deeper into the mire

On the sensitivity of harmony to tuning:

>It depends what you're listening to. If the chord occurs as a result of
>simultaneous melodic lines in a piece of music, I don't think you're right,
>unless you'd allow 20 cents to be considered a few cents. If you're listening-

>for beats, then yes, you're exactly right.

I was thinking in terms of 2 or 3 cents. You do have to listen
pretty hard for the difference, but it's there. By the time
you get to 20 cents, the detuning is instantly obvious with the
most sensitive timbres. I wouldn't allow a tonic to be out of
tune by this amount.

>3, 5 and 7 are successive odd numbers, so I don't think one has to resort to
>12tet to explain the increasing dissonance of their respective intervals
>(which I clearly hear).

I wouldn't like to say whether 4:5:6 or 4:6:7 is more consonant
in my schismic tuning. Both, though, are a lot better than
4:5:7. Either this is a result of the poor 7/5, or the odd
limit is not a perfect discriminator of consonance. If my
soundcard's behaving itself, 7/5 should only be out by 5 cents.
The dissonance of 4:5:7 is very obvious. I'm using the "Chorused
Piano" preset at the moment, but I hear the same effect with
others.

In 31tet, the 7/4 has an unpleasant ring to it, which may be a
result of its purity. "Vibraphone" illustrates this well. I
conclude, therefore, that the minutiae of tuning are more
important than the difference between the residual dissonances
of the 5- and 7-axes. This is all subjective, so you're
welcome to disagree with me. I expect to see some evidence of
what you're listening to, though.

A definition of consonance you ask? I can only give an example.
the purity an interval of 702 cents posesses that a randomly
chosen interval does not -- that's consonance.

>Right. If an n:1 frequency ratio is tuned too purely in sustained harmonic
>timbres, the upper note will disappear.

In 12TET the octaves are, of course, pure. I'm sure this has
a lot to do with why octaves are treated differently to other
overtones, but I don't want to get into that at the moment.

I was more thinking that the blurred sound of 12TET is suited
to some music -- easy listening, cocktail jazz, that sort of
thing. Also, I sometimes hear unpleasant things in the upper
harmonics with the nearest I can get to JI.

I wouldn't go this far, though:

> But there's nothing conceptually wrong, as far as I can see, with saying
> that two intervals whose difference is less than the ear's accuracy are
> musically the same. This is the only way Canright's "all intervals" can be
> "just".

The range of musical similarity is _much_ wider than the accuracy
of the ear for the tuning of chords.


The following statement of Paul Erlich's is entirely correct:

>... the probability that two physical strings are tuned
>to a rational inteval is zero, as Cantor showed...

As a general rule, there's no such thing in the real world as
integer relationships of continuous quantities. Entrainment
only goes so far.

However, by the same argument, this is wrong:

> _Bowed_
> strings, however, if bowed at a constant speed and pressure, are
> perfectly harmonic, since they undergo exactly the same motion (analysed
> by Helmholtz) in each slip-stick cycle.

Never say "perfect" -- say _how_ imperfect!

Turning to this,

> It must be something else, then, that causes additive patches to sound more
> like violins when you detune the harmonics.

I suggest that "something else" is vibrato.


SMTPOriginator: tuning@eartha.mills.edu
From: William Sethares
Subject: a definition of sensory dissonance
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