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Encyclopedia of Microtonal Music Theory

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[Paul Erlich]

The weighting scheme determines how much weight is given to each [musical] 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. [As] this caused more importance to be put on the tuning of the thirds than on the fifth, Manuel [Op de Coul] 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:

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.

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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.

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[Dave Keenan]

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.

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