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.