Decimal lattices

As promised, here's Blackjack on my 7-limit/neutral-third lattice

0^
/
/
2^----5^
/ \/
/ /\
1^----4^-/--7^-\--0
/ \ / / \
/ \ / / \
0^----3^----6^----9^----2-----5
/ \/ \/ /
/ /\ /\ /
2^----5^-/--8^-\--1--/--4-----7-----0v
/ \ / / \ /
/ \ / / \ /
1^----4^----7^----0-----3-----6-----9
/ \/ \/ /
/ /\ /\ /
0^----3^----6^-/--9^ \--2--/--5-----8
/ \ / / \ /
/ \ / / \ /
5^----8^----1-----4-----7-----0v
\/ \/ /
/\ /\ /
0--\--3--/--6-----9
\/ /
/\ /
5-----8
/
/
0v

Not all points are connected up to avoid clutter. Here's the 31 note
(Canasta?) scale:

0^

2^--5^
/ \
1^ 4^/ 7^\ 0
/ \
0^--3^--6^--9^--2---5
/ \ / \ /
2^ 5^/ 8^\ 1 / 4 \ 7 / 0v
/ \ / \ /
1^--4^--7^--0---3---6---9---2v--5v
/ \ / \ / \ /
0^ 3^ 6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v 0w
/ \ / \ / \ /
5^--8^--1---4---7---0v--3v--6v--9v
\ / \ / \ /
0 \ 3 / 6 \ 9 / 2v\ 5v/ 8v
\ / \ / \ /
5---8---1v--4v--7v--0w
/
0v/ 3v 6v 9v
/
5v--8v

0w

If you took away all the lines, you'd be left with a grid. Hence this
really is a lattice! A step to the right is 3s, a step down is 5s.

The following is the 11-limit tonality diamond around 5

0^
/
3^^-6^^-----2^/
/ \ / \
/ 1^\---/-7^\-0
/ \ / \
0^--3^--6^--9^--2---5
/ \ / \ /
3^^-6^^-----2^ / 8^\ / \ 7-/-0v
\ / / \ / \ / /
8^^-1^------7^--0---3---6-------2v/
\ / / \ / \ / /
0^--3^--6^--9^--2 / 5 \ 8 / 1v\ 4v/-7v--0w
/ \ / / \ / \ /
/ 8^\ / 7---0v--3v------9v
/ \ / / \ /
0---3---6 / 2v\ / 8v------4w
/ \ / \ /
5---8---1v--4v--7v--0w
/ \ /
0v--3v------9v/
/ /
/ 8v------4w
/
0w

That would have been clearer if I'd connected up the 5 in the middle,
but you'll take what you're given and be grateful. This should be a 43
note Partch scale with 5 as the 1/1:

4m^-----0^^
\ /
\ /
\ /
4^^-7^^-0^
/ \
4m^-----0^^-3^^-6^^-----2^\
\ / \ / \ / / \ \
\ / \8^^/ 1^\ 4^/-7^\-0
\ / \ / \ / \
4^^-7^^-0^--3^--6^--9^--2---5
/ \ / \ / \ /
0^^-3^^-6^^---/-2^\ / 8^\ 1 / 4 \ 7 / 0v
/ / \ / \ / \ / /
/ 8^^-1^--4^--7^--0---3---6---9---2v/
/ \ / \ / \ / \ / /
0^--3^\ 6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v--0w
\ \ / \ / \ / \ / / \
\ 8^--1---4---7---0v--3v--6v--9v/ \
\ / \ / \ / \ / \
0---3 / 6 \ 9 / 2v\ / 8v\-----4w--7w--0vw
/ / \ / \ / \
5---8---1v--4v--7v--0w--3w--6w
/ \ / / \
0v--3v--6v--9v\ / / \
\ / \ / / \
\ 8v------4w--7w--0vw-----6vw
\ /
0w--3w--6w
/ \
/ \
/ \
0vw-----6vw

Harry didn't use the same 43 notes for his whole life. I don't know where
I copied this from, probably Genesis. It'd be interesting to see if
another version fills in more of the gaps.

7vv (11/10) is lower than 6^^ (10/9) and 3vv (9/5) is lower than 2^^
(20/11). These the only exceptions to the rule that any 1 is lower than
any 2, any 2 is lower than any 3, etc (assuming you get the octaves
right). They're also the only enharmonic pairs in 41=.

Graham

--- In tuning@y..., graham@m... wrote:>
>
> Harry didn't use the same 43 notes for his whole life. I don't
know where
> I copied this from, probably Genesis. It'd be interesting to see
if
> another version fills in more of the gaps.

As far as I know, Partch never proposed a different 43-tone scale.
Earlier in his life he had a 29-tone scale (just the 11-limit
Tonality Diamond) and then a 37-note scale briefly. Later he objected
that people put too much importance on the number 43, it was just an
arbitrary place to stop in the potential infinitude of 11-limit JI,
which was itself kind of an arbitrary place to stop (though Partch
does seem to feel that 25-limit would be too high for any kind of
consonance).

