S/K notational specificity

Question for anyone who can point me to an archive or offer a quick answer: What is the smallest acceptable notational - intonational error in the proposed Secor / Keenan notation system, and on what grounds has this smallest margin of error been agreed upon?

thanks,
Aaron

--- In tuning-math@yahoogroups.com, pitchcolor@a... wrote:
> Question for anyone who can point me to an archive or offer a quick
answer: What is the smallest acceptable notational - intonational
error in the proposed Secor / Keenan notation system, and on what
grounds has this smallest margin of error been agreed upon?
>

Hi Aaron,

Thanks for your interest. I expect you meant what is the _largest_
acceptable error, but were at the same time thinking what is the
smallest pitch change that can be notated, or how small is the largest
error.

We have not allowed
any approximation greater than a cent, and many are less than half a
cent. And in fact there is no approximation at all until one goes
beyond 11-odd-limit in the single-symbol and double-symbol forms of
the notation, or beyond the 31-prime-limit in the multi-symbol form.

It is exact in all equal and linear temperaments. This is possible
because the symbols are not defined as referring to specific numbers
of cents but as giving the best approximation of various ratios
relative to a chain of fifths. The size of those fifths is a variable
that must be specified by the composer. If it's in an ET then it is
usually enough to say which ET.

These notational fifths are always between 3/5 and 4/7 octave but are
usually much closer to just. Non-octave scales can probably also be
accomodated by specifying a stretched or compressed "octave", the
limits of which have not been decided but will need to be close to 1:2.

On what grounds has the half-schisma/one-cent maximum error been
agreed on?

At first it was simply political. If there were to be no errors ever,
you would need an infinite number of symbols, or potentially infinite
strings of a finite number of symbols. So there had to be _some_
approximations. And while we personally could tolerate errors of two
to three cents, we wanted the notation to be acceptable to strict
just-intonationists, so we decided to try to match or better the
accuracy of the notation promoted by Johnny Reinhard of the AFMM,
which uses integers representing cents.

Much later we found we wanted to introduce symbols for the 5-schisma
anyway. As the smallest ratio to be notated, our maximum error becomes
half of this. But as I said, in most cases we can be exact, because we
are not constrained to notating relative to 12-ET or 24-ET.