shoptalk

I've had a 20 equal guitar for about ten years now...

Note that 20 equal solves the open-string tuning conundrum for
guitarist who'd like all fourths between the strings (for ease of
shuffling fingerings across different registers) while retaining an
open-string layout were you have the low and the high string tuned to
the same pitch.

Letting E be 7/20 you get,

E 420
A 900
D 180
G 660
B 1140
E 420

My favorite modal subsets of 20 I found by ear in the V or mixolydian
mode on this guitar -- 3323414.

This also happens to be the diatonic inside a 12-out-of-20 subset
(note that 20 equal has the smallest diatonic scale step at 60c out of
any 12-out-of-N subset as well). This is handy in that it allows a
consistent traditional notation relative to 12 equal. (I've had some
correspondences with Margo Schulter about interesting diatonics in 20
equal, and maybe I can nudge her into posting a bit on them by
including this little parenthetical hello!)

Here's the rotations...

0 3 7 8 12 15 18 20
0 4 5 9 12 15 17 20
0 1 5 8 11 13 16 20
0 4 7 10 12 15 19 20
0 3 6 8 11 15 16 20
0 3 5 8 12 13 17 20
0 2 5 9 10 14 17 20

Note that this seven note subset has the entire intervallic resources
of 20 equal folded into its structure. (Robert Walker and I did some
work together on these types of uniquely or maximally compact subsets
as well, and I think Robert had a post and music examples that are
still in the tank relating to these, Hi Robert!)

Here's the inverted mirror rotations...

0 4 7 9 12 15 19 20
0 3 5 8 11 15 16 20
0 2 5 8 12 13 17 20
0 3 6 10 11 15 18 20
0 3 7 8 12 15 17 20
0 4 5 9 12 14 17 20
0 1 5 8 10 13 16 20

These scales are exactly the type I was trying to get at in the "trust
music" thread.

The path has been cleared for seven note scales, you literally can't
go wrong. What's special about scales like these is that they bath the
familiar in pungent hues of the new and the exotic.

Speaking very broadly, the sound of these modes is like that of
7-limit diatonic scales bent through the unique mood of 20 equal with
its ancestral genealogy in 5 equal.

But don't take my word for any of this, check them out for
yourselves... I think you'll be pleasantly surprised,

--Dan Stearns

> From: "D.Stearns" <STEARNS@CAPECOD.NET>
> Subject: shoptalk
>
> Note that 20 equal solves the open-string tuning conundrum for
> guitarist who'd like all fourths between the strings (for ease of
> shuffling fingerings across different registers) while retaining an
> open-string layout were you have the low and the high string tuned to
> the same pitch.
>

NICE solution to that little-known guitar conundrum! I propose
20tet as the intonational standard for the rest of the millenium!

> My favorite modal subsets of 20 I found by ear in the V or mixolydian
> mode on this guitar -- 3323414.
>
> This also happens to be the diatonic inside a 12-out-of-20 subset
> (note that 20 equal has the smallest diatonic scale step at 60c out of
> any 12-out-of-N subset as well).

I understood the first comment, and here is a sort-of-a-lattice
that I worked out to help understand it.

F Ab B D
480 780 1080 180

C Eb F# A
0 300 600 900

G Bb C# E
720 1020 120 420

What exactly do you mean by the "smallest diatonic step" statement?

Bob Valentine

Hi Rob V.,

By that I just meant that 20 is the N that would give the smallest
diatonic or modal stepsize of any diatonic subset of 12-out-of-N.

And yes, 20 is the first ET where those guitar-friendly tempered
fourths also spell EADGBE.

I wonder what an "optimization" of that particular little conundrum
might look like that unhinges the octave?

Sounds like a fun job for Dave K.

--Dan Stearns