Golden magic: Difference between revisions
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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: | 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" | {| class="wikitable" | ||
|Scale Degree | |'''Scale Degree''' | ||
|Ratio | |'''Ratio''' | ||
|Golden Magic | |'''Golden Magic''' | ||
|Detempered | |'''Detempered''' | ||
|Error | |'''Error''' | ||
|'''Color Notation''' | |||
|- | |- | ||
|0 | |0 | ||
| Line 16: | Line 17: | ||
|0.000 | |0.000 | ||
|0.000 | |0.000 | ||
|w1 | |||
|- | |- | ||
|1 | |1 | ||
| Line 22: | Line 24: | ||
|35.697 | |35.697 | ||
| -0.134 | | -0.134 | ||
|zz2 | |||
|- | |- | ||
|2 | |2 | ||
| Line 28: | Line 31: | ||
|53.273 | |53.273 | ||
|4.270 | |4.270 | ||
|1o1 | |||
|- | |- | ||
|3 | |3 | ||
| Line 34: | Line 38: | ||
|87.676 | |87.676 | ||
|5.430 | |5.430 | ||
|1ur1 | |||
|- | |- | ||
|4 | |4 | ||
| Line 40: | Line 45: | ||
|116.234 | |116.234 | ||
| -1.149 | | -1.149 | ||
|1oz2 | |||
|- | |- | ||
|5 | |5 | ||
| Line 46: | Line 52: | ||
|150.637 | |150.637 | ||
|0.012 | |0.012 | ||
|1u2 | |||
|- | |- | ||
|6 | |6 | ||
| Line 52: | Line 59: | ||
|168.213 | |168.213 | ||
|4.415 | |4.415 | ||
|rr1 | |||
|- | |- | ||
|7 | |7 | ||
| Line 58: | Line 66: | ||
|203.910 | |203.910 | ||
|4.281 | |4.281 | ||
|w2 | |||
|- | |- | ||
|8 | |8 | ||
| Line 64: | Line 73: | ||
|231.174 | |231.174 | ||
| -1.004 | | -1.004 | ||
|r2 | |||
|- | |- | ||
|9 | |9 | ||
| Line 70: | Line 80: | ||
|266.871 | |266.871 | ||
| -1.137 | | -1.137 | ||
|z3 | |||
|- | |- | ||
|10 | |10 | ||
| Line 76: | Line 87: | ||
|284.447 | |284.447 | ||
|3.266 | |3.266 | ||
|1or2 | |||
|- | |- | ||
|11 | |11 | ||
| Line 82: | Line 94: | ||
|320.144 | |320.144 | ||
|3.133 | |3.133 | ||
|1oz3 | |||
|- | |- | ||
|12 | |12 | ||
| Line 88: | Line 101: | ||
|347.408 | |347.408 | ||
| -2.152 | | -2.152 | ||
|1o3 | |||
|- | |- | ||
|13 | |13 | ||
| Line 94: | Line 108: | ||
|381.811 | |381.811 | ||
| -0.992 | | -0.992 | ||
|1ur3 | |||
|- | |- | ||
|14 | |14 | ||
| Line 100: | Line 115: | ||
|417.508 | |417.508 | ||
| -1.126 | | -1.126 | ||
|1uz4 | |||
|- | |- | ||
|15 | |15 | ||
| Line 106: | Line 122: | ||
|435.084 | |435.084 | ||
|3.278 | |3.278 | ||
|r3 | |||
|- | |- | ||
|16 | |16 | ||
| Line 112: | Line 129: | ||
|470.781 | |470.781 | ||
|3.144 | |3.144 | ||
|z4 | |||
|- | |- | ||
|17 | |17 | ||
| Line 118: | Line 136: | ||
|498.045 | |498.045 | ||
| -2.141 | | -2.141 | ||
|w4 | |||
|- | |- | ||
|18 | |18 | ||
| Line 124: | Line 143: | ||
|533.742 | |533.742 | ||
| -2.274 | | -2.274 | ||
|zz5 | |||
|- | |- | ||
|19 | |19 | ||
| Line 130: | Line 150: | ||
|551.318 | |551.318 | ||
|2.129 | |2.129 | ||
|1o4 | |||
|- | |- | ||
|20 | |20 | ||
| Line 136: | Line 157: | ||
|585.721 | |585.721 | ||
|3.289 | |3.289 | ||
|1ur4 | |||
|- | |- | ||
