I've ignore tetrachordality for too long, it's time I figured out how to

make it happen in any linear temperament. Paul, I had hoped you had more of

a handle on it than I did. It seems not.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > Are "tetrachordal alterations" only possible when the interval of

> > repetition is some whole-number fraction of octave?

>

> In an MOS, the interval of repetition is _always_ some whole-number

> fraction of the interval of equivalence.

I meant "fraction" in the popular sense, as not including the whole. But

no, I now believe tetrachordal (not necc. omnitetrachordal) alterations are

always possible, it's just a question of finding the minimal alteration

that makes it tetrachordal and just how big that alteration turns out to be

in each case.

It's not clear to me why a tetrachordal scale in some linear temperament,

must have the same number of notes as a MOS in that temperament, but I'll

assume it for now. Know any good reason(s)?

I'd measure the magnitude of a particular alteration (the alteredness?) by

how many notes would have to be added to the original MOS, by consecutive

generators, keeping all chains the same, before you can contain the

tetrachordal scale. e.g. Your tetrachordal decatonic has an alteredness of

2, the minimum possible with a half-octave period.

> > How do you do them, in general?

>

> I don't know if there's a general way, but you understand what

> omnitetrachorality is, right?

Yes. But I wasn't considering _omni_tetrachordality. That looks too hard

for now. Do any popular or historical scales have it?

> "Alteration" simply means re-shuffling

> the step sizes in an MOS or hyper-MOS.

> >

> > What would be a "tetrachordal alteration" of Blackjack?

>

> Don't know if there is one! Can you make a blackjack-like scale

> omnitetrachordal?

God knows!

But I had previously failed to appreciate that Blackjack (a 21 note chain

of Miracle generators = secors = 116.7c) is tetrachordal. duh! You get

disjunct 9-step tetrachords when you start from the point of symmetry.

Ok. Lets look at the 10 note Miracle MOS. What's the minimum alteration to

make it tetrachordal. This just amounts to asking: What is the 10 note

tetrachordal Miracle scale that spans the fewest secors.

I'll notate my tetrachords as the conventional C...F and G...C. Since the

tetrachords must share exactly one note (C) and must be melodically

identical, a 10-note tetrachordal scale must have 5 notes in each

tetrachord and 1 note outside the tetrachords. (The possibility of 3 notes

outside the tetrachord can be ignored because even Blackjack has only 2 and

we expect to get out of it with fewer generators than Blackjack).

Here's what we start with

5 1

C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C<

5 1

That's a long enough chain of secors (Blackjack) where we will number the

notes of each tetrachord of our 10-note scale in pitch order. One

tetrachord above and one below. We'll then mark the note that's outside the

tetrachords with an "X". We'll try to stay as close to the center of the

chain as possible.

Note that X can go on either Gb^ or F#v. So here are eight minimal

possibilities. They all span 16 notes in the chain.

5 1 2 3 4

C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C<

X 5 X 1 2 3 4

4 5 1 2 3

C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C<

X 4 5 X 1 2 3

3 4 5 1 2

C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C<

X 3 4 5 X 1 2

2 3 4 5 1

C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C<

X 2 3 4 5 X 1

Actually, not all eight are distinct. Some are simply transposed. There are

really only 5. 3 of which are disjunct-tetrachordal in 2 positions.

Here's what they look like melodically (in steps of 72-tET). Vertical bar

"|" is used to show tetrachords.

7 7 7 9|5 7|7 7 7 9

7 7 9 7|5 7|7 7 9 7 7 7 7 9|7 5|7 7 7 9

7 9 7 7|5 7|7 9 7 7 7 7 9 7|7 5|7 7 9 7

9 7 7 7|5 7|9 7 7 7 7 9 7 7|7 5|7 9 7 7

9 7 7 7|7 5|9 7 7 7

The middle one (the most even) looks like a very interesting detempering of

your tetrachordal (pentachordal) decatonic.

It has

2 of 4:5:6:7

2 of 1/(7:6:5:4)

2 of 4:5:6

2 of 1/(6:5:4)

2 of 3:7:9:21 ASS

1 of 3:9:11:33 ASS

and probably some necessarily-tempered ASSes.