>
> 7vv (11/10) is lower than 6^^ (10/9) and 3vv (9/5) is lower than
2^^
> (20/11). These the only exceptions to the rule that any 1 is lower
than
> any 2, any 2 is lower than any 3, etc (assuming you get the octaves
> right). They're also the only enharmonic pairs in 41=.
>
Exactly. Which is why Wilson mapped each of these pairs to a single
key in his keyboard program for Partch's scale. Partch's 43-tone
scale is a 41-tone periodicity block with a unison-vector variant
allowed for two of the tones -- see
http://www.ixpres.com/interval/td/erlich/partchpblock.htm

Paul wrote:

> As far as I know, Partch never proposed a different 43-tone scale.
> Earlier in his life he had a 29-tone scale (just the 11-limit
> Tonality Diamond) and then a 37-note scale briefly. Later he
objected
> that people put too much importance on the number 43, it was just an
> arbitrary place to stop in the potential infinitude of 11-limit JI,
> which was itself kind of an arbitrary place to stop (though Partch
> does seem to feel that 25-limit would be too high for any kind of
> consonance).

I thought he chose a different 43 notes in a later essay. But as
nobody's posted to say so yet, maybe not. Well, the 29 notes I've
done, which notes were "missing" in the 37? The 43 aren't quite
arbitrary because of the approximation to an equal interval scale,
hence the periodicity block.

> > 7vv (11/10) is lower than 6^^ (10/9) and 3vv (9/5) is lower than
> 2^^
> > (20/11). These the only exceptions to the rule that any 1 is
lower
> than
> > any 2, any 2 is lower than any 3, etc (assuming you get the
octaves
> > right). They're also the only enharmonic pairs in 41=.
> >
> Exactly. Which is why Wilson mapped each of these pairs to a single
> key in his keyboard program for Partch's scale. Partch's 43-tone
> scale is a 41-tone periodicity block with a unison-vector variant
> allowed for two of the tones -- see
> http://www.ixpres.com/interval/td/erlich/partchpblock.htm

That's the page I was thinking of! Thanks.

These are the unison vectors you settled on there:

(-4 0 1 -1)
( 2 -1 2 -1)
(-5 1 2 0)
(-2 2 0 -1)

Here's a translation from some vectors consistent with the 41-note
MOS:

(1 0 0 1)(-2 2 0 -1)
(0 1 1 1)(-1 1 1 1)
(1 1 0 1)( 5 0 0 -2)
(1 0 0 0)(-2 -2 1 0)

The top one is 100:99, the others I gave before for the 31/41/72
family.

Incidentally, I thought before that 100:99 approximated to "q" in
decimal terms, which is wrong. It's actually 4q-s.

Here's a translation from some more MOS-consistent vectors

(1 0 0 1)(-2 -2 1 0)
(0 1 0 0)( 2 -1 2 -1)
(0 0 1 1)(-3 -1 2 1)
(0 0 0 1)(-2 2 0 -1)

I've moved the 100:99 to the bottom to match your ordering. The other
three are still general to the 31/41/72 family.

So one of your unison vectors also works for the "miracle" scale. The
other two that aren't 100:99 can be divided by 100:99 to become
vectors that do work.

I'm note sure if this means your unison vectors are unusually close to
the 31/41/72 family. Any set of 41-note unison vectors can be
transformed into any other set. And I had a lot of freedom in
choosing my other three to make this simple.

I'm also not sure that this has anything to say about how close
Partch's scale is to this 41-note MOS. The lattice I gave before is
enough to show they don't match. But I'm very dubious about the
connection between any given periodicity block and a particular set of
unison vectors.

On that web page you say "translating a note of a periodicity block by
one or two unison vectors does not change its important properties".
But as any set of unison vectors can be generated from any other, this
seems to mean the "important properties" are independent of any
particular choice. So being a periodicity block simply means
approximating to the relevant equal tuning with no notes missing and
none duplicated? That should be obvious from Wilson's keyboard
layout.

Erv's paper is in Xenharmonikon 3, part 2 at
<http://www.anaphoria.com/wilson.html>. So does the scale approximate
to an exact fifth-based MOS? It seems to be a better match than to
the 31/41/72 family, at any rate.

Graham

Erv refers to it as a 41-tone Constant structure with two alternates. No need for it to be any
more complex than that

Graham Breed wrote:

> > Exactly. Which is why Wilson mapped each of these pairs to a single
> > key in his keyboard program for Partch's scale. Partch's 43-tone
> > scale is a 41-tone periodicity block with a unison-vector variant
> > allowed for two of the tones --
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., "Graham Breed" <graham@m...> wrote:

> > http://www.ixpres.com/interval/td/erlich/partchpblock.htm
>
> That's the page I was thinking of! Thanks.
>
>
> These are the unison vectors you settled on there:
>
> (-4 0 1 -1)
> ( 2 -1 2 -1)
> (-5 1 2 0)
> (-2 2 0 -1)
>
>
> Here's a translation from some vectors consistent with the 41-note
> MOS:

What does that mean?
>
> (1 0 0 1)(-2 2 0 -1)
> (0 1 1 1)(-1 1 1 1)
> (1 1 0 1)( 5 0 0 -2)
> (1 0 0 0)(-2 -2 1 0)
>
> The top one is 100:99, the others I gave before for the 31/41/72
> family.
>
> Incidentally, I thought before that 100:99 approximated to "q" in
> decimal terms, which is wrong. It's actually 4q-s.
>
> Here's a translation from some more MOS-consistent vectors

???
>
> (1 0 0 1)(-2 -2 1 0)
> (0 1 0 0)( 2 -1 2 -1)
> (0 0 1 1)(-3 -1 2 1)
> (0 0 0 1)(-2 2 0 -1)
>
> I've moved the 100:99 to the bottom to match your ordering. The
other
> three are still general to the 31/41/72 family.
>
> So one of your unison vectors also works for the "miracle" scale.
The
> other two that aren't 100:99 can be divided by 100:99 to become
> vectors that do work.
>
> I'm note sure if this means your unison vectors are unusually close
to
> the 31/41/72 family.

