Definition of consistency

It appears that Paul R. hasn't caught on to the consistency thread, although
he certainly understands the concept. Since even the Great McLaren's
interest was piqued by one of my posts on consistency, maybe it's worth
reviewing. Unfortunately, all of my e-mail from that far back is lost, so
instead I'll review Manuel's post in Tuning Digest 776:

Manuel wrote:
>I have created a list of equal temperaments up to about 650 tones
>per octave with their consistency limits as was explained by Paul
>Erlich's post of 13 June 1996. However it's not restricted to
>integer divisions, all scale step sizes that give a different
>consistency limit were calculated. Initially I thought this could
>make a nice graph, step size horizontally, limit vertically,
>but the function behaves very wildly and there are very many points.
>Not included in the file are points where both left and right side
>limits are smaller than 5.
>This is the filename:
>ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt

>Another larger file
>ftp://ella.mills.edu/ccm/tuning/papers/cons_limit_bounds.txt

>contains the step size bounds for each consistency limit of 3 to 18,
>also up to 650-tET.

Here's a definition of consistency: Given an odd number n, an octave-based
equal temperament is consistent within the n-limit if, for any odd numbers
a, b, and c such that 0a:b plus the number of steps that best approximates b:c is equal to the
number of steps that best approximates a:c.

No equal temperament can be adequately notated using the symbols of just
intonation (Pythagorean scale plus syntonic commas) if it is not consistent
within the 5-limit. Paul Rapoport disagrees. I guess we'll just have to
agree to disagree.

I found 111-tet by looking at consist_limits.txt.


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