The limits of the ear

I hope I haven't generated any harsh feelings with these flaming posts. I
just enjoy 'em. Mr. Erlich can rest assured that I have nothing but the
highest respect for his work. His was one of my favorite cuts on the tape
swap and has got me wanting more.

He also may or may not wish to know that one of the first tunings I'm going
to map to my generalized keyboard (when I get it) will be 22 equal.

>Today is my last day as a full-time employee of an investment company. I
>already spend almost all my free time playing music...

So how is that working out? Are you able to resist the temptation of
playing with your 12 tone friends enough to go home and be microtonal?

Lastly, I will say again that altough I used a posting of Mr. Erlich's as an
example, I was addressing the community at large with an issue at large.
Anyway, here's my response...

>Now these intervals are getting closer and closer to 3/2, and after just
>a few of them even the best ear can't tell the difference between them
>and 3/2. Yet according to Carl, 3/2 is more consonant than any of these
>ratios. Thus we must conclude that Carl's definition of consonance has
>little to do with anything you can actually hear.

The idea of defining Consonance as a product of the sizes of the numbers in
a ratio wasn't Just something I dreamt up. It is littered all over the
literature, especially the JI Primer. But there's always something vague
like "strong implied preference" involved.

It is told straight in Denny Genovese's thesis "The Harmonic Series as a
Practical Approach to Just Intonation", in chapter 2, "Quantification of
Dissonance and an Acoustic Definition of Consonance".

To paraphrase: If you've got two frequencies represented by a ratio in
lowest terms whose decimal value is between 1 and 2, then the period of the
composite waveform of these two frequencies is the product of the two
numbers in the ratio. I believe this makes a more useful definition of
Consonance than any other I've ever heard. If somebody's got one they think
I havn't heard, please share.

This definition is independent of what a person may or may not hear, and it
is independent of how they might like or dislike what they hear. For those
of you who do not view these as assets, don't use this definition.

The ear has a limit of resolution, just like anything else. That the one
example given by Mr. Erlich is beyond the ear's resolution does not mean
that this definition has no practical application. Clorox is poison but
lots of folk find it useful to chlorinate their water. I think the
practical usefulness of the definition is obvious.

While I can't hear the difference between a 3/2 and a 300001/200001, I can
hear the difference between a 19/16 and a 19723/16565. I'm listening to it
right now. But I've been told that an error of 2 cents is not significant
when comparing tempered intervals to just ones.

Working within the practical limits of human hearing, the perception of
subtle mis-tuning is very sensitive to the timbres used, the voicing of the
intervals, how high the identities are, and how they are used in combination
with other intervals.

If your timbres have a high degree of inharmonicity, you'll loose
resolution. Don't use bowed strings for high resolution work. If your
interval is down in the lowest octave of the piano, don't expect to hear a
10 cent difference. You're much less likely to notice a 3/2 2 cents off
than a 11/7 mistuned by 2 cents. You can't hear the difference between a
3/2 and 700 cents in a melody, but in an otherwise just triad, it sticks out
like a sore thumb.

So my original point was: When comparing an equal-step tuning to a just
tuning on a broad, theoretical level, define Consonance on a broad
theoretical level.

My other original point was: Equal step tunings have nothing to apologize
for. I don't view them as imitating just tunings. They're a different
breed of cat. The kind of music that makes sense in an equal temperament
doesn't make sense in JI and vice versa. It's like apples and oranges.

I used to think that an equal step tuning was like having all the pitches
that you needed to modulate in every key of a just tuning, except you used a
single average pitch instead of two or three or four just ones. So you
could make music on this huge matrix of a just tuning with less notes and
worse sound. But the difference is much deeper than that. I can't explain
it. The Pauls (Erlich and Hahn) turned me on to this, and their ideas on
this matter are much farther along than mine.


And I never said dissonance was bad,

Carl


Aline Surman writes...

>I always enjoy the discussions about ratios and eq temps. It is true
>that being "in tune" is a mathematical phenomenon of frequencies matching
>up perfectly...I agree that when we say eq temps approximate a pure
>ratio, it doesn't correlate in a way because as soon as you are off of
>the ratio, the new note is NOT the ratio, so it IS something else.

Thanks for the support! I know writing these posts is beneficial for me.
It's nice to know they are of interest to others as well.


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