Scala analysis: sqrtphi
Sqrtphi[23], the 23-note MOS of the 49&72 temperament in sqrt(phi) tuning
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 49.635 cents 49.635440 2: 99.271 cents 99.270890 3: 148.906 cents 148.906330 4: 218.003 cents 218.003370 5: 267.639 cents 267.638810 6: 317.274 cents 317.274260 7: 366.910 cents 366.909700 8: 416.545 cents 416.545150 9: 466.181 cents 466.180590 10: 515.816 cents 515.816040 11: 565.451 cents 565.451480 12: 634.549 cents 634.548520 13: 684.184 cents 684.183960 14: 733.819 cents 733.819410 15: 783.455 cents 783.454850 16: 833.090 cents 833.090300 17: 882.726 cents 882.725740 18: 932.361 cents 932.361190 19: 981.997 cents 981.996630 20: 1051.094 cents 1051.093670 21: 1100.729 cents 1100.729110 22: 1150.365 cents 1150.364560 23: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 49.6354 cents 49.6354 1: 49.6354 cents 49.6354 2: 49.6355 cents 49.6355 3: 49.6354 cents 49.6354 4: 69.0970 cents 69.0970 5: 49.6354 cents 49.6354 6: 49.6355 cents 49.6355 7: 49.6354 cents 49.6354 8: 49.6355 cents 49.6355 9: 49.6354 cents 49.6354 10: 49.6355 cents 49.6355 11: 49.6354 cents 49.6354 12: 69.0970 cents 69.0970 13: 49.6354 cents 49.6354 14: 49.6355 cents 49.6355 15: 49.6354 cents 49.6354 16: 49.6355 cents 49.6355 17: 49.6354 cents 49.6354 18: 49.6355 cents 49.6355 19: 49.6354 cents 49.6354 20: 69.0970 cents 69.0970 21: 49.6354 cents 49.6354 22: 49.6354 cents 49.6354 23: 49.6354 cents 49.6354
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 20 49.63544 cents 1: 3 69.09704 cents 2: 17 99.27089 cents 2: 6 118.73248 cents 3: 14 148.90633 cents 3: 9 168.36793 cents 4: 11 198.54178 cents 4: 12 218.00337 cents 5: 8 248.17722 cents 5: 15 267.63881 cents 6: 5 297.81267 cents 6: 18 317.27426 cents 7: 2 347.44811 cents 7: 21 366.90970 cents 8: 22 416.54515 cents 8: 1 436.00674 cents 9: 19 466.18059 cents 9: 4 485.64219 cents 10: 16 515.81604 cents 10: 7 535.27763 cents 11: 13 565.45148 cents 11: 10 584.91308 cents 12: 10 615.08693 cents 12: 13 634.54852 cents 13: 7 664.72237 cents 13: 16 684.18396 cents 14: 4 714.35782 cents 14: 19 733.81941 cents 15: 1 763.99326 cents 15: 22 783.45485 cents 16: 21 833.09030 cents 16: 2 852.55189 cents 17: 18 882.72574 cents 17: 5 902.18734 cents 18: 15 932.36119 cents 18: 8 951.82278 cents 19: 12 981.99663 cents 19: 11 1001.45823 cents 20: 9 1031.63207 cents 20: 14 1051.09367 cents 21: 6 1081.26752 cents 21: 17 1100.72911 cents 22: 3 1130.90296 cents 22: 20 1150.36456 cents Highest number of different intervals for one interval class: 2 Average number of different intervals per interval class: 2.00000 = 2
