Scala analysis: xen15-gilson-generalized-pythagorean-11-8-24
Generalized Pythagorean Scale, 11/8 stacked 24=11+13 times
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 64.497 cents 64.497366 2: 128.995 cents 128.994732 3: 161.861 cents 161.861481 4: 226.359 cents 226.358847 5: 19487171/16777216 259.225597 6: 323.723 cents 323.722963 7: 161051/131072 356.589712 8: 421.087 cents 421.087078 9: 1331/1024 453.953827 10: 518.451 cents 518.451193 11: 11/8 551.317942 undecimal semi-augmented fourth 12: 615.815 cents 615.815308 13: 680.313 cents 680.312674 14: 713.179 cents 713.179424 15: 777.677 cents 777.676790 16: 214358881/134217728 810.543539 17: 875.041 cents 875.040905 18: 1771561/1048576 907.907654 19: 972.405 cents 972.405020 20: 14641/8192 1005.271769 21: 1069.769 cents 1069.769135 22: 121/64 1102.635885 two (undecimal semi-augmented fourth) 23: 1167.133 cents 1167.133251 24: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 32.8667 cents 32.8667 1: 64.4974 cents 64.4974 2: 64.4974 cents 64.4974 3: 32.8667 cents 32.8667 4: 64.4974 cents 64.4974 5: 32.8667 cents 32.8667 6: 64.4974 cents 64.4974 7: 32.8667 cents 32.8667 8: 64.4974 cents 64.4974 9: 32.8667 cents 32.8667 10: 64.4974 cents 64.4974 11: 32.8667 cents 32.8667 12: 64.4974 cents 64.4974 13: 64.4974 cents 64.4974 14: 32.8668 cents 32.8668 15: 64.4974 cents 64.4974 16: 32.8667 cents 32.8667 17: 64.4974 cents 64.4974 18: 32.8667 cents 32.8667 19: 64.4974 cents 64.4974 20: 32.8667 cents 32.8667 21: 64.4974 cents 64.4974 22: 32.8667 cents 32.8667 23: 64.4974 cents 64.4974 24: 32.8667 cents 32.8667
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 11 32.86675 cents 1: 13 64.49737 cents 2: 22 97.36412 cents 2: 2 128.99473 cents 3: 9 130.23086 cents 3: 15 161.86148 cents 4: 20 194.72823 cents 4: 4 226.35885 cents 5: 7 227.59498 cents 5: 17 19487171/16777216 259.226 cents 6: 18 292.09235 cents 6: 6 323.72296 cents 7: 5 324.95910 cents 7: 19 161051/131072 356.590 cents 8: 16 389.45646 cents 8: 8 421.08708 cents 9: 3 422.32321 cents 9: 21 1331/1024 453.954 cents 10: 14 486.82058 cents 10: 10 518.45119 cents 11: 1 519.68733 cents 11: 23 11/8 551.318 cents undecimal semi-augmented fourth 12: 12 584.18469 cents 12: 12 615.81531 cents 13: 23 648.68206 cents 13: 1 680.31267 cents 14: 10 681.54881 cents 14: 14 713.17942 cents 15: 21 746.04617 cents 15: 3 777.67679 cents 16: 8 778.91292 cents 16: 16 214358881/134217728 810.544 cents 17: 19 843.41029 cents 17: 5 875.04090 cents 18: 6 876.27704 cents 18: 18 1771561/1048576 907.908 cents 19: 17 940.77440 cents 19: 7 972.40502 cents 20: 4 973.64115 cents 20: 20 14641/8192 1005.272 cents 21: 15 1038.13852 cents 21: 9 1069.76913 cents 22: 2 1071.00527 cents 22: 22 121/64 1102.636 cents two (undecimal semi-augmented fourth) 23: 13 1135.50263 cents 23: 11 1167.13325 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 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0.0 : 64.5 129.0 161.9 226.4 259.2 323.7 356.6 421.1 454.0 518.5 551.3 615.8 680.3 713.2 777.7 810.5 875.0 907.9 972.4 1005.3 1069.8 1102.6 1167.1 1200.0 64.5 : 64.5 97.4 161.9 194.7 259.2 292.1 356.6 389.5 454.0 486.8 551.3 