Scala analysis: Groenewald_simplified_Bach
Journal: Ars Organi Volume#57, Issue 1, March(2009) p.39
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 90.225 cents 90.225000 2: 189.250 cents 189.250000 3: 294.135 cents 294.135000 4: 386.606 cents 386.606000 5: 498.045 cents 498.045000 6: 588.270 cents 588.270000 7: 693.175 cents 693.175000 8: 792.180 cents 792.180000 9: 887.275 cents 887.275000 10: 996.090 cents 996.090000 11: 1086.808 cents 1086.808000 12: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 113.1920 cents 113.1920 1: 90.2250 cents 90.2250 2: 99.0250 cents 99.0250 3: 104.8850 cents 104.8850 4: 92.4710 cents 92.4710 5: 111.4390 cents 111.4390 6: 90.2250 cents 90.2250 7: 104.9050 cents 104.9050 8: 99.0050 cents 99.0050 9: 95.0950 cents 95.0950 10: 108.8150 cents 108.8150 11: 90.7180 cents 90.7180 12: 113.1920 cents 113.1920
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 90.22500 cents 1: 1 90.71800 cents 1: 1 92.47100 cents 1: 1 95.09500 cents 1: 1 99.00500 cents 1: 1 99.02500 cents 1: 1 104.88500 cents 1: 1 104.90500 cents 1: 1 108.81500 cents 1: 1 111.43900 cents 1: 1 113.19200 cents 2: 1 189.25000 cents 2: 1 194.10000 cents 2: 1 195.13000 cents 2: 1 197.35600 cents 2: 1 199.53300 cents 2: 1 201.66400 cents 2: 1 203.41700 cents 2: 5 203.91000 cents 3: 4 294.13500 cents 3: 1 294.62800 cents 3: 1 296.38100 cents 3: 1 299.00500 cents 3: 1 302.44200 cents 3: 1 302.91500 cents 3: 1 306.56900 cents 3: 1 308.79500 cents 3: 1 312.72500 cents 4: 1 386.60600 cents 4: 1 389.23000 cents 4: 1 393.16000 cents 4: 1 393.63300 cents 4: 1 399.02000 cents 4: 1 399.04000 cents 4: 1 402.95000 cents 4: 1 405.57400 cents 4: 1 407.32700 cents 4: 3 407.82000 cents 5: 6 498.04500 cents 5: 1 498.53800 cents 5: 1 499.79800 cents 5: 1 500.66900 cents 5: 1 501.97500 cents 5: 1 503.92500 cents 5: 1 506.82500 cents 6: 1 588.27000 cents 6: 1 588.76300 cents 6: 1 590.51600 cents 6: 1 593.14000 cents 6: 1 597.05000 cents 6: 1 597.07000 cents 6: 1 602.93000 cents 6: 1 602.95000 cents 6: 1 606.86000 cents 6: 1 609.48400 cents 6: 1 611.23700 cents 6: 1 611.73000 cents 7: 1 693.17500 cents 7: 1 696.07500 cents 7: 1 698.02500 cents 7: 1 699.33100 cents 7: 1 700.20200 cents 7: 1 701.46200 cents 7: 6 701.95500 cents 8: 3 792.18000 cents 8: 1 792.67300 cents 8: 1 794.42600 cents 8: 1 797.05000 cents 8: 1 800.96000 cents 8: 1 800.98000 cents 8: 1 806.36700 cents 8: 1 806.84000 cents 8: 1 810.77000 cents 8: 1 813.39400 cents 9: 1 887.27500 cents 9: 1 891.20500 cents 9: 1 893.43100 cents 9: 1 897.08500 cents 9: 1 897.55800 cents 9: 1 900.99500 