Modulatoy topology in 22-TET

Here are some thoughts about how to modulate in 22-TET, especially with
Paul Erlich's decatonic scales. (Recently there was a thread in the
newsgroup rec.music.theory - but Paul insisted that I write into this group, so
here I am.)
In rec.music.theory, Dr. Matt used the word "modulatory topology" - a really
great word ! It was not a priori clear what the word means, though - I give an
ad-hoc definition here.
Topology deals with "neighbourhoods" and "distances", and in case of
music, "modulatory topology" or "distance between tonalities" can e.g. mean
how easily a modulation between the tonalities goes or how dramatic it
sounds (C major and G major should have a smaller "distance" than C major
and E major or C major and Gb major.)

A very simple criterion for a distance between tonalities is how many chords
they have in common. Such chords are candidates to use in the "neutral
phase" of a modulation (the first phase according to Schönberg's modulation
model), moreover, there is overall less change (and hence less surprise and
dramatics) when the tonality changes

In case of the diatonic scale in 12-TET, two tonalities have the more chords
in common the closer they are along the circle of fifths. Common chords
exist for a distance of maximally 2 fifths in either direction, the largest
intersection of 4 common chords being reached by a distance of one fifth.
Hence, the circle of fifths can be seen as a natural visualization of the
"modulatory topology" of the diatonic triad system.

Now, how about ET22? All the concepts (they are, after all, quite simple, not
to say trivial) can be applied directly. I did it for Paul Erlich's pentachordal
decatonic scale - with a little surprising result!

Two transposes of the decatonic major scale at the distance of fifth (which in
ET 22 contains 13 steps) have - exactly like in case of the diatonic scale in
12-TET - 4 chords in common. However, this is not the maximum! The
maximum of 6 common chords is reached by distances of 2 (half tone) OR
11 steps (tritone). Hence the diagram for the modulatory topology of the
pentachordal decatonic scale in ET22 is not a circle of fifths, but a 2-
dimensional structure, best visualized as two concentric circles of 11 points
each (the half tone steps) with radial connections between the inner to the
outer (the tritone steps).
An interesting coincidence is that 2 and 11 happen to be the prime numbers
that compose 22, and the modulatory topology as above is also a
visualization of the decomposition of Z22 into Z2xZ11. A coincidence it is -
you can create whatever topology you like if you choose the basic chords
appropriately.

Any one of the tuning punks ever thought about this or even used it?

Hans Straub

--- In tuning-math@y..., "hs" <straub@d...> wrote:

> Now, how about ET22? All the concepts (they are, after all, quite
simple, not
> to say trivial) can be applied directly. I did it for Paul Erlich's
pentachordal
> decatonic scale - with a little surprising result!
>
> Two transposes of the decatonic major scale at the distance of
fifth (which in
> ET 22 contains 13 steps) have - exactly like in case of the
diatonic scale in
> 12-TET - 4 chords in common. However, this is not the maximum! The
> maximum of 6 common chords is reached by distances of 2 (half tone)
OR
> 11 steps (tritone). Hence the diagram for the modulatory topology
of the
> pentachordal decatonic scale in ET22 is not a circle of fifths, but
a 2-
> dimensional structure, best visualized as two concentric circles of
11 points
> each (the half tone steps) with radial connections between the
inner to the
> outer (the tritone steps).

thanks, hans! this topology is already implied in the appendix to my
paper, isn't it? just roll the page around so that the bottom meets
the top, and you have it!

>thanks, hans! this topology is already implied in the appendix to my
>paper, isn't it? just roll the page around so that the bottom meets
>the top, and you have it!

Oops - looks like I should get the habit of readiong reports to their _very_
end...

--- In tuning-math@y..., "hs" <straub@d...> wrote:
> >thanks, hans! this topology is already implied in the appendix to
my
> >paper, isn't it? just roll the page around so that the bottom meets
> >the top, and you have it!
>
> Oops - looks like I should get the habit of readiong reports to
their _very_
> end...

you can also see in that diagram how the *symmetrical* modes fit
nicely into the overall modulatory topology -- see it?

