Talk:Pajara: Difference between revisions

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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 [https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm 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. [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 13:56, 16 October 2022 (UTC)

I suggest labeling the tuning with eigenminzo 48/35 the 7-odd 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 [https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm 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. [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 13:56, 16 October 2022 (UTC)

: On the page for [[Minimax tuning]], it says:

: <blockquote>However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.</blockquote>

: I'm not sure which tuning would be minimax by this definition, though it seems like it probably wouldn't be the 7/6 eigenmonzo tuning.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 02:01, 11 January 2026 (UTC)

:: I think the limit approach makes more sense. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:33, 27 February 2026 (UTC)

::: What do you mean by "the limit approach"? [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 09:18, 1 March 2026 (UTC)

:::: I mean using the ''p''-norm as ''p'' approaches infinity. Alternatively, discarding the bounding intervals and balancing the rest would make sense too. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:12, 1 March 2026 (UTC) (last edited [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:14, 1 March 2026 (UTC))

::::: Why are least-squares tunings being called "minimax" here? The most consistent tiebreaker should be second largest error (largest error except for 7/5~10/7). [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 03:07, 2 March 2026 (UTC)

:::::: That's what the [[Target tuning]] page says, as originally written by [[Gene Ward Smith]]. However, the second-largest error approach gives a much simpler result, and I agree that it should be used instead.

:::::: And it's also Gene who added the original minimax tunings to this page, inconsistently to his own definition.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:58, 2 March 2026 (UTC)

::::::: Should we change the minimax page too then? [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 06:04, 2 March 2026 (UTC)

:::::::: I think we should, but keep Gene's method as a historical note. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:23, 2 March 2026 (UTC)

== Unclear/questionable paragraph ==

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 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 [https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm 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. [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 13:56, 16 October 2022 (UTC)
+I suggest labeling the tuning with eigenminzo 48/35 the 7-odd 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 [https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm 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. [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 13:56, 16 October 2022 (UTC)
+: On the page for [[Minimax tuning]], it says:
+: <blockquote>However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.</blockquote>
+: I'm not sure which tuning would be minimax by this definition, though it seems like it probably wouldn't be the 7/6 eigenmonzo tuning.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 02:01, 11 January 2026 (UTC)
+:: I think the limit approach makes more sense. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:33, 27 February 2026 (UTC)
+::: What do you mean by "the limit approach"? [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 09:18, 1 March 2026 (UTC)
+:::: I mean using the ''p''-norm as ''p'' approaches infinity. Alternatively, discarding the bounding intervals and balancing the rest would make sense too. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:12, 1 March 2026 (UTC) (last edited [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:14, 1 March 2026 (UTC))
+::::: Why are least-squares tunings being called "minimax" here? The most consistent tiebreaker should be second largest error (largest error except for 7/5~10/7). [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 03:07, 2 March 2026 (UTC)
+:::::: That's what the [[Target tuning]] page says, as originally written by [[Gene Ward Smith]]. However, the second-largest error approach gives a much simpler result, and I agree that it should be used instead.
+:::::: And it's also Gene who added the original minimax tunings to this page, inconsistently to his own definition.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:58, 2 March 2026 (UTC)
+::::::: Should we change the minimax page too then? [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 06:04, 2 March 2026 (UTC)
+:::::::: I think we should, but keep Gene's method as a historical note. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:23, 2 March 2026 (UTC)
 == Unclear/questionable paragraph ==