Golden magic: Difference between revisions

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'''Golden magic''' is a tuning of [[magic]] temperament where the 19 tone and 22 tone MOS's have step size ratios equal to the golden ratio. It is very closely approximated by 104edo and 167edo.
'''Golden magic''' is a tuning of [[magic]] temperament where the 19 tone and 22 tone MOS's have step size ratios equal to the golden ratio. It is very closely approximated by 104edo and 167edo. The generator of golden magic is roughly 380.819 cents.


==Errors relative to just intonation==
==Errors relative to just intonation==


Golden magic approximates simple intervals in the 2,3,7,11 subgroup very closely.
Golden magic approximates simple intervals in the 2,3,7,11 subgroup very closely and 5 somewhat less closely (in fact 5 has the highest error of any identity in the 11 limit). Here's the 41 tone MOS with a JI detempering in the 2,3,7,11 subgroup:
{| class="wikitable"
|Scale Degree
|Ratio
|Golden Magic
|Detempered
|Error
|-
|0
|1/1
|0.000
|0.000
|0.000
|-
|1
|49/48
|35.563
|35.697
| -0.134
|-
|2
|33/32
|57.543
|53.273
|4.270
|-
|3
|81/77
|93.106
|87.676
|5.430
|-
|4
|77/72
|115.085
|116.234
| -1.149
|-
|5
|12/11
|150.649
|150.637
|0.012
|-
|6
|54/49
|172.628
|168.213
|4.415
|-
|7
|9/8
|208.191
|203.910
|4.281
|-
|8
|8/7
|230.171
|231.174
| -1.004
|-
|9
|7/6
|265.734
|266.871
| -1.137
|-
|10
|33/28
|287.713
|284.447
|3.266
|-
|11
|77/64
|323.276
|320.144
|3.133
|-
|12
|11/9
|345.256
|347.408
| -2.152
|-
|13
|96/77
|380.819
|381.811
| -0.992
|-
|14
|14/11
|416.382
|417.508
| -1.126
|-
|15
|9/7
|438.362
|435.084
|3.278
|-
|16
|21/16
|473.925
|470.781
|3.144
|-
|17
|4/3
|495.904
|498.045
| -2.141
|-
|18
|49/36
|531.468
|533.742
| -2.274
|-
|19
|11/8
|553.447
|551.318
|2.129
|-
|20
|108/77
|589.010
|585.721
|3.289
|-
|21
|77/54
|610.990
|614.279
| -3.289
|-
|22
|16/11
|646.553
|648.682
| -2.129
|-
|23
|72/49
|668.532
|666.258
|2.274
|-
|24
|3/2
|704.096
|701.955
|2.141
|-
|25
|32/21
|726.075
|729.219
| -3.144
|-
|26
|14/9
|761.638
|764.916
| -3.278
|-
|27
|11/7
|783.618
|782.492
|1.126
|-
|28
|77/48
|819.181
|818.189
|0.992
|-
|29
|18/11
|854.744
|852.592
|2.152
|-
|30
|128/77
|876.724
|879.856
| -3.133
|-
|31
|56/33
|912.287
|915.553
| -3.266
|-
|32
|12/7
|934.266
|933.129
|1.137
|-
|33
|7/4
|969.829
|968.826
|1.004
|-
|34
|16/9
|991.809
|996.090
| -4.281
|-
|35
|49/27
|1027.372
|1031.787
| -4.415
|-
|36
|11/6
|1049.351
|1049.363
| -0.012
|-
|37
|144/77
|1084.915
|1083.766
|1.149
|-
|38
|154/81
|1106.894
|1112.324
| -5.430
|-
|39
|64/33
|1142.457
|1146.727
| -4.270
|-
|40
|96/49
|1164.437
|1164.303
|0.134
|-
|41
|2/1
|1200.000
|1200.000
|0.000
|}

Revision as of 18:07, 12 April 2022

Golden magic is a tuning of magic temperament where the 19 tone and 22 tone MOS's have step size ratios equal to the golden ratio. It is very closely approximated by 104edo and 167edo. The generator of golden magic is roughly 380.819 cents.

Errors relative to just intonation

Golden magic approximates simple intervals in the 2,3,7,11 subgroup very closely and 5 somewhat less closely (in fact 5 has the highest error of any identity in the 11 limit). Here's the 41 tone MOS with a JI detempering in the 2,3,7,11 subgroup:

Scale Degree Ratio Golden Magic Detempered Error
0 1/1 0.000 0.000 0.000
1 49/48 35.563 35.697 -0.134
2 33/32 57.543 53.273 4.270
3 81/77 93.106 87.676 5.430
4 77/72 115.085 116.234 -1.149
5 12/11 150.649 150.637 0.012
6 54/49 172.628 168.213 4.415
7 9/8 208.191 203.910 4.281
8 8/7 230.171 231.174 -1.004
9 7/6 265.734 266.871 -1.137
10 33/28 287.713 284.447 3.266
11 77/64 323.276 320.144 3.133
12 11/9 345.256 347.408 -2.152
13 96/77 380.819 381.811 -0.992
14 14/11 416.382 417.508 -1.126
15 9/7 438.362 435.084 3.278
16 21/16 473.925 470.781 3.144
17 4/3 495.904 498.045 -2.141
18 49/36 531.468 533.742 -2.274
19 11/8 553.447 551.318 2.129
20 108/77 589.010 585.721 3.289
21 77/54 610.990 614.279 -3.289
22 16/11 646.553 648.682 -2.129
23 72/49 668.532 666.258 2.274
24 3/2 704.096 701.955 2.141
25 32/21 726.075 729.219 -3.144
26 14/9 761.638 764.916 -3.278
27 11/7 783.618 782.492 1.126
28 77/48 819.181 818.189 0.992
29 18/11 854.744 852.592 2.152
30 128/77 876.724 879.856 -3.133
31 56/33 912.287 915.553 -3.266
32 12/7 934.266 933.129 1.137
33 7/4 969.829 968.826 1.004
34 16/9 991.809 996.090 -4.281
35 49/27 1027.372 1031.787 -4.415
36 11/6 1049.351 1049.363 -0.012
37 144/77 1084.915 1083.766 1.149
38 154/81 1106.894 1112.324 -5.430
39 64/33 1142.457 1146.727 -4.270
40 96/49 1164.437 1164.303 0.134
41 2/1 1200.000 1200.000 0.000
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