This article is about my experience building several guitars based on just intonation rather than 12-tone equal temperament (12-TET). My aim is to show that just intonation is a viable alternative to 12-TET, even on fretted instruments, and that with a little careful design, one may build a guitar that can play familiar, western music with better intonation than would be possible in 12-TET.
Secondarily, I hope to provide a decent introduction to the mathematical foundations of tonal music. That said, I should also note that there's really nothing "original" in this article, from the standpoint of music theory. I am merely trying to draw attention to what is already known. However, much of this knowledge is hard to find if you don't already know what you're looking for, and it isn't as well known as it should be.
I should also note that I'm not an expert on music theory in general, and only a casual guitarist. This article assumes a basic familiarity with musical terminology (octaves, semitones, names of intervals, scales, major, minor, etc..) and basic math. The underlying math of just intonation is conceptually simpler than 12-TET, since it's based on fraction multiplication rather than logarithms.
This isn't intended as a tutorial for building guitars in general. For that, see my cigar box guitar tutorial, or any good book on the subject.
In order to present just intonation as an alternative to equal temperament, I will first describe that which it is an alternative to.
Nearly all modern music is composed and performed in 12-tone equal temperament (12-TET), in which the octave is logarithmically divided into twelve equal-spaced intervals, of which seven are used in any major scale. The frequency of any note is the same as it's lower neighbor multiplied by the twelfth root of 2, and the resulting musical interval is called a semitone. (In tuning nomenclature, a semitone is an interval of 100 cents -- there are 1200 cents in an octave.) This may seem somewhat arbitrary, and it is, but there are good reasons for the popularity of equal temperament in general, and 12-TET in particular.
Regarding equal temperament first: ET makes it possible to change key at any time, which is something we've come to expect from modern music. Also, and of perhaps greater concern to guitarists, ET is a natural fit for fretted instruments. Typically, any given fret will cross all strings at a right angle, so it follows that the spacing of the frets should relate to the tuning of the strings in a straightforward way, so that every intersection of string and fret should result in a usable note. The easiest way to accomplish this is with a regular grid, like a guitar.
There is a particular reason why we use 12-TET, and not, say, 10-TET or 15-TET. It has to do with the physics of musical instruments and a nice mathematical coincidence.
In most tonal music, there are certain tones that sound good together, and certain tones that don't. The ones that do generally have frequencies that are some integer ratio with respect to each other. For instance, if one string vibrates twice in the same time it takes another string to vibrate three times, we call the interval formed by the two notes a perfect fifth (being the interval between the root and the fifth note of the major scale). Note that the major scale was around long before equal temperament. The whole diatonic major scale is sometimes described by whole number ratios:
| 1:1 | I - unison (tonic) |
| 9:8 | II - major second |
| 5:4 | III - major third |
| 4:3 | IV - perfect fourth (subdominant) |
| 3:2 | V - perfect fifth (dominant) |
| 5:3 | VI - major sixth |
| 15:8 | VII - major seventh |
| 2:1 | IIX or I - octave |
This way of describing musical intervals as ratios is the basis of just intonation.
These notes are all tuned relative to the unison, which is often C (261.6 hz), but can be any note, or even any arbitrary frequency. (A440 has only been a standard reference pitch for about a hundred years.)
Of the ratios shown above, only the octave (i.e. a doubling of frequency) is represented exactly in 12-TET. The perfect fourth and perfect fifth, very important intervals in nearly all tonal music, happen to line up almost exactly as the tone five semitones and seven semitones above the tonic. The difference is about 2 cents (a cent being 1/100th of a semitone in 12-TET; an octave is 1200 cents). 2 cents is generally below the threshold of human perception, so fifths and fourths sound very good in 12-TET.
As an example of how the math works out, suppose C at 261.6hz is the root of the key, and we want to find the frequency of the perfect fifth of the scale (G). We get 261.6hz * (3/2) = 392.4hz as the ideal frequency in just intonation. In 12-TET, we get 261.6hz * 2^(7/12) = 391.95hz. That's pretty close.
One neat thing about pretending that a fifth is exactly seven semitones is that if you go up in pitch by a perfect fifth twelve times, you visit every note and return where you started (12 and 7 sharing no common factors). This is the "circle of fifths". The same is true of fourths.
What about the other intervals? Most don't fare so well in 12-TET. The major second is off by about four cents from 9:8, which is not bad. The minor third and the major sixth, however, are off by more than 15 cents from the tones they approximate, which is quite a lot. (Interestingly, they're off by equal amounts in opposite directions.) The major third and minor sixth are off by almost 14 cents. The Major seventh is off by almost 12 cents.
Going outside the major scale, the minor seventh, if we take it to represent the interval 7:4 (which is debatable), is off by over 31 cents!
