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'''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.
'''Arcturus''' is the [[non-octave]] [[rank]]-2 [[regular temperament]] of the 3.5.7 [[subgroup]] that [[tempering out|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 scalesig|2L 5s<3/1>|link=1}} [[MOS]] 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.
 
{{tdlink|No-twos subgroup temperaments #Arcturus}}
 
== Etymology ==
This temperament is named after the star {{w|Arcturus}}, following a series of non-octave temperaments that are named after stars.
{{todo|add etymology|inline=1|text=Add name (person who coined the term) and year (when it was coined).}}


== Chords ==
== 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.
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 ==
== 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:
Below is a list of MOS families which present it completely (however smearily) using a generator of 845.3 to 951.0{{c}}:
 
[[Sub-Arcturus|Mini chromatic]]
 
[[Anti-Arcturus|Anti-chromatic]]
 
{{Scale tree|11L 2s (3/1-equivalent)|depth=6}}


[[Super-Arcturus_15L_2s|Mini enharmonic]]
{| class="wikitable" style="text-align: center;"
|-
! colspan="7" | Generator
! Cents<br>Hekts
! L
! s
! 2g
! Notes
|-
| 6\13
|
|
|
|
|
|
| 877.825<br>600
| 146.304<br>100
| 0
| 1755.651<br>1200
| {{nowrap|L {{=}} 1|s {{=}} 0}}
|-
|
|
|
|
|
|
| 43\93
| 879.399<br>601.075
| 143.158<br>97.8495
| 20.451<br>13.9785
| 1758.797<br>1202.151
| {{nowrap|L {{=}} 7|s {{=}} 1}}
|-
|
|
|
|
|
| 37\80
|
| 879.654<br>601.25
| 142.647<br>97.5
| 23.774<br>16.25
| 1759.38<br>1202.5
| {{nowrap|L {{=}} 6|s {{=}} 1}}
|-
|
|
|
|
|
|
| 68\147
| 879.816<br>601.3605
| 142.323<br>97.279
| 25.877<br>17.687
| 1759.632<br>1202.721
|
|-
|
|
|
|
| 31\67
|
|
| 880.009<br>601.4925
| 141.937<br>97.015
| 28.387<br>19.403
| 1760.081<br>1202.985
| {{nowrap|L {{=}} 5|s {{=}} 1}}
|-
|
|
|
|
|
|
| 87\188
| 880.16<br>601.596
| 141.634<br>96.8085
| 30.35<br>20.745
| 1760.32<br>1203.191
|
|-
|
|
|
|
|
| 56\121
|
| 880.243<br>601.653
| 141.468<br>96.694
| 31.437<br>21.488
| 1760.487<br>1203.306
|
|-
|
|
|
|
|
|
| 81\175
| 880.3335<br>601.714
| 141.288<br>96.571
| 32.605<br>22.286
| 1760.667<br>1203.429
|
|-
|
|
|
| 25\54
|
|
|
| 880.535<br>601.852
| 140.886<br>96.296
| 35.221<br>24.074
| 1761.069<br>1203.704
| {{nowrap|L {{=}} 4|s {{=}} 1}}
|-
|
|
|
|
|
|
| 94\203
| 880.708<br>601.97
