Re: 13/31 generator

> Paul
> > Bob

> Yup. The generator of this scale is 13/31 oct. or approximately a 7:9.
> >
> > Despite having lots of common ingrediants,
> > I couldn't figure out a way that
> > LssLssLs could map to 72 and preserve these
> > identities.
> >
> You must be making use of some of 31-tET's unison vectors that are
> not quite unisons in 72-tET. I'm sure we could figure this out if we
> wanted to (do you want to?).
>

You are right and no, I don't have to find the mapping. I was a little
surprised that it didn't pop out, but your point about unison vectors
is exactly on target. Although I didn't work it out, I have no doubts
that 31 and 72 will have a different way of connecting '3' to the
other terms the same way they differ in connecting to '5'.

This revealed to me some assumptions I'd had in this thread which were
also too 31 oriented. I had sort of assumed that an EDO that could
function as a meantone would have the augmented second map to the
"best 7/6" and the augmented sixth to the "best 7/4". A cursory glance
at a few showed that this was not the case.

Bob Valentine

Paul E, wrote:
> > Yup. The generator of this scale is 13/31 oct. or approximately a
7:9.

You mean 11/31 oct. 13/31 oct is a meantone fourth.

I see that around a 425.7 cent (approx 7:9) is a good generator for
11-limit-sans-5's, _if_ you don't mind a minimax error of 9.4 cents.
This is also the optimum for 9-limit-sans-5's and 7-limit-sans-5's.

If you don't care about the 4:9 or 4:11 (which first appear at 9 and
11 notes), the minimax error can go down to 8.8 cents with a generator
of about 426.3 cents. However the 2:3's are sounding pretty bad by
then, being 7 cents narrow.

11/31 oct is 425.8 cents.

It has improper MOS at 5, 8 and 11 notes and proper at 14. The first
optimum is represented in EDOs 17,31,48,79. The second in EDO's
17,31,45,76. Neither the 5-tone, 8-tone nor 11-tone are approximated
by anything I can find in the Scala archive.

I certainly wouldn't call them quasi-just, but they are
stil interesting.

Bob Valentine wrote:

> > > Despite having lots of common ingrediants,
> > > I couldn't figure out a way that
> > > LssLssLs could map to 72 and preserve these
> > > identities.

No it can't because 98:99 must vanish. i.e. as you showed 7:11 is the
same as 9:14.

It has a melodic approximation in Miracle (and hence in 72-EDO) with a
generator which is 4 miracle generators (28/72 oct or 466.7 cents).
The pentatonic fits in Blackjack and the octatonic fits in Canasta.
But they do not preserve the harmonic identities.

In Miracle, the pentatonic becomes proper. Both the penatatonic and
octatonic contain quasi-just 6:7's and 8:9's but no other subsets of
1-3-7-9.

-- Dave Keenan