Some of these intervals work out better on equal-tempered scales other that 12-TET. 19-TET and 31-TET are the two most popular alternatives. I have tried both, and they both "work" musically. 19-TET tends to sound a little strange, whereas 31-TET can sound fairly normal. However, on a guitar, 31 frets per octave requires very close spacing. It is by no means unplayable, but it is awkward. 31-TET may perhaps be best suited for something with a long fingerboard, like an electric bass.
One assumption common to 12-TET, 19-TET, and 31-TET is that the octave must come out exactly, even if nothing else does. It may be that there are equal temperaments that have a slightly-out-of-tune octave which are better matches for the other intervals. (Someday, maybe, I'll write a program to search for such a scale.)
That's enough background on ET. Just intonation, by contrast, does not require notes to be equally spaced. Rather, the pitch of each note is based on a whole-number ratio relative to the tonic of some particular key, just like the ratios we gave earlier for the major scale. Unlike ET, there isn't a definitive list of notes that must be present in order to qualify as just intonation. Any set of whole-number ratio will do, so there are an infinite (or more specifically, countably infinite) collection of notes to choose from.
Another difference is that in ET, you can change key at any time and still have the same set of notes available. In just intonation, some notes will only be available in certain keys. (Electronic synthesis and fretless instruments are exceptions to this, as they can, in theory, play any note within their pitch range.) In practice, however, we do retain at least some flexibility, as will be seen later.
Though we may lose some of the flexibility of ET, we gain in precision -- by being able to play notes exactly, we can play melodies and chords with greater consonance, and we can play notes that can't even be approximated in ET.
I should explain a little bit of what I mean by consonance. In general terms, it means the tendency of some combination of notes to go together in a pleasing way. Intervals that express tension are less consonant (or more dissonant), and if they sound like nothing but badly out of tune notes, we describe them as very dissonant. There are physical reasons why certain sounds are more consonant than others. I will give one explanation, but there may be others.
On most instruments, the sound of a single note is not just one exact frequency. Rather, it is a combination of frequencies that are multiples of the base frequency or "fundamental" of the note. For instance, if you play an A on a piano, and that note has a fundamental frequency of 440hz, you will also, at the same time, hear tones at 880hz, 1320hz, 1760hz, 220hz, 2640hz, 3080hz, etc on up. These are called harmonics. (On a piano, it's actually a bit more complicated than this due to the Railsback curve, but that's something you usually only need to worry about if you're tuning a piano.)
If you play an interval of two notes, what you really hear is the fundamental of both notes along with all their harmonics. This collection of tones might not go well with each other. Tones that are far apart are okay, and tones that coincide exactly are also okay. However, if two tones almost coincide but not quite, the difference of the two frequencies emerges as an audible tone distinct from the other tones, called a beat frequency. For instance, tones at 440hz and 443hz will produce a 3hz beat frequency, which will sound harsh, or dissonant.
Applying this example to a musical interval, if two notes are a perfect fifth apart -- say, 200hz and 300z, then the third harmonic of the one (200hz * 3 = 600hz) will coincide with the second harmonic of the other (300hz * 2 = 600hz). This sounds very pleasant. If one note is a little out of tune, the harmonics won't coincide properly and the result is less pleasant. This is one reason why music can sound better when notes are properly aligned with each other based on ratios of relatively small integers, rather than multiples of the twelfth root of two, which is a bit off for every interval but the octave.
Besides being more in tune, there is another benefit to just intonation that is harder to define but perhaps just as important. In equal temperament, I am often frustrated by the arbitrary-seeming rules handed down to us as music theory. To know that a major triad is composed of a note, and the note four semitones above that, and a third note three semitones above the second, is very simple and practical, but it doesn't tell me what a major triad really is. Why does a major triad sound better than some other collection of notes? I've heard it said that music is very mathematical, but where's the math? It seems like just a bunch of rules memorized by rote. We're told to do such-and-such because it sounds better that way, but there's no indication of why. When we describe intervals by mathematical values like "5:4", rather than arbitrary seeming names like "major third", it becomes clear what's going on. If this doesn't make sense yet, I hope it will in due course.
When building a guitar based on just intonation, there are two big design issues that must be solved before making the fingerboard. These are: where do we place the frets, and how will the strings be tuned relative to each other?
These are related problems, since the combination of the two will determine what notes can be played on the instrument. In general, we don't want our fingerboard to have an unreasonable number of frets, or to have frets that are too close together to be playable, and neither do we want an unusually large number of strings -- probably no more than six or seven. At the same time, we want to include all the notes that we're likely to want to play, and we want to avoid notes that aren't musically useful.
