source file: m1450.txt
Date: Wed, 17 Jun 1998 12:12:24 -0500 (CDT)

Subject: RE: Commas and Consistency

From: Paul Hahn <Paul-Hahn@library.wustl.edu>

On Sun, 14 Jun 1998, Paul H. Erlich wrote:
> Barbershop harmony seems to be able to take the septimal comma, which
> would be 27 cents in just intonation, and by adjusting the (harmonic
> and/or melodic) intervals in the comma's context, and/or shrinking the
> comma itself, make it an effective "performance comma".

I'm curious about this.  I assume Paul E. is talking about the
reinterpretation of the traditional dominant seventh chord as a 4:5:6:7
tetrad.  However, this would imply an interval of 16/9 between the root
and the seventh, like so:

		5/4

16/9  [4/3]  1/1   3/2

and I'm not sure why one would choose 16/9 when that makes that pitch
totally unrelated (within the 5-limit) to any of the other pitches in
the chord.  Well, I mean, obviously it's the distance between the 4/3
and the 3/2 (3/4) of the key, critical pitches if you're using the dom7
actually as a _dominant_ 7th.  But considered purely as an isolated
sonority, within the 5-limit I would tune it thus:

   5/4

1/1   3/2

	 9/5

(Arrgh!  This is why I prefer ETs.  In an ET in which the 81/80
vanishes, these two are the same and you don't have to worry about it.)

I'm not very familiar with the barbershop style.  Do they generally use
the dominant 7th as V7 of the key, as opposed to jazz harmony which uses
it as a sound color that can occur almost anywhere, functionally?

The point is, given this interpretation the distance to the 4:5:6:7 is
36/35, which is the septimal comma plus the syntonic comma or nearly 49
cents, a significantly larger interval to swallow as a comma--though
still not as big as the 648/625 I mentioned in a previous message.

> >Now, the reason I started looking at this is because I have been 
> >trying to work out some ideas about consistency.   It occurred to 
> >me, as both Pauls reactions seem to confirm, that the ancient notion 
> >of a 'comma' is (kind of) the original root of the idea of 
> >consistency. 
> 
> I don't see it.

I'm not sure, but I _may_ see it--if we require that a comma, meaning
any just interval that vanishes in a particular ET, be smaller (in its
just version) than any of the non-vanishing intervals in that ET--which
seems reasonable to me--that has certain implications as to the
approximation errors of the primary intervals in relation to stepsize,
and hence to consistency.  It's not a _direct_ relationship (complexity,
or "n-ary-ness" as Patrick and I have been calling it in private
correspondence, comes into it as well, plus some other stuff too), but
it is a close one.

> >(i) choose the n-ET and the m-limit to be considered; 
> >(ii) list the intervals which belong to the primary m-limit; 
> >(iii) find the primary m-limit interval which is least well
> >approximated by the chosen n- ET; 
> >(iv) add this interval to itself until the deviation from its 
> >intended equivalent is greater than (1200/2n) cents; call the number 
> >of additions at this point t. 
> >(v) then n-ET is level-(t-1) consistent at the m-limit.
> 
> This certainly doesn't work for level 1 consistency. Is this the thing
> that happens to work for higher levels? If so, how are you defining
> "primary" and "number of additions"?

We went over this when I first posted my algorithm to the list--the
difficulty comes in how the roundoff error of, say, the 5/3 is
calculated.  Rather than calculate the size of the 5/3 in steps of the
ET and finding the roundoff error directly, one must calculate the
roundoff error of the 5/4 and the 3/2 separately, and then subtract
them.  Given that, the algorithm works for level 1 as well as higher
levels of consistency.  It was easier to see from the pseudocode I
posted months back.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
    O
   /\        "Churchill? Can he run a hundred balls?"
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