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0.0 : 49.6 99.3 148.9 218.0 267.6 317.3 366.9 416.5 466.2 515.8 565.5 634.5 684.2 733.8 783.5 833.1 882.7 932.4 982.0 1051.1 1100.7 1150.4 1200.0 49.6 : 49.6 99.3 168.4 218.0 267.6 317.3 366.9 416.5 466.2 515.8 584.9 634.5 684.2 733.8 783.5 833.1 882.7 932.4 1001.5 1051.1 1100.7 1150.4 1200.0 99.3 : 49.6 118.7 168.4 218.0 267.6 317.3 366.9 416.5 466.2 535.3 584.9 634.5 684.2 733.8 783.5 833.1 882.7 951.8 1001.5 1051.1 1100.7 1150.4 1200.0 148.9 : 69.1 118.7 168.4 218.0 267.6 317.3 366.9 416.5 485.6 535.3 584.9 634.5 684.2 733.8 783.5 833.1 902.2 951.8 1001.5 1051.1 1100.7 1150.4 1200.0 218.0 : 49.6 99.3 148.9 198.5 248.2 297.8 347.4 416.5 466.2 515.8 565.5 615.1 664.7 714.4 764.0 833.1 882.7 932.4 982.0 1031.6 1081.3 1130.9 1200.0 267.6 : 49.6 99.3 148.9 198.5 248.2 297.8 366.9 416.5 466.2 515.8 565.5 615.1 664.7 714.4 783.5 833.1 882.7 932.4 982.0 1031.6 1081.3 1150.4 1200.0 317.3 : 49.6 99.3 148.9 198.5 248.2 317.3 366.9 416.5 466.2 515.8 565.5 615.1 664.7 733.8 783.5 833.1 882.7 932.4 982.0 1031.6 1100.7 1150.4 1200.0 366.9 : 49.6 99.3 148.9 198.5 267.6 317.3 366.9 416.5 466.2 515.8 565.5 615.1 684.2 733.8 783.5 833.1 882.7 932.4 982.0 1051.1 1100.7 1150.4 1200.0 416.5 : 49.6 99.3 148.9 218.0 267.6 317.3 366.9 416.5 466.2 515.8 565.5 634.5 684.2 733.8 783.5 833.1 882.7 932.4 1001.5 1051.1 1100.7 1150.4 1200.0 466.2 : 49.6 99.3 168.4 218.0 267.6 317.3 366.9 416.5 466.2 515.8 584.9 634.5 684.2 733.8 783.5 833.1 882.7 951.8 1001.5 1051.1 1100.7 1150.4 1200.0 515.8 : 49.6 118.7 168.4 218.0 267.6 317.3 366.9 416.5 466.2 535.3 584.9 634.5 684.2 733.8 783.5 833.1 902.2 951.8 1001.5 1051.1 1100.7 1150.4 1200.0 565.5 : 69.1 118.7 168.4 218.0 267.6 317.3 366.9 416.5 485.6 535.3 584.9 634.5 684.2 733.8 783.5 852.6 902.2 951.8 1001.5 1051.1 1100.7 1150.4 1200.0 634.5 : 49.6 99.3 148.9 198.5 248.2 297.8 347.4 416.5 466.2 515.8 565.5 615.1 664.7 714.4 783.5 833.1 882.7 932.4 982.0 1031.6 1081.3 1130.9 1200.0 684.2 : 49.6 99.3 148.9 198.5 248.2 297.8 366.9 416.5 466.2 515.8 565.5 615.1 664.7 733.8 783.5 833.1 882.7 932.4 982.0 1031.6 1081.3 1150.4 1200.0 733.8 : 49.6 99.3 148.9 198.5 248.2 317.3 366.9 416.5 466.2 515.8 565.5 615.1 684.2 733.8 783.5 833.1 882.7 932.4 982.0 1031.6 1100.7 1150.4 1200.0 783.5 : 49.6 99.3 148.9 198.5 267.6 317.3 366.9 416.5 466.2 515.8 565.5 634.5 684.2 733.8 783.5 833.1 882.7 932.4 982.0 1051.1 1100.7 1150.4 1200.0 833.1 : 49.6 99.3 148.9 218.0 267.6 317.3 366.9 416.5 466.2 515.8 584.9 634.5 684.2 733.8 783.5 833.1 882.7 932.4 1001.5 1051.1 1100.7 1150.4 1200.0 882.7 : 49.6 99.3 168.4 218.0 