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

Paul Erlich wrote:

>
> >
> 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

I confused by this. I meant that any type of tetrachord between D# and G# is repeated at A#-D#

Also those between e-a are reaped between b-e. It contains at least of one example of all the
superparticular trichords possible as discussed by Ptolemy.

>
>
> 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?

they are actually pentads 1-3-5-7-9

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

--- In tuning@egroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
>
> Paul Erlich wrote:
>
> >
> > >
> > 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
>
> I confused by this. I meant that any type of tetrachord between D#
and G# is repeated at A#-D#
>
> Also those between e-a are reaped between b-e.

Right -- that follows trivially from the hexachords being identical.
The hexachords being identical means that the 12-tone octave species
listed above all contain two identical hexachords either a 4:3 apart
or a 3:2 apart (oops, left that second possibility out before).