Superparticular Pentatonics

David Canright

This article first appeared in 1/1, the Journal of the Just Intonation Network, Volume 9, Number 1, p.10 (1995).

In a good pentatonic scale, it is impossible to play a wrong note. The relative simplicity of five- tone scales makes them good candidates for melodic explorations, especially when dealing with intervals new to one's experience. The limited choices in pentatonic scales seem to lead to strong melodies, avoiding diatonic cliches. (See "Pentatonics I Have Known" for more discussion and several examples.)

What makes a good pentatonic? Intervals that sound good! A pentatonic contains 20 intervals, not all necessarily different (excluding unisons and assuming "octave equivalence"): 5 each of one-step intervals ("seconds"), two-step intervals ("thirds"), three-step intervals ("fourths"), and four-step intervals ("fifths"). The terminology in quotes is probably more confusing than useful, in that pentatonic four-step intervals would, in a diatonic context, be called "sixths" or "sevenths", so is best avoided. This article examines pentatonics whose one- step intervals are superparticular, that is, of the form (n+1)/n.

Among all just intervals, superparticular intervals have been preferred by such notables as Ptolemy and Lou Harrison. The reasons for preferring superparticular ratios are not entirely clear, but one special property concerns difference tones. As Dudley Duncan discussed in "Why Superparticular," when two tones related by a superparticular interval sound together, the primary difference tone is the fundamental of the harmonic series to which both tones belong. (For other just intervals, the primary difference tone is another tone in the harmonic series, not the fundamental.) Another special property is that a superparticular interval is the "simplest interval of its size." For example, the major third 5/4 is arguably simpler than 11/9 or 9/7 or 81/64 or any other interval in the same ball park (that is, nearer to 5/4 than to 4/3 or 6/5).

My inspiration for seeking out superparticular pentatonics was Table 2-3 in John Chalmers's Divisions of the Tetrachord. This table, attributed to I. E. Hofmann, listed all possible tetrachords whose steps are superparticular, 26 in all. At first I was surprised that there were not an infinite number of superparticular tetrachords; after all, there are an infinite number of superparticular intervals. But on reflection, I realized that the limiting factor was that exactly 3 superparticular steps must make up the 4/3, and while there is no shortage of tiny superparticular intervals, there are only a few big ones. This suggested the general algorithm for dividing a given interval into a certain number of superparticular steps.

For example, say we want to divide a 5/4 into two superparticular steps. The largest superparticular interval that will fit into the 5/4 is 6/5, leaving 25/24, which is also superparticular, so one division is 6/5 25/24. The next largest superparticular that will fit into 5/4 is 7/6, leaving 15/14; this makes a second superparticular division. Trying the next largest, 8/7, leaves 35/32, but that's not superparticular. The next works, giving the third division 9/8 10/9. The next largest after 9/8 is 10/9, but that will just give the last case (in a different order) so we're done. Altogether we get three divisions of 5/4 into two superparticular steps, and each could be taken in two different orders.

As another example, let's divide 2/1 into three superparticular steps, or in other terms, let's find superparticular triads. The largest superparticular that could fit is 3/2, leaving 4/3 to be split into two steps. Proceeding as above gives three ways to do that, into 5/4 16/15, or 6/5 10/9, or 7/6 8/7. The next largest step that could fit into 2/1 is 4/3, leaving 3/2 to be divided into two steps, which can be done only two ways: 4/3 9/8, or 5/4 6/5. Then the next largest step that fits is 5/4, but that just gives two of the previous cases (in different orders), hence we're done. So altogether there are five different ways to divide a 2/1 into three superparticular steps, disregarding the order of the steps. For each such division, we could order the three steps in 3! = 3*2*1 = 6 different ways (except the 4/3 4/3 9/8 division has two steps the same, so only 3!/2! = 3 different orders); if we count all possible orderings we get 4*6+1*3 = 27 different triads. Then again, we may consider different inversions (cyclic permutations) of a triad to be the same, and a triad has three inversions, for a total of 27/3 = 9 different triads.

These examples should make clear that the general procedure for dividing a given interval (call it r) into some number (call it n) of superparticular steps is recursive. You first try the biggest superparticular step (say s) that will fit in r, then you need to find how to divide what's left, r/s, into n-1 steps. Then try the next biggest step, etc. until the first step is small enough that n of them are smaller than r, and you're done. I used the Maple mathematics software to program the algorithm (it could be done in any programming language); the Appendix describes the program in detail. Applying this procedure to finding all superparticular divisions of 4/3 into 3 steps does indeed give the 26 different ways tabulated by Hofmann.