cents 9: 1 903.61900 cents 9: 1 905.37200 cents 9: 4 905.86500 cents 10: 5 996.09000 cents 10: 1 996.58300 cents 10: 1 998.33600 cents 10: 1 1000.46700 cents 10: 1 1002.64400 cents 10: 1 1004.87000 cents 10: 1 1005.90000 cents 10: 1 1010.75000 cents 11: 1 1086.80800 cents 11: 1 1088.56100 cents 11: 1 1091.18500 cents 11: 1 1095.09500 cents 11: 1 1095.11500 cents 11: 1 1100.97500 cents 11: 1 1100.99500 cents 11: 1 1104.90500 cents 11: 1 1107.52900 cents 11: 1 1109.28200 cents 11: 2 1109.77500 cents Highest number of different intervals for one interval class: 12 Average number of different intervals per interval class: 9.27273 = 102/11
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 0.0 : 90.2 189.3 294.1 386.6 498.0 588.3 693.2 792.2 887.3 996.1 1086.8 1200.0 90.2 : 99.0 203.9 296.4 407.8 498.0 603.0 702.0 797.0 905.9 996.6 1109.8 1200.0 189.3 : 104.9 197.4 308.8 399.0 503.9 602.9 698.0 806.8 897.6 1010.7 1101.0 1200.0 294.1 : 92.5 203.9 294.1 399.0 498.0 593.1 702.0 792.7 905.9 996.1 1095.1 1200.0 386.6 : 111.4 201.7 306.6 405.6 500.7 609.5 700.2 813.4 903.6 1002.6 1107.5 1200.0 498.0 : 90.2 195.1 294.1 389.2 498.0 588.8 702.0 792.2 891.2 996.1 1088.6 1200.0 588.3 : 104.9 203.9 299.0 407.8 498.5 611.7 702.0 801.0 905.9 998.3 1109.8 1200.0 693.2 : 99.0 194.1 302.9 393.6 506.8 597.1 696.1 801.0 893.4 1004.9 1095.1 1200.0 792.2 : 95.1 203.9 294.6 407.8 498.0 597.1 702.0 794.4 905.9 996.1 1101.0 1200.0 887.3 : 108.8 199.5 312.7 402.9 502.0 606.9 699.3 810.8 901.0 1005.9 1104.9 1200.0 996.1 : 90.7 203.9 294.1 393.2 498.0 590.5 702.0 792.2 897.1 996.1 1091.2 1200.0 1086.8: 113.2 203.4 302.4 407.3 499.8 611.2 701.5 806.4 905.4 1000.5 1109.3 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 90.2250 .. 113.1920 cents 22.96700 cents 90.22500 cents 2: 189.2500 .. 203.9100 cents 14.66000 cents 76.05800 cents 3: 294.1350 .. 312.7250 cents 18.59000 cents 90.22500 cents 4: 386.6060 .. 407.8200 cents 21.21400 cents 73.88100 cents 5: 498.0450 .. 506.8250 cents 8.78000 cents 90.22500 cents 6: 588.2700 .. 611.7300 cents 23.46000 cents 81.44500 cents 7: 693.1750 .. 701.9550 cents 8.78000 cents 81.44500 cents 8: 792.1800 .. 813.3940 cents 21.21400 cents 90.22500 cents 9: 887.2750 .. 905.8650 cents 18.59000 cents 73.88100 cents 10: 996.0900 .. 1010.7500 cents 14.66000 cents 90.22500 cents 11: 1086.8080 .. 1109.7750 cents 22.96700 cents 76.05800 cents
SHOW DATA
