RE: Odd vs. Prime

>a) For describing music tuned in Just Intonation
>b) For describing music in root-of-two equal-step tunings

>For "a" I prefer prime, and for "b" I prefer odd. Now I say I use "limit"
>for "describing music", NOT for creating an acoustical theory of interval
>perception. I admit the two are intimately tied up, and I will address the
>latter only as the former requires.

Here's what I've always said: "a" is preferable for describing the
resources of a given system of Just Intonations, and "b" is preferable
for describing the maximum interval complexity in simultaneities
considered consonant in a given style. The latter is very tied up with
an acoustical theory of interval perception.

>>"An arbitrary power of two produces an effect of equivalence." This must
>>of course include the case where the power is zero, and the equivalence
>>is greatest. The equivalence evidently falls off as the power increases.
>
>>But what about steely coldness? Is it there when the power is zero? Does
>>a unison contain all potential interval qualities, to a greater degree
>>than the intervals themselves? It would seem hard to argue that way.

>..And I wonder if he is talking powers or factors. Factors are involved
>in the ratio of frequencies in a given interval, and powers are involved in
>the stacking of any interval. The idea is that adding a factor of 2 to an
>interval produces a new interval with a certain "sameness" to the first,
>because 2 is the "sameness factor". And stacking any interval will produce
>a new interval with a certain "sameness" to the first, simply because
>stacking doesn't add any new factors of any kind.

>If he's means what he says ("powers"), then his first paragraph is correct,
>but his second is not, as "steely coldness" is an attribute of the factor
>3, not the power 3.

>If he means factors instead of "powers", then I suppose his whole thing is
>correct. Although I do not see why he insists on making the unison into
>some kind of "white light", I am willing to oblige. As far as factor
>determines which partials line up, and all of the partials in a unison line
>up, the analogy holds...

I'm not sure what it is in my wording that was problematic. Perhaps you
could suggest an alternative wording for each of the interpretations you
mention, and I could endorse one, allowing us to proceed?

>Mideval music choral music has ratios of 3 and 9, but not of 5 or 7. So is
>it 9-limit? Now you might say that we can just as well have music with 7's
>and no 5's, creating an ill-defined prime limit. And true, we can have
>such a music, but we don't. Except for isolated experiments with
>fixed-pitch instruments (I believe Fokker did work with such tunings),

And don't forget LaMonte Young!!!

>music has evolved by prime limit. Pull out a CD of English tudor music, as
>sung by the King's Singers, for example, and try to add the 7's. You
>won't. They didn't, not for hundreds of years. They've got ratios of 25
>and everything else needed to modulate around the 5-limit, throwing commas
>around like frisbees, but no 7's. Barbershop's got 7's and 28's, and 63's
>a-plenty. And 9's, and 18's, and 27's. But no 11's. Never will you hear
>an 11 used harmonically in Barbershop music. And the first time you do,
>you'll be hearing them again soon and often :~)

Again, the odd-limit definition applies not when considering all
intervals present, but when considering which intervals can be
considered consonant. In all your examples, the higher composites are
dissonant.

>The reason, I suspect, that Paul Erlich fights for odd limits: He is a man
>of equal temperaments :~)

Equal temperaments or not, that has nothing to do with this issue.

>The idea that 9's do not become harmonically
>significan't until we have 7's does not hold in my experience. Try playing
>just 4-5-6-8-9 chords and see if the 9 doesn't serve the same purpose as it
>does in a 4-5-6-7-8-9 chord.

>That is not at all a valid interpretaion of anything I've tried to say.