-0.2175 c. 677: 85 93 103 115 85 93 103 SP SD: 0.0830 c. M: 0.1665 c. 829: 104 114 126 141 104 114 126 SP SD: 0.0696 c. M: 0.0957 c. 836: 105 115 127 142 105 115 127 SP SD: 0.0801 c. M:-0.1482 c. 1019: 128 140 155 173 128 140 155 SP SD: 0.0794 c. M: 0.1282 c. 1171: 147 161 178 199 147 161 178 SP SD: 0.0088 c. M: 0.0136 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: 10.5889 cents 4 x 5 x 6 x 7:8 S SD: 21.4668 cents 5 x 6:7 x 8:9:10 S SD: 18.5993 cents 6 x 7:8:9:10:11:12 S SD: 11.0514 cents 7:8 x 9 x 11:13:14 SD: 27.0494 cents 8:9:10:11:12:13:14:16 S SD: 16.4097 cents 9:10:11:12:14:15:16:18 S SD: 12.3338 cents 10:11:12:13:15:16:18:20 SD: 8.6204 cents 14:15:17:19:21:23:25:28 SD: 7.2409 cents 19:21:23:25:29:31:34:38 SD: 6.8541 cents 20:22:24:27:30:33:36:40 S SD: 4.2266 cents 24:26:29:32:36:39:43:48 SD: 3.1918 cents 30:33:36:40:45:49:54:60 SD: 2.1030 cents 44:48:53:59:66:72:79:88 SD: 1.7969 cents 54:59:65:72:81:88:97:108 SD: 1.4234 cents 66:72:79:88:99:108:119:132 SD: 0.7511 cents 90:98:108:120:135:147:162:180 SD: 0.6483 cents 110:120:132:147:165:180:198:220 SD: 0.5615 cents 120:131:144:160:180:196:216:240 SD: 0.4895 cents 174:190:209:232:261:285:313:348 SD: 0.4395 cents 186:203:223:248:279:304:335:372 SD: 0.4131 cents 210:229:252:280:315:344:378:420 SD: 0.2794 cents 240:262:288:320:360:393:432:480 SD: 0.2428 cents 264:288:317:352:396:432:475:528 SD: 0.1876 cents 330:360:396:440:495:540:594:660 S SD: 0.0000 cents