Don't know what that would mean.

> Any set of 41-note unison vectors can be
> transformed into any other set.

Within certain reasonable limits, yes.

> And I had a lot of freedom in
> choosing my other three to make this simple.
>
> I'm also not sure that this has anything to say about how close
> Partch's scale is to this 41-note MOS.

Which 41-note MOS? The one generated by the amazing generator?

> The lattice I gave before is
> enough to show they don't match.

If the answer to the last question was "yes", the answer to this
question is "well, I'm not surprised".

> But I'm very dubious about the
> connection between any given periodicity block and a particular set
of
> unison vectors.

Huh?
>
> On that web page you say "translating a note of a periodicity block
by
> one or two unison vectors does not change its important properties".
> But as any set of unison vectors can be generated from any other,
this
> seems to mean the "important properties" are independent of any
> particular choice. So being a periodicity block simply means
> approximating to the relevant equal tuning with no notes missing and
> none duplicated?

Often but not always. For example the schisma and diesis as unison
vectors give a 24-tone PB which approximates 12-tET and not 24-tET.

> That should be obvious from Wilson's keyboard
> layout.

What should be obvious?
>
> Erv's paper is in Xenharmonikon 3, part 2 at
> <http://www.anaphoria.com/wilson.html>. So does the scale
approximate
> to an exact fifth-based MOS? It seems to be a better match than to
> the 31/41/72 family, at any rate.

You mean a better match than to the amazing generator based MOS?

Paul wrote:

> > These are the unison vectors you settled on there:
> >
> > (-4 0 1 -1)
> > ( 2 -1 2 -1)
> > (-5 1 2 0)
> > (-2 2 0 -1)
> >
> >
> > Here's a translation from some vectors consistent with the 41-note
> > MOS:
>
> What does that mean?

Should have been "transformation" rather than "translation" here and
below. It means you multiply two matrices together to get a third.

> > (1 0 0 1)(-2 2 0 -1)
> > (0 1 1 1)(-1 1 1 1)
> > (1 1 0 1)( 5 0 0 -2)
> > (1 0 0 0)(-2 -2 1 0)

> > Here's a translation from some more MOS-consistent vectors
>
> ???

This is a simple one. The second down and the one on the bottom stay.
Add the bottom one to the top one and the second one up.

> > (1 0 0 1)(-2 -2 1 0)
> > (0 1 0 0)( 2 -1 2 -1)
> > (0 0 1 1)(-3 -1 2 1)
> > (0 0 0 1)(-2 2 0 -1)

> > I'm note sure if this means your unison vectors are unusually close
> to
> > the 31/41/72 family.
>
> Don't know what that would mean.

It's a simple transformation matrix: all ones and zeros, mostly zeros.
Is this unusual? I don't know.

> > Any set of 41-note unison vectors can be
> > transformed into any other set.
>
> Within certain reasonable limits, yes.

What limits?

> > And I had a lot of freedom in
> > choosing my other three to make this simple.
> >
> > I'm also not sure that this has anything to say about how close
> > Partch's scale is to this 41-note MOS.
>
> Which 41-note MOS? The one generated by the amazing generator?

Yes.

> > The lattice I gave before is
> > enough to show they don't match.
>
> If the answer to the last question was "yes", the answer to this
> question is "well, I'm not surprised".

Good, at least somebody's understanding something.

> > But I'm very dubious about the
> > connection between any given periodicity block and a particular set
> of
> > unison vectors.
>
> Huh?

Fokker says "here's a periodicity block for these unison vectors" but
there's nothing I can see to match the one to the other, except that the
numbers of notes have to match.

> > On that web page you say "translating a note of a periodicity block
> by
> > one or two unison vectors does not change its important properties".
> > But as any set of unison vectors can be generated from any other,
> this
> > seems to mean the "important properties" are independent of any
> > particular choice. So being a periodicity block simply means
> > approximating to the relevant equal tuning with no notes missing and
> > none duplicated?
>
> Often but not always. For example the schisma and diesis as unison
> vectors give a 24-tone PB which approximates 12-tET and not 24-tET.

Hmm. I'd say that was a 12-tone PB where the determinant gave the wrong
value.

> > That should be obvious from Wilson's keyboard
> > layout.
>
> What should be obvious?

Partch's scale approximates 41-equal with no notes missing and two
duplications.

> > Erv's paper is in Xenharmonikon 3, part 2 at
> > <http://www.anaphoria.com/wilson.html>. So does the scale
> approximate
> > to an exact fifth-based MOS? It seems to be a better match than to
> > the 31/41/72 family, at any rate.
>
> You mean a better match than to the amazing generator based MOS?

Yes.

