neutral triads

[Graham Breed:]
"That's either an ambiguous interval between a major and minor third,
or an approximation to 11/9, depending on whether you believe in
11-limit harmony or not"

The 7L 3s (though I'm thinking of it as a scale in its own right)
would seem to have a neutral third triadic analogue in:

5/4------15/8
/ \ / \
/ \ / \
/ \ / \
/ \ / \
16/9-------4/3-------1/1-------3/2-------9/8
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
64/45-----16/15------8/5

as:

11/9------11/6
/ \ / \
/ \ / \
/ \ / \
/ \ / \
16/9-------4/3-------1/1-------3/2-------9/8
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
16/11-----12/11-----18/11

where a:

*---*---*---*---*
/ \ / \ / \ / \ /
*---*---*---*---*---*---*---*---*
/ \ / \ / \ / \ /
*---*---*---*---*

'works.'

Dan

In-Reply-To: <940062676.11571@onelist.com>
Dan Stearns, Digest 356.5, wrote:

> The 7L 3s (though I'm thinking of it as a scale in its own right)
> would seem to have a neutral third triadic analogue in:
>
> 5/4------15/8
> / \ / \
> / \ / \
> / \ / \
> / \ / \
> 16/9-------4/3-------1/1-------3/2-------9/8
> / \ / \ /
> / \ / \ /
> / \ / \ /
> / \ / \ /
> 64/45-----16/15------8/5
>
> as:
>
> 11/9------11/6
> / \ / \
> / \ / \
> / \ / \
> / \ / \
> 16/9-------4/3-------1/1-------3/2-------9/8
> / \ / \ /
> / \ / \ /
> / \ / \ /
> / \ / \ /
> 16/11-----12/11-----18/11

Why? Besides, it's a poor analog. In a 7L+3s scale, the interval
between the notes you've written as 18/11 and 11/9 is an
approximation to 3/2 (or 3/4). The JI mapping you use, and the
5-limit "analog", don't have this. So in what sense is this
"analog" more analogous than, say,

16/9-------4/3-------1/1-------3/2-------9/8
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
/ \ / \ / \ / \ /
64/45-----16/15------8/5-------6/5-------9/5

?

> where a:
>
> *---*---*---*---*
> / \ / \ / \ / \ /
> *---*---*---*---*---*---*---*---*
> / \ / \ / \ / \ /
> *---*---*---*---*
>
> 'works.'

In what way??? And what size thirds are you using?

I wrote:

> > where a:
> >
> > *---*---*---*---*
> > / \ / \ / \ / \ /
> > *---*---*---*---*---*---*---*---*
> > / \ / \ / \ / \ /
> > *---*---*---*---*
> >
> > 'works.'

Graham Breed responded:

> In what way??? And what size thirds are you using?

I'm substituting the 5/4 (in a 4:5:6 triad) with an 11/9. So, using
24e as an example for both:

08---22---12---02---16
/ \ / \ / \ / \ /
16---06---20---10---00---14---04---18---08
/ \ / \ / \ / \ /
08---22---12---02---16

Becomes:

07---21---11---01---15
/ \ / \ / \ / \ /
16---06---20---10---00---14---04---18---08
/ \ / \ / \ / \ /
09---23---13---03---17

Dan.

Dan Stearns wrote,

>[Graham Breed:]
>"That's either an ambiguous interval between a major and minor third,
>or an approximation to 11/9, depending on whether you believe in
>11-limit harmony or not"

>The 7L 3s (though I'm thinking of it as a scale in its own right)
>would seem to have a neutral third triadic analogue in:

> 5/4------15/8
> / \ / \
> / \ / \
> / \ / \
> / \ / \
> 16/9-------4/3-------1/1-------3/2-------9/8
> / \ / \ /
> / \ / \ /
> / \ / \ /
> / \ / \ /
> 64/45-----16/15------8/5

> as:

> 11/9------11/6
> / \ / \
> / \ / \
> / \ / \
> / \ / \
> 16/9-------4/3-------1/1-------3/2-------9/8
> / \ / \ /
> / \ / \ /
> / \ / \ /
> / \ / \ /
> 16/11-----12/11-----18/11

I have reservations about this diagram becuase strictly speaking, the point of
the triangular lattice diagrams is to show all consonant intervals (in this case,
those of 11-limit (Partchian) JI). So I would diagram those pitches like this:

11/9------11/6
/ \`._ / \`._
/ \ `._/ \ `._
/ \ /`._ \ `._
/ \ / `._\ `._
16/9=======4/3=-=-=-=1/1=-=-=-=3/2=======9/8
`._ \`._ / \`._ / \ /
`._ \ `._/ \ `._/ \ /
`._ \ /`._ \ /`._ \ /
`._\ / `._\ / `._\ /
16/11=====12/11=====18/11

