Scala analysis: fokker_12
Fokker's 7-limit 12-tone just scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 15/14 119.442808 major diatonic semitone 2: 9/8 203.910002 major whole tone 3: 7/6 266.870906 septimal minor third 4: 5/4 386.313714 major third 5: 4/3 498.044999 perfect fourth 6: 45/32 590.223716 diatonic tritone 7: 3/2 701.955001 perfect fifth 8: 45/28 821.397809 septimale wide minor sixth 9: 5/3 884.358713 major sixth, BP sixth 10: 7/4 968.825906 harmonic seventh 11: 15/8 1088.268715 classic major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 16/15 111.7313 minor diatonic semitone 1: 15/14 119.4428 major diatonic semitone 2: 21/20 84.4672 minor semitone 3: 28/27 62.9609 Archytas' 1/3-tone 4: 15/14 119.4428 major diatonic semitone 5: 16/15 111.7313 minor diatonic semitone 6: 135/128 92.1787 major chroma, major limma 7: 16/15 111.7313 minor diatonic semitone 8: 15/14 119.4428 major diatonic semitone 9: 28/27 62.9609 Archytas' 1/3-tone 10: 21/20 84.4672 minor semitone 11: 15/14 119.4428 major diatonic semitone 12: 16/15 111.7313 minor diatonic semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 28/27 62.961 cents Archytas' 1/3-tone 1: 2 21/20 84.467 cents minor semitone 1: 1 135/128 92.179 cents major chroma, major limma 1: 3 16/15 111.731 cents minor diatonic semitone 1: 4 15/14 119.443 cents major diatonic semitone 2: 2 49/45 147.428 cents BP minor semitone 2: 2 10/9 182.404 cents minor whole tone 2: 4 9/8 203.910 cents major whole tone 2: 4 8/7 231.174 cents septimal whole tone 3: 4 7/6 266.871 cents septimal minor third 3: 2 32/27 294.135 cents Pythagorean minor third 3: 3 6/5 315.641 cents minor third 3: 2 135/112 323.353 cents septimal wide minor third 3: 1 60/49 350.617 cents smaller approximation to neutral third 4: 4 56/45 378.602 cents septimal narrow major third 4: 4 5/4 386.314 cents major third 4: 4 9/7 435.084 cents septimal major third, BP third 5: 2 21/16 470.781 cents narrow fourth 5: 8 4/3 498.045 cents perfect fourth 5: 1 27/20 519.551 cents acute fourth 5: 1 135/98 554.527 cents 6: 4 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 6: 2 45/32 590.224 cents diatonic tritone 6: 2 64/45 609.776 cents 2nd tritone 6: 4 10/7 617.488 cents Euler's tritone 7: 1 196/135 645.473 cents 7: 1 40/27 680.449 cents grave fifth 7: 8 3/2 701.955 cents perfect fifth 7: 2 32/21 729.219 cents wide fifth 8: 4 14/9 764.916 cents septimal minor sixth 8: 4 8/5 813.686 cents minor sixth 8: 4 45/28 821.398 cents septimale wide minor sixth 9: 1 49/30 849.383 cents larger approximation to neutral sixth 9: 2 224/135 876.647 cents narrow septimal major sixth 9: 3 5/3 884.359 cents major sixth, BP sixth 9: 2 27/16 905.865 cents Pythagorean major sixth 9: 4 12/7 933.129 cents septimal major sixth 10: 4 7/4 968.826 cents harmonic seventh 10: 4 16/9 996.090 cents Pythagorean minor seventh 10: 2 9/5 1017.596 cents just minor seventh, BP seventh 10: 2 90/49 1052.572 cents 11: 4 28/15 1080.557 cents grave major seventh 11: 3 15/8 1088.269 cents classic major seventh 11: 1 