Scala analysis: xen15-chalmers-triadic-diamond-81-64-tetrachord
Upper tetrachord 256/243 * 2187/2048 * 32/27 of triadic diamond for M=81/64, D=3/2
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 2: 9/8 203.910002 major whole tone 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 32/27 294.1350 Pythagorean minor third 1: 256/243 90.2250 limma, Pythagorean minor second 2: 2187/2048 113.6850 apotome 3: 32/27 294.1350 Pythagorean minor third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 256/243 90.225 cents limma, Pythagorean minor second 1: 1 2187/2048 113.685 cents apotome 1: 1 32/27 294.135 cents Pythagorean minor third 2: 1 9/8 203.910 cents major whole tone 2: 1 8192/6561 384.360 cents Pythagorean diminished fourth 2: 1 81/64 407.820 cents Pythagorean major third 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 : 90.2 203.9 498.0 90.2 : 113.7 407.8 498.0 203.9: 294.1 384.4 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 294.1350 cents 9/8, 203.9100 cents 256/243, 90.2250 cents 2: 203.9100 .. 407.8200 cents 9/8, 203.9100 cents 243/256,-90.2250 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 32/27, 294.1350 cents Average / Smallest step : 1.840011 Largest / Average step : 1.771737 Largest / Smallest step : 3.260017 Median interval of one step : 2187/2048, 113.6850 cents, amount: 1 Least squares average step : 142.29857 cents, oct.: 426.89571 cents Scale is not proper Scale has trivalence property Scale is a mode of a 276-tone equal temperament with octave 4/3 degrees: 50 113 276 Least number of segments generator : 139 of 250.827 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 163 of 294.135 cents and inv. generated superset size: 4 = 1 more = 133.333% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : SML Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 90.22500 cents Scale diversity : 1.099647 Lumma stability : 0.362317 Lumma impropriety factor : 0.181158 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 : 531441/524288, 23.4600 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 Steps are strictly increasing Formal octave complements present : 1 = 33.3333% 2/1 octave complements present : 0 = 0.0000% Limited inverse transpositions with margin 180.4500 cents: 2 -- Rational properties -- Prime limit : 3 Odd number limit : 6561 (O: 2187 U: 6561) Highest odd numerator or denominator: 243 Scale harmonicity : 0.033708 Average absolute harmonicity : 0.127053 Specific harmonicity : 0.170818 Fundamental : 1/1944, -10.9248 octaves, 0.1346 Hz. Guide tone : 2304, 11.1699 octaves, 602785.302 Hz. Exponens Consonantiae : 4.478976E+06, 22.09474 octaves Euler's gradus suavitatis : 26 Sum of Mann's harmonic distance : 261.5, average 87.16667 Mersenne's string divisions : 59381 Sum of van Prooijen's expressibility: 3.81697, average 1.27232 Sum of Tenney's harmonic distance : 7.73036, average 2.57679 Vogel's harmonic complexity : 16.33333 Wille's k value : 1093 Wilson's harmonic complexity : 21 Rectangular lattice diameter : 7 Triangular lattice diameter : 7 Lattice compactness : 41.35606, average 6.89268 Lattice compactness (without 2's) : 22.00000, average 3.66667 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -3 .. 8 2.33333 13 3: -5 .. 2 -1.33333 8 3 Average exponent except 2's :-4 / 3 =-1.33333 Average absolute exponent except 2's: 8 / 3 = 2.66667
FIT/MODE
3: 0 1 2 N G T3 SD: 56.4995 c. M: 90.2250 c. 4: 1 1 2 P M ME S SD: 32.7143 c. M:-45.1125 c. 5: 1 1 3 N M DE S SD: 6.0573 c. M:-9.3840 c. 12: 2 3 7 N T3 SD: 4.6589 c. M: 7.2175 c. 17: 3 4 10 N T3 SD: 1.5070 c. M: 2.3347 c. 22: 4 5 13 N T3 SD: 0.2121 c. M:-0.3286 c. 83: 15 19 49 N T3 SD: 0.1400 c. M: 0.2169 c. 105: 19 24 62 N T3 SD: 0.0662 c. M: 0.1026 c. 127: 23 29 75 N T3 SD: 0.0180 c. M: 0.0279 c. 149: 27 34 88 N T3 SD: 0.0160 c. M:-0.0248 c. 276: 50 63 163 N T3 SD: 0.0004 c. M:-0.0005 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: 70.0332 cents 6 x 7:8 S SD: 31.4805 cents 7 x 8:9 S SD: 34.3053 cents 8 x 9:11 SD: 26.6365 cents 9 x 10:12 S SD: 10.7531 cents 10 x 11:13 SD: 29.3037 cents 11 x 12:15 S SD: 32.9836 cents 12:13:14:16 S SD: 26.4608 cents 13:14:15:17 SD: 22.3628 cents 14:15:16:19 SD: 16.7861 cents 15:16:17:20 SD: 8.3384 cents 18:19:20:24 S SD: 7.2567 cents 24:25:27:32 SD: 6.5175 cents 33:35:37:44 SD: 4.3413 cents 39:41:44:52 SD: 2.0425 cents 57:60:64:76 SD: 1.2220 cents 72:76:81:96 SD: 1.1260 cents 96:101:108:128 SD: 0.7754 cents 111:117:125:148 SD: 0.6528 cents 129:136:145:172 SD: 0.6504 cents 168:177:189:224 SD: 0.0403 cents