Number of notes : 12 -- Interval properties -- Smallest interval : 90.22500 cents, class 1 Average step (divided formal octave): 100.0000 cents Largest one step interval : 113.19200 cents Average / Smallest step : 1.108340 Largest / Average step : 1.131920 Largest / Smallest step : 1.254553 Median interval of one step : 99.01500 cents, amount: 0 Most common interval of one step : 90.22500 cents, amount: 2 Least squares average step : 99.08973 cents, oct.: 1189.07681 cents Scale is strictly proper Step pattern alph. order: ABCDEAFGHIJK Step pattern size order : AFGCJAHEDIBK Scale is a Constant Structure, by a margin of 73.88100 cents Scale diversity : 1.137662 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.836765 Rothenberg efficiency : 0.208829 redundancy: 0.791171 Efficiency x scale size : 2.505952 Number of different interval sizes : 102 = 9.27273 / class Number of one step interval sizes : 11 Highest interval variety : 12 Mean interval variety : 9.27273 = 102/11 Median interval variety : 7 Lowest interval variety : 7 Smallest interval difference : 0.02000 cents Most common intervals : 498.04500 cents & inv., amount: 6 Most common triad is 0.0 498.045 701.955 cents, amount: 5 Number of recognisable fifths : 12, average 700.0000 cents Scale is a well-temperament. It has 7 different sizes of fifth Scale has monotonic third-sizes over circle of fifths Best fifths form a closed circle Formal octave complements present : 1 = 8.3333% Limited transpositions with margin 23.4600 cents: 2 5 7 10 Limited inverse transpositions with margin 22.9670 cents: 1 2 4 6 7 9 11 -- Rational properties -- Prime limit : 2 (not all pitches rational)
FIT/MODE
12: 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 9.2435 c. M:-13.3940 c. Twelve-tone Chromatic (1/11-comma) 31: 2 3 3 2 3 2 3 2 3 3 2 3 SP M ME SD: 8.9953 c. M: 17.9865 c. Meantone Chromatic (53/220-comma) 38: 3 3 3 3 4 3 3 3 3 4 2 4 SP D3 SD: 7.9950 c. M:-14.4363 c. 40: 3 3 4 3 4 3 3 3 4 3 3 4 SP D3 SD: 8.2454 c. M:-12.7250 c. 41: 3 3 4 3 4 3 4 3 3 4 3 4 SP M ME SD: 5.9918 c. M: 13.6402 c. 49: 4 4 4 4 4 4 4 4 4 5 3 5 SP D3 SD: 6.5156 c. M: 9.2570 c. 50: 4 4 4 4 5 4 4 4 4 5 3 5 SP D3 SD: 6.1586 c. M:-11.9100 c. 53: 4 4 5 4 5 4 5 4 4 5 4 5 SP M ME SD: 3.6881 c. M:-8.7118 c. 57: 4 5 5 4 6 4 5 5 4 5 5 5 SP D3 SD: 5.2314 c. M:-7.9288 c. 62: 5 5 5 5 6 4 6 5 5 5 5 6 SP D3 SD: 4.7849 c. M: 8.9932 c. 65: 5 5 6 5 6 5 6 5 5 6 5 6 SP M ME SD: 3.1027 c. M:-8.3635 c. 77: 6 6 7 6 7 6 7 6 6 7 6 7 SP M ME SD: 3.3803 c. M:-8.1237 c. 78: 6 6 7 6 7 6 7 6 7 7 6 7 SP D3 SD: 4.1656 c. M: 7.5646 c. 81: 6 7 7 6 8 6 7 6 7 7 6 8 SP SD: 3.7885 c. M: 6.9948 c. 83: 6 7 7 7 7 7 7 7 6 8 6 8 SP D3 SD: 3.6685 c. M: 6.4787 c. 88: 7 7 8 6 9 6 8 7 7 8 7 8 SP SD: 3.6306 c. M:-6.5005 c. 