This past March 15 composer Ben Johnston celebrated his 90th birthday. He is one of those composers who name is likely to be familiar to anyone who has studied the past century of modernism, particularly as practiced by American composers. However, it would be safe to assume that only a relatively small fraction of those who know his name have had much exposure to his music. Furthermore, if we exclude the recording that the Kronos Quartet made of his arrangement of “Amazing Grace,” we can probably diminish that “relatively small” fraction to “almost negligible.”

Nevertheless, for about fourteen years the Kepler Quartet (violinists Sharan Leventhal and Eric Segnitz, violist Brek Renzelman, and cellist Karl Lavine) have been working hard to reverse this trend. Over that period they have committed themselves to learning to play and then record Johnston’s ten string quartets. This resulted in three volumes of single-CD albums, the last of which was released by New World Records this past Friday. These recordings do not cover the quartets in numerical order. Volume 1 consists of the second, third, fourth, and ninth, Volume 2 presents the first, fifth, and tenth, and Volume 3 closes out the series with the sixth, seventh, and eighth. None of the individual volumes present the quartets in numerical order.

In approaching Johnston’s music, it helps to appreciate that he has been guided by a keen sense of mathematics. Whether his knowledge of mathematics came from academic study of the topic or whether it is the result of his posing himself problems and then thinking them through to solutions is not important. The best mathematics tends to emerge as a result of intuition that is quickly and effectively refined through disciplined thinking; and the discipline that Johnston has brought to his work is nothing short of awe-inspiring. It is evident as early as his first quartet, composed in 1959, in which his approach to nine variations on a twelve-tone row taps into properties of permutation groups that Milton Babbitt tried to investigate with his own “academically qualified” background as a mathematician but fared far less successfully.

However, Johnston’s work became more focused and much more interesting when he decided to explore the possibility of making music with pitch classes based on the overtones of natural harmonics. My guess is that, at this point, several readers are likely to raise eyebrows and ask (at least in their heads), “Aren’t you talking about physics, rather than mathematics?” I would replay that, while it is true that the properties of vibrating strings and columns of air is, strictly speaking, the domain of physics, the domain of pitch classes and their organization into either scales or twelve-tone rows is the domain of mathematics.

An interval between two pitch classes is defined by the *ratio* of the frequencies of the two pitch classes. The simplest such interval is, of course, the octave, whose ratio is 2:1 (i.e. the frequency of the higher note is twice that of the lower). Ratios of integers correspond to intervals between pitches found in the series of natural harmonics, and these are the intervals that attracted Johnston’s attention. Unfortunately, the only one of those ratios that can be reproduced on most pianos is the octave. Because the octave is divided into twelve equal parts, none of the other intervals can be expressed as a ratio of integers, Those intervals can only be represented as *irrational numbers*, all of which can be reduced to multiples of the twelfth root of two.

Thus, in the span of time between the composition of his second quartet in 1964 and the composition of his tenth quartet in 1995, Johnston experimented with and deployed a variety of different techniques for composition based on intervals defined by integer ratios. These involved a variety of different approaches, many of which began with folk songs based on the pentatonic scale as a point of departure. When asked what a pentatonic scale is, most people reply by saying it is what you get when you restrict yourself to the black keys of a piano keyboard. This is true but misleading. The pentatonic scale is formed by taking the 3:2 ratio of the perfect fifth and traversing it through the first five pitch classes in the circle of fifths. Starting on C, the pitches would be C-G-D-A-E.

Many instruments for folk music are limited to playing the pentatonic scale. The bagpipe is one of the best examples. Thus, Johnston’s decision to write a set of variations on “Amazing Grace” (the result being his fourth quartet) was motivated by the fact that the scale for “Amazing Grace” is pentatonic. The variations, in turn, are based on 7-limit tuning, a tuning system in which all intervals are represented by ratios involving only the integers from 1 to 7. The most interesting of those ratios if 7:4, which is a slightly flatter version of the minor seventh (which some have insisted is the “true” version of the “blue seventh”).

Johnston’s next major move was to work with the integers from 1 to 16, and the tenth quartet amounts to a summa of his exploratory efforts. He called the scale consisting of the pitch classes of the eighth through sixteenth harmonics the “otonal” scale. The problem, however, was that the intervals would keep getting smaller as one ascended through this scale, while, in the more familiar scales and modes, they are more evenly spaced (but not perfectly so). Johnston thus postulated a “utonal” scale, which inverts the “otonal” scale with intervals that go down, rather than up. This made for a more even distribution of pitch classes; but it also through the mathematical foundations out the window, at least where the integer ratios of physical overtones are concerned.

On the other hand Johnston’s interest in inversion led to a fortuitous side effect. In many respects the tenth is the most accessible of the quartets in the full canon. That is because just about any theme that Johnston invents gets presented in inversion after having been introduced in its “prime” form. The result is an overall rhetoric that is highly reminiscent of the music of Béla Bartók (who never met a theme that he could not improve by inverting). Thus, while many of the Johnston quartets draw upon folk sources to provide the listener with a sense of familiarity, then tenth quartet achieves the same effect in a more abstract domain, simply because Johnston proved himself as being just as effective in working with inversion as Bartók had been.

The real challenge, however, comes earlier with the seventh quartet, composed in 1984. Kepler did not undertake the recording of this quartet until their final volume, and it is easy to appreciate why. Johnston organized the quartet around a gamut of 1200 distinct pitch classes. Say what you will; but this was just too much of a good thing (or, perhaps more politely, a *reductio ad absurdum* of a good idea).

This is where mathematics must give way to psychoacoustics, because auditory perception is capable of going to great lengths to simplify complexity. This is most evident in a phenomenon known as “categorical perception.” Simply put, one is more likely to hear a “strange interval” as a “normal” one that is “slightly out of tune;” and, because most minds are saturated with the sounds of equal-tempered tuning on a piano, the criterion for “normal” based on equal-tempered tuning tends to be very deeply ingrained. Thus, while the seventh quartet was almost certainly the most difficult piece that Kepler had to play as part of their project, it is unclear that just about any “listening brain” would detect the full complexity of their undertaking. Indeed, it is unlikely that the quartet musicians would have been able to provide themselves with enough ear training to satisfy every last letter of Johnston’s specifications.

I must now conclude with a disclaimer. In my home town of San Francisco I was fortunate enough to listen to a performance of the tenth quartet prepared by the Del Sol String Quartet. All four members of the ensemble (violinists Benjamin Kreith and Rick Shinozaki, violist Charlton Lee, and cellist Kathryn Bates) contributed to a lecture-demonstration to provide listeners with some orientation. Not only did they review the basic principles behind Johnston’s theory, but also they illustrated it by performing excerpts from the composition. As a result, I could approach the Kepler recordings as a listener informed by experience in the sonorities of non-standard intervals. It is therefore difficult for me to judge how the organization of content across the three volumes of the Kepler release may support (or undermine) the ability of the inexperienced listener to get to know just what Johnston has done in these pieces, let alone how he did it. Nevertheless, the result is still a valuable document, particularly for those not fortunate enough to be in a city that has a string quartet interested in performing Johnston’s music.