RE: Commas and Consistency

>In Intervals, Scales and Temperaments (p133) Hugh Boyle defines the
>syntonic comma (81/80, or 21.506 cents) in a couple of the usual ways,
>and then goes on:

>'In musical performance, this comma should not be thought of as the
>problem it presents to the tuner of keyboard instruments, but as a
>somewhat elastic interval depending, as it must, on the ear's ability
>to estimate intervals under varying circumstances. Further, if the
>musical instinct of the artist for melody or for concord compels him
>to make a small adjustment of this kind in the width if his intervals,
>then any subsequent readjustment felt necessary can easily be
>absorbed, without offending the ear, in a discord, which, because of
>its dissonance, is lacking in definition.'

For flexibly-pitched instruments, that's fairly on the money.

>Therefore, going from what Pauls H and E wrote - is a *disappearing*
>comma something which belongs exclusively to the logic of a
>temperament? Or is the phrase 'disappearing comma' sometimes used
>equally to refer to intonational 'disappearance' in commatic
>performance adjustments?

The latter meaning strikes me as more useful, and the former is just a
special case of the latter.

>I'll try. Is there an historically accepted limit to the size of
>what will work as an effective 'performance comma' - (that is, from a
>traditional harmonic point of view)? Is that a question which is too
>subjective to worry about? I have always assumed it could not be more
>than about 21-24 cents. I've tried some experiments with various
>conventional and unconventional progressions, and in differing
>temperaments and timbres, but so far they are inconclusive.

Barbershop harmony seems to be able to take the septimal comma, which
would be 27 cents in just intonation, and by adjusting the (harmonic
and/or melodic) intervals in the comma's context, and/or shrinking the
comma itself, make it an effective "performance comma".

>Now, the reason I started looking at this is because I have been
>trying to work out some ideas about consistency. It occurred to
>me, as both Pauls reactions seem to confirm, that the ancient notion
>of a 'comma' is (kind of) the original root of the idea of
>consistency.

I don't see it.

>(i) choose the n-ET and the m-limit to be considered;
>(ii) list the intervals which belong to the primary m-limit;
>(iii) find the primary m-limit interval which is least well
>approximated by the chosen n- ET;
>(iv) add this interval to itself until the deviation from its
>intended equivalent is greater than (1200/2n) cents; call the number
>of additions at this point t.
>(v) then n-ET is level-(t-1) consistent at the m-limit.

This certainly doesn't work for level 1 consistency. Is this the thing
that happens to work for higher levels? If so, how are you defining
"primary" and "number of additions"?

>If we replace the expression (1200/2n) with (1200 x 21506/10000n) we
>might track 'consistency' relative to the sytonic comma - which I
>have assumed is the traditionally acceptable ambit of intonational
>wandering in 'commatic situations'. But I wanted to know if my
>assumption was grounded.

It strikes me that you're drawing a false connection between commas and
consistency, and therefore trying to kill one bird with two stones. If
the comma vanishes consistently, then all commatic situations will be
completely well-behaved by traditional criteria. If the comma doesn't
vanish, consistency (of any kind) or no, the rules that govern the
general performance situation apply. The consistency of a tuning,
relative to half a step or any other measure, is neither sufficient nor
necessary to guarantee that the commatic situations can be handled.

>> commas constructed from n-limit intervals are uniquely defined in all

>>tunings consistent within the n-limit. Given consistency, the
>>determination of the size of a comma in an ET is fairly trivial.

>and I wonder if Paul would give us a couple of examples of what he
>means?

For the syntonic comma, calculate the difference between four 3/2s up
and a 5/4 down. If the tuning is 5-limit consistent, then this will be
the same as the (absolute) difference between three 3/2s up and a 5/3
down. In 19-tone equal temperament, four 3/2s up is 6 steps (neglecting
octaves), and that is the 5/4, so the comma is 0 steps. In 22-tone equal
temperament, four 3/2s up is 8 steps, but the 5/4 is 7 steps, hence the
comma is 1 step.