Inconsistencies in n-EDO approximations of the m-limit (was: Sims Wikipedia article)

On Tue, 06 Dec 2005 "wallyesterpaulrus" wrote:
>
[snip]
>
> ... Gene, you may be interested to know that Ezra was not
> aware of the inconsistencies of 72-equal beyond the 17-limit until I
> pointed them out to him. In 72-equal, the best approximation of 8:13
> plus the best approximation of 13:17 [corrected to 13:19] does not
> equal the best approximation of 8:17 [corrected to 8:19] ...

Hi all,

That's interesting ... I think I can finally (!) phrase
this question correctly, and invite your thoughts:

Is there a good general approach for finding out
the inconsistencies in approximation by n-EDO
of all intervals up to the odd m-limit (m & n both
odd)?

Regards,
Yahya

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--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> On Tue, 06 Dec 2005 "wallyesterpaulrus" wrote:
> >
> [snip]
> >
> > ... Gene, you may be interested to know that Ezra was not
> > aware of the inconsistencies of 72-equal beyond the 17-limit
until I
> > pointed them out to him. In 72-equal, the best approximation of
8:13
> > plus the best approximation of 13:17 [corrected to 13:19] does
not
> > equal the best approximation of 8:17 [corrected to 8:19] ...
>
>
> Hi all,
>
> That's interesting ... I think I can finally (!) phrase
> this question correctly, and invite your thoughts:
>
> Is there a good general approach for finding out
> the inconsistencies in approximation by n-EDO
> of all intervals up to the odd m-limit (m & n both
> odd)?

Why would n have to be odd, or why would it make a difference? Above,
n=72.

Let me illustrate the general approach with the example of 72-equal.

Express each odd number up to m's ratio (to 1) in steps of the ET
(EDO implies no approximations to ratios, so I won't use that
acronym):

3 is approximated by 114.12 steps
5 is approximated by 167.18 steps
7 is approximated by 202.13 steps
9 is approximated by 228.23 steps
11 is approximated by 249.08 steps
13 is approximated by 266.43 steps
15 is approximated by 281.3 steps
17 is approximated by 294.3 steps
19 is approximated by 305.85 steps

Next, write down what fraction of a step each one overshoots or
undershoots (in the latter case, use a negative value) its best
approximation in the ET. These fractions will all be between -0.5 and
0.5:

3 overshoots by 0.12 steps
5 overshoots by 0.18 steps
7 overshoots by 0.13 steps
9 overshoots by 0.23 steps
11 overshoots by 0.08 steps
13 overshoots by 0.43 steps
15 overshoots by 0.3 steps
17 overshoots by 0.3 steps
19 overshoots by -0.15 steps

Now, if and only if the difference between the highest and lowest of
these values is greater than 0.5, we have inconsistency. In this case,

0.43 - (-0.15) = 0.58, so 72-equal is inconsistent in the 19-limit.

The numbers above also make it clear that 72-equal is consistent in
the 17-limit.

Thanks to Paul Hahn for this algorithm.

This table by Paul Hahn:

http://library.wustl.edu/~manynote/consist3.txt

shows the greatest error, in cents, of each ET in each odd limit in
which it is consistent. Since 72-equal has 8 numbers to the right of
the "|", we know that it's consistent up to an odd limit of 2*8 + 1 =
17. We can also see from this table that within the 11-odd-limit, 72-
equal commits no error 4 cents or greater . . .