Scala analysis: xen18-schulter-symmetrical
A JI version of a symmetrical scale in 17-WT
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 14/13 128.298245 2/3-tone 2: 7/6 266.870906 septimal minor third 3: 4/3 498.044999 perfect fourth 4: 3/2 701.955001 perfect fifth 5: 21/13 830.253246 6: 7/4 968.825906 harmonic seventh 7: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 8/7 231.1741 septimal whole tone 1: 14/13 128.2982 2/3-tone 2: 13/12 138.5727 tridecimal 2/3-tone 3: 8/7 231.1741 septimal whole tone 4: 9/8 203.9100 major whole tone 5: 14/13 128.2982 2/3-tone 6: 13/12 138.5727 tridecimal 2/3-tone 7: 8/7 231.1741 septimal whole tone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 14/13 128.298 cents 2/3-tone 1: 2 13/12 138.573 cents tridecimal 2/3-tone 1: 1 9/8 203.910 cents major whole tone 1: 2 8/7 231.174 cents septimal whole tone 2: 2 7/6 266.871 cents septimal minor third 2: 1 63/52 332.208 cents 2: 1 16/13 359.472 cents tridecimal neutral third 2: 2 26/21 369.747 cents 2: 1 9/7 435.084 cents septimal major third, BP third 3: 1 21/16 470.781 cents narrow fourth 3: 4 4/3 498.045 cents perfect fourth 3: 1 18/13 563.382 cents tridecimal augmented fourth 3: 1 39/28 573.657 cents 4: 1 56/39 626.343 cents 4: 1 13/9 636.618 cents tridecimal diminished fifth 4: 4 3/2 701.955 cents perfect fifth 4: 1 32/21 729.219 cents wide fifth 5: 1 14/9 764.916 cents septimal minor sixth 5: 2 21/13 830.253 cents 5: 1 13/8 840.528 cents tridecimal neutral sixth 5: 1 104/63 867.792 cents 5: 2 12/7 933.129 cents septimal major sixth 6: 2 7/4 968.826 cents harmonic seventh 6: 1 16/9 996.090 cents Pythagorean minor seventh 6: 2 24/13 1061.427 cents tridecimal neutral seventh 6: 2 13/7 1071.702 cents 16/3-tone Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 4.33333 = 13/3
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 0.0 : 128.3 266.9 498.0 702.0 830.3 968.8 1200.0 128.3 : 138.6 369.7 573.7 702.0 840.5 1071.7 1200.0 266.9 : 231.2 435.1 563.4 702.0 933.1 1061.4 1200.0 498.0 : 203.9 332.2 470.8 702.0 830.3 968.8 1200.0 702.0 : 128.3 266.9 498.0 626.3 764.9 996.1 1200.0 830.3 : 138.6 369.7 498.0 636.6 867.8 1071.7 1200.0 968.8 : 231.2 359.5 498.0 729.2 933.1 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: 470.7809 .. 573.6568 cents 52/49, 102.8758 cents 49/48, 35.6968 cents 4: 626.3432 .. 729.2191 cents 52/49, 102.8758 cents 1568/1521, 52.6865 cents 5: 764.9159 .. 933.1291 cents 54/49, 168.2132 cents 49/48, 35.6968 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 : 13/12, 138.5727 cents, amount: 2 Least squares average step : 166.63012 cents, oct.: 1166.41084 cents Scale is strictly proper Step pattern alph. order: ABCDABC Step pattern size order : ABDCABD Scale is a Constant Structure, by a margin of 35.69681 cents Scale diversity : 1.485107 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.376725 Rothenberg efficiency : 0.421769 redundancy: 0.578231 Efficiency x scale size : 2.952381 Number of different interval sizes : 26 = 4.33333 / class Number of one step interval sizes : 4 Highest interval variety : 5 Mean interval variety : 4.33333 = 13/3 Median interval variety : 4 Lowest interval variety : 4 Smallest interval difference : 169/168, 10.2744 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 4 Number of recognisable fifths : 4, average 701.9550 cents Scale is Dorian octave species of abstract tetrachord Scale contains two identical disjunct tetrachords Scale contains a complete diamond : 1 3 All steps are superparticular (epimoric) Length of longest harmonic part : 2 at degree 2 Length of longest subharmonic part : 2 at degree 0 Formal octave complements present : 3 = 42.8571% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval classes 3 and 4 combined Limited inverse transpositions with margin 168.2132 cents: 6 -- Rational