615.8 648.7 713.2 746.0 810.5 843.4 907.9 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 129.0 : 32.9 97.4 130.2 194.7 227.6 292.1 325.0 389.5 422.3 486.8 551.3 584.2 648.7 681.5 746.0 778.9 843.4 876.3 940.8 973.6 1038.1 1071.0 1135.5 1200.0 161.9 : 64.5 97.4 161.9 194.7 259.2 292.1 356.6 389.5 454.0 518.5 551.3 615.8 648.7 713.2 746.0 810.5 843.4 907.9 940.8 1005.3 1038.1 1102.6 1167.1 1200.0 226.4 : 32.9 97.4 130.2 194.7 227.6 292.1 325.0 389.5 454.0 486.8 551.3 584.2 648.7 681.5 746.0 778.9 843.4 876.3 940.8 973.6 1038.1 1102.6 1135.5 1200.0 259.2 : 64.5 97.4 161.9 194.7 259.2 292.1 356.6 421.1 454.0 518.5 551.3 615.8 648.7 713.2 746.0 810.5 843.4 907.9 940.8 1005.3 1069.8 1102.6 1167.1 1200.0 323.7 : 32.9 97.4 130.2 194.7 227.6 292.1 356.6 389.5 454.0 486.8 551.3 584.2 648.7 681.5 746.0 778.9 843.4 876.3 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 356.6 : 64.5 97.4 161.9 194.7 259.2 323.7 356.6 421.1 454.0 518.5 551.3 615.8 648.7 713.2 746.0 810.5 843.4 907.9 972.4 1005.3 1069.8 1102.6 1167.1 1200.0 421.1 : 32.9 97.4 130.2 194.7 259.2 292.1 356.6 389.5 454.0 486.8 551.3 584.2 648.7 681.5 746.0 778.9 843.4 907.9 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 454.0 : 64.5 97.4 161.9 226.4 259.2 323.7 356.6 421.1 454.0 518.5 551.3 615.8 648.7 713.2 746.0 810.5 875.0 907.9 972.4 1005.3 1069.8 1102.6 1167.1 1200.0 518.5 : 32.9 97.4 161.9 194.7 259.2 292.1 356.6 389.5 454.0 486.8 551.3 584.2 648.7 681.5 746.0 810.5 843.4 907.9 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 551.3 : 64.5 129.0 161.9 226.4 259.2 323.7 356.6 421.1 454.0 518.5 551.3 615.8 648.7 713.2 777.7 810.5 875.0 907.9 972.4 1005.3 1069.8 1102.6 1167.1 1200.0 615.8 : 64.5 97.4 161.9 194.7 259.2 292.1 356.6 389.5 454.0 486.8 551.3 584.2 648.7 713.2 746.0 810.5 843.4 907.9 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 680.3 : 32.9 97.4 130.2 194.7 227.6 292.1 325.0 389.5 422.3 486.8 519.7 584.2 648.7 681.5 746.0 778.9 843.4 876.3 940.8 973.6 1038.1 1071.0 1135.5 1200.0 713.2 : 64.5 97.4 161.9 194.7 259.2 292.1 356.6 389.5 454.0 486.8 551.3 615.8 648.7 713.2 746.0 810.5 843.4 907.9 940.8 1005.3 1038.1 1102.6 1167.1 1200.0 777.7 : 32.9 97.4 130.2 194.7 227.6 292.1 325.0 389.5 422.3 486.8 551.3 584.2 648.7 681.5 746.0 778.9 843.4 876.3 940.8 973.6 1038.1 1102.6 1135.5 1200.0 810.5 : 64.5 97.4 161.9 194.7 259.2 292.1 356.6 389.5 454.0 518.5 551.3 615.8 648.7 713.2 746.0 810.5 843.4 907.9 940.8 1005.3 1069.8 1102.6 1167.1 1200.0 875.0 : 32.9 97.4 130.2 194.7 227.6 292.1 325.0 389.5 454.0 486.8 551.3 584.2 648.7 681.5 746.0 778.9 843.4 876.3 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 907.9 : 64.5 97.4 161.9 194.7 259.2 292.1 356.6 421.1 454.0 518.5 551.3 615.8 648.7 713.2 746.0 810.5 843.4 907.9 972.4 1005.3 1069.8 1102.6 1167.1 1200.0 972.4 : 32.9 97.4 130.2 194.7 227.6 292.1 356.6 389.5 454.0 486.8 551.3 584.2 648.7 681.5 746.0 778.9 843.4 907.9 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 1005.3: 64.5 97.4 161.9 194.7 259.2 323.7 356.6 421.1 454.0 518.5 551.3 615.8 648.7 713.2 746.0 810.5 875.0 907.9 972.4 1005.3 1069.8 1102.6 1167.1 1200.0 1069.8: 32.9 97.4 130.2 194.7 259.2 292.1 356.6 389.5 454.0 486.8 551.3 584.2 648.7 681.5 746.0 810.5 843.4 907.9 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 1102.6: 64.5 97.4 161.9 226.4 259.2 323.7 356.6 421.1 454.0 518.5 551.3 615.8 648.7 713.2 777.7 810.5 875.0 907.9 972.4 1005.3 1069.8 1102.6 1167.1 1200.0 1167.1: 32.9 97.4 161.9 194.7 259.2 292.1 356.6 389.5 454.0 486.8 551.3 584.2 648.7 713.2 746.0 810.5 843.4 907.9 940.8 1005.3 1038.1 1102.6 1135.