Scala analysis: qm3a
Qm(3) 10-note quasi-miracle scale, mode A
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 116.667 cents 116.666667 2: 266.667 cents 266.666667 3: 383.333 cents 383.333333 4: 500.000 cents 500.000000 5: 583.333 cents 583.333333 6: 700.000 cents 700.000000 7: 816.667 cents 816.666667 8: 966.667 cents 966.666667 9: 1083.333 cents 1083.333333 10: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 116.6667 cents 116.6667 1: 116.6667 cents 116.6667 2: 150.0000 cents 150.0000 3: 116.6667 cents 116.6667 4: 116.6667 cents 116.6667 5: 83.3333 cents 83.3333 6: 116.6667 cents 116.6667 7: 116.6667 cents 116.6667 8: 150.0000 cents 150.0000 9: 116.6667 cents 116.6667 10: 116.6667 cents 116.6667
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 1 83.33333 cents 1: 7 116.66667 cents 1: 2 150.00000 cents 2: 2 200.00000 cents 2: 4 233.33333 cents 2: 4 266.66667 cents 3: 3 316.66667 cents 3: 1 350.00000 cents 3: 6 383.33333 cents 4: 2 433.33333 cents 4: 2 466.66667 cents 4: 6 500.00000 cents 5: 1 550.00000 cents 5: 4 583.33333 cents 5: 4 616.66667 cents 5: 1 650.00000 cents 6: 6 700.00000 cents 6: 2 733.33333 cents 6: 2 766.66667 cents 7: 6 816.66667 cents 7: 1 850.00000 cents 7: 3 883.33333 cents 8: 4 933.33333 cents 8: 4 966.66667 cents 8: 2 1000.00000 cents 9: 2 1050.00000 cents 9: 7 1083.33333 cents 9: 1 1116.66667 cents Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.11111 = 28/9
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 0.0 : 116.7 266.7 383.3 500.0 583.3 700.0 816.7 966.7 1083.3 1200.0 116.7 : 150.0 266.7 383.3 466.7 583.3 700.0 850.0 966.7 1083.3 1200.0 266.7 : 116.7 233.3 316.7 433.3 550.0 700.0 816.7 933.3 1050.0 1200.0 383.3 : 116.7 200.0 316.7 433.3 583.3 700.0 816.7 933.3 1083.3 1200.0 500.0 : 83.3 200.0 316.7 466.7 583.3 700.0 816.7 966.7 1083.3 1200.0 583.3 : 116.7 233.3 383.3 500.0 616.7 733.3 883.3 1000.0 1116.7 1200.0 700.0 : 116.7 266.7 383.3 500.0 616.7 766.7 883.3 1000.0 1083.3 1200.0 816.7 : 150.0 266.7 383.3 500.0 650.0 766.7 883.3 966.7 1083.3 1200.0 966.7 : 116.7 233.3 350.0 500.0 616.7 733.3 816.7 933.3 1050.0 1200.0 1083.3: 116.7 233.3 383.3 500.0 616.7 700.0 816.7 933.3 1083.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 83.3333 .. 150.0000 cents 66.66667 cents 83.33333 cents 2: 200.0000 .. 266.6667 cents 66.66667 cents 50.00000 cents 3: 316.6667 .. 383.3333 cents 66.66667 cents 50.00000 cents 4: 433.3333 .. 500.0000 cents 66.66667 cents 50.00000 cents 5: 550.0000 .. 650.0000 cents 100.00000 cents 50.00000 cents 6: 700.0000 .. 766.6667 cents 66.66667 cents 50.00000 cents 7: 816.6667 .. 883.3333 cents 66.66667 cents 50.00000 cents 8: 933.3333 .. 1000.0000 cents 66.66667 cents 50.00000 cents 9: 1050.0000 .. 1116.6667 cents 66.66667 cents 50.00000 cents
SHOW DATA
Number of notes : 10 -- Interval properties -- Smallest interval : 83.33333 cents, class 1 Average step (divided formal octave): 120.0000 cents Largest one step interval : 150.00000 cents Average / Smallest step : 1.440000 Largest / Average step : 1.250000 Largest / Smallest step : 1.800000 Median interval of one step : 116.66667 cents, amount: 7 Most common interval of one step : 116.66667 cents, amount: 7 Least squares average step : 119.78355 cents, oct.: 1197.83550 cents Scale is strictly proper Scale is a mode of a 72-tone equal temperament with octave 2/1 degrees: 7 16 23 30 35 42 49 58 65 72 Least number of segments generator : 37 of 616.667 cents and inv. number of contiguous generator circle segments: 1 Shortest superset generator : 65 of 1083.333 cents and inv. generated superset size: 16 = 6 more = 160.000% Number of contiguous 1-step segments: 0 Scale is fully connected major-minor triadic in 72-tone equal temperament Step pattern alph. order: ABAACAABAA Step pattern size order : MLMMSMMLMM Interval vector is [ 0 0 0 0 1 0 7 0 2 0 0 2 0 4 0 4 0 0 3 0 1 0 6 0 0 2 0 