89: 7 7 8 7 8 7 8 7 7 8 7 8 SP M ME SD: 3.8829 c. M:-7.9486 c. 90: 7 7 8 7 8 7 8 7 8 8 7 8 SP D3 SD: 3.7277 c. M:-6.5253 c. 91: 7 7 8 7 9 7 8 7 7 9 6 9 SP SD: 4.1890 c. M:-6.1078 c. 92: 7 8 8 7 8 7 8 8 7 8 7 9 SP D3 SD: 3.6773 c. M:-6.4022 c. 93: 7 8 8 7 9 7 8 7 8 8 7 9 SP SD: 3.4588 c. M:-5.2784 c. 94: 7 8 8 7 9 7 8 8 7 9 7 9 SP D3 SD: 2.5753 c. M: 6.4239 c. 102: 8 8 9 8 9 8 9 8 8 10 7 10 SP SD: 3.0287 c. M: 4.9221 c. 106: 8 9 9 8 10 8 9 9 8 10 8 10 SP D3 SD: 1.7910 c. M: 4.2561 c. 118: 9 10 10 9 11 9 10 10 9 11 9 11 SP D3 SD: 1.6540 c. M:-3.9703 c. 130: 10 11 11 10 12 10 11 11 10 12 10 12 SP D3 SD: 1.9573 c. M:-4.5962 c. 135: 10 11 12 10 13 10 12 11 11 12 10 13 SP SD: 1.8395 c. M: 4.3838 c. 137: 10 12 12 10 13 10 12 11 11 13 10 13 SP SD: 2.3606 c. M: 3.8588 c. 139: 10 12 12 11 13 10 12 12 11 12 11 13 SP SD: 2.1269 c. M: 3.8941 c. 145: 11 12 13 11 13 11 13 12 11 13 11 14 SP SD: 2.1008 c. M:-3.7960 c. 146: 11 12 13 11 14 11 12 12 12 13 11 14 SP SD: 2.0467 c. M:-3.5108 c. 147: 11 12 13 11 14 11 13 12 12 13 11 14 SP SD: 1.2764 c. M: 2.9325 c. 151: 11 13 13 12 14 11 13 13 12 13 12 14 SP SD: 2.1061 c. M: 2.8078 c. 158: 12 13 14 12 15 11 14 13 13 14 12 15 SP SD: 1.8595 c. M: 3.4599 c. 159: 12 13 14 12 15 12 14 13 13 14 12 15 SP SD: 1.1485 c. M:-3.2910 c. 173: 13 14 15 14 16 13 15 14 14 16 13 16 SP SD: 1.6789 c. M: 2.8055 c. 176: 13 15 15 14 16 13 16 14 14 16 13 17 SP SD: 1.5671 c. M: 2.7171 c. 183: 14 15 16 14 17 14 16 15 14 17 14 17 SP SD: 1.3128 c. M: 2.0291 c. 204: 15 17 18 16 19 15 18 17 16 18 16 19 SP SD: 1.3832 c. M: 1.9897 c. 211: 16 17 19 16 20 15 19 17 17 19 16 20 SP SD: 1.3515 c. M: 2.4880 c. 212: 16 17 19 16 20 16 18 18 17 19 16 20 SP SD: 1.2293 c. M: 2.6090 c. 224: 17 18 20 17 21 17 19 19 18 20 17 21 SP SD: 1.1333 c. M: 2.1036 c. 229: 17 19 20 18 21 17 20 19 18 21 17 22 SP SD: 1.1495 c. M: 2.0918 c. 241: 18 20 21 19 22 18 21 20 19 22 18 23 SP SD: 0.8310 c. M:-1.7757 c. 253: 19 21 22 20 23 19 22 21 20 23 19 24 SP SD: 0.7451 c. M:-2.3268 c. 261: 20 21 23 20 24 20 23 21 21 24 19 25 SP SD: 1.1076 c. M: 1.7505 c. 265: 20 22 23 20 25 20 23 22 21 24 20 25 SP SD: 0.6048 c. M: 1.7003 c. 277: 21 23 24 21 26 21 24 23 22 25 21 26 SP SD: 0.7054 c. M:-1.3637 c. 306: 23 25 27 24 28 23 27 25 24 28 23 29 SP SD: 0.6989 c. M:-1.6293 c. 318: 24 26 28 24 30 24 28 26 25 29 24 30 SP SD: 0.6602 c. M: 1.7003 c. 330: 25 27 29 25 31 25 29 27 26 30 25 31 SP SD: 0.6550 c. M:-1.3705 c. 363: 27 30 32 28 34 27 32 30 28 33 28 34 SP SD: 0.8648 c. M: 1.3246 c. 369: 28 30 32 29 34 28 32 31 29 33 28 35 SP SD: 0.7795 c. M: 1.4521 c. 376: 28 31 33 29 35 28 33 