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.