Scala analysis: xen15-gilson-generalized-pythagorean-13-8-7
Generalized Pythagorean Scale, 13/8 stacked 7=5+2 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2197/2048 121.582985 2: 4826809/4194304 243.165971 3: 169/128 481.055324 4: 371293/262144 602.638309 5: 13/8 840.527662 tridecimal neutral sixth 6: 28561/16384 962.110647 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 32768/28561 237.8894 1: 2197/2048 121.5830 2: 2197/2048 121.5830 3: 32768/28561 237.8894 4: 2197/2048 121.5830 5: 32768/28561 237.8894 6: 2197/2048 121.5830 7: 32768/28561 237.8894
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 4 2197/2048 121.583 cents 1: 3 32768/28561 237.889 cents 2: 1 4826809/4194304 243.166 cents 2: 6 16/13 359.472 cents tridecimal neutral third 3: 5 169/128 481.055 cents 3: 2 524288/371293 597.362 cents 4: 2 371293/262144 602.638 cents 4: 5 256/169 718.945 cents two (tridecimal neutral third) 5: 6 13/8 840.528 cents tridecimal neutral sixth 5: 1 8388608/4826809 956.834 cents 6: 3 28561/16384 962.111 cents 6: 4 4096/2197 1078.417 cents 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 6 7 0.0 : 121.6 243.2 481.1 602.6 840.5 962.1 1200.0 121.6 : 121.6 359.5 481.1 718.9 840.5 1078.4 1200.0 243.2 : 237.9 359.5 597.4 718.9 956.8 1078.4 1200.0 481.1 : 121.6 359.5 481.1 718.9 840.5 962.1 1200.0 602.6 : 237.9 359.5 597.4 718.9 840.5 1078.4 1200.0 840.5 : 121.6 359.5 481.1 602.6 840.5 962.1 1200.0 962.1 : 237.9 359.5 481.1 718.9 840.5 1078.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 121.5830 .. 237.8894 cents 67108864/62748517, 116.3064 cents 2197/2048, 121.5830 cents 2: 243.1660 .. 359.4723 cents 67108864/62748517, 116.3064 cents 5.27662 cents 3: 481.0553 .. 597.3617 cents 67108864/62748517, 116.3064 cents 2197/2048, 121.5830 cents 4: 602.6383 .. 718.9447 cents 67108864/62748517, 116.3064 cents 5.27662 cents 5: 840.5277 .. 956.8340 cents 67108864/62748517, 116.3064 cents 2197/2048, 121.5830 cents 6: 962.1106 .. 1078.4170 cents 67108864/62748517, 116.3064 cents 5.27662 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 2197/2048, 121.5830 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 32768/28561, 237.8894 cents Average / Smallest step : 1.409972 Largest / Average step : 1.387688 Largest / Smallest step : 1.956601 Median interval of one step : 2197/2048, 121.5830 cents, amount: 4 Most common interval of one step : 2197/2048, 121.5830 cents, amount: 4 Least squares average step : 163.12097 cents, oct.: 1141.84682 cents Scale is strictly proper Scale has Myhill's property chromas: 67108864/62748517, 116.3064 cents and 5.27662 cents, ratio: 22.041841 generators: 359.47234 cents and 5 of 840.52766 cents ET pair of linear temperament : 10&227 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 227-tone equal temperament with octave 2/1 degrees: 23 46 91 114 159 182 227 generators: 68 and 159 Number of contiguous 1-step segments: 0 Step pattern alph. order: AABABAB Step pattern size order : SSLSLSL Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 5.27662 cents Scale diversity : 1.196392 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.418468 Rothenberg efficiency : 0.761905 redundancy: 0.238095 Efficiency x scale size : 5.333333 Number of different interval sizes : 12 = 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 : 5.27662 cents Most common intervals : 16/13, 359.4723 cents & inv., amount: 6 Scale is a chain of 3 triads 0.0 359.472 718.945 cents Most common triad is 0.0 359.472 840.528 cents, amount: 5 Number of recognisable fifths : 5, average 718.9447 cents Number of appreciable fifths : 0 Best fifths form a closed circle Best minor thirds form a closed circle Scale is a complete genus musicum : 2^m 13^6 Formal octave complements present : 1 = 14.2857% Limited inverse transpositions : 2 Inversional symmetry on degrees : 1 