Adventures in Just Intonation Guitar

An on-going account of an experiment

by Dante Rosati

Feb.12, 1999

After playing guitar in standard equal temperament for close to thirty years, I was finally moved to take the frets off an old guitar and see what would happen. What finally convinced me was listening to a recording of Lou Harrison playing his music, "Cinna", on a justly tuned piano. The sounds and harmonies are fascinating and beautiful, as well as strange to ears used to equal temperament. Now, its not like I've never heard anything outside of ET. I've written Csound computer pieces using Phi ratios, prime numbers and frequencies of the sun's spherical harmonics. I play blues on the guitar which uses microtonal pitch bending, as well as sitar which also uses microtones. But I think there is a difference here - in both the blues and indian classical music microtones are heard as intervals, rather than harmonically. In computer music, the pitch continuum is sliced and diced any way imaginable, and the spectra of the sounds themselves can be constructed however you want as well. To me having a harmonic and contrapuntal instrument in an alternate tuning is something different. Here intervals are discrete without having to bend notes across a continuous pitch space, and exactly tuned intervals can be combined harmonically and contrapuntally. In addition, there is something about a flesh and blood (or should I say wood and string) instrument that is lacking in a synth keyboard or computer sound. That is why while I occasionally make forays into digitally synthesized music, I always come back to my box with six strings.

As William Sethares has shown, consonance and dissonance are a function of the spectra of the constituent sounds. When I wrote PHI, I used the same ratios for all parameters: the partials in the sounds, the envelopes of the partials, the ratios between sounds, the durations, volumes and the section lengths. This is only possible on computer, since there are no vibrating bodies in nature that produce only Phi related partials. A guitar string, on the other hand, is a classic vibrating body which produces integer multiple partials, and the consonance or dissonance of various intervals is the result of this.

The frets of an ET guitar impose a tuning which does not completely "resonate" with the sound of the string itself. Without going into the history of tuning and temperament (see here or here) I will only say that equal temperament is an idealized Pythagorean tuning, and tons of great music has been played, and continues to be played, using it. But Lou Harrison's piano gave me a glimpse of something else.....

O.K, so I get a pair of pliers and a screwdriver and pry the frets off of a one-time great classical guitar that I haven't used for a decade or so. Its got a bunch of cracks but maybe I can do something about them too at the same time. The frets come out without too much trouble, although a few fragments of the fingerboard flake off as I pull them out. Next time I will remove frets a little more gently.

After a good cleaning and a little sanding, I fill the grooves with wood putty, and fill in the flaked bits too. After it dries I sand and then put some more putty on. Another sanding produces a nice smooth blank fingerboard. I paint it white with some left over housepaint so that I can put easily seen marks on it with a pencil.

My idea at this point was to leave it fretless - why have to limit yourself to any one tuning? Violinists dont need frets, after all. I put the strings on, tune them up and start playing - thud, thud thud. Instead of nice clear notes, all I hear is muffled thuds produced by stopping the strings with my fingers. Hmmm.. maybe I just have to get used to it. In any case, I find that I can play pretty well even without the frets, it's not so bad. Having experimented alot with harmonics, I can hear the difference between a justly tuned third and an equal tempered one. I try using my fingernail to stop the string, and now the sound is alot better. Sliding around on the strings, I sound like an Indian sarod player. That's cool, but I want to play chords. They are a little harder fretless because on a fretted guitar you can stretch into a fret and get the right note. Now, I have to stretch each finger to its exact position, which is tricky.

Right away I am made aware of the question of the tuning of the open strings. Should I leave the standard tuning? When I improvise Indian style on guitar I usually use low D tuning so the bottom three strings can be a kind of tambour. Sometimes I tune the third string to f# also. So I think what Im moving towards is a open tuning. Then I think, what if I tune the string in fifths and fourths? That is : 1/1, 3/2, 2/1, 3/1, 4/1, 6/1? This is the most general 3-limit tuning which will accomodate other tunings on the strings themselves. I tune the sixth string way down so that the first string is not going to snap. With the sixth tuned to somewhere between C and C#, the first string is high, but holds up. So the guitar is tuned C-G-C-G-C-G. I mark the edge of the fingerboard at 1/2 the scale length, 1/3, 1/4, 1/5, just to orient myself. I use red for 1/2 and 1/4, green for 1/3 and blue for 1/5 and 2/5. The whole guitar resonates powerfully with the open tuning. I like it.

After playing with this set up for awhile I can see that in order to stay fretless I'm going to have to use my fingernail and play sarod style. Otherwise I'm going to have to put some frets on in order to play chords. I decide to go for the frets. But now I have to decide: how many frets and where?

