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RE: Ghosttones and LucyTuning

🔗Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul)

2/18/1997 4:39:56 PM
From: PAULE

Thanks to Matt Nathan from the cross-posting from sci.physics.acoustics. I
have been familiar with these facts, including the integer harmonicity of
sustained bowed strings and consistently blown wind/brass tones, for some
time now, and have been posting to that effect recently. I'm glad to have
more authority on my side, but it doesn't change my opinions and probably
won't change the extent to which they are ignored. :)

Matt, I pursued the entire line of discussion with Charles Lucy years ago on
rec.music.compose. The man is clearly interested only in promulgating his
opinions -- any factual evidence that "supports" them he will gladly jump
on, while all contradictory evidence or semblance of mathematical reasoning
he evades through obfuscatory ramblings. The whole pi business is utter
nonsense. Its value can only reside in its aesthetic appeal to musicians who
are used to a major third being 1/3 octave. Now, you tell them, it's 1/pi
octave. Easily enough understood, and as much mathematics as most musicians
would want.

Despite all this, LucyTuning is about the best tuning around for the
existing diatonic, triadic repertoire. Let me try to explain why.

First, it is a meantone tuning. This means that 5-limit just intonation is
approached without any commas to deal with. For anyone who denies the
importance of this, take a piece of diatonic, triadic music and try to
render it in just intonation. Inevitably, different-sized whole tones, comma
shifts during sustained notes, out-of-tune chords, and/or wandering tonics
arise. Now in meantone, everything is just as easy as good old 12-equal
(unless you start using enharmonic equivalents, since meantone is usually an
infinite system with no enharmonic equivalents), but the maximum deviation
from JI is reduced by a factor of 3-4.


Now meantone temperaments in general have a fifth from 692-700 cents.
LucyTuning is almost exactly 3/10 comma meantone (.3005 comma, to be exact).
His perfect fifth is thus flat by 3/10 comma; it is 695.493 cents. Now if
you take all 5-limit just intervals and calculate the meantone tuning that
minimizes the mean squared error (which is appropriate if you think
dissonance is a function whose first derivative is zero and second
derivative is positive at just intervals (I do)), you get a perfect fifth of
696.165 cents. If you use a weighted mean, with weights inversely
proportional to the limit (following Partch who said higher-limit intervals
need to be tuned more accurately in proportion to their limit), you get a
perfect fifth of 696.019 cents. The optimal tuning for any real musical
situation (as long as it is diatonic and triadic) will probably fall
somewhere between these two tunings (a very narrow range!).

Lucy is obviously very close. The 19-equal fifth is 694.737 cents, and the
31-equal fifth is 696.774 cents. Lucy is thus doing considerably better than
19-equal, and about as well as 31-equal. (50-equal would be almost perfect).
It's no wonder that an eminent scholar of days past (John Harrison)
advocated the pi-based tuning (LucyTuning). Another eminent scholar of days
past, Christiaan Huygens, advocated 31-equal.

However, with 31-equal you get consistent and very accurate representations
through the 11-limit, not just the 5-limit. Although an infinite tuning
system like LucyTuning can of course approximate any pitch with any degree
of accuarcy, you may need a very large number of notes to get all the
harmonies you want. With 31-equal, you never need more than, uh, 31.

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