Arcturus
Arcturus is a no-twos temperament which tempers out the comma 15625/15309. Having an ~5/3 as a generator, this temperament is the application of the Pythagorean principle of tuning a stack of the next higher prime number and then factoring out powers of the equivalence to tritave composition. However, a heptatonic MOS (2L 5s) will not suffice to produce an understandable rendition of it because a very close ~5/3 generates a L:s ratio between 4:1 and 5:1, which is beginning to get too lopsided to still be a complete presentation of a temperament.
Chords
Arcturus contains the triad 5:7:9 (used in Bohlen-Pierce harmony) and the triad 27:35:45 which divides 5/3 into two nearly-equal parts.
Tuning spectrum
Below is a list of MOS families which present it completely (however smearily) using a generator of 845.3 to 951.0 cents:
| Generator | cents
hekts |
L | s | 2g | Notes | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 6\13 | 877.825
600 |
146.304
100 |
0 | 1755.651
1200 |
L=1 s=0 | ||||||
| 43\93 | 879.399
601.075 |
143.158
97.8495 |
20.451
13.9785 |
1758.797
1202.151 |
L=7 s=1 | ||||||
| 37\80 | 879.654
601.25 |
142.647
97.5 |
23.774
16.25 |
1759.38
1202.5 |
L=6 s=1 | ||||||
| 68\147 | 879.816
601.3605 |
142.323
97.279 |
25.877
17.687 |
1759.632
1202.721 |
|||||||
| 31\67 | 880.009
601.4925 |
141.937
97.015 |
28.387
19.403 |
1760.081
1202.985 |
L=5 s=1 | ||||||
| 87\188 | 880.16
601.596 |
141.634
96.8085 |
30.35
20.745 |
1760.32
1203.191 |
|||||||
| 56\121 | 880.243
601.653 |
141.468
96.694 |
31.437
21.488 |
1760.487
1203.306 |
|||||||
| 81\175 | 880.3335
601.714 |
141.288
96.571 |
32.605
22.286 |
1760.667
1203.429 |
|||||||
| 25\54 | 880.535
601.852 |
140.886
96.296 |
35.221
24.074 |
1761.069
1203.704 |
L=4 s=1 | ||||||
| 94\203 | 880.708
601.97 |
140.5385
96.059 |
37.477
25.616 |
1761.4165
1203.971 |
|||||||
| 69\149 | 880.711
602.013 |
140.413
95.973 |
38.294
26.1745 |
1761.542
1204.027 |
|||||||
| 113\244 | 880.823
602.049 |
140.308
95.902 |
38.9745
26.639 |
1761.647
1204.098 |
|||||||
| 44\95 | 880.9055
602.105 |
140.144
95.7895 |
40.041
27.368 |
1761.811
1204.2105 |
L=7 s=2 | ||||||
| 107\231 | 880.992
602.1645 |
139.971
95.671 |
41.168
28.1385 |
1761.984
1204.329. |
|||||||
| 63\136 | 881.053
602.206 |
139.85
95.588 |
41.955
28.6765 |
1762.105
1204.412 |
|||||||
| 82\177 | 881.132
602.26 |
139.692
95.48 |
42.982
22.034 |
1762.263
1204.52 |
|||||||
| 19\41 | 881.394
602.439 |
139.167
95.122 |
46.389
31.707 |
1762.788
1204.878 |
L=3 s=1 | ||||||
| 89\192 | 881.635
602.604 |
138.684
94.792 |
49.53
33.854 |
1763.271
1205.208 |
|||||||
| 70\151 | 881.701
602.649 |
138.553
94.702 |
50.383
25.828 |
1763.402
1205.298 |
|||||||
| 121\261 | 881.794
602.682 |
138.4565
89.655 |
51.01
34.866 |
1763.4985
1205.362 |
|||||||
| 51\110 | 881.8155
602.727 |
138.324
94.5455 |
51.8715
35.4545 |
1763.631
1205.4545 |
|||||||
| 134\289 | 881.875
602.768 |
138.204
94.464 |
52.649
35.986 |
1763.751
1205.536 |
|||||||
| 83\179 | 881.912
602.793 |
138.131
94.413 |
53.172
36.313 |
1763.824
1205.586 |
|||||||
| 115\248 | 881.955
602.823 |
138.045
94.355 |
53.684
36.6935 |
1763.91
1205.645 |
|||||||
| 32\69 | 882.066
602.899 |
137.823
94.203 |
55.129
37.681 |
1764.132
1205.797 |
L=5 s=2 | ||||||
| 109\235 | 882.183
602.979 |
137.588
94.043 |
56.654
38.723 |
1764.367
1205.957 |
|||||||
| 77\166 | 882.232
603.012 |
137.491
93.976 |
57.288
39.157 |
1764.464
1206.024 |
|||||||
| 122\263 | 882.276
603.042 |
137.404
93.916 |
57.854
39.544 |