Now for some more lattices. Here's a corrected and expanded 11-limit
diamond

0^
/
3^^-6^^-----2^/
\ / /
8^^-1^------7^--0
/ \ / \
0^--3^/-6^--9^--2-\-5
/ \
3^^-6^^-----2^------8^ 7---0v
/ \ / \ / \ /
8^^/ 1^\ / 7^\ 0 / 3 \ 6 2v/
/ \ / \ / \ /
0^--3^--6^--9^--2---5---8---1v--4v--7v--0w
/ \ / \ / \ / \ /
3^^-6^^-----2^ / 8^\ / \ 7 / 0v\ 3v/ \ 9v/
\ / / \ / \ / \ / \ /
8^^-1^------7^--0---3---6-------2v------8v------4w
\ / / \ / \ /
3^--6^--9^--2 / 5 \ 8 / 1v\ 4v/ 7v 0w
/ \ / \ /
8^ 7---0v--3v------9v 2w
/ \ /
3---6 / 2v\ / 8v------4w
/ \ / \ /
8---1v--4v--7v--0w
/ \ /
3v------9v/ 2w
/
8v------4w

Here's the similar for the Partch 43 note scale:

4m^-0^^
\
\
\
4^^-7^^
/ \
4m^-----0^^-3^^-6^^ \ 2^
\ / \ / \
\ / \ 8^^-1^--4^--7^
\ / \ / \ /
4^^-7^^-0^--3^/ 6^\ 9^/ 2
/ \ /
4m^-----0^^-3^^-6^^-----2^------8^--1---4---7
\ / \ / \ / \ / \ / \
\ / \8^^/ 1^\ 4^/ 7^\ 0 / 3 \ 6 / 9 \ 2v
\ / \ / \ / \ / \ / \
4^^-7^^-0^--3^--6^--9^--2---5---8---1v--4v--7v
/ \ / \ / \ / \ / \ /
0^^-3^^-6^^---/-2^\ / 8^\ 1 / 4 \ 7 / 0v\ 3v/ 6v\ 9v/ 2w
/ / \ / \ / \ / / \ \ / \ /
/ 8^^-1^--4^--7^--0---3---6---9---2v------8v------4w--7w
/ \ / \ / \ / \ / / \
0^ 3^\ 6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v--0w--3w--6w
\ \ / \ / \ / \ / / \ /
\ 8^--1---4---7---0v--3v--6v--9v--2w\ /
\ / \ / \ / \ / / \ /
0---3 / 6 \ 9 / 2v\ / 8v\ / 4w--7w--0vw
/ \ / \ / \ /
5---8---1v--4v--7v--0w--3w--6w
\ / / \
0v--3v--6v--9v\ 2w/ / \
\ / \ / / \
\ 8v------4w--7w--0vw-----6vw
\ /
0w--3w--6w
/ \
/ \
/ \
0vw-----6vw

This is a variation with 27/20 swapped for 15/11, 81/80 for 45/44, 40/27
for 22/15 and 160/81 for 88/45. All ratios are simpler than the ones they
substitute for, and fifths are preserved. It means taking them from 11/10
instead of 10/9.

4^^-7^^
/ \
3^^ 6^^ 9^^ 2^
/ \
8^^-1^--4^--7^
/ \ /
4^^-7^^-0^--3^/ 6^\ 9^/ 2
/ \ /
3^^-6^^-9^^-2^--5---8^--1---4---7
/ \ / \ / \ / \
8^^/ 1^\ 4^/ 7^\ 0 / 3 \ 6 / 9 \ 2v
/ \ / \ / \ / \
4^^-7^^-0^--3^--6^--9^--2---5---8---1v--4v--7v
/ \ / \ / \ / \ / \ /
3^^-6^^-9^^-2^\ 5^/ 8^\ 1 / 4 \ 7 / 0v\ 3v/ 6v\ 9v/ 2w
/ / \ / \ / \ / \ / \ /
/ 8^^-1^--4^--7^--0---3---6---9---2v--5v--8v--1w--4w--7w
/ / \ / \ / \ / \ /
0^/ 3^\ 6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v--0w--3w--6w
/ \ / \ / \ / \ / /
5^--8^--1---4---7---0v--3v--6v--9v--2w
\ / \ / \ / \ / /
0 \ 3 / 6 \ 9 / 2v\ 5v/ 8v\ 1w/ 4w--7w
\ / \ / \ / \ /
5---8---1v--4v--7v--0w--3w--6w
/ \ / \ /
0v/ 3v\ 6v/ 9v\ 2w/
/ \ / \ /
5v--8v--1w--4w--7w
\ /
0w--3w--6w

That's more compact, and all notes are contained within 45 generated
notes: the minimum to contain the 11-limit diamond. It still doesn't
contain the 41-note MOS. I think it does if you swap 4^^ and 6w for 5^^
and 5w.

7^^

(3^^)6^^-9^^-2^
/
/ 8^^-1^--4^--7^
/ / \ /
7^^-0^--3^/ 6^\ 9^/ 2
/ \ /
(3^^)6^^-9^^-2^--5---8^--1---4---7
/ \ / \ / \ / \
8^^/ 1^\ 4^/ 7^\ 0 / 3 \ 6 / 9 \ 2v
/ \ / \ / \ / \
7^^-0^--3^--6^--9^--2---5---8---1v--4v--7v
/ \ / \ / \ / \ / \ /
(3^^)6^^-9^^-2^\ 5^/ 8^\ 1 / 4 \ 7 / 0v\ 3v/ 6v\ 9v/ 2w
/ / \ / \ / \ / \ / \ /
5^^-8^^-1^--4^--7^--0---3---6---9---2v--5v--8v--1w--4w--7w
/ / \ / \ / \ / \ / \
0^/ 3^\ 6^/ 9^\ 2 / 5 \ 8 / 1v\ 4v/ 7v\-0w--3w
/ \ / \ / \ / \ / / \
5^--8^--1---4---7---0v--3v--6v--9v--2w--5w
\ / \ / \ / /
0 \ 3 / 6 \ 9 / 2v\ 5v/ 8v--1w--4w-(7w)
\ / \ / \ /
5---8---1v--4v--7v--0w--3w
/ \ / \ /
0v/ 3v\ 6v/ 9v\ 2w/ 5w
/ \ / \ /
5v--8v--1w--4w-(7w)
/
0w/-3w
/
5w

