Re: Tetrachordal alterations

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.