Scala analysis: xen18-ayers-table-19
3 Iterated Harmonic Means and 3 Iterated Subcontraries to the Harmonic Mean
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 8/7 231.174094 septimal whole tone 2: 4/3 498.044999 perfect fourth 3: 17/12 603.000409 2nd septendecimal tritone 4: 8/5 813.686286 minor sixth 5: 5/3 884.358713 major sixth, BP sixth 6: 61/33 1063.611862 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 66/61 136.3881 1: 8/7 231.1741 septimal whole tone 2: 7/6 266.8709 septimal minor third 3: 17/16 104.9554 17th harmonic 4: 96/85 210.6859 5: 25/24 70.6724 classic chromatic semitone, minor chroma 6: 61/55 179.2531 7: 66/61 136.3881
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 25/24 70.672 cents classic chromatic semitone, minor chroma 1: 1 17/16 104.955 cents 17th harmonic 1: 1 66/61 136.388 cents 1: 1 61/55 179.253 cents 1: 1 96/85 210.686 cents 1: 1 8/7 231.174 cents septimal whole tone 1: 1 7/6 266.871 cents septimal minor third 2: 1 305/264 249.926 cents 2: 1 20/17 281.358 cents septendecimal augmented second 2: 2 6/5 315.641 cents minor third 2: 1 528/427 367.562 cents 2: 1 119/96 371.826 cents 2: 1 4/3 498.045 cents perfect fourth 3: 2 5/4 386.314 cents major third 3: 1 244/187 460.611 cents 3: 1 48/35 546.815 cents septimal semi-augmented fourth 3: 1 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 3: 1 17/12 603.000 cents 2nd septendecimal tritone 3: 1 88/61 634.433 cents 4: 1 61/44 565.567 cents 4: 1 24/17 597.000 cents 1st septendecimal tritone 4: 1 10/7 617.488 cents Euler's tritone 4: 1 35/24 653.185 cents septimal semi-diminished fifth 4: 1 187/122 739.389 cents 4: 2 8/5 813.686 cents minor sixth 5: 1 3/2 701.955 cents perfect fifth 5: 1 192/119 828.174 cents 5: 1 427/264 832.438 cents 5: 2 5/3 884.359 cents major sixth, BP sixth 5: 1 17/10 918.642 cents septendecimal diminished seventh 5: 1 528/305 950.074 cents 6: 1 12/7 933.129 cents septimal major sixth 6: 1 7/4 968.826 cents harmonic seventh 6: 1 85/48 989.314 cents 6: 1 110/61 1020.747 cents 6: 1 61/33 1063.612 cents 6: 1 32/17 1095.045 cents 17th subharmonic 6: 1 48/25 1129.328 cents classic diminished octave Highest number of different intervals for one interval class: 7 Average number of different intervals per interval class: 6.33333 = 19/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 231.2 498.0 603.0 813.7 884.4 1063.6 1200.0 231.2 : 266.9 371.8 582.5 653.2 832.4 968.8 1200.0 498.0 : 105.0 315.6 386.3 565.6 702.0 933.1 1200.0 603.0 : 210.7 281.4 460.6 597.0 828.2 1095.0 1200.0 813.7 : 70.7 249.9 386.3 617.5 884.4 989.3 1200.0 884.4 : 179.3 315.6 546.8 813.7 918.6 1129.3 1200.0 1063.6: 136.4 367.6 634.4 739.4 950.1 1020.7 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 70.6724 .. 266.8709 cents 28/25, 196.1985 cents 25/24, 70.6724 cents 2: 249.9256 .. 498.0450 cents 352/305, 248.1194 cents 305/308,-16.9453 cents 3: 386.3137 .. 634.4331 cents 352/305, 248.1194 cents 15/16,-111.7313 cents 4: 565.5669 .. 813.6863 cents 352/305, 248.1194 cents 3721/3872,-68.8663 cents 5: 701.9550 .. 950.0744 cents 352/305, 248.1194 cents 15/16,-111.7313 cents 6: 933.1291 .. 