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

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