243/128 or 256/135?

I'm currently reworking and experimenting with my all-purpose 53-tone
5-limit scale, and just have a quick question. It concerns the 49-step
interval in 53-tone, or the Pythgorean major seventh.

The general rule is to choose the smaller interval, or at least I've gotten
that impression. For the inverse of the aforementioned interval, the 4-step
Pythagorean minor second, I replaced 256/243 with 128/125. For the sake of
balance, that would mean I would put the 5-limit interval in the 43-step
interval, which would be the slightly larger-factor interval, 256/135.

So what would you do in this situation? (The differece between the two
intervals is the schisma.)

~DaW~

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> I'm currently reworking and experimenting with my all-purpose 53-
tone
> 5-limit scale, and just have a quick question. It concerns the 49-
step
> interval in 53-tone, or the Pythgorean major seventh.
>
> The general rule is to choose the smaller interval, or at least
I've gotten
> that impression.

I'm not sure what you mean. General rule for what problem? Smaller in
cents?

> For the inverse of the aforementioned interval, the 4-step
> Pythagorean minor second, I replaced 256/243 with 128/125.

You mean 135/128?

> For the sake of
> balance, that would mean I would put the 5-limit interval in the 43-
step
> interval, which would be the slightly larger-factor interval,
256/135.

I wouldn't consider that a larger-factor interval, because you're
focusing on the octave span _above_ the 1/1, but for symmetry, you
should also focus on the octave span _below_ the 1/1, in which case
256/135 would correspond to 128/135, and 243/128 (which I assume was
your original value??) would correspond to 243/256.

In any case, I don't think there is such a general rule anyway -- one
prefers to maximize the degree which the notes are connected _to one
another_ in the lattice.

> So what would you do in this situation? (The differece between the
two
> intervals is the schisma.)

I would draw a lattice and see.

From: "paulerlich" <paul@stretch-music.com>

> --- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

> > The general rule is to choose the smaller interval, or at least
> I've gotten
> > that impression.
>
> I'm not sure what you mean. General rule for what problem? Smaller in
> cents?

Bad wording on my part there. I meant the interval with the smallest
numerators and denominators, whatever that's called.

> > For the inverse of the aforementioned interval, the 4-step
> > Pythagorean minor second, I replaced 256/243 with 128/125.
>
> You mean 135/128?

You're right; 128/125 is the 2-step interval (triple augmented sixth).

> > For the sake of
> > balance, that would mean I would put the 5-limit interval in the 43-
> step
> > interval, which would be the slightly larger-factor interval,
> 256/135.

> I would draw a lattice and see.

I just did that, and decided on 256/125, since it's more symmetrical that
way. Now in 665-tone, there would have both intervals....

~DaW~

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

> I just did that, and decided on 256/125, since it's more
symmetrical that
> way. Now in 665-tone, there would have both intervals....

Danny, 612-tone will be far more appropriate for such a thing -- 665-
tone only excels in the 3-limit, while 612-tone excels in the 5-
limit. You might want to take a look at 118-tone before jumping all
the way to 612, if you're working in the 5-limit.