2 0 6 0 0 1 0 4 0 ] Scale is a Constant Structure, by a margin of 50.00000 cents Scale diversity : 1.418000 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.472222 Rothenberg efficiency : 0.447857 redundancy: 0.552143 Efficiency x scale size : 4.478571 Number of different interval sizes : 28 = 3.11111 / class Number of one step interval sizes : 3 Highest interval variety : 4 Mean interval variety : 3.11111 = 28/9 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 33.33333 cents Most common intervals : 116.66667 cents & inv., amount: 7 Number of recognisable fifths : 6, average 700.0000 cents Scale contains two identical disjunct pentachords Formal octave complements present : 7 = 70.0000% Limited inverse transpositions : 9 Inversional symmetry on intervals : 4-5 9-10 -- Rational properties -- Prime limit : 2 (not all pitches rational)
FIT/MODE
10: 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 17.1594 c. M: 26.6667 c. 19: 2 2 2 2 1 2 2 2 2 2 SP M ME SD: 10.3531 c. M: 19.2982 c. 21: 2 3 2 2 1 2 2 3 2 2 P SD: 12.2567 c. M:-19.0476 c. 22: 2 3 2 2 2 2 2 3 2 2 SP D3 SD: 9.1035 c. M:-16.6667 c. Alternate Pentachordal Major 27: 3 3 3 2 2 3 2 4 2 3 P SD: 12.2977 c. M:-16.6667 c. 28: 3 3 3 3 2 2 3 4 2 3 P SD: 11.9760 c. M:-19.0476 c. 29: 3 3 3 3 2 3 3 3 3 3 SP M ME SD: 9.7549 c. M: 18.3908 c. 31: 3 4 3 3 2 3 3 4 3 3 SP SD: 2.7676 c. M:-4.3011 c. 41: 4 5 4 4 3 4 4 5 4 4 SP SD: 2.0926 c. M: 3.2520 c. 72: 7 9 7 7 5 7 7 9 7 7 SP SD: 0.0000 c. M:-0.0000 c. Quasi-Miracle-10
FIT/HARMONIC
1 x x x x x x x x x 2 S SD: 0.0000 cents 2 x x x x x 3 x x x 4 S SD: 0.9775 cents 3 x x x 4 x x 5 x x 6 S SD: 22.5734 cents 4 x x 5 x x 6 x 7 x 8 S SD: 1.0418 cents 5 x 6 x x 7 x 8:9 x 10 S SD: 14.1448 cents 6 x 7 x 8 x 9:10 x 11:12 S SD: 12.6314 cents 7 x 8:9 x 10 x 11:12:13:14 S SD: 12.3985 cents 8 x 9:10 x 11:12:13:14:15:16 S SD: 9.3328 cents 9:10 x 11:12:13 x 14:16:17:18 S SD: 13.8703 cents 10:11:12 x 13:14:15:16:17:19:20 SD: 11.0709 cents 11:12:13:14:15 x 16:18:19:21:22 SD: 11.0241 cents 12:13:14:15:16:17:18:19:21:22:24 SD: 4.9874 cents 20:21:23:25:27:28:30:32:35:37:40 SD: 4.8902 cents 23:25:27:29:31:32:34:37:40:43:46 SD: 4.7788 cents 24:26:28:30:32:34:36:38:42:45:48 SD: 3.6849 cents 28:30:33:35:37:39:42:45:49:52:56 SD: 2.9969 cents 30:32:35:37:40:42:45:48:52:56:60 SD: 2.5832 cents 40:43:47:50:53:56:60:64:70:75:80 SD: 2.1099 cents 48:51:56:60:64:67:72:77:84:90:96 SD: 1.4861 cents 60:64:70:75:80:84:90:96:105:112:120 SD: 0.7928 cents 113:121:132:141:151:158:169:181:198:211:226 SD: 0.7507 cents 114:122:133:142:152:160:171:183:199:213:228 SD: 0.6606 cents 132:141:154:165:176:185:198:212:231:247:264 SD: 0.6576 cents 138:148:161:172:184:193:207:221:241:258:276 SD: 0.6522 cents 143:153:167:178:191:200:214:229:250:267:286 SD: 0.6494 cents 144:154:168:180:192:202:216:231:252:269:288 SD: 0.5724 cents 158:169:184:197:211:221:237:253:276:295:316 SD: 0.5473 cents 162:173:189:202:216:227:243:260:283:303:324 SD: 0.5025 cents 173:185:202:216:231:242:259:277:302:323:346 SD: 0.4962 cents 174:186:203:217:232:244:261:279:304:325:348 SD: 0.4264 cents 197:211:230:246:263:276:295:316:344:368:394 SD: 0.4063 cents 215:230:251:268:287:301:322:345:376:402:430 SD: 0.3423 cents 227:243:265:283:303:318:340:364:397:424:454 SD: 0.3301 cents 229:245:267:286:306:321:343:367:400:428:458 SD: 0.3161 cents 245:262:286:306:327:343:367:393:428:458:490 SD: 0.2819 cents 259:277:302:323:346:363:388:415:453:484:518 SD: 0.2712 cents 275:294:321:343:367:385:412:441:481:514:550 SD: 0.2545 cents 287:307:335:358:383:402:430:460:502:537:574 SD: 0.2272 cents 347:371:405:433:463:486:520:556:606:649:694 SD: 0.2194 cents 385:412:449:480:514:539:577:617:673:720:770 SD: 0.2050 cents 415:444:484:518:554:581:622:665:725:776:830 SD: 0.1523 cents 504:539:588:629:673:706:755:808:881:942:1008 SD: 0.1171 cents 516:552:602:644:689:723:773:827:902:965:1032 SD: 0.1085 cents