RE:Equal Divisions -- of What?

Paul Erlich writes:

> picturing frequency space logarithmically
> makes more sense. Indeed, it is more natural, given that nature gave us
> basilar membranes on which equal lengths correspond to equal portions of
> log-frequency space. So it is odd that David [has]
> JI in mind as the most natural tuning. David, care to reconsider?

Quite. That was indeed an enlightening posting! - clear and directly to the
point. Man down, call the paramedics!

One thing I'm wondering about is this: the goal of all of your calculations
in the "long answer" (and I greatly appreciate that you took the time to give
it) was equal spacing. Experience shows us clearly that equal spacings make
for smoother sounding melodic intervals than those that JI scales provide.
But they don't tend to provide the most consonant harmonies, without fudging
and compromising, right? because, while we do perceive pitch logarithmically,
the instruments we use generally have harmonic overtones which beat against
each other if more than one ET pitch is played at once. Evidently we are
left with a no-win situation - pick whether you want the best melodic
intervals or the most consonant chords unless you want to abandon a firm set
of pitches, using dynamic retuning via software or extraordinary virtuostity
on a fretless instrument. Yes, I know this has been bantered about for ages.
What I'm saying is, while your demonstration of a natural foundation for ETs
is well done, it still leaves the quandry of melody vs. harmony.

Just for kicks, I made a sound, using additive sine wave synthesis, composed
of a fundamental and a few dozen 12-tET-tuned overtones - the only harmonic
ones were the octaves. Not surprisingly, it a was musically useless sound
even using 12-tET scales; it didn't sound like a single tone, and chords made
with it only made a hideous racket. I recognize that this is not an
indictment of ETs. I just had to see.


> Finally, consider the geometric series
>
> . .x^-9, x^-8, x^-7, x^-6, x^-5, x^-4, x^-3, x^-2, x^-1, x, x^2, x^3,
> x^4, x^5, x^6, x^7, x^8, x^9, . . .
>
> where x>0 and x does not equal 1. Whether the units of measurement are
> string length or wavelength or time period, as above, or frequency, as in
> the 800:900:1000 Hz example, the result is the same: equal temperament!

Alright. I figured out how to use the geometric series shown above to
acheive octave-repeating ET by supplying x with the nth root of two, where n
is the number of divisions/octave. But I don't understand what you mean by
putting a series of integrally-related frequencies such as 800:900:1000 Hz
into the equation. If you apply the geometric series to a _single_ frequency
you get ET. But I can't seem to get the 800:900:1000 Hz _series_ to make ET,
no matter what I do to it.

And the harmonic series is what we're dealing with when we make chords with
tonal sounds. You know, talented singers who know nothing of the theory or
math behind pitches and scales (say, most bluegrass singers) tune up their
harmonies JI-wise automatically if the chord sustains long enough let them
find the pocket. And with good reason - it sounds "right," as well it should
for those who know musical-acoustical principles.

Hmm. Nothing new here, I know. Are we going in circles?

Oh, speaking of circles, what happens if you treat the octave as one, and
apply geometry there as well? LucyTuning! Ah, let's save that for next
time, after my bandages are off from this one.

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