Scala analysis: xen15-gilson-generalized-pythagorean-7-4-5
Generalized Pythagorean Scale, 7/4 stacked 5=4+1 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2401/2048 275.303626 2: 343/256 506.477719 3: 49/32 737.651813 4: 7/4 968.825906 harmonic seventh 5: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 8/7 231.1741 septimal whole tone 1: 2401/2048 275.3036 2: 8/7 231.1741 septimal whole tone 3: 8/7 231.1741 septimal whole tone 4: 8/7 231.1741 septimal whole tone 5: 8/7 231.1741 septimal whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 8/7 231.174 cents septimal whole tone 1: 1 2401/2048 275.304 cents 2: 3 64/49 462.348 cents 2 septatones or septatonic major third 2: 2 343/256 506.478 cents 3: 2 512/343 693.522 cents 3 septatones or septatonic fifth 3: 3 49/32 737.652 cents 4: 1 4096/2401 924.696 cents 4 septatones or septatonic major sixth 4: 4 7/4 968.826 cents harmonic seventh 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 4 5 0.0 : 275.3 506.5 737.7 968.8 1200.0 275.3 : 231.2 462.3 693.5 924.7 1200.0 506.5 : 231.2 462.3 693.5 968.8 1200.0 737.7 : 231.2 462.3 737.7 968.8 1200.0 968.8 : 231.2 506.5 737.7 968.8 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 231.1741 .. 275.3036 cents 16807/16384, 44.1295 cents 8/7, 231.1741 cents 2: 462.3482 .. 506.4777 cents 16807/16384, 44.1295 cents 131072/117649, 187.0446 cents 3: 693.5223 .. 737.6518 cents 16807/16384, 44.1295 cents 131072/117649, 187.0446 cents 4: 924.6964 .. 968.8259 cents 16807/16384, 44.1295 cents 131072/117649, 187.0446 cents
SHOW DATA
Number of notes : 5 -- Interval properties -- Smallest interval : 8/7, 231.1741 cents, class 1 Average step (divided formal octave): 240.0000 cents Largest one step interval : 2401/2048, 275.3036 cents Average / Smallest step : 1.038179 Largest / Average step : 1.147098 Largest / Smallest step : 1.190893 Median interval of one step : 8/7, 231.1741 cents, amount: 4 Most common interval of one step : 8/7, 231.1741 cents, amount: 4 Least squares average step : 243.20942 cents, oct.: 1216.04710 cents Scale is strictly proper Scale has Myhill's property chromas: 16807/16384, 44.1295 cents and 131072/117649, 187.0446 cents, ratio: 4.238535 generators: 231.17409 cents and 4 of 968.82591 cents ET pair of linear temperament : 26&109 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 571-tone equal temperament with octave 2/1 degrees: 131 241 351 461 571 generators: 110 and 461 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABBBB Step pattern size order : LSSSS Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 187.04456 cents Scale diversity : 1.168393 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.852902 Rothenberg efficiency : 0.800000 redundancy: 0.200000 Efficiency x scale size : 4.000000 Number of different interval sizes : 8 = 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 : 16807/16384, 44.1295 cents Most common intervals : 8/7, 231.1741 cents & inv., amount: 4 Scale is a chain of 2 triads 0.0 506.478 968.826 cents Number of recognisable fifths : 2, average 693.5223 cents Best fifths form a closed circle Best minor thirds form a closed circle Scale is a complete genus musicum : 2^m 7^4 