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 32.8667 .. 64.4974 cents 31.63062 cents 32.86675 cents 2: 97.3641 .. 128.9947 cents 31.63062 cents 32.86675 cents 3: 130.2309 .. 161.8615 cents 31.63062 cents 1.23613 cents 4: 194.7282 .. 226.3588 cents 31.63062 cents 32.86675 cents 5: 227.5950 .. 259.2256 cents 31.63062 cents 1.23613 cents 6: 292.0923 .. 323.7230 cents 31.63062 cents 32.86675 cents 7: 324.9591 .. 356.5897 cents 31.63062 cents 1.23613 cents 8: 389.4565 .. 421.0871 cents 31.63062 cents 32.86675 cents 9: 422.3232 .. 453.9538 cents 31.63062 cents 1.23613 cents 10: 486.8206 .. 518.4512 cents 31.63062 cents 32.86675 cents 11: 519.6873 .. 551.3179 cents 31.63062 cents 1.23613 cents 12: 584.1847 .. 615.8153 cents 31.63062 cents 32.86675 cents 13: 648.6821 .. 680.3127 cents 31.63062 cents 32.86675 cents 14: 681.5488 .. 713.1794 cents 31.63062 cents 1.23613 cents 15: 746.0462 .. 777.6768 cents 31.63062 cents 32.86675 cents 16: 778.9129 .. 810.5435 cents 31.63062 cents 1.23613 cents 17: 843.4103 .. 875.0409 cents 31.63062 cents 32.86675 cents 18: 876.2770 .. 907.9077 cents 31.63062 cents 1.23613 cents 19: 940.7744 .. 972.4050 cents 31.63062 cents 32.86675 cents 20: 973.6412 .. 1005.2718 cents 31.63062 cents 1.23613 cents 21: 1038.1385 .. 1069.7691 cents 31.63062 cents 32.86675 cents 22: 1071.0053 .. 1102.6359 cents 31.63062 cents 1.23613 cents 23: 1135.5026 .. 1167.1333 cents 31.63062 cents 32.86675 cents
SHOW DATA
Number of notes : 24 -- Interval properties -- Smallest interval : 32.86675 cents, class 1 Average step (divided formal octave): 50.0000 cents Largest one step interval : 64.49737 cents Average / Smallest step : 1.521294 Largest / Average step : 1.289947 Largest / Smallest step : 1.962390 Median interval of one step : 64.49737 cents, amount: 13 Most common interval of one step : 64.49737 cents, amount: 13 Least squares average step : 50.85101 cents, oct.: 1220.42434 cents Scale is strictly proper Scale has Myhill's property chromas: 31.63062 cents and 1.23613 cents, ratio: 25.588381 generators: 11 of 551.31794 cents and 648.68206 cents ET pair of linear temperament : 11&13 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 1935-tone equal temperament with octave 2/1 degrees: 104 208 261 365 418 522 575 679 732 836 889 993 1097 1150 1254 1307 1411 1464 1568 1621 1725 1778 1882 1935 generators: 889 and 1046 Number of contiguous 1-step segments: 0 Step pattern alph. order: AABABABABABAABABABABABAB Step pattern size order : LLSLSLSLSLSLLSLSLSLSLSLS Scale is a Constant Structure, by a margin of 1.23613 cents Scale diversity : 1.142485 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.393747 Rothenberg efficiency : 0.694444 redundancy: 