1129.3276 cents 28/25, 196.1985 cents 305/308,-16.9453 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 25/24, 70.6724 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 7/6, 266.8709 cents Average / Smallest step : 2.425678 Largest / Average step : 1.556747 Largest / Smallest step : 3.776167 Median interval of one step : 61/55, 179.2531 cents, amount: 1 Least squares average step : 182.10339 cents, oct.: 1274.72376 cents Scale is not proper Step pattern alph. order: ABCDEFG Step pattern size order : FGBEADC Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 6.00082 cents Scale diversity : 1.716361 Lumma stability : 0.117787 Lumma impropriety factor : 0.271850 Rothenberg efficiency : 0.312925 redundancy: 0.687075 Efficiency x scale size : 2.190476 Number of different interval sizes : 38 = 6.33333 / class Number of one step interval sizes : 7 Highest interval variety : 7 Mean interval variety : 6.33333 = 19/3 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 50813/50688, 4.2641 cents Number of recognisable fifths : 1, average 701.9550 cents Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 2 at degree 0 Formal octave complements present : 1 = 14.2857% Scale is n-1 differentially coherent in interval classes 1 and 2 combined -- Rational properties -- Prime limit : 61 Odd number limit : 427 (O: 427 U: 427) Highest odd numerator or denominator: 61 Scale harmonicity : 0.012650 Fundamental : 1/4620, -12.1737 octaves, 0.0566 Hz. Guide tone : 41480, 15.3401 octaves, 10852228.449 Hz. Exponens Consonantiae : 1.916376E+08, 27.51381 octaves Euler's gradus suavitatis : 108 Sum of Mann's harmonic distance : 84.5, average 12.07143 Mersenne's string divisions : 14883960 Sum of van Prooijen's expressibility: 5.73594, average 0.81942 Sum of Tenney's harmonic distance : 11.52002, average 1.64572 Vogel's harmonic complexity : 35.85714 Wille's k value : 3272 Wilson's harmonic complexity : 109 Rectangular lattice diameter : 7 Triangular lattice diameter : 2 Lattice compactness : 72.00933, average 2.57176 Lattice compactness (without 2's) : 36.86688, average 1.31667 Number of different primes : 7 Prime exponents' range, average, count, tones@limit: 2: -2 .. 3 1.00000 11 1 3: -1 .. 0 -0.57143 4 1 5: -1 .. 1 0.00000 2 2 7: -1 .. 0 -0.14286 1 1 11: -1 .. 0 -0.14286 1 17: 0 .. 1 0.14286 1 1 61: 0 .. 1 0.14286 1 1 Average exponent except 2's :-4 / 7 =-0.57143 Average absolute exponent except 2's: 10 / 7 = 1.42857 Scale is weakly epimorphic with val: <7 11 15 19 23 30 42| Scale is weakly epimorphic with val: <7 11 15 19 24 30 43| Scale is weakly epimorphic with val: <7 11 15 20 23 30 43| Scale is weakly epimorphic with val: <7 11 17 19 23 30 42| Scale is weakly epimorphic with val: <7 11 17 19 24 30 43| Scale is weakly epimorphic with val: <7 11 17 20 23 30 43| Scale is weakly epimorphic with val: <7 12 15 20 24 30 41| Scale is weakly epimorphic with val: <7 12 15 20 25 30 42| Scale is weakly epimorphic with val: <7 12 16 20 23 29 41| Scale is weakly epimorphic with val: <7 12 16 20 24 29 42| Scale is weakly epimorphic with val: <7 12 16 20 25 29 43| Scale is JI-epimorphic with val: <7 12 17 20 23 29 41| Scale is JI-epimorphic with val: <7 12 17 20 24 29 42| Scale is JI-epimorphic with val: <7 12 17 20 25 29 43|
FIT/MODE