If we know we're only going to play in a certain scale, such as C major, we can get by with as few as two frets and four strings -- maybe less. If we want to play in multiple scales over a wide tone range, we will need more frets and more strings.
The easiest way to ensure that our guitar has all the notes we want is simply to imagine the guitar having one string, with the open string as the tonic note, and place a fret for every note (except the tonic) in our chosen scale. Later, when we add more strings tuned to different notes, each new string will interact with the tuning of that string to introduce some new notes we didn't specifically want and (we hope) some duplicates of the ones we do.
Wikipedia has a surprisingly comprehensive list of candidate notes (including impressive-sounding names) we could include in our scale. Which should we include, and which should we leave out? This is largely a matter of taste, but we can narrow it down a bit by asking ourselves what kind of music we want to play.
We get the tonic for free as the open string. If we want to play western music (and just about anything else), then the six other notes of the major scale is a good start, and if we throw in three more notes we get the natural minor scale as well. While we're at it, we can include the minor second and augmented fourth (i.e. the tritone) to get a complete chromatic scale.
Unfortunately, there isn't one universally-accepted just chromatic scale. For instance, the augmented fourth could be 7:5, or maybe it's 10:7. In cases like this, the easy thing is to include both. They're far enough apart not to be a problem.
The minor seventh is a problem though: it could be 7:4, 16:9, or 9:5, depending on who you ask. 7:4 and 9:5 are far enough apart (barely) to be individually playable if we were to include both, but 16:9 lands right between them and it's definitely too close to 9:5. My initial solution was just to omit 16:9. However, I now prefer to include the 16:9 and omit the other two. (16:9 is the "complement" of 9:8, which is a more important note in western music than the complements of the other two, which are 8:7 and 10:9. A good rule of thumb appears to be that if a note is important, then its complement is equally important. By complement, I mean the interval that you add to an existing interval to make an octave. More on that later.) Another workaround when neighboring notes are uncomfortably close is to just place a fret at the average, and then treat them as equivalent. That note would then no longer be a just note, but rather a tempered note. I have so far managed to avoid doing this.
The octave is another very good note to have (it is particularly useful, not just for music but also for setting the intonation). Musically, it is the same as the tonic, but it would seem weird not to include it in our list of desired notes.
Now that we have the western scale more-or-less covered, we might want to include some other notes from foreign scales, or even make up our own scale. How do we search for likely notes?
Considering that all of the notes in the major scale are ratios of small whole numbers, we could just search exhaustively for all possible ratios involving numbers less than some threshold. (Fortunately, many ratios are equivalent, such as 2:1 and 4:2. We can throw out any ratios whose numbers aren't relatively prime, which is most of them.) That is what I did for my first just intonation guitar: I included all ratios of numbers 10 or less, plus a few extras (the minor second, 16:15; the lesser undecimal neutral second, 12:11; the septimal major sixth, 12:7). I overlooked the major seventh (15:8) unfortunately, and wished that I hadn't, but otherwise it worked out well, even if the fingerboard was a little crowded.
One interesting way of categorizing notes is by their prime limit. The prime limit can be found by reducing each interval to its prime factors, and taking the biggest one. For instance:
The major scale is composed of notes with a prime limit of 5 or less. Keep in mind that even with a prime limit of 3, the total number of possible notes is infinite: we can get away with notes like 1024:729, for instance. Including a few 7-limit notes is not a bad idea. I kind of like 7:6 -- aptly named the subminor third. (It's useful for constructing dominant seventh chords.) Some of the 11-limit intervals may also be nice to have, but you probably won't miss them unless you're specifically interested in composing 11-limit music. ( composed music with 11-limit intervals -- after that, even he gave up.)
In my most recent guitars, I adapted the scale used by Dante Rosati (from whose website I got the idea of making a just-intonation guitar in the first place), with a few notes omitted. Here are some More detailed notes. (Though I borrowed his scale, my design differs quite a bit from his in that he uses partial frets made from 1cm lengths of wire, each of which only touches a single string, whereas I use standard frets that go all the way across.)
Once we've decided our scale, we need to know where to actually place the frets on the fingerboard. For this, I wrote a simple computer program, but it should be straightforward (though tedious) with a calculator.
For some interval of the form a:b, its distance from the nut is: s*(1-(b/a)), where "s" is the scale length (the distance from the nut to the bridge). 25 1/2" (647.7 mm) is a common guitar scale length.
If all of the preceding discussion seems overly abstract, here's a simpler way to choose fret locations which yields similar results: First, start with the scale length, which we'll call S. If we place a fret at the midpoint between the nut and the bridge, or a distance of S/2 from the bridge, we get the octave (2:1) fret.
Next, we divide the scale length into thirds, placing a fret S*1/3 and S*2/3 from the nut. That gives us our perfect fifth fret (3:2), and another fret that's an octave and a fifth (3:1).