Inversional symmetry on intervals : 4-5 -- Rational properties -- Prime limit : 13 Odd number limit : 4826809 (O: 4826809 U: 4826809) Highest odd numerator or denominator: 4826809 Scale harmonicity : 0.006413 Average absolute harmonicity : 0.156588 Specific harmonicity : 0.032098 Fundamental : 1/4194304, -22.0000 octaves, 0.0001 Hz. Guide tone : 9653618, 23.2026 octaves, 2525633266.450 Hz. Exponens Consonantiae : 4.049021E+13, 45.20264 octaves Euler's gradus suavitatis : 96 Sum of Mann's harmonic distance : 4852030.5, average 693147.21429 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 23.39281, average 3.34183 Sum of Tenney's harmonic distance : 46.27109, average 6.61016 Vogel's harmonic complexity : 50.42857 Wille's k value : 2413404 Wilson's harmonic complexity : 78 Rectangular lattice diameter : 6 Triangular lattice diameter : 6 Lattice compactness : 298.17667, average 10.64917 Lattice compactness (without 2's) : 77.00000, average 2.75000 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -22 .. 1 -10.57143 76 1 13: 0 .. 6 3.00000 21 6 Average exponent except 2's : 21 / 7 = 3.00000 Average absolute exponent except 2's: 21 / 7 = 3.00000 Scale is weakly epimorphic with val: <7 24| Scale is weakly epimorphic with val: <7 25| Scale is JI-epimorphic with val: <7 26| = patent Scale is weakly epimorphic with val: <7 27| Scale is weakly epimorphic with val: <7 29|
FIT/MODE
7: 0 1 1 1 1 1 2 N T3 SD: 108.1130 c. M: 154.8134 c. 8: 1 1 1 1 2 0 2 P D3 S SD: 42.0947 c. M: 62.1106 c. 9: 1 1 2 1 1 1 2 P M ME S SD: 37.7843 c. M:-64.0284 c. Pseudo Pelog, Pelogic-7, Antiionian 10: 1 1 2 1 2 1 2 P M ME S SD: 1.9025 c. M: 3.1660 c. 117: 12 12 23 12 23 12 23 SP M DE S SD: 1.7955 c. M:-2.9879 c. 127: 13 13 25 13 25 13 25 SP M DE S SD: 1.5043 c. M:-2.5033 c. 137: 14 14 27 14 27 14 27 SP M DE S SD: 1.2556 c. M:-2.0895 c. 147: 15 15 29 15 29 15 29 SP M DE S SD: 1.0408 c. M:-1.7320 c. 157: 16 16 31 16 31 16 31 SP M DE S SD: 0.8533 c. M:-1.4200 c. 167: 17 17 33 17 33 17 33 SP M DE S SD: 0.6883 c. M:-1.1454 c. 177: 18 18 35 18 35 18 35 SP M DE S SD: 0.5419 c. M:-0.9018 c. 187: 19 19 37 19 37 19 37 SP M DE S SD: 0.4112 c. M:-0.6843 c. 197: 20 20 39 20 39 20 39 SP M DE S SD: 0.2938 c. M:-0.4889 c. 207: 21 21 41 21 41 21 41 SP M DE S SD: 0.1877 c. M:-0.3123 c. 217: 22 22 43 22 43 22 43 SP M DE S SD: 0.0913 c. M:-0.1520 c. 227: 23 23 45 23 45 23 45 SP M DE S SD: 0.0035 c. M:-0.0058 c.
FIT/HARMONIC
1 x x x x x x 2 S SD: 0.0000 cents 2 x x x 3 x x 4 S SD: 49.6583 cents 3 x x 4 x 5 x 6 S SD: 15.6695 cents 4 x x 5:6 x 7:8 S SD: 34.3556 cents 5 x 6 x 7:8:9:10 S SD: 19.4493 cents 6 x 7:8:9:10 x 12 S SD: 22.4816 cents 7 x 8:9:10:11:12:14 S SD: 13.6277 cents 8 x 9 x 11:13:14:16 SD: 12.9922 cents 9 x 10:12:13:15:16:18 SD: 15.1025 cents 10:11:12:13:14:16:17:20 SD: 14.8967 cents 11:12:13:15:16:18:19:22 SD: 13.2537 cents 12:13:14:16:17:20:21:24 SD: 7.9611 cents 13:14:15:17:18:21:23:26 SD: 7.3489 cents 15:16:17:20:21:24:26:30 SD: 6.8652 cents 19:20:22:25:27:31:33:38 SD: 5.2393 cents 24:26:28:32:34:39:42:48 SD: 4.9165 cents 26:28:30:34:37:42:45:52 SD: 3.6952 cents 27:29:31:36:38:44:47:54 SD: 3.0649 cents 38:41:44:50:54:62:66:76 SD: 2.7469 cents 40:43:46:53:57:65:70:80 SD: 2.0598 cents 53:57:61:70:75:86:92:106 SD: 1.2953 cents 81:87:93:107:115:132:141:162 SD: 1.1840 cents 93:100:107:123:132:151:162:186 SD: 0.9301 cents 94:101:108:124:133:153:164:188 SD: 0.7982 cents 112:120:129:148:159:182:195:224 SD: 0.7753 cents 113:121:130:149:160:184:197:226 SD: 0.7588 cents 125:134:144:165:177:203:218:250 SD: 0.3734 cents 206:221:237:272:292:335:359:412 SD: 0.2855 cents 219:235:252:289:310:356:382:438 SD: 0.2720 cents 272:292:313:359:385:442:474:544 SD: 0.2695 cents 312:335:359:412:442:507:544:624 SD: 0.2394 cents 331:355:381:437:469:538:577:662 SD: 0.1375 cents 550:590:633:726:779:894:959:1100 SD: 0.1104 cents