Steps are decreasing Formal octave complements present : 1 = 20.0000% Scale is n-1 differentially coherent in interval class 1 Limited inverse transpositions : 1 Inversional symmetry on degrees : 3 Inversional symmetry on intervals : 0-1 -- Rational properties -- Prime limit : 7 Odd number limit : 2401 (O: 2401 U: 2401) Highest odd numerator or denominator: 2401 Scale harmonicity : 0.018817 Average absolute harmonicity : 0.233083 Specific harmonicity : 0.091007 Fundamental : 1/2048, -11.0000 octaves, 0.1277 Hz. Guide tone : 4802, 12.2294 octaves, 1256325.965 Hz. Exponens Consonantiae : 9.834496E+06, 23.22942 octaves Euler's gradus suavitatis : 37 Sum of Mann's harmonic distance : 2571.5, average 514.30000 Mersenne's string divisions : 791490447 Sum of van Prooijen's expressibility: 8.45098, average 1.69020 Sum of Tenney's harmonic distance : 16.57879, average 3.31576 Vogel's harmonic complexity : 20.00000 Wille's k value : 1200 Wilson's harmonic complexity : 28 Rectangular lattice diameter : 4 Triangular lattice diameter : 4 Lattice compactness : 92.11549, average 6.14103 Lattice compactness (without 2's) : 30.00000, average 2.00000 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -11 .. 1 -5.00000 27 1 7: 0 .. 4 2.00000 10 4 Average exponent except 2's : 10 / 5 = 2.00000 Average absolute exponent except 2's: 10 / 5 = 2.00000 Scale is weakly epimorphic with val: <5 13| Scale is JI-epimorphic with val: <5 14| = patent
FIT/MODE
5: 1 1 1 1 1 SP B ME I SD: 21.6190 c. M: 35.3036 c. 14: 3 3 3 2 3 SP M ME S SD: 21.0049 c. M:-33.7768 c. 16: 4 3 3 3 3 SP M ME S SD: 15.1234 c. M:-24.6964 c. 21: 5 4 4 4 4 SP M ME S SD: 6.3752 c. M:-10.4107 c. 26: 6 5 5 5 5 SP M ME S SD: 0.9917 c. M:-1.6195 c. 57: 13 11 11 11 11 SP M DE S SD: 0.9917 c. M: 1.6194 c. 83: 19 16 16 16 16 SP M DE S SD: 0.3704 c. M: 0.6048 c. 109: 25 21 21 21 21 SP M DE S SD: 0.0455 c. M: 0.0743 c. 353: 81 68 68 68 68 SP M DE S SD: 0.0309 c. M:-0.0505 c. 462: 106 89 89 89 89 SP M DE S SD: 0.0129 c. M:-0.0210 c. 571: 131 110 110 110 110 SP M DE S SD: 0.0017 c. M:-0.0029 c. 1822: 418 351 351 351 351 SP M DE S SD: 0.0011 c. M: 0.0018 c. 2393: 549 461 461 461 461 SP M DE S SD: 0.0004 c. M: 0.0007 c.
FIT/HARMONIC
1 x x x x 2 S SD: 0.0000 cents 2 x x 3 x 4 S SD: 17.8484 cents 3 x 4 x 5:6 S SD: 28.2957 cents 4:5 x 6:7:8 S SD: 29.1521 cents 5:6:7:8:9:10 H SD: 24.9545 cents 6:7:8:9:11:12 SD: 17.7794 cents 9:11:12:14:16:18 SD: 16.4400 cents 11:13:15:17:19:22 SD: 8.6941 cents 12:14:16:18:21:24 S SD: 7.5272 cents 17:20:23:26:30:34 SD: 4.6238 cents 23:27:31:35:40:46 SD: 3.7077 cents 24:28:32:37:42:48 SD: 3.3465 cents 35:41:47:54:61:70 SD: 3.0860 cents 36:42:48:55:63:72 SD: 2.5113 cents 41:48:55:63:72:82 SD: 1.8189 cents 47:55:63:72:82:94 SD: 1.2491 cents 64:75:86:98:112:128 SD: 1.0183 cents 88:103:118:135:154:176 SD: 0.8977 cents 100:117:134:153:175:200 SD: 0.7548 cents 115:135:154:176:201:230 SD: 0.6800 cents 132:155:177:202:231:264 SD: 0.6551 cents 139:163:186:213:243:278 SD: 0.6290 cents 145:170:194:222:254:290 SD: 0.6029 cents 156:183:209:239:273:312 SD: 0.2791 cents 209:245:280:320:366:418 SD: 0.2437 cents 256:300:343:392:448:512 SD: 0.1442 cents 412:483:552:631:721:824 SD: 0.0700 cents