Dividing the fingerboard into fourths gives us our perfect fourth fret (4:3), and our 2nd octave fret (4:1), if we have enough space for it on our fingerboard. The S*2/4 is the same as our octave fret (2:1), so we can skip that one.
We can keep going like this, dividing the fingerboard into fifths, sixths, sevenths, on up the harmonic series. You'll probably want to stop at 10, and then selectively add a couple others like the minor second and major sevenths. This is essentially just another way of thinking about scale layout that will achieve similar results.
Installing the frets on a just-intonation guitar is not substantially different than installing frets on an equal tempered guitar, so I'm not going to cover that here. If you opt to go with partial frets, though, it may complicate things a little. I don't have much experience with partial frets, but it seems like it might open up some possibilities.
Along with fret placement, we also need to know how the strings will be tuned. Unlike frets, you can easily change your mind and re-tune the guitar as many times as you like, though you may have to change strings if you make drastic changes.
We have a lot of options here. We could use:
One consideration for an instrument with a range that spans multiple octaves is that we'd like our scale to repeat at the octave -- if we go up or down an octave, we'd like the same notes to be available. This is trivially easy in 12-TET, since any set of intervals that adds up to 12 semitones is an octave. In just intonation, it's a little harder to construct an octave out of intervals. For instance, if we go up from the tonic by a major third (5:4) three times, we arrive at 125:64 instead of 2:1. There is, in fact, no way to construct an octave by stacking similar intervals. We can, however, construct an octave by stacking dis-similar intervals.
The rule for stacking intervals is that you treat them as fractions, and multiply to go up and divide to go down. For instance, suppose you start at the tonic (1:1) and go up by a major second (9:8) and then up again by a major fourth (4:3) -- what note do you land on? 1/1 * 9/8 * 4/3 = 36/24 = 3*12/2*12 = 3/2, which is a perfect fifth. So, a major second and a perfect fourth add together to make a perfect fifth.
Here are some useful equalities:
Notationally, I'm using addition of intervals to represent multiplication of the fractions. I think it's more intuitive to think composing intervals as addition.
There are many more equalities like these, but for now we'll just note that any interval can be turned upside down and multiplied by two to get a complementary interval that together with the original interval make up an octave. (Here is a more complete exploration of this idea by Harry Partch, from Dante Rosati's website.)
Having a scale that repeats at the octave along with irregular fret spacings means that we'll probably want to have some strings that match other strings, but an octave up or down.
Dante Rosati's approach was to tune his guitar in alternating fourths and fifths. Thus his strings are (from treble to bass):
The tonic is C, so this is a CGCGCG tuning.
I used this tuning on an acoustic just-intoned guitar, and it works very nicely. Strumming it open gets you a C5 chord (C without the major 3rd), which sounds very nice. Playing melodies is pretty straightforward, and it's not too hard to remember where the important notes are.
The downside of this tuning, as I found, is that a perfect fifth difference between adjacent strings is a pretty big jump. Scales require a bit more hand movement than I like, and most of the chords (more on chords in the next section) were physically difficult due to the long reach required.
Also, consider our ability to transpose from one key to another. We know that, for each note corresponding to the open tuning of a particular string, we can play a whole scale without leaving that string -- all the notes are there. Therefore, with a CGCGCG tuning, we can play all the notes of our chosen scale in the key of C and the key of G. So, we can start in C, then transpose a perfect fifth up or (equivalently) a perfect fourth down, and then return back to C.
If we want to transpose up by a fourth from C to F, though, it's not so easy. In order to solve both the problem of difficult chords and to add more options for key changes, I decided on a tuning system that repeats every three strings, starting at the tonic, then up a fourth, then up a major second to the fifth from the tonic, then it reaches the octave with another fourth.
Here it is on a six string. I find it convenient to think of the 3rd lowest string (i.e. the D string in standard tuning) as the tonic.
If we wanted a full 2-octave range, like a standard 6-string guitar, we might add a seventh bass string:
For my electric just-intonation guitar, I went with six strings, but I think the seventh string would be a very nice addition. However, I found that tuning the low 2:3 string down to 1:2 works well.
I've been pretty happy with my 4th/2nd/4th tuning. Chords are much easier to play, and melodies don't require as much unnecessary hand movement.
Constructing a table of notes for the whole finger board is pretty easy -- for each string, just multiply the note corresponding to that open string by the list of notes in our scale.
