source file: m1424.txt
Date: Fri, 22 May 1998 14:17:55 -0400

Subject: Reply to Graham Breed

From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>

>>> For exactness, state the signed errors, and you can work
>>> other intervals out from that.

>> Isn't that what Carl Lumma said? But no, you guys are wrong, and that's
>> the whole reason for the consistency concept. Wendy Carlos, Yunik and
>> Swift, and others also seem to have missed out on the importance of
>> consistency.

>I think Carl Lumma (mistakes aside) takes the best
>approximation for each interval, and tries to describe them in
>terms of prime intervals.  I really _mean_ to take the best
>approximations to prime intervals (not necessarily prime
>numbers) and define the rest from this.  If the scale is
>inconsistent, one of the intervals will have an error greater
>than half the step size.  If that error is acceptable, so is
>the scale.  I'm not wrong.

Sure. I'm glad you think the prime intervals don't have to be prime
numbers. It bears pointing out that depending on which "prime intervals"
you choose, you will end up with different approximations to the rest.
Different tunings will have different "prime intervals" which lead to an
optimal set of compatible approximations -- for the 5-limit, for
example, some ETs will yield greater accuracy if 5/4 and 3/2 are chosen
as the prime intervals, some if 5/3 and 3/2 are chosen, and some if 5/4
and 5/3 are chosen. Consistency simply means that all choices lead to
the same result. That's how I interpreted your statement "For exactness
 . ." above.

>Consistency makes sense if you happen to be working within an
>odd limit.  Expressing the consistency level as a fraction
>[(step size) / (2 * worst error)] is more precise

Of course that's only correct if the worst error is 1/3 of the step size
or less (Paul Hahn originally pointed this out).