TUNING digest 561

> > The 12 tone equal temperament scale is derived from the overtone
> > series of a vibrating string.
>
> Of course most of us tuning listers will be prompt to point out that 12TET
> doesn't provide a *really great* harmonic series approximation. The 5th
> harmonic is way sharp and it misses some really neat harmonics - 7, 11, and
13
> for example. It does, however, provide a fairly EFFICIENT mapping to the
> harmonics it does match - matching the first 5 harmonics reasonably well on
as a
> coarse resolution as 100 cents, is in my view an impressive feat.

Thanks for replying. I'm afraid I'm waaaaay back at Tuning 101.

Would you (or anybody on the list) mind elaborating scale construction base on
partial frequency formulas? i.e., Given the really neat partial freq ratios
that mclaren described, how do I construct a scale from them?

I realize that we can do NonJust Equal Temperament, Just Non Equal Temperament
(what others are there?). There also seems to be the option of octave
transposition. For the harmonic series, and the cylindrical vibration mode
series, transposing partials somehow into an octave seems really important,
because the series increases too rapidly to construct a scale merely composed
of partials.

For 12TET, did someone merely lay out the harmonic series to some order, octave
transpose them, and then make a best fit to a 2^(n/12) scale (discarding
partials 7,11,13 in this case)?

Thanks a bujillion,

Marcus

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> Thanks for the explanation of LCMs, but why is this important musically?

I thought I mentioned that in the message before my last one on that topic,
but perhaps not.

Marion McCoskey views the LCM of the numbers in the extended ratio for a
chord as a good indicator of the harmonic complexity of the chord. So the LCM
of 60 for either 4:5:6 major or 10:12:15 minor chords is a fairly low LCM,
implying that it's a fairly simple chord, harmonically speaking.

If you turn that into a septimal dominant seventh chord (4:5:6:7), the LCM
goes up by a factor of seven to 420, suggesting that it's a much more
harmonically complex - tense - chord. If you tune the upper interval of that to
the usual 6:5 minor third instead, the LCM goes way up again 20:25:30:36 to 900,
again, correctly predicting that that chord has a yet more complex sonority.

Some people on the list have questioned whether LCM is the best choice of
harmonic complexity, which is after all, a somewhat subjective thing. But I get
the impression that most listers accept it as a reasonable model.

Speaking for myself, it strikes me as reasonable, but I suspect that it might
be the numbers might be a little more appropriate if they were the log of the
LCM. Using the common log for example, the 4:5:6 would have a complexity of
1.8, the 4:5:6:7 a complexity of 2.62, and the 20:25:30:36 dominant seventh
chord a harmonic complexity of 2.95. That strikes me as a little more in
proportion to the complexity as I hear them. Or perhaps even better would be
the square root: 7.75, 20.5, and 30 respectively.


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