Talk:Fractional monzo

Would it be possible to give an example of an eigenmonzo in use? As a non-mathematician I can't make sense of this page, but I see eigenmonzos talked about and shared all the time and I would like to not be left out :) --Cmloegcmluin (talk) 18:54, 16 April 2021 (UTC)

The main use of these is to represent complex numbers to infinite precision in a smaller space than typing out an endless sequence of nonrepeating digits after the decimal place. A simple example would be demonstrating the difference between 1/3 comma meantone and 19edo. 1/3 comma meantone takes three generators to reach 10/3, which means the fifth's eigenmonzo is |1/3 -1/3 1/3>. (2x5/3)^(1/3) Meanwhile 19edo's 5th has an eigenmonzo of |11/19>, as it's a simple fraction of a power of 2. --Yourmusic Productions (talk) 19:31, 16 April 2021 (UTC)

Thanks for the explanation. Well that makes sense to me, but what you've just described seems to only be a fractional monzo, the idea described in the introduction section of this page. It then goes on to define eigenmonzo as something that builds upon that concept. At least that's what it seems like to me. If fractional monzo = eigenmonzo, if it's that simple, then I think the page could be made a bit clearer. --Cmloegcmluin (talk) 22:11, 16 April 2021 (UTC)

Often the way with maths. Simple concepts that lead to very complicated implications when you iterate upon them. The clever stuff comes when you combine several fractional monzos to create an eigenmonzo that hits lots of near-just intervals in a small number of notes, as in miracle. Getting the numbers for all those various minmax and least squares tunings properly computed can be a real pain. --Yourmusic Productions (talk) 08:03, 17 April 2021 (UTC)

Hm. Okay. Well I'm still confused about anything beyond "fractional monzo" then and I reiterate my original ask which was for a specific example or two. Thanks for trying to explain! --Cmloegcmluin (talk) 20:23, 17 April 2021 (UTC)

Frobenius norm

The article says, "Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the Frobenius tuning, which is the same as the unweighted RMS tuning which can be found using the pseudoinverse. If r is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(r)..."

I am testing my understanding of this concept on 5-limit meantone, with mapping [⟨1 1 0] ⟨0 1 4]⟩. According to the first sentence of this, I understand that the Frobenius tuning would be √(1² + 1² + 0² + 0² + 1² + 4²) = √(1 + 1 + 1 + 16) = √19. However, according to the last sentence of this, I understand that the Frobenius tuning would be √2, which is a different result. Which is correct? I think the paragraph could be revised to make the answer clearer. --Cmloegcmluin (talk) 22:46, 22 July 2021 (UTC)

I'm still seeking clarity on this issue, if anyone can provide it. Thank you! --Cmloegcmluin (talk) 10:09, 26 November 2022 (UTC)

The Frobenius norm is said with respect to the projection map, not the temperament map. The projection map of meantone in the Frobenius tuning is [⟨17/33 16/33 -4/33], ⟨16/33 17/33 4/33], ⟨-4/33 4/33 32/33]], whose norm is indeed sqrt (2). FloraC (talk) 13:48, 26 November 2022 (UTC)

Oh, wow! That's great: [math]\sqrt{\strut (\frac{17}{33})^2 + (\frac{16}{33})^2 + (\frac{-4}{33})^2 + (\frac{16}{33})^2 + (\frac{17}{33})^2 + (\frac{4}{33})^2 + (\frac{-4}{33})^2 + (\frac{4}{33})^2 + (\frac{32}{33})^2} = \sqrt{2}[/math]. Thank you so much for clearing that up. Now I finally understand and accept the name "Frobenius". Maybe the article doesn't need any revision — maybe its target audience is better equipped to discern this — but FWIW this was not clear to me without your help. --Cmloegcmluin (talk) 19:55, 26 November 2022 (UTC)

Talk:Fractional monzo

Frobenius norm

See also link to Projection article

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