Having this general procedure, I decided to find all superparticular pentatonics (divisions of 2/1 into 5 steps). I was rather taken aback to find that there are 876 of them! The first one the algorithm came up with is 3/2 5/4 17/16 257/256 65536/65535, which is not musically useful as a pentatonic. For one thing, the smallest interval, 65536/65535, is only 0.026 cents, which is smaller than human pitch discrimination! For another, the largest interval, 3/2, is much bigger than one would expect for a "step" of a pentatonic; the whole scale is poorly proportioned. In fact, of the 876 superparticular pentatonics, 593 of them have a largest step of 3/2, and another 229 of them have a largest step of 4/3, which is still pretty large to be considered a step in a pentatonic. (Another 46 have 5/4 as largest, 7 have 6/5, and only 1 has 7/6 as largest. There are 308 different smallest steps possible.) So for Table 1, I only include those I consider musically useful as pentatonics, whose largest steps are smaller than 4/3, and whose smallest steps are at least as large as a Ptolemaic comma, 81/80 (21.5 cents), though none of those remaining actually have a step quite that small. Of course, this eliminates some interesting possibilities, such as the scale 1/1 9/8 5/4 11/8 3/2, and superparticular pentatonics similar to LaMonte Young's "dream chords," such as 4/3 25/24 26/25 27/26 4/3.

Those restrictions reduce the number to 40 pentatonics, an almost digestible quantity, until one thinks about all the different orderings. The divisions with all five steps different (#2, 4, 5, 6, 9, 10, 12, 16, 18, 23, 27, 38) have 5! = 120 different orderings, those with a pair the same have 5!/2! = 60 orderings, those with two pairs (#1, 3, 13, 34) have 5!/(2!*2!) = 30, and the one with three of a kind (#20) gets 5!/3! = 20; in this sense, Table 1 represents 2960 pentatonic scales. If one counts the five different cyclic permutations (modes) of a scale as one then we get 24 scales for divisions with all steps different, 12 with a pair, 6 with two pairs, and 4 with a triplet, for a total of 592 different scales in Table 1.

There are a number of ways to classify, evaluate, and measure scales, by which to make comparisons of all these possibilities. Here I will discuss only a few; for many more see John Chalmers's book. Perhaps the most basic comparison is in terms of step size. For example, a pentatonic whose smallest step is a whole tone will have a different character than another containing one or more semitones; the small steps give a scale a sort of inherent tension. Hence in Table 1 the pentatonics are arranged in order of smallest step size, from largest to smallest, or "mellowest" to "tensest." Another property based on interval size is called "propriety"; a scale is "proper" if all the one-step intervals are smaller than all the two-step intervals, which in turn are smaller than all the three-step intervals, etc. Propriety depends on the ordering of the steps (independent of cyclic permutations), for example, pentatonic #3 in the given order is improper, since one of the two-step intervals, 6/5 * 6/5 = 36/25, is larger than one of the three-step intervals, 10/9 * 10/9 * 9/8 = 25/18, though other orderings of the same steps (that separate the two 6/5's) give proper scales. One could also judge a scale by the total number of superparticular intervals it contains; this depends on step order but not on cyclic permutation. Of course, not all of the 20 intervals of a pentatonic can be superparticular, because the octave complement of a superparticular interval is not superparticular (except 3/2 and 4/3).

A basic harmonic property of a set of intervals is the set of prime factors involved; the highest prime factor is often called the prime limit of a scale. The underlying premise is that any just interval can be analyzed in terms of prime harmonies, the (octave-reduced) intervals of the prime numbers in the harmonic series, and that each prime harmony has a unique character that it contributes to any composite interval of which it forms a part. For example, pentatonic #1, which involves primes 3 and 7, combines the power of perfect fifths (3/2) with the bluesiness of natural sevenths (7/4), but does not have any of the sweetness of major thirds (5/4). The prime factors (excluding 2) of each pentatonic are given in brackets in Table 1. Traditional harmony is based on primes 3 and 5, and possibly 7 for dominant-seventh chords; harmonies using 11, 13, and beyond will definitely sound unconventional.

A more detailed form of harmonic analysis involves plotting the notes of the scale on a harmonic lattice, with one dimension for each different prime harmony (in other words, each prime is a harmonic dimension). Here the order of the steps is important, although cyclic permutations give the same structure. For example, when the steps of #3 are taken in the order 9/8 10/9 6/5 10/9 6/5, the resulting scale is 1/1 9/8 5/4 3/2 5/3 (a "major" pentatonic), which is diagrammed harmonically as


     5/3   5/4

           1/1   3/2   9/8

where the horizontal direction shows the 3/2 prime harmonies and the vertical shows 5/4. Another common order for #3 is 6/5 10/9 9/8 6/5 10/9 giving the scale 1/1 6/5 4/3 3/2 9/5 ("minor" version) with the structure

     4/3   1/1   3/2

                 6/5   9/5

which in a sense is just the inverse of the other scale. A third order is 6/5 10/9 9/8 10/9 6/5 giving 1/1 6/5 4/3 3/2 5/3 ("dorian" perhaps?) or

     5/3

     4/3   1/1   3/2

                 6/5

The other three (cyclically distinct) orderings give different harmonic structures that are less harmonically compact, that is, not as well connected by prime harmonies.

