Scala analysis: xen18-ayers-table-05
6 Generalized Arithmetic Means between 1/1 and 2/1
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: 9/7 435.084095 septimal major third, BP third 3: 10/7 617.487807 Euler's tritone 4: 11/7 782.492036 undecimal augmented fifth 5: 12/7 933.129094 septimal major sixth 6: 13/7 1071.701755 16/3-tone 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 14/13 128.2982 2/3-tone 1: 8/7 231.1741 septimal whole tone 2: 9/8 203.9100 major whole tone 3: 10/9 182.4037 minor whole tone 4: 11/10 165.0042 Ptolemy's second, 4/5-tone 5: 12/11 150.6371 undecimal neutral second, 3/4-tone 6: 13/12 138.5727 tridecimal 2/3-tone 7: 14/13 128.2982 2/3-tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 14/13 128.298 cents 2/3-tone 1: 1 13/12 138.573 cents tridecimal 2/3-tone 1: 1 12/11 150.637 cents undecimal neutral second, 3/4-tone 1: 1 11/10 165.004 cents Ptolemy's second, 4/5-tone 1: 1 10/9 182.404 cents minor whole tone 1: 1 9/8 203.910 cents major whole tone 1: 1 8/7 231.174 cents septimal whole tone 2: 1 7/6 266.871 cents septimal minor third 2: 1 13/11 289.210 cents tridecimal minor third 2: 1 6/5 315.641 cents minor third 2: 1 11/9 347.408 cents undecimal neutral third 2: 1 16/13 359.472 cents tridecimal neutral third 2: 1 5/4 386.314 cents major third 2: 1 9/7 435.084 cents septimal major third, BP third 3: 1 14/11 417.508 cents undecimal diminished fourth or major third 3: 1 13/10 454.214 cents tridecimal semi-diminished fourth 3: 2 4/3 498.045 cents perfect fourth 3: 1 11/8 551.318 cents undecimal semi-augmented fourth 3: 1 18/13 563.382 cents tridecimal augmented fourth 3: 1 10/7 617.488 cents Euler's tritone 4: 1 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 4: 1 13/9 636.618 cents tridecimal diminished fifth 4: 1 16/11 648.682 cents undecimal semi-diminished fifth 4: 2 3/2 701.955 cents perfect fifth 4: 1 20/13 745.786 cents tridecimal semi-augmented fifth 4: 1 11/7 782.492 cents undecimal augmented fifth 5: 1 14/9 764.916 cents septimal minor sixth 5: 1 8/5 813.686 cents minor sixth 5: 1 13/8 840.528 cents tridecimal neutral sixth 5: 1 18/11 852.592 cents undecimal neutral sixth 5: 1 5/3 884.359 cents major sixth, BP sixth 5: 1 22/13 910.790 cents tridecimal major sixth 5: 1 12/7 933.129 cents septimal major sixth 6: 1 7/4 968.826 cents harmonic seventh 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 1 9/5 1017.596 cents just minor seventh, BP seventh 6: 1 20/11 1034.996 cents large minor seventh 6: 1 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 6: 1 24/13 1061.427 cents tridecimal neutral seventh 6: 1 13/7 1071.702 cents 16/3-tone Highest number of different intervals for one interval class: 7 Average number of different intervals per interval class: 6.66667 = 20/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 231.2 435.1 617.5 782.5 933.1 1071.7 1200.0 231.2 : 203.9 386.3 551.3 702.0 840.5 968.8 1200.0 435.1 : 182.4 347.4 498.0 636.6 764.9 996.1 1200.0 617.5 : 165.0 315.6 454.2 582.5 813.7 1017.6 1200.0 782.5 : 150.6 289.2 417.5 648.7 852.6 1035.0 1200.0 933.1 : 138.6 266.9 498.0 702.0 884.4 1049.4 1200.0 1071.7: 128.3 359.5 563.4 745.8 910.8 1061.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 128.2982 .. 231.1741 cents 52/49, 102.8758 cents 14/13, 128.2982 cents 2: 266.8709 .. 435.0841 cents 54/49, 168.2132 cents 49/48, 35.6968 cents 3: 417.5080 .. 617.4878 cents 55/49, 199.9798 cents 98/99,-17.5761 cents 4: 582.5122 .. 782.4920 cents 55/49, 199.9798 cents 49/50,-34.9756 cents 5: 764.9159 .. 933.1291 cents 54/49, 168.2132 cents 98/99,-17.5761 cents 6: 968.8259 .. 1071.7018 cents 52/49, 102.8758 cents 49/48, 35.6968 cents