FRETTING ABOUT RATIOS

Since a guitar string produces whole number partials, I wanted to use a tuning which would work with this rather than against it.

Just intonation is the system of tuning which most directly reflects the stucture of the sounds of a vibrating string. There are certain intervals which are highly consonant because of the way the partials of the constituent sounds match up. The most consonant interval is of course the unison (1/1), and after that the octave (2/1). Then comes the perfect fifth (3/2) and the fourth (4/3). These are the only consonances that were recognized by the Greek theorists and they form the basis of their tetrachord system. All the other notes and intervals were considered dissonant to varying degrees, and that is why there are a cornucopia of possible notes within the tetrachord.

All ratios can be thought of as intervals within the harmonic series. Thus 3/2 can be thought of as the interval between the second and third partial, 4/3 as the interval between the third and fourth partial, etc. Thus, the harmonic series, as exhibited by a vibrating string, has encoded within it all possible intervals that can be expressed as ratios of whole numbers. Roughly speaking, the smaller the numbers involved, the more consonant the interval.

Therefore, in choosing a set of ratios within an octave which will make up your vocabulary of tones, you start with the lowest integer ratios and work your way up the series. The question then becomes where to stop and how many of the ratios to include. The more ratios you use, the more sounds you will have at your disposal. However, there are a few things to consider. First, frets can only be so close together before it becomes difficult or impossible to play. Also, there is a limit to what our ears can hear. There's no point in having two different ratios in your scale if, for all practical purposes, they are indistinguishable. So, you can generate a scale using rules, but there comes a point where the reality of execution becomes a limiting factor.

There is a special connection between prime numbers and ratios. All ratios may be reduced to prime numbers. For example 9/8 can be expressed as 3*3/2*2*2. We see that 3 is the largest prime number here, so 9/8 is called a "3-limit ratio". Ratios belonging to the same limit can be thought of as members of a family or species. The familiar 3-limit ratios are 3/2 (fifth), 4/3 (fourth), 9/8 (whole step, also the space between a fifth and a fourth), but of course there are many others (in fact, an infinite number). 5-limit ratios include 5/4 (major third), 8/5 (minor sixth) 6/5 (minor third) and 5/3 (major sixth). These are fundamental intervals of the kind of music that we are accustomed to hearing. With the introduction of 7-limit ratios, we begin to leave familiar waters. These ratios have a sound all their own. This also applies to 11-limit, 13-limit and onwards. However, the major consonances (lowest number ratios) function as such strong basins of attraction that high-prime ratios, if they stray too close, will be heard as mistuned consonances or simply as indistinguishable. So while there are an infinite number of possible ratios, the number of usefully distinguishable ratios is definitly finite.

So, one parameter of choice is what prime limit are you going to use? The other parameter is how "high" are you going to go in the harmonic series. That is, 16/9 is a 3-limit ratio but it is much higher up in the harmonic series than 3/2 or 4/3. I believe that it is the interplay of these two factors: prime limit, and "height" in the harmonic series that influences what we hear as consonance or "color".

If you study the harmonic series (which may also be thought of as the series of integers) it becomes apparent that octaves form the framework. 1/1 - 2/1 - 4/1 - 8/1 - 16/1..etc. Within this framework, the other ratios make their appearance. Thus, the first octave (1/1-2/1) is "empty". The next octave (2/1-4/1), contains 3/2 and 4/3. The next octave (4/1-8/1) introduces 5/4, 6/5, 7/6 , 7/5, 7/4, 8/7, and 8/5. Higher octaves contain more and more new ratios, as well as all the ratios from previous octaves. So it seems natural to group ratios by octave limits. We can speak of all the ratios in the second, third or fourth octave, etc. In addition, ratios within an octave limit can be grouped as to their prime limit as well. We see that up to the fourth octave limit (8/1) there are 3, 5, and 7-limit ratios. The next octave (up to 16/1) introduces 11 and 13 limit ratios.

So we can make a scale of ratios of prime limit "P" and octave limit "O".

A scale where P=5 and O=3 (that is 2^3 = 8/1) would consist of the following ratios:

1/1 6/5 5/4 4/3 3/2 8/5 5/3 2/1

or:

unison, minor third, major third, fourth, fifth, minor sixth, major sixth, octave.

If instead we use P=7 and O=3 we would have:

2/1 -octave

7/4 -septimal or harmonic seventh

5/3 -major sixth

8/5 -minor sixth

3/2 -perfect fifth

7/5 -septimal tritone

4/3 -perfect fourth

5/4 -major third

6/5 -minor third

7/6 -septimal minor third

8/7 -septimal whole tone

1/1 -unison

This is an 11 note scale which I'm sure is quite seviceable, but somewhat limited. There is too big of an empty space between 1/1 and 8/7, and also between 7/4 and 2/1. The major seventh (15/8) is SO incredibly beautiful that I wouldn't want to be without it in my scale. So it seems that I must go beyond P=7 and O=3.