Now if we treat 121:120 as a unison vector, 11/9 can be interpreted as 27/22,
11/6 as 81/44, 18/11 as 44/27, 12/11 as 88/81, and 16/11 as 352/243, and we
can extend the diagram like this:

352/243===88/81=-=44/27=-=11/9======11/6
`._ \`._ / \`._ / \`._
`._ \ `._/ \ `._/ \ `._
`._ \ /`._ \ /`._ \ `._
`._\ / `._\ / `._\ `._
16/9=======4/3=-=-=-=1/1=-=-=-=3/2=======9/8
`._ \`._ / \`._ / \`._ / \`._
`._ \ `._/ \ `._/ \ `._/ \ `._
`._ \ /`._ \ /`._ \ /`._ \ `._
`._\ / `._\ / `._\ / `._\ `._
16/11=====12/11=-=-=18/11=-=27/22===81/44

which would repeat itself infinitely since the top row and bottom row are
equivalent. Is that what you were getting at with this:

> where a:
>
> *---*---*---*---*
> / \ / \ / \ / \ /
> *---*---*---*---*---*---*---*---*
> / \ / \ / \ / \ /
> *---*---*---*---*
>
> 'works.'

?

[Paul Erlich:]
>Now if we treat 121:120 as a unison vector, 11/9 can be interpreted
as 27/22 [--SNIP--] Is that what you were getting at with this

No, I think that your first 11-limit diagram, but as all the pitches
of a 1/1, 12/11, 9/8, 11/9, 4/3, 16/11, 3/2, 18/11, 16/9, 11/6, 2/1,
7L 3s where the 243/242 disappears:

*.----*.----*.----*.----*.
/ \`._/ \`._/ \`._/ \`._/ \`._
/ \ /`._\ /`._\ /`._\ /`._\ `._
*.----*.----*.----*.---`*.---`*----`*----`*----`*
`._ \`._/ \`._/ \`._/ \`._/ \ /
`._\ /`._\ /`._\ /`._\ /`._\ /
`*----`*----`*----`*----`*

was perhaps more of what I was trying to show (though I probably
shouldn't have used a lattice in the attempt to illustrate this idea
in the first place).

Dan

I wrote,

>>Now if we treat 121:120 as a unison vector, 11/9 can be interpreted
>>as 27/22 [--SNIP--] Is that what you were getting at with this

Dan Stearns wrote,

>No, I think that your first 11-limit diagram, but as all the pitches
>of a 1/1, 12/11, 9/8, 11/9, 4/3, 16/11, 3/2, 18/11, 16/9, 11/6, 2/1,
>7L 3s where the 243/242 disappears:

> *.----*.----*.----*.----*.
> / \`._/ \`._/ \`._/ \`._/ \`._
> / \ /`._\ /`._\ /`._\ /`._\ `._
>*.----*.----*.----*.---`*.---`*----`*----`*----`*
> `._ \`._/ \`._/ \`._/ \`._/ \ /
> `._\ /`._\ /`._\ /`._\ /`._\ /
> `*----`*----`*----`*----`*
>
>was perhaps more of what I was trying to show (though I probably
>shouldn't have used a lattice in the attempt to illustrate this idea
>in the first place).

I made a mistake and said 120:121 instead of 243:242. But your diagram is
different from mine -- for example, you have 9 notes in a chain of fifths.
What pitches are those?

[Paul H. Erlich:]
>I made a mistake and said 120:121 instead of 243:242. But your
diagram is different from mine -- for example, you have 9 notes in a
chain of fifths. What pitches are those?

Just a chain of 3/2s centered on 1/1 (128/64, 32/27, 16/9, 4/3, 1/1,
3/2, 9/8, 27/16, 81/64), which really was only meant to show the 7L 3s
as two plains - a 3(-limit) and an 11(-limit): 1/1, 12/11 (33/32), 9/8
(32/27), 11/9 (128/99), 4/3 (81/64), 16/11 (11/8), 3/2 (128/81), 18/11
(99/64), 16/9 (27/16), 11/6 (64/63), 2/1.

Dan

[Paul H. Erlich:]
>>I made a mistake and said 120:121 instead of 243:242. But your
>>diagram is different from mine -- for example, you have 9 notes in a
>>chain of fifths. What pitches are those?

