Reply to Carl Lumma

I've posted at length on definitions of consonance and dissonance (as
well as discussions of melody and modulation) in previous Tuning
Digests. The issue is complicated but that is no reason to retreat to a
simpler, less useful definition. Your definition makes no sense, because
you'll never come up with an answer for irrational intervals. How do you
propose to "measure [the mistuning of an equal temperament] in cents
deviation _and_ consonance* deviation" when you can't define the
consonance of an irrational interval? Presumably, you want to use
rational approximations, but how do you decide whether to use 25/21,
44/37, 1785/1501, 10754/9043, or 19723/16585? You seem to like the
latter, but if you really go to that level of accuracy, you face a very
large danger that some of your so-called JI perfect fifths will turn out
to be closer to 30001/20001, in which case you'll have to call the
equal-tempered minor third more consonant that the just perfect fifth!


Actually, if you want to get into silly mathematical discussions, the
probability that two physical strings are tuned to a rational inteval is
zero, as Cantor showed that the cardinality of the reals is greater than
that of the rationals. The probability that the strings are tuned to an
n-tone-equal-tempered interval is also zero, since the algebraic numbers
have the same cardinality as the rationals (which have the same
cardinality as the integers).

The only definition I care about is one with a psychoacoustic
correlative. Otherwise we're not talking about music, we're talking
about abstract marks on a piece of paper or computer screen. My
calculation of beat rates was an attempt to show that the one
psychoacoustic correlate you did mention does not lead to a first-order
qualitative difference between 19/16 and 2^(1/4). Personally I think
19/16 in a high register has a peculiar stability due to its denominator
being octave-equivalent with the fundamental (the brain often fills in
the missing fundamental with a "virtual pitch" and any combination tones
will only be in agreement with the virtual pitch in the case of JI).

Oh, I finally saw that cream cheese commercial.


SMTPOriginator: tuning@eartha.mills.edu
From: "Paul H. Erlich"
Subject: Inharmonicity (again!)
PostedDate: 27-10-97 22:23:36
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