Scala analysis: 457_D15
Diatonic tetrachord 9/8 * 12/11 * 88/81, Avicenna
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 9/8 203.910002 major whole tone 2: 27/22 354.547060 neutral third, Zalzal wosta of al-Farabi 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 88/81 143.4979 2nd undecimal neutral second 1: 9/8 203.9100 major whole tone 2: 12/11 150.6371 undecimal neutral second, 3/4-tone 3: 88/81 143.4979 2nd undecimal neutral second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 88/81 143.498 cents 2nd undecimal neutral second 1: 1 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 1 9/8 203.910 cents major whole tone 2: 1 32/27 294.135 cents Pythagorean minor third 2: 1 11/9 347.408 cents undecimal neutral third 2: 1 27/22 354.547 cents neutral third, Zalzal wosta of al-Farabi 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 : 203.9 354.5 498.0 203.9: 150.6 294.1 498.0 354.5: 143.5 347.4 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 143.4979 .. 203.9100 cents 729/704, 60.4121 cents 88/81, 143.4979 cents 2: 294.1350 .. 354.5471 cents 729/704, 60.4121 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 88/81, 143.4979 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.156916 Largest / Average step : 1.228263 Largest / Smallest step : 1.420996 Median interval of one step : 12/11, 150.6371 cents, amount: 1 Least squares average step : 171.93851 cents, oct.: 515.81553 cents Scale is strictly proper Scale has trivalence property Scale is a mode of a 1954-tone equal temperament with octave 4/3 degrees: 800 1391 1954 Least number of segments generator : 979 of 249.532 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 1847 of 470.772 cents and inv. generated superset size: 45 = 42 more = 1500.000% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : LMS Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 90.22500 cents Scale diversity : 1.348744 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.757403 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 : 243/242, 7.1391 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 Steps are strictly decreasing 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 Limited inverse transpositions with margin 53.2729 cents: 1 -- Rational properties -- Prime limit : 11 Odd number limit : 81 (O: 27 U: 81) Highest odd numerator or denominator: 27 Scale harmonicity : 0.029543 Average absolute harmonicity : 0.123691 Specific harmonicity : 0.148314 Fundamental : 1/264, -8.0444 octaves, 0.9910 Hz. Guide tone : 108, 6.7549 octaves, 28255.561 Hz. Exponens Consonantiae : 2.851200E+04, 14.79928 octaves Euler's gradus suavitatis : 24 Sum of Mann's harmonic distance : 36.5, average 12.16667 Mersenne's string divisions : 833 Sum of van Prooijen's expressibility: 2.86273, average 0.95424 Sum of Tenney's harmonic distance : 5.71030, average 1.90343 Vogel's harmonic complexity : 16.66667 Wille's k value : 148 Wilson's harmonic complexity : 23 Rectangular lattice diameter : 5 Triangular lattice diameter : 4 Lattice compactness : 22.53771, average 3.75628 Lattice compactness (without 2's) : 14.69960, average 2.44993 Number of different primes : 3 Prime exponents' range, average, count, tones@limit: 2: -3 .. 2 -0.66667 6 3: -1 .. 3 1.33333 6 2 11: -1 .. 0 -0.33333 1 1 Average exponent except 2's : 3 / 3 = 1.00000 Average absolute exponent except 2's: 7 / 3 = 2.33333 Scale is JI-epimorphic with val: <7 11 24| Scale is weakly epimorphic with val: <7 11 27| Scale is weakly epimorphic with val: <7 12 25|
FIT/MODE
3: 1 1 1 SP B ME I SD: 25.4496 c. M: 37.8950 c. 7: 3 2 2 SP M ME S SD: 5.5500 c. M:-9.5379 c. 10: 4 3 3 SP M ME S SD: 4.3592 c. M: 5.9156 c. 17: 7 5 5 SP M DE S SD: 1.8510 c. M: 2.9859 c. 39: 16 12 11 SP T3 SD: 1.7622 c. M:-3.0237 c. 41: 17 12 12 SP M DE S SD: 1.9917 c. M:-2.5965 c. 42: 17 13 12 SP T3 SD: 1.5080 c. M: 2.3204 c. 49: 20 15 14 SP T3 SD: 0.7812 c. M:-1.1994 c. 56: 23 17 16 SP T3 SD: 0.7860 c. M:-1.1994 c. 59: 24 18 17 SP T3 SD: 0.7595 c. M: 1.3154 c. 66: 27 20 19 SP T3 SD: 0.1179 c. M: 0.1643 c. 149: 61 45 43 SP T3 SD: 0.1348 c. M: 0.2332 c. 205: 84 62 59 SP T3 SD: 0.1328 c. M:-0.1670 c. 215: 88 65 62 SP T3 SD: 0.0795 c. M: 0.1243 c. 281: 115 85 81 SP T3 SD: 0.0618 c. M: 0.0838 c. 347: 142 105 100 SP T3 SD: 0.0599 c. M: 0.0991 c. 354: 145 107 102 SP T3 SD: 0.0530 c. M:-0.0915 c. 420: 172 127 121 SP T3 SD: 0.0306 c. M:-0.0513 c. 486: 199 147 140 SP T3 SD: 0.0206 c. M:-0.0282 c. 701: 287 212 202 SP T3 SD: 0.0109 c. M: 0.0186 c. 767: 314 232 221 SP T3 SD: 0.0104 c. M: 0.0167 c. 1187: 486 359 342 SP T3 SD: 0.0042 c. M:-0.0073 c. 1253: 513 379 361 SP T3 SD: 0.0041 c. M:-0.0069 c. 1888: 773 571 544 SP T3 SD: 0.0043 c. M: 0.0066 c. 1954: 800 591 563 SP T3 SD: 0.0018 c. M: 0.0022 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: 31.7667 cents 5 x 6:7 S SD: 46.4983 cents 6:7 x 8 S SD: 31.4805 cents 7:8 x 9 S SD: 34.3053 cents 8:9:10:11 H SD: 20.6751 cents 9:10:11:12 H SD: 7.5534 cents 30:34:37:40 SD: 5.1205 cents 39:44:48:52 SD: 2.3218 cents 48:54:59:64 SD: 0.8899 cents 105:118:129:140 SD: 0.8636 cents 153:172:188:204 SD: 0.8143 cents 159:179:195:212 SD: 0.5703 cents 168:189:206:224 SD: 0.5091 cents 207:233:254:276 SD: 0.3264 cents 216:243:265:288 SD: 0.1979 cents 264:297:324:352 SD: 0.0000 cents