In theory, Just Intonation knows no bounds. That is, if Just Intonation includes all intervals whose frequency ratios are ratios of whole numbers, then it includes an infinity of intervals, as there are an infinite number of rational numbers, even if you only count those between one and two. How is one to get a handle on this limitless realm of harmonic possibilities? One approach that I have found particularly helpful is to consider only five tones at a time: pentatonic scales.

Why five tones? Why not seven, or three, or twelve, or forty-three? There are several reasons. Pentatonic scales are universal; they have been used all over the world for centuries. Five-tone scales are sufficiently complex to be interesting without being overwhelming. But the main reason is that a good pentatonic scale can serve as a solid basis for both melody and harmony.

Five tones can give enough variety to yield great melodies. As I listen more and more from a pentatonic viewpoint, I hear that many, many good melodies, particularly vocal melodies, are strictly pentatonic, although they are often harmonized in a more complicated way. And many more melodies are essentially pentatonic, using other tones only for ornamentation or as passing tones. A good pentatonic scale can give great melodic freedom, since all the notes work well together. (This point is well illustrated in the song "Blacknotes" by Graham Nash.) This makes pentatonics particularly good for improvisation.

Five tones can also all work together harmonically, as an extended chord. The effect can be either clearly tonal or harmonically ambiguous as to which tone is the "tonic." This is especially the case when the scale contains more than one good-sounding triad -- then the one scale can be treated in different modes, depending on which tone is emphasized as the reference point. Another possibility is a scale that sounds good as a harmonic whole without any clear "tonic" at all! (More about that later.)

For my own composing, I have started to think of pentatonic scales as the tonal skeleton for a piece, joined by modulating through related pentatonics. This framework can then be filled in and ornamented, including whatever other tones are needed for the desired effect.

Now, to get more specific, I will discuss several particular pentatonic scales. I make no attempt at any sort of completeness. Rather, what follows is an annotated list of several different scales, more or less in the order in which I discovered them in my search for just-intoned pentatonic resources.

The classic example of a pentatonic scale, familiar from a variety of contexts, is: 1/1, 9/8, 5/4, 3/2, 5/3. This was my starting point in the quest to understand what makes a good pentatonic. The first characteristic that makes this scale so good is the lack of small intervals. I think this point, although obvious, is very important. Large intervals make for strong melodies, small intervals make tense harmonies. (Some cultures take this idea so far as to consider as the ideal a scale of five equal, tempered steps, as its smallest step is the largest possible in a pentatonic, but I shan't discuss the seductions and disappointments of temperament here.) Don't get me wrong. Small intervals can be beautiful; I like pelog as much as the next guy. But for all the tones of a pentatonic to work well together simultaneously, small, dissonant intervals must be excluded.

The other obvious strong point of this scale is the harmonic structure, the way all the notes relate to each other. This is best shown diagrammatically, with one direction representing steps of 3:2 and the other steps of 5:4:

5/3 5/4 1/1 3/2 9/8This shows the embedded major tried as an L shape (1/1, 5/4, 3/2). The inverted (rotated 180 degrees) L of the minor tried (5/3, 1/1, 5/4) is also apparent, the two interlocking in a symmetric Z shape, which is a major 6th chord, or a minor 7th chord, depending on how you look at it.

Even in this simple pentatonic, we run up against the comma problem. As shown above, the 9/8 gives a perfect 3:2 from the 3/2, but the 5/3 gives an "imperfect" 40:27 relative to the 9/8. On the other hand if we choose a 10/9 instead of the 9/8 (in effect inverting the scale) then we still get a 40:27 relative to the 3/2. But even with this somewhat dissonant interval, the scale works quite well as a whole, because all the other intervals are so harmonious.

The scale can also be reinterpreted in terms of the 3/2 tone, giving a different mode: 1/1, 10/9, 4/3, 3/2, 5/3. And its inverse becomes minor relative to the old 5/3: 1/1, 6/5, 4/3, 3/2, 9/5. So there are at least three different harmonious pentatonic scales, all from the same structure of intervals.

The only way to eliminate the comma problem from this type of scale, that is, to get all the "fifths" perfect, is to go Pythagorean:

1/1 3/2 9/8 27/16 81/64Of course the 81:64 gives a very different effect than a 5:4. However, this simple linear harmonic structure yields four strong modes, in order progressively less "major", all the way to 1/1, 32/27, 4/3, 3/2, 16/9.

Because the first scale involves only the primes 3 and 5, (2 being taken for granted) it is a 5-limit scale. In the sense that different primes are independent harmonic dimensions, this scale is two-dimensional. The Pythagorean scale is then one-dimensional. (For convenience, I'll call the first scale the "first 3x5 scale".) The first 3x5 scale differs from the Pythagorean scale only in two tones lowered by a syntonic comma (81:80). Below are shown two other 2-D pentatonics, this time 3x7, that differ from the Pythagorean by septimal commas (64:63). Accordingly, they too sound somewhat familiar, with harmonies of 3:2 dominating.

