Badness space?

I've been thinking about equation 102 in my Prime Errors and Complexities PDF (on page 20, the latest versinon's in the files section of this list). Roughly ASCIIfied, it's

B(M0, M1) = B(M0)*B(M1)*rin(M0, M1)

where B is error*complexity badness, M0 and M1 are weighted mappings, and rin is a function named by analogy with sin. The thing is, what would happen if you take that rin to actually be a sine? The result is that this formula looks like it describes an area.

We already know that scalar complexity can be interpreted as the magnitude of the hypervolume between the weighted mappings of the equal temperaments defining a regular temperament. From this formula it looks like badness is also the magnitude of a hypervolume, but in a different space. From reverse engineering "rin" it turns out that the cosine of the angle between two equal temperaments in this space is equal to the correlation coefficient of the weighted mappings.

http://mathworld.wolfram.com/CorrelationCoefficient.html

So, to find the correct geometry to describe badness, we need to look for a space where cosines are the same as correlation coefficients. From searching google, it looks like this is an existing concept! I found this:

http://www.ams.org/bull/1925-31-01/S0002-9904-1925-04003-3/S0002-9904-1925-04003-3.pdf

http://tinyurl.com/3dunmm

which I'll try and digest. It depends on the averages being zero, but you can always stretch the octaves of equal temperaments to make that so. Also this:

http://links.jstor.org/sici?sici=0003-1305%28199302%2947%3A1%3C30%3AAGIOPC%3E2.0.CO%3B2-5&size=LARGE&origin=JSTOR-enlargePage

http://tinyurl.com/3a5neq

which appears to be spot on, but I don't have access to the full article.

Anyway, if complexity and simple badness both have geometric interpretations, it follows that so does error.

Graham

Hi Graham,

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

http://links.jstor.org/sici?sici=0003-1305%28199302%2947%3A1%3C30%3AAGIOPC%3E2.0.CO%3B2-5&size=LARGE&origin=JSTOR-enlargePage
>
> http://tinyurl.com/3a5neq
>
> which appears to be spot on, but I don't have access to the
> full article.

The JSTOR license is quite generous: if you can get online
on a computer at any library which has a JSTOR account,
you will have access to JSTOR via their account (the
library is obligated to do so).

-monz
http://tonalsoft.com
Tonescape microtonal music software
email: joemonz(AT)yahoo.com