Hibernal: Difference between revisions
mNo edit summary |
mNo edit summary |
||
| Line 5: | Line 5: | ||
For the tempering process, I will use [[erac]]s to show error accumulation. On a basic level, 25/21 * 19/15 is equated to >3/2, tempering out <190/189. This >3 is in the Archy range, so <<64/63 can be tempered out to add >7. At this point, tempering becomes less obvious. For 25/21 to be accurate, >>25 would be required to offset the very sharp Archy >>21. Assuming >>25 splits into (>5)^2, an accurate 19/15 would need an intolerably sharp >>19 to offset >>15. In my numerous attempts to turn Hibernal into a rank-3 2.3.5.7.19 temperament, I came to the conclusion that I have to make it straddle-5 to make any sense, with a roughly accurate 5 and an extra sharp >>5, equated to 19/15. It may look silly, 32edo actually works that way in my experience. This is my definitive 32edo temperament. | For the tempering process, I will use [[erac]]s to show error accumulation. On a basic level, 25/21 * 19/15 is equated to >3/2, tempering out <190/189. This >3 is in the Archy range, so <<64/63 can be tempered out to add >7. At this point, tempering becomes less obvious. For 25/21 to be accurate, >>25 would be required to offset the very sharp Archy >>21. Assuming >>25 splits into (>5)^2, an accurate 19/15 would need an intolerably sharp >>19 to offset >>15. In my numerous attempts to turn Hibernal into a rank-3 2.3.5.7.19 temperament, I came to the conclusion that I have to make it straddle-5 to make any sense, with a roughly accurate 5 and an extra sharp >>5, equated to 19/15. It may look silly, 32edo actually works that way in my experience. This is my definitive 32edo temperament. | ||
I use the Oceanfront mapping of 13/10, resulting in a straddled 13 as well. The following table equates it to 22/17. The exact distribution of eracs varies, but | I use the Oceanfront mapping of 13/10, resulting in a straddled 13 as well. The following table equates it to 22/17. The exact distribution of eracs varies, but 2.>3.5.>>5.>7.>19 always works out, and may be extended to include 11.13-.13+.17 which reflects the unpredictability of tuning 13. | ||
}} | }} | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 80: | Line 80: | ||
|} | |} | ||
A simple rank-2 reduction uses Superpyth mappings for 25/21 and 19/15 in place of 6/5 and 5/4, seen in edos like 86, 59, 32, and 69. Other useful edos may include | A simple rank-2 reduction uses Superpyth mappings for 25/21 and 19/15 in place of 6/5 and 5/4, seen in edos like 86, 59, 32, and 69. | ||
Another basic reduction is equating (25/21)^4 to 2/1, making a quadter-octave temperament. Edos include 32, 44, 56, 76, 100, and 108. | |||
Other useful edos may include 47, 71, 79, 83, 91, 106, and 111. | |||
{{Cat|Temperaments}} | {{Cat|Temperaments}} | ||
Revision as of 14:14, 7 July 2026
Hibernal is a set of temperaments based on stacking a 25/21 minor third and 19/15 major third. 25 and 15 are both divisible by 5, resulting in 63:75:95, which is also delta-rational +3+5. The name is the adjective form of "winter" after the 32edo song that uses it extensively, Winter's Mortal Hope.
For the tempering process, I will use eracs to show error accumulation. On a basic level, 25/21 * 19/15 is equated to >3/2, tempering out <190/189. This >3 is in the Archy range, so <<64/63 can be tempered out to add >7. At this point, tempering becomes less obvious. For 25/21 to be accurate, >>25 would be required to offset the very sharp Archy >>21. Assuming >>25 splits into (>5)^2, an accurate 19/15 would need an intolerably sharp >>19 to offset >>15. In my numerous attempts to turn Hibernal into a rank-3 2.3.5.7.19 temperament, I came to the conclusion that I have to make it straddle-5 to make any sense, with a roughly accurate 5 and an extra sharp >>5, equated to 19/15. It may look silly, 32edo actually works that way in my experience. This is my definitive 32edo temperament.
I use the Oceanfront mapping of 13/10, resulting in a straddled 13 as well. The following table equates it to 22/17. The exact distribution of eracs varies, but 2.>3.5.>>5.>7.>19 always works out, and may be extended to include 11.13-.13+.17 which reflects the unpredictability of tuning 13.
| -2 | -1 | 0 | 1 | 2 | |
| -4 | 1137 | 346 | 756 | 1165 | 374 |
| -3 | 648 | 1057 | 267 | 676 | 1085 |
| -2 | 159 | 569 | 978 | 187 | 596 |
| -1 | 870 | 80 | 489 | 898 | 107 |
| 0 | 382 | 791 | 0 | 409 | 818 |
| 1 | 1093 | 302 | 711 | 1120 | 330 |
| 2 | 604 | 1013 | 222 | 631 | 1041 |
| 3 | 115 | 524 | 933 | 143 | 552 |
| 4 | 826 | 35 | 444 | 854 | 63 |
A simple rank-2 reduction uses Superpyth mappings for 25/21 and 19/15 in place of 6/5 and 5/4, seen in edos like 86, 59, 32, and 69.
Another basic reduction is equating (25/21)^4 to 2/1, making a quadter-octave temperament. Edos include 32, 44, 56, 76, 100, and 108.
Other useful edos may include 47, 71, 79, 83, 91, 106, and 111.