>Just a chain of 3/2s centered on 1/1 (128/64,

should be 128/81

>32/27, 16/9, 4/3, 1/1,
>3/2, 9/8, 27/16, 81/64),
>which really was only meant to show the 7L 3s
>as two plains - a 3(-limit) and an 11(-limit): 1/1, 12/11 (33/32), 9/8
>(32/27), 11/9 (128/99), 4/3 (81/64), 16/11 (11/8), 3/2 (128/81), 18/11
>(99/64), 16/9 (27/16), 11/6 (64/63), 2/1.

I don't understand your notation (the parenthetical ratios are 90 cents off
their leftward neighbors), and still don't see how the scale could be said
to have anything resembling a chain of nine 3/2s.

[Paul H. Erlich:]
>should be 128/81

Right.

> I don't understand your notation (the parenthetical ratios are 90
cents off their leftward neighbors),

I did it that in an attempt to show that the extra intervals on the
diagram were just the same class 'chromatic' intervals of the 7L 3s,
as a 1/1, 12/11, 9/8, 11/9, 4/3, 16/11, 3/2, 18/11, 16/9, 11/6, 2/1:

1/1 12/11 9/8 11/9 4/3 16/11 3/2 18/11 16/9 11/6 2/1
1/1 33/32 121/108 11/9 4/3 11/8 3/2 44/27 121/72 11/6 2/1
1/1 88/81 32/27 128/99 4/3 16/11 128/81 44/27 16/9 64/33 2/1
1/1 12/11 144/121 27/22 162/121 16/11 3/2 18/11 216/121 81/44 2/1
1/1 12/11 9/8 27/22 4/3 11/8 3/2 18/11 27/16 11/6 2/1
1/1 33/32 9/8 11/9 121/96 11/8 3/2 99/64 121/72 11/6 2/1
1/1 12/11 32/27 11/9 4/3 16/11 3/2 44/27 16/9 64/33 2/1
1/1 88/81 121/108 11/9 4/3 11/8 121/81 44/27 16/9 11/6 2/1
1/1 33/32 9/8 27/22 81/64 11/8 3/2 18/11 27/16 81/44 2/1
1/1 12/11 144/121 27/22 4/3 16/11 192/121 18/11 216/121 64/33 2/1

if the 243/242 were to disappear.

>and still don't see how the scale could be said to have anything
resembling a chain of nine 3/2s.

Well again, not the 10-note scale, but rather the whole 'chromatic'
set (minus the 243/242).

Dan

Dan,

What is your "whole 'chromatic' set"? Are you constructing all the modes of
Graham's scale on a common 1/1? If so, that is what Carl would call the
"diamond" of Graham's scale -- in other words, you're taking the cross-set
of the scale with its own inverse. Another way of putting it is that you're
taking every interval occuring anywhere in Graham's scale and constructing
that interval above and below 1/1. Although you seem to be saying that now,
there seems to be no indication that you were constructing a larger scale in
any of your previous messages. You should try to be more clear. Also,
calling the diamond of a scale its 'chromatic' set seems like misleading
terminology -- for example, the diamond of the pentachordal decatonic scale
in 22-tET would be missing the 1/22 oct. and 21/22 oct. above the tonic,
while the 'chromatic' set in this case would be more rightly considered to
be the entirety of 22-tET, wouldn't you think?

-Paul

[Paul Erlich:]
>Are you constructing all the modes of Graham's scale on a common 1/1?

Yes.

>Although you seem to be saying that now, there seems to be no
indication that you were constructing a larger scale in any of your
previous messages

Yes, that's what I've been 'saying' right along.

>You should try to be more clear.

Well (believe it or not) I do try (sometimes harder than others), but
I should also add that "what Carl would call the diamond," or "the
cross-set of the scale with its own inverse," are not necessarily any
clearer to me... One of the reasons why I post (and very much enjoy
the TD) is that it gives me a better idea of what what interest me is
actually "called," or perhaps how what I know sits in a more
standardized body of tuning theory - in other words, I *do* want to
learn, but (for better or worse), I also don't want to iron out every
wrinkle in the way *I* approach things... But I'm a big boy, and if I
post something that's wrong, I can handle being told so (in fact, I
kinda like to know).

>Also, calling the diamond of a scale its 'chromatic' set seems like
misleading terminology -- for example, the diamond of the pentachordal
decatonic scale in 22-tET would be missing the 1/22 oct. and 21/22
oct. above the tonic, while the 'chromatic' set in this case would be
more rightly considered to be the entirety of 22-tET, wouldn't you
think?

I think your absolutely right, and I understand this, and that's why I
had used the single quotation markers (and next time I just might just
say: the cross-set of the scale with its own inverse...).

Dan

>Well (believe it or not) I do try (sometimes harder than others), but
>I should also add that "what Carl would call the diamond," or "the
>cross-set of the scale with its own inverse," are not necessarily any
>clearer to me...