256/135 1107.821 cents octave - major chroma 11: 2 40/21 1115.533 cents acute major seventh 11: 2 27/14 1137.039 cents septimal major seventh Highest number of different intervals for one interval class: 5 Average number of different intervals per interval class: 4.18182 = 46/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 119.4 203.9 266.9 386.3 498.0 590.2 702.0 821.4 884.4 968.8 1088.3 1200.0 119.4 : 84.5 147.4 266.9 378.6 470.8 582.5 702.0 764.9 849.4 968.8 1080.6 1200.0 203.9 : 63.0 182.4 294.1 386.3 498.0 617.5 680.4 764.9 884.4 996.1 1115.5 1200.0 266.9 : 119.4 231.2 323.4 435.1 554.5 617.5 702.0 821.4 933.1 1052.6 1137.0 1200.0 386.3 : 111.7 203.9 315.6 435.1 498.0 582.5 702.0 813.7 933.1 1017.6 1080.6 1200.0 498.0 : 92.2 203.9 323.4 386.3 470.8 590.2 702.0 821.4 905.9 968.8 1088.3 1200.0 590.2 : 111.7 231.2 294.1 378.6 498.0 609.8 729.2 813.7 876.6 996.1 1107.8 1200.0 702.0 : 119.4 182.4 266.9 386.3 498.0 617.5 702.0 764.9 884.4 996.1 1088.3 1200.0 821.4 : 63.0 147.4 266.9 378.6 498.0 582.5 645.5 764.9 876.6 968.8 1080.6 1200.0 884.4 : 84.5 203.9 315.6 435.1 519.6 582.5 702.0 813.7 905.9 1017.6 1137.0 1200.0 968.8 : 119.4 231.2 350.6 435.1 498.0 617.5 729.2 821.4 933.1 1052.6 1115.5 1200.0 1088.3: 111.7 231.2 315.6 378.6 498.0 609.8 702.0 813.7 933.1 996.1 1080.6 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 62.9609 .. 119.4428 cents 405/392, 56.4819 cents 28/27, 62.9609 cents 2: 147.4281 .. 231.1741 cents 360/343, 83.7460 cents 686/675, 27.9853 cents 3: 266.8709 .. 350.6169 cents 360/343, 83.7460 cents 49/48, 35.6968 cents 4: 378.6022 .. 435.0841 cents 405/392, 56.4819 cents 686/675, 27.9853 cents 5: 470.7809 .. 554.5269 cents 360/343, 83.7460 cents 49/48, 35.6968 cents 6: 582.5122 .. 617.4878 cents 50/49, 34.9756 cents 686/675, 27.9853 cents 7: 645.4731 .. 729.2191 cents 360/343, 83.7460 cents 686/675, 27.9853 cents 8: 764.9159 .. 821.3978 cents 405/392, 56.4819 cents 49/48, 35.6968 cents 9: 849.3831 .. 933.1291 cents 360/343, 83.7460 cents 686/675, 27.9853 cents 10: 968.8259 .. 1052.5719 cents 360/343, 83.7460 cents 49/48, 35.6968 cents 11: 1080.5572 .. 1137.0391 cents 405/392, 56.4819 cents 686/675, 27.9853 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 28/27, 62.9609 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 15/14, 119.4428 cents Average / Smallest step : 1.588287 Largest / Average step : 1.194428 Largest / Smallest step : 1.897095 Median interval of one step : 16/15, 111.7313 cents, amount: 3 Most common interval of one step : 15/14, 119.4428 cents, amount: 4 Least squares average step : 99.08924 cents, oct.: 1189.07093 cents Scale is strictly proper Step pattern alph. order: ABCADEDACBAD Step pattern size order : EBAEDCDEABED Scale is a Constant Structure, by a margin of 27.98529 cents Scale diversity : 1.454756 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.363851 Rothenberg efficiency : 0.341077 redundancy: 0.658923 Efficiency x scale size : 4.092929 Number of different interval sizes : 46 = 4.18182 / class Number of one step