[tuning] innovative fretboards

V. D'e. -

I think you may actually have almost all the info you need. Let's suppose
you're going to use a scale length of 650mm. Divide 650 by 1.05. Your first
fret would be 619.04761905mm - call it 619.0 unless you're using an
electron microscope - from the bridge, or 650.0 - 619.0 = 31.0mm from the
nut. _Ordinarily, you'd continue on that way, plugging the answer of each
division into the next, thus:

619.04761905 / 1.05 = 589.569161, the 2nd fret position;
589.569161 / 1.05 = 561.49443905, the 3rd fret position;
561.49443905 / 1.05 = 534.75660861, etc.

Of course, your problem is different, and I don't know whether you have to
divide 619.04761905 by your next ratio of frequencies, or if you have to go
back and start with 650mm again, and run through the math until you get to
the fret you're figuring out. That is, if your ratio for the 2nd fret were
1.065, you'd figure:

650 / 1.065 = 610.3286385, the 1st fret position, which we ignore, except
to
plug it into the next calculation:
610.3286385 / 1.065 = 573.0785338, the 2nd fret position, the one we're
after.

After all this, sadly, you have to compensate all your figures. We've been
assuming that nut, frets and bridge are in a plane. Of course they're not.
When the player depresses the string, the vibrating length of the string
lies at an angle from the _top of the bridge _down to the plane of the
fretboard, or more precisely, the plane of the tops of the frets. The
string length is the hypotenuse of a very skinny right triangle whose other
sides are that ideal length you just spent so much time calculating, and
the height from the top-of-fret plane to the top of the bridge. You could
do all that math, too, and after you did, you might want to consider that
when the string is depressed, it stretches. Or you might not.

Hope this doesn't leave you more depressed than the string.

BTW, I've been known to spout off before I really know what I'm talking
about - if anyone out there would care to check both my theory and the
practicality of it all, I'd much appreciate it.

- Chris Parker,
St. Paul, MN USA

I prophetically wrote:

>I've been known to spout off before I really know what I'm talking
about...

And sure enough, looking into the matter a bit further, I find that all
that stuff about the fretted string making an angle with the fretboard
doesn't have to affect your calculations one bit. What you do, apparently,
is go ahead and make that neck according to your ideal-world calculations.
Then just _move the bridge_ further from the nut. The amount to move it
varies with string thickness, which is why good bridges aren't straight
across.

Rik Middleton, author of _The Guitar Maker's Workshop_, from which I'm
getting all this, and hopefully not garbling it, suggests the following
compensations for a standard classical guitar:

high E 2.5 mm
B 3.0
G 4.0
D 3.0
A 3.5
low E 4.0 mm.

The book is published by Crowwood, BTW, and is full of good advise and
step-by-step directions for building a classical guitar.

- Chris

Sounds like a fascinating project for a Java or C+++ programmer.

>After all this, sadly, you have to compensate all your figures. We've been
>assuming that nut, frets and bridge are in a plane. Of course they're not.
>When the player depresses the string, the vibrating length of the string
>lies at an angle from the _top of the bridge _down to the plane of the
>fretboard, or more precisely, the plane of the tops of the frets. The
>string length is the hypotenuse of a very skinny right triangle whose other
>sides are that ideal length you just spent so much time calculating, and
>the height from the top-of-fret plane to the top of the bridge. You could
>do all that math, too, and after you did, you might want to consider that
>when the string is depressed, it stretches. Or you might not.

John Thaden
Little Rock, Arkansas, USA
http://www.flash.net/~jjthaden

> Sounds like a fascinating project for a Java or C+++ programmer.
Oh no, now there's a C+++ too? Sorry couldn't resist.

Anyway, this thing has been programmed already. First in Basic, by
Eduardo Sabat and later in Ada, by me, in Scala (the command SHOW STRINGLEN).
It calculates fret positions for one string at a time, taking the
string stretching into consideration. But getting the parameters
right is rather cumbersome. Also, string stiffness is not taken into
account and I don't know if its influence can be neglected.

Manuel Op de Coul coul@ezh.nl

Regarding programs for calculating fret placement:

>Anyway, this thing has been programmed already. First in Basic, by
>Eduardo Sabat and later in Ada, by me, in Scala (the command SHOW STRINGLEN).
>It calculates fret positions for one string at a time, taking the
>string stretching into consideration. But getting the parameters
>right is rather cumbersome. Also, string stiffness is not taken into
>account and I don't know if its influence can be neglected.
>
>Manuel Op de Coul coul@ezh.nl

These programs consider both the stringlength correction (hypotenuse of the
triangle) and the additional tautness (stretching) of the string? I wonder
if the tautness correction is independent of string gauge, i.e., the
starting tension on the string?

As for string stiffness, am I mistaken that stiffness only affects the
pitch of the partials and not the fundamental? (I suppose this is a
difficult question to test since, to do so, one would need to compare two
strings of identical per-length mass but rather different stiffness).

By the way, I neglected to acknowledge the immense help provided to me by
Manuel's huge database of extant tuning scales/schemes, during my recent
effort to sweeten the tuning of some existing MIDI files. Thanks! FYI,
the file is available at
ftp://ella.mills.edu/ccm/tuning/software/scales/scales.zip

John Thaden
Little Rock, Arkansas, USA
http://www.flash.net/~jjthaden

"The obvious mathematical breakthrough would be development of an
"easy way to factor large prime numbers." Bill Gates, The Road Ahead,
Viking/Penguin (1995)

John Thaden wrote:

>"The obvious mathematical breakthrough would be development of an
>"easy way to factor large prime numbers." Bill Gates, The Road Ahead,
>Viking/Penguin (1995)

Was Bill Gates serious, or did he mean "composite"? In either case I wonder
how the editor let that one by.

