Question on John Chalmers' "Divisions of the Tetrachord"

Hi Folks,

Can anyone help me understand Chapter 6, pages 116 and 117 of John
Chalmers' "Divisions of the Tetrachord"? I always thought that any three
connected tones of a hexany formed some sort of consonant structure. I
read on page 117 that "in the hexagram or octahedral representation, the
three tone sets appear as triangular faces or facets. The triads of
(diagram) 6-21 are tabulated in (table) 6-22." Table 6-22 gives 1.5
3.5 5.7 as one of the "essential consonant chords of the hexany",
namely the 1 3 7. But in the diagram these three tones are not fully
connected. In fact four out of the eight chords listed are not fully
connected.

I've read the chapter inside out and upside down without understanding
what I'm missing. Any help would be most appreciated.

Kind Regards

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

Table 6-22 gives 1.5
> 3.5 5.7 as one of the "essential consonant chords of the hexany",
> namely the 1 3 7. But in the diagram these three tones are not fully
> connected. In fact four out of the eight chords listed are not fully
> connected.
>
> I've read the chapter inside out and upside down without understanding
> what I'm missing. Any help would be most appreciated.

It should be connected; maybe the diagram just doesn't show it? Or perhaps it shows utonalities as connected, as you read it, at least, but not otonalities? What does it show for 7/5-21/5-7, 1/5-1-7/5 or
1/5-3-21/5?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> It should be connected; maybe the diagram just doesn't show it? Or
>perhaps it shows utonalities as connected, as you read it, at least,
>but not otonalities?

that's not geometrically possible, is it? i mean, as long as all the
consonant intervals in the utonalities are connected, then *all* the
consonant intervals are connected, thus all the consonant intervals
in the otonalities are connected . . .