The notes in brackets lie outside the MOS, so you get two diagrams for
the price of one. This is still only 6 notes different from Partch's 43
note scale. It may be a good JI scale to use in relation to the amazing
generator, but I don't think it has any real advantages compared to the
one before. You could use all 45 generated notes that cover the tonality
diamond.

Graham

--- In tuning@y..., graham@m... wrote:
> Paul wrote:
>
> > > These are the unison vectors you settled on there:
> > >
> > > (-4 0 1 -1)
> > > ( 2 -1 2 -1)
> > > (-5 1 2 0)
> > > (-2 2 0 -1)
> > >
> > >
> > > Here's a translation from some vectors consistent with the 41-
note
> > > MOS:
> >
> > What does that mean?
>
> Should have been "transformation" rather than "translation" here
and
> below. It means you multiply two matrices together to get a third.

I understand that, but what do you mean by "consistent with the 41-
note MOS?"
>
> > > (1 0 0 1)(-2 2 0 -1)
> > > (0 1 1 1)(-1 1 1 1)
> > > (1 1 0 1)( 5 0 0 -2)
> > > (1 0 0 0)(-2 -2 1 0)
>
>
> > > Here's a translation from some more MOS-consistent vectors
> >
> > ???
>
> This is a simple one. The second down and the one on the bottom
stay.
> Add the bottom one to the top one and the second one up.
>
> > > (1 0 0 1)(-2 -2 1 0)
> > > (0 1 0 0)( 2 -1 2 -1)
> > > (0 0 1 1)(-3 -1 2 1)
> > > (0 0 0 1)(-2 2 0 -1)
>
Again, I was confused about what you meant by "MOS-consistent".
>
> > > I'm note sure if this means your unison vectors are unusually
close
> > to
> > > the 31/41/72 family.
> >
> > Don't know what that would mean.
>
> It's a simple transformation matrix: all ones and zeros, mostly
zeros.
> Is this unusual? I don't know.

Again, my confusion lies elsewhere.
>
> > > Any set of 41-note unison vectors can be
> > > transformed into any other set.
> >
> > Within certain reasonable limits, yes.
>
> What limits?

Well, if I chose unison vectors that were too large, this wouldn't
necessarily work.

>
> > > But I'm very dubious about the
> > > connection between any given periodicity block and a particular
set
> > of
> > > unison vectors.
> >
> > Huh?
>
> Fokker says "here's a periodicity block for these unison vectors"
but
> there's nothing I can see to match the one to the other, except
that the
> numbers of notes have to match.

The JI ratios of a Fokker periodicity block are uniquely determined
(if there are an odd number of notes) by placing the central pitch at
1/1 and including those pitches, and only those pitches, enclosed in
a parallelepiped formed by using the unison vectors as edges.

> >
> > Often but not always. For example the schisma and diesis as
unison
> > vectors give a 24-tone PB which approximates 12-tET and not 24-
tET.
>
> Hmm. I'd say that was a 12-tone PB where the determinant gave the
wrong
> value.

Why would you say that?

>
> > > That should be obvious from Wilson's keyboard
> > > layout.
> >
> > What should be obvious?
>
> Partch's scale approximates 41-equal with no notes missing and two
> duplications.

I thought you were saying something else.
>
>
> This is a variation with 27/20 swapped for 15/11, 81/80 for 45/44,
40/27
> for 22/15 and 160/81 for 88/45. All ratios are simpler

By what measure?

>
> That's more compact, and all notes are contained within 45
generated
> notes: the minimum to contain the 11-limit diamond.

45 notes generated by the amazing generator? You've got to be
clearer, my friend!

> It still doesn't
> contain the 41-note MOS.

Generated by the amazing generator, and not by the fifth, I
presume . . . please say so if that's what you mean.

Paul wrote:

> As far as I know, Partch never proposed a different 43-tone scale.

I thought I'd finish my clean room implementation before checking this.
Manuel's scale archive includes the file partch_43a.scl with this comment
at the top:

"From "Exposition on Monophony" 1933, unp. see Ayers, 1/1 vol. 9(2)"

That sounds like it's earlier than Genesis. Whatever, the differences
with partch_43.scl are:

81/80 is replaced by 49/48

27/20 is replaced by 15/11

40/27 is replaced by 22/15

160/81 is replaced by 96/49

In decimal terms, this is the same as the first alternative I gave. It's
not the same in JI as I went for 45/44 over 49/48, but that's incidental.

