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Hi
Welcome to my talk page! -- Overthink (talk) 03:40, 21 September 2025 (UTC)
23-limit in 159edo
Hello! I see you've been working on a 159edo well temperament. I should mention that when I work in that tuning system and I use the 23-limit, I choose either the no-17's or the no-19's form, with the former being viable up to the 29-odd-limit, and the latter being viable up to the 27-odd-limit. I think you'd do well to take stock of the 2.3.11 subgroup in 159edo as that is the skeleton of what I work with, though I will add the 5-prime and I'm admittedly trying to learn how to add the 7-prime and the 13-prime without stacking them multiple times. I've also literally invented a microtonal cadence in the 2.3.5.11 subgroup that might be of interest. --Aura (talk) 05:37, 19 October 2025 (UTC)
- I believe a 159-note mos of tribilo (2.3.11 nexus) temperament may be useful, and it could contain better approximations of intervals outside of 2.3.11 as well. --Overthink (talk) 06:10, 24 October 2025 (UTC)
- You'd be right about that, but you'd also be right if you decided on a 159-note MOS of frameshift. Fortunately, 159edo itself tempers out both the nexus comma and the frameshift comma. --Aura (talk) 17:37, 25 October 2025 (UTC)
- It would be nice if the 2.3.5.11 subgroup was more connected. Prime 5 is equated to -8 fifths octave reduced, which is relatively close, but three 11/8s reach the original chain of fifths +28/-25 fifths away, which is almost exactly the opposite side of the circle. Three 11/10's return only a single fifth down, but unfortunately intervals with 3 factors of 5 are (just barely) inconsistent. There doesn't seem to be a very simple way to connect 2.3.11 to prime 5, but think I found a solution: Prime 17! The 4 intervals closest to the 12edo semitone in 159edo are 256/243, 18/17, 17/16, and 2187/2048. Prime 17 could be quite useful in classical music, such as in the 17:20:24 diminished chord. Prime 5 is found much more easily and consistently, as simply 3/2 minus 3 17/16s, and nothing of this simplicity is in 2.3.5.11. I believe it would be a good idea to devise a system of 2.3.17 analogous to alpharabian tuning, and you could easily just add back prime 11. --Overthink (talk) 04:24, 16 November 2025 (UTC)
- I believe a 12edo-based classification of intervals based on 2.3.5.17 Term temperament may be good, and for prime 11 use an 24edo-based classification from hemiterm. Note that 159edo supports both.--Overthink (talk) 05:07, 16 November 2025 (UTC)
- Or maybe just start with simpler intervals of 2.3.17.--Overthink (talk) 05:17, 16 November 2025 (UTC)
- One way I've noticed for connecting primes 11 and 5, or rather combinations of 3 and 11 with combinations of 3 and 5, is to equate three instances of 243/242 with 81/80. Prime 17 is also a good addition, and it simplifies certain 11-based gestures in 159edo by 17/16 being equated with two instances of 33/32. I should note that 159edo supports both of these options. --Aura (talk) 06:12, 16 November 2025 (UTC)
- A third option I just found for connecting primes 3, 5 and 11 is to equate two instances of 81/80 with 4096/3993. You are right that none of these are particularly simple, but I still find these methods to be highly valuable- especially with prime 17 coming in to simplify things. --Aura (talk) 07:00, 16 November 2025 (UTC)