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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: I have to agree that "being supported by all patent vals that support pajara save 12edo" seems like an arbitrary fact and not a specific advantage. Can you explain why it is useful? – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 01:25, 23 November 2025 (UTC)

:: I honestly don't see why everyone sees patent vals as so arbitrary. To me the patent val essentially is the edo, non-patent vals really just exist to define wedgies/ET joins and that's it. -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 06:44, 23 November 2025 (UTC)

::: There's a difference between an edo and its patent val; the val describes the corresponding rank-1 temperament. See [[EDO vs ET]]. Sometimes it's better to use the second-best approximation for a prime harmonic; for example, it's much better to use the 34d val than the patent val when using prime [[7/1|7]] in 34edo. In the patent val all of [[7/6]], [[7/5]], [[9/7]], [[14/11]], [[14/13]], [[15/14]], and their [[octave complement]]s are inconsistent, while in the 34d val only [[7/4]] and [[8/7]] are inconsistent. The ratios between harmonics are just as important as the harmonics themselves, and even more so when one realizes that there's much more of them. Also, the 34d val is much more useful than the 32edo or 54edo patent val.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 09:14, 25 December 2025 (UTC)

@@ Line 4: / Line 4: @@
 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 ==
@@ Line 51: / Line 71: @@
 : I have to agree that "being supported by all patent vals that support pajara save 12edo" seems like an arbitrary fact and not a specific advantage. Can you explain why it is useful? – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 01:25, 23 November 2025 (UTC)
+:: I honestly don't see why everyone sees patent vals as so arbitrary. To me the patent val essentially is the edo, non-patent vals really just exist to define wedgies/ET joins and that's it. -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 06:44, 23 November 2025 (UTC)
+::: There's a difference between an edo and its patent val; the val describes the corresponding rank-1 temperament. See [[EDO vs ET]]. Sometimes it's better to use the second-best approximation for a prime harmonic; for example, it's much better to use the 34d val than the patent val when using prime [[7/1|7]] in 34edo. In the patent val all of [[7/6]], [[7/5]], [[9/7]], [[14/11]], [[14/13]], [[15/14]], and their [[octave complement]]s are inconsistent, while in the 34d val only [[7/4]] and [[8/7]] are inconsistent. The ratios between harmonics are just as important as the harmonics themselves, and even more so when one realizes that there's much more of them. Also, the 34d val is much more useful than the 32edo or 54edo patent val.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 09:14, 25 December 2025 (UTC)