interval sizes : 5 Highest interval variety : 5 Mean interval variety : 4.18182 = 46/11 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 225/224, 7.7115 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 8 Most common triad is 0.0 498.045 701.955 cents, amount: 4 Number of recognisable fifths : 8, average 701.9550 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 Length of longest harmonic part : 2 at degree 3 Length of longest subharmonic part : 2 at degree 6 Formal octave complements present : 3 = 25.0000% Scale is differentially coherent in interval class 1 Scale is differentially coherent in interval classes 2 and 3 combined Scale is n-1 differentially coherent in interval classes 3 and 4 combined Scale is n-1 differentially coherent in interval classes 4 and 5 combined Scale is n-1 differentially coherent in interval classes 5 and 6 combined Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 7 Odd number limit : 135 (O: 135 U: 135) Highest odd numerator or denominator: 45 Scale harmonicity : 0.023826 Average absolute harmonicity : 0.185235 Specific harmonicity : 0.145768 Fundamental : 1/672, -9.3923 octaves, 0.3893 Hz. Guide tone : 1260, 10.2992 octaves, 329648.212 Hz. Exponens Consonantiae : 8.467200E+05, 19.69153 octaves Euler's gradus suavitatis : 30 Sum of Mann's harmonic distance : 137.5, average 11.45833 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 10.65523, average 0.88794 Sum of Tenney's harmonic distance : 20.22336, average 1.68528 Vogel's harmonic complexity : 17.58333 Wille's k value : 472 Wilson's harmonic complexity : 28 Rectangular lattice diameter : 6 Triangular lattice diameter : 4 Lattice compactness : 230.30858, average 2.95267 Lattice compactness (without 2's) : 138.83998, average 1.78000 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -5 .. 2 -1.41667 23 1 3: -1 .. 2 0.50000 12 3 5: 0 .. 1 0.50000 6 4 7: -1 .. 1 0.00000 4 4 Average exponent except 2's : 12 / 12 = 1.00000 Average absolute exponent except 2's: 22 / 12 = 1.83333 Scale is weakly epimorphic with val: <12 17 28 32| Scale is weakly epimorphic with val: <12 19 28 32| Scale is JI-epimorphic with val: <12 19 28 34| = patent Scale is weakly epimorphic with val: <12 21 26 31| Scale is weakly epimorphic with val: <12 21 26 37| Scale is weakly epimorphic with val: <12 21 28 32|
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 17.3045 c. M:-33.1291 c. Twelve-tone Chromatic (1/11-comma) 19: 2 1 1 2 2 1 2 2 1 1 2 2 P D3 SD: 12.1295 c. M: 21.8027 c. 22: 2 2 1 2 2 2 2 2 1 2 2 2 SP B ME SD: 8.5838 c. M:-14.2718 c. Jubilee-12 31: 3 2 2 3 3 2 3 3 2 2 3 3 SP D3 SD: 5.8410 c. M: 10.3616 c. Fokker 12-tone 41: 4 3 2 4 4 3 4 4 2 3 4 4 SP SD: 3.6199 c. M: 6.3099 c. 53: 5 4 3 5 5 4 5 5 3 4 5 5 SP SD: 3.3283 c. M: 6.3035 c. 72: 7 5 4 7 7 5 7 7 4 5 7 7 SP SD: 3.4095 c. M: 6.8904 c. Smith's Prism 77: 8 5 4 8 7 6 7 8 4 5 8 7 SP SD: 2.8690 c. M:-5.2325 c. 89: 9 6 5 9 8 7 8 9 5 6 9 8 SP SD: 2.8484 c. M:-5.5289 c. 94: 9 7 5 9 9 7 9 9 5 7 9 9 SP SD: 2.6738 c. M: 4.5492 c. 99: 10 7 5 10 9 8 9 10 