Absolute TOP error

We've been doing the weighted TOP error as the error in cents divided
by the log base two of the product of the numerator and the
denominator. This is good for most purposes, but if we stick to the
same log for the ratio (both cents, or both log base two, etc.) then
we get something with a meaning independent of unit/log base choice.
It can be described as the logarithm, base N = the product of
numerator and denominator, of the error; Log_N(E). The reciprocal is
Log_E(N); it is how many steps of size E (the error) are required to
get to N (product of numerator and denominator.) In TOP tuning, there
is a minimum value for this which defines the relative error. The same
remarks apply for NOT tuning and the product of the numerator and
denominator of the odd part of the interval.

If T is the TOP error by the definition we've been using, then 1200/T
is the minimum number of error-sized steps needed to get to the
product of numerator and denominator. For (5, 7, 11-limit) meantone
that would be 706.497 steps, for instance; miracle would be 1901.701
and ennealimmal 32987.408.

>We've been doing the weighted TOP error as the error in cents divided
>by the log base two of the product of the numerator and the
>denominator. This is good for most purposes, but if we stick to the
>same log for the ratio (both cents, or both log base two, etc.) then
>we get something with a meaning independent of unit/log base choice.
>It can be described as the logarithm, base N = the product of
>numerator and denominator, of the error; Log_N(E). The reciprocal is
>Log_E(N); it is how many steps of size E (the error) are required to
>get to N (product of numerator and denominator.) In TOP tuning, there
>is a minimum value for this which defines the relative error. The same
>remarks apply for NOT tuning and the product of the numerator and
>denominator of the odd part of the interval.
>
>If T is the TOP error by the definition we've been using, then 1200/T
>is the minimum number of error-sized steps needed to get to the
>product of numerator and denominator. For (5, 7, 11-limit) meantone
>that would be 706.497 steps, for instance; miracle would be 1901.701
>and ennealimmal 32987.408.

I think I understand some of this. How is it absolute? It still
sounds weighted to me.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I think I understand some of this. How is it absolute? It still
> sounds weighted to me.

It's a pure number; no units of cents or whatever log base you use are
used.