How to tune a guitar - Calculations

8 June 2003

Maths

When describing my guitar tuning method, I didn’t use much theory as the basis, but preferred experience. Nevertheless it is also interesting to apply some tuning theory and mathematics in hindsight, to see what frequencies and intervals the tuning method results in, and to try to explain why it sounds as good as it does.

Anecdote

First, an anecdotal aside:
I was already interested in tuning problems at a young age, in 1972 or thereabouts. And I thought I understood it rather well. I recognised the advantages of even temperament, but also thought it was pretty easy to design a pure tuning based on interval frequency ratios:

octave2:1
fifth3:2
major third5:4
minor third6:5
major second9:8 or 10:9

That leaves some arbitrary choices to be made, but that seems easy. So yesterday, in 2003 at the age of 48, I did that again, and set up a table like I did so many times before, just for the fun of it.

noteratio
rel. to a
freq.cents
rel. to c
a1:1440 Hz884.359
e3:2330 Hz386.314
d2:3293.33 Hz182.404
c#5:4275 Hz70.672
c6:5264 Hz0.000
f4:5352 Hz498.045
f#5:6366.67 Hz573.433
b9:8495 Hz1088.269
g9:10396 Hz701.955
g#3:2
*
5:4
412.5 Hz772.627
eb3:2
*
15:16
309.375 Hz274.582
bb16:15469.333 Hz1082.430

Or sorted by pitch, instead of by decision moment, it becomes:

noteratio
rel. to a
freq.cents
rel. to c
c6:5264 Hz0.000
c#5:4275 Hz70.672
d2:3293.33 Hz182.404
eb3:2
*
15:16
309.375 Hz274.582
e3:2330 Hz386.314
f4:5352 Hz498.045
f#5:6366.67 Hz573.433
g9:10396 Hz701.955
g#3:2
*
5:4
412.5 Hz772.627
a1:1440 Hz884.359
bb16:15469.333 Hz1082.430
b9:8495 Hz1088.269

Or sorted to "next fifth" order, to make it comparable with Pierre Lewis’s Java Tuner (difference in cents, between interval between this and next note, and the pure 5th, added in first column):

deviationnoteratio
rel. to a
freq.cents
rel. to c
0.0c6:5264 Hz0.000
-21.5g9:10396 Hz701.955
0.0d2:3293.33 Hz182.404
0.0a1:1440 Hz884.359
0e3:2330 Hz386.314
-21.5b9:8495 Hz1088.269
0.0f#5:6366.67 Hz573.433
0.0c#5:4275 Hz70.672
0.0g#3:2
*
5:4
412.5 Hz772.627
19.6eb3:2
*
15:16
309.375 Hz274.582
0.0bb16:15469.333 Hz1082.430
0.0f4:5352 Hz498.045

Entering these frequency values into the applet reveals that I did what all those others, hundreds of years ago, did before me, except that they did it much smarter: try to hide that wolf somewhere, preferably where it isn’t too conspicuous. In my tuning, designed on paper with a calculator, without actually hearing it, the wolf is conspicuous: in the example piecelet in C major, it already appears in the G-chord just before the end. Not a very smart choice indeed. Live and learn.

So this applet (also try this extended version) was a real eye-opener, well, er, ear-opener to me.

The strange thing though, is that when I play that tune (Pierre Lewis’s example piecelet in C major, that is) on the guitar (after rewriting the score, because I find reading the bass key quite difficult!), a guitar tuned following my own recommendations, there is no wolf in sight! Apparently, my guitar tuning method does not result in what I always thought would be a nice pure tuning, but isn’t: this one’s better than that.
(Not surprising of course, because playing in a pure tuning on a fretted instrument is possible only if you restrict yourself to a single key).

This makes me curious, I’d really like to know why my tuning works, so I’ll pick up that calculator again.


Guitar maths

So back to guitar tuning we go. What I’m actually doing there, to put it simply, is using octaves, an occasional fifth, and the even temperament of fret placement. That can be expected to result in a mixture of pure and even temperament, which should be pretty close to even temperament.
Let’s see if that’s so.

Below are the frequencies that are set by using the method, also expressed in cents relative to 110 Hz, modulo 1200:
(The frequency of d' on the b string is calculated by first going, from the 165 Hz e on the d string, to the open d string, going down by two 100 cent even temperament steps, and then one octave up: the result is 293.996577 Hz)

fret
string012345
e' 330
b 294 330
g 220
d 165
A 110
E 82.5
fret
string012345
e' 702
b 502 702
g 0
d 702
A 0
E 702

Now assuming that each fret adds almost exactly 100 cents - and they should when using high quality strings at the tension the guitar builder expected you would - we get:

fret
string012345
e' 702 802 902 1002 1102 2
b 202 302 402 502 602 702
g 1000 1100 0 100 200 300
d 502 602 702 802 902 1002
A 0 100 200 300 400 500
E 702 802 902 1002 1102 2

So it now turns out that what I really designed, without knowing it, is a manual way of tuning a guitar to even temperament, give or take 2 cents.

That means that tuning each of the open strings to a piano that was well-tuned in even temperament, or to an electronic tuning device, or even to all six notes of a pitch-pipe if it is accurate enough (the one I had many years ago wasn’t, perhaps modern ones are), although I didn’t recommend these methods, should work just as well or even better than mine!

So this whole exercise was an elaborate way of finding out the obvious! But I enjoyed it and don’t regret it.    I hope you do and don’t too.


Copyright © 2003 R. Harmsen

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