>> >thanks, hans! this topology is already implied in the appendix to my
>> >paper, isn't it? just roll the page around so that the bottom meets
>> >the top, and you have it!
>>
>> Oops - looks like I should get the habit of readiong reports to
>their _very_
>> end...
>
>you can also see in that diagram how the *symmetrical* modes fit
>nicely into the overall modulatory topology -- see it?

Errm... After having a closer look at the diagrams, I think not. The diagram
axes are fifth, major third and harmonic seventh: I see no hint that the
easiest modulation steps are semitone and tritone.
The dimension of the whole diagram has no importance in this aspect - in
ET12, you probably can find chord systems with easiest modulation steps
4 and 3 (I already found one with 4).
Or maybe there is a way of looking at the diagrams I havn't found yet?

--- In tuning-math@y..., "hs" <straub@d...> wrote:
> >> >thanks, hans! this topology is already implied in the appendix
to my
> >> >paper, isn't it? just roll the page around so that the bottom
meets
> >> >the top, and you have it!
> >>
> >> Oops - looks like I should get the habit of readiong reports to
> >their _very_
> >> end...
> >
> >you can also see in that diagram how the *symmetrical* modes fit
> >nicely into the overall modulatory topology -- see it?
>
> Errm... After having a closer look at the diagrams, I think not.
The diagram
> axes are fifth, major third and harmonic seventh:

no, not those diagrams!

>I see no hint that the
> easiest modulation steps are semitone and tritone.

i said the appendix!

> The dimension of the whole diagram has no importance in this
aspect - in
> ET12, you probably can find chord systems with easiest modulation
steps
> 4 and 3 (I already found one with 4).

that would be systems related to the augmented scale (3 1 3 1 3 1).
for easiest modulation step 3, try systems related to the diminished
scale (1 2 1 2 1 2 1 2).

>no, not those diagrams!
>
>>I see no hint that the
>> easiest modulation steps are semitone and tritone.
>
>i said the appendix!

OK - found it now. Indeed!

Hi!

The following is quoted from an old message #4498 that was posted
over a year ago to tuning-math by Hans Straub. Please read the
original message before proceeding.

So Hans Straub says:

> Now, how about ET22? All the concepts (they are, after all, quite
simple, not
> to say trivial) can be applied directly. I did it for Paul Erlich's
pentachordal
> decatonic scale - with a little surprising result!
>
> Two transposes of the decatonic major scale at the distance of
fifth (which in
> ET 22 contains 13 steps) have - exactly like in case of the
diatonic scale in
> 12-TET - 4 chords in common. However, this is not the maximum! The
> maximum of 6 common chords is reached by distances of 2 (half tone)
OR
> 11 steps (tritone). Hence the diagram for the modulatory topology
of the
> pentachordal decatonic scale in ET22 is not a circle of fifths, but
a 2-
> dimensional structure, best visualized as two concentric circles of
11 points
> each (the half tone steps) with radial connections between the
inner to the
> outer (the tritone steps).
> An interesting coincidence is that 2 and 11 happen to be the prime
numbers
> that compose 22, and the modulatory topology as above is also a
> visualization of the decomposition of Z22 into Z2xZ11. A
coincidence it is -
> you can create whatever topology you like if you choose the basic
chords
> appropriately.
>
> Any one of the tuning punks ever thought about this or even used it?
>
> Hans Straub

I don't understand this because I get 2 common chords with distance
of 13 steps, 3 with distance of 2 steps and 4 with distance of 11
steps!

Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> Hi!
>
> The following is quoted from an old message #4498 that was posted
> over a year ago to tuning-math by Hans Straub. Please read the
> original message before proceeding.
>
> So Hans Straub says:
>
> > Now, how about ET22? All the concepts (they are, after all, quite
> simple, not
> > to say trivial) can be applied directly. I did it for Paul
Erlich's
> pentachordal
> > decatonic scale - with a little surprising result!
> >
> > Two transposes of the decatonic major scale at the distance of
> fifth (which in
> > ET 22 contains 13 steps) have - exactly like in case of the
> diatonic scale in
> > 12-TET - 4 chords in common. However, this is not the maximum!
The
> > maximum of 6 common chords is reached by distances of 2 (half
tone)
> OR
> > 11 steps (tritone). Hence the diagram for the modulatory topology
> of the
> > pentachordal decatonic scale in ET22 is not a circle of fifths,
but
> a 2-
> > dimensional structure, best visualized as two concentric circles
of
> 11 points
> > each (the half tone steps) with radial connections between the
> inner to the
> > outer (the tritone steps).
> > An interesting coincidence is that 2 and 11 happen to be the
prime
> numbers
> > that compose 22, and the modulatory topology as above is also a
> > visualization of the decomposition of Z22 into Z2xZ11. A
> coincidence it is -
> > you can create whatever topology you like if you choose the basic
> chords
> > appropriately.
> >
> > Any one of the tuning punks ever thought about this or even used
it?
> >
> > Hans Straub
>
> I don't understand this because I get 2 common chords with distance
> of 13 steps, 3 with distance of 2 steps and 4 with distance of 11
> steps!
>
> Kalle

kalle, maybe you're thinking tetrads while hans was thinking of some
other kind of chord? thanks for pointing this out!

> kalle, maybe you're thinking tetrads while hans was thinking of
some
> other kind of chord? thanks for pointing this out!

Hi Paul, I think you're right. It's weird that Hans didn't mention
the chords he was thinking.

Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > kalle, maybe you're thinking tetrads while hans was thinking of
> some
> > other kind of chord? thanks for pointing this out!
>
> Hi Paul, I think you're right. It's weird that Hans didn't mention
> the chords he was thinking.
>

Uh, long time ago!
I did not mention the chords because Paul had already specified them -
tetrads, of course.
Hmm - you say you get a different result? I will have a look at Paul's
paper again to check.

OK, I checked it over now - and found out that Kalle is right.
In details:
Two pentachord major scales at the distance of 11 steps have 4
tetrads in common (I, IV, VI and IX), while two pentachord major
scales at the distance of 2 steps have 3 tetrads in common (one of
II/III, VI/VII and VIII/IX each) and two at at the distance of 9 or
13 (which, BTW, is 11 + 2 and 11 - 2) have 2 in common (one of I/VII
and IV/VIII each).

So, if we use only modulations following the "easiest" path, they do
not lead all around the circle of fifths as in Z12 but oscillate
forever between just 2 tonalities...

But if we include the next-easiest step, we still get the 2-
dimensional structure of Z11 x Z2 I described! (Just not with equal
distances - the tritone distance is smaller.)

--- In tuning-math@yahoogroups.com, "hstraub64" <straub@d...> wrote:
> OK, I checked it over now - and found out that Kalle is right.
> In details:
> Two pentachord major scales at the distance of 11 steps have 4
> tetrads in common (I, IV, VI and IX), while two pentachord major
> scales at the distance of 2 steps have 3 tetrads in common (one of
> II/III, VI/VII and VIII/IX each) and two at at the distance of 9 or
> 13 (which, BTW, is 11 + 2 and 11 - 2) have 2 in common (one of I/VII
> and IV/VIII each).
>
> So, if we use only modulations following the "easiest" path, they do
> not lead all around the circle of fifths as in Z12 but oscillate
> forever between just 2 tonalities...
>
> But if we include the next-easiest step, we still get the 2-
> dimensional structure of Z11 x Z2 I described! (Just not with equal
> distances - the tritone distance is smaller.)

you could also look at the number of *notes* in common, and get back
to the two distances being the same.