But I don't see any sense in which any of these tetrachordal scales are an

"alteration" of the 10 note MOS. It seems to me that tetrachordal scale

creation (in a given temperament) is not related to MOS scale creation (in

the same temperament) in any way.

Here's the most compact (on the chain) inversionally-symmetric 7-note

tetrachordal scale in Miracle.

3 4 1 2

C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C<

3 4 1 2

Steps 7 16 7|12|7 16 7

Here's the most even one that fits in Blackjack.

2 4 1 3

C> D[ Eb< Ev F Gb^ G> A[ Bb< Bv C Db^ D> E[ F< F#v G Ab^ A> B[ C<

2 4 1 3

Steps 9 12 9|12|9 12 9

Which is of course the neutral thirds MOS, or Mohajira.

These are even less of an alteration of a MOS.

-- Dave Keenan

Brisbane, Australia

http://dkeenan.com

--- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote:

> I've ignore tetrachordality for too long, it's time I figured out

how to

> make it happen in any linear temperament. Paul, I had hoped you had

more of

> a handle on it than I did. It seems not.

>

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > > Are "tetrachordal alterations" only possible when the interval

of

> > > repetition is some whole-number fraction of octave?

> >

> > In an MOS, the interval of repetition is _always_ some whole-

number

> > fraction of the interval of equivalence.

>

> I meant "fraction" in the popular sense, as not including the

whole. But

> no, I now believe tetrachordal (not necc. omnitetrachordal)

alterations are

> always possible, it's just a question of finding the minimal

alteration

> that makes it tetrachordal and just how big that alteration turns

out to be

> in each case.

>

> It's not clear to me why a tetrachordal scale in some linear

temperament,

> must have the same number of notes as a MOS in that temperament,

but I'll

> assume it for now. Know any good reason(s)?

Because the unison vectors define a set of N equivalence classes,

which we typically map to letters of the alphabet, etc. Melodies, and

even harmonies, will be understood in terms of patterns in this set

of N. If you add or eliminate notes, you'll be ruining these patterns.

In other words, the tetrachordal scale should be a periodicity block

(not necessarily of the Fokker parallelogram type).

>

> Yes. But I wasn't considering _omni_tetrachordality. That looks too

hard

> for now. Do any popular or historical scales have it?

The most popular ones do -- the diatonic and pentatonic.

>

> Here's what they look like melodically (in steps of 72-tET).

Vertical bar

> "|" is used to show tetrachords.

>

> 7 7 7 9|5 7|7 7 7 9

> 7 7 9 7|5 7|7 7 9 7 7 7 7 9|7 5|7 7 7 9

> 7 9 7 7|5 7|7 9 7 7 7 7 9 7|7 5|7 7 9 7

> 9 7 7 7|5 7|9 7 7 7 7 9 7 7|7 5|7 9 7 7

> 9 7 7 7|7 5|9 7 7 7

I might not call these tetrachordal. According to my paper, the

disjunction is supposed to contain some pattern of scale steps found

with the tetrachords themselves. But in this case, the 5/72 interval

isn't found in the tetrachords.

Perhaps we can call these "weakly tetrachordal".

> But I don't see any sense in which any of these tetrachordal scales

are an

> "alteration" of the 10 note MOS.

Each note should either agree with the corresponding note in the MOS

or be a chromatic unison vector different.

> It seems to me that tetrachordal scale

> creation (in a given temperament) is not related to MOS scale

creation (in

> the same temperament) in any way.

I see the periodicity block concept as underlying both.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., David C Keenan <D.KEENAN@U...> wrote:

> In other words, the tetrachordal scale should be a periodicity block

> (not necessarily of the Fokker parallelogram type).

Ok. That makes sense.

> > Yes. But I wasn't considering _omni_tetrachordality. That looks

too

> hard

> > for now. Do any popular or historical scales have it?

>

> The most popular ones do -- the diatonic and pentatonic.

Oh yeah. Silly me.

