weighing schemes

[Brett Barbaro]:
> My original weighting scheme multiplied the error in each
> interval by the interval's "limit", so that errors were measured in
cents*limit. > This caused more importance to be put on the tuning of the
thirds than on the > fifth, Manuel suggested a weighting scheme in which
the errors were
>measured in cents/limit, which caused more importance to be put on the
>tuning of the fifth. Make sense?

This seems like a chicken-and-egg sort of thing to me. Correct me if i
don't understand this right, but your scheme would place more importance in
tuning the highest "limit" presented in a system, whereas the other
emphasizes the lowest "limit" represented. That is, if the strict error is
our measuring stick for the moment.

For me, it makes sense to have the higher numbers presented more
accurately, since i can either sense a fifth or i can't, the 3:2 being so
fundamentally engraved/engrained/inbrained in my head. But then again, if
the most "familiar" intervals (lowest ones) are off, then there's no hope
in getting any sense out of a piece of music. So there has to be a balance
somehow. It seems the attainment of this very balance has been the goal of
theorists for a while now... So each scheme pulls towards the higher/lower
"limits" to be approximated. To achieve a balance, perhaps you can somehow
(and here i show my artist's badge, not that of a theorist!) merge the two
? SOMEHOW ? Otherwise it seems we're just going in circles...
but it's a fun circle nonetheless.

Ara S. wrote,

>This seems like a chicken-and-egg sort of thing to me. Correct me if i
>don't understand this right, but your scheme would place more importance in
>tuning the highest "limit" presented in a system, whereas the other
>emphasizes the lowest "limit" represented. That is, if the strict error is
>our measuring stick for the moment.

>For me, it makes sense to have the higher numbers presented more
>accurately, since i can either sense a fifth or i can't, the 3:2 being so
>fundamentally engraved/engrained/inbrained in my head. But then again, if
>the most "familiar" intervals (lowest ones) are off, then there's no hope
>in getting any sense out of a piece of music. So there has to be a balance
>somehow. It seems the attainment of this very balance has been the goal of
>theorists for a while now... So each scheme pulls towards the higher/lower
>"limits" to be approximated. To achieve a balance, perhaps you can somehow
>(and here i show my artist's badge, not that of a theorist!) merge the two
>? SOMEHOW ? Otherwise it seems we're just going in circles...
>but it's a fun circle nonetheless.

I usually use equal-weighting to evaluate tuning systems, to acheive this
balance. Mann's book (Analytic Study of Harmonic Intervals), after all the
series and mediants are calculated, falls back on experimental results where
for all consonant intervals, less than 20 cents mistuning was generally
tolerated, and 30 cents mistuning considered quite unacceptable. Using
equal-weighted RMS, I find, for example, that the 7-limit is approximated
better and better by the following sequence of equal temperaments, in which
no additional ETs can be inserted: 9, 10, 12, 15, 19, 22, 27, 31=62, 68, 72.
I also found that the optimal meantone temperament has a perfect fifth of
2-2*log(3)+7*log(5) steps in 26-tone equal temperament, where the logs are
in base 2. I posted the derivation of that some time ago. That's 696.1648
cents. I just discovered that that's 7/26-comma meantone temperament! I wish
I had realized that before, so I could have mentioned it in my paper.

Proof: An untempered perfect fifth would be 26*log(3)-26 steps in 26-tone
equal temperament. A comma would be 26*4*log(3)-26*log(5)-4*26 steps. 7/26
of that is 28*log(3)-7*log(5)-28 steps. Subtract that from 26*log(3)-26 and
you get 2-2*log(3)+7*log(5) steps.

So, between this and the derivation I posted a while back, we have proved
that the meantone tuning with the smallest equal-weighted RMS error in the
three 5-limit intervals is 7/26-comma meantone temperament.

Using maximum error instead of RMS error, 1/4-comma meantone is best when
equal-weighting is used. But maximum error ignores the second-worst error
and third-worst error, so I prefer RMS.

P.S. RMS is root-mean-square, the standard statistical measure of error. It
is the square root of the mean of the squares of the errors of the
individual intervals (in this case, the perfect fifth, major third, and
minor third).

>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>

>I just discovered that that's 7/26-comma meantone temperament! I wish
>I had realized that before, so I could have mentioned it in my paper.
>
WOOHOO! could be a short paper itself.

>P.S. RMS is root-mean-square, the standard statistical measure of error. It
>is the square root of the mean of the squares of the errors of the
>individual intervals (in this case, the perfect fifth, major third, and
>minor third).
>

numbers, numbers, numbers.

-Ara S.