Talk:Height
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Height is not dissonance
Not a big fan of this opening:
- The height is a tool to measure the dissonance of JI intervals.
It measures the complexity. This might be related to dissonance, but not in an obvious way.
-Sintel (talk) 14:20, 11 April 2022 (UTC)
- Agreed. As far as I can tell, "height" and "complexity" are synonymous. If they're not, then it may be a good idea to explain the difference on the page. I don't understand what the motivation is for using the term "height" when we already have the descriptive and in-common-use term "complexity". --Cmloegcmluin (talk) 15:56, 11 April 2022 (UTC)
- Sorry, but I'm a little confused by your previous post; I don't understand whether you're saying that what I had just said was naive or that what you say next is naive. Because I'm confused by that, I'm not sure whether you agree with me or not. In either case, what you say next reads to me as a defense of "complexity" over "height", because many of the "heights" we use in xen are indeed measurements of multidimensional objects: prime-count vectors representing JI intervals. So that's another good point, and one that I hadn't considered before. --Cmloegcmluin (talk) 14:56, 12 April 2022 (UTC)
- 'I don't understand what the motivation is for using the term "height" when we already have the descriptive and in-common-use term "complexity".'
- It's borrowed from mathematics: Wikipedia: Height function ResonantFrequencies (talk) 19:22, 30 December 2022 (UTC)
gradus suavitatis
Does Euler's gradus suavitatis count? What about the other metrics listed in Scala? (DEPTH, ENTROPY, GRADUS, HARMON, HEIGHT, MANN, MAX, PROOIJEN, RHSM, RECTANGULAR, TENNEY, TE_NORM, TRIANGLE, TR_LOG, VOGEL, WEIL, WILSON) ResonantFrequencies (talk) 19:30, 30 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. – 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.
– 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. --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. – Sintel🎏 (talk) 11:19, 26 April 2025 (UTC)
- I'd love to do away with heights and stick to complexities. --Cmloegcmluin (talk) 02:38, 30 April 2025 (UTC)