Hibernal

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.

g_

I used Hibernal in 32edo for a while because I loved the melodic properties of its blackdye, without knowing why it sounded more concordant than expected. I initially guessed it was 19/16 and 14/11, but I tried slight variations of the minor and major triads and realized that I was hearing the well of 19/15 instead.

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 is something like 2.>3.5.>>5.>7.>19.(11.<<13.13.17).

19-limit Hibernal Stradle-5-13, JI tuning in cents (generators 95/63 and 19/15), primes highlighted

	-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. Other useful edos may include 44, 47, 56, 64, 71, 76, 79, 83, 91, 100, 106, 108, and 111.