SHOW DATA
Number of notes : 7 -- Interval properties -- Smallest interval : 14/13, 128.2982 cents, class 1 Average step (divided formal octave): 171.4286 cents Largest one step interval : 8/7, 231.1741 cents Average / Smallest step : 1.336172 Largest / Average step : 1.348516 Largest / Smallest step : 1.801849 Median interval of one step : 11/10, 165.0042 cents, amount: 1 Least squares average step : 182.71164 cents, oct.: 1278.98149 cents Scale is not proper Step pattern alph. order: ABCDEFG Step pattern size order : GFEDCBA Scale is sum-free (all different intervals) Scale is a Constant Structure, by a margin of 17.57613 cents Scale diversity : 1.628017 Lumma stability : 0.273325 Lumma impropriety factor : 0.058440 Rothenberg efficiency : 0.299320 redundancy: 0.700680 Efficiency x scale size : 2.095238 Number of different interval sizes : 40 = 6.66667 / class Number of one step interval sizes : 7 Highest interval variety : 7 Mean interval variety : 6.66667 = 20/3 Median interval variety : 6 Lowest interval variety : 6 Smallest interval difference : 169/168, 10.2744 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 2 Most common triad is 0.0 498.045 701.955 cents, amount: 1 Number of recognisable fifths : 2, average 701.9550 cents All steps are superparticular (epimoric) Steps are strictly decreasing Scale is harmonic, base degree is : 0 Length of longest subharmonic part : 1 at degree 0 Scale is an arithmetic sequence and a class 1 isoharmonic scale Formal octave complements present : 1 = 14.2857% Scale is differentially coherent -- Rational properties -- Prime limit : 13 Odd number limit : 13 (O: 13 U: 13) Highest odd numerator or denominator: 13 Scale harmonicity : 0.015301 Average absolute harmonicity : 0.189809 Specific harmonicity : 0.119348 Fundamental : 1/7, -2.8074 octaves, 37.3751 Hz. Guide tone : 51480, 15.6517 octaves, 13468484.102 Hz. Exponens Consonantiae : 3.603600E+05, 18.45908 octaves Euler's gradus suavitatis : 40 Sum of Mann's harmonic distance : 54.0, average 7.71429 Mersenne's string divisions : 232560 Sum of van Prooijen's expressibility: 5.64487, average 0.80641 Sum of Tenney's harmonic distance : 11.46347, average 1.63764 Vogel's harmonic complexity : 7.42857 Wille's k value : 6 Wilson's harmonic complexity : 42 Rectangular lattice diameter : 6 Triangular lattice diameter : 2 Lattice compactness : 58.95166, average 2.10542 Lattice compactness (without 2's) : 39.97933, average 1.42783 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: 0 .. 3 1.00000 7 1 3: 0 .. 2 0.42857 3 5: 0 .. 1 0.14286 1 7: -1 .. 0 -0.85714 6 4 11: 0 .. 1 0.14286 1 1 13: 0 .. 1 0.14286 1 1 Average exponent except 2's : 0 / 7 = 0.00000 Average absolute exponent except 2's: 12 / 7 = 1.71429 Scale is weakly epimorphic with val: <7 11 16 19 24 27| Scale is JI-epimorphic with val: <7 11 16 20 24 26| = patent Scale is weakly epimorphic with val: <7 11 17 19 23 27| Scale is weakly epimorphic with val: <7 11 17 20 23 26| Scale is weakly epimorphic with val: <7 12 15 20 23 25|