|21 | |21 | ||
| Line 142: | Line 164: | ||
|614.279 | |614.279 | ||
| -3.289 | | -3.289 | ||
|1oz5 | |||
|- | |- | ||
|22 | |22 | ||
| Line 148: | Line 171: | ||
|648.682 | |648.682 | ||
| -2.129 | | -2.129 | ||
|1u5 | |||
|- | |- | ||
|23 | |23 | ||
| Line 154: | Line 178: | ||
|666.258 | |666.258 | ||
|2.274 | |2.274 | ||
|rr4 | |||
|- | |- | ||
|24 | |24 | ||
| Line 160: | Line 185: | ||
|701.955 | |701.955 | ||
|2.141 | |2.141 | ||
|w5 | |||
|- | |- | ||
|25 | |25 | ||
| Line 166: | Line 192: | ||
|729.219 | |729.219 | ||
| -3.144 | | -3.144 | ||
|r5 | |||
|- | |- | ||
|26 | |26 | ||
| Line 172: | Line 199: | ||
|764.916 | |764.916 | ||
| -3.278 | | -3.278 | ||
|z6 | |||
|- | |- | ||
|27 | |27 | ||
| Line 178: | Line 206: | ||
|782.492 | |782.492 | ||
|1.126 | |1.126 | ||
|1or5 | |||
|- | |- | ||
|28 | |28 | ||
| Line 184: | Line 213: | ||
|818.189 | |818.189 | ||
|0.992 | |0.992 | ||
|1oz6 | |||
|- | |- | ||
|29 | |29 | ||
| Line 190: | Line 220: | ||
|852.592 | |852.592 | ||
|2.152 | |2.152 | ||
|1u6 | |||
|- | |- | ||
|30 | |30 | ||
| Line 196: | Line 227: | ||
|879.856 | |879.856 | ||
| -3.133 | | -3.133 | ||
|1ur6 | |||
|- | |- | ||
|31 | |31 | ||
| Line 202: | Line 234: | ||
|915.553 | |915.553 | ||
| -3.266 | | -3.266 | ||
|1uz7 | |||
|- | |- | ||
|32 | |32 | ||
| Line 208: | Line 241: | ||
|933.129 | |933.129 | ||
|1.137 | |1.137 | ||
|r6 | |||
|- | |- | ||
|33 | |33 | ||
| Line 214: | Line 248: | ||
|968.826 | |968.826 | ||
|1.004 | |1.004 | ||
|z7 | |||
|- | |- | ||
|34 | |34 | ||
| Line 220: | Line 255: | ||
|996.090 | |996.090 | ||
| -4.281 | | -4.281 | ||
|w7 | |||
|- | |- | ||
|35 | |35 | ||
| Line 226: | Line 262: | ||
|1031.787 | |1031.787 | ||
| -4.415 | | -4.415 | ||
|zz8 | |||
|- | |- | ||
|36 | |36 | ||
| Line 232: | Line 269: | ||
|1049.363 | |1049.363 | ||
| -0.012 | | -0.012 | ||
|1o7 | |||
|- | |- | ||
|37 | |37 | ||
| Line 238: | Line 276: | ||
|1083.766 | |1083.766 | ||
|1.149 | |1.149 | ||
|1ur7 | |||
|- | |- | ||
|38 | |38 | ||
| Line 244: | Line 283: | ||
|1112.324 | |1112.324 | ||
| -5.430 | | -5.430 | ||
|1oz8 | |||
|- | |- | ||
|39 | |39 | ||
| Line 250: | Line 290: | ||
|1146.727 | |1146.727 | ||
| -4.270 | | -4.270 | ||
|1u8 | |||
|- | |- | ||
|40 | |40 | ||
| Line 256: | Line 297: | ||
|1164.303 | |1164.303 | ||
|0.134 | |0.134 | ||
|rr7 | |||
|- | |- | ||
|41 | |41 | ||
| Line 262: | Line 304: | ||
|1200.000 | |1200.000 | ||
|0.000 | |0.000 | ||
|w8 | |||
|} | |} | ||
The highest error of any interval in the 1,3,7,9,11 tonality diamond is 9/8, which is 4.281 cents sharp. | The highest error of any interval in the 1,3,7,9,11 tonality diamond is 9/8, which is 4.281 cents sharp. | ||
Revision as of 18:17, 12 April 2022
Golden magic is a tuning of magic temperament where the 19 tone, 22 tone and 41 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 | Color Notation |
