I am a founding member of the Just Intonation Network, an international organization of composers, musicians, and theorists interested in acoustically pure tunings based on the harmonic series, and have published several articles in their journal 1/1, including the following:
* Superparticular Pentatonics * Fibonacci Gamelan Rhythms * Harmonic-Melodic Diagrams * A Tour Up The Harmonic Series * A Justly-Tuned Guitar * On Piano Retuning * Pentatonics I Have Known * Rational Notation
A JUSTLY-TUNED GUITAR
This article first appeared in 1/1, the Journal of the Just Intonation Network, Volume 2, Number 2, p.8 (1986).
* Introduction * Choice Of Frets * Playing The Thing * Conclusions * Details Of Construction * Table
Several years ago, I decided I needed a versatile instrument to play music using pure harmonies. My classical guitar seemed a natural choice to modify to play in Just Intonation, for several reasons: I like the sound of the instrument, it is portable and relatively easy to play, and it is capable of both melodies and harmonies. I originally intended that it should be a satisfactory solo instrument, for pieces of complexity comparable to the standard classical guitar repertoire. I also wanted to have access to a very wide variety of just intervals. These two desires, unfortunately, have proved incompatible; in satisfying the latter I precluded the former.
Choice Of Frets
One of the guitars with interchangeable fingerboards created by Tom Stone (belonging to Jonathan Glasier) had greatly impressed me with its versatility and ease of use. But, as I had more time than money, I decided to modify my own guitar, and I felt that an interchangeable-fingerboard system was beyond my level of craftsmanship. Thus limited to a single fingerboard, it was important to optimize my tonal resources.
Some guitars have partial frets across only certain strings, staggered with other partial frets across others, to give the desired scale in a particular tuning. This makes playing easier than if all the frets crossed all the strings, which would give many notes outside the desired scale. However, this scheme is limited to a particular tuning of the strings. This struck me as a serious drawback, since I envisioned choosing a tuning according to the needs of the composition, so I used only complete frets.
Ideally, any desired tone should be available on any string, but practically there is a limit to how close the frets can get and still be playable. (Other people have removed such limitations by using fretless guitars, but on a classical guitar the resulting sound is damped, somewhat like pizzicato violin, and I wanted clear ringing tones.) Thus, my choice of frets was a balance between wanting all tones and needing space for my fingers. To help choose, I laid out all the frets I was considering, to scale, on a piece of paper. I left extra space below the 2/1 and 3/2 frets to make them particularly easy to play.
To guide my choices out of the infinitude of possible intervals, I knew I wanted a harmonic scale of overtones 8-16:
1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 2/1
so all these frets were essential. (I marked these frets with abalone dots on the fingerboard to give some visual points of reference.)
To get those tones on strings tuned to 3/2, 5/4, 7/4, 9/8, or 4/3 (keeping in mind various possible tunings) I included the following frets (grouped according to the string requiring them):
13/12 7/6 4/3 5/3 11/6 11/10 6/5 7/5 8/5 9/5 8/7 9/7 10/7 11/7 12/7 10/9 11/9 13/9 14/9 16/9 33/32 21/16 27/16
The frets needed to get the 13/8 and 15/8 tones on some of the strings are lacking (specifically, frets 13/10, 13/7, 15/14, 39/32, and 45/32) in order to minimize crowding on the fingerboard. The only one of those I really miss now is the 45/32 fret, but it would be unplayably close to the 7/5, so when I need it I play on the 7/5 and bend up a little. (Originally I decided to exclude the 14/9 fret as too close to the 11/7, but years of playing made me want it so much I finally added it.)
In addition, I included two Pythagorean intervals:
and some other miscellaneous intervals from Partch's system to fill up some gaps:
21/20 16/15 27/20 32/21 40/21
As a result, there are 38 frets in the first 2:1, plus 14 more up to the 3:1. The number of different possible tones depends on the particular tuning. In any tuning I can get several overtone scales (not all complete) and a bewildering variety of other scales. Indeed, just about any scale I'm likely to want could be obtained with an appropriate tuning. For example, to get Partch's 43-tone scale, the tuning should include strings tuned to 1/1, 16/11, 16/5, and 8/5 (or 3/2), and so could be tuned to a utonality.
