Quintads of the 9-limit diamond

I uploaded a graph of the ten quintads of the 9-limit tonality
diamond, where there is an edge drawn if two quintads share an
interval. The quintads are nameed in terms of the cubic lattice notation.

It's in the photos section, but you should be able to get it from this
url also:

http://tinyurl.com/4n2k7

If you click on "full size", you should also get the full-sized view.

>I uploaded a graph of the ten quintads of the 9-limit tonality
>diamond, where there is an edge drawn if two quintads share an
>interval. The quintads are nameed in terms of the cubic lattice
>notation.
>
>It's in the photos section, but you should be able to get it
>from this url also:
>
>http://tinyurl.com/4n2k7
>
>If you click on "full size", you should also get the
>full-sized view.

Ask and ye shall recieve. Except the 7-limit stellated
hexany is not immediately evident here. Can you tell me
which vertices it includes? And a two-sentence refresher
on cubic lattice notation would be great.

-Carl

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I uploaded a graph of the ten quintads of the 9-limit tonality
> diamond, where there is an edge drawn if two quintads share an
> interval. The quintads are nameed in terms of the cubic lattice
notation.
>

It's a pretty picture, but:

(a) There's no need to invent a new term for a chord of five notes
(assuming that's what you mean). "Pentad" already exists in English,
and has been used on this list for more than a decade. English uses
the Greek numeric prefixes, not Latin, for this purpose: dyad,
triad, tetrad, pentad, hexad, heptad, ogdoad. That last one is
somewhat surprising and is pronounced OG-doe-ad (OG-dough-ad), _not_
OG-dode.

(b) I have no idea what pentads are indicated by your "cubic lattice
notation". Why not use a musical notation and list all the notes?
Either by ratios or by the letters A to G plus accidentals.

Here's one form of the 9-limit diamond in (unreduced) ratios and
Sagittal ASCII-shorthand notation with 1/1 as G.

5/5
5/6 6/5
5/7 6/6 7/5
5/8 6/7 7/6 8/5
5/9 6/8 7/7 8/6 9/5
6/9 7/8 8/7 9/6
7/9 8/8 9/7
8/9 9/8
9/9

G
E\ Bb/
C#r G Dbc
B\ Ef Bbt Eb/
A\ D G C F/
C Ft Af D
Ebt G Bf
F A
G

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > I uploaded a graph of the ten quintads of the 9-limit tonality
> > diamond, where there is an edge drawn if two quintads share an
> > interval. The quintads are nameed in terms of the cubic lattice
> notation.
> >
>
> It's a pretty picture, but:
>
> (a) There's no need to invent a new term for a chord of five notes
> (assuming that's what you mean). "Pentad" already exists in English,
> and has been used on this list for more than a decade.

"Pentad" is not in the Monzopedia, and seems to have been rarely used.
In that respect in seems like "quintad".

English uses
> the Greek numeric prefixes, not Latin, for this purpose: dyad,
> triad, tetrad, pentad, hexad, heptad, ogdoad. That last one is
> somewhat surprising and is pronounced OG-doe-ad (OG-dough-ad), _not_
> OG-dode.

Ogdoad sound like it is referring to Gnostic theology, but pentad,
hexad and heptad will do.

> (b) I have no idea what pentads are indicated by your "cubic lattice
> notation". Why not use a musical notation and list all the notes?

This way made more sense to me, because it tells you where it is in
the lattice.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Ask and ye shall recieve. Except the 7-limit stellated
> hexany is not immediately evident here. Can you tell me
> which vertices it includes?

I thought I had mentioned that the stellated hexany was a 2x2x2 chord
cube; the verticies in question therfore are [a b c], -1 <= a,b,c <= 0.
Not, incidentally, the kind of thing which can be seen clearly, or
even murkily, using other notations for the chords.

And a two-sentence refresher
> on cubic lattice notation would be great.

The lattice of tetrads works as follows: [a b c] refers to a utonal
tetrad if a+b+c is even, then the root of the tetrad is
r = 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2); the tetrad itself of
course then is r*[1,3,5,7]. If a+b+c is odd, we have a utonal tetrad.
The root (not guide tone!) of the tetrad is the root r of the
corresponding otonal tetrad [a+1 b c], and the tetrad itself of course
then is r*[1,3,3/5,3/7]. To get pentads, you merely take instead
r*[1,3,5,7,9] or r*[1,3,3/5,3/7,1/3] instead. Of course the latter no
longer has much of a claim to having 1 as a root, but that does not
matter.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Ogdoad sound like it is referring to Gnostic theology, but pentad,
> hexad and heptad will do.

Yeah. "Ogdoad" is pretty wierd. I think folks here adopted it from
Erv Wilson, who uses it to refer to 15-limit complete chords. But I
just checked my Shorter Oxford and "octad" is also perfectly valid.

> This way made more sense to me, because it tells you where it is in
> the lattice.

Thanks for trying to explain your notation in
/tuning/topicId_55743.html#55752
but I'm afraid my brain just keeps sliding off the outside of it.

I wonder if it might not be easier to see the common notes by using
the diamond keyboard layout itself, with worm-holes connecting the
duplicate notes.

>> And a two-sentence refresher
>> on cubic lattice notation would be great.
>
>The lattice of tetrads works as follows: [a b c] refers to a utonal
>tetrad if a+b+c is even,

You mean otonal?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> And a two-sentence refresher
> >> on cubic lattice notation would be great.
> >
> >The lattice of tetrads works as follows: [a b c] refers to a utonal
> >tetrad if a+b+c is even,
>
> You mean otonal?

Right, sorry. Maybe I should just go back to "major" and "minor".