Scala analysis: 456_D15
Diatonic tetrachord 2187/2048 * 65536/59049 * 9/8, Anonymous
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2187/2048 113.685006 apotome 2: 32/27 294.134997 Pythagorean minor third 3: 4/3 498.044999 perfect fourth
SHOW/INTERVAL
0: 9/8 203.9100 major whole tone 1: 2187/2048 113.6850 apotome 2: 65536/59049 180.4500 Pythagorean diminished third 3: 9/8 203.9100 major whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 2187/2048 113.685 cents apotome 1: 1 65536/59049 180.450 cents Pythagorean diminished third 1: 1 9/8 203.910 cents major whole tone 2: 1 32/27 294.135 cents Pythagorean minor third 2: 1 19683/16384 317.595 cents Pythagorean augmented second 2: 1 8192/6561 384.360 cents Pythagorean 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 : 113.7 294.1 498.0 113.7: 180.4 384.4 498.0 294.1: 203.9 317.6 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 113.6850 .. 203.9100 cents 256/243, 90.2250 cents 2187/2048, 113.6850 cents 2: 294.1350 .. 384.3600 cents 256/243, 90.2250 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 2187/2048, 113.6850 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.460307 Largest / Average step : 1.228263 Largest / Smallest step : 1.793640 Median interval of one step : 65536/59049, 180.4500 cents, amount: 1 Least squares average step : 156.86357 cents, oct.: 470.59071 cents Scale is strictly proper Scale has trivalence property Scale is a mode of a 276-tone equal temperament with octave 4/3 degrees: 63 163 276 Least number of segments generator : 139 of 250.827 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 163 of 294.135 cents and inv. generated superset size: 5 = 2 more = 166.667% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABC Step pattern size order : SML Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 90.22500 cents Scale diversity : 1.252674 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.637683 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 : 531441/524288, 23.4600 cents Number of recognisable fifths : 0 Number of recognisable fourths : 0 Best major thirds form a closed circle Best minor thirds form a closed circle Steps are strictly increasing Formal octave complements present : 1 = 33.3333% 2/1 octave complements present : 0 = 0.0000% Limited inverse transpositions with margin 66.7650 cents: 1 -- Rational properties -- Prime limit : 3 Odd number limit : 59049 (O: 19683 U: 59049) Highest odd numerator or denominator: 2187 Scale harmonicity : 0.023438 Average absolute harmonicity : 0.108305 Specific harmonicity : 0.120603 Fundamental : 1/55296, -15.7549 octaves, 0.0047 Hz. Guide tone : 69984, 16.0947 octaves, 18309603.562 Hz. Exponens Consonantiae : 3.869835E+09, 31.84963 octaves Euler's gradus suavitatis : 37 Sum of Mann's harmonic distance : 2150.5, average 716.83333 Mersenne's string divisions : 249865 Sum of van Prooijen's expressibility: 5.24833, average 1.74944 Sum of Tenney's harmonic distance : 10.66687, average 3.55562 Vogel's harmonic complexity : 21.66667 Wille's k value : 29524 Wilson's harmonic complexity : 30 Rectangular lattice diameter : 10 Triangular lattice diameter : 10 Lattice compactness : 58.84328, average 9.80721 Lattice compactness (without 2's) : 31.00000, average 5.16667 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -11 .. 5 -1.33333 18 3: -3 .. 7 1.00000 11 3 Average exponent except 2's : 3 / 3 = 1.00000 Average absolute exponent except 2's: 11 / 3 = 3.66667
FIT/MODE
3: 1 1 1 SP B ME I SD: 37.3026 c. M:-52.3300 c. 4: 1 1 2 P M ME S SD: 26.7852 c. M: 45.1125 c. 5: 1 2 2 SP M ME S SD: 8.5664 c. M: 14.0760 c. 12: 3 4 5 SP T3 SD: 6.5886 c. M:-10.8262 c. Ute tritonic, Peruvian tritonic 2 13: 3 5 5 SP M DE S SD: 7.1690 c. M:-12.3542 c. 14: 3 5 6 SP T3 SD: 6.8173 c. M: 9.5379 c. 17: 4 6 7 SP T3 SD: 2.1313 c. M:-3.5021 c. 22: 5 8 9 SP T3 SD: 0.3000 c. M: 0.4930 c. 83: 19 30 34 SP T3 SD: 0.1980 c. M:-0.3253 c. 105: 24 38 43 SP T3 SD: 0.0936 c. M:-0.1539 c. 127: 29 46 52 SP T3 SD: 0.0254 c. M:-0.0418 c. 149: 34 54 61 SP T3 SD: 0.0226 c. M: 0.0372 c. 276: 63 100 113 SP 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 5 x S SD: 92.1787 cents 5 x 6:7 S SD: 43.5810 cents 6 x 7:8 S SD: 13.6320 cents 7 x 8:9 S SD: 44.5201 cents 8 x 9:11 SD: 52.3893 cents 9:10:11:12 H SD: 28.9833 cents 10:11:12:13 H SD: 23.6111 cents 11:12:13:15 SD: 17.9610 cents 12:13:14:16 S SD: 12.3050 cents 15:16:18:20 S SD: 7.1983 cents 27:29:32:36 SD: 3.3424 cents 42:45:50:56 SD: 3.2080 cents 48:51:57:64 SD: 3.1201 cents 59:63:70:79 SD: 2.5157 cents 60:64:71:80 SD: 1.1128 cents 75:80:89:100 SD: 0.9715 cents 87:93:103:116 SD: 0.8582 cents 102:109:121:136 SD: 0.6693 cents 135:144:160:180 S SD: 0.6512 cents 162:173:192:216 SD: 0.0163 cents