31 30 34 28 36 SP SD: 0.7688 c. M: 1.7016 c. 383: 29 31 34 29 36 29 33 32 30 35 29 36 SP SD: 0.6853 c. M: 1.2604 c. 388: 29 32 34 30 36 29 34 32 31 35 29 37 SP SD: 0.5169 c. M: 1.2410 c. 395: 30 32 35 30 37 30 34 33 31 36 30 37 SP SD: 0.6695 c. M:-1.0971 c. 400: 30 33 35 31 37 30 35 33 32 36 30 38 SP SD: 0.3664 c. M: 0.8080 c. 506: 38 42 44 39 47 38 44 42 40 46 38 48 SP SD: 0.3227 c. M: 0.6849 c. 518: 39 43 45 40 48 39 45 43 41 47 39 49 SP SD: 0.2882 c. M:-0.7114 c. 559: 42 46 49 43 52 42 49 46 44 51 42 53 SP SD: 0.2933 c. M: 0.6918 c. 571: 43 47 50 44 53 43 50 47 45 52 43 54 SP SD: 0.1971 c. M: 0.4099 c. 848: 64 70 74 65 79 64 74 70 67 77 64 80 SP SD: 0.2403 c. M:-0.4092 c. 959: 72 79 84 74 89 72 84 79 76 87 72 91 SP SD: 0.2286 c. M: 0.6766 c. 971: 73 80 85 75 90 73 85 80 77 88 73 92 SP SD: 0.1704 c. M: 0.5052 c. 983: 74 81 86 76 91 74 86 81 78 89 74 93 SP SD: 0.1801 c. M:-0.3726 c. 1089: 82 90 95 84 101 82 95 90 86 99 82 103 SP SD: 0.1634 c. M: 0.3066 c. 1224: 92 101 107 94 114 92 107 101 97 111 92 116 SP SD: 0.1827 c. M: 0.5335 c. 1236: 93 102 108 95 115 93 108 102 98 112 93 117 SP SD: 0.1390 c. M: 0.4002 c. 1248: 94 103 109 96 116 94 109 103 99 113 94 118 SP SD: 0.1470 c. M: 0.2695 c. 1359: 102 112 119 105 126 102 119 112 108 123 103 128 SP SD: 0.1430 c. M: 0.2875 c. 1453: 109 120 127 112 135 109 127 120 115 132 110 137 SP SD: 0.1654 c. M: 0.2826 c. 1465: 110 121 128 113 136 110 128 121 116 133 111 138 SP SD: 0.1130 c. M: 0.2057 c. 1477: 111 122 129 114 137 111 129 122 117 134 112 139 SP SD: 0.0998 c. M:-0.2604 c. 1624: 122 134 142 125 151 122 142 134 129 147 123 153 SP SD: 0.0929 c. M:-0.1634 c. 1865: 140 154 163 144 173 140 163 154 148 169 141 176 SP SD: 0.1059 c. M: 0.1991 c. 1877: 141 155 164 145 174 141 164 155 149 170 142 177 SP SD: 0.0874 c. M:-0.1814 c. 2036: 153 168 178 157 189 153 178 168 161 185 154 192 SP SD: 0.0788 c. M: 0.2416 c. 2048: 154 169 179 158 190 154 179 169 162 186 155 193 SP SD: 0.0658 c. M: 0.1656 c. 2195: 165 181 192 169 204 165 192 181 174 199 166 207 SP SD: 0.0414 c. M: 0.0928 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: 4.3900 cents 3 x x x x 4 x x x 5 x x 6 S SD: 0.9721 cents 4 x x x 5 x x 6 x x 7 x 8 S SD: 7.1611 cents 5 x x 6 x x 7 x 8 x 9 x 10 S SD: 7.5385 cents 6 x x 7 x 8 x 9 x 10 x 11:12 S SD: 7.8723 cents 7 x 8 x 9 x 10 x 11:12 x 13:14 S SD: 12.2775 cents 8 x 9 x 10 x 11:12 x 13:14:15:16 S SD: 8.4671 cents 9 x 10 x 11:12:13 x 14:15:16:17:18 S SD: 7.7582 cents 10 x 11:12 x 13:14:15:16:17:18:19:20 S SD: 7.4737 cents 11 x 12:13:14:15 x 16:17:18:20:21:22 S SD: 10.5981 cents 