Scala analysis: xen15-chalmers-triadic-diamond-40-33-tetrachord
Upper tetrachord 320/297 * 3267/3200 * 40/33 of triadic diamond for M=40/33, D=3/2
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 320/297 129.130769 2: 11/10 165.004228 Ptolemy's second, 4/5-tone 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 40/33 333.0408 1: 320/297 129.1308 2: 3267/3200 35.8735 3: 40/33 333.0408
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 3267/3200 35.873 cents 1: 1 320/297 129.131 cents 1: 1 40/33 333.041 cents 2: 1 11/10 165.004 cents Ptolemy's second, 4/5-tone 2: 1 99/80 368.914 cents 2: 1 12800/9801 462.172 cents Highest number of different intervals for one interval class: 3 Average number of different intervals per interval class: 3.00000 = 3
SHOW/LINE/CENTS INTERVALS
1 2 3 0.0 : 129.1 165.0 498.0 129.1: 35.9 368.9 498.0 165.0: 333.0 462.2 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 35.8735 .. 333.0408 cents 128000/107811, 297.1673 cents 3267/3200, 35.8735 cents 2: 165.0042 .. 462.1715 cents 128000/107811, 297.1673 cents 363/400,-168.0365 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 3267/3200, 35.8735 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 40/33, 333.0408 cents Average / Smallest step : 4.627795 Largest / Average step : 2.006088 Largest / Smallest step : 9.283765 Median interval of one step : 320/297, 129.1308 cents, amount: 1 Least squares average step : 139.51959 cents, oct.: 418.55876 cents Scale is not proper Scale has trivalence property Scale is a mode of a 1319-tone equal temperament with octave 4/3 degrees: 342 437 1319 Least number of segments generator : 660 of 249.211 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1300 of 490.871 cents and inv. generated superset size: 24 = 21 more = 800.000% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : MSL Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 35.87346 cents Scale diversity : 0.701654 Lumma stability : 0.144057 Lumma impropriety factor : 0.337392 Rothenberg efficiency : 0.666667 redundancy: 0.333333 Efficiency x scale size : 2.000000 Number of different interval sizes : 6 = 3.00000 / class Number of one step interval sizes : 3 Highest interval variety : 3 Mean interval variety : 3.00000 = 3 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 3267/3200, 35.8735 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 Formal octave complements present : 1 = 33.3333% 2/1 octave complements present : 0 = 0.0000% -- Rational properties -- Prime limit : 11 Odd number limit : 9801 (O: 3267 U: 9801) Highest odd numerator or denominator: 297 Scale harmonicity : 0.015585 Average absolute harmonicity : 0.093098 Specific harmonicity : 0.081153 Fundamental : 1/2970, -11.5362 octaves, 0.0881 Hz. Guide tone : 3520, 11.7814 octaves, 920921.990 Hz. Exponens Consonantiae : 1.045440E+07, 23.31761 octaves Euler's gradus suavitatis : 42 Sum of Mann's harmonic distance : 322.5, average 107.50000 Mersenne's string divisions : 12957 Sum of van Prooijen's expressibility: 3.99127, average 1.33042 Sum of Tenney's harmonic distance : 8.09848, average 2.69949 Vogel's harmonic complexity : 24.33333 Wille's k value : 1633 Wilson's harmonic complexity : 41 Rectangular lattice diameter : 7 Triangular lattice diameter : 5 Lattice compactness : 27.10233, average 4.51705 Lattice compactness (without 2's) : 14.03548, average 2.33925 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -1 .. 