| 140.5385<br>96.059
| 37.477<br>25.616
| 1761.4165<br>1203.971
|
|-
|
|
|
|
|
| 69\149
|
| 880.711<br>602.013
| 140.413<br>95.973
| 38.294<br>26.1745
| 1761.542<br>1204.027
|
|-
|
|
|
|
|
|
| 113\244
| 880.823<br>602.049
| 140.308<br>95.902
| 38.9745<br>26.639
| 1761.647<br>1204.098
|
|-
|
|
|
|
| 44\95
|
|
| 880.9055<br>602.105
| 140.144<br>95.7895
| 40.041<br>27.368
| 1761.811<br>1204.2105
| {{nowrap|L {{=}} 7|s {{=}} 2}}
|-
|
|
|
|
|
|
| 107\231
| 880.992<br>602.1645
| 139.971<br>95.671
| 41.168<br>28.1385
| 1761.984<br>1204.329.
|
|-
|
|
|
|
|
| 63\136
|
| 881.053<br>602.206
| 139.85<br>95.588
| 41.955<br>28.6765
| 1762.105<br>1204.412
|
|-
|
|
|
|
|
|
| 82\177
| 881.132<br>602.26
| 139.692<br>95.48
| 42.982<br>22.034
| 1762.263<br>1204.52
|
|-
|
|
| 19\41
|
|
|
|
| 881.394<br>602.439
| 139.167<br>95.122
| 46.389<br>31.707
| 1762.788<br>1204.878
| {{nowrap|L {{=}} 3|s {{=}} 1}}
|-
|
|
|
|
|
|
| 89\192
| 881.635<br>602.604
| 138.684<br>94.792
| 49.53<br>33.854
| 1763.271<br>1205.208
|
|-
|
|
|
|
|
| 70\151
|
| 881.701<br>602.649
| 138.553<br>94.702
| 50.383<br>25.828
| 1763.402<br>1205.298
|
|-
|
|
|
|
|
|
| 121\261
| 881.794<br>602.682
| 138.4565<br>89.655
| 51.01<br>34.866
| 1763.4985<br>1205.362
|
|-
|
|
|
|
| 51\110
|
|
| 881.8155<br>602.727
| 138.324<br>94.5455
| 51.8715<br>35.4545
| 1763.631<br>1205.4545
|
|-
|
|
|
|
|
|
| 134\289
| 881.875<br>602.768
| 138.204<br>94.464
| 52.649<br>35.986
| 1763.751<br>1205.536
|
|-
|
|
|
|
|
| 83\179
|
| 881.912<br>602.793
| 138.131<br>94.413
| 53.172<br>36.313
| 1763.824<br>1205.586
|
|-
|
|
|
|
|
|
| 115\248
| 881.955<br>602.823
| 138.045<br>94.355
| 53.684<br>36.6935
| 1763.91<br>1205.645
|
|-
|
|
|
| 32\69
|
|
|
| 882.066<br>602.899
| 137.823<br>94.203
| 55.129<br>37.681
| 1764.132<br>1205.797
| {{nowrap|L {{=}} 5|s {{=}} 2}}
|-
|
|
|
|
|
|
| 109\235
| 882.183<br>602.979
| 137.588<br>94.043
| 56.654<br>38.723
| 1764.367<br>1205.957
|
|-
|
|
|
|
|
| 77\166
|
| 882.232<br>603.012
| 137.491<br>93.976
| 57.288<br>39.157
| 1764.464<br>1206.024
|
|-
|
|
|
|
|
|
| 122\263
| 882.276<br>603.042
| 137.404<br>93.916
| 57.854<br>39.544
| 1764.551<br>1206.084
|
|-
|
|
|
|
| 45\97
|
|
| 882.35<br>603.093
| 137.2545<br>93.814
| 58.823<br>40.206
| 1764.7005<br>1203.185
| {{nowrap|L {{=}} 7|s {{=}} 3}}
|-
|
|
|
|
|
|
| 103\222
| 882.439<br>603.153
| 137.078<br>93.694
| 59.972<br>40.991
| 1764.877<br>1206.306
|
|-
|
|
|
|
|
| 58\125
|