Graham

--- In tuning@y..., graham@m... wrote:
> Paul wrote:
>
> > As far as I know, Partch never proposed a different 43-tone
scale.
>
> I thought I'd finish my clean room implementation before checking
this.
> Manuel's scale archive includes the file partch_43a.scl with this
comment
> at the top:
>
> "From "Exposition on Monophony" 1933, unp. see Ayers, 1/1 vol. 9(2)"
>
> That sounds like it's earlier than Genesis. Whatever, the
differences
> with partch_43.scl are:
>
> 81/80 is replaced by 49/48
>
> 27/20 is replaced by 15/11
>
> 40/27 is replaced by 22/15
>
> 160/81 is replaced by 96/49
>
> In decimal terms, this is the same as the first alternative I
gave. It's
> not the same in JI as I went for 45/44 over 49/48, but that's
incidental.
>
So there _was_ another 43-tone scale! Can any of the Partch
cognoscenti out there check this?

Anyway, I might have chosen a different set of unison vectors if this
putative earlier version had been presented instead. Again, the
difference between these different sets of unison vectors is only
material if you adopt Fokker's convention of using the vectors as the
edges of a block with a parallelopiped shape. As discussed with Paul
Hahn, and for the 2-D case in the "Excursion", other shapes can be
equally logical from a JI point of view.

Paul wrote:

> > Should have been "transformation" rather than "translation" here
> and
> > below. It means you multiply two matrices together to get a third.
>
> I understand that, but what do you mean by "consistent with the 41-
> note MOS?"

"The" MOS refers to the amazing generator. That's what we're discussing
now. The title is "decimal lattices" and there's nothing decimal about
a fifth-generated MOS.

So, now that's out of the way, by consistent I mean that if you take out
the vector that closes the cycle at 41, you have three vectors that define
the amazing generator. The MOS is already periodic by these vectors, so a
periodicity block based on them is likely to be consistent with it.

> > > > Any set of 41-note unison vectors can be
> > > > transformed into any other set.
> > >
> > > Within certain reasonable limits, yes.
> >
> > What limits?
>
> Well, if I chose unison vectors that were too large, this wouldn't
> necessarily work.

Wouldn't it? Why not? This is vector spaces, surely, sets of orthonormal
bases are interchangable.

> The JI ratios of a Fokker periodicity block are uniquely determined
> (if there are an odd number of notes) by placing the central pitch at
> 1/1 and including those pitches, and only those pitches, enclosed in
> a parallelepiped formed by using the unison vectors as edges.

That always works, does it (generalized to however many dimensions)? And
for Fokker, it'll be a square lattice.

> > > Often but not always. For example the schisma and diesis as
> unison
> > > vectors give a 24-tone PB which approximates 12-tET and not 24-
> tET.
> >
> > Hmm. I'd say that was a 12-tone PB where the determinant gave the
> wrong
> > value.
>
> Why would you say that?

It defines an ET by

( 1 0 0) (1)
(-15 8 1)H' = (0)Oct
( 7 0 -3) (0)

and solving that equation gives

1(12)
H' = --(19)Oct
12(28)

Taking the determinant of the defining matrix or unison vectors is a quick
way of finding out how many notes the resulting ET will have. But it
looks like it can sometimes be a multiple, I hadn't seen an example
before. As a rectangle it would be 24 notes, but you can still divide it
into two different PBs of 12 notes.

> > This is a variation with 27/20 swapped for 15/11, 81/80 for 45/44,
> 40/27
> > for 22/15 and 160/81 for 88/45. All ratios are simpler
>
> By what measure?

Smaller numbers.

Graham

--- In tuning@y..., graham@m... wrote:
>
> > > > > Any set of 41-note unison vectors can be
> > > > > transformed into any other set.
> > > >
> > > > Within certain reasonable limits, yes.
> > >
> > > What limits?
> >
> > Well, if I chose unison vectors that were too large, this
wouldn't
> > necessarily work.
>
> Wouldn't it? Why not?

For example, if you interchanged some columns in the matrix, you'd
still get a "periodicity block" with 41 notes, but you wouldn't be
able to transform it, using its "unison vectors", to the 41-note
periodicity block we're actually interested in.
>
> > The JI ratios of a Fokker periodicity block are uniquely
determined
> > (if there are an odd number of notes) by placing the central
pitch at
> > 1/1 and including those pitches, and only those pitches, enclosed
in
> > a parallelepiped formed by using the unison vectors as edges.
>
> That always works, does it (generalized to however many dimensions)?

I've made a computer program that can do this in any number of
dimensions -- just transform the lattice using the inverse of the
matrix and all the points within a unit cube, with the origin in the
center, are the points in the Fokker periodicity block.

> And
> for Fokker, it'll be a square lattice.

Sure, but it doesn't matter if you use a triangular lattice
instead . . . the parallelepiped will still be a parallelepiped.
>
> > > > Often but not always. For example the schisma and diesis as
> > unison
> > > > vectors give a 24-tone PB which approximates 12-tET and not
24-
> > tET.
> > >
> > > Hmm. I'd say that was a 12-tone PB where the determinant gave
the
> > wrong
> > > value.
> >
> > Why would you say that?
>
> It defines an ET by
>
> ( 1 0 0) (1)
> (-15 8 1)H' = (0)Oct
> ( 7 0 -3) (0)
>
> and solving that equation gives
>
> 1(12)
> H' = --(19)Oct
> 12(28)
>
> Taking the determinant of the defining matrix or unison vectors is
a quick
> way of finding out how many notes the resulting ET will have. But
it
> looks like it can sometimes be a multiple, I hadn't seen an example
> before. As a rectangle it would be 24 notes, but you can still
divide it
> into two different PBs of 12 notes.