267.6 317.3 366.9 416.5 466.2 535.3 584.9 634.5 684.2 733.8 783.5 833.1 882.7 951.8 1001.5 1051.1 1100.7 1150.4 1200.0 932.4 : 49.6 118.7 168.4 218.0 267.6 317.3 366.9 416.5 485.6 535.3 584.9 634.5 684.2 733.8 783.5 833.1 902.2 951.8 1001.5 1051.1 1100.7 1150.4 1200.0 982.0 : 69.1 118.7 168.4 218.0 267.6 317.3 366.9 436.0 485.6 535.3 584.9 634.5 684.2 733.8 783.5 852.6 902.2 951.8 1001.5 1051.1 1100.7 1150.4 1200.0 1051.1: 49.6 99.3 148.9 198.5 248.2 297.8 366.9 416.5 466.2 515.8 565.5 615.1 664.7 714.4 783.5 833.1 882.7 932.4 982.0 1031.6 1081.3 1130.9 1200.0 1100.7: 49.6 99.3 148.9 198.5 248.2 317.3 366.9 416.5 466.2 515.8 565.5 615.1 664.7 733.8 783.5 833.1 882.7 932.4 982.0 1031.6 1081.3 1150.4 1200.0 1150.4: 49.6 99.3 148.9 198.5 267.6 317.3 366.9 416.5 466.2 515.8 565.5 615.1 684.2 733.8 783.5 833.1 882.7 932.4 982.0 1031.6 1100.7 1150.4 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 49.6354 .. 69.0970 cents 19.46160 cents 49.63544 cents 2: 99.2709 .. 118.7325 cents 19.46159 cents 30.17385 cents 3: 148.9063 .. 168.3679 cents 19.46160 cents 30.17385 cents 4: 198.5418 .. 218.0034 cents 19.46159 cents 30.17385 cents 5: 248.1772 .. 267.6388 cents 19.46159 cents 30.17385 cents 6: 297.8127 .. 317.2743 cents 19.46159 cents 30.17386 cents 7: 347.4481 .. 366.9097 cents 19.46159 cents 30.17385 cents 8: 416.5452 .. 436.0067 cents 19.46159 cents 49.63545 cents 9: 466.1806 .. 485.6422 cents 19.46160 cents 30.17385 cents 10: 515.8160 .. 535.2776 cents 19.46159 cents 30.17385 cents 11: 565.4515 .. 584.9131 cents 19.46160 cents 30.17385 cents 12: 615.0869 .. 634.5485 cents 19.46159 cents 30.17385 cents 13: 664.7224 .. 684.1840 cents 19.46159 cents 30.17385 cents 14: 714.3578 .. 733.8194 cents 19.46159 cents 30.17386 cents 15: 763.9933 .. 783.4548 cents 19.46159 cents 30.17385 cents 16: 833.0903 .. 852.5519 cents 19.46159 cents 49.63545 cents 17: 882.7257 .. 902.1873 cents 19.46160 cents 30.17385 cents 18: 932.3612 .. 951.8228 cents 19.46159 cents 30.17385 cents 19: 981.9966 .. 1001.4582 cents 19.46160 cents 30.17385 cents 20: 1031.6321 .. 1051.0937 cents 19.46160 cents 30.17384 cents 21: 1081.2675 .. 1100.7291 cents 19.46159 cents 30.17385 cents 22: 1130.9030 .. 1150.3646 cents 19.46160 cents 30.17385 cents
SHOW DATA