0.305556 Efficiency x scale size : 16.666667 Number of different interval sizes : 46 = 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 : 1.23613 cents Most common intervals : 11/8, 551.3179 cents & inv., amount: 23 Scale is a chain of 12 triads 0.0 64.497 648.682 cents Most common triad is 0.0 551.318 648.682 cents, amount: 22 Number of recognisable fifths : 14, average 713.1794 cents Scale contains two identical disjunct "tetrachords" of 11 notes Formal octave complements present : 1 = 4.1667% Limited inverse transpositions : 13 Inversional symmetry on intervals : 6-7 18-19 -- Rational properties -- Prime limit : 11 (not all pitches rational)
FIT/MODE
24: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 17.6902 c. M: 30.3127 c. 25: 1 2 0 2 0 2 0 2 0 2 0 2 1 1 1 1 1 1 1 1 1 1 1 1 N D3 SD: 13.2432 c. M: 23.3179 c. 26: 1 2 1 1 1 1 1 1 1 1 1 1 2 0 2 1 1 1 1 1 1 1 1 1 P D3 SD: 12.1649 c. M:-22.7539 c. 28: 2 1 1 1 1 2 0 2 1 1 1 1 2 1 1 1 1 1 2 0 2 1 1 1 N D3 SD: 11.9188 c. M:-21.2169 c. 30: 2 1 1 2 0 2 1 2 0 2 1 1 2 1 1 1 2 1 1 1 2 1 1 1 N D3 SD: 11.3027 c. M: 19.2256 c. 32: 2 1 1 2 1 2 1 1 1 2 1 1 2 1 2 1 1 1 2 1 2 0 2 1 N D3 SD: 10.7038 c. M:-18.4103 c. 35: 2 2 1 2 1 1 1 2 1 2 1 2 2 1 2 1 2 0 2 1 2 1 2 1 N D3 SD: 10.0707 c. M: 16.4791 c. 36: 2 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 P D3 SD: 9.4995 c. M: 15.8153 c. 37: 2 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 P M ME SD: 0.4484 c. M:-0.7684 c. 505: 27 27 14 27 14 27 14 27 14 27 14 27 27 14 27 14 27 14 27 14 27 14 27 14 SP M DE SD: 0.4136 c. M: 0.7087 c. 542: 29 29 15 29 15 29 15 29 15 29 15 29 29 15 29 15 29 15 29 15 29 15 29 15 SP M DE SD: 0.3548 c. M: 0.6079 c. 579: 31 31 16 31 16 31 16 31 16 31 16 31 31 16 31 16 31 16 31 16 31 16 31 16 SP M DE SD: 0.3034 c. M: 0.5199 c. 616: 33 33 17 33 17 33 17 33 17 33 17 33 33 17 33 17 33 17 33 17 33 17 33 17 SP M DE SD: 0.2583 c. M: 0.4425 c. 653: 35 35 18 35 18 35 18 35 18 35 18 35 35 18 35 18 35 18 35 18 35 18 35 18 SP M DE SD: 0.2182 c. M: 0.3739 c. 690: 37 37 19 37 19 37 19 37 19 37 19 37 37 19 37 19 37 19 37 19 37 19 37 19 SP M DE SD: 0.1825 c. M: 0.3127 c. 727: 39 39 20 39 20 39 20 39 20 39 20 39 39 20 39 20 39 20 39 20 39 20 39 20 SP M DE SD: 0.1504 c. M: 0.2577 c. 764: 41 41 21 41 21 41 21 41 21 41 21 41 41 21 41 21 41 21 41 21 41 21 41 21 SP M DE SD: 0.1214 c. M: 0.2080 c. 801: 43 43 22 43 22 43 22 43 22 43 22 43 43 22 43 22 43 22 43 22 43 22 43 22 SP M DE SD: 0.0950 c. M: 0.1629 c. 838: 45 45 23 45 23 45 23 45 23 45 23 45 45 23 45 23 45 23 45 23 45 23 45 23 SP M DE SD: 0.0710 c. M: 0.1217 c. 875: 47 47 24 47 24 47 24 47 24 47 24 47 47 24 47 24 47 24 47 24 47 24 47 24 SP M DE SD: 0.0491 c. M: 0.0841 c. 912: 49 49 25 49 25 49 25 49 25 49 25 49 49 25 49 25 49 25 49 25 49 25 49 25 SP M DE SD: 0.0289 c. M: 0.0495 c. 949: 51 51 26 51 26 51 26 51 26 51 26 51 51 26 51 26 51 26 51 26 51 26 51 26 SP M DE SD: 0.0103 c. M: 0.0176 c. 986: 53 53 27 53 27 53 27 53 27 53 27 53 53 27 53 27 53 27 53 27 53 27 53 27 SP M DE SD: 0.0069 c. M:-0.0119 c. 1935: 104 104 53 104 53 104 53 104 53 104 53 104 104 53 104 53 104 53 104 53 104 53 104 53 SP M DE SD: 0.0015 c. M: 0.0026 c.
