Talk:Pajara

Minimax tunings

Although I'd like to have any excuse for the 7th harmonic to be more in tune, I don't think it makes much sense to call the tuning with eigenmonzo 7/6 the "7-odd-limit minimax".

If I understand correctly, this generator size was chosen because it balances 10/7, which is 17.5 cents flat, with 6/5, which is 17.5 cents sharp, but in pajara 10/7 is always 17.5 cents flat, so there is no point in balancing it. With the same reasoning you can say that the tuning where 5/4 is eigenmonzo, with a fifth of 706.843 cents, is the 7-odd limit minimax, because it balances the two worst intervals – 10/7 with 7/4, which are both, again, 17.5 cents sharp/flat respectively. If you take the term "minimax" literally, i.e. "The tuning in which the maximal error of any consonance is minimal", you get that any tuning between them is also a minimax, because in all of them the maximal error is 17.5 cents – that of 10/7.

I suggest labeling the tuning with eigenminzo 48/35 the 7-off limit minimax, because that's where 6/5 and 8/7 are balanced. That's also what you get if you take the minimum of the p-norm (Wikipedia) of the errors when p approaches infinity. I haven't checked but some other odd-limit minimax tunings may be changed too. Roeesi (talk) 13:56, 16 October 2022 (UTC)