1764.551
1206.084 |
|||||||
| 45\97 | 882.35
603.093 |
137.2545
93.814 |
58.823
40.206 |
1764.7005
1203.185 |
L=7 s=3 | ||||||
| 103\222 | 882.439
603.153 |
137.078
93.694 |
59.972
40.991 |
1764.877
1206.306 |
|||||||
| 58\125 | 882.507
603.2 |
136.941
93.6 |
60.863
41.6 |
1765.014
1206.4 |
|||||||
| 71\153 | 882.607
603.268 |
136.742
93.464 |
62.155
42.484 |
1765.213
1206.536 |
|||||||
| 13\28 | 883.0505
603.571 |
135.854
92.857 |
67.93
46.429 |
1766.101
1207.143 |
L=2 s=1 | ||||||
| 72\155 | 883.489
603.871 |
134.9775
92.258 |
73.624
50.323 |
1766.9775
1207.742 |
|||||||
| 59\127 | 883.585
603.937 |
134.784
92.126 |
74.88
51.181 |
1767.171
1207.574 |
|||||||
| 105\226 | 883.652
603.982 |
134.652
92.035 |
75.742
51.77 |
1767.303
1207.964 |
|||||||
| 46\99 | 883.737
604.04 |
134.482
91.919 |
76.847
52.525 |
1767.473
1208.081 |
L=7 s=4 | ||||||
| 125\269 | 883.808
604.089 |
134.339
91.822 |
77.775
53.16 |
1767.616
1208.178 |
|||||||
| 79\170 | 883.85
604.118 |
134.256
91.765 |
78.316
53.529 |
1767.699
1208.235 |
|||||||
| 112\241 | 883.896
604.149 |
134.163
91.701 |
78.919
53.942 |
1767.792
1208.299 |
|||||||
| 33\71 | 884.007
604.225 |
133.94
91.549 |
80.364
54.93 |
1768.0145
1208.451 |
L=5 s=3 | ||||||
| 119\256 | 884.112
604.297 |
133.731
91.406 |
81.725
55.859 |
1768.224
1208.594 |
|||||||
| 86\185 | 884.152
604.324 |
133.651
91.351 |
82.247
56.216 |
1768.304
1208.649 |
|||||||
| 139\299 | 884.186
604.348 |
133.582
91.304 |
82.694
56.522 |
1768.373
1208.696 |
Golden Arcturus is near here | ||||||
| 53\114 | 884.24
604.386 |
133.4705
91.228 |
83.419
57.0175 |
1768.4845
1208.772 |
|||||||
| 126\271 | 884.303
604.428 |
133.347
91.144 |
84.219
57.565 |
1768.608
1208.856 |
|||||||
| 73\157 | 884.3485
604.459 |
133.258
91.083 |
84.8005
57.962 |
1768.697
1208.917 |
5/3-Pythagorean is near here | ||||||
| 93\200 | 884.409
604.5 |
133.137
91 |
85.588
58.5 |
1768.818
1209 |
|||||||
| 20\43 | 884.63
604.651 |
132.6945
90.698 |
88.463
60.465 |
1769.2605
1209.302 |
L=3 s=2 | ||||||
| 87\187 | 884.867
604.813 |
132.2215
90.374 |
91.538
62.567 |
1769.7335
1209.626 |
|||||||
| 67\144 | 884.937
604.861 |
132.08
90.278 |
92.456
63.194 |
1769.875
1209.722 |
|||||||
| 114\245 | 884.991
604.898 |
131.972
90.204 |
93.157
52.6735 |
1769.983
1209.896 |
|||||||
| 47\101 | 885.068
604.9505 |
131.819
90.099 |
94.156
64.356 |
1770.136
1209.901 |
L=7 s=5 | ||||||
| 121\260 | 885.141
605 |
131.674
90 |
95.098
65 |
1770.281
1210 |
|||||||
| 74\159 | 885.187
605.031 |
131.582
89.937 |
95.696
65.409 |
1770.373
1210.063 |
|||||||
| 101\217 | 885.242
605.069 |
131.4715
89.862 |
96.4125
65.899 |
1770.4835
1210.138 |
|||||||
| 27\58 | 885.393
605.172 |
131.169
89.655 |
98.377
67.241 |
1770.786
1210.345 |
L=4 s=3 | ||||||
| 88\189 | 885.566
605.291 |
130.822
89.418 |
100.6325
68.783 |
1771.133
1210.582 |
|||||||
| 61\131 | 885.643
605.3435 |
130.669
89.313 |
101.631
69.466 |
1771.286
1210.687 |
|||||||
| 95\204 | 885.714
605.392 |
130.526
89.216 |
102.556
70.098 |
1771.429
1210.784 |
|||||||
| 34\73 | 885.842
605.4795 |
130.271
89.041 |
104.217
71.233 |
1771.684
1210.959 |
L=5 s=4 | ||||||
| 75\161 | 886.004
605.59 |
129.947
88.82 |
106.3205
72.671 |
1772.008
1211.18 |
|||||||
| 41\88 | 886.138
605.682 |
129.679
88.636 |
108.065
73.864 |
1772.276
1211.364 |
L=6 s=5 | ||||||
| 48\103 | 886.348
605.825 |
129.259
88.3495 |
110.7935
75.728 |
1772.696
1211.6505 |
L=7 s=6 | ||||||
| 7\15 | 887.579
606.667 |
126.797
86.667 |
1775.158
1213.333 |
L=1 s=1 | |||||||