12:13 x 14:15:16:17:18:19:20:21:22:24 S SD: 6.7667 cents 13:14 x 15:16:17:18:19:21:22:23:24:26 SD: 9.1425 cents 14:15:16:17:18:19:20:21:22:23:25:26:28 SD: 8.2287 cents 15:16:17:18:19:20:21:22:24:25:27:28:30 SD: 5.3383 cents 16:17:18:19:20:21:22:24:25:27:28:30:32 SD: 5.3376 cents 17:18:19:20:21:23:24:25:27:28:30:32:34 SD: 4.4897 cents 23:24:26:27:29:31:32:34:36:38:41:43:46 SD: 4.4272 cents 24:25:27:28:30:32:34:36:38:40:43:45:48 SD: 3.5729 cents 27:28:30:32:34:36:38:40:43:45:48:51:54 SD: 3.2405 cents 32:34:36:38:40:43:45:48:51:53:57:60:64 SD: 2.7963 cents 33:35:37:39:41:44:46:49:52:55:59:62:66 SD: 2.2997 cents 36:38:40:43:45:48:51:54:57:60:64:67:72 SD: 2.1796 cents 43:45:48:51:54:57:60:64:68:72:76:81:86 SD: 2.1603 cents 45:47:50:53:56:60:63:67:71:75:80:84:90 SD: 1.9664 cents 52:55:58:62:65:69:73:78:82:87:92:97:104 SD: 1.7774 cents 53:56:59:63:66:71:74:79:84:88:94:99:106 SD: 1.7665 cents 54:57:60:64:68:72:76:81:85:90:96:101:108 SD: 1.5921 cents 55:58:61:65:69:73:77:82:87:92:98:103:110 SD: 1.4231 cents 59:62:66:70:74:79:83:88:93:99:105:111:118 SD: 1.4225 cents 60:63:67:71:75:80:84:90:95:100:107:112:120 SD: 1.2968 cents 69:73:77:82:86:92:97:103:109:115:123:129:138 SD: 1.0217 cents 76:80:85:90:95:101:107:113:120:127:135:142:152 SD: 0.9959 cents 87:92:97:103:109:116:122:130:137:145:155:163:174 SD: 0.9599 cents 93:98:104:110:116:124:131:139:147:155:165:174:186 SD: 0.8459 cents 96:101:107:114:120:128:135:143:152:160:171:180:192 SD: 0.6824 cents 112:118:125:133:140:149:157:167:177:187:199:210:224 SD: 0.5613 cents 136:143:152:161:170:181:191:203:215:227:242:255:272 SD: 0.5382 cents 148:156:165:175:185:197:208:221:234:247:263:277:296 SD: 0.4783 cents 172:181:192:204:215:229:242:257:272:287:306:322:344 SD: 0.4637 cents 183:193:204:217:229:244:257:273:289:306:325:343:366 SD: 0.3868 cents 205:216:229:243:256:273:288:306:324:342:364:384:410 SD: 0.3786 cents 207:218:231:245:259:276:291:309:327:346:368:388:414 SD: 0.3321 cents 243:256:271:288:304:324:341:363:384:406:432:455:486 SD: 0.2596 cents 260:274:290:308:325:347:365:388:411:434:462:487:520 SD: 0.2063 cents 339:357:378:402:424:452:476:506:536:566:603:635:678 SD: 0.1829 cents 372:392:415:441:465:496:523:555:588:621:661:697:744 SD: 0.1703 cents 432:455:482:512:540:576:607:645:683:721:768:809:864 SD: 0.1360 cents 536:565:598:635:670:715:753:800:847:895:953:1004:1072 SD: 0.1336 cents 544:573:607:645:680:725:764:812:860:908:967:1019:1088 SD: 0.1291 cents 563:593:628:667:704:751:791:840:890:940:1001:1055:1126 SD: 0.1260 cents 579:610:646:686:724:772:813:864:915:967:1029:1085:1158 SD: 0.1163 cents