JI vs. ET for Ease of Exploration

Marion wrote a very thoughtful reply to my question of why he finds just
intonation systems easier to explore systematically than equal-temperaments. I
just now finally got a chance to read it.

One of his main points is that temperament introduces ambiguity: If our ears
are presented with an irrational frequency relationship, we are confronted with
a question of how our ears interpret it.

Marion didn't cite this specific example, but I suspect it's an illustration
of this sort of concern: Suppose a 12TET composition plays a diminished seventh
chord, such as a B D F Ab. How will our ears interpret each of the minor thirds
and augmented seconds in that chord? One possibility would be as a 5:6:7:8
chord. There are other possibilities of course, including the subharmonic
equivalent of that chord, or as all of its minor thirds being 6:5s, or a 6:5
atop a 7:6 atop a 6:5.

Each of those possibilities have a distinctly different feeling to our ears.
The 12TET version of that chord, being an approximation of all of them at the
same time, is an ambiguous mishmash of all of those possibilities. It doesn't
present any of those subtly but fascinatingly different sensations very clearly
to our ears. So which sensation our ears will attribute to that chord is hard
to judge.

Marion's conclusion, again if I understood him correctly, is that that sort
of ambiguity complicates a systematic exploration. We're all a lot better off
taking the simple, honest approach: Just way exactly what we mean in the first
place - use the exact pitch relationship. Perhaps later we can deal with the
effects of approximation.

That strikes me as a reasonable view. But let me present another quite
different view.

That view is that when you play a B D F Ab diminished seventh chord in 12TET,
you ARE saying exactly what you mean. If you present our ears with a stack of
ambiguous representations of 6:5, 7:6, or other minor third formulations, our
ears make no attempt to resolve the ambiguity. I have no doubt at all that when
we present these sorts of interval distinction explicitly, our ears attribute
subtle meanings to them. I also have no doubt that tuning systems that preserve
those distinctions are a very valuable area of musical pursuit. But if you
avoid those distinctions, I don't see that our ears make a much effort to CREATE
them. As far as I can tell our ears just view the intervals in that chord just
as a generic m3/A2 pitch relationship, without further detail. Certainly I
doubt if we expend too much effort to create those distinctions over the course
of a fraction of the second that chords often sound in many realistic musical
applications.

And another factor is worth bearing in mind: Our 12TET-oriented culture
presents the illusion that there are no such distinctions, such as between 6:5
and 7:6. So the burden of proof, so to speak, is on us. Our music has to
explicitly motivate its audience to make that distinction. Certainly, as I
believe Marion is pointing out, that can be done more effectively in exact
representation rather than in equal-tempered approximation. But I believe that
there is meaning to music that doesn't even attempt to make those distinctions.


Meanwhile, there are certainly many clearly meaningful ways to group equal
temperaments for systematic exploration. And these ambiguities are often at the
root of the meaningfulness of classifications. For example, the "normal"
resolution of B D F Ab, is to have the B go up to C, the F go down the E, and
the Ab go down to G. In the 1800s especially, a popular technique was to
suddenly reinterpret B D F Ab into B D F G#. In that nomenclature the expected
resolution is to have the G# go up to an A, the D go down to a C#, and the F go
down to E. This is one kind of "tritone substitution". Because of the
nondistinct, ambiguous, equal distribution of the notes in that chord, you can
quickly but seamlessly - almost mysteriously - modulate from the key of C Major
to the somewhat remote key of A Major. Or similarly to the much more remote key
of F# major.

Because of this ambiguity then, any equal-temperament of a multiple of four
steps per octave has the potential to be able to achieve this surprising and
musically useful effect. And there are other useful bases for systematizing
equal temperaments, like multiples of five or seven, or tunings that have
traditional-sounding major scales, and so forth.

So, the distinction between whether just-intonation systems are easier to
systematically explore than equal-temperaments, probably boils down to what
sorts of musical effects you want to explore and thus interpret the results. I
suppose that's not terribly surprising, is it? If you're interested in
exploring the exciting distinctions between the septimal sensations of intervals
like 7:6 and 7:4 and their 5- or 11-limit counterparts, then you're best off
using those intervals in their most precise form. But if you're interested in
exploring surprising contrapuntal effects that simplified, ambiguous
representations of chords make possible, then you're probably better off
thinking in terms of temperaments.


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