source file: mills2.txt
Date: Thu, 19 Sep 1996 12:55:13 -0700

Subject: RE: Consistency generalized

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

On Thu, 19 Sep 1996, PAULE wrote:
> Paul H.,
> 
> 	    I believe your original definition leads to consistency for level 
> N>1.5. For 1<N<1.5, there is no guarantee that the approximating intervals 
> will be consistent with one another.

Yes, I noted that in my "erratum" followup message.  However, the
algorithm I just posted in response to John Chalmers is correct.

> 	    I don't think the concept of consistency has much to do with whether a 
> particular JI interval is approximated well enough to be used as a harmonic 
> consonance.

I don't think that's exactly what I meant, or even what I said.  But
anyway, see below.

[bigsnip]
> 	    What consistency offers is a supplement to considerations of how good 
> the approximations to JI are. Essentially, composing with a consistent 
> tuning will be no more difficult than composing in JI, while in inconsistent 
> tunings, complications may arise if one attempts to always use the best 
> approximations to JI intervals.

Here's where we part company, I'm afraid.  I find level 1 consistency to
be far too low a standard.  Very weird things can happen in level 1
consistent tunings.  A simple example: 5TET is consistent at the
5-limit.  A 16/15, represented as a 4/3 less a 5/4, becomes 2-2=0 steps,
or a unison.  However, a 25/24, represented as a 5/4 less a 6/5, becomes
2-1=1 step.  In other words, a larger interval becomes a unison while a
smaller interval does not.  Maybe this doesn't bother you, but for me it
causes cognitive dissonance.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
       O
      /\             "Foul?  What the hell for?"
     -\-\-- o        "Because you are chalking your cue with the 3-ball."

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