Lou Harrison: Explorations in Just Intonation and Temperaments on the Keyboard

May 11, 2017

Lou Harrison (1917-2003) was one of the most influential American composers of the twentieth century and a creator of complex, yet elegant and beautiful music. He was also a visionary experimentalist who worked in a wide range of musical styles.

A student of composer-pianist Henry Cowell, and a collaborator with both John Cage and Charles Ives, he pioneered the syntheses of modernist idioms with classical musics from around the world and through history (including Asian and historical European styles). Harrison was also one of the earliest composers of percussion music, and he and his partner William Colvig built many of their own instruments. One of the things that most concerned him, however, for much of his compositional life, was the investigation of tuning.

Harrison (along with Harry Partch and many others) helped return the music of the twentieth and twenty-first centuries to a pitch-world not constrained by the twelve pitches of mainstream equal temperament. Harrison looked backward—into the Just, Meantone and Well-tempered traditions of historical European tunings—as well as outward—to gamelan (primarily the slendro and pelog modes of central Java) and the tuning systems of Korean and Japanese court traditions. And he looked forward, with the old, yet still revolutionary idea that the principles of tuning are malleable in real time and need not require a fixed notion of “scale” (in what he called Free Style).

“Crossing the Pacific on a freighter I began a little list of pentatonic modes, some of them already known to me & some of them noted as possibilities for future investigation. I suddenly realized that European music theory lacks any usable classification or naming system for penta-modes” (Music Primer, p. 26)

For Harrison, the building of new instruments and the modification of existing ones was a crucial and enduring endeavor. Though he’s widely known for the several impressive gamelan and gamelan-inspired instruments he built with Colvig in the ‘70s and ‘80s, Harrison had been tinkering with instruments in other ways for many years. In 1943, having moved to New York City from California, and finding himself in need of practicing late at night without bothering close-quartered neighbors, Harrison decided to build a clavichord, a centuries-old keyboard instrument that utilizes small metal strikers (called tangents) to hit the strings, resulting in a quiet and intimate sound. In this early project, the composer demonstrated an uncanny aptitude to instrument design as well as iteration: When the simple, box-shaped first model warped due to excessive string tension, Harrison came up with an alternative plan, shaped like a piano, and with the strings running underneath the soundboard, which proved a more durable solution that could also be folded down for ease of transport.

In 1947, Harrison suffered a devastating mental breakdown and embarked on a long recovery process, which eventually led to the formation of a new compositional voice. The exploration of various tuning systems from different cultures, places, and times provided a creative springboard for him; learning how to tune (and retune) keyboards was a practical path into this new world. Two pieces from early in this period showcase Harrison’s varied approach to tuning, even when dealing with the constraints of a twelve-note keyboard. In the incidental music for Cinna, Harrison specified a single-strung piano with thumbtacks inserted in the hammers (commonly known as a tack piano, and frequently used as a makeshift harpsichord at the time). This yielded a lighter, silvery timbre well suited for precise tuning requirements, and evocative of Southeast Asian gamelan to Harrison’s ears. The tuning for Cinna’s five movements affords a wide variety of interval sizes, all tuned in just ratios and ranging in quality from serene to strident.

Conversely, each of Harrison’s four Strict Songs (originally for eight baritones and orchestra, but existing in various vocal arrangements) utilizes a different pentatonic mode. These modes combine chromatically, in an ingenious intonational crossword puzzle, to fit a conventional keyboard. In this way Harrison is able to employ a variety of modal flavors and tonal centers in his piece without compromising playability (confounding the widely-held assumption that one cannot readily modulate in just intonation). The piano (and harp) are the only fixed-pitch instruments in the ensemble. These serve as intonational signposts for the other musicians and singers, allowing them to become familiar with and eventually internalize the mathematically pure intervals of the piece.
Strict Songs — Modes
I. G A B D E G
1:1 9:8 5:4 3:2 5:3 2:1
This is the “prime pentatonic” mode, widely found across many different musical cultures. It has no semitones, and uses both the “large” and “small” whole-tone (204¢ and 182¢, respectively).

II. E F# G B C E

1:1 9:8 6:5 3:2 8:5 2:1
This mode alternates large and small “steps” (for example, 9:8 to 6:5).

III. Eb F Ab Bb Db Eb

1:1 9:8 21:16 3:2 7:4 2:1
Another anhemitonic tuning, this time with septimal intervals (derived from the seventh overtone) for an entirely different feel.


1:1 5:4 4:3 3:2 15:8 2:1
This last tuning also features alternating wide and small steps—the wide “steps” are actually the pure major third=386¢.


G Ab A Bb B C Db D Eb E F F#
1:1 28:27 9:8 32:27 5:4 4:3 112:81 3:2 128:81 5:3 16:9 15:8

“Just Intonation is the best intonation” (Music Primer)

Since his adoption of just tunings in the late early 1950s, Harrison became a vocal proponent of their aesthetic and sensual superiority. In interviews, essays, and musical compositions, Harrison has often analogized twelve-tone equal temperament with some of the noxious by-products of industrialization and civilization, such as mechanization, noise, and pollution. Perhaps the most emblematic example of this is in one of his greatest and most well-known works, Pacifika Rondo, a multi-movement composition for a cross-cultural, Pan-Pacific ensemble composed almost entirely in just intonation. In the penultimate movement—“Hatred of the Filthy Bomb”—Harrison wrote a grotesque, militaristic march, made especially ominous and strident by the explicit requirement of being tuned in 12-tone equal temperament (12TET). The movement also expresses Harrison’s profound anxiety about the nuclear experiments of the 1960s, and the perennial threat of war.