The first 3x7 scale looks like (3:2 horizontal, 7:4 vertical):

7/6 7/4 4/3 1/1 3/2I find the sound of this compact harmonic structure quite beautiful, as well as its inverse: 1/1, 8/7, 4/3, 3/2, 12/7. The 21:16 between 4/3 and 7/4 does not sound discordant in context. There are nice modes based on the 7/6 (1/1, 8/7, 9/7, 3/2, 12/7) and on the 4/3 (1/1, 9/8, 21/16, 3/2, 7/4) as well as on the 4/3 and 8/7 of the inverse.

The second 3x7 scale is:

7/4 21/16 1/1 3/2 8/7Here, the 7:4 harmony is more dominant, and the 21:16 shows up twice, giving a different mood with a less clear "tonic". The 147:128 interval (240 cents) between 8/7 and 21/16 is only 8 cents sharper than an 8:7, and so effectively adds yet another septimal relation, albeit noticeably off. And the 49:32, only 36 cents sharper than 3:2, surprises the ear. (This scale makes a good just approximation of the equi-pentatonic mentioned earlier.) The inverse (1/1, 8/7, 21/16, 3/2, 12/7) is nice too.

All of these scales were shaped by avoidance of small intervals. In the 3x5 case, these are 16:15, 25:24, and 128:125 (81:80 being unreachable by four harmonically contiguous steps). For the 3x7 case, they are 49:48, 64:63, and 1029:1024. On careful examination of the 3x5 harmonic plane, I was surprised to find another pentatonic that fit my criteria (other than a scale like 1/1, 9/8, 5/4, 3/2, 27/16):

5/3 4/3 1/1 3/2 6/5This one is different from the others so far in its sequence of step sizes. And it is symmetric, with a major and a minor triad joined at the 1/1, suggesting a skeletal dorian scale in sound. The 25:18 interval between the extremes sounds striking but still pleasant. In the mode based on the 4/3, this is essentially a 5-limit version of the first harmonic pentatonic discussed below. (Here I mean harmonic in the sense of a set of tones taken from relatively low in a single harmonic series.)

I also looked at the 5x7 plane in my pentatonic search. Here the constraining small intervals are 50:49, 128:125, and 256:245; besides which, one cannot go very far without encountering large-number intervals. The first result:

7/5 7/4 8/5 1/1 5/4sounds rather strange, but still harmonious, even though it lacks any approximation of a 3:2. The uncommon steps 28:25 (196 cents) and 35:32 (155 cents) sound neither consonant nor dissonant to my ear. The 32:25 suggests a rough "major third" even though it is 41 cents sharper than 5:4. However, all the other nice intervals give this pentatonic and its inverse a unique, if nebulous, character.

The second 5x7 scale:

7/4 8/5 1/1 5/4 8/7is similar in effect, but symmetric, with one less 5:4.

In contrast with the two-dimensional (2-D) scales above, which were motivated by analogy with the first 3x5 scale, there are also several nice pentatonics utilizing some subset of the harmonic series. Obviously, the first harmonic pentatonic to consider is harmonics 5 through 9, your basic dominant seventh/ninth chord. I'll try to diagram it below in perspective 3-D, 3:2 horizontal, 5:4 vertical, 7:4 diagonal:

5/4 7/4 1/1 3/2 9/8Of course, this scale has no clunker intervals; it all works great together. And even though the whole implies the 1/1 fundamental, it still contains two other modes of clear tonal character. The mode based on the 3/2 septimal minor triad is very strong. That based on the 5/4 diminished triad works well also, with the steps getting smaller going up the scale. As a 3-D 7-limit pentatonic, this is useful as an intermediate step in modulating between some of the above 2-D scales. The undertone version (inversion) has a wholly different character, with three interesting modes: 1/1, 6/5, 4/3, 3/2, 12/7; and 1/1, 9/8, 9/7, 3/2, 9/5; and 1/1, 7/6, 7/5, 14/9, 7/4.

One idea that has always intrigued me is that of a scale in which each interval is unique, that is, a scale where, upon hearing an interval, one knows unambiguously which two degrees of the scale are sounding. Any scale like that gives the maximum number of different intervals for a given number of tones. One way of coming up with such a scale is to make sure each tone (other than 1/1) involves a different prime number. For a harmonic pentatonic, this means going at least to harmonic 11, giving a 4-D scale. Two such are: 1/1, 5/4, 11/8, 3/2, 7/4; and 1/1, 7/6, 3/2, 5/3, 11/6; the latter lacking the implied fundamental of the series. I find the second one particularly charming. Another 4-D pentatonic, with no semblance of a 3:2, is harmonics 7, 8, 10, 11, 13, which is similar to the first 5x7 scale above in character. (Admittedly, the 14:13 step is almost "small", but doesn't sound too dissonant.) And finally, one odd scale that is 5-D, lacking a clear 1/1 and any 3:2, harmonics 7, 9, 10, 11, 13. This one really stretches the ear.

I hope this personal tour of some just-intoned pentatonic possibilities has been stimulating. The scales discussed above vary widely in sound, from the familiar to the shockingly different, but all are, in my opinion, good, useful pentatonics. In fact, I have used most of them in compositions, and all of them in improvisation on my refretted guitar. While I have tried to describe some of their features, the only way to get any clear idea what all these various pentatonics are really like is to play with them and listen to them.

David Canright -- DCanright@NPS.edu