When O=4 (up to 16/1), P can be as high as 13. But this produces too many ratios to be of practical use on a guitar with a 65cm scale length. It turns out that P=7, O=4 produces just the right balance between too many ratios and two few, while including all the tasty ones:

2/1 octave

15/8 classic major seventh

9/5 just minor seventh

16/9 Pythagorean minor seventh

7/4 harmonic seventh

12/7 septimal major sixth

5/3 major sixth

8/5 minor sixth

14/9 septimal minor sixth

3/2 perfect fifth

10/7 Euler's tritone

7/5 septimal tritone

4/3 perfect fourth

9/7 septimal major third

5/4 major third

6/5 minor third

7/6 septimal minor third

8/7 septimal whole tone

9/8 major whole tone

10/9 minor whole tone

16/15 minor diatonic semitone

1/1 unison

Now there is actually one more ratio that comes out of P=7, O=4: and that is 15/14. However, its inversion, 28/15 is outside the O=4 limit and all the other ratios have their inversions within the limit. In addition, 15/14 is only about 8 cents different from 16/15 and can barely be distinguished. Not to mention that the frets for the two ratios would be way too close. For all these reasons, it has been expunged.

Notice that the successive ratios are all superparticular or epimore ratios, widely held since the time of the Greek theorists to be especially nifty.

Speaking of nifty, check out the symmetry:

So, this is the scale I decided on. Only fretting the guitar and playing around with this scale will prove if it is musically viable. So its time to move from theory to practice.

PUTTING ON THE FRETZ

Putting regular frets on a guitar might be within my limited woodworking abilities, but a just fretboard requires partial frets, and I wouldn't even know how to saw partial fret grooves, at least not in my living room. After mulling it over I hit on the idea of using regular ole' #16 Ga. wire and epoxy. It ain't no big thing to cut 1cm pieces and glue them on. Having done three strings so far, it works great and is not so permanent that I can't consider changing my plans. But I'm getting ahead of myself...

First I had to mark off the locations. Now I always thought you just measured the scale lenghth and computed from there. But its not so easy to tell exactly where the string begins or ends as it lies across a bridge or nut. So, knowing that harmonics never lie, I thought to find the 1/2 point of the strings by using the harmonic. I did so with the first string, good enough. But rather than assuming that was all I needed to cut a wire that would span the whole fretboard, on a whim I checked the octave harmonics on the other strings as well and marked them. Imagine my surprise when they were all in a slightly different spot! Then it dawned on me (duh), thats why electric guitars have a separate adjustable bridge for each string - the varying thickness and tensions of the strings change the way notes are produced. So, if you want to have one fret for all six strings, each string has to be a slightly different scale length so they will be in tune. But my classical guitar, and all other classical guitars I have seen, have ONE bridge for all six strings, AND one fret for all six strings. What does this mean? It means all these guitars are out of tune! Yikes! Suddenly it all made sense - whenever I check the octave harmonic against the 12th fret note on a classical guitar there is always one or two that seem not to match up. I had always ascribed this to an untrue string, or had just glossed over it because I had no explanation for it. Now I know why, but its not pretty! Now, it seems to me that it wouldn't be all that difficult to make a bone that had angled segments for each string that would correct this, and I have a hazy memory of seeing just such a bridge somewhere, but its definitly not the norm. So many guitars, some with high price tags, and all of them out of tune!

It was clear that measurment was not going to give me the accuracy of intonation that was the whole reason for this endeavor to begin with. What to do? It just so happens that there is a program by the name of JIcalc that is the very soul of Pythagoras incarnate. Yes, it will tell you fret positions, but better still it will play you the ratios you want. That way, I can, using my fingernail, find that exact location of the ratio in question and mark it. And I can do this on each string and place the frets for that string just where they need to be. So, every string has its own frets.

Now, finding the exact spot for each ratio is a meditative exercise in itself, and great ear training. I mark the spots, go away for awhile or a day, come back and check them again. After doing this a few times, I glue my wires. Then I color in the frets to help orient myself red for 2 and 3-limit, green for 5-limit and blue for 7-limit. So far I have the first three strings done, and I play with what I have. A couple of the frets here and there are pretty close together, but still managable. The intervals are beautiful. When I had one string done, it was a matter of hearing each ratio against the drone of the open strings. Now with two and then three strings fretted, other intervals and triads are being heard. It just a matter of hours and hours of improvisation and experimentation to hear what sounds like what with what. So far its been heaven.