RE: Catch All

Carl Lumma wrote,

>What isn't clear is just what the claim is as to what mistuning like this
>does to interval perception. Whatever it is, you seem to feel it's big
>enough to cause problems in ET's over 35/oct.

Well, the only real problem in ET's over 34/oct is that consistency is
not really relevant anymore, since there might be an adequate
approximation for a given interval which is not the closest
approximation. My paper, for the sake of brevity, sticks with the simple
ETs, with at most one decent approximation to each just interval. There
may be some wonderful scales in 41tET for all I know.

>And then there's the matter
>of the thing being based on pure tones. What about tones with partials?

Good point -- the answer is something that somehow slipped out of the
latest version of my paper, but was there before. Sorry! Basically, the
0.6%-1.2% deals with a certain optimal frequency range. In other ranges,
the discrimination was much more coarse. From the original version og
the paper (1994): "Musical tones normally have uypper partials within
this range, and since harmonic partials are integer multiples of their
fundamentals, they will yield the same ratio-interpretation for their
fundamentals as would the fundamentals transposed into this range.
(This is why we did not worry about overtones in the first place)."
Carl, perhaps you could do me a favor and stick something like this into
the .pdf version of my paper? You have already helped make it more clear
than the version that's going to be in Xenharmonikon 17 (Hey John
Chalmers, how's it going?).

>Okay. Gary, representing the prime-number advocates (of which, I will be
>sure to add, he is not necessarily one), said that steely coldness was
>often considered to be related to the FACTOR of 3. You start out by saying
>"an arbitrary power of 2 produces an effect of equivalence",

I put that phrase in quotes because I wasn't the one who said it!

>and then
>finish up asking if a power of 3 can produce steely coldness. But it is a
>factor of three that is supposed to produce steely coldness.

No, according to Gary, a factor of 3 will only produce steely coldness
if there are no higher prime factors in the ratio. Powers of 3 are
candidates, since they only have factors of 3.

>Any change in power (stacking the interval two, or however many times) is
>supposed to produce equivalence, falling off as stacking brings the
>compound interval close in size to an interval with different prime factors

This equivalence, if it exists at all (I don't think it does), is
clearly of a different kind than that produced by factors of two, and
the latter is what Dave Hill was initially referring to at the beginning
of this whole discussion. Gary tried to argue that it may just be a
special case of the former, but the fact that different intervals of
arbitrarily high prime limit can be similar to one another due to
differing only by a factor of 2, shows that this equivalence is of a
very different sort.

>Don't I use this idea when talking about music evolving by prime limit?
>Not really. Because you will NEVER hear a higher prime being used
>harmonically in the earlier style. It's not a question of what they
>considered it because they *didn't* consider it.

I think the fact that the tritone in major nearly forms a 4:5:6:7 with
the dominant, and that the tritones in minor nearly form an
8:10:12:14:17 with the dominant, were not inconsequential for the
development of tonal harmony. (Unfortunately, in my decatonic scales in
22tET, the characteristic dissonances (analogues to the tritone) fail to
approximate higher harmonics over any consonant chord in the scale).

When I play diatonic music in 26tET, something interesting happens.
Instead of the tritone in the major mode sounding like the 5th and 7th
harmonics of the dominant, it sounds like the 8th and 11th harmonics of
the subdominant. In some ways, in the major mode in 26tET, the
subdominant functions as the dominant.

>Sort of like you could tell what prime limit music on a 33 is, playing at
>45 rpms, but not what odd limit the same music was without hearing it at
>actual speed?

See, I have no problem with this, since the melodic and harmonic
vocabularies both figure into the pitch sets that are heard, and the
pitch set determines the character of the music to a large extent. I
already stated where I think the odd-limit concept is useful.

>The 12-tone 5/4 is treated as a consonance. I wonder what Mr. Erlich
>thinks about this, since it very close both in size and derrivation to the
>81/64.

I think Partch was right that an interval's allowance for mistuning, as
regards the interval's ability to be recognized as the just ratio, is
inversely proportional to the ODD limit. 81/64, if recognizable at all
(I don't think so), has an EXTREMELY small allowance for mistuning.

>He doesn't seem to have a problem considering a 7 cent sharp 3/2
>consonant!
>This 3/2, to my ear, makes the triads of 22 no better than those of 12,
>despite the better third.

Most people who have made the comparison (including me) consider 19tET,
whose 3/2 is 7.5 cents flat, to have better triads than 12tET. Do you
disagree?

>Even if the "association of certain intervals with the tuning systems in
>which they occur" (that is, I asume, some sort of cultural conditioning) is
>the only way we can pick prime factors apart from odd ones in music, are we
>not still performing a prime factorization on what we hear?

I guess I would have to concur in that case.