Without further ado, here's the full fingerboard of my electric 6-string just-intonation guitar:
String Tuning
| 0.0 | 38.89 | 62.23 | 69.14 | 88.90 | 103.72 | 124.46 | 155.58 | 177.80 | 186.69 | 207.43 | 233.36 | 248.92 | 266.70 | 276.58 | 290.41 | 311.15 | 345.72 | 363.01 | 373.38 | 388.94 | 400.05 | 414.87 |
| 2:1 | 32:15 | 20:9 | 9:4 | 7:3 | 12:5 | 5:2 | 8:3 | 14:5 | 20:7 | 3:1 | 16:5 | 10:3 | 7:2 | 18:5 | 15:4 | 4:1 | 9:2 | 24:5 | 5:1 | 16:3 | 28:5 | 6:1 |
| 3:2 | 8:5 | 5:3 | 27:16 | 7:4 | 9:5 | 15:8 | 2:1 | 21:10 | 15:7 | 9:4 | 12:5 | 5:2 | 21:8 | 27:10 | 45:16 | 3:1 | 27:8 | 18:5 | 15:4 | 4:1 | 21:5 | 9:2 |
| 4:3 | 64:45 | 40:27 | 3:2 | 14:9 | 8:5 | 5:3 | 16:9 | 28:15 | 40:21 | 2:1 | 32:15 | 20:9 | 7:3 | 12:5 | 5:2 | 8:3 | 3:1 | 16:5 | 10:3 | 32:9 | 56:15 | 4:1 |
| 1:1 | 16:15 | 10:9 | 9:8 | 7:6 | 6:5 | 5:4 | 4:3 | 7:5 | 10:7 | 3:2 | 8:5 | 5:3 | 7:4 | 9:5 | 15:8 | 2:1 | 9:4 | 12:5 | 5:2 | 8:3 | 14:5 | 3:1 |
| 3:4 | 4:5 | 5:6 | 27:32 | 7:8 | 9:10 | 15:16 | 1:1 | 21:20 | 15:14 | 9:8 | 6:5 | 5:4 | 21:16 | 27:20 | 45:32 | 3:2 | 27:16 | 9:5 | 15:8 | 2:1 | 21:10 | 9:4 |
| 2:3 | 32:45 | 20:27 | 3:4 | 7:9 | 4:5 | 5:6 | 8:9 | 14:15 | 20:21 | 1:1 | 16:15 | 10:9 | 7:6 | 6:5 | 5:4 | 4:3 | 3:2 | 8:5 | 5:3 | 16:9 | 28:15 | 2:1 |
For comparison, here is what this looks like as a completed fingerboard:
The table may be somewhat spacialy misleading, as the column alignment doesn't represent the true spacing between frets. For the most part, the frets are far enough away from each other that they won't present a playability issue with the exception of the 10:9 and 9:8 frets, which are only about 7mm apart, or about a quarter inch. Unfortunately, these are both important notes (as will be seen in the section on chords). The result of the closeness is that the 9:8 note sounds rather dull - it's difficult to play without muting the string somewhat with my fingertip, which is too large to fit in that space. For lack of a suitable workaround, I just accept this as a limitation.
The distance from the nut in millimeters is given on the top row. (This a 25-inch scale, or 622.3mm.) The open strings and the octave fret (half-way from the nut to the bridge) are highlighted. Note that I kept going after the octave and added a few notes that I thought were probably musically useful.
What suprised me was just how many usable notes there are. They aren't all winners; I can't imagine when I'm going to need, say, 64:45 or 56:15. On the whole, though, there are a lot of useful notes, and the major and minor scales show up quite prominently. That will prove very convenient as we move on to the next topic, chord construction.
This is where it starts getting interesting, and the elegance of the major and minor scales become apparent.
The major scale has seven unique notes, from which we can build three major triads, three minor triads, and a diminished triad. I'm going to pretend the diminished triad isn't there, because I haven't yet figured out how to deal with diminished triads. Major triads, though, are pretty simple. They're a set of notes with ratios of 4:5:6. Since notes that differ only by octave are considered equivalent, we could also write this as 2:3:5, or 4:6:10, or 5:8:10, or whatever, but I'll stick to 4:5:6 for now. Interestingly, these are all harmonics of the note two octaves below the root -- we can think of it as 1:4:5:6 if we like, and we can even stick in the missing harmonics to get 1:2:3:4:5:6. This would still be a major chord. If we add a seven onto a major triad we get, I believe, a dominant seventh (an accidentally appropriate name): 4:5:6:7.
Minor triads are a bit more subtle. They're composed of notes with the ratio of 10:12:15. The connection between them is less obvious than the notes in a major triad, but they can be re-formulated to make the pattern more clear: 60/6 : 60/5 : 60/4. As with major triads, they all have something in common with a certain other "virtual" note, but this time it's a note that is much higher -- two octaves and a fifth above the root.
