Scala analysis: xen18-schulter-pythagorean
12-note Pythagorean tuning
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 2187/2048 113.685006 apotome 2: 9/8 203.910002 major whole tone 3: 32/27 294.134997 Pythagorean minor third 4: 81/64 407.820003 Pythagorean major third 5: 4/3 498.044999 perfect fourth 6: 729/512 611.730005 Pythagorean tritone 7: 3/2 701.955001 perfect fifth 8: 6561/4096 815.640007 Pythagorean augmented fifth 9: 27/16 905.865003 Pythagorean major sixth 10: 16/9 996.089998 Pythagorean minor seventh 11: 243/128 1109.775004 Pythagorean major seventh 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 256/243 90.2250 limma, Pythagorean minor second 1: 2187/2048 113.6850 apotome 2: 256/243 90.2250 limma, Pythagorean minor second 3: 256/243 90.2250 limma, Pythagorean minor second 4: 2187/2048 113.6850 apotome 5: 256/243 90.2250 limma, Pythagorean minor second 6: 2187/2048 113.6850 apotome 7: 256/243 90.2250 limma, Pythagorean minor second 8: 2187/2048 113.6850 apotome 9: 256/243 90.2250 limma, Pythagorean minor second 10: 256/243 90.2250 limma, Pythagorean minor second 11: 2187/2048 113.6850 apotome 12: 256/243 90.2250 limma, Pythagorean minor second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 7 256/243 90.225 cents limma, Pythagorean minor second 1: 5 2187/2048 113.685 cents apotome 2: 2 65536/59049 180.450 cents Pythagorean diminished third 2: 10 9/8 203.910 cents major whole tone 3: 9 32/27 294.135 cents Pythagorean minor third 3: 3 19683/16384 317.595 cents Pythagorean augmented second 4: 4 8192/6561 384.360 cents Pythagorean diminished fourth 4: 8 81/64 407.820 cents Pythagorean major third 5: 11 4/3 498.045 cents perfect fourth 5: 1 177147/131072 521.505 cents Pythagorean augmented third 6: 6 1024/729 588.270 cents Pythagorean diminished fifth 6: 6 729/512 611.730 cents Pythagorean tritone 7: 1 262144/177147 678.495 cents Pythagorean diminished sixth 7: 11 3/2 701.955 cents perfect fifth 8: 8 128/81 792.180 cents Pythagorean minor sixth 8: 4 6561/4096 815.640 cents Pythagorean augmented fifth 9: 3 32768/19683 882.405 cents Pythagorean diminished seventh 9: 9 27/16 905.865 cents Pythagorean major sixth 10: 10 16/9 996.090 cents Pythagorean minor seventh 10: 2 59049/32768 1019.550 cents Pythagorean augmented sixth 11: 5 4096/2187 1086.315 cents Pythagorean diminished octave 11: 7 243/128 1109.775 cents Pythagorean major seventh 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 0.0 : 113.7 203.9 294.1 407.8 498.0 611.7 702.0 815.6 905.9 996.1 1109.8 1200.0 113.7 : 90.2 180.4 294.1 384.4 498.0 588.3 702.0 792.2 882.4 996.1 1086.3 1200.0 203.9 : 90.2 203.9 294.1 407.8 498.0 611.7 702.0 792.2 905.9 996.1 1109.8 1200.0 294.1 : 113.7 203.9 317.6 407.8 521.5 611.7 702.0 815.6 905.9 1019.6 1109.8 1200.0 407.8 : 90.2 203.9 294.1 407.8 498.0 588.3 702.0 792.2 905.9 996.1 1086.3 1200.0 498.0 : 113.7 203.9 317.6 407.8 498.0 611.7 702.0 815.6 905.9 996.1 1109.8 1200.0 611.7 : 90.2 203.9 294.1 384.4 498.0 588.3 702.0 792.2 882.4 996.1 1086.3 1200.0 702.0 : 113.7 203.9 294.1 407.8 498.0 611.7 702.0 792.2 905.9 996.1 1109.8 1200.0 815.6 : 90.2 180.4 294.1 384.4 498.0 588.3 678.5 792.2 882.4 996.1 1086.3 1200.0 905.9 : 90.2 203.9 294.1 407.8 498.0 588.3 702.0 792.2 905.9 996.1 1109.8 1200.0 996.1 : 113.7 203.9 317.6 407.8 498.0 611.7 702.0 815.6 905.9 1019.6 1109.8 1200.0 1109.8: 90.2 203.9 294.1 384.4 498.0 588.3 702.0 792.2 905.9 996.1 1086.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 113.6850 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 2: 180.4500 .. 203.9100 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 3: 294.1350 .. 317.5950 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 4: 384.3600 .. 407.8200 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 5: 498.0450 .. 521.5050 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 6: 588.2700 .. 