19-tone Fret Lengths

Bobby Lee said:
> After reading in this group about 19TET (I'd never heard of it until a week
> ago!), I've come to realize that it would be fairly easy to retune my
> diatonic steel to it. I'd have to paint myself a 19 tone fretboard - that
> sounds like fun, actually. Tuning the strings, pedals and knee levers would
> be a snap with my Korg tuner.

I'm afraid I'm very ignorant about the pedal steel. However, here are
the 19-tone equal fret lengths for a guitar with a scale length of 25
inches. These frets give a stretched octave of 1202.7 cents, which is
what I recommend for the best harmony. These figures do not include the
necessary compensation (necessary on guitars at any rate) which would
_decrease_ these values by something on the order of 0.07 inches (more
for the bass strings, less for the treble). I give only the first 19
frets, since the little-used higher frets would perhaps be too close
together to play easily, and are, as I understand, less used in any
event. However, I'm not a guitar player, so someone else on the list who
has actually built a guitar or other stringed instrument will probably
be a better source of info. However, I can comment on fret placement and
compensation, if you have questions.

Fret number from nut Fret distance from nut in inches
1 0.90
2 1.76
3 2.60
4 3.40
5 (minor third) 4.18
6 (major third) 4.92
7 5.65
8 (fourth) 6.34
9 7.01
10 7.66
11 (fifth) 8.28
12 8.88
13 (minor sixth) 9.46
14 (major sixth) 10.02
15 10.55
16 11.07
17 11.57
18 12.05
19 (octave) 12.52

I apologize to those to whom what follows is elementary. Here is how to
find fret lengths in any temperament you may desire:
1. Convert from the cents of the interval you're interested in, to the
equivalent power of 10, that is to say to the logarithm (this is what a
logarithm is, a power of 10; for example the log of 100 is 2, because we
must raise 10 to the second power, i. e. 10 squared, to get 100)
2. Raise 10 to this power to get the decimal ratio of the cent interval
3. Divide total string length (or scale length, i.e. the length of
string from nut to bridge) by this decimal ratio; this will give you the
part of the string that will sound the desired interval, i.e. that will
sound the note above the note of the open string. I wish I could give a
diagram here...
4. Finally, to find the distance from the nut, subtract the distance
from the bridge from the total length of the string from nut to bridge

For example, to find the fret sounding the pitch 696.3 cents above the
note sounded by an open string of length 25 inches:
1. 696.3/(1200/log(2)) (divide 1200 by the log of 2, then take the
result and divide 1200 by that result) = .174672654984
2. 10^.174672654984 (raise 10 to the .174672654984 power) =
1.49510830872
3. 25 inches/1.49510830872 (divide the scale length 25 inches by
the result)
=16.7211966212 inches from the bridge
4. 25 inches - 16.7211966212 inches = 8.27880337882 inches from nut,
which we can round to 8.28 inches, as in the above table

These are not difficult calculations, and anyone with a pocket
calculator can perform them.

This does _not_ take into account compensation however, which typically
requires that the scale length be lengthened by 1 to 5 millimeters
(according to Fletcher & Rossing: Physics of Musical Instruments, p.
228). The bass strings particularly require to be lengthened, hence the
angled bridge concept. Electric basses are said to require even more
compensation. If this compensation is not made, fretted notes will be
sharper than open ones. Perhaps I should do a post on compensation, and
see if someone with more experience than I have (I'm not a guitar player
or maker) has some more thoughts on the subject. But there are some
makers, I am told, including even some professional guitar-makers, who
do not properly compensate, so their instruments do not give terribly
accurate results. Some of the compensation techniques are patented, I
believe.

The log of 2 refers to the definition of the cent, and is not to be
changed merely because we have stretched the octave.


SMTPOriginator: tuning@eartha.mills.edu
From: Gregg Gibson
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PostedDate: 18-12-97 01:54:49
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