Arcturus: Difference between revisions

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Line 34: Line 34:
| 0
| 0
| 1755.651<br />1200
| 1755.651<br />1200
| L=1 s=0
| {{nowrap|L {{=}} 1|s {{=}} 0}}
|-
|-
|  
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Line 47: Line 47:
| 20.451<br />13.9785
| 20.451<br />13.9785
| 1758.797<br />1202.151
| 1758.797<br />1202.151
| L=7 s=1
| {{nowrap|L {{=}} 7|s {{=}} 1}}
|-
|-
|  
|  
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| 23.774<br />16.25
| 23.774<br />16.25
| 1759.38<br />1202.5
| 1759.38<br />1202.5
| L=6 s=1
| {{nowrap|L {{=}} 6|s {{=}} 1}}
|-
|-
|  
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Line 86: Line 86:
| 28.387<br />19.403
| 28.387<br />19.403
| 1760.081<br />1202.985
| 1760.081<br />1202.985
| L=5 s=1
| {{nowrap|L {{=}} 5|s {{=}} 1}}
|-
|-
|  
|  
Line 138: Line 138:
| 35.221<br />24.074
| 35.221<br />24.074
| 1761.069<br />1203.704
| 1761.069<br />1203.704
| L=4 s=1
| {{nowrap|L {{=}} 4|s {{=}} 1}}
|-
|-
|  
|  
Line 190: Line 190:
| 40.041<br />27.368
| 40.041<br />27.368
| 1761.811<br />1204.2105
| 1761.811<br />1204.2105
| L=7 s=2
| {{nowrap|L {{=}} 7|s {{=}} 2}}
|-
|-
|  
|  
Line 242: Line 242:
| 46.389<br />31.707
| 46.389<br />31.707
| 1762.788<br />1204.878
| 1762.788<br />1204.878
| L=3 s=1
| {{nowrap|L {{=}} 3|s {{=}} 1}}
|-
|-
|  
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Line 346: Line 346:
| 55.129<br />37.681
| 55.129<br />37.681
| 1764.132<br />1205.797
| 1764.132<br />1205.797
| L=5 s=2
| {{nowrap|L {{=}} 5|s {{=}} 2}}
|-
|-
|  
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| 58.823<br />40.206
| 58.823<br />40.206
| 1764.7005<br />1203.185
| 1764.7005<br />1203.185
| L=7 s=3
| {{nowrap|L {{=}} 7|s {{=}} 3}}
|-
|-
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| 67.93<br />46.429
| 67.93<br />46.429
| 1766.101<br />1207.143
| 1766.101<br />1207.143
| L=2 s=1
| {{nowrap|L {{=}} 2|s {{=}} 1}}
|-
|-
|  
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| 76.847<br />52.525
| 76.847<br />52.525
| 1767.473<br />1208.081
| 1767.473<br />1208.081
| L=7 s=4
| {{nowrap|L {{=}} 7|s {{=}} 4}}
|-
|-
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| 80.364<br />54.93
| 80.364<br />54.93
| 1768.0145<br />1208.451
| 1768.0145<br />1208.451
| L=5 s=3
| {{nowrap|L {{=}} 5|s {{=}} 3}}
|-
|-
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| 88.463<br />60.465
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| 1769.2605<br />1209.302
| 1769.2605<br />1209.302
| L=3 s=2
| {{nowrap|L {{=}} 3|s {{=}} 2}}
|-
|-
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| 94.156<br />64.356
| 94.156<br />64.356
| 1770.136<br />1209.901
| 1770.136<br />1209.901
| L=7 s=5
| {{nowrap|L {{=}} 7|s {{=}} 5}}
|-
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| 98.377<br />67.241
| 98.377<br />67.241
| 1770.786<br />1210.345
| 1770.786<br />1210.345
| L=4 s=3
| {{nowrap|L {{=}} 4|s {{=}} 3}}
|-
|-
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| 104.217<br />71.233
| 104.217<br />71.233
| 1771.684<br />1210.959
| 1771.684<br />1210.959
| L=5 s=4
| {{nowrap|L {{=}} 5|s {{=}} 4}}
|-
|-
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| 108.065<br />73.864
| 108.065<br />73.864
| 1772.276<br />1211.364
| 1772.276<br />1211.364
| L=6 s=5
| {{nowrap|L {{=}} 6|s {{=}} 5}}
|-
|-
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| 110.7935<br />75.728
| 110.7935<br />75.728
| 1772.696<br />1211.6505
| 1772.696<br />1211.6505
| L=7 s=6
| {{nowrap|L {{=}} 7|s {{=}} 6}}
|-
|-
| 7\15
| 7\15
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|  
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| 887.579<br />606.667
| 887.579<br />606.667
| colspan="2" 126.797<br />86.667
| colspan="2" | 126.797<br />86.667
| 1775.158<br />1213.333
| 1775.158<br />1213.333
| L=1 s=1
| {{nowrap|L {{=}} 1|s {{=}} 1}}
|}
|}


Revision as of 01:12, 24 November 2024

This page on a regular temperament, temperament collection, or aspect of regular temperament theory is being revised for clarity as part of WikiProject TempClean.

Arcturus is the non-octave rank-2 regular temperament of the 3.5.7 subgroup that tempers out the Arcturus 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⟨3/1⟩) 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.

For technical data, see No-twos subgroup temperaments#Arcturus.

Etymology

This temperament is named after the star Arcturus, following a series of non-octave temperaments that are named after stars.

Todo: add etymology

Add name (person who coined the term) and year (when it was coined).

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

Scales

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