Scala analysis: xen18-ayers-table-63
Didymos' Chromatic Tetrachord
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 6/5 315.641287 minor third 2: 5/4 386.313714 major third 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 6/5 315.6413 minor third 2: 25/24 70.6724 classic chromatic semitone, minor chroma 3: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 1 16/15 111.731 cents minor diatonic semitone 1: 1 6/5 315.641 cents minor third 2: 1 10/9 182.404 cents minor whole tone 2: 1 5/4 386.314 cents major third 2: 1 32/25 427.373 cents classic diminished fourth 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 : 315.6 386.3 498.0 315.6: 70.7 182.4 498.0 386.3: 111.7 427.4 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 70.6724 .. 315.6413 cents 144/125, 244.9689 cents 25/24, 70.6724 cents 2: 182.4037 .. 427.3726 cents 144/125, 244.9689 cents 25/27,-133.2376 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 25/24, 70.6724 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 6/5, 315.6413 cents Average / Smallest step : 2.349077 Largest / Average step : 1.901282 Largest / Smallest step : 4.466258 Median interval of one step : 16/15, 111.7313 cents, amount: 1 Least squares average step : 184.45741 cents, oct.: 553.37222 cents Scale is not proper Scale has trivalence property Scale is a mode of a 1783-tone equal temperament with octave 4/3 degrees: 1130 1383 1783 Least number of segments generator : 892 of 249.162 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1477 of 412.570 cents and inv. generated superset size: 28 = 25 more = 933.333% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : LSM Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 70.67243 cents Scale diversity : 0.976978 Lumma stability : 0.283799 Lumma impropriety factor : 0.267521 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 : 128/125, 41.0589 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 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 : 1 = 33.3333% 2/1 octave complements present : 0 = 0.0000% Scale is differentially coherent in interval class 1 Scale is n-1 differentially coherent in interval class 2 -- Rational properties -- Prime limit : 5 Odd number limit : 25 (O: 25 U: 25) Highest odd numerator or denominator: 5 Scale harmonicity : 0.045181 Average absolute harmonicity : 0.144223 Specific harmonicity : 0.237388 Fundamental : 1/60, -5.9069 octaves, 4.3604 Hz. Guide tone : 60, 5.9069 octaves, 15697.534 Hz. Exponens Consonantiae : 3.600000E+03, 11.81378 octaves Euler's gradus suavitatis : 17 Sum of Mann's harmonic distance : 13.5, average 4.50000 Mersenne's string divisions : 693 Sum of van Prooijen's expressibility: 1.87506, average 0.62502 Sum of Tenney's harmonic distance : 3.85733, average 1.28578 Vogel's harmonic complexity : 10.33333 Wille's k value : 37 Wilson's harmonic complexity : 16 Rectangular lattice diameter : 4 Triangular lattice diameter : 2 Lattice compactness : 16.63797, average 2.77300 Lattice compactness (without 2's) : 9.30056, average 1.55009 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -2 .. 2 0.33333 5 3: -1 .. 1 0.00000 2 1 5: -1 .. 1 0.00000 2 2 Average exponent except 2's : 0 / 3 = 0.00000 Average absolute exponent except 2's: 4 / 3 = 1.33333 Scale is weakly epimorphic with val: <6 10 15| Scale is weakly epimorphic with val: <6 11 15|
FIT/MODE
3: 2 0 1 N G T3 SD: 32.7379 c. M: 54.2837 c. 4: 2 1 1 P M ME S SD: 39.1637 c. M: 66.6188 c. 5: 3 1 1 N M DE S SD: 11.9676 c. M: 16.8143 c. 8: 5 1 2 N T3 SD: 7.7967 c. M: 12.7800 c. 13: 8 2 3 N T3 SD: 5.5980 c. M: 9.1521 c. 14: 9 2 3 N T3 SD: 3.8987 c. M:-5.0074 c. 22: 14 3 5 N T3 SD: 1.1276 c. M: 1.4608 c. 49: 31 7 11 N T3 SD: 0.3214 c. M: 0.5516 c. 71: 45 10 16 N T3 SD: 0.2914 c. M: 0.5042 c. 120: 76 17 27 N T3 SD: 0.2261 c. M: 0.3288 c. 134: 85 19 30 N T3 SD: 0.2100 c. M:-0.2828 c. 183: 116 26 41 N T3 SD: 0.0918 c. M:-0.1474 c. 232: 147 33 52 N T3 SD: 0.0706 c. M:-0.1005 c. 254: 161 36 57 N T3 SD: 0.0345 c. M:-0.0487 c. 303: 192 43 68 N T3 SD: 0.0367 c. M: 0.0484 c. 486: 308 69 109 N T3 SD: 0.0178 c. M:-0.0298 c. 740: 469 105 166 N T3 SD: 0.0080 c. M:-0.0116 c. 1043: 661 148 234 N T3 SD: 0.0051 c. M: 0.0065 c. 1529: 969 217 343 N T3 SD: 0.0047 c. M: 0.0065 c. 1783: 1130 253 400 N T3 SD: 0.0009 c. M:-0.0014 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 5 x S SD: 0.0000 cents 5:6 x 7 S SD: 42.2336 cents 6:7 x 8 S SD: 24.3852 cents 7:8:9 x S SD: 48.7680 cents 8 x 10:11 S SD: 26.6365 cents 9:11 x 12 SD: 15.8833 cents 10:12 x 13 S SD: 21.9155 cents 11:13:14:15 SD: 18.8130 cents 12:14:15:16 S SD: 16.2568 cents 13 x 16:17 SD: 21.5092 cents 14:17:18:19 SD: 20.3777 cents 15:18:19:20 S SD: 7.6435 cents 20:24:25:27 SD: 7.1688 cents 24:29:30:32 SD: 3.9936 cents 36:43:45:48 SD: 2.6779 cents 45:54:56:60 S SD: 2.5705 cents 60:72:75:80 S SD: 0.0000 cents