This is because the product of the schisma and the diesis is the
square of the syntonic comma. In any logical tuning system, if the
square of a given comma vanishes, then the comma itself must either
vanish or be a half-octave. I'd hate to use a syntonic comma as a
half-octave!

I wonder if this impacts Joe Monzo's analysis of the shruti system as
the 24-tone periodicity block in question (with two notes omitted)?

--- In tuning@y..., graham@m... wrote:

/tuning/topicId_22064.html#22106

> Paul wrote:
>
> > As far as I know, Partch never proposed a different 43-tone
> > scale.
>
> I thought I'd finish my clean room implementation before
> checking this. Manuel's scale archive includes the file
> partch_43a.scl with this comment at the top:
>
> "From "Exposition on Monophony" 1933, unp. see Ayers,
> 1/1 vol. 9(2)"
>
> That sounds like it's earlier than Genesis. Whatever,
> the differences with partch_43.scl are: <...etc.>

I don't know if "proposed" would be the correct word, because
at this stage of his career I think his use of JI scales was
quite personal, but Partch did indeed experiment with *several*
different (and different-sized) scales before finally settling
on the 43-tone one presented in his book _Genesis of a Music_.

I don't have it handy right now, but Richard Kassel's
1996 dissertation _The Evolution of Harry Partch's Monophony_
gives a good rundown of all of them as they appeared in
the various early drafts of _Exposition on Monophony_.

IIRC, he used as little as 29 (the 11-limit tonality diamond)
and as many as 55. But I could be misremembering these numbers...

What's even more interesting to me is that based on surviving
sketches of his _Li-Po Lyrics_, Partch apparently thought
vaguely in terms of 36-EDO when making his initial sketches
of the parts for "intoning voice" (at least at this early
stage, early 1930s), then adjusted the pitches to conform to
his JI scales.

This is a *very* overlooked aspect of Partchiana.

-monz
http://www.monz.org
"All roads lead to n^0"

Paul wrote:

> For example, if you interchanged some columns in the matrix, you'd
> still get a "periodicity block" with 41 notes, but you wouldn't be
> able to transform it, using its "unison vectors", to the 41-note
> periodicity block we're actually interested in.

Not using an integer transformation matrix. But you can in all cases if
you allow fractions in there, which brings us to the problem of
periodicity blocks with a multiple of the number of notes we want.

> I've made a computer program that can do this in any number of
> dimensions -- just transform the lattice using the inverse of the
> matrix and all the points within a unit cube, with the origin in the
> center, are the points in the Fokker periodicity block.

I've found this paper <http://www.xs4all.nl/~huygensf/doc/fokkerpb.html>.
Are the examples in there generated using this method? It looks like
they could be, although they didn't make sense last time I saw it.

But the thing is, although a single PB can always be generated from a set
of UVs, is the inverse true?

> > Taking the determinant of the defining matrix or unison vectors is
> a quick
> > way of finding out how many notes the resulting ET will have. But
> it
> > looks like it can sometimes be a multiple, I hadn't seen an example
> > before. As a rectangle it would be 24 notes, but you can still
> divide it
> > into two different PBs of 12 notes.
>
> This is because the product of the schisma and the diesis is the
> square of the syntonic comma. In any logical tuning system, if the
> square of a given comma vanishes, then the comma itself must either
> vanish or be a half-octave. I'd hate to use a syntonic comma as a
> half-octave!

After reflecting on this, I agree with everything you said. The PB must
have 24 notes, but it defines 12-equal.

So the general case will be that a scale will be a PB if it has the same
number of approximations to each step in the relevant ET?

> I wonder if this impacts Joe Monzo's analysis of the shruti system as
> the 24-tone periodicity block in question (with two notes omitted)?

So the 24 notes would be a normal 12 note scale with all notes
comma-doubled. I can see how that relates to the shrutis, so such
examples do have to be considered.

Okay, now for some more lattices. Here's D'alessandro (1989 version):

1^ (4^) 7^ 3 6 9 2v 8v 4w 0vw

3^------9^------5---8---1v--4v-(7v)-0w--3w--6w-----(2vw)
/ \ / \ / / \ / \ / \ /
(2^) / 8^\ 1 / 4 \(7)/ 0v/ 3v\ 6v/ 9v\ 2w/ 5w\ /1vw
/ \ / \ / / \ / \ / \ /
4^------0-------6-------2v--5v--8v--1w------7w-(0vw)3vw

Notes in brackets are also bracketed in the diagram. Apply the last
Miracle unison vector, and you get

(2^)
/ \
1^--4^/ 7^\ 0
/ / \
/ 9^------5
/ / \ /
(2^)-----8^--1 / 4 \(7)/ 0v
/ \ /
1^--4^--7^--0---3---6---9---2v--5v
/ \ / \ / \ \ / \
/ \ 9^/ \ 5 / 8 \ 1v\ 4v/(7v) 0w
/ \ / \ / \ \ / \
(2^)-----8^--1---4--(7)--0v--3v--6v--9v--2w--5w
/ \ / \ / \ / \ / / \ /
1^ 4^/ 7^\ 0 / 3 \ 6 / 9 \ 2v/ 5v\ 8v/ 1w/ 4w\ 7w/ 0vw
/ \ / \ / \ / \ / / \ /
3^------9^------5---8---1v--4v-(7v)-0w--3w--6w----(2vw)
/ \ / \ / \ / \ / \ /
(2^) / 8^\ 1 / 4 \(7)/ 0v\ 3v/ 6v\ 9v/ 2w\ 5w/ 1vw
/ \ / \ / \ / \ / \ /
4^--7^--0---3---6---9---2v--5v--8v--1w--4w--7w-0vw-3vw
\ / \ / \ / \ / \ /
9^\ / 5 \ 8 / 1v\ 4v/(7v) 0w/ 3w\ 6w/ 2vw
\ / \ / \ / \ / \ /
4--(7)--0v--3v--6v--9v--2w--5w------1vw
/ \ / \ / \ /
9 / 2v\ 5v/ 8v\ 1w/ 4w\ 7w/0vw 3vw
/ \ / \ / \ /
4v-(7v)-0w--3w--6w-----2vw
\ / \ /
9v\ 2w/ 5w\ /1vw
\ / \ /
4w--7w-0vw-3vw