611.7300 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 7: 678.4950 .. 701.9550 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 8: 792.1800 .. 815.6400 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 9: 882.4050 .. 905.8650 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents 10: 996.0900 .. 1019.5500 cents 531441/524288, 23.4600 cents 256/243, 90.2250 cents 11: 1086.3150 .. 1109.7750 cents 531441/524288, 23.4600 cents 134217728/129140163, 66.7650 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 256/243, 90.2250 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 2187/2048, 113.6850 cents Average / Smallest step : 1.108340 Largest / Average step : 1.136850 Largest / Smallest step : 1.260017 Median interval of one step : 256/243, 90.2250 cents, amount: 7 Most common interval of one step : 256/243, 90.2250 cents, amount: 7 Least squares average step : 100.54740 cents, oct.: 1206.56880 cents Scale is strictly proper Scale has Myhill's property chromas: 531441/524288, 23.4600 cents and 134217728/129140163, 66.7650 cents, ratio: 2.845906 generators: 5 of 498.04500 cents and 7 of 701.95500 cents ET pair of linear temperament : 12&41 Scale is maximally even for L / S <= 2 Scale is distributional even Scale is a mode of a 665-tone equal temperament with octave 2/1 degrees: 63 113 163 226 276 339 389 452 502 552 615 665 generators: 276 and 389 Number of contiguous 1-step segments: 0 Step pattern alph. order: ABBABABABBAB Step pattern size order : LSSLSLSLSSLS Scale is a Constant Structure, by a margin of 66.76499 cents Ratio Whole / Half tone : 2.260017 Scale diversity : 1.154991 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.784950 Rothenberg efficiency : 0.722222 redundancy: 0.277778 Efficiency x scale size : 8.666667 Number of different interval sizes : 22 = 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 : 531441/524288, 23.4600 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 11 Scale is a chain of 6 triads 0.0 90.225 701.955 cents Most common triad is 0.0 498.045 701.955 cents, amount: 10 Number of recognisable fifths : 11, average 701.9550 cents Scale contains two identical disjunct hexachords Best fifths form a closed circle Scale is a complete genus musicum : 2^m 3^11 Scale contains a complete diamond : 1 3 9 27 Formal octave complements present : 7 = 58.3333% Scale is n-1 differentially coherent in interval classes 4 and 5 combined Scale is n-1 differentially coherent in interval class 5 Scale is n-1 differentially coherent in interval classes 6 and 7 combined Scale is n-1 differentially coherent in interval class 7 Limited inverse transpositions : 11 Inversional symmetry on intervals : 5-6 11-12 -- Rational properties -- Prime limit : 3 Odd number limit : 177147 (O: 177147 U: 177147) Highest odd numerator or denominator: 6561 Scale harmonicity : 0.021583 Average absolute harmonicity : 0.173941 Specific harmonicity : 0.121175 Fundamental : 1/110592, -16.7549 octaves, 0.0024 Hz. Guide tone : 209952, 17.6797 octaves, 54928810.686 Hz. Exponens Consonantiae : 2.321901E+10, 34.43459 octaves Euler's gradus suavitatis : 40 Sum of Mann's harmonic distance : 8404.0, average 700.33333 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 20.03909, average 1.66992 Sum of Tenney's harmonic distance : 39.60604, average 3.30050 Vogel's harmonic complexity : 25.08333 Wille's k value : 88573 Wilson's harmonic complexity : 33 Rectangular lattice diameter : 11 Triangular lattice diameter : 11 Lattice compactness : 615.48341, average 7.89081 Lattice compactness (without 2's) : 328.00000, average 4.20513 Number of different primes : 2 Prime exponents' range, average, count, tones@limit: 2: -12 .. 5 -3.41667 65 1 3: -3 .. 8 2.50000 42 11 Average exponent except 2's : 30 / 12 = 2.50000 Average absolute exponent except 2's: 42 / 12 = 3.50000 Scale is weakly epimorphic with val: <12 17| Scale is JI-epimorphic with val: <12 19| = patent
