Scala analysis: 454_D15
Diatonic tetrachord 256/243 * 9/8 * 9/8, Pythagoras?
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 256/243 90.224996 limma, Pythagorean minor second 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: 256/243 90.2250 limma, Pythagorean minor second 2: 9/8 203.9100 major whole tone 3: 9/8 203.9100 major whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 256/243 90.225 cents limma, Pythagorean minor second 1: 2 9/8 203.910 cents major whole tone 2: 2 32/27 294.135 cents Pythagorean minor third 2: 1 81/64 407.820 cents Pythagorean major third Highest number of different intervals for one interval class: 2 Average number of different intervals per interval class: 2.00000 = 2
SHOW/LINE/CENTS INTERVALS
1 2 3 0.0 : 90.2 294.1 498.0 90.2 : 203.9 407.8 498.0 294.1: 203.9 294.1 498.0 498.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 203.9100 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents 2: 294.1350 .. 407.8200 cents 2187/2048, 113.6850 cents 256/243, 90.2250 cents
SHOW DATA
Number of notes : 3 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 166.0150 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.840011 Largest / Average step : 1.228263 Largest / Smallest step : 2.260017 Median interval of one step : 9/8, 203.9100 cents, amount: 2 Most common interval of one step : 9/8, 203.9100 cents, amount: 2 Least squares average step : 155.18786 cents, oct.: 465.56357 cents Scale is strictly proper Scale has Myhill's property chromas: 2187/2048, 113.6850 cents and 531441/524288, 23.4600 cents, ratio: 4.845906 generators: 203.91000 cents and 2 of 294.13500 cents ET pair of linear temperament : 34&44 Scale is distributional even Scale is a mode of a 276-tone equal temperament with octave 4/3 degrees: 50 163 276 generators: 113 and 163 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABB Step pattern size order : SLL Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 90.22500 cents Scale diversity : 1.001178 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.543475 Rothenberg efficiency : 0.888889 redundancy: 0.111111 Efficiency x scale size : 2.666667 Number of different interval sizes : 4 = 2.00000 / class Number of one step interval sizes : 2 Highest interval variety : 2 Mean interval variety : 2.00000 = 2 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 256/243, 90.2250 cents Most common intervals : 9/8, 203.9100 cents & inv., amount: 2 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 increasing 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 : 1 Inversional symmetry on degrees : 2 Inversional symmetry on intervals : 0-1 -- Rational properties -- Prime limit : 3 Odd number limit : 243 (O: 81 U: 243) Highest odd numerator or denominator: 243 Scale harmonicity : 0.046875 Average absolute harmonicity : 0.112694 Specific harmonicity : 0.221198 Fundamental : 1/243, -7.9248 octaves, 1.0766 Hz. Guide tone : 256, 8.0000 octaves, 66976.145 Hz. Exponens Consonantiae : 6.220800E+04, 15.92481 octaves Euler's gradus suavitatis : 19 Sum of Mann's harmonic distance : 282.5, average 94.16667 Mersenne's string divisions : 206087 Sum of van Prooijen's expressibility: 4.29409, average 1.43136 Sum of Tenney's harmonic distance : 8.80954, average 2.93651 Vogel's harmonic complexity : 11.00000 Wille's k value : 40 Wilson's harmonic complexity : 15 Rectangular lattice diameter : 5 Triangular lattice diameter : 4 Lattice compactness : 31.92321, average 5.32053 Lattice compactness (without 2's) : 17.00000, average 2.83333 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: 2 .. 8 5.00000 15 3: -5 ..-1 -3.00000 9 3 Average exponent except 2's :-9 / 3 =-3.00000 Average absolute exponent except 2's: 9 / 3 = 3.00000 Scale is JI-epimorphic with val: <7 11| = patent
FIT/MODE
3: 1 1 1 SP B ME I SD: 48.9222 c. M:-75.7900 c. 4: 1 1 2 P M ME S SD: 32.7143 c. M: 45.1125 c. 5: 1 2 2 SP M ME S SD: 6.0573 c. M:-9.3840 c. 12: 2 5 5 SP M DE S SD: 4.6589 c. M: 7.2175 c. 17: 3 7 7 SP M DE S SD: 1.5070 c. M: 2.3347 c. 22: 4 9 9 SP M DE S SD: 0.2121 c. M:-0.3286 c. 83: 15 34 34 SP M DE S SD: 0.1400 c. M: 0.2169 c. 105: 19 43 43 SP M DE S SD: 0.0662 c. M: 0.1026 c. 127: 23 52 52 SP M DE S SD: 0.0180 c. M: 0.0279 c. 149: 27 61 61 SP M DE S SD: 0.0160 c. M:-0.0248 c. 276: 50 113 113 SP M DE S SD: 0.0004 c. M:-0.0005 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: 35.4885 cents 10:11:12:13 H SD: 29.7688 cents 11:12:13:15 SD: 24.0082 cents 12:13:14:16 S SD: 18.5017 cents 15:16:18:20 S SD: 10.1382 cents 17:18:20:23 SD: 9.8783 cents 18:19:21:24 SD: 9.1575 cents 21:22:25:28 SD: 4.1275 cents 38:40:45:51 SD: 3.8432 cents 39:41:46:52 SD: 3.0350 cents 54:57:64:72 SD: 1.1260 cents 75:79:89:100 SD: 0.7265 cents 114:120:135:152 SD: 0.6714 cents 129:136:153:172 SD: 0.5929 cents 168:177:199:224 SD: 0.3246 cents 189:199:224:252 SD: 0.3221 cents 243:256:288:324 SD: 0.0000 cents