2vw

That's close to the 41 note Miracle MOS. You can see the gaps in the
lattice, some of which are filled in by extra notes Erv added. I think
you need 43 notes from a chain of Miracle generators to contain the whole
scale. The extremes are 5.7 (1^) and 3.9.11 (3vw). Next to them are the
bonus notes 2^ and 2vw. Remove either of them, and you can get a subset
of the 41-note Miracle MOS plus one extra note.

Here's one of the "blues scales" I came up with a few years back

7^^-----3^
\ / \
\ / \
\ / \
0-------6-------2v

2-------8-------4v

2 and 2v are considered equivalent notes. 0-2-4v-6-8-0 is close to
5-equal, and looks like it with this notation.

A pentachordal decatonic:

9^^
/ \
8^^-----4^------0
\ / / \ \ / \
\ 6^/ 9^\ 2 / 5 \
\ / \ \ / / \
1-----\-7-/-----3v
\ /
2v

The pairs (9^,9^^) and (2,2v) are equivalent in 22-equal, so decimal names
match decatonic degrees. Finally a symmetric decatonic

9^^-----5^------1
/ \ / \ /
/ 4^--7^--0---3---6
/ / \ \ / / \ \ / /
6^/ 9^\ 2 / 5 \ 8 /
/ \ / \ /
1-------7-------3v

Graham

--- In tuning@y..., graham@m... wrote:
> Paul wrote:
>
> > For example, if you interchanged some columns in the matrix, you'd
> > still get a "periodicity block" with 41 notes, but you wouldn't be
> > able to transform it, using its "unison vectors", to the 41-note
> > periodicity block we're actually interested in.
>
> Not using an integer transformation matrix. But you can in all cases if
> you allow fractions in there,

Not sure how this would work, but the point is this:
All "reasonable" 41-tone periodicity blocks all map the entire 11-limit lattice to the
integers 0 through 40 in exactly the same way. Some "unreasonable" sets of unison
vectors would lead to blocks that give a different mapping.

> > I've made a computer program that can do this in any number of
> > dimensions -- just transform the lattice using the inverse of the
> > matrix and all the points within a unit cube, with the origin in the
> > center, are the points in the Fokker periodicity block.
>
> I've found this paper <http://www.xs4all.nl/~huygensf/doc/fokkerpb.html>.
> Are the examples in there generated using this method?

Yes, but with a slight complication ensuing when the block has an even number of notes
(Fokker is obsessed with symmetry).

> But the thing is, although a single PB can always be generated from a set
> of UVs, is the inverse true?

Not sure what you mean.
>
> So the general case will be that a scale will be a PB if it has the same
> number of approximations to each step in the relevant ET?

It doesn't work out that nicely.
>
> Okay, now for some more lattices.

This Mac uses a proportional font -- I'll have to look later.

> > Okay, now for some more lattices.
>
> This Mac uses a proportional font -- I'll have to look later.

Copy the lattice to the clipboard, launch SimpleText, select an equal width font, paste. :-)

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

> Paul wrote:

> > Not using an integer transformation matrix. But you can in all
cases if
> > you allow fractions in there,
>
> Not sure how this would work, but the point is this:
> All "reasonable" 41-tone periodicity blocks all map the entire
11-limit lattice to the
> integers 0 through 40 in exactly the same way. Some "unreasonable"
sets of unison
> vectors would lead to blocks that give a different mapping.

Oh, yes, that might happen. If it's a different approximation, I call
it a different temperament.

With fractions in the transformation matrix, it'd mean you got the
right unison vectors by combining the complex ones, but not in their
lowest terms.

> > But the thing is, although a single PB can always be generated
from a set
> > of UVs, is the inverse true?
>
> Not sure what you mean.

Can you start with an arbitrary periodicity block and uniquely assign
a set of unison vectors to it?

> > Okay, now for some more lattices.
>
> This Mac uses a proportional font -- I'll have to look later.

Hopefully your web browser will obey the <PRE> tag, so try

<http://x31eq.com/decimal_lattice.htm>

if you haven't already.

Graham

--- In tuning@y..., "Graham Breed" <graham@m...> wrote:

> Can you start with an arbitrary periodicity block and uniquely
assign
> a set of unison vectors to it?

No -- there are a lot of possible "shapes" of periodicity blocks, and
only the parallelopipeds with one note in the center will have a
unique set of unison vectors assigned to them according to the Fokker
method. In 3-D, one might want to use hexagonal prisms or rhombic
dodecahedra, and not have one note exactly in the center.