Now I would like to discuss a few of the specific pentatonics in Table 1, mainly near the top of the list. The first is the closest to five equal steps, and gives just versions of the Indonesian "slendro" scale; some of the resulting scales were discussed by Jacques Dudon in "Seven-limit Slendro Mutations." One such ordering would be 7/6 8/7 9/8 7/6 8/7 (a mode of Dudon's "N" scale), giving the scale 1/1 7/6 4/3 3/2 7/4, one of my favorites for improvisation. And, like #3 discussed above, the bottom pair of steps and/or the top pair can be reversed while still retaining a strong harmonic structure. This pentatonic is the only one on the list based on primes 3 & 7 without using 5.

The second pentatonic (in reverse order) gives the scale 1/1 6/5 7/5 8/5 9/5; this is five consecutive steps of the harmonic series, 5 to 10, so the frequency differences from one tone to the next are all equal. Another (cyclic) way to look at it is as a dominant-seventh ninth chord (with the ninth brought down into the octave): 1/1 9/8 5/4 3/2 7/4, which makes clear its strongly tonal nature. Of course, the opposite order gives the subharmonic version. Again, if we look at it as 1/1 7/6 4/3 3/2 5/3, we can reverse the bottom and/or top pairs of steps or even swap the two pairs and still keep fairly strong harmonic structure with the 4/3 1/1 3/2 backbone. Other orderings (and since all five steps are different, there are a lot of 'em) are less tonal.

Pentatonic #3 was discussed previously; this is what might be called the "standard" pentatonic, being based on traditional harmony, with no semitones. Many traditional melodies are based on the "major" and "minor" versions.

The fourth one can be arranged to give 1/1 5/4 11/8 3/2 7/4 (another favorite of mine), whose tones are all prime harmonies (the 1/1 represents the prime 2). Here, not only are all the steps different, but all 20 intervals of the scale are different, and this is true regardless of the ordering of the steps. (The other pentatonics with this property are #6, 18, 23, and 38.) This is the first using the prime 11, definitely an adventurous harmony, and the smallest step, 12/11 (150.6 cents), is halfway between a whole step and a half step. Number 6 is similar to #4 in many ways, having the prime scale 1/1 5/4 3/2 13/8 7/4, all intervals different, involving adventurous 13, and with a smallest step of 13/12 (138.6 cents).

With #5, it seems difficult to make a scale with a nice harmonic structure; perhaps the best is 1/1 5/4 3/2 5/3 11/6. The smallest step of #7, 15/14 (119.4 cents), is getting into the semitone range; one nice scale of #7 is 1/1 7/6 5/4 3/2 7/4. The next is somewhat similar. Pentatonic #9 includes the scale 1/1 5/4 11/8 3/2 15/8, which has a nice character, similar to some Indonesian "pelog" scales; the 15/11 interval (537.0 cents) is enough larger than a perfect fourth that it doesn't sound dissonant, just unusual.

The first pentatonic with a semitone that uses only traditional harmony (3 & 5) is # 10, one scale of which is 1/1 5/4 4/3 3/2 5/3; #13 has two semitones, and includes the scale 1/1 9/8 6/5 3/2 8/5 that the breeze plays hauntingly on the Woodstock windchimes (of "Olympos") on my porch. Traditional-harmony scales with small semitones of 25/24 (70.7 cents) are #20, the only one with three steps the same, and #21, which also has a larger semitone. (Terry Riley makes good use of the 25/24 steps of his 5-limit piano tuning on the album "The Harp of New Albion.")

On the non-traditional side, there are two pentatonics that cannot have any perfect fifths, being free of the prime factor 3: #22 and #36, though the latter, with a step of 56/55 (31.2 cents) may be too tense to use. The highest-limit harmony occurs in # 25, which involves the prime 31. And as for the rest, well, I would love to hear what you discover in your explorations!

References

Table 1

Table 2-3 after Hofmann

Appendix: Superparticular Subdivision Program in Maple

Note: Since writing this article on superparticular pentatonics, I have generated comparable tables for superparticular hexatonic (6-tone) scales (380 scales, 19KB) and superparticular heptatonic (7-tone) scales (2908 scales, 166KB). For more information, see the description.

David Canright -- DCanright@NPS.edu