| 882.507<br>603.2
| 136.941<br>93.6
| 60.863<br>41.6
| 1765.014<br>1206.4
|
|-
|
|
|
|
|
|
| 71\153
| 882.607<br>603.268
| 136.742<br>93.464
| 62.155<br>42.484
| 1765.213<br>1206.536
|
|-
|
| 13\28
|
|
|
|
|
| 883.0505<br>603.571
| 135.854<br>92.857
| 67.93<br>46.429
| 1766.101<br>1207.143
| {{nowrap|L {{=}} 2|s {{=}} 1}}
|-
|
|
|
|
|
|
| 72\155
| 883.489<br>603.871
| 134.9775<br>92.258
| 73.624<br>50.323
| 1766.9775<br>1207.742
|
|-
|
|
|
|
|
| 59\127
|
| 883.585<br>603.937
| 134.784<br>92.126
| 74.88<br>51.181
| 1767.171<br>1207.574
|
|-
|
|
|
|
|
|
| 105\226
| 883.652<br>603.982
| 134.652<br>92.035
| 75.742<br>51.77
| 1767.303<br>1207.964
|
|-
|
|
|
|
| 46\99
|
|
| 883.737<br>604.04
| 134.482<br>91.919
| 76.847<br>52.525
| 1767.473<br>1208.081
| {{nowrap|L {{=}} 7|s {{=}} 4}}
|-
|
|
|
|
|
|
| 125\269
| 883.808<br>604.089
| 134.339<br>91.822
| 77.775<br>53.16
| 1767.616<br>1208.178
|
|-
|
|
|
|
|
| 79\170
|
| 883.85<br>604.118
| 134.256<br>91.765
| 78.316<br>53.529
| 1767.699<br>1208.235
|
|-
|
|
|
|
|
|
| 112\241
| 883.896<br>604.149
| 134.163<br>91.701
| 78.919<br>53.942
| 1767.792<br>1208.299
|
|-
|
|
|
| 33\71
|
|
|
| 884.007<br>604.225
| 133.94<br>91.549
| 80.364<br>54.93
| 1768.0145<br>1208.451
| {{nowrap|L {{=}} 5|s {{=}} 3}}
|-
|
|
|
|
|
|
| 119\256
| 884.112<br>604.297
| 133.731<br>91.406
| 81.725<br>55.859
| 1768.224<br>1208.594
|
|-
|
|
|
|
|
| 86\185
|
| 884.152<br>604.324
| 133.651<br>91.351
| 82.247<br>56.216
| 1768.304<br>1208.649
|
|-
|
|
|
|
|
|
| 139\299
| 884.186<br>604.348
| 133.582<br>91.304
| 82.694<br>56.522
| 1768.373<br>1208.696
| Golden Arcturus is near here
|-
|
|
|
|
| 53\114
|
|
| 884.24<br>604.386
| 133.4705<br>91.228
| 83.419<br>57.0175
| 1768.4845<br>1208.772
|
|-
|
|
|
|
|
|
| 126\271
| 884.303<br>604.428
| 133.347<br>91.144
| 84.219<br>57.565
| 1768.608<br>1208.856
|
|-
|
|
|
|
|
| 73\157
|
| 884.3485<br>604.459
| 133.258<br>91.083
| 84.8005<br>57.962
| 1768.697<br>1208.917
| style="text-align:center;" | 5/3-Pythagorean is near here
|-
|
|
|
|
|
|
| 93\200
| 884.409<br>604.5
| 133.137<br>91
| 85.588<br>58.5
| 1768.818<br>1209
|
|-
|
|
| 20\43
|
|
|
|
| 884.63<br>604.651
| 132.6945<br>90.698
| 88.463<br>60.465
| 1769.2605<br>1209.302
| {{nowrap|L {{=}} 3|s {{=}} 2}}
|-
|
|
|
|
|
|
| 87\187
| 884.867<br>604.813
| 132.2215<br>90.374
| 91.538<br>62.567
| 1769.7335<br>1209.626
|
|-
|
|
|
|
|
| 67\144
|
| 884.937<br>604.861
| 132.08<br>90.278
| 92.456<br>63.194
