Scala analysis: piaji-helmholtz
"PiaJI" scale tempered to TOP helmholtz
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 91.262 cents 91.262030 2: 203.508 cents 203.508210 3: 294.770 cents 294.770240 4: 386.032 cents 386.032270 5: 498.278 cents 498.278450 6: 610.525 cents 610.524640 7: 701.787 cents 701.786670 8: 793.049 cents 793.048690 9: 905.295 cents 905.294880 10: 996.557 cents 996.556910 11: 1087.819 cents 1087.818930 12: 1200.065 cents 1200.065120
SHOW/INTERVAL
0: 112.2462 cents 112.2462 1: 91.2620 cents 91.2620 2: 112.2462 cents 112.2462 3: 91.2620 cents 91.2620 4: 91.2620 cents 91.2620 5: 112.2462 cents 112.2462 6: 112.2462 cents 112.2462 7: 91.2620 cents 91.2620 8: 91.2620 cents 91.2620 9: 112.2462 cents 112.2462 10: 91.2620 cents 91.2620 11: 91.2620 cents 91.2620 12: 112.2462 cents 112.2462
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 7 91.26203 cents 1: 5 112.24618 cents 2: 3 182.52406 cents 2: 8 203.50821 cents 2: 1 224.49237 cents 3: 9 294.77024 cents 3: 3 315.75440 cents 4: 4 386.03227 cents 4: 8 407.01642 cents 5: 1 477.29429 cents 5: 9 498.27845 cents 5: 2 519.26261 cents 6: 6 589.54048 cents 6: 6 610.52464 cents 7: 2 680.80251 cents 7: 9 701.78667 cents 7: 1 722.77083 cents 8: 8 793.04869 cents 8: 4 814.03285 cents 9: 3 884.31072 cents 9: 9 905.29488 cents 10: 1 975.57275 cents 10: 8 996.55691 cents 10: 3 1017.54106 cents 11: 5 1087.81893 cents 11: 7 1108.80309 cents Highest number of different intervals for one interval class: 3 Average number of different intervals per interval class: 2.36364 = 26/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 91.3 203.5 294.8 386.0 498.3 610.5 701.8 793.0 905.3 996.6 1087.8 1200.1 91.3 : 112.2 203.5 294.8 407.0 519.3 610.5 701.8 814.0 905.3 996.6 1108.8 1200.1 203.5 : 91.3 182.5 294.8 407.0 498.3 589.5 701.8 793.0 884.3 996.6 1087.8 1200.1 294.8 : 91.3 203.5 315.8 407.0 498.3 610.5 701.8 793.0 905.3 996.6 1108.8 1200.1 386.0 : 112.2 224.5 315.8 407.0 519.3 610.5 701.8 814.0 905.3 1017.5 1108.8 1200.1 498.3 : 112.2 203.5 294.8 407.0 498.3 589.5 701.8 793.0 905.3 996.6 1087.8 1200.1 610.5 : 91.3 182.5 294.8 386.0 477.3 589.5 680.8 793.0 884.3 975.6 1087.8 1200.1 701.8 : 91.3 203.5 294.8 386.0 498.3 589.5 701.8 793.0 884.3 996.6 1108.8 1200.1 793.0 : 112.2 203.5 294.8 407.0 498.3 610.5 701.8 793.0 905.3 1017.5 1108.8 1200.1 905.3 : 91.3 182.5 294.8 386.0 498.3 589.5 680.8 793.0 905.3 996.6 1087.8 1200.1 996.6 : 91.3 203.5 294.8 407.0 498.3 589.5 701.8 814.0 905.3 996.6 1108.8 1200.1 1087.8: 112.2 203.5 315.8 407.0 498.3 610.5 722.8 814.0 905.3 1017.5 1108.8 1200.1 1200.1
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 91.2620 .. 112.2462 cents 20.98415 cents 91.26203 cents 2: 182.5241 .. 224.4924 cents 41.96831 cents 70.27788 cents 3: 294.7702 .. 315.7544 cents 20.98416 cents 70.27787 cents 4: 386.0323 .. 407.0164 cents 20.98415 cents 70.27787 cents 5: 477.2943 .. 519.2626 cents 41.96832 cents 70.27787 cents 6: 589.5405 .. 610.5246 cents 20.98416 cents 70.27787 cents 7: 680.8025 .. 722.7708 cents 41.96832 cents 70.27787 cents 8: 793.0487 .. 814.0328 cents 20.98416 cents 70.27786 cents 9: 884.3107 .. 905.2949 cents 20.98416 cents 70.27787 cents 10: 975.5728 .. 1017.5411 cents 41.96831 cents 70.27787 cents 11: 1087.8189 .. 1108.8031 cents 20.98416 cents 70.27787 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 91.26202 cents, class 1 Average step (divided formal octave): 100.0054 cents Largest one step interval : 112.24618 cents Average / Smallest step : 1.095806 Largest / Average step : 1.122401 Largest / Smallest step : 1.229933 Median interval of one step : 91.26203 cents, amount: 7 Most common interval of one step : 91.26203 cents, amount: 7 Least squares average step : 99.72026 cents, oct.: 1196.64309 cents Scale is strictly proper Scale is 3-distributional even Scale is a mode of a 1144-tone equal temperament with octave 1200.065 cents degrees: 87 194 281 368 475 582 669 756 863 950 1037 1144 Least number of segments generator : 573 of 601.082 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 669 of 701.786 cents and inv. generated superset size: 15 = 3 more = 125.000% Number of contiguous 1-step segments: 0 Step pattern alph. order: