Convert a 12-tET guitar to JI by only changing strings?

I wonder if this is a new idea and whether it is at all feasible? If you
see anyone trying to patent it in the future, remember you read it here
first, in a public forum.

So you want to convert your standard 12-tET guitar to 5-limit JI, or
meantone or whatever (so long as it has 12 notes per octave and doesn't
differ too wildly from 12-tET). You wander down to your music store and
select the appropriate packet of strings, come home and fit them, and away
you go.

How could this magic be acheived?

By having unevenly ground strings. Strings whose diameter varies between
frets.

Consider this standard 5-limit JI tuning

C# G#
A E B F#
F C G D
Eb Bb

Tune the open strings as E A D G B E on this lattice.

Consider that we want the first fret to change the pitch by different
amounts for each string. We'll ignore the fact that pressing the string
down increases the tension as well as shortening (decreasing the mass).

In 12-tET every string gets it's mass reduced by the 12th root of 2 which
is approximately 16/17. In our JI tuning we will want the E D and B strings
reduced by 15/16, the G string reduced by 24/25 and the A string reduced by
25/27. So the part of the E D and B strings above the first fret would be
slighly fatter so it represents 1/16 of the mass of the string instead of
1/17, and the part of the G string above the first fret would be slightly
narrower so it is only 1/25 of the mass of the string (about 18% smaller
diameter).

Similarly we need the E and A strings to have 1/9 of their mass above the
second fret while the D G and B strings have 1/10, and so on.

Now we just need to find someone to make these things!

Regards,
-- Dave Keenan
http://dkeenan.com

I wrote:

"In 12-tET every string gets it's mass reduced by the 12th root of 2 which
is approximately 16/17. In our JI tuning we will want the E D and B strings
reduced by 15/16, the G string reduced by 24/25 and the A string reduced by
25/27. So the part of the E D and B strings above the first fret would be
slighly fatter so it represents 1/16 of the mass of the string instead of
1/17, and the part of the G string above the first fret would be slightly
narrower so it is only 1/25 of the mass of the string (about 18% smaller
diameter)."

It should have been:

In 12-tET every string gets it's mass reduced by the 12th root of 2 which
is approximately 17/18. In our JI tuning we will want the E D and B strings
reduced by 15/16, the G string reduced by 24/25 and the A string reduced by
25/27. So the part of the E D and B strings above the first fret would be
slighly fatter so it represents 1/16 of the mass of the string instead of
1/18, and the part of the G string above the first fret would be slightly
narrower so it is only 1/25 of the mass of the string (about 15% smaller
diameter).

Regards,
-- Dave Keenan
http://dkeenan.com

Hi David

What a deal! No messing with fret changes? Can the strings needed to
accomplish this procedure be manufactured? Very interesting concept.
Good work!

Wally

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
> I wrote:
>
> "In 12-tET every string gets it's mass reduced by the 12th root of 2
which
> is approximately 16/17. In our JI tuning we will want the E D and B
strings
> reduced by 15/16, the G string reduced by 24/25 and the A string
reduced by
> 25/27. So the part of the E D and B strings above the first fret
would be
> slighly fatter so it represents 1/16 of the mass of the string
instead of
> 1/17, and the part of the G string above the first fret would be
slightly
> narrower so it is only 1/25 of the mass of the string (about 18%
smaller
> diameter)."
>
> It should have been:
>
> In 12-tET every string gets it's mass reduced by the 12th root of 2
which
> is approximately 17/18. In our JI tuning we will want the E D and B
strings
> reduced by 15/16, the G string reduced by 24/25 and the A string
reduced by
> 25/27. So the part of the E D and B strings above the first fret
would be
> slighly fatter so it represents 1/16 of the mass of the string
instead of
> 1/18, and the part of the G string above the first fret would be
slightly
> narrower so it is only 1/25 of the mass of the string (about 15%
smaller
> diameter).
>
> Regards,
> -- Dave Keenan
> http://dkeenan.com

[Dave Keenan:]
>"In 12-tET every string gets it's mass reduced by the 12th root of 2
>which is approximately 16/17. In our JI tuning we will want the E D and
>B strings reduced by 15/16, the G string reduced by 24/25 and the A
>string reduced by 25/27. So the part of the E D and B strings above the
>first fret would be slighly fatter so it represents 1/16 of the mass of
>the string instead of 1/17, and the part of the G string above the
>first fret would be slightly narrower so it is only 1/25 of the mass of
>the string (about 18% smaller diameter)."

>It should have been:

>In 12-tET every string gets it's mass reduced by the 12th root of 2
>which is approximately 17/18. In our JI tuning we will want the E D and
>B strings reduced by 15/16, the G string reduced by 24/25 and the A
>string reduced by 25/27. So the part of the E D and B strings above the
>first fret would be slighly fatter so it represents 1/16 of the mass of
>the string instead of 1/18, and the part of the G string above the
>first fret would be slightly narrower so it is only 1/25 of the mass of
>the string (about 15% smaller diameter).

Dave, you have an interesting idea, but... first, the effect of
nonlinear string density is itself nonlinear: put a heavy section in the
middle of a vibrating length and it modifies the frequency much more
than a density modification near the end of a vibrating length. I think
you'd be hard pressed to make a significant frequency change just by
tweaking the density of a fret-distance at the very end of a vibrating
string.

(this could be easily verified by clamping a small fishing sinker to the
middle of a guitar string, then successively closer to the bridge. I do
not have a guitar, or a sinker, but I'm quite certain of the results.)

Also, I'm pretty sure you'd get nonharmonic partials as a result of such
a trick, and they'd be different for each note you played. Perhaps
interesting? It'd be fun to hear.

And, just to be a complete and total Captain Bringdown (sorry!), such
a scheme would be extremely sensitive to a given guitar's string length,
and each string would have to be carefully aligned - even quite small
absolute tuning changes could pull a string significantly off where it's
supposed to align.

JdL

[Dave Keenan:]
>By what proportion would you have to increase the diameter of the
>string uniformly above the first fret in order to lower its pitch by
>29 cents (the difference between 25/24 and a 12-tET semitone)?

Good question! Setting up the second order differential equation is
easy enough, but solving it for nonlinear string density, I don't know
how to do. The ideal way to answer the question would be to model the
string in a finite element program capable of dynamic analysis, which,
unfortunately, I don't have access to at the moment.

Here's a wacky thought: could piano strings be made to have more
harmonic overtones (offsetting the effect of string stiffness, at least
partially) by careful invocation of nonlinear string density? If so,
would the result sound more pleasing, or less? Has this ever been done?

JdL