Scala analysis: 08_o8
Mode 8 of the harmonic series.
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: 15/8 1088.268715 classic major seventh 8: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 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: 15/14 119.4428 major diatonic semitone 8: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 16/15 111.731 cents minor diatonic semitone 1: 1 15/14 119.443 cents major diatonic semitone 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 2: 1 8/7 231.174 cents septimal whole tone 2: 1 15/13 247.741 cents tridecimal 5/4-tone 2: 1 7/6 266.871 cents septimal minor third 2: 1 13/11 289.210 cents tridecimal minor third 2: 2 6/5 315.641 cents minor third 2: 1 11/9 347.408 cents undecimal neutral third 2: 1 5/4 386.314 cents major third 3: 1 16/13 359.472 cents tridecimal neutral third 3: 1 5/4 386.314 cents major third 3: 1 14/11 417.508 cents undecimal diminished fourth or major third 3: 1 9/7 435.084 cents septimal major third, BP 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 4: 1 4/3 498.045 cents perfect fourth 4: 1 15/11 536.951 cents undecimal augmented fourth 4: 1 18/13 563.382 cents tridecimal augmented fourth 4: 1 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 4: 1 10/7 617.488 cents Euler's tritone 4: 1 13/9 636.618 cents tridecimal diminished fifth 4: 1 22/15 663.049 cents undecimal diminished fifth 4: 1 3/2 701.955 cents perfect fifth 5: 1 16/11 648.682 cents undecimal semi-diminished fifth 5: 2 3/2 701.955 cents perfect fifth 5: 1 20/13 745.786 cents tridecimal semi-augmented fifth 5: 1 14/9 764.916 cents septimal minor sixth 5: 1 11/7 782.492 cents undecimal augmented fifth 5: 1 8/5 813.686 cents minor sixth 5: 1 13/8 840.528 cents tridecimal neutral sixth 6: 1 8/5 813.686 cents minor sixth 6: 1 18/11 852.592 cents undecimal neutral sixth 6: 2 5/3 884.359 cents major sixth, BP sixth 6: 1 22/13 910.790 cents tridecimal major sixth 6: 1 12/7 933.129 cents septimal major sixth 6: 1 26/15 952.259 cents tridecimal semi-augmented sixth 6: 1 7/4 968.826 cents harmonic seventh 7: 1 16/9 996.090 cents Pythagorean minor seventh 7: 1 9/5 1017.596 cents just minor seventh, BP seventh 7: 1 20/11 1034.996 cents large minor seventh 7: 1 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 7: 1 24/13 1061.427 cents tridecimal neutral seventh 7: 1 13/7 1071.702 cents 16/3-tone 7: 1 28/15 1080.557 cents grave major seventh 7: 1 15/8 1088.269 cents classic major seventh Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 7.42857 = 52/7
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 0.0 : 203.9 386.3 551.3 702.0 840.5 968.8 1088.3 1200.0 203.9 : 182.4 347.4 498.0 636.6 764.9 884.4 996.1 1200.0 386.3 : 165.0 315.6 454.2 582.5 702.0 813.7 1017.6 1200.0 551.3 : 150.6 289.2 417.5 537.0 648.7 852.6 1035.0 1200.0 702.0 : 138.6 266.9 386.3 498.0 702.0 884.4 1049.4 1200.0 840.5 : 128.3 247.7 359.5 563.4 745.8 910.8 1061.4 1200.0 968.8 : 119.4 231.2 435.1 617.5 782.5 933.1 1071.7 1200.0 1088.3: 111.7 315.6 498.0 663.0 813.7 952.3 1080.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 111.7313 .. 