So overwhelming are the possibilities that I've used almost exclusively a single tuning chosen for simplicity, an open major chord (on D):
1/1 3/2 1/1 5/4 3/2 1/1
The tones available on each string in this tuning are shown in the Table, and the fingerboard is shown in the Photo
Playing The Thing
Having more than three times the normal number of frets, my guitar is very difficult to play. First there is the necessity of learning what all the frets are by ratio, and then figuring out what tones they give on each string. Also, the fingering of tones takes extreme precision; some of the frets are only 4 millimeters apart (center to center). However, all the tones can be played clearly (even though my fingers are 15 mm wide), with sufficient accuracy of finger placement. This required accuracy means that some stretches that might be relatively simple on a standard guitar become impossible on mine. Barred chords are out of the question on some frets but clear on others; 8 mm seems the minimum required separation. Furthermore, all music must be memorized, since it is necessary to watch where the fingers go.
On the other hand, the closeness of frets makes sliding along a string a much smoother effect. Single melodies can be played this way using the ears rather than the eyes to determine where the slides end.
As a result of the difficulties of playing, I am unable to play solo pieces of the complexity I originally imagined. The music that I do play falls into three categories. Most often I play melodic improvisations, alone or with other musicians, exploring one or more scales, sometimes with a fixed cycle of tonal changes. Also, I play some simple solo compositions, but as this is more demanding, I do it less often. Finally, I play pieces I've composed to record on several tracks on tape, allowing me to play duets, trios, etc., where each part is played on my guitar. This yields music which satisfies my taste for complexity but which cannot be performed live.
Apart from playing music, this guitar is superb for demonstrating just intervals and scales with a familiar, acoustic sound. Having so many different tones available facilitates comparisons. As a result I've come to know the distinct sounds of various just intervals; my ear has improved tremendously. One result is that tuning up now takes a long time. The strings must be as accurately tuned as possible, otherwise the advantage of the precise harmonies is nullified.
All things considered, I'd rather have a guitar with interchangeable fingerboards. I'd have several simple fingerboards using partial frets for a standard tuning to get particular scales of perhaps 12 to 18 tones per octave. These would make much of my music (that which doesn't wander too far tonally) much easier to play, perhaps even without staring at the frets. But I would also have one fingerboard much like the one I have now, to play music that uses many different tones or that uses particularly unusual scales. And since I am currently limited to the one fixed fingerboard (Tom Stone is out of business), I'm glad it's so versatile.
Details Of Construction
I began with a 1968 Giannini classical guitar from Brazil. With the strings off and the nut carefully removed, I heated the fingerboard with an electric iron to melt the glue holding it on, then pried it up, a bit at a time, using a table knife. Once the old fingerboard was off, I used it as a model in shaping a plain rosewood blank I had gotten from a guitar shop and sanded down to the right thickness. On the new blank, I cut a straight edge parallel to the length to use as a guide.
Then I laid out the frets with a carpenter's square and a metric tape measure. I would clamp the square in position for one fret, then use it as a guide in sawing the fret groove, using a Disston dovetail saw with all the kerf hammered out to get just the right thickness. Only after all the grooves were sawn did I taper the fingerboard blank to fit the neck. The curved end I cut with a jigsaw.
Then I hammered in all the frets and trimmed the ends. Pounding in frets is a delicate business, best done with a jeweler's chasing hammer, but I used a regular claw hammer and hit too hard in the middle, so the ends of the frets tended to stick up. Then I glued the fingerboard onto the neck, and made my second mistake: the board I used over the fingerboard to clamp it down wasn't wide enough, so the edges of the fingerboard actually warped away from the neck a little, from the moisture of the glue (aliphatic resin wood glue). Finally, I filed the frets level and smoothed their edges, rounded them with a fret file, and polished them with fine sandpaper, then steel wool.
(The warp of the fingerboard meant that the sides of the frets had to be filed down much lower than the middles to make the tops even.) After I had finished, a couple of the fret ends popped up, pulling up wood chips. (I hadn't realized that cutting fret grooves so close together would weaken the wood.) Recently I fixed this problem with a little epoxy, but that will make refretting extremely difficult if the frets ever wear out.
As for determining the positions of the frets, simple theory would predict that to get an interval of, say, 5:4 would need 4/5 of the open string length, so the fret goes 1/5 of the open length from the nut. However, in real guitars, a little adjustment needs to be made to account for pressing down the string, etc. In effect, one places the frets as if the string were slightly shorter than its actual length, but this effect changes both very close to the nut and above about one-third of the string. So I made measurements on the original frets and compared with the theoretical positions to estimate how to adjust theory to practice. The results sound just right.
ASSOCIATE PROFESSOR DAVID CANRIGHT
* Mailing Address: Mathematics Department, Code MA/Ca Naval Postgraduate School Monterey, CA 93943 USA * Email Address: DCanright@NPS.Navy.mil * Office: 339 Glasgow Hall * Phone: (408) 656-2782 * Fax: (408) 656-2355