Scala analysis: 458_D15
Diatonic tetrachord 13/12 * 9/8 * 128/117, Avicenna
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 13/12 138.572661 tridecimal 2/3-tone 2: 39/32 342.482663 39th harmonic, Zalzal wosta of Ibn Sina 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 128/117 155.5623 1: 13/12 138.5727 tridecimal 2/3-tone 2: 9/8 203.9100 major whole tone 3: 128/117 155.5623
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 13/12 138.573 cents tridecimal 2/3-tone 1: 1 128/117 155.562 cents 1: 1 9/8 203.910 cents major whole tone 2: 1 32/27 294.135 cents Pythagorean minor third 2: 1 39/32 342.483 cents 39th harmonic, Zalzal wosta of Ibn Sina 2: 1 16/13 359.472 cents tridecimal neutral 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 : 138.6 342.5 498.0 138.6: 203.9 359.5 498.0 342.5: 155.6 294.1 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 138.5727 .. 203.9100 cents 27/26, 65.3373 cents 13/12, 138.5727 cents 2: 294.1350 .. 359.4723 cents 27/26, 65.3373 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 13/12, 138.5727 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.198036 Largest / Average step : 1.228263 Largest / Smallest step : 1.471502 Median interval of one step : 128/117, 155.5623 cents, amount: 1 Least squares average step : 165.54807 cents, oct.: 496.64421 cents Scale is strictly proper Scale has trivalence property Scale is a mode of a 1319-tone equal temperament with octave 4/3 degrees: 367 907 1319 Least number of segments generator : 660 of 249.211 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 682 of 257.518 cents and inv. generated superset size: 25 = 22 more = 833.333% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : SLM Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 90.22500 cents Scale diversity : 1.355785 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.737625 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 : 512/507, 16.9897 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 Best fifths form a closed circle Best major thirds form a closed circle Best minor thirds form a closed circle 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 n-1 differentially coherent in interval class 1 -- Rational properties -- Prime limit : 13 Odd number limit : 117 (O: 39 U: 117) Highest odd numerator or denominator: 39 Scale harmonicity : 0.028996 Average absolute harmonicity : 0.095034 Specific harmonicity : 0.122810 Fundamental : 1/96, -6.5850 octaves, 2.7253 Hz. Guide tone : 156, 7.2854 octaves, 40813.588 Hz. Exponens Consonantiae : 1.497600E+04, 13.87036 octaves Euler's gradus suavitatis : 24 Sum of Mann's harmonic distance : 51.5, average 17.16667 Mersenne's string divisions : 12425 Sum of van Prooijen's expressibility: 3.18213, average 1.06071 Sum of Tenney's harmonic distance : 6.36852, average 2.12284 Vogel's harmonic complexity : 14.00000 Wille's k value : 58 Wilson's harmonic complexity : 19 Rectangular lattice diameter : 3 Triangular lattice diameter : 3 Lattice compactness : 24.95884, average 4.15981 Lattice compactness (without 2's) : 9.06450, average 1.51075 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -5 .. 2 -1.66667 9 3: -1 .. 1 -0.33333 3 1 13: 0 .. 1 0.66667 2 2 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: <7 10 27| Scale is JI-epimorphic with val: <7 11 26| Scale is weakly epimorphic with val: <7 11 29| Scale is weakly epimorphic with val: <8 11 30|
FIT/MODE
3: 1 1 1 SP B ME I SD: 16.9543 c. M:-27.4423 c. 7: 2 3 2 SP M ME I SD: 7.9542 c. M:-13.2638 c. 10: 3 4 3 SP M ME I SD: 7.1956 c. M:-10.8408 c. 15: 4 6 5 SP T3 SD: 6.8907 c. M: 10.4527 c. 17: 5 7 5 SP M DE I SD: 6.9524 c. M:-9.0785 c. 18: 5 7 6 SP T3 SD: 6.0363 c. M: 10.4527 c. 19: 5 8 6 SP T3 SD: 4.4465 c. M: 7.5082 c. 22: 6 9 7 SP T3 SD: 2.3071 c. M: 2.9065 c. 25: 7 10 8 SP T3 SD: 2.2588 c. M: 3.8121 c. 29: 8 12 9 SP T3 SD: 0.8922 c. M: 1.1809 c. 32: 9 13 10 SP T3 SD: 0.8686 c. M:-1.5025 c. 54: 15 22 17 SP T3 SD: 0.7219 c. M: 1.2296 c. 61: 17 25 19 SP T3 SD: 0.2825 c. M:-0.4336 c. 115: 32 47 36 SP T3 SD: 0.2007 c. M: 0.3474 c. 144: 40 59 45 SP T3 SD: 0.1382 c. M: 0.2268 c. 176: 49 72 55 SP T3 SD: 0.0672 c. M:-0.0876 c. 237: 66 97 74 SP T3 SD: 0.0779 c. M:-0.1234 c. 320: 89 131 100 SP T3 SD: 0.0541 c. M: 0.0767 c. 381: 106 156 119 SP T3 SD: 0.0059 c. M: 0.0090 c. 1319: 367 540 412 SP T3 SD: 0.0041 c. M: 0.0059 c. 1700: 473 696 531 SP T3 SD: 0.0021 c. M: 0.0035 c. 2081: 579 852 650 SP T3 SD: 0.0012 c. M: 0.0019 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: 43.8311 cents 5 x 6:7 S SD: 44.3147 cents 6 x 7:8 S SD: 37.8059 cents 7:8 x 9 S SD: 55.9891 cents 8:9:10:11 H SD: 31.6722 cents 9:10:11:12 H SD: 14.7023 cents 12:13:15:16 SD: 14.6104 cents 13:14:16:17 S SD: 13.0142 cents 14:15:17:19 SD: 12.2259 cents 21:23:26:28 SD: 11.0621 cents 22:24:27:29 SD: 8.7087 cents 23:25:28:31 SD: 6.5612 cents 24:26:29:32 SD: 4.9535 cents 36:39:44:48 SD: 1.6418 cents 60:65:73:80 SD: 0.9873 cents 96:104:117:128 SD: 0.0000 cents