Commas and Consistency

Re commas:

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.'

I'm not quoting this because I think it's marvellously illuminating.
Unfortunately I haven't got a copy here of Blackwood's book at the
moment, which (as I recall) is more interesting on the topic.

However, my questions about 'disappearing commas' stem from the fact
that various writers use the term 'comma' (as above) to refer,
naturally enough, to two facets of the same thing. We talk about a
fixed temperament, such that, as Paul H put it:

>An interval vanishes in a particular ET (or quasi-ET like a
>well-temperament) if it is represented by 0 scale degrees.

And we talk about a comma as an 'elastic' interval, the acceptable
leeway employed by a violinist or clarinetist when taking a tone
(for example) as either major (9/8) or minor (10/9) (the difference
between the two being 81/80) in different situations, or some
approximation of either. A 'comma' is an anomaly
specifically resulting from combinations of just intervals (or
equivalent tempered intervals); it is also (more or less) the
ambit of an interval within which performers adjust their intonation
in certain circumstances. Well, this is the way I've always thought
of it. OK?

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? ('disappearing' in the sense of 'going
unnoticed')? I'm not clear on this - and maybe that's why my question
wasn't clear to Paul E. I wrote:

>>Could anyone tell me - is there a unique theoretical limit at which
>>a comma (of different kinds?) may be said to disappear?

Paul E replied:

>I don't understand this--can you rephrase?

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.

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 know that the following has been discussed on the list before, but
I'll just try to explain what I was trying (unsuccessfully) to work
out. Both Pauls have been very kind in explaining some of this to me
off-list. Paul H has also explained a simple algorithm to show
whether an n-ET is level-p consistent at the m-limit. The algorithm
goes something like this: (I'm dumping a bit of pre-written stuff
here) -

(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.

Thus inconsistency occurs when the combination of t m-limit ratios
diverges by more than half a step in n-ET from the expected
consistent combination of n-ET scale-steps. This corresponds to our
intuitive notion of 'consistency' in everyday terms. But a more
stringent criterion (from the point of view of intonation) might be
substituted - for example, the criterion could be changed to 1/3rd of
a step (by changing (1200/2n) to (1200/3n)) (as has been discussed
before). Obviously, doing this changes the meaning of 'consistency'.

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.

Well, that's an inconclusive explanation of what I was trying to get
at. I found both Pauls' responses interesting - but unfortunately I
soon get into difficulties making sense of their maths. I'm sorry to
be a useless correspondent in this respect, and also sorry that I
don't have adequate time at the moment to try harder.

However - I was particularly interested in this bit (by Paul E):

> 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?

Patrick O-L