Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
weighting
The weighting scheme determines how much weight is given to each
interval in determining the magnitude of the error. 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?
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.
- Erlich
I use RMS because if two tunings have the same worst error, the
second-worst errors should allow one to judge one tuning better than the
other.
RMS still makes the worst errors more important than the second-worst
errors, while MAD (mean absolute deviation) puts equal weight on all
errors.
- Erlich
I certainly would not consider MAD, and agree that RMS is better than MAD,
but if two tunings have the same worst error, I *do* just use the
second-worst errors to compare them. e.g. as I did in comparing certain
tunings having the half-octave as their approximation to 5:7. There is no
need to go to RMS.
The problem with RMS is that, even though it is not as bad in this regard
as MAD, it may still consider a tuning that has most ratios rendered very
accurately but one rendered badly, to be as good as another tuning where
all the errors are moderate. This is not how humans (me at least) perceive
these things.
I suspect RMS gives a very broad-bottomed parabola when applied to the
meantone spectrum, whatever the odd-limit being considered, and whatever
the weighting.
-Keenan
[from
Paul Erlich]
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