Number of notes : 23 -- Interval properties -- Smallest interval : 49.63544 cents, class 1 Average step (divided formal octave): 52.1739 cents Largest one step interval : 69.09704 cents Average / Smallest step : 1.051142 Largest / Average step : 1.324360 Largest / Smallest step : 1.392091 Median interval of one step : 49.63544 cents, amount: 20 Most common interval of one step : 49.63544 cents, amount: 20 Least squares average step : 52.18291 cents, oct.: 1200.20704 cents Scale is strictly proper Scale has Myhill's property chromas: 19.46160 cents and 30.17384 cents, ratio: 1.550430 generators: 8 of 416.54515 cents and 15 of 783.45485 cents ET pair of linear temperament : 23&49 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 1233-tone equal temperament with octave 2/1 degrees: 51 102 153 224 275 326 377 428 479 530 581 652 703 754 805 856 907 958 1009 1080 1131 1182 1233 generators: 428 and 805 Number of contiguous 1-step segments: 0 Step pattern alph. order: AAABAAAAAAABAAAAAAABAAA Step pattern size order : SSSLSSSSSSSLSSSSSSSLSSS Scale is a Constant Structure, by a margin of 30.17384 cents Scale diversity : 1.181077 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.643204 Rothenberg efficiency : 0.695652 redundancy: 0.304348 Efficiency x scale size : 16.000000 Number of different interval sizes : 44 = 2.00000 / class Number of one step interval sizes : 2 Highest interval variety : 2 Mean interval variety : 2.00000 = 2 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 19.46159 cents Most common intervals : 416.54515 cents & inv., amount: 22 Scale is a chain of 11 triads 0.0 416.545 833.090 cents Most common triad is 0.0 416.545 783.455 cents, amount: 21 Number of recognisable fifths : 4, average 714.3578 cents Number of appreciable fifths : 0 Best minor thirds form a closed circle Formal octave complements present : 23 = 100.0000% Inversional symmetry on degrees : 0 Inversional symmetry on intervals : 11-12 -- Rational properties -- Prime limit : 2 (not all pitches rational)
FIT/MODE
23: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 5.6128 c. M:-9.3077 c. 49: 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 SP M ME I SD: 1.4501 c. M:-2.4048 c. 72: 3 3 3 4 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 4 3 3 3 SP M ME I SD: 0.8061 c. M:-1.3367 c. 121: 5 5 5 7 5 5 5 5 5 5 5 7 5 5 5 5 5 5 5 7 5 5 5 SP M DE I SD: 0.1076 c. M:-0.1784 c. 314: 13 13 13 18 13 13 13 13 13 13 13 18 13 13 13 13 13 13 13 18 13 13 13 SP M DE I SD: 0.1019 c. M:-0.1690 c. 435: 18 18 18 25 18 18 18 18 18 18 18 25 18 18 18 18 18 18 18 25 18 18 18 SP M DE I SD: 0.0436 c. M:-0.0723 c. 556: 23 23 23 32 23 23 23 23 23 23 23 32 23 23 23 23 23 23 23 32 23 23 23 SP M DE I SD: 0.0107 c. M:-0.0178 c. 677: 28 28 28 39 28 28 28 28 28 28 28 39 28 28 28 28 28 28 28 39 28 28 28 SP M DE I SD: 0.0104 c. M:-0.0173 c. 1233: 51 51 51 71 51 51 51 51 51 51 51 71 51 51 51 51 51 51 51 71 51 51 51 SP M DE I SD: 0.0009 c. M:-0.0015 c.
