Scala analysis: 110_E13
Enharmonic tetrachord 21/20 * 64/63 * 5/4, Pachymeres
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 21/20 84.467193 minor semitone 2: 16/15 111.731285 minor diatonic semitone 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 5/4 386.3137 major third 1: 21/20 84.4672 minor semitone 2: 64/63 27.2641 septimal comma, Archytas' comma 3: 5/4 386.3137 major third
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 64/63 27.264 cents septimal comma, Archytas' comma 1: 1 21/20 84.467 cents minor semitone 1: 1 5/4 386.314 cents major third 2: 1 16/15 111.731 cents minor diatonic semitone 2: 1 80/63 413.578 cents wide major third 2: 1 21/16 470.781 cents narrow 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 : 84.5 111.7 498.0 84.5 : 27.3 413.6 498.0 111.7: 386.3 470.8 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 27.2641 .. 386.3137 cents 315/256, 359.0496 cents 64/63, 27.2641 cents 2: 111.7313 .. 470.7809 cents 315/256, 359.0496 cents 64/75,-274.5824 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 64/63, 27.2641 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 5/4, 386.3137 cents Average / Smallest step : 6.089145 Largest / Average step : 2.326981 Largest / Smallest step : 14.169323 Median interval of one step : 21/20, 84.4672 cents, amount: 1 Least squares average step : 128.71891 cents, oct.: 386.15673 cents Scale is not proper Scale has trivalence property Scale is a mode of a 1881-tone equal temperament with octave 4/3 degrees: 319 422 1881 Least number of segments generator : 941 of 249.155 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1439 of 381.014 cents and inv. generated superset size: 36 = 33 more = 1200.000% 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 27.26409 cents Scale diversity : 0.434450 Lumma stability : 0.109484 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 : 64/63, 27.2641 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 -- Rational properties -- Prime limit : 7 Odd number limit : 63 (O: 21 U: 63) Highest odd numerator or denominator: 21 Scale harmonicity : 0.035690 Average absolute harmonicity : 0.112549 Specific harmonicity : 0.168201 Fundamental : 1/60, -5.9069 octaves, 4.3604 Hz. Guide tone : 336, 8.3923 octaves, 87906.190 Hz. Exponens Consonantiae : 2.016000E+04, 14.29921 octaves Euler's gradus suavitatis : 21 Sum of Mann's harmonic distance : 39.5, average 13.16667 Mersenne's string divisions : 8897 Sum of van Prooijen's expressibility: 2.97543, average 0.99181 Sum of Tenney's harmonic distance : 6.08264, average 2.02755 Vogel's harmonic complexity : 9.33333 Wille's k value : 31 Wilson's harmonic complexity : 18 Rectangular lattice diameter : 4 Triangular lattice diameter : 3 Lattice compactness : 22.45407, average 3.74234 Lattice compactness (without 2's) : 9.83182, average 1.63864 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -2 .. 4 1.33333 8 3: -1 .. 1 -0.33333 3 1 5: -1 .. 0 -0.66667 2 1 7: 0 .. 