6 2.33333 9 3: -3 .. 0 -1.33333 4 1 5: -1 .. 1 0.00000 2 11: -1 .. 1 0.00000 2 2 Average exponent except 2's :-4 / 3 =-1.33333 Average absolute exponent except 2's: 8 / 3 = 2.66667 Scale is weakly epimorphic with val: <7 10 15 24| Scale is weakly epimorphic with val: <7 10 15 25| Scale is weakly epimorphic with val: <7 10 16 25| Scale is weakly epimorphic with val: <7 10 16 26| Scale is weakly epimorphic with val: <7 10 17 26| Scale is weakly epimorphic with val: <7 10 17 27| Scale is weakly epimorphic with val: <7 10 18 27| Scale is weakly epimorphic with val: <8 11 15 26| Scale is weakly epimorphic with val: <8 11 15 27| Scale is weakly epimorphic with val: <8 11 16 27|
FIT/MODE
3: 1 0 2 N G T3 SD: 21.3031 c. M:-36.8842 c. 5: 1 1 3 N M DE S SD: 26.0903 c. M:-34.2138 c. 6: 2 0 4 N G T3 SD: 21.3031 c. M:-36.8842 c. 7: 2 0 5 N G T3 SD: 15.1541 c. M: 22.7057 c. 8: 2 1 5 N T3 SD: 12.8446 c. M:-21.7626 c. 9: 2 1 6 N T3 SD: 10.6705 c. M: 18.4541 c. 11: 3 1 7 N T3 SD: 10.0697 c. M:-16.1030 c. 12: 3 1 8 N T3 SD: 2.7302 c. M: 4.6195 c. 15: 4 1 10 N T3 SD: 2.2040 c. M:-3.6812 c. 27: 7 2 18 N T3 SD: 0.5836 c. M:-1.0108 c. 85: 22 6 57 N T3 SD: 0.5594 c. M: 0.9423 c. 97: 25 7 65 N T3 SD: 0.6005 c. M: 0.7687 c. 100: 26 7 67 N T3 SD: 0.4289 c. M: 0.6494 c. 112: 29 8 75 N T3 SD: 0.2899 c. M: 0.4715 c. 127: 33 9 85 N T3 SD: 0.2364 c. M: 0.2964 c. 139: 36 10 93 N T3 SD: 0.1335 c. M: 0.1836 c. 166: 43 12 111 N T3 SD: 0.0690 c. M: 0.1191 c. 320: 83 23 214 N T3 SD: 0.0326 c. M:-0.0497 c. 486: 126 35 325 N T3 SD: 0.0093 c. M: 0.0140 c. 999: 259 72 668 N T3 SD: 0.0091 c. M:-0.0137 c. 1319: 342 95 882 N T3 SD: 0.0041 c. M:-0.0060 c. 1805: 468 130 1207 N T3 SD: 0.0014 c. M:-0.0022 c.
FIT/HARMONIC
1 x x 2 S SD: 701.9550 cents 2 x x 3 S SD: 203.9100 cents 3 x x 4 S SD: 0.0000 cents 4 x x 5 S SD: 111.7313 cents 5 x 6:7 S SD: 86.3514 cents 6 x 7:8 S SD: 50.9333 cents 7 x 8:9 S SD: 45.6687 cents 8 x 9:11 SD: 32.9836 cents 9 x 10:12 S SD: 8.6997 cents 10 x 11:13 SD: 21.9155 cents 11 x 12:15 S SD: 20.7369 cents 12:13 x 16 SD: 4.7209 cents 13:14 x 17 SD: 16.8138 cents 14:15 x 19 SD: 16.0686 cents 17:18:19:23 SD: 16.0158 cents 18:19:20:24 S SD: 13.1865 cents 20 x 22:27 SD: 10.7531 cents 21 x 23:28 SD: 3.7554 cents 22 x 24:29 SD: 12.2259 cents 23:25 x 31 SD: 12.0625 cents 24:26 x 32 SD: 4.7209 cents 33 x 36:44 SD: 7.1836 cents 34:37 x 45 SD: 10.7364 cents 35:38:39:47 SD: 9.5814 cents 36:39:40:48 S SD: 6.5987 cents 37:40:41:49 SD: 6.0874 cents 38:41:42:51 SD: 4.7494 cents 39:42:43:52 SD: 1.3720 cents 51:55:56:68 SD: 1.1581 cents 90:97:99:120 SD: 0.1803 cents 489:527:538:652 SD: 0.1794 cents 531:572:584:708 SD: 0.1572 cents 570:614:627:760 SD: 0.1329 cents