203.9100 cents 135/128, 92.1787 cents 16/15, 111.7313 cents 2: 231.1741 .. 386.3137 cents 35/32, 155.1396 cents 64/63, 27.2641 cents 3: 359.4723 .. 551.3179 cents 143/128, 191.8456 cents 64/65,-26.8414 cents 4: 498.0450 .. 701.9550 cents 9/8, 203.9100 cents 32/33,-53.2729 cents 5: 648.6821 .. 840.5277 cents 143/128, 191.8456 cents 32/33,-53.2729 cents 6: 813.6863 .. 968.8259 cents 35/32, 155.1396 cents 64/65,-26.8414 cents 7: 996.0900 .. 1088.2687 cents 135/128, 92.1787 cents 64/63, 27.2641 cents
SHOW DATA
Number of notes : 8 -- Interval properties -- Smallest interval : 16/15, 111.7313 cents, class 1 Average step (divided formal octave): 150.0000 cents Largest one step interval : 9/8, 203.9100 cents Average / Smallest step : 1.342507 Largest / Average step : 1.359400 Largest / Smallest step : 1.825004 Median interval of one step : 144.60486 cents, amount: 0 Least squares average step : 160.15581 cents, oct.: 1281.24651 cents Scale is not proper Step pattern alph. order: ABCDEFGH Step pattern size order : HGFEDCBA Scale is not a Constant Structure Scale diversity : 1.621543 Lumma stability : 0.231659 Lumma impropriety factor : 0.133524 Rothenberg efficiency : 0.285714 redundancy: 0.714286 Efficiency x scale size : 2.285714 Number of different interval sizes : 48 = 6.85714 / class Number of one step interval sizes : 8 Highest interval variety : 8 Mean interval variety : 7.42857 = 52/7 Median interval variety : 7 Lowest interval variety : 7 Smallest interval difference : 225/224, 7.7115 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 3 Number of recognisable fifths : 3, 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 = 12.5000% Scale is differentially coherent -- Rational properties -- Prime limit : 13 Odd number limit : 15 (O: 15 U: 15) Highest odd numerator or denominator: 15 Scale harmonicity : 0.015071 Average absolute harmonicity : 0.220376 Specific harmonicity : 0.129827 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 : 54.0, average 6.75000 Mersenne's string divisions : 25741485 Sum of van Prooijen's expressibility: 6.30686, average 0.78836 Sum of Tenney's harmonic distance : 11.72540, average 1.46567 Vogel's harmonic complexity : 7.62500 Wille's k value : 7 Wilson's harmonic complexity : 42 Rectangular lattice diameter : 6 Triangular lattice diameter : 2 Lattice compactness : 85.91018, average 2.38639 Lattice compactness (without 2's) : 49.52520, average 1.37570 Number of different primes : 6 Prime exponents' range, average, count, tones@limit: 2: -3 .. 1 -2.00000 18 1 3: 0 .. 2 0.50000 4 2 5: 0 .. 1 0.25000 2 2 7: 0 .. 1 0.12500 1 1 11: 0 .. 1 0.12500 1 1 13: 0 .. 1 0.12500 1 1 Average exponent except 2's : 9 / 8 = 1.12500 Average absolute exponent except 2's: 9 / 8 = 1.12500 Scale is weakly epimorphic with val: <8 13 20 22 27 31| Scale is weakly epimorphic with val: <8 13 20 23 27 30|
FIT/MODE