FIT/HARMONIC
1 .. 2 match: 1 H SD: 0.0000 cents 2 .. 4 match: 2 H SD: 5.6122 cents 3 .. 6 match: 3 H SD: 7.4776 cents 4 .. 8 match: 4 H SD: 7.9934 cents 5 .. 10 match: 5 H SD: 6.9287 cents 6 .. 12 match: 6 H SD: 5.5380 cents 7 .. 14 match: 7 H SD: 4.2497 cents 8 .. 16 match: 8 H SD: 6.4125 cents 9 .. 18 match: 9 H SD: 4.5703 cents 10 .. 20 match: 10 H SD: 3.9047 cents 11 .. 22 match: 11 H SD: 6.5431 cents 12 .. 24 match: 12 H SD: 4.5961 cents 13 .. 26 match: 13 H SD: 4.5683 cents 14 .. 28 match: 14 H SD: 3.5799 cents 15 .. 30 match: 15 H SD: 4.4433 cents 16 .. 32 match: 16 H SD: 4.6113 cents 17 .. 34 match: 16 S SD: 4.2131 cents 18 .. 36 match: 17 S SD: 3.6900 cents 19 .. 38 match: 18 S SD: 2.9956 cents 20 .. 40 match: 20 H SD: 3.3110 cents 21 .. 42 match: 20 SD: 2.8984 cents 22 .. 44 match: 19 SD: 3.4980 cents 23 .. 46 match: 21 SD: 2.5710 cents 24 .. 48 match: 18 SD: 2.7570 cents 25 .. 50 match: 21 SD: 2.5527 cents 26 .. 52 match: 21 SD: 2.4315 cents 27 .. 54 match: 22 SD: 2.4617 cents 28 .. 56 match: 22 SD: 2.4862 cents 29 .. 58 match: 24 SD: 1.8859 cents 31 .. 62 match: 23 SD: 2.4653 cents 32 .. 64 match: 23 SD: 2.3202 cents 33 .. 66 match: 21 SD: 2.0311 cents 34 .. 68 match: 22 SD: 2.2082 cents 35 .. 70 match: 22 SD: 1.7423 cents 37 .. 74 match: 23 SD: 1.7134 cents 40 .. 80 match: 23 SD: 1.7318 cents 43 .. 86 match: 24 SD: 1.4507 cents 48 .. 96 match: 24 SD: 1.3897 cents 49 .. 98 match: 24 SD: 1.2280 cents 57 .. 114 match: 24 SD: 1.1998 cents 61 .. 122 match: 24 SD: 1.1221 cents 62 .. 124 match: 24 SD: 1.0660 cents 69 .. 138 match: 24 SD: 1.0130 cents 70 .. 140 match: 24 SD: 1.0007 cents 72 .. 144 match: 24 SD: 0.9976 cents 73 .. 146 match: 24 SD: 0.9217 cents 74 .. 148 match: 24 SD: 0.8347 cents 77 .. 154 match: 24 SD: 0.8196 cents 79 .. 158 match: 24 SD: 0.7984 cents 80 .. 160 match: 24 SD: 0.7318 cents 94 .. 188 match: 24 SD: 0.7150 cents 100 .. 200 match: 24 SD: 0.6644 cents 101 .. 202 match: 24 SD: 0.6368 cents 102 .. 204 match: 24 SD: 0.6112 cents 106 .. 212 match: 24 SD: 0.5331 cents 126 .. 252 match: 24 SD: 0.5228 cents 130 .. 260 match: 24 SD: 0.5037 cents 131 .. 262 match: 24 SD: 0.3511 cents 166 .. 332 match: 24 SD: 0.3299 cents 184 .. 368 match: 24 SD: 0.3126 cents 206 .. 412 match: 24 SD: 0.2710 cents 235 .. 470 match: 24 SD: 0.2430 cents 243 .. 486 match: 24 SD: 0.2197 cents 283 .. 566 match: 24 SD: 0.2159 cents 286 .. 572 match: 24 SD: 0.2007 cents 293 .. 586 match: 24 SD: 0.1999 cents 310 .. 620 match: 24 SD: 0.1930 cents 316 .. 632 match: 24 SD: 0.1828 cents 337 .. 674 match: 24 SD: 0.1711 cents 367 .. 734 match: 24 SD: 0.1662 cents 389 .. 778 match: 24 SD: 0.1508 cents 427 .. 854 match: 24 SD: 0.1448 cents 429 .. 858 match: 24 SD: 0.1376 cents 447 .. 894 match: 24 SD: 0.1308 cents 450 .. 900 match: 24 SD: 0.1276 cents 451 .. 902 match: 24 SD: 0.1198 cents 501 .. 1002 match: 24 SD: 0.1143 cents 530 .. 1060 match: 24 SD: 0.1129 cents 531 .. 1062 match: 24 SD: 0.1025 cents 559 .. 1118 match: 24 SD: 0.0991 cents 573 .. 1146 match: 24 SD: 0.0945 cents