Scala analysis: thirteenlim
Thirteen-limit otonal chord
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 9/8 203.910002 major whole tone 2: 5/4 386.313714 major third 3: 11/8 551.317942 undecimal semi-augmented fourth 4: 3/2 701.955001 perfect fifth 5: 13/8 840.527662 tridecimal neutral sixth 6: 7/4 968.825906 harmonic seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 8/7 231.1741 septimal whole tone 1: 9/8 203.9100 major whole tone 2: 10/9 182.4037 minor whole tone 3: 11/10 165.0042 Ptolemy's second, 4/5-tone 4: 12/11 150.6371 undecimal neutral second, 3/4-tone 5: 13/12 138.5727 tridecimal 2/3-tone 6: 14/13 128.2982 2/3-tone 7: 8/7 231.1741 septimal whole 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 : 203.9 386.3 551.3 702.0 840.5 968.8 1200.0 203.9 : 182.4 347.4 498.0 636.6 764.9 996.1 1200.0 386.3 : 165.0 315.6 454.2 582.5 813.7 1017.6 1200.0 551.3 : 150.6 289.2 417.5 648.7 852.6 1035.0 1200.0 702.0 : 138.6 266.9 498.0 702.0 884.4 1049.4 1200.0 840.5 : 128.3 359.5 563.4 745.8 910.8 1061.4 1200.0 968.8 : 231.2 435.1 617.5 782.5 933.1 1071.7 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 : 170.38504 cents, oct.: 1192.69525 cents Scale is not proper Step pattern alph. order: ABCDEFG Step pattern size order : FEDCBAG 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) Scale is harmonic, base degree is : 6 Length of longest subharmonic part : 1 at degree 0 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 : 0.015071 Average absolute harmonicity : 0.240019 Specific harmonicity : 0.128303 Fundamental : 1/8, -3.0000 octaves, 32.7032 Hz. Guide tone : 90090, 16.4591 octaves, 23569847.178 Hz. Exponens Consonantiae : 7.207200E+05, 19.45908 octaves Euler's gradus suavitatis : 41 Sum of Mann's harmonic distance : 42.5, average 6.07143 Mersenne's string divisions : 1119195 Sum of van Prooijen's expressibility: 5.13077, average 0.73297 Sum of Tenney's harmonic distance : 9.64622, average 1.37803 Vogel's harmonic complexity : 7.57143 Wille's k value : 6 Wilson's harmonic complexity : 42 Rectangular lattice diameter : 6 Triangular lattice diameter : 2 Lattice compactness : 67.82232, average 2.42223 Lattice compactness (without 2's) : 38.08641, average 1.36023 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: -3 .. 1 -1.85714 15 1 3: 0 .. 2 0.42857 3 2 5: 0 .. 1 0.14286 1 1 7: 0 .. 1 0.14286 1 1 11: 0 .. 1 0.14286 1 1 13: 0 .. 