5 7 10 9 SP SD: 1.8803 c. M:-3.7157 c. 118: 12 8 6 12 11 9 11 12 6 8 12 11 SP SD: 1.4828 c. M: 2.7242 c. 130: 13 9 7 13 12 10 12 13 7 9 13 12 SP SD: 0.8444 c. M:-1.7951 c. 171: 17 12 9 17 16 13 16 17 9 12 17 16 SP SD: 0.3646 c. M: 0.7500 c. 301: 30 21 16 30 28 23 28 30 16 21 30 28 SP SD: 0.3297 c. M:-0.6911 c. 441: 44 31 23 44 41 34 41 44 23 31 44 41 SP SD: 0.1864 c. M:-0.3709 c. 612: 61 43 32 61 57 47 57 61 32 43 61 57 SP SD: 0.1092 c. M: 0.2042 c. 783: 78 55 41 78 73 60 73 78 41 55 78 73 SP SD: 0.1274 c. M: 0.2435 c. 913: 91 64 48 91 85 70 85 91 48 64 91 85 SP SD: 0.1177 c. M:-0.1977 c. 935: 93 66 49 93 87 72 87 93 49 66 93 87 SP SD: 0.0953 c. M:-0.1581 c. 1106: 110 78 58 110 103 85 103 110 58 78 110 103 SP SD: 0.0562 c. M: 0.0938 c. 1407: 140 99 74 140 131 108 131 140 74 99 140 131 SP SD: 0.0516 c. M:-0.0801 c. 1547: 154 109 81 154 144 119 144 154 81 109 154 144 SP SD: 0.0516 c. M:-0.0978 c. 1718: 171 121 90 171 160 132 160 171 90 121 171 160 SP SD: 0.0363 c. M: 0.0747 c. 1848: 184 130 97 184 172 142 172 184 97 130 184 172 SP SD: 0.0312 c. M:-0.0569 c. 2019: 201 142 106 201 188 155 188 201 106 142 201 188 SP SD: 0.0250 c. M: 0.0467 c.
FIT/HARMONIC
1 x x x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x x 3 x x x x 4 S SD: 0.0000 cents 3 x x x x 4 x x x 5 x x 6 S SD: 0.0000 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 0.0000 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 13.9657 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 6.4843 cents 7 x 8 x 9 x 10 x 11 x 12:13:14 S SD: 11.8903 cents 8 x 9 x 10 x 11:12:13 x 14:15:16 S SD: 5.4193 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 10.0771 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 10.0060 cents 11:12 x 13:14:15 x 17:18 x 19:21:22 SD: 10.5925 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 4.6752 cents 13:14 x 15:16:17:18:20:21:22:23:24:26 S SD: 7.4922 cents 14:15:16 x 18:19:20:21 x 23:25:26:28 SD: 8.3201 cents 15:16:17:18:19:20:21:23:24:25:26:28:30 SD: 5.9089 cents 16:17:18:19:20:21:23:24:26:27:28:30:32 SD: 5.3803 cents 17:18:19:20:21:23:24:26:27:28:30:32:34 SD: 5.3284 cents 18:19:20:21:23:24:25:27:29:30:32:34:36 SD: 5.2476 cents 20:21 x 23:25:27:28:30:32:33:35:38:40 SD: 5.1857 cents 23:25:26:27:29:31:32:35:37:38:40:43:46 SD: 4.3427 cents 24:26:27:28:30:32:34:36:39:40:42:45:48 S SD: 2.4932 cents 40:43:45:47:50:53:56:60:64:67:70:75:80 SD: 1.8565 cents 48:51:54:56:60:64:68:72:77:80:84:90:96 SD: 1.6318 cents 54:58:61:63:68:72:76:81:87:90:95:101:108 SD: 1.5680 cents 56:60:63:65:70:75:79:84:90:93:98:105:112 SD: 1.1971 cents 71:76:80:83:89:95:100:107:114:118:124:133:142 SD: 1.1703 cents 72:77:81:84:90:96:101:108:116:120:126:135:144 SD: 0.5703 cents 96:103:108:112:120:128:135:144:154:160:168:180:192 SD: 0.3341 cents 168:180:189:196:210:224:236:252:270:280:294:315:336 SD: 0.1527 cents 336:360:378:392:420:448:473:504:540:560:588:630:672 SD: 0.1526 cents 408:437:459:476:510:544:574:612:656:680:714:765:816 SD: 0.1006 cents 504:540:567:588:630:672:709:756:810:840:882:945:1008 SD: 0.0509 cents