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 8.3327 c. M: 15.6400 c. Twelve-tone Chromatic (1/11-comma) 29: 3 2 2 3 2 3 2 3 2 2 3 2 SP M ME SD: 6.3647 c. M:-11.9462 c. Twelve-tone Chromatic (1/14-comma positive) 41: 4 3 3 4 3 4 3 4 3 3 4 3 SP M ME SD: 2.0630 c. M:-3.8722 c. Twelve-tone Chromatic (1/44-comma positive) 53: 5 4 4 5 4 5 4 5 4 4 5 4 SP M ME SD: 0.2907 c. M: 0.5457 c. Meantone Chromatic (1/315-comma) 200: 19 15 15 19 15 19 15 19 15 15 19 15 SP M DE SD: 0.1918 c. M:-0.3600 c. Twelve-tone Chromatic (1/478-comma positive) 253: 24 19 19 24 19 24 19 24 19 19 24 19 SP M DE SD: 0.0907 c. M:-0.1703 c. 306: 29 23 23 29 23 29 23 29 23 23 29 23 SP M DE SD: 0.0247 c. M:-0.0463 c. 359: 34 27 27 34 27 34 27 34 27 27 34 27 SP M DE SD: 0.0219 c. M: 0.0411 c. 665: 63 50 50 63 50 63 50 63 50 50 63 50 SP M DE SD: 0.0005 c. M: 0.0009 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: 7.1688 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 8.6813 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 8.4440 cents 6 x x 7 x 8 x 9 x 10:11 x 12 S SD: 10.5986 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 9.9093 cents 8 x 9 x 10:11 x 12:13 x 14:15:16 S SD: 8.9493 cents 9 x 10:11 x 12:13 x 14:15:16:17:18 S SD: 9.3172 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.3366 cents 11:12 x 13:14:15:16:17:18:19:20:21:22 S SD: 9.8052 cents 12:13 x 14:15:16:17:18:19:20:21:23:24 SD: 5.5924 cents 13:14:15 x 16:17 x 20:21:22:23:25:26 SD: 9.1265 cents 14:15:16:17:18:19:20:21:22:24:25:27:28 SD: 6.9224 cents 15:16:17:18:19:20:21:23:24:25:27:28:30 SD: 5.7226 cents 16:17:18:19:20:21:23:24:26:27:28:30:32 SD: 4.8537 cents 18:19:20:21:23:24:26:27:29:30:32:34:36 SD: 4.6708 cents 22:23:25:26:28:29:31:33:35:37:39:42:44 SD: 4.3920 cents 26:28:29:31:33:35:37:39:42:44:46:49:52 SD: 3.0150 cents 30:32:34:36:38:40:43:45:48:51:53:57:60 SD: 2.7019 cents 33:35:37:39:42:44:47:50:53:56:59:63:66 SD: 2.4998 cents 38:41:43:45:48:51:54:57:61:64:68:72:76 SD: 2.2675 cents 42:45:47:50:53:56:60:63:67:71:75:80:84 SD: 1.6428 cents 48:51:54:57:61:64:68:72:77:81:85:91:96 SD: 1.3755 cents 64:68:72:76:81:85:91:96:103:108:114:122:128 SD: 1.3635 cents 70:75:79:83:89:93:100:105:112:118:124:133:140 SD: 1.3062 cents 71:76:80:84:90:95:101:107:114:120:126:135:142 SD: 1.1405 cents 80:85:90:95:101:107:114:120:128:135:142:152:160 SD: 1.0217 cents 86:92:97:102:109:115:122:129:138:145:153:163:172 SD: 0.9282 cents 87:93:98:103:110:116:124:131:139:147:155:165:174 SD: 0.8454 cents 90:96:101:107:114:120:128:135:144:152:160:171:180 SD: 0.6720 cents 118:126:133:140:149:157:168:177:189:199:210:224:236 SD: 0.5772 cents 120:128:135:142:152:160:171:180:192:203:213:228:240 SD: 0.5689 cents 135:144:152:160:171:180:192:203:216:228:240:256:270 SD: 0.5243 cents 144:154:162:171:182:192:205:216:231:243:256:273:288 SD: 0.4984 cents 158:169:178:187:200:211:225:237:253:267:281:300:316 SD: 0.4869 cents 166:177:187:197:210:221:236:249:266:280:295:315:332 SD: 0.4830 cents 174:186:196:206:220:232:248:261:279:294:309:330:348 SD: 0.4721 cents 176:188:198:209:223:235:251:264:282:297:313:334:352 SD: 0.4594 cents 189:202:213:224:239:252:269:284:303:319:336:359:378 SD: 0.4297 cents 206:220:232:244:261:275:293:309:330:348:366:391:412 SD: 0.3778 cents 222:237:250:263:281:296:316:333:356:375:395:421:444 SD: 0.3378 cents 234:250:263:277:296:312:333:351:375:395:416:444:468 SD: 0.2787 cents 264:282:297:313:334:352:376:396:423:446:469:501:528 SD: 0.2246 cents 297:317:334:352:376:396:423:446:476:501:528:564:594 SD: 0.2210 cents 336:359:378:398:425:448:478:504:538:567:597:638:672 SD: 0.2127 cents 342:365:385:405:433:456:487:513:548:577:608:649:684 SD: 0.1989 cents 368:393:414:436:466:491:524:552:589:621:654:699:736 SD: 0.1987 cents 384:410:432:455:486:512:547:576:615:648:683:729:768 SD: 0.1073 cents