7: 1 2 1 1 0 1 1 N T3 SD: 45.5630 c. M:-82.7139 c. 8: 1 2 1 1 1 1 1 P M ME S SD: 43.7402 c. M: 81.1741 c. 9: 2 2 1 1 1 1 1 N D3 S SD: 36.1614 c. M:-63.6663 c. 10: 2 2 1 2 0 2 1 N T3 S SD: 21.8475 c. M: 44.3587 c. 12: 2 3 1 2 1 2 1 N T3 SD: 19.7871 c. M:-36.3881 c. Nohkan Flute scale: Japan 15: 3 3 1 3 1 2 2 N SD: 20.7584 c. M: 43.0004 c. 16: 3 4 1 3 1 2 2 N SD: 13.7925 c. M:-26.9550 c. 19: 4 4 2 3 1 3 2 N S SD: 14.5651 c. M:-28.5785 c. 20: 4 4 2 4 1 3 2 N SD: 15.2041 c. M:-26.3137 c. 22: 4 5 2 4 1 3 3 N SD: 12.6868 c. M: 27.2482 c. 24: 5 5 2 4 2 3 3 N SD: 11.8592 c. M:-18.8259 c. 26: 5 6 2 5 1 4 3 N SD: 8.0503 c. M:-17.0829 c. 28: 5 7 2 5 2 4 3 N SD: 11.1104 c. M: 16.8884 c. 31: 6 7 3 5 2 5 3 N SD: 10.2963 c. M:-20.2591 c. 32: 6 7 3 6 2 4 4 N SD: 10.1154 c. M:-15.6413 c. 34: 6 8 3 6 2 5 4 N SD: 7.8540 c. M: 19.4094 c. 37: 7 8 4 6 2 6 4 N SD: 8.0485 c. M:-13.2158 c. 38: 7 9 3 7 2 6 4 N SD: 6.7537 c. M: 10.1215 c. 41: 8 9 4 7 2 6 5 N SD: 6.7320 c. M:-11.6337 c. 42: 8 9 4 7 3 6 5 N SD: 7.5482 c. M: 13.6863 c. 43: 8 10 4 7 3 6 5 N SD: 6.6029 c. M:-10.9531 c. 46: 9 10 4 8 3 7 5 N SD: 3.6810 c. M:-5.9534 c. 53: 10 12 5 9 3 8 6 N SD: 3.7023 c. M:-8.3203 c. 62: 12 14 5 11 4 9 7 N SD: 3.2517 c. M:-5.9639 c. 68: 13 15 6 12 4 10 8 N SD: 2.8829 c. M: 4.7883 c. 72: 14 16 6 13 4 11 8 N SD: 2.2914 c. M:-3.0548 c. 87: 17 19 8 15 5 13 10 N SD: 2.1814 c. M:-3.8961 c. 99: 19 22 9 17 6 15 11 N SD: 1.8245 c. M:-3.0602 c. 115: 22 26 10 20 7 17 13 N SD: 1.8057 c. M:-2.8246 c. 125: 24 28 11 22 7 19 14 N SD: 1.5033 c. M:-2.3137 c. 140: 27 31 12 25 8 21 16 N SD: 1.3662 c. M: 3.0004 c. 149: 29 33 13 26 9 22 17 N SD: 1.2603 c. M:-2.3830 c. 167: 32 37 15 29 10 25 19 N SD: 1.2010 c. M: 2.2366 c. 171: 33 38 15 30 10 26 19 N SD: 1.1915 c. M:-3.0548 c. 183: 35 41 16 32 11 27 21 N SD: 0.9103 c. M: 1.6659 c. 193: 37 43 17 34 11 29 22 N SD: 0.8135 c. M: 1.4572 c. 229: 44 51 20 40 14 34 26 N SD: 0.7773 c. M: 1.4592 c. 239: 46 53 21 42 14 36 27 N SD: 0.5925 c. M: 0.9739 c. 255: 49 57 22 45 15 38 29 N SD: 0.4902 c. M:-0.7785 c. 342: 66 76 30 60 20 51 39 N SD: 0.3406 c. M:-0.5084 c. 422: 81 94 37 74 25 63 48 N SD: 0.3946 c. M: 0.8423 c. 494: 95 110 43 87 29 74 56 N SD: 0.3047 c. M: 0.5713 c. 571: 110 127 50 100 34 85 65 N SD: 0.2316 c. M:-0.4049 c. 581: 112 129 51 102 34 87 66 N SD: 0.1919 c. M: 0.3656 c. 643: 124 143 56 113 38 96 73 N SD: 0.1856 c. M:-0.2447 c. 764: 147 170 67 134 45 114 87 N SD: 0.1681 c. M: 0.2840 c. 836: 161 186 73 147 49 125 95 N SD: 0.1084 c. M:-0.1893 c. 1065: 205 237 93 187 63 159 121 N SD: 0.1316 c. M: 0.1882 c. 1152: 222 256 101 202 68 172 131 N SD: 0.0955 c. M: 0.1446 c. 1224: 236 272 107 215 72 183 139 N SD: 0.0921 c. M:-0.1985 c. 1407: 271 313 123 247 83 210 160 N SD: 0.0480 c. M:-0.0763 c. 1988: 383 442 174 349 117 297 226 N SD: 0.0327 c. M: 0.0571 c.
FIT/HARMONIC
1 x x x x x x 2 S SD: 0.0000 cents 2 x x 3 x x x 4 S SD: 49.4773 cents 3 x 4 x x 5 x 6 S SD: 0.0000 cents 4 x 5:6 x 7 x 8 S SD: 42.8738 cents 5:6 x 7:8 x 9:10 S SD: 19.6692 cents 6:7:8:9 x 10:11:12 S SD: 17.6928 cents 7:8:9:10:11:12:13:14 H SD: 12.4467 cents 8:9 x 11:13 x 15:16 SD: 13.7736 cents 9:10:12:13:14:15:17:18 SD: 12.1961 cents 11:13:15:16 x 18:20:22 SD: 15.6317 cents 12:14:16:17:19:20:22:24 SD: 6.0709 cents 15:17:20:21:24:25:28:30 SD: 4.3255 cents 21:24:28:30:34:35:39:42 SD: 3.7664 cents 27:31:36:38:43:45:50:54 SD: 2.3356 cents 48:55:64:68:77:80:89:96 SD: 1.1847 cents 72:82:96:102:115:120:133:144 SD: 0.9763 cents 84:96:112:119:134:140:155:168 SD: 0.8558 cents 99:113:132:140:158:165:183:198 SD: 0.8267 cents 105:120:140:149:168:175:194:210 SD: 0.4312 cents 120:137:160:170:192:200:222:240 SD: 0.3279 cents 225:257:300:319:360:375:416:450 SD: 0.2437 cents 252:288:336:357:403:420:466:504 SD: 0.1561 cents 357:408:476:506:571:595:660:714 SD: 0.1536 cents 420:480:560:595:672:700:776:840 SD: 0.1159 cents 468:535:624:663:749:780:865:936 SD: 0.0970 cents 540:617:720:765:864:900:998:1080 SD: 0.0729 cents 588:672:784:833:941:980:1087:1176 SD: 0.0565 cents