Now that we have these formulas down, we can start building chords. We'll assume that the tonic (1:1) is C, and show the basic chords in C major and C harmonic minor.
These are built from the notes of the major scale, except that we build the second chord off of a root of 10:9 rather than 9:8, to make it work out a little nicer. I will follow the convention of writing major chords (4:5:6) with capital roman numerals, and minor chords (10:12:15) in lower-case.
Major chords:
Harmonic minor chords:
If we ignore the diminished chords, which I don't know what to do with, the other chords seem to fit nicely into place. The chords in the C major scale only require 8 distinct notes in order for the major and minor chords to work out exactly. Traditionally, the major scale only has 7 notes, but we only had to introduce the 10:9 note as an alternative to the 9:8 that would normally be used in its place.
This was a big surprise to me when I saw it, and confirms that the major scale isn't just some arbitrary collection of notes but rather they are a nice clean solution to a difficult constraint satisfaction problem. The harmonic minor works out pretty nicely as well. Both scales have chords with reasonable fingerings and they don't require notes that are particularly strange.
Deciding on the best voicing for a particular chord is largely a matter of preference, but here are some reasonable fingerings for the chords in C major:
I (C)
ii (d)
iii (e)
IV (F)
V (G)
vi (a)
Rather than give the actual note, I figured it was easier to understand if I labeled each fret according to what note it would be on the 1:1 string, where "x" means "don't play this string" and "1:1" means play it open.
It is interesting that we could play these six chords on a guitar with only three frets.
These basically work, but G major unfortunately suffers a bit, because it requires pressing down three strings on the 9:8 fret at the same time, which, as I mentioned earlier, is very close to the 10:9 fret. Consequently, G sounds rather dull and muted.
We can construct a G major from the notes 3:2, 15:8, and 9:4 anywhere on the fingerboard. It would be nice if we could use the 3:2 note on the 1:1 string, which conveniently has a 9:4 note (a fifth up from 3:2) two strings up on the same fret, but there is no 15:8 note on the 4:3 string to fill in the major third we need for our triad -- we would need to have included a 45:32 fret in our scale for that to have worked out. However, if we run our fret position calculation given above for 45:32, we see that it should be 179.78 mm from the nut. We already have a 7:5 fret that is 177.80 mm from the nut, a difference of less than 2 millimeters, which means it's 7.71 cents flat (if my math is right). It's not perfect, but it's not terribly far off either, and it gives us a much more playable G chord:
V (G)
The chords in C minor I'm not going to include, because I haven't come up with good fingerings for all of them. In theory, we could use the same C major chords for A natural minor, since the keys of C major and A natural minor are commonly regarded as having the same notes. However, I don't believe that C major and A natural minor are equivalent in just intonation the same way they are in 12-TET.
Here is a clip of how these chords (and a few variations) sound on my electric.
This is just a starting point, there are many other chords that are commonly used in modern music: dominant sevenths, suspended chords, etc...
Building a guitar in just intonation is, aside from fret placement, very much like building any other kind of guitar. One could even modify an existing instrument, with difficulty varying by instrument (an electric with a bolt-on neck would probably be pretty easy, an acoustic guitar looks like it would be pretty difficult unless you're good at undoing glue joints, or willing to fill the existing fret slots and re-saw new ones).
When building an instrument from scratch, it's a good idea to pay special attention to intonation, and to design the instrument in such a way that it can be easily adjusted. A more precise tuning system isn't much help if the intonation is off. On acoustics, I'm a big fan of movable bridges. Electrics, on the other hand, usually have individually-adjustable saddles.
One big surprise on my electric was that the open strings were much flatter than they were supposed to be relative to the fretted notes. This may have been due to the fact that pressing down on the string causes it to go sharp by a little bit, and when placing frets one should include some small correcting factor to compensate, or I may have just installed the nut in slightly the wrong place. In the end, I just removed the nut, cut a bit off the end of the fingerboard, and re-attached the nut a little closer in. This was a big improvement.
Testing intonation, it turns out, is easy with the right tools. Conventional guitar tuners work ok for basic tuning to get in the ballpark, but they're always going to be a little off, since they're usually designed for 12-TET. I've found that an oscilloscope works very well. The trick is to adjust the horizontal sweep to match the tonic note (1:1), which you'll probably want to set to middle C or something like that. (My oscilloscope's built-in horizontal sweep tends to drift on its own, so I use a laptop generating a sawtooth wave as an external source for the X-axis.) Now, if you play 1:1, you should expect to see a standing wave. If it drifts one way or the other, tune the string to make it stationary. Interestingly, most of the other notes, if the frets are properly aligned, result in a standing wave as well. If they drift a bit to the right or left, it usually means the intonation is off and you need to make some adjustments (if possible).