John Link

**************************************************************************
To purchase the CD "Live at Saint Peter's" by the JOHN LINK VOCAL QUINTET,
visit WWW.JOHNLINKMUSIC.COM, or write to johnlink@con2.com.
**************************************************************************

As far as guitar intonation -- I wouldn't get too fancy with these
calculations. On my Martin acoustic, the action (string height at 12th fret)
is about three times higher in the summer than in the winter. An adjustable
bridge (as on some of my electrics) would be a godsend for the intonation
and certainly far more valuable than any tweaking of fret positions.

John Thaden wrote:

> These programs consider both the stringlength correction (hypotenuse of the
> triangle) and the additional tautness (stretching) of the string? I wonder
> if the tautness correction is independent of string gauge, i.e., the
> starting tension on the string?

That's right. I found it a nice mathematical exercise. But like Paul said,
the practical value is limited. For instruments with a high bridge it may
be somewhat useful. When I applied it to my sitar, I found the input parameters,
like string elasticity, difficult to measure accurately. The method assumes
constant elasticity, which may not be true for nonmetal strings. If the frets
are movable then it's better to tune acoustically.

> As for string stiffness, am I mistaken that stiffness only affects the
> pitch of the partials and not the fundamental?

My thought was that when the string is bent on the fret, the stiffness might
affect the effective stringlength a little bit, e.g. shorten it, but I don't
know that.

Manuel Op de Coul coul@ezh.nl

--- In tuning@egroups.com, "James C. Parker" <ChrisParker1@c...>
wrote:
>
> V. D'e. -
>
> I think you may actually have almost all the info you need. Let's
suppose
> you're going to use a scale length of 650mm. Divide 650 by 1.05.
Your first
> fret would be 619.04761905mm - call it 619.0 unless you're using an
> electron microscope - from the bridge, or 650.0 - 619.0 = 31.0mm
from the
> nut. _Ordinarily, you'd continue on that way, plugging the answer
of each
> division into the next, thus:
>
> 619.04761905 / 1.05 = 589.569161, the 2nd fret position;
> 589.569161 / 1.05 = 561.49443905, the 3rd fret position;
> 561.49443905 / 1.05 = 534.75660861, etc.
>
> Of course, your problem is different, and I don't know whether you
have to
> divide 619.04761905 by your next ratio of frequencies, or if you
have to go
> back and start with 650mm again, and run through the math until you
get to
> the fret you're figuring out. That is, if your ratio for the 2nd
fret were
> 1.065, you'd figure:
>
> 650 / 1.065 = 610.3286385, the 1st fret position, which we ignore,
except
> to
> plug it into the next calculation:
> 610.3286385 / 1.065 = 573.0785338, the 2nd fret position, the one
we're
> after.
>
> After all this, sadly, you have to compensate all your figures.
We've been
> assuming that nut, frets and bridge are in a plane. Of course
they're not.
> When the player depresses the string, the vibrating length of the
string
> lies at an angle from the _top of the bridge _down to the plane of
the
> fretboard, or more precisely, the plane of the tops of the frets.
The
> string length is the hypotenuse of a very skinny right triangle
whose other
> sides are that ideal length you just spent so much time
calculating, and
> the height from the top-of-fret plane to the top of the bridge. You
could
> do all that math, too, and after you did, you might want to
consider that
> when the string is depressed, it stretches. Or you might not.
>
> Hope this doesn't leave you more depressed than the string.
>
> BTW, I've been known to spout off before I really know what I'm
talking
> about - if anyone out there would care to check both my theory and
the
> practicality of it all, I'd much appreciate it.
>
> - Chris Parker,
> St. Paul, MN USA

Thanks alot Chris,

I looked a little deeper and found the formula I need:
L - length of string; x - frequency ratio
L/1-1/x = fret positon

So my first fret mearsurement would be:
L/1-1/1.05
L/21

Simple. I'm almost ready to build my fretboard.

Since I'm taking the time to cut a fretboard to play pure just
harmonies, my dilemna now is intonation. Before I go ahead, purchase
a blank fretboard, and bring out the hacksaw, I want to make certain
that I am placing frets correctly. I want to produce the notes I have
on paper as exactly as I can. I know there much contemplation on this
topic. I need someone to summarize all the errors that occur in
stringed instruments, and how to diminish them. Here's what I know:

1. A string has mass. Therefore it's total length is shorter than it
measures. This is because it is more rigid the closer it reaches both
fulcrums. It doesn't vigbrate freely.

2. A string has tension. Depressing a string causes it's tension to
increase, causing errors in intonation.

Someone said that I am able to measure the frets to "perfect world
conditions" and then simply adjust the bridge to compensate for
intonation errors. I need some conformation.

I have seen replacement guitar nuts, that are not straight. Earvana
is one company that pushes this product as an intonation solver.

Another company named Dingwall, makes bass guitars with something
called a fanned fret system. They have extented the low bass string,
so that the frets change angle along the entire neck. They have done
this both to improve intonation and tone quality. What is a good
string length per diameter?

There is much to understand here. If anybody can straighten me out,
it would be appreciated.

ps. I've found another fret board builder, who is working along
similar lines as myself. Check out:
http://www.organic design.org/peterson/customers/catler_ji.html