symmetry

Here's a question for ya'll...I just read some liner notes, where it
was said that the tuning of a particular stringed instrument (fretted)
came about by superimposing geometrical shapes on the fingerboard. I
know this is certainly not new (Erv Wilson, SJ Taylor and others have
used these concepts) but I've never thought much about it till now...my
question is: by doing that, what sort of symmetry will arise in the
tuning in a mathematical sense? Do 5ths, 3rds and other intervals appear
more/less often in certain shapes, as opposed to others (the Golden Mean
and triangles and all that)? A lot of guitarists use fingerboard shapes
to create their concept of the notes on the neck...Howard Roberts wrote
a series about this in Guitar Player some years ago, and called it
"Sonic Shapes." Now, I'm curious as to how this can be used with non 12
tunings...in fact, the new piece I'm working on uses this concept
throughout, and the fact that the strings are tuned to a non 12, purely
tuned scale (on my fretless electric) makes the shapes sound totally
unlike my normal ideas, leading to new and unexpected melodic and
rhythmic phrases...the tuning itself is leading the way...Hstick

Neil -- if you use equally-spaced frets, such that the distance between nut
and bridge is divided into an integer number of equal distances, you get a
subharmonic scale on each string. For example, six equal parts will get you
the scale 1/1 6/5 3/2 2/1 3/1 6/1, and twelve equal parts, the scale 1/1
12/11 6/5 4/3 3/2 12/7 2/1 12/5 3/1 4/1 6/1 12/1. If you then tune the
strings to a harmonic series, you get a tonality diamond guitar (with
one-finger otonal chords).

>Neil -- if you use equally-spaced frets, such that the distance between nut
>and bridge is divided into an integer number of equal distances, you get a
>subharmonic scale on each string. For example, six equal parts will get you
>the scale 1/1 6/5 3/2 2/1 3/1 6/1, and twelve equal parts, the scale 1/1
>12/11 6/5 4/3 3/2 12/7 2/1 12/5 3/1 4/1 6/1 12/1. If you then tune the
>strings to a harmonic series, you get a tonality diamond guitar (with
>one-finger otonal chords).

Which is the tuning Denny gave me for my slide guitar. Of course, you
can't accurately stop different strings at different places at the same
time on a slide guitar. . . and I still like my slide guitar.

-Carl