1 0.33333 1 1 Average exponent except 2's :-2 / 3 =-0.66667 Average absolute exponent except 2's: 6 / 3 = 2.00000 Scale is weakly epimorphic with val: <6 10 16 21| Scale is weakly epimorphic with val: <6 11 17 21| Scale is weakly epimorphic with val: <7 10 15 18| Scale is weakly epimorphic with val: <7 10 15 21| Scale is JI-epimorphic with val: <7 11 15 19| Scale is weakly epimorphic with val: <7 11 15 22| 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 19| Scale is weakly epimorphic with val: <7 11 18 22| Scale is weakly epimorphic with val: <7 11 19 21| Scale is weakly epimorphic with val: <7 12 15 20| Scale is weakly epimorphic with val: <8 11 18 18| Scale is weakly epimorphic with val: <8 11 18 21| Scale is weakly epimorphic with val: <8 12 15 22| Scale is weakly epimorphic with val: <8 12 17 20|
FIT/MODE
3: 0 0 3 N M DE S SD: 80.8674 c. M: 111.7313 c. 4: 1 0 3 N G T3 SD: 24.2683 c. M:-40.0441 c. 5: 1 0 4 N G T3 SD: 11.1986 c. M:-15.1418 c. 12: 2 1 9 N T3 SD: 7.4265 c. M:-12.7800 c. 13: 2 1 10 N T3 SD: 4.8920 c. M: 7.8449 c. 17: 3 1 13 N T3 SD: 3.7186 c. M:-5.4558 c. 18: 3 1 14 N T3 SD: 1.0397 c. M: 1.4597 c. 53: 9 3 41 N T3 SD: 0.5999 c. M:-1.0336 c. 71: 12 4 55 N T3 SD: 0.3360 c. M:-0.5042 c. 89: 15 5 69 N T3 SD: 0.3232 c. M: 0.5270 c. 94: 16 5 73 N T3 SD: 0.3220 c. M: 0.4659 c. 112: 19 6 87 N T3 SD: 0.3239 c. M: 0.5605 c. 129: 22 7 100 N T3 SD: 0.3031 c. M:-0.4707 c. 147: 25 8 114 N T3 SD: 0.1420 c. M:-0.2343 c. 165: 28 9 128 N T3 SD: 0.0400 c. M:-0.0495 c. 401: 68 22 311 N T3 SD: 0.0292 c. M:-0.0494 c. 566: 96 31 439 N T3 SD: 0.0127 c. M:-0.0209 c. 731: 124 40 567 N T3 SD: 0.0100 c. M:-0.0165 c. 1150: 195 63 892 N T3 SD: 0.0096 c. M: 0.0161 c. 1297: 220 71 1006 N T3 SD: 0.0099 c. M:-0.0123 c. 1315: 223 72 1020 N T3 SD: 0.0048 c. M: 0.0079 c. 1480: 251 81 1148 N T3 SD: 0.0045 c. M: 0.0077 c. 1881: 319 103 1459 N T3 SD: 0.0033 c. M:-0.0045 c. 2046: 347 112 1587 N T3 SD: 0.0005 c. M:-0.0008 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 x 7 SD: 84.4672 cents 6 x 7:8 S SD: 77.5698 cents 7 x 8:9 S SD: 67.5105 cents 8 x 9:11 SD: 53.2328 cents 9 x 10:12 S SD: 35.3362 cents 10 x 11:13 SD: 34.4934 cents 11 x 12:15 S SD: 27.5105 cents 12 x 13:16 SD: 13.4207 cents 13 x 14:17 SD: 18.7389 cents 14 x 15:19 SD: 15.7988 cents 15 x 16:20 S SD: 0.0000 cents 16 x 17:21 SD: 14.0467 cents 17 x 18:23 SD: 14.1599 cents 18:19 x 24 SD: 4.5679 cents 19:20 x 25 S SD: 11.6682 cents 20:21 x 27 SD: 10.7531 cents 21:22 x 28 SD: 1.9651 cents 22:23 x 29 SD: 10.5817 cents 24:25:26:32 SD: 10.0596 cents 27:28:29:36 SD: 8.2061 cents 30 x 32:40 S SD: 0.0000 cents 31 x 33:41 SD: 7.2235 cents 32 x 34:43 SD: 7.5403 cents 33 x 35:44 SD: 4.9323 cents 34 x 36:45 S SD: 9.0345 cents 35:37 x 47 SD: 8.5087 cents 36:38 x 48 SD: 4.5679 cents 37:39 x 49 SD: 6.7504 cents 39:41:42:52 SD: 5.5670 cents 41:43:44:55 SD: 5.0063 cents 42:44:45:56 SD: 2.8851 cents 50 x 53:67 SD: 6.9349 cents 51 x 54:68 SD: 6.3883 cents 52:55 x 69 SD: 7.5713 cents 53:56:57:71 SD: 6.5552 cents 54:57:58:72 SD: 5.0222 cents 55:58:59:73 SD: 4.8802 cents 56:59:60:75 SD: 4.1297 cents 57:60:61:76 SD: 2.3829 cents 60:63:64:80 S SD: 0.0000 cents