That's why I immediately preceded that with "Are you constructing all the
modes of Graham's scale on a common 1/1 If so, that is what Carl would call
. . .?" and immediately followed that with "Another way of putting it is
that you're taking every interval occuring anywhere in Graham's scale and
constructing that interval above and below 1/1". So not only did I make
myself clear to you, but I named a few equivalent concepts from Partch's and
Wilson's terminologies, so that you'll be more likely to understand those
when they come up, and not feel as if you're being alienated by some
exclusive language.

>(and next time I just might just
>say: the cross-set of the scale with its own inverse...).

Sure, but since you said that was not clear to you, let me explain what a
cross-set is. To make the cross-set of an m-tone scale with an n-tone scale,
each expressed as a set of ratios from 1/1, you'll need to make an m-by-n
table. Label each column (at the top) with one of the ratios from the first
scale -- you should then have a header row containing the whole scale. Label
each row (on the left) with the ratios from the second scale -- now you'll
have a header column containing the inverted scale. Now each entry in the
table will be the product of the ratio for that column times the ratio for
that row.

If you try doing this for a scale and its own inverse, you'll see that you
end up with the same ratios you get when constructing all the modes of a
scale on a common 1/1. In fact, each row in your table will contain the
ratios for one mode (though you might have to transpose things by an
octave). Carl calls this (the cross-set of a scale with its own inverse) a
diamond because Partch's diamonds are a table of this sort, where the
starting "scale" is simply the first few odd harmonics, transposed to within
one octave.

Dan wrote,

>No, I think that your first 11-limit diagram, but as all the pitches
>of a 1/1, 12/11, 9/8, 11/9, 4/3, 16/11, 3/2, 18/11, 16/9, 11/6, 2/1,
>7L 3s where the 243/242 disappears:

> *.----*.----*.----*.----*.
> / \`._/ \`._/ \`._/ \`._/ \`._
> / \ /`._\ /`._\ /`._\ /`._\ `._
>*.----*.----*.----*.---`*.---`*----`*----`*----`*
> `._ \`._/ \`._/ \`._/ \`._/ \ /
> `._\ /`._\ /`._\ /`._\ /`._\ /
> `*----`*----`*----`*----`*
>
>was perhaps more of what I was trying to show (though I probably
>shouldn't have used a lattice in the attempt to illustrate this idea
>in the first place).

If you were really conceiving of the union of all the modes of Graham's
scale as a single, larger scale, then the lattice is perfectly appropriate
and a valid depiction as you drew it. Completed, (to account for 243:242
equivalency and including 9:8 connectors) it would look like this . . .

*=====*.����*.����*.����*.����*.����*.����*.����*.====*.
`._ \`._/ \`._/ \`._/ \`._/ \`._/ \`._/ \`._/ \`._
`._\ /`._\ /`._\ /`._\ /`._\ /`._\ /`._\ /`._\ `._
*.===`*.���`*.���`*.���`@.���`*����`*����`*====`*.
`._ \`._/ \`._/ \`._/ \`._/ \`._ \`._/ \`._/ \`._
`._\ /`._\ /`._\ /`._\ /`._\ /`._\ /`._\ /`._\ `._

`*====`*����`*����`*����`*����`*����`*����`*����`*=====*

. . . infinitely repeated since the top and bottom rows are equivalent.

But the text above your diagram leaves out any reference to creating a
larger scale out of the union of the modes, and seems to clearly indicate
that you intend to show the "pitches" of Graham's 10-tone scale. If you
really meant "intervals" and not "pitches", then I can understand what you
did, though unless you really intend to go on and create the larger,
"diamond" scale out of those intervals, then I would say the lattice diagram
would not be appropriate, since all 19 intervals can be seen in the 10-tone
lattice diagram I posted, while your diagram is appropriate for a 19-tone
scale.

[Paul H. Erlich:]
>but I named a few equivalent concepts from Partch's and Wilson's
terminologies, so that you'll be more likely to understand those when
they come up,

And that's why (when the fist start flying) I always got something
good to say about Paul Erlich.

>Sure, but since you said that was not clear to you,

Actually, I said "are not necessarily any clearer," and I really meant
it in more of a figurative, rather than literal, sense... In other
words, unless these types of standardized terminology's are either
ultra-generic, or I really really feel like my grasp of the term lines
up comfortably with the thing as I understand it (or better yet use
it), I feel very hesitant, or uncomfortable citing them. (However, I
have little doubt that this is more indicative of the ways in which I
learn and deal with these things, as opposed to the ways in which
they're presented and used here at the TD.)

Dan