properties -- Prime limit : 13 Odd number limit : 63 (O: 63 U: 63) Highest odd numerator or denominator: 21 Scale harmonicity : 0.023939 Average absolute harmonicity : 0.242638 Specific harmonicity : 0.118866 Fundamental : 1/156, -7.2854 octaves, 1.6771 Hz. Guide tone : 84, 6.3923 octaves, 21976.547 Hz. Exponens Consonantiae : 1.310400E+04, 13.67772 octaves Euler's gradus suavitatis : 27 Sum of Mann's harmonic distance : 50.0, average 7.14286 Mersenne's string divisions : 4594590 Sum of van Prooijen's expressibility: 5.08060, average 0.72580 Sum of Tenney's harmonic distance : 9.92500, average 1.41786 Vogel's harmonic complexity : 18.28571 Wille's k value : 136 Wilson's harmonic complexity : 26 Rectangular lattice diameter : 4 Triangular lattice diameter : 3 Lattice compactness : 62.61661, average 2.23631 Lattice compactness (without 2's) : 38.57775, average 1.37778 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -2 .. 2 0.00000 8 1 3: -1 .. 1 0.00000 4 2 7: 0 .. 1 0.57143 4 2 13: -1 .. 0 -0.28571 2 2 Average exponent except 2's : 2 / 7 = 0.28571 Average absolute exponent except 2's: 10 / 7 = 1.42857 Scale is weakly epimorphic with val: <7 11 19 24| Scale is JI-epimorphic with val: <7 11 20 26| = patent Scale is weakly epimorphic with val: <7 12 18 29|
FIT/MODE
7: 1 1 1 1 1 1 1 SP B ME I SD: 42.1803 c. M:-75.9862 c. Lebeng 9: 1 1 2 1 1 1 2 P M ME S SD: 25.8869 c. M: 35.4926 c. Pseudo Pelog, Pelogic-7, Antiionian 10: 1 1 2 2 1 1 2 P D3 S SD: 15.1895 c. M: 26.8709 c. 17: 2 2 3 3 2 2 3 SP D3 S SD: 12.5102 c. M:-19.4094 c. 19: 2 2 4 3 2 2 4 P SD: 11.0581 c. M: 21.4575 c. 22: 2 3 4 4 2 3 4 P SD: 10.8205 c. M: 19.2073 c. Raga Kanakangi 26: 3 3 5 4 3 3 5 SP SD: 7.4729 c. M:-10.1633 c. 36: 4 4 7 6 4 4 7 SP SD: 2.5963 c. M:-5.0351 c. 58: 6 7 11 10 6 7 11 SP SD: 2.5674 c. M: 4.1603 c. 77: 8 9 15 13 8 9 15 SP SD: 2.4717 c. M: 4.2792 c. 82: 9 9 16 14 9 9 16 SP SD: 2.6193 c. M:-3.8931 c. 84: 9 10 16 14 9 10 16 SP SD: 2.3328 c. M:-4.5577 c. 94: 10 11 18 16 10 11 18 SP SD: 0.7637 c. M:-1.3869 c. 130: 14 15 25 22 14 15 25 SP SD: 0.5759 c. M:-0.9325 c. 224: 24 26 43 38 24 26 43 SP SD: 0.5047 c. M:-0.9862 c. 234: 25 27 45 40 25 27 45 SP SD: 0.4179 c. M:-0.6091 c. 270: 29 31 52 46 29 31 52 SP SD: 0.4265 c. M:-0.8579 c. 328: 35 38 63 56 35 38 63 SP SD: 0.3960 c. M:-0.6863 c. 337: 36 39 65 57 36 39 65 SP SD: 0.3600 c. M: 0.5797 c. 347: 37 40 67 59 37 40 67 SP SD: 0.3442 c. M: 0.5885 c. 364: 39 42 70 62 39 42 70 SP SD: 0.3045 c. M:-0.5160 c. 441: 47 51 85 75 47 51 85 SP SD: 0.2202 c. M: 0.4071 c. 477: 51 55 92 81 51 55 92 SP SD: 0.1360 c. M: 0.2724 c. 571: 61 66 110 97 61 66 110 SP SD: 0.0654 c. M: 0.1307 c. 1048: 112 121 202 178 112 121 202 SP SD: 0.0742 c. M: 0.1236 c. 1178: 126 136 227 200 126 136 227 SP SD: 0.0587 c. M:-0.0874 c. 1412: 151 163 272 240 151 163 272 SP SD: 0.0300 c. M:-0.0584 c. 1983: 212 229 382 337 212 229 382 SP SD: 0.0079 c. M:-0.0117 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: 18.0352 cents 4 x x 5:6 x 7:8 S SD: 27.9328 cents 5 x 6:7 x 8:9:10 S SD: 22.0602 cents 6 x 7:8:9:10:11:12 S SD: 16.1706 cents 7 x 8:9 x 11:12:14 SD: 18.7550 cents 8 x 9:11:12:13:14:16 SD: 13.8520 cents 10:11:12:13:15:16:18:20 SD: 13.0149 cents 11:12:13:15:17:18:19:22 SD: 11.2431 cents 12:13:14:16:18:19:21:24 SD: 5.1692 cents 24:26:28:32:36:39:42:48 S SD: 2.0757 cents 36:39:42:48:54:58:63:72 SD: 1.6074 cents 54:58:63:72:81:87:95:108 SD: 1.6006 cents 60:65:70:80:90:97:105:120 SD: 1.4808 cents 66:71:77:88:99:107:116:132 SD: 1.4164 cents 76:82:89:101:114:123:133:152 SD: 1.3988 cents 78:84:91:104:117:126:137:156 SD: 0.9043 cents 102:110:119:136:153:165:179:204 SD: 0.8475 cents 114:123:133:152:171:184:200:228 SD: 0.8011 cents 120:129:140:160:180:194:210:240 SD: 0.4836 cents 132:142:154:176:198:213:231:264 SD: 0.3787 cents 156:168:182:208:234:252:273:312 S SD: 0.0000 cents