| 0 | 1/1 | 0.000 | 0.000 | 0.000 | w1 |
| 1 | 49/48 | 35.563 | 35.697 | -0.134 | zz2 |
| 2 | 33/32 | 57.543 | 53.273 | 4.270 | 1o1 |
| 3 | 81/77 | 93.106 | 87.676 | 5.430 | 1ur1 |
| 4 | 77/72 | 115.085 | 116.234 | -1.149 | 1oz2 |
| 5 | 12/11 | 150.649 | 150.637 | 0.012 | 1u2 |
| 6 | 54/49 | 172.628 | 168.213 | 4.415 | rr1 |
| 7 | 9/8 | 208.191 | 203.910 | 4.281 | w2 |
| 8 | 8/7 | 230.171 | 231.174 | -1.004 | r2 |
| 9 | 7/6 | 265.734 | 266.871 | -1.137 | z3 |
| 10 | 33/28 | 287.713 | 284.447 | 3.266 | 1or2 |
| 11 | 77/64 | 323.276 | 320.144 | 3.133 | 1oz3 |
| 12 | 11/9 | 345.256 | 347.408 | -2.152 | 1o3 |
| 13 | 96/77 | 380.819 | 381.811 | -0.992 | 1ur3 |
| 14 | 14/11 | 416.382 | 417.508 | -1.126 | 1uz4 |
| 15 | 9/7 | 438.362 | 435.084 | 3.278 | r3 |
| 16 | 21/16 | 473.925 | 470.781 | 3.144 | z4 |
| 17 | 4/3 | 495.904 | 498.045 | -2.141 | w4 |
| 18 | 49/36 | 531.468 | 533.742 | -2.274 | zz5 |
| 19 | 11/8 | 553.447 | 551.318 | 2.129 | 1o4 |
| 20 | 108/77 | 589.010 | 585.721 | 3.289 | 1ur4 |
| 21 | 77/54 | 610.990 | 614.279 | -3.289 | 1oz5 |
| 22 | 16/11 | 646.553 | 648.682 | -2.129 | 1u5 |
| 23 | 72/49 | 668.532 | 666.258 | 2.274 | rr4 |
| 24 | 3/2 | 704.096 | 701.955 | 2.141 | w5 |
| 25 | 32/21 | 726.075 | 729.219 | -3.144 | r5 |
| 26 | 14/9 | 761.638 | 764.916 | -3.278 | z6 |
| 27 | 11/7 | 783.618 | 782.492 | 1.126 | 1or5 |
| 28 | 77/48 | 819.181 | 818.189 | 0.992 | 1oz6 |
| 29 | 18/11 | 854.744 | 852.592 | 2.152 | 1u6 |
| 30 | 128/77 | 876.724 | 879.856 | -3.133 | 1ur6 |
| 31 | 56/33 | 912.287 | 915.553 | -3.266 | 1uz7 |
| 32 | 12/7 | 934.266 | 933.129 | 1.137 | r6 |
| 33 | 7/4 | 969.829 | 968.826 | 1.004 | z7 |
| 34 | 16/9 | 991.809 | 996.090 | -4.281 | w7 |
| 35 | 49/27 | 1027.372 | 1031.787 | -4.415 | zz8 |
| 36 | 11/6 | 1049.351 | 1049.363 | -0.012 | 1o7 |
| 37 | 144/77 | 1084.915 | 1083.766 | 1.149 | 1ur7 |
| 38 | 154/81 | 1106.894 | 1112.324 | -5.430 | 1oz8 |
| 39 | 64/33 | 1142.457 | 1146.727 | -4.270 | 1u8 |
| 40 | 96/49 | 1164.437 | 1164.303 | 0.134 | rr7 |
| 41 | 2/1 | 1200.000 | 1200.000 | 0.000 | w8 |
The highest error of any interval in the 1,3,7,9,11 tonality diamond is 9/8, which is 4.281 cents sharp.
Golden Kite Guitar
A guitar fretting very similar to the Kite guitar is possible with the 41 tone golden magic MOS -- frets can be set to every other note of the 41 tone MOS. To make 11 limit chord fingerings more ergonomic across the neck (since 11 limit intervals require relatively huge stretches near the nut on a Kite guitar), one can fret the guitar to the full 41 tone MOS up to the twelfth fret and transition to 2\41 increments beyond that.
A tuning that consists of alternating downmajor thirds at 380.819c and upmajor thirds at 438.362c is recommended as it makes the stretches needed to play 11 limit chords on a Kite guitar more manageable.