8: 1 1 1 1 1 1 1 1 SP B ME I SD: 75.3436 c. M: 101.9550 c. 9: 1 2 1 1 1 1 1 1 P M ME SD: 35.5422 c. M: 70.5767 c. 10: 2 1 2 1 1 1 1 1 N D3 SD: 24.5892 c. M:-48.6821 c. 14: 2 2 2 2 2 1 2 1 P D3 SD: 27.8523 c. M: 43.4566 c. 15: 3 2 2 2 2 1 2 1 N SD: 23.8056 c. M:-39.4723 c. 17: 3 2 3 2 2 2 1 2 N SD: 18.2225 c. M: 33.3725 c. 19: 3 3 3 2 2 2 2 2 N SD: 14.4035 c. M: 21.4575 c. 22: 4 3 3 3 2 3 2 2 N SD: 11.0922 c. M: 22.3458 c. 24: 4 4 3 3 3 2 3 2 N SD: 9.9368 c. M: 18.8259 c. 28: 5 4 4 3 4 3 2 3 N SD: 12.5029 c. M:-16.8884 c. 29: 5 4 4 4 3 3 3 3 N SD: 11.1660 c. M: 17.1018 c. 30: 5 5 4 4 3 3 3 3 N SD: 9.6834 c. M:-18.0450 c. 31: 5 5 4 4 4 3 3 3 N SD: 6.7661 c. M:-11.0852 c. 41: 7 6 6 5 5 4 4 4 N SD: 4.5199 c. M:-8.2528 c. 46: 8 7 6 6 5 5 5 4 N SD: 4.5589 c. M:-7.3835 c. 53: 9 8 7 7 6 6 5 5 N SD: 3.4889 c. M: 7.9217 c. 57: 10 8 8 7 7 6 6 5 N SD: 5.1263 c. M: 7.3663 c. 58: 10 9 8 7 7 6 6 5 N SD: 5.6136 c. M:-8.2830 c. 61: 10 10 8 8 7 6 6 6 N SD: 5.4109 c. M: 7.1887 c. 63: 11 9 9 8 7 7 6 6 N SD: 3.3257 c. M:-5.6138 c. 72: 12 11 10 9 8 8 7 7 N SD: 3.7167 c. M: 7.1943 c. 74: 13 11 10 9 9 8 7 7 N SD: 3.6286 c. M:-6.9008 c. 77: 13 12 10 10 9 8 8 7 N SD: 2.7873 c. M: 5.8634 c. 87: 15 13 12 11 10 9 9 8 N SD: 1.7650 c. M: 3.3087 c. 128: 22 19 18 16 15 13 13 12 N SD: 2.0943 c. M:-3.2223 c. 130: 22 20 18 16 15 14 13 12 N SD: 1.1472 c. M:-2.5282 c. 181: 31 27 25 23 21 19 18 17 N SD: 1.1922 c. M: 1.7833 c. 183: 31 28 25 23 21 20 18 17 N SD: 0.8150 c. M:-1.6659 c. 217: 37 33 30 27 25 23 22 20 N SD: 0.9010 c. M:-1.6774 c. 224: 38 34 31 28 26 24 22 21 N SD: 0.5333 c. M:-0.8170 c. 270: 46 41 37 34 31 29 27 25 N SD: 0.3859 c. M:-0.6202 c. 441: 75 67 61 55 51 47 44 41 N SD: 0.4029 c. M:-1.0630 c. 494: 84 75 68 62 57 53 49 46 N SD: 0.1604 c. M:-0.4049 c. 764: 130 116 105 96 88 82 76 71 N SD: 0.1851 c. M:-0.2840 c. 882: 150 134 121 111 102 94 88 82 N SD: 0.1792 c. M: 0.2975 c. 925: 157 141 127 116 107 99 92 86 N SD: 0.1786 c. M:-0.2809 c. 1012: 172 154 139 127 117 108 101 94 N SD: 0.1485 c. M:-0.2688 c. 1084: 184 165 149 136 125 116 108 101 N SD: 0.1575 c. M: 0.3063 c. 1106: 188 168 152 139 128 118 110 103 N SD: 0.1373 c. M:-0.3403 c. 1236: 210 188 170 155 143 132 123 115 N SD: 0.1166 c. M:-0.2490 c. 1299: 221 197 179 163 150 139 129 121 N SD: 0.1608 c. M:-0.2470 c. 1308: 222 199 180 164 151 140 130 122 N SD: 0.1349 c. M: 0.2403 c. 1376: 234 209 189 173 159 147 137 128 N SD: 0.1123 c. M:-0.1700 c. 1395: 237 212 192 175 161 149 139 130 N SD: 0.1020 c. M: 0.2238 c. 1506: 256 229 207 189 174 161 150 140 N SD: 0.1035 c. M:-0.1775 c. 1578: 268 240 217 198 182 169 157 147 N SD: 0.0920 c. M: 0.2235 c. 1665: 283 253 229 209 192 178 166 155 N SD: 0.0899 c. M: 0.1773 c. 1802: 306 274 248 226 208 193 179 168 N SD: 0.1004 c. M: 0.1444 c. 1826: 310 278 251 229 211 195 182 170 N SD: 0.0999 c. M: 0.1860 c. 1848: 314 281 254 232 213 198 184 172 N SD: 0.0980 c. M: 0.2679 c. 1889: 321 287 260 237 218 202 188 176 N SD: 0.0604 c. M:-0.0849 c. 2190: 372 333 301 275 253 234 218 204 N SD: 0.0508 c. M: 0.0851 c.
FIT/HARMONIC
1 x x x x x x x 2 S SD: 0.0000 cents 2 x x x 3 x x x 4 S SD: 0.0000 cents 3 x x 4 x 5 x x 6 S SD: 22.9956 cents 4 x 5 x 6 x 7 x 8 S SD: 0.0000 cents 5 x 6:7 x 8:9 x 10 S SD: 19.0438 cents 6:7 x 8:9:10 x 11:12 S SD: 16.8629 cents 7:8:9:10 x 11:12:13:14 S SD: 15.9189 cents 8:9:10:11:12:13:14:15:16 H SD: 0.0000 cents