Besides being a very precise tuning aid, the oscilloscope is fun to watch in its own right.
Another lesson I learned is that tremolo systems can be more trouble than they're worth. They can be a lot of fun, but it generally just knocks everything out of tune. If you really want a tremolo arm, then I'd recommend planning to install a roller nut or a locking nut and a roller bridge.
I have built both an electric and a box-shaped acoustic just intoned guitar (in addition to some prototypes), and I'm pretty happy with them both. (Update: I now also have a nylon-string just intonation cigar box guitar and an XV-300 from GFS that I've converted over to just intonation.) Electrics and acoustics each seem to benefit in different ways from the improved consonance of just intonation. The subtle differences between various intervals seem more pronounced on the acoustic. The electric seems more sensitive to proper intonation -- if it's off, chords sound wrong, but if it's dead on they sound great.
Chords in general have a "hollow" sound -- though thirds do give a major or minor feel to a chord, they tend not to add the "edgy" feel they do in 12-TET. This could be good or bad, depending on what you're aiming for. One could make a reasonable case that the somewhat out-of-tune thirds add a lot to the character of 12-TET, and to take that away detracts from the music. On the other hand, with heavy overdrive, a perfectly in-tune major chords sounds very good.
Melodies sound more expressive, in my view.
Being stuck in one key and not really being able to use barre chords can be kind of a drag. Playing with other people is a little harder when you can't just change key to whatever everyone else is playing. (I really don't have much experience playing with other people -- I'm not sure if a just intonation guitar would clash with other 12-TET instruments or not, even in the same key.)
With an electric, it may be possible to transpose into any key with a pitch-shifter effects pedal.
In the end, I'd really need to either improve my musical ability or put one of these guitars into the hands of an accomplished musician to really explore the benefits of just intonation. In the meantime, I think this exercise has helped me a great deal in understanding the mathematical underpinnings of modern music, and I would recommend it for anyone wanting to make a guitar with a different kind of sound.
Here is a collection of short audio clips, which demonstrate how the tuning system sounds:
(It turns out that old hymns are a pretty good way to familiarize oneself to a new tuning system.)
These really aren't the greatest examples, but they should at least suffice to show that "normal" music is doable. I'll probably be adding more sound clips over time.
Pierce's The Science of Musical Sound is a very good overview of the things we've figured out about the way we perceive music.
Holmholtz (1821-1894) did some amazing research to lay the foundations of what we know about the perception of music.
Partch (1901-1974) applied Helmholtz's research to compose new music, based on the mathematical foundations of just intonation, but deliberately distancing itself from the influences of modern western music. He had to build new instruments to play his music.
Johnston studied under Partch for a time, and adopted just intonation for his own compositions, but unlike Partch he did not disdain western music. Rather, he used just intonation along with modern music theory and contemporary musical influences to write new music. He wrote a re-harmonization of Amazing Grace that's worth hearing. His book is more or less a collection of essays. One of the points he makes that has stuck with me is that by throwing out the approximations of ET and playing the exact note we intended, we make it possible to make musical distinctions that weren't possible before. Effectively, when we drain the swamp, a whole lot of new possibilities for subtlety and musical complexity arises.
Hopkin's book I included because it covers the fundamental mechanics of how musical instruments work, from the perspective of someone who isn't interested so much in copying existing instruments, as building new, unconventional ones.