> > Here's what they look like melodically (in steps of 72-tET).

> Vertical bar

> > "|" is used to show tetrachords.

> >

> > 7 7 7 9|5 7|7 7 7 9

> > 7 7 9 7|5 7|7 7 9 7 7 7 7 9|7 5|7 7 7 9

> > 7 9 7 7|5 7|7 9 7 7 7 7 9 7|7 5|7 7 9 7

> > 9 7 7 7|5 7|9 7 7 7 7 9 7 7|7 5|7 9 7 7

> > 9 7 7 7|7 5|9 7 7 7

>

> I might not call these tetrachordal. According to my paper, the

> disjunction is supposed to contain some pattern of scale steps found

> with the tetrachords themselves. But in this case, the 5/72 interval

> isn't found in the tetrachords.

>

> Perhaps we can call these "weakly tetrachordal".

Hmm. So in the case where the disjunction consists of a single step,

you insist that that step (always an approx 8:9) appears in the

tetrachords, before calling a scale tetrachordal?

Isn't this just an additional desirable property because it tends to

make the scale tetrachordal in more rotations.

"Weakly tetrachordal" is still tetrachordal, right? Wouldn't it be

better if we simply said "tetrachordal in n rotations". Of course the

minimum for n is 3 (other than 0). And when n is the number of notes

in the scale it is omnitetrachordal.

I'll abbreviate both tetrachord and tetrachordal as "Tc" in future I'm

getting tired of typing it.

I suppose one could have the disjunction containing only intervals

from the Tc but still only have 3 Tc rotations. But is the important

thing about this the minimising of different step sizes, rather than

having the steps in the disjunction the same as in the Tc?

> > But I don't see any sense in which any of these tetrachordal

scales

> are an

> > "alteration" of the 10 note MOS.

>

> Each note should either agree with the corresponding note in the MOS

> or be a chromatic unison vector different.

Right. So my alteredness metric was wrong (based on width on the

chains) it should instead be melodically based. e.g. How many notes

differ from the MOS, and by how many cents.

> I see the periodicity block concept as underlying both.

Yes. I get it now.

-- Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > I might not call these tetrachordal. According to my paper, the

> > disjunction is supposed to contain some pattern of scale steps

found

> > with the tetrachords themselves. But in this case, the 5/72

interval

> > isn't found in the tetrachords.

> >

> > Perhaps we can call these "weakly tetrachordal".

>

> Hmm. So in the case where the disjunction consists of a single

step,

> you insist that that step (always an approx 8:9) appears in the

> tetrachords, before calling a scale tetrachordal?

Right.

>

> Isn't this just an additional desirable property because it tends

to

> make the scale tetrachordal in more rotations.

No.

>

> "Weakly tetrachordal" is still tetrachordal, right?

Weakly.

> Wouldn't it be

> better if we simply said "tetrachordal in n rotations". Of course

the

> minimum for n is 3 (other than 0). And when n is the number of

notes

> in the scale it is omnitetrachordal.

This can all still be "weakly" or "strongly".

>

> I'll abbreviate both tetrachord and tetrachordal as "Tc" in future

I'm

> getting tired of typing it.

>

> I suppose one could have the disjunction containing only intervals

> from the Tc but still only have 3 Tc rotations.

Yes, like Mohajira.

> But is the important

> thing about this the minimising of different step sizes, rather

than

> having the steps in the disjunction the same as in the Tc?

I think the disjuction should smoothly connect the tetrachords,

rather than sounding "different".

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > Hmm. So in the case where the disjunction consists of a single

> step,

> > you insist that that step (always an approx 8:9) appears in the

> > tetrachords, before calling a scale tetrachordal?

>

> Right.

Does John Chalmers agree with this?

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > > Hmm. So in the case where the disjunction consists of a single

> > step,

> > > you insist that that step (always an approx 8:9) appears in the

> > > tetrachords, before calling a scale tetrachordal?

> >

> > Right.

>

> Does John Chalmers agree with this?

Oops! OK, let's call his "tetrachordal" and mine "strongly

tetrachordal" or something.