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 75.3336 c. M: 103.2021 c. Lebeng 8: 1 2 1 1 1 1 1 P M ME S SD: 37.3053 c. M: 81.1741 c. 9: 2 1 2 1 1 1 1 N D3 S SD: 27.3629 c. M:-49.1789 c. 10: 2 2 1 2 1 1 1 N D3 SD: 30.4707 c. M:-57.5080 c. 11: 2 2 2 1 2 1 1 N D3 SD: 25.7427 c. M:-48.6891 c. 12: 2 2 2 2 1 2 1 P D3 S SD: 25.9488 c. M: 35.0841 c. Lydian Augmented, Lydian sharp 5 14: 3 2 2 2 2 2 1 P T3 SD: 20.8842 c. M:-42.5840 c. 16: 3 3 2 2 2 2 2 P D3 S SD: 21.3497 c. M: 33.1291 c. 17: 3 3 3 2 2 2 2 N S SD: 13.4648 c. M: 19.4094 c. 25: 5 4 4 3 3 3 3 N SD: 12.1468 c. M: 21.1291 c. 26: 5 4 4 4 3 3 3 P SD: 11.3572 c. M: 19.6995 c. 27: 5 5 4 4 3 3 3 N SD: 8.6362 c. M:-17.5080 c. 28: 5 5 4 4 4 3 3 P SD: 11.0231 c. M: 17.4878 c. 29: 6 5 4 4 4 3 3 N SD: 12.4393 c. M:-20.0883 c. 30: 6 5 4 5 3 4 3 N SD: 11.6842 c. M:-17.5080 c. 31: 6 5 5 4 4 4 3 N SD: 6.8083 c. M:-12.1692 c. 35: 7 6 5 5 4 4 4 N SD: 7.1831 c. M:-10.6302 c. 37: 7 6 6 5 5 4 4 N SD: 6.2559 c. M: 13.4625 c. 41: 8 7 6 6 5 5 4 N SD: 5.7389 c. M:-11.2251 c. 46: 9 8 7 6 6 5 5 N SD: 5.3187 c. M:-8.5991 c. 55: 11 9 8 8 7 6 6 N SD: 4.8394 c. M:-8.8259 c. 57: 11 10 8 8 7 7 6 N SD: 4.7934 c. M:-7.0212 c. 58: 11 10 9 8 7 7 6 N SD: 2.9028 c. M:-4.1603 c. 72: 14 12 11 10 9 8 8 N SD: 2.2201 c. M: 5.0351 c. 94: 18 16 14 13 12 11 10 N SD: 2.4316 c. M: 4.7218 c. 103: 20 17 16 14 13 12 11 N SD: 1.8653 c. M: 4.0161 c. 113: 22 19 17 16 14 13 12 N SD: 1.7900 c. M:-3.3487 c. 121: 23 21 18 17 15 14 13 N SD: 1.6933 c. M: 3.0749 c. 130: 25 22 20 18 16 15 14 N SD: 1.1147 c. M:-2.1233 c. 198: 38 34 30 27 25 23 21 N SD: 0.7951 c. M:-1.2795 c. 224: 43 38 34 31 28 26 24 N SD: 0.8601 c. M: 1.4164 c. 253: 49 43 38 35 32 29 27 N SD: 0.8949 c. M:-1.2795 c. 270: 52 46 41 37 34 31 29 N SD: 0.3325 c. M: 0.5906 c. 494: 95 84 75 68 62 57 53 N SD: 0.3535 c. M: 0.4838 c. 535: 103 91 81 74 67 62 57 N SD: 0.3310 c. M: 0.6654 c. 571: 110 97 87 78 72 66 61 N SD: 0.3056 c. M: 0.7057 c. 653: 126 111 99 90 82 75 70 N SD: 0.3264 c. M:-0.4442 c. 665: 128 113 101 92 83 77 71 N SD: 0.3185 c. M:-0.6659 c. 670: 129 114 102 92 84 77 72 N SD: 0.3127 c. M: 0.6570 c. 684: 132 116 104 94 86 79 73 N SD: 0.1935 c. M:-0.4049 c. 805: 155 137 122 111 101 93 86 N SD: 0.1674 c. M: 0.3450 c. 814: 157 138 124 112 102 94 87 N SD: 0.1904 c. M:-0.3089 c. 908: 175 154 138 125 114 105 97 N SD: 0.1759 c. M: 0.3072 c. 935: 180 159 142 129 117 108 100 N SD: 0.1757 c. M:-0.3957 c. 954: 184 162 145 131 120 110 102 N SD: 0.1511 c. M:-0.2724 c. 1012: 195 172 154 139 127 117 108 N SD: 0.1580 c. M:-0.2988 c. 1075: 207 183 163 148 135 124 115 N SD: 0.1350 c. M:-0.2647 c. 1084: 209 184 165 149 136 125 116 N SD: 0.1387 c. M:-0.2244 c. 1178: 227 200 179 162 148 136 126 N SD: 0.1007 c. M: 0.1703 c. 1236: 238 210 188 170 155 143 132 N SD: 0.0962 c. M:-0.1429 c. 1448: 279 246 220 199 182 167 155 N SD: 0.0946 c. M: 0.1716 c. 1506: 290 256 229 207 189 174 161 N SD: 0.0484 c. M: 0.0984 c. 1776: 342 302 270 244 223 205 190 N SD: 0.0633 c. M: 0.0930 c. 2190: 422 372 333 301 275 253 234 N SD: 0.0435 c. M:-0.0791 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: 40.2685 cents 3 x 4 x x 5 x 6 S SD: 26.5469 cents 4 x 5 x 6:7 x 8 S SD: 25.1732 cents 5:6 x 7:8 x 9:10 S SD: 22.1436 cents 6:7:8 x 9:10:11:12 S SD: 20.1399 cents 7:8:9:10:11:12:13:14 H SD: 0.0000 cents