1 0.14286 1 1 Average exponent except 2's : 7 / 7 = 1.00000 Average absolute exponent except 2's: 7 / 7 = 1.00000 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: 34.6910 c. M:-59.7455 c. Lebeng 9: 1 2 1 1 1 1 2 N D3 S SD: 37.1085 c. M: 70.5767 c. 10: 2 1 2 1 1 1 2 N D3 SD: 26.1005 c. M:-48.6821 c. 13: 2 2 2 1 2 1 3 N SD: 29.2097 c. M: 55.8012 c. 14: 2 2 2 2 2 1 3 P T3 SD: 28.1047 c. M: 43.4566 c. 15: 3 2 2 2 2 1 3 N SD: 22.4460 c. M:-39.4723 c. 16: 3 2 2 2 2 2 3 P D3 I SD: 18.0128 c. M: 26.9550 c. 17: 3 3 2 2 2 2 3 N S SD: 17.1579 c. M:-37.2157 c. 19: 3 3 3 2 2 2 4 N SD: 14.3775 c. M: 21.4575 c. 22: 4 3 3 3 2 3 4 P SD: 11.8160 c. M: 22.3458 c. 24: 4 4 3 3 3 2 5 N SD: 9.6533 c. M: 18.8259 c. 26: 4 4 4 3 3 3 5 P SD: 11.0774 c. M: 19.2946 c. 28: 5 4 4 3 4 3 5 P SD: 11.7533 c. M:-16.8884 c. 29: 5 4 4 4 3 3 6 N SD: 10.9774 c. M: 17.1018 c. 30: 5 5 4 4 3 3 6 N SD: 9.8690 c. M:-18.0450 c. 31: 5 5 4 4 4 3 6 N SD: 7.0397 c. M:-11.0852 c. 37: 6 6 5 5 4 4 7 N SD: 6.0145 c. M:-11.5585 c. 41: 7 6 6 5 5 4 8 N SD: 4.3899 c. M:-8.2528 c. 46: 8 7 6 6 5 5 9 N SD: 3.9956 c. M: 5.7451 c. 53: 9 8 7 7 6 6 10 N SD: 3.6879 c. M: 7.9217 c. 63: 11 9 9 8 7 7 12 N SD: 3.4217 c. M:-5.6138 c. 72: 12 11 10 9 8 8 14 N SD: 3.5082 c. M: 7.1943 c. 77: 13 12 10 10 9 8 15 N SD: 2.8077 c. M: 5.8634 c. 87: 15 13 12 11 10 9 17 N SD: 1.8126 c. M: 3.3087 c. 94: 16 14 13 12 11 10 18 N SD: 1.8115 c. M: 3.3350 c. 128: 22 19 18 16 15 13 25 N SD: 2.2200 c. M:-3.2223 c. 130: 22 20 18 16 15 14 25 N SD: 1.1712 c. M:-2.5282 c. 176: 30 27 24 22 20 19 34 N SD: 1.2423 c. M:-2.3226 c. 178: 30 27 25 22 21 19 34 N SD: 1.6215 c. M:-2.1690 c. 181: 31 27 25 23 21 19 35 N SD: 1.2199 c. M: 1.7833 c. 183: 31 28 25 23 21 20 35 N SD: 0.8659 c. M:-1.6659 c. 217: 37 33 30 27 25 23 42 N SD: 0.8629 c. M:-1.6774 c. 224: 38 34 31 28 26 24 43 N SD: 0.4906 c. M:-0.8170 c. 270: 46 41 37 34 31 29 52 N SD: 0.3395 c. M:-0.5344 c. 494: 84 75 68 62 57 53 95 N SD: 0.1714 c. M:-0.4049 c. 764: 130 116 105 96 88 82 147 N SD: 0.1808 c. M:-0.2840 c. 925: 157 141 127 116 107 99 178 N SD: 0.1806 c. M:-0.2809 c. 1012: 172 154 139 127 117 108 195 N SD: 0.1220 c. M:-0.2476 c. 1178: 200 179 162 148 136 126 227 N SD: 0.1504 c. M: 0.2356 c. 1195: 203 182 164 150 138 128 230 N SD: 0.1408 c. M:-0.2972 c. 1236: 210 188 170 155 143 132 238 N SD: 0.1208 c. M:-0.2490 c. 1282: 218 195 176 161 148 137 247 N SD: 0.1204 c. M:-0.2697 c. 1376: 234 209 189 173 159 147 265 N SD: 0.1136 c. M:-0.1700 c. 1395: 237 212 192 175 161 149 269 N SD: 0.1027 c. M: 0.2238 c. 1506: 256 229 207 189 174 161 290 N SD: 0.0880 c. M:-0.1405 c. 1802: 306 274 248 226 208 193 347 N SD: 0.0924 c. M: 0.1364 c. 1889: 321 287 260 237 218 202 364 N SD: 0.0582 c. M:-0.0849 c. 2190: 372 333 301 275 253 234 422 N SD: 0.0510 c. M: 0.0851 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: 0.0000 cents 3 x x 4 x 5 x 6 S SD: 22.9956 cents 4 x 5 x 6 x 7:8 S SD: 0.0000 cents 5 x 6:7 x 8:9:10 S SD: 19.0438 cents 6:7 x 8:9:10:11:12 S SD: 20.5544 cents 7:8:9:10 x 11:12:14 S SD: 18.3657 cents 8:9:10:11:12:13:14:16 S SD: 0.0000 cents