Scala analysis: xen18-ayers-table-64
Archytas' Enharmonic Tetrachord
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 5/4 386.313714 major third 2: 9/7 435.084095 septimal major third, BP third 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 28/27 62.9609 Archytas' 1/3-tone 1: 5/4 386.3137 major third 2: 36/35 48.7704 septimal diesis, 1/4-tone 3: 28/27 62.9609 Archytas' 1/3-tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 36/35 48.770 cents septimal diesis, 1/4-tone 1: 1 28/27 62.961 cents Archytas' 1/3-tone 1: 1 5/4 386.314 cents major third 2: 1 16/15 111.731 cents minor diatonic semitone 2: 1 9/7 435.084 cents septimal major third, BP third 2: 1 35/27 449.275 cents septimal semi-diminished fourth, 9/4-tone 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 : 386.3 435.1 498.0 386.3: 48.8 111.7 498.0 435.1: 63.0 449.3 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 48.7704 .. 386.3137 cents 175/144, 337.5433 cents 36/35, 48.7704 cents 2: 111.7313 .. 449.2746 cents 175/144, 337.5433 cents 64/75,-274.5824 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 36/35, 48.7704 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 5/4, 386.3137 cents Average / Smallest step : 3.404013 Largest / Average step : 2.326981 Largest / Smallest step : 7.921072 Median interval of one step : 28/27, 62.9609 cents, amount: 1 Least squares average step : 196.47264 cents, oct.: 589.41791 cents Scale is not proper Scale has trivalence property Scale is a mode of a 1052-tone equal temperament with octave 4/3 degrees: 816 919 1052 Least number of segments generator : 527 of 249.496 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 585 of 276.955 cents and inv. generated superset size: 30 = 27 more = 1000.000% 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 48.77038 cents Scale diversity : 0.509188 Lumma stability : 0.195847 Lumma impropriety factor : 0.551321 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 : 245/243, 14.1905 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 Limited inverse transpositions with margin 337.5433 cents: 1 -- Rational properties -- Prime limit : 7 Odd number limit : 35 (O: 35 U: 35) Highest odd numerator or denominator: 9 Scale harmonicity : 0.034861 Average absolute harmonicity : 0.132452 Specific harmonicity : 0.185923 Fundamental : 1/84, -6.3923 octaves, 3.1146 Hz. Guide tone : 180, 7.4919 octaves, 47092.602 Hz. Exponens Consonantiae : 1.512000E+04, 13.88417 octaves Euler's gradus suavitatis : 21 Sum of Mann's harmonic distance : 16.0, average 5.33333 Mersenne's string divisions : 1008 Sum of van Prooijen's expressibility: 2.13033, average 0.71011 Sum of Tenney's harmonic distance : 4.17955, average 1.39318 Vogel's harmonic complexity : 12.33333 Wille's k value : 52 Wilson's harmonic complexity : 21 Rectangular lattice diameter : 5 Triangular lattice diameter : 3 Lattice compactness : 17.85478, average 2.97580 Lattice compactness (without 2's) : 11.26205, average 1.87701 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -2 .. 2 0.00000 4 3: -1 .. 2 0.33333 3 1 5: 0 .. 1 0.33333 1 1 7: -1 .. 