Odd limits/ Sensitivity

[Paul Erlich:]
>> I think Partch was right that an interval's allowance
>> for mistuning, as regards the interval's ability to be
>> recognized as the just ratio, is inversely proportional
>> to the ODD limit. 81/64, if recognizable at all (I don't
>> think so), has an EXTREMELY small allowance for
>> mistuning.

[Carl Lumma:]
> Ah. Well, that works. Thing is, "sensitivity to mistuning"
> can mean so many things. I'm just now seeing how
> you're using it here, and I can't say that it is in any way
> incorrect.

> There is a type of "sensitivity to mistuning" that does not
> obey the inverse to odd limit rule. For this type of
> sensitivity, it seems that the 3/2 is much more sensitive
> than the 7/4. Further, it seems to matter to this type of
> sensitivity wether an interval is mis-tuned sharp or flat.

I think we're touching on something quite important here.
It seems to me that these two different types of mistuning
sensitivity may relate directly to thinking in terms of equal
temperament or ratios. (This is possibly incorrect, though.)

The fact that a 3/2 is simultaneous more and less sensitive
to mistuning also seems to have something to do with
the fact that, because 3 is the next-strongest prime quality
after the octave (2), it can be both absorbed into the strength
of 2, in which case mistuning is not much of a problem, or
can stand out as a powerful new identity, in which case
mistuning will be very obvious.

I have said before that a basis of my theories is that we
will always try to "break down" any complex situation
into its most easily understood components. If periodic
frequency vibrations can be understood in terms of
how they divide into 2s and 3s, this is a very easy
comparison for our brains to make, thus even a small
deviation will stand out very noticeably. The case is
much more complex for, say, an accurately-tuned 7/4.
It would not be as easy to notice a deviation from a
ratio with 7 as a factor, simply because 7 is a much
more complex comparison to make than 2, 3, or even 5,
and each higher prime is correspondingly more complex.

Joseph L. Monzo
monz@juno.com


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