| 1769.875<br>1209.722
|
|-
|
|
|
|
|
|
| 114\245
| 884.991<br>604.898
| 131.972<br>90.204
| 93.157<br>52.6735
| 1769.983<br>1209.896
|
|-
|
|
|
|
| 47\101
|
|
| 885.068<br>604.9505
| 131.819<br>90.099
| 94.156<br>64.356
| 1770.136<br>1209.901
| {{nowrap|L {{=}} 7|s {{=}} 5}}
|-
|
|
|
|
|
|
| 121\260
| 885.141<br>605
| 131.674<br>90
| 95.098<br>65
| 1770.281<br>1210
|
|-
|
|
|
|
|
| 74\159
|
| 885.187<br>605.031
| 131.582<br>89.937
| 95.696<br>65.409
| 1770.373<br>1210.063
|
|-
|
|
|
|
|
|
| 101\217
| 885.242<br>605.069
| 131.4715<br>89.862
| 96.4125<br>65.899
| 1770.4835<br>1210.138
|
|-
|
|
|
| 27\58
|
|
|
| 885.393<br>605.172
| 131.169<br>89.655
| 98.377<br>67.241
| 1770.786<br>1210.345
| {{nowrap|L {{=}} 4|s {{=}} 3}}
|-
|
|
|
|
|
|
| 88\189
| 885.566<br>605.291
| 130.822<br>89.418
| 100.6325<br>68.783
| 1771.133<br>1210.582
|
|-
|
|
|
|
|
| 61\131
|
| 885.643<br>605.3435
| 130.669<br>89.313
| 101.631<br>69.466
| 1771.286<br>1210.687
|
|-
|
|
|
|
|
|
| 95\204
| 885.714<br>605.392
| 130.526<br>89.216
| 102.556<br>70.098
| 1771.429<br>1210.784
|
|-
|
|
|
|
| 34\73
|
|
| 885.842<br>605.4795
| 130.271<br>89.041
| 104.217<br>71.233
| 1771.684<br>1210.959
| {{nowrap|L {{=}} 5|s {{=}} 4}}
|-
|
|
|
|
|
|
| 75\161
| 886.004<br>605.59
| 129.947<br>88.82
| 106.3205<br>72.671
| 1772.008<br>1211.18
|
|-
|
|
|
|
|
| 41\88
|
| 886.138<br>605.682
| 129.679<br>88.636
| 108.065<br>73.864
| 1772.276<br>1211.364
| {{nowrap|L {{=}} 6|s {{=}} 5}}
|-
|
|
|
|
|
|
| 48\103
| 886.348<br>605.825
| 129.259<br>88.3495
| 110.7935<br>75.728
| 1772.696<br>1211.6505
| {{nowrap|L {{=}} 7|s {{=}} 6}}
|-
| 7\15
|
|
|
|
|
|
| 887.579<br>606.667
| colspan="2" | 126.797<br>86.667
| 1775.158<br>1213.333
| {{nowrap|L {{=}} 1|s {{=}} 1}}
|}


[[Super-Arcturus_17L_2s|Enharmonic]]
== Scales ==
* {{mos scalesig|9L 2s<3/1>|link=1}} (mini chromatic, aka sub-Arcturus)
* {{mos scalesig|11L 2s<3/1>|link=1}} (anti-chromatic, aka anti-Arcturus)
* {{mos scalesig|15L 2s<3/1>|link=1}} (mini enharmonic, aka super-Arcturus 15L 2s)
* {{mos scalesig|17L 2s<3/1>|link=1}} (enharmonic, aka super-Arcturus 17L 2s)
* {{mos scalesig|2L 17s<3/1>|link=1}} (anti-enharmonic, aka trans-Arcturus 2L 7s)


[[Trans-Arcturus_enneadecatonic|Anti-enharmonic]]
[[Category:Arcturus| ]] <!-- main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Nonoctave]]
[[Category:Non-octave temperaments]]
Retrieved from "https://en.xen.wiki/w/Arcturus"