FIT/HARMONIC
1 .. 2 match: 1 H SD: 0.0000 cents 2 .. 4 match: 2 H SD: 8.8855 cents 3 .. 6 match: 3 H SD: 5.9486 cents 4 .. 8 match: 4 H SD: 7.3561 cents 5 .. 10 match: 5 H SD: 8.4672 cents 6 .. 12 match: 6 H SD: 4.2094 cents 7 .. 14 match: 7 H SD: 5.0192 cents 8 .. 16 match: 8 H SD: 4.8003 cents 9 .. 18 match: 9 H SD: 5.4064 cents 10 .. 20 match: 10 H SD: 5.2994 cents 11 .. 22 match: 11 H SD: 4.7152 cents 12 .. 24 match: 12 H SD: 4.5754 cents 13 .. 26 match: 13 H SD: 3.8282 cents 14 .. 28 match: 14 H SD: 4.3291 cents 15 .. 30 match: 14 S SD: 3.6202 cents 16 .. 32 match: 16 H SD: 3.4810 cents 17 .. 34 match: 17 H SD: 3.4484 cents 18 .. 36 match: 18 H SD: 3.0039 cents 19 .. 38 match: 19 H SD: 3.5677 cents 20 .. 40 match: 19 SD: 3.4139 cents 21 .. 42 match: 20 SD: 2.5849 cents 22 .. 44 match: 21 S SD: 3.5339 cents 23 .. 46 match: 21 SD: 2.7435 cents 24 .. 48 match: 21 SD: 2.9832 cents 25 .. 50 match: 22 SD: 2.6358 cents 26 .. 52 match: 22 SD: 2.8715 cents 27 .. 54 match: 21 SD: 3.3211 cents 28 .. 56 match: 23 SD: 3.0317 cents 29 .. 58 match: 23 SD: 1.9662 cents 34 .. 68 match: 23 SD: 1.9581 cents 35 .. 70 match: 23 SD: 1.9093 cents 36 .. 72 match: 23 SD: 1.8186 cents 42 .. 84 match: 23 SD: 1.7929 cents 43 .. 86 match: 23 SD: 1.7188 cents 44 .. 88 match: 23 SD: 1.7085 cents 45 .. 90 match: 23 SD: 1.5784 cents 47 .. 94 match: 23 SD: 1.5263 cents 52 .. 104 match: 23 SD: 1.3760 cents 54 .. 108 match: 23 SD: 1.2194 cents 55 .. 110 match: 23 SD: 1.1336 cents 65 .. 130 match: 23 SD: 1.0216 cents 66 .. 132 match: 23 SD: 0.8938 cents 80 .. 160 match: 23 SD: 0.8427 cents 81 .. 162 match: 23 SD: 0.8022 cents 84 .. 168 match: 23 SD: 0.7823 cents 89 .. 178 match: 23 SD: 0.7123 cents 98 .. 196 match: 23 SD: 0.6881 cents 101 .. 202 match: 23 SD: 0.6167 cents 110 .. 220 match: 23 SD: 0.6120 cents 113 .. 226 match: 23 SD: 0.6021 cents 114 .. 228 match: 23 SD: 0.5964 cents 119 .. 238 match: 23 SD: 0.5522 cents 126 .. 252 match: 23 SD: 0.5430 cents 132 .. 264 match: 23 SD: 0.5053 cents 133 .. 266 match: 23 SD: 0.4812 cents 135 .. 270 match: 23 SD: 0.4524 cents 136 .. 272 match: 23 SD: 0.4279 cents 165 .. 330 match: 23 SD: 0.3509 cents 178 .. 356 match: 23 SD: 0.3423 cents 199 .. 398 match: 23 SD: 0.2905 cents 210 .. 420 match: 23 SD: 0.2827 cents 233 .. 466 match: 23 SD: 0.2099 cents 269 .. 538 match: 23 SD: 0.2041 cents 305 .. 610 match: 23 SD: 0.1900 cents 334 .. 668 match: 23 SD: 0.1541 cents 388 .. 776 match: 23 SD: 0.1411 cents 411 .. 822 match: 23 SD: 0.1317 cents 456 .. 912 match: 23 SD: 0.1170 cents 479 .. 958 match: 23 SD: 0.1119 cents 508 .. 1016 match: 23 SD: 0.1116 cents 544 .. 1088 match: 23 SD: 0.1067 cents 576 .. 1152 match: 23 SD: 0.1028 cents 580 .. 1160 match: 23 SD: 0.0975 cents