20-tet and the 13:16:19, 1/(19:16:13) triads

The [2,9] scale in 20 equal has been written about in the past because
of certain numerical properties relative to the [2,5] in 12 equal.

However, an interesting rational interpretation exists for this scale
as well based on a chain of five 13:16:19 triads,

18 4 9 15 20 6 11 17 2 8 13

416/361-----32/19-----16/13----304/169--2888/2197
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
676/361----26/19------1/1------19/13----361/338--6859/4394

This gives a comma of 47525504/47045881 at 17.56021 cents.

Note that this identity has a Navarro like quality of increasing by
the same whole number, three in this case, and that 20-tet is
consistent through

13:16:19:22:25:28:31:34:37

Also note that a meantone interpretation would be virtually
indistinguishable from 20-tet.

--Dan Stearns

> From: "D.Stearns" <STEARNS@CAPECOD.NET>
> Subject: 20-tet and the 13:16:19, 1/(19:16:13) triads
>
> The [2,9] scale in 20 equal has been written about in the past because
> of certain numerical properties relative to the [2,5] in 12 equal.

Hi Dan,

What a precipitous post, I was just looking at various 11's
this morning (as part of a continuing study of potential MOS
tunings).

I have been much more focused on fitting in some primary
low-limit consonances and seeing what else comes out. This
approach, the "rationalization of an EDO" is a complete
flip on the thing, and certainly just as valid. (Generally
anytime I find something that looks like 31EDO I recognize
it... and I recognize how many MOS fit into EDOs WITHOUT
having a particularly well supported low-limit
interpretation...)

>
> However, an interesting rational interpretation exists for this scale
> as well based on a chain of five 13:16:19 triads,
>

This is pretty cool. I played with the 12-o-o-20 diatonic last week,
and I'll try to wrap my ears around this.

Bob Valentine