[tuning-math] Modulatoy topology in 22-TET
- From: "hs" <straub@xxxxxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Sun, 30 Jun 2002 13:50:41 +0200
Here are some thoughts about how to modulate in 22-TET, especially with=20
Paul Erlich's decatonic scales. (Recently there was a thread in the=20
newsgroup rec.music.theory - but Paul insisted that I write into this group=
, so=20
here I am.)
In rec.music.theory, Dr. Matt used the word "modulatory topology" - a reall=
y=20
great word ! It was not a priori clear what the word means, though - I give=
an=20
ad-hoc definition here.=20
Topology deals with "neighbourhoods" and "distances", and in case of=20
music, "modulatory topology" or "distance between tonalities" can e.g. mean=
=20
how easily a modulation between the tonalities goes or how dramatic it=20
sounds (C major and G major should have a smaller "distance" than C major=20
and E major or C major and Gb major.)
A very simple criterion for a distance between tonalities is how many chord=
s=20
they have in common. Such chords are candidates to use in the "neutral=20
phase" of a modulation (the first phase according to Sch=F6nberg's modulati=
on=20
model), moreover, there is overall less change (and hence less surprise and=
=20
dramatics) when the tonality changes
In case of the diatonic scale in 12-TET, two tonalities have the more chord=
s=20
in common the closer they are along the circle of fifths. Common chords=20
exist for a distance of maximally 2 fifths in either direction, the largest=
=20
intersection of 4 common chords being reached by a distance of one fifth.=20
Hence, the circle of fifths can be seen as a natural visualization of the=20
"modulatory topology" of the diatonic triad system.
Now, how about ET22? All the concepts (they are, after all, quite simple, n=
ot=20
to say trivial) can be applied directly. I did it for Paul Erlich's pentach=
ordal=20
decatonic scale - with a little surprising result!
Two transposes of the decatonic major scale at the distance of fifth (which=
in=20
ET 22 contains 13 steps) have - exactly like in case of the diatonic scale =
in=20
12-TET - 4 chords in common. However, this is not the maximum! The=20
maximum of 6 common chords is reached by distances of 2 (half tone) OR=20
11 steps (tritone). Hence the diagram for the modulatory topology of the=20
pentachordal decatonic scale in ET22 is not a circle of fifths, but a 2-
dimensional structure, best visualized as two concentric circles of 11 poin=
ts=20
each (the half tone steps) with radial connections between the inner to the=
=20
outer (the tritone steps).
An interesting coincidence is that 2 and 11 happen to be the prime numbers=
=20
that compose 22, and the modulatory topology as above is also a=20
visualization of the decomposition of Z22 into Z2xZ11. A coincidence it is =
-=20
you can create whatever topology you like if you choose the basic chords=20
appropriately.
Any one of the tuning punks ever thought about this or even used it?
Hans Straub
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