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.
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:
| octave | 2:1 |
| fifth | 3:2 |
| major third | 5:4 |
| minor third | 6:5 |
| major second | 9:8 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.
| note | ratio rel. to a | freq. | cents rel. to c |
|---|---|---|---|
| a | 1:1 | 440 Hz | 884.359 |
| e | 3:2 | 330 Hz | 386.314 |
| d | 2:3 | 293.33 Hz | 182.404 |
| c# | 5:4 | 275 Hz | 70.672 |
| c | 6:5 | 264 Hz | 0.000 |
| f | 4:5 | 352 Hz | 498.045 |
| f# | 5:6 | 366.67 Hz | 573.433 |
| b | 9:8 | 495 Hz | 1088.269 |
| g | 9:10 | 396 Hz | 701.955 |
| g# | 3:2 * 5:4 | 412.5 Hz | 772.627 |
| eb | 3:2 * 15:16 | 309.375 Hz | 274.582 |
| bb | 16:15 | 469.333 Hz | 1082.430 |
Or sorted by pitch, instead of by decision moment, it becomes:
| note | ratio rel. to a | freq. | cents rel. to c |
|---|---|---|---|
| c | 6:5 | 264 Hz | 0.000 |
| c# | 5:4 | 275 Hz | 70.672 |
| d | 2:3 | 293.33 Hz | 182.404 |
| eb | 3:2 * 15:16 | 309.375 Hz | 274.582 |
| e | 3:2 | 330 Hz | 386.314 |
| f | 4:5 | 352 Hz | 498.045 |
| f# | 5:6 | 366.67 Hz | 573.433 |
| g | 9:10 | 396 Hz | 701.955 |
| g# | 3:2 * 5:4 | 412.5 Hz | 772.627 |
| a | 1:1 | 440 Hz | 884.359 |
| bb | 16:15 | 469.333 Hz | 1082.430 |
| b | 9:8 | 495 Hz | 1088.269 |
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):
| deviation | note | ratio rel. to a | freq. | cents rel. to c |
|---|---|---|---|---|
| 0.0 | c | 6:5 | 264 Hz | 0.000 |
| -21.5 | g | 9:10 | 396 Hz | 701.955 |
| 0.0 | d | 2:3 | 293.33 Hz | 182.404 |
| 0.0 | a | 1:1 | 440 Hz | 884.359 |
| 0 | e | 3:2 | 330 Hz | 386.314 |
| -21.5 | b | 9:8 | 495 Hz | 1088.269 |
| 0.0 | f# | 5:6 | 366.67 Hz | 573.433 |
| 0.0 | c# | 5:4 | 275 Hz | 70.672 |
| 0.0 | g# | 3:2 * 5:4 | 412.5 Hz | 772.627 |
| 19.6 | eb | 3:2 * 15:16 | 309.375 Hz | 274.582 |
| 0.0 | bb | 16:15 | 469.333 Hz | 1082.430 |
| 0.0 | f | 4:5 | 352 Hz | 498.045 |
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.
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)
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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 | ||||||
|---|---|---|---|---|---|---|
| string | 0 | 1 | 2 | 3 | 4 | 5 |
| 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