ABAABBAABAAB Step pattern size order : SLSSLLSSLSSL Scale is a Constant Structure, by a margin of 70.27786 cents Scale diversity : 1.201920 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.737712 Rothenberg efficiency : 0.493519 redundancy: 0.506481 Efficiency x scale size : 5.922222 Number of different interval sizes : 26 = 2.36364 / class Number of one step interval sizes : 2 Highest interval variety : 3 Mean interval variety : 2.36364 = 26/11 Median interval variety : 2 Lowest interval variety : 2 Smallest interval difference : 20.98415 cents Number of recognisable fifths : 9, average 701.7867 cents Number of recognisable fourths : 9, average 498.2784 cents Scale contains two identical disjunct hexachords Best fifths form a closed circle Formal octave complements present : 7 = 58.3333% 2/1 octave complements present : 0 = 0.0000% -- Rational properties --
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 7.4533 c. M:-13.9894 c. Twelve-tone Chromatic (1/11-comma) 29: 2 3 2 2 3 3 2 2 3 2 2 3 SP D3 SD: 7.2449 c. M: 13.5983 c. 41: 3 4 3 3 4 4 3 3 4 3 3 4 SP D3 SD: 2.9430 c. M: 5.5238 c. 53: 4 5 4 4 5 5 4 4 5 4 4 5 SP D3 SD: 0.5891 c. M: 1.1057 c. 118: 9 11 9 9 11 11 9 9 11 9 9 11 SP D3 SD: 0.2288 c. M:-0.4294 c. 171: 13 16 13 13 16 16 13 13 16 13 13 16 SP D3 SD: 0.0247 c. M: 0.0464 c. 631: 48 59 48 48 59 59 48 48 59 48 48 59 SP D3 SD: 0.0227 c. M:-0.0426 c. 802: 61 75 61 61 75 75 61 61 75 61 61 75 SP D3 SD: 0.0126 c. M:-0.0236 c. 973: 74 91 74 74 91 91 74 74 91 74 74 91 SP D3 SD: 0.0060 c. M:-0.0113 c. 1144: 87 107 87 87 107 107 87 87 107 87 87 107 SP D3 SD: 0.0014 c. M:-0.0027 c.
FIT/HARMONIC
1 x x x x x x x x x x x 2 S SD: 0.0651 cents 2 x x x x x x 3 x x x x 4 S SD: 0.0902 cents 3 x x x x 4 x x x 5 x x 6 S SD: 6.9792 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 6.9333 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 9.1408 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 8.6533 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 9.4393 cents 8 x 9 x 10:11 x 12:13 x 14:15:16 S SD: 9.5500 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.0475 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.8980 cents 11 x 12:13:14:15:16 x 17:19:20:21:22 SD: 11.1685 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 6.9031 cents 13:14:15 x 16:17 x 20:21:22:23:24:26 SD: 9.6457 cents 14:15:16:17:18:19:20:21:22:24:25:26:28 S SD: 7.3823 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.9931 cents 16:17:18:19:20:21:23:24:25:27:28:30:32 SD: 4.1273 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.6067 cents 31:33:35:37:39:41:44:46:49:52:55:58:62 SD: 2.9848 cents 32:34:36:38:40:43:46:48:51:54:57:60:64 SD: 2.4816 cents 33:35:37:39:41:44:47:49:52:56:59:62:66 SD: 2.3943 cents 38:40:43:45:47:51:54:57:60:64:68:71:76 SD: 2.2633 cents 40:42:45:47:50:53:57:60:63:67:71:75:80 SD: 2.0824 cents 48:51:54:57:60:64:68:72:76:81:85:90:96 SD: 1.4702 cents 64:67:72:76:80:85:91:96:101:108:114:120:128 SD: 1.2372 cents 71:75:80:84:89:95:101:106:112:120:126:133:142 SD: 1.1809 cents 72:76:81:85:90:96:102:108:114:121:128:135:144 SD: 1.0776 cents 80:84:90:95:100:107:114:120:126:135:142:150:160 SD: 0.9934 cents 90:95:101:107:112:120:128:135:142:152:160:169:180 SD: 0.9323 cents 96:101:108:114:120:128:137:144:152:162:171:180:192 SD: 0.6513 cents 112:118:126:133:140:149:159:168:177:189:199:210:224 SD: 0.5512 cents 128:135:144:152:160:171:182:192:202:216:228:240:256 SD: 0.5211 cents 144:152:162:171:180:192:205:216:228:243:256:270:288 SD: 0.3842 cents 168:177:189:199:210:224:239:252:266:283:299:315:336 SD: 0.3596 cents 184:194:207:218:230:245:262:276:291:310:327:345:368 SD: 0.3418 cents 248:261:279:294:310:331:353:372:392:418:441:465:496 SD: 0.3071 cents 258:272:290:306:322:344:367:387:408:435:459:484:516 SD: 0.2778 cents 279:294:314:331:349:372:397:418:441:471:496:523:558 SD: 0.2736 cents 280:295:315:332:350:373:398:420:443:472:498:525:560 SD: 0.2712 cents 296:312:333:351:370:395:421:444:468:499:526:555:592 SD: 0.1918 cents 312:329:351:370:390:416:444:468:493:526:555:585:624 SD: 0.1582 cents 464:489:522:550:580:619:660:696:734:783:825:870:928 SD: 0.1399 cents 480:506:540:569:600:640:683:720:759:810:854:900:960 SD: 0.1116 cents