Re: [tuning] 12-tone tunings for 7-limit harmony (for Joseph Pehr son et al)

"Paul H. Erlich" wrote:

> Kraig Grady wrote,
>
> >yes if one takes the generalized lattice and you have one cube on top of
> another as what would be the following (if you ran the >5/4 vertical and the
> 3/2 horizontal).
>
> Sorry - I lost you. Would you taking a step back and rephrasing this?

I was just translating this scale into what I know as the generalized lattice with the 5/4
above, the 3/2 at the horizontal position and the 7 at the diagonal. If we take the scale you
quoted, it appears as one cube on top of another. Some of the rotations I will just map out.

>
>
> From here you just have to rotate your factors in each of the possible
> permutations to generate all sort of variations with like properties

>
>
> Except that rotating factors wreaks havoc on the melodic properties such as
> Constant Structures and tetrachordality (I'm still waiting to hear your take
> on the latter regarding the remarks on your Centaur page).

It seems that scales with two fifths in a row would have more of a chance of being in a
Constant structure than one with two fifths . But I will map them out and we will see.
I believe on my page I show where the repeating tetrachords exist.

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

--- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
>
> "Paul H. Erlich" wrote:
>
> > Kraig Grady wrote,
> >
> >yes if one takes the generalized lattice and you have one cube > >
>on top of
> > another as what would be the following (if you ran the >5/4
>>vertical and the
> > 3/2 horizontal).
> >
> > Sorry - I lost you. Would you taking a step back and rephrasing
this?
>
> I was just translating this scale into what I know as the
>generalized lattice with the 5/4
> above, the 3/2 at the horizontal position and the 7 at the
>diagonal. If we take the scale you
> quoted, it appears as one cube on top of another.

That's not quite right. One note would be different. (The two-cube
scale you're talking about is the 3557 Euler-Fokker genus. In the
triangular lattice, the two point that stick out would be diagonally
opposite one another, rather than both on the left or both on the
right.)
>
> >Some of the rotations I will just map out.

> >
> > From here you just have to rotate your factors in each of the
possible
> > permutations to generate all sort of variations with like
properties
>
> >
> >
> > Except that rotating factors wreaks havoc on the melodic
properties such as
> > Constant Structures and tetrachordality (I'm still waiting to
hear your take
> > on the latter regarding the remarks on your Centaur page).
>
> It seems that scales with two fifths in a row would have more of a
chance of being in a
> Constant structure than one with two fifths .

Not necessarily (this case being an example).

But I will map them out and we will see.
> I believe on my page I show where the repeating tetrachords exist.

I'll look again.

--- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
>
> "Paul H. Erlich" wrote:
>
> >
> >(I'm still waiting to hear your take
> > on the latter regarding the remarks on your Centaur page).
>
> I believe on my page I show where the repeating tetrachords exist.
>
Ah yes they are what I would have called hexachords in my paper
(since there are 6 notes spanning a 4:3). So these six octave species
of the Centaur scale have identical hexachords a 4:3 apart:

D#-D#
E-E
G#-G#
A#-A#
A-A
B-B

Although in my paper I ask for all octave species to
be "tetrachordal", half of the octave species being tetrachordal is
not too bad for a JI scale.

The Centaur scale

5/3-------5/4------15/8
/|\ /|\ / \
/ | \ / | \ / \
14/9-------7/6-------7/4 \ / \
`. /,' \`.\ /,'/ \`.\ / \
4/3-----\-1/1-/---\-3/2-------9/8
\ | / \ |
\|/ \|
7/5------21/20

only has 3 consonant 7-limit tetrads. Challenge: can a scale with 4
of them be as "tetrachordal" as Centaur?