| 2:1 | 32:15 | 20:9 | 9:4 | 7:3 | 12:5 | 5:2 | 8:3 | 45:16 | 128:45 | 3:1 | 16:5 | 10:3 | 32:9 | 15:4 | 4:1 | 64:15 | 9:2 | 24:5 | 5:1 | 16:3 | 6:1 | 32:5 | 20:3 |
| 3:2 | 8:5 | 5:3 | 27:16 | 7:4 | 9:5 | 15:8 | 2:1 | 135:64 | 32:15 | 9:4 | 12:5 | 5:2 | 8:3 | 45:16 | 3:1 | 16:5 | 27:8 | 18:5 | 15:4 | 4:1 | 9:2 | 24:5 | 5:1 |
| 4:3 | 64:45 | 40:27 | 3:2 | 14:9 | 8:5 | 5:3 | 16:9 | 15:8 | 256:135 | 2:1 | 32:15 | 20:9 | 64:27 | 5:2 | 8:3 | 128:45 | 3:1 | 16:5 | 10:3 | 32:9 | 4:1 | 64:15 | 40:9 |
| 1:1 | 16:15 | 10:9 | 9:8 | 7:6 | 6:5 | 5:4 | 4:3 | 45:32 | 64:45 | 3:2 | 8:5 | 5:3 | 16:9 | 15:8 | 2:1 | 32:15 | 9:4 | 12:5 | 5:2 | 8:3 | 3:1 | 16:5 | 10:3 |
| 3:4 | 4:5 | 5:6 | 27:32 | 7:8 | 9:10 | 15:16 | 1:1 | 135:128 | 16:15 | 9:8 | 6:5 | 5:4 | 4:3 | 45:32 | 3:2 | 8:5 | 27:16 | 9:5 | 15:8 | 2:1 | 9:4 | 12:5 | 5:2 |
| 2:3 | 32:45 | 20:27 | 3:4 | 7:9 | 4:5 | 5:6 | 8:9 | 15:16 | 128:135 | 1:1 | 16:15 | 10:9 | 32:27 | 5:4 | 4:3 | 64:45 | 3:2 | 8:5 | 5:3 | 16:9 | 2:1 | 32:15 | 20:9 |
| 1:2 | 8:15 | 5:9 | 9:16 | 7:12 | 3:5 | 5:8 | 2:3 | 45:64 | 32:45 | 3:4 | 4:5 | 5:6 | 8:9 | 15:16 | 1:1 | 16:15 | 9:8 | 6:5 | 5:4 | 4:3 | 3:2 | 8:5 | 5:3 |
| 2:1 | 32:15 | 20:9 | 9:4 | 7:3 | 12:5 | 5:2 | 8:3 | 45:16 | 128:45 | 3:1 | 16:5 | 10:3 | 24:7 | 32:9 | 18:5 | 15:4 | 4:1 | 64:15 | 9:2 | 24:5 | 5:1 | 16:3 | 6:1 | 32:5 | 20:3 | 64:9 | 15:2 | 8:1 |
| 3:2 | 8:5 | 5:3 | 27:16 | 7:4 | 9:5 | 15:8 | 2:1 | 135:64 | 32:15 | 9:4 | 12:5 | 5:2 | 18:7 | 8:3 | 27:10 | 45:16 | 3:1 | 16:5 | 27:8 | 18:5 | 15:4 | 4:1 | 9:2 | 24:5 | 5:1 | 16:3 | 45:8 | 6:1 |
| 4:3 | 64:45 | 40:27 | 3:2 | 14:9 | 8:5 | 5:3 | 16:9 | 15:8 | 256:135 | 2:1 | 32:15 | 20:9 | 16:7 | 64:27 | 12:5 | 5:2 | 8:3 | 128:45 | 3:1 | 16:5 | 10:3 | 32:9 | 4:1 | 64:15 | 40:9 | 128:27 | 5:1 | 16:3 |
| 1:1 | 16:15 | 10:9 | 9:8 | 7:6 | 6:5 | 5:4 | 4:3 | 45:32 | 64:45 | 3:2 | 8:5 | 5:3 | 12:7 | 16:9 | 9:5 | 15:8 | 2:1 | 32:15 | 9:4 | 12:5 | 5:2 | 8:3 | 3:1 | 16:5 | 10:3 | 32:9 | 15:4 | 4:1 |
| 2:1 | 32:15 | 20:9 | 9:4 | 75:32 | 12:5 | 5:2 | 8:3 | 45:16 | 128:45 | 3:1 | 16:5 | 10:3 | 256:75 | 32:9 | 18:5 | 15:4 | 4:1 | 64:15 | 9:2 | 24:5 | 5:1 | 16:3 | 6:1 | 32:5 | 20:3 | 64:9 | 15:2 | 8:1 |
| 3:2 | 8:5 | 5:3 | 27:16 | 225:128 | 9:5 | 15:8 | 2:1 | 135:64 | 32:15 | 9:4 | 12:5 | 5:2 | 64:25 | 8:3 | 27:10 | 45:16 | 3:1 | 16:5 | 27:8 | 18:5 | 15:4 | 4:1 | 9:2 | 24:5 | 5:1 | 16:3 | 45:8 | 6:1 |
| 4:3 | 64:45 | 40:27 | 3:2 | 25:16 | 8:5 | 5:3 | 16:9 | 15:8 | 256:135 | 2:1 | 32:15 | 20:9 | 512:225 | 64:27 | 12:5 | 5:2 | 8:3 | 128:45 | 3:1 | 16:5 | 10:3 | 32:9 | 4:1 | 64:15 | 40:9 | 128:27 | 5:1 | 16:3 |
| 1:1 | 16:15 | 10:9 | 9:8 | 75:64 | 6:5 | 5:4 | 4:3 | 45:32 | 64:45 | 3:2 | 8:5 | 5:3 | 128:75 | 16:9 | 9:5 | 15:8 | 2:1 | 32:15 | 9:4 | 12:5 | 5:2 | 8:3 | 3:1 | 16:5 | 10:3 | 32:9 | 15:4 | 4:1 |