Just Interval Progressions

I have enumerated all the progressions, derived from the consonances, by
which one can arrive at the first 61 just intervals. For example, here
are the progressions by which one arrives at 16:15 (112 cents, the
diatonic semitone, the diminished minor tone:)

4/3 x 8/5
4/3 x 4/5
8/5 x 4/3
8/5 x 2/3
32/25 x 5/3
32/25 x 5/6
16/9 x 6/5
16/9 x 3/5
64/45 x 3/2
64/45 x 3/4
128/75 x 5/4
128/75 x 5/8

Similar enumerations were made many centuries ago, but always so far as
I can discover, from a strictly diatonic and/or fifth-based standpoint.
My method uses all the consonances, and is equally well-adapted to serve
as a theoretical basis for chromatic and enharmonic music.

The modern just intonationists have performed many notable and
instructive calculations involving the 5-limit, although their
doctrinaire insistence on the sacredness of the ratios, their obsession
with the 7-limit (and still more indefensibly, their musical
castles-in-the-air concerning the purely dissonant and quite
imperceptible 9-, 11-, 13-limits, etc.) and their refusal to seriously
consider the problems of adapting these ratios to practical music, has
vitiated much of their work. Still their approach is an honorable
antidote to the crudities of the dominant 12-tone-equal ethos.


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From: Gregg Gibson
Subject: Septimal Intervals
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Gary Morrison, at least, seems to be listening to the same chords
as me:

> That's interesting. I've always thought of 4:6:7 as fairly
> self-content, but 4:5:7 is most likely dissonant. (Then again dissonance
> is context-related whereas discordance is not, but that's a different
> discussion altogether.)

However, I did actually specify 31 equal, where the situation is
more complicated. In lieu of Gregg Gibson discovering why, I'll
respond to his other points:

> The perfect fifth 3:2 is comprised of the minor third 6:5 and major
> third 5:4. This in turn means that the consonant chords of the 3-limit
> and those of the 5-limit are congruent, that is to say, they can exist
> together without producing dissonant intervals.

Yeah, yeah, yeah. This is taking a feature of 5-limit and
turning it into a principle. I don't buy it.

> But when we reach the septimals, we find that the perfect fifth 3:2 is
> comprised of 7:6 and the undoubtedly dissonant 9:7. The other possible
> septimal bisection of the fifth, 8:7 and 21:16, also includes an
> undoubted dissonance (21:16) although the closeness of 8:7 makes it too,
> undoubtedly dissonant.

Yes, bisecting the fifth is a bad idea with septimals.

> Now a chord with one dissonant interval, is dissonant. Someone may try
> to deny this, but I am afraid he will fail.

I believe your fears are unjustified.

> There do exist a few septimal triads that seem to escape this argument,
> e.g. 5:6:7, 4:5:7, 4:6:7. But a few triads do not begin to compare with
> the richness and variety of the harmonies of the senario.

Exactly what are you trying to say, Gregg? You come up with an
"undeniable" argument, and then deny it! Or, am I
misunderstanding? To my ears, these chords are certainly more
concordant than their constituent dyads. Three examples are quite
enough to disprove that little theory.

What is "the senario(sic)" supposed to mean? If 5-limit
consonance, we are talking about 6 triads within the octave --
3:4:5, 4:5:6, 5:6:8, 1/3:1/4:1/5, 1/4:1/5:1/6 and 1/5:1/6:1/8.
Granted, septimal chords don't work well in inverted forms with an
8/7. However, I am prepared to admit the utonal versions. That
gives us ... 6 chords. So, we've doubled the number available
from the 5-limit.

> If follows that the septimals, even if one admits them to be weakly
> consonant - which I do _not_ - certainly give rise, for the most part,
> to undoubtedly dissonant chords. They have only the most tenuous
> relation with the harmony of the senario, which on the contrary, forms a
> closed, well-ordered system.

There is certainly doubt as to the first statement. Unless you
consider all chords containing a septimal interval, when it is
true but septimals are no different to quintals in this respect.
There is no question that the 7-limit is a different sonority to
the 5-limit. Why this is an argument for rejecting 7-limit
consonances is beyond me. Well-ordered systems are really
dreadfully early 20th Century.

> All this was observed centuries ago. Mersenne is full of highly
> ingenious arguments regarding the septimals - nihil sub sole novum.

My ignorance on the subject prevents me form getting into a
historical discussion.

> So today, some people still take it as a personal
> insult if an interval they fancy is called dissonant, and - very
> absurdly - try to make consonance and dissonance purely relative ideas
> with no basis in physics.

Certainly, consonance and dissonance are relative concepts. By
which I mean that a chord is only consonant or dissonant relative
to another chord. Why is this absurd?

> But I - just as about 99% of all the Western musical theorists who have
> ever lived - do really hate and wish to banish a certain narrow class of
> dissonances - those which are so close to consonances that they beat
> badly, but yet are melodically confounded with the consonances.

Well, that's Western musical theorists for you. Do these include
the people who advocated 12 note well temperaments?

We're clearly going to get nowhere in a consonance/dissonance
argument. I will, however, make the following assertions.
7-limit chords have a character which makes them closely akin to
5-limit consonances. 7-limit chords are sensitive to tuning and
so, if they are to be used, they must be considered in choosing
a temperament. A well tuned 4:6:7 chord fulfils all the usual
criteria of concordance. Some 5-limit chords, spread over more
than an octave, sound frankly dissonant -- see Helmholtz for more
on this. In the right circumstances, a 7-limit chord can function
as a consonance.


SMTPOriginator: tuning@eartha.mills.edu
From: gbreed@cix.compulink.co.uk (Graham Breed)
Subject: Pythagoreanism
PostedDate: 19-12-97 18:47:23
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