0 -0.33333 1 1 Average exponent except 2's : 1 / 3 = 0.33333 Average absolute exponent except 2's: 5 / 3 = 1.66667 Scale is weakly epimorphic with val: <6 10 15 19| Scale is weakly epimorphic with val: <6 10 15 22| Scale is weakly epimorphic with val: <6 11 15 20| Scale is weakly epimorphic with val: <6 11 17 19| Scale is weakly epimorphic with val: <7 10 17 18| Scale is weakly epimorphic with val: <7 10 17 21| Scale is JI-epimorphic with val: <7 11 15 20| Scale is weakly epimorphic with val: <7 11 16 18| Scale is weakly epimorphic with val: <7 11 16 21| Scale is weakly epimorphic with val: <7 11 18 20| Scale is weakly epimorphic with val: <7 11 19 21| Scale is weakly epimorphic with val: <7 12 15 21| Scale is weakly epimorphic with val: <7 12 17 20| Scale is weakly epimorphic with val: <7 12 18 21| Scale is weakly epimorphic with val: <8 11 17 19| Scale is weakly epimorphic with val: <8 11 19 21| Scale is weakly epimorphic with val: <8 12 15 21| Scale is weakly epimorphic with val: <8 12 18 21| Scale is weakly epimorphic with val: <8 12 19 19| Scale is weakly epimorphic with val: <8 12 19 22|
FIT/MODE
3: 2 0 1 N G T3 SD: 67.2480 c. M: 103.0541 c. 4: 3 0 1 N G T3 SD: 36.2940 c. M: 61.5503 c. 5: 4 0 1 N G T3 SD: 22.2863 c. M: 36.6481 c. 6: 5 0 1 N G T3 SD: 20.2231 c. M:-28.7238 c. 7: 5 1 1 N M DE S SD: 18.2703 c. M: 30.5673 c. 8: 6 1 1 N M DE S SD: 7.3897 c. M: 12.7800 c. 9: 7 1 1 N M DE S SD: 4.4428 c. M:-7.6226 c. 17: 13 2 2 N M DE S SD: 4.0348 c. M: 5.4558 c. 22: 17 2 3 N T3 SD: 2.9821 c. M: 4.9543 c. 23: 18 2 3 N T3 SD: 2.3081 c. M:-3.4606 c. 31: 24 3 4 N T3 SD: 0.8624 c. M: 1.3030 c. 40: 31 4 5 N T3 SD: 0.4493 c. M:-0.7053 c. 71: 55 7 9 N T3 SD: 0.3075 c. M: 0.5042 c. 103: 80 10 13 N T3 SD: 0.3043 c. M:-0.5174 c. 111: 86 11 14 N T3 SD: 0.2679 c. M: 0.4410 c. 134: 104 13 17 N T3 SD: 0.1848 c. M:-0.2287 c. 174: 135 17 22 N T3 SD: 0.0583 c. M:-0.1005 c. 245: 190 24 31 N T3 SD: 0.0543 c. M: 0.0747 c. 388: 301 38 49 N T3 SD: 0.0490 c. M:-0.0635 c. 419: 325 41 53 N T3 SD: 0.0218 c. M: 0.0376 c. 459: 356 45 58 N T3 SD: 0.0235 c. M: 0.0304 c. 593: 460 58 75 N T3 SD: 0.0236 c. M: 0.0296 c. 633: 491 62 80 N T3 SD: 0.0102 c. M:-0.0168 c. 878: 681 86 111 N T3 SD: 0.0100 c. M: 0.0169 c. 1052: 816 103 133 N T3 SD: 0.0032 c. M: 0.0049 c. 1685: 1307 165 213 N T3 SD: 0.0029 c. M:-0.0037 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 4:5 x x S SD: 0.0000 cents 5:6 x 7 S SD: 55.0665 cents 6 x x 8 S SD: 0.0000 cents 7 x 9 x SD: 0.0000 cents 8:10 x 11 S SD: 26.6365 cents 9:11 x 12 SD: 19.4529 cents 10 x 13 x SD: 19.1299 cents 11 x 14:15 SD: 21.3458 cents 12:15 x 16 S SD: 0.0000 cents 13:16:17 x SD: 19.8841 cents 14 x 18:19 SD: 15.3211 cents 15:19 x 20 SD: 11.4653 cents 16:20 x 21 S SD: 13.6320 cents 17:21:22:23 SD: 11.4782 cents 18 x 23:24 SD: 5.3599 cents 19:24 x 25 SD: 14.6155 cents 20:25:26:27 S SD: 9.5944 cents 21:26:27:28 SD: 5.5223 cents 22 x 28:29 SD: 13.2325 cents 23:29:30:31 SD: 11.5253 cents 24:30:31:32 S SD: 2.6655 cents 26 x 33:35 SD: 13.9058 cents 27:34:35:36 SD: 6.3650 cents 28:35:36:37 S SD: 5.1756 cents 32:40:41:43 SD: 4.9191 cents 36:45:46:48 S SD: 3.5732 cents 39:49:50:52 SD: 3.3799 cents 45:56:58:60 SD: 2.9382 cents 48:60:62:64 S SD: 2.6655 cents 56:70:72:75 S SD: 2.5705 cents 60:75:77:80 SD: 1.0697 cents 84:105:108:112 S SD: 0.0000 cents