User:Dave Keenan/Wolkenstein height

[Published on April 1st]

Portrait of Italian composer Oswald von Wolkenstein with the emblem of the order of the Knights of the Jar on his sash and hanging from his collar

Wolkenstein height is named after the Italian composer Oswald von Wolkenstein, 1376-1445, who was famous for his brawls, adultery, vengeful litigation and ribald poetry. I named it after Oswald because it's a complete bastard of a height function, obtained by closing one eye and augmenting the monzo with an additional entry that completely cancels out the effect of all the other entries when you take the taxicab norm. That's only possible because I defined a new kind of number whose absolute value is negative.

For any temperament, the Wolkenstein tuning is the one that minimizes the maximum of the error weighted by the Wolkenstein height, on all rationals mapped by that temperament.

Shame about the phonological similarity between "Wolkenstein" and "Weil", "Wilson", "Kees" and "Tenney", but hey you probably can't remember which of those is which anyway. They could have been given obvious names like odd-limit, integer-limit, product-complexity and sopfr, but where's the fun in that? Or maybe the eponyms refer to the logs of those things? Who can remember? And they could have been called "complexities", but "height" doesn't have any other meaning in music theory, does it?

A new kind of number

You may be aware that "imaginary numbers" are defined so that when you square them you get a negative number, despite the fact that when you square any ordinary number, even a negative one, you always get a positive result.

Weil, there's some similarity between squaring a number and taking its absolute value, so I invented a new kind of number such that when you take its absolute value, you get a negative number. I call them clinically depressed numbers because what else would you call a number that thinks its absolute value is negative?

The simplest imaginary number is the one whose square is -1, namely 𝑖. Likewise the simplest clinically depressed number is the one whose absolute value is -1. I've given it the Unicode symbol 항, which is called "ARCHAIC LETTER KRAPPA WITH OGONEK", because that's what popped up when I drew a few random squiggles in ShapeCatcher.

|항| = -1 and |항a| = |a항| = -|a|

Abbreviations

To avoid confusion with other tunings beginning with "W", such as Weil and Wilson, we use the abbreviation "W" for Wolkenstein. And of course there is a Euclidean version, "WE", that you can easily compute using the Moore-Penrose pseudoinverse.

Proof

The skewered matrix for Wolkenstein tuning is

𝑊 =
⎡ 1  0  0 ⎤
⎢ 0  1  0 ⎥
⎢ 0  0  1 ⎥
⎣ 항 항 항 ⎦

not to be confused with the weight matrix 𝑊.

Consider the monzo 𝐢 = [a b c⟩.

When we multiply it by our 𝑊 matrix we turn it into a monzarella:

𝑊𝐢 = [a b c (항a+항b+항c)⟩

Now we take its Minkowski norm of order 1, and obtain:

‖𝑊𝐢‖₁ = |a| + |b| + |c| + |항a+항b+항c|

= |a| + |b| + |c| - |a| - |b| - |c|

= 0

So everything has a Wolkenstein complexity of zero, which means that its simplicity is infinite and therefore everything is perfectly tuned. Oh wait! No. The other thing. Everything gets tuned to a unison. But at least it's beatless. They were a good band, back in the 60's — The Beatless.

Actually, both the Taxicab and Euclidean versions of the Wolkenstein tuning sound like complete krapola, but who cares about that. Don't you love the clever math, and the ease of computation?