Talk:Height: Difference between revisions

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: Could you provide some materials on what it is? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:49, 31 December 2022 (UTC)

: The gradus suavitatis is indeed a proper height function. It might be worth adding because of its historical significance. Not sure about the other ones, though a lot of those (tenney, te_norm, weil, wilson) are already described as height functions on the wiki. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 13:43, 25 April 2025 (UTC)

== Counterexample ==

Not sure how relevant this is, but I was wondering why we have "height" and "complexity" as different things. A counterexample might clear up the difference. Consider the total number of prime factors of a number. This is some kind of complexity measure. It's really just the unweighted l_1 norm when expressed in vector form, so it seems quite sensible. For example 5/4 = 2^-2 * 5^1, so h(5/4) = 3. Similarly we have h(81/80) = 9. This satisfies all of the criteria except finiteness, all prime numbers p have h(p/1) = 1. So there are infinitely many rationals for which h(x) <= C.

So this defines a complexity which is not a height.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 13:41, 25 April 2025 (UTC)

: Agreed. The [[Complexity]] page will also require an update, as it literally states that the complexity of an interval is called "height", whereas this counterexample shows that heights are a subset of interval complexity measures. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 20:45, 25 April 2025 (UTC)

:: Actually, I'm not really sure if we should be talking about height functions at all, since those properties aren't really used anywhere! The only place heights actuall show up is in the proof for Dirichlet (logflat) badness, where it's actually a height on temperaments, not intervals! And then the actual height functions we do end up using are actually just norms on some vector space. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 11:19, 26 April 2025 (UTC)

::: I'd love to do away with heights and stick to complexities. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 02:38, 30 April 2025 (UTC)

@@ Line 27: / Line 27: @@
 : Could you provide some materials on what it is? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:49, 31 December 2022 (UTC)
+: The gradus suavitatis is indeed a proper height function. It might be worth adding because of its historical significance. Not sure about the other ones, though a lot of those (tenney, te_norm, weil, wilson) are already described as height functions on the wiki. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 13:43, 25 April 2025 (UTC)
+== Counterexample ==
+Not sure how relevant this is, but I was wondering why we have "height" and "complexity" as different things. A counterexample might clear up the difference. Consider the total number of prime factors of a number. This is some kind of complexity measure. It's really just the unweighted l_1 norm when expressed in vector form, so it seems quite sensible. For example 5/4 = 2^-2 * 5^1, so h(5/4) = 3. Similarly we have h(81/80) = 9. This satisfies all of the criteria except finiteness, all prime numbers p have h(p/1) = 1. So there are infinitely many rationals for which h(x) <= C.
+So this defines a complexity which is not a height.
+– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 13:41, 25 April 2025 (UTC)
+: Agreed. The [[Complexity]] page will also require an update, as it literally states that the complexity of an interval is called "height", whereas this counterexample shows that heights are a subset of interval complexity measures. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 20:45, 25 April 2025 (UTC)
+:: Actually, I'm not really sure if we should be talking about height functions at all, since those properties aren't really used anywhere! The only place heights actuall show up is in the proof for Dirichlet (logflat) badness, where it's actually a height on temperaments, not intervals! And then the actual height functions we do end up using are actually just norms on some vector space. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 11:19, 26 April 2025 (UTC)
+::: I'd love to do away with heights and stick to complexities. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 02:38, 30 April 2025 (UTC)