As a composer, Harrison was adaptable and generous. To accommodate musicians, he often allowed for alternative, equal-tempered performances of his pieces. Another compromise arose from his serious study of historical unequal temperaments, which are a sort of bridge between the uncompromising acoustical rigor of just intonation, and the eventual versatility of 12TET. Unequal temperaments—beginning in mean-tone tunings, and culminating in the great well temperaments of Werckmeister, Young, and others—favor the consonance of certain keys and intervals at the expense of others, while striving for an overall “acceptability” across all twelve keys. They also offer a novel solution to the so-called “historical tuning problem,” which postulates that having two or more prime factors in the tuning guarantees that eventually pitches will diverge, since the powers of two different prime numbers will never match. Harrison was fond of reminding people that Bach wrote two books of preludes and fugues for a “well-tempered clavier,” not an “equal tempered” one. Indeed, 12TET did not become widespread until well into the 19th century; much of the music of Mozart, Beethoven, and even Chopin assumes a new and sometimes revelatory character when heard in historical temperaments.

Fig. 1. Harrison’s hand-drawn chart showing the intervals used in the Kirnberger II temperament and how to tune them.

For his Piano Concerto (written for Keith Jarrett between 1983 and 1985), Harrison chose a famous temperament developed by Johann Kirnberger, a student of J.S. Bach. Kirnberger II, as it’s known today, uses the key of C major as its center, and features three purely-tuned major thirds (over C, G, and D), at the expense of wider, Pythagorean thirds opposite on the circle of fifths. In Kirnberger II, no interval strays from 12TET by more than about 9 cents (see Fig. 1). (At the end of his life, Harrison kept his own piano, which had once belonged to Henry Cowell, tuned to Kirnberger II). In the Concerto, the listener is invited into a harmonic world that is immediately more familiar to ears accustomed to equal temperament, yet the modulations sound a bit more exciting than mere transpositions, because the rearrangement of pitches actually yields slightly different-sized intervals. For additional large-scale effect, Harrison withholds the truly “in tune” key of C Major for much of the work, until the ecstatic, expansive third movement, which sounds positively blissful in the temperament.

“After only a brief study of intervals it becomes clear that there are two ways of composing with them: 1) arranging them into a fixed mode, or gamut & then composing within that structure. This is Strict Style, & is the vastly predominant world method. However, another way is possible — 2) to freely assemble, or compose with whatever intervals one feels that he needs as he goes along. This is Free Style, & I used this method first in my Simfony in Free Style. (Music Primer, p. 6)

One of Harrison’s most extraordinary, but perhaps least understood contributions was his exploration of the idea of Free Style. In Free Style tuning there is no intonational terra firma, no scales, nor gamuts. Each interval is tuned from a previously occurring pitch, resulting in the drift of absolute pitch over time.

In a sense, this is exactly what occurs during an unaccompanied choral performance, provided the singers are not deliberately trying to stay in equal-temperament or attempting to end up on the same absolute pitch as they started. The composer Ben Johnston, for example, used to have his students analyze renaissance motets, assuming that singers were trying to sing purely-tuned thirds and fifths. Because the concern in that context is “local” rather than global tuning, singers maintaining the smoothness of just intonation will end up, over a short time, on a different absolute pitch than when they started. Figure 2 shows a very simple melodic example for a solo voice. The ratios (intervals) between pitches are simple just intonation ratios, easy to tune and sing by ear. Their “size” (how wide or narrow) is given below each one in cents deviations from their nearest 12TET equivalents. In other words, the first interval, the ascending 5:4 major third from C to E is 14 cents narrow (approximately one-seventh of a semitone). Let’s sing the melody, accumulating tuning differences as we go.

This melody is short and only concerned with horizontal (melodic) intonation; we don’t even have to worry about harmonic relations. Yet even with just one small chromatic alteration (a temporary tonicization) we end up 10 cents flat of where we started. It’s easy to see how this would pose a problem when combining instruments capable of adjusting pitch on the fly (such as the human voice, fretless strings, and trombones) with ones “stuck” with fixed pitch (such as fretted strings and keyboard instruments). Indeed, the eventual standardization of different tuning systems into equal temperament might have arisen in part as a convenient approach to smooth these discrepancies. Such smoothing, however, came at the cost of much beautiful detail and color.

Harrison took this idea of consecutive tuning very far, though he used it in only a few pieces: Simfony in Free Style (1955), Political Primer (1958), At the Tomb of Charles Ives (1963), and A Phrase for Arion’s Leap (1974), which we’ll discuss below. The paucity of examples is almost entirely due to how difficult these pieces are to perform—even today, but especially for the time in which they were written. Tomb… was premiered in 1970 and released on LP by Lukas Foss and the Brooklyn Philharmonic in 1983; more recently the American Festival of Microtonal Music revived the piece in 2004. To date, the Simfony remains realized only in electronic form.

Harrison’s idea of Free Style makes the wandering of pitch not just an artifact of musical practice, but an explicit and fertile compositional exploration, freeing music from the constraints of scales. Let’s look at the short (and beautiful) score for A Phrase for Arion’s Leap (see Fig. 3). A recording of the piece was made by Harrison, Colvig, and their friend and frequent collaborator, Richard Dee. One reason they were able to make the recording, besides the brevity of the piece, is that though A Phrase for Arion’s Leap is composed in Free Style, the instruments are all fixed pitch (a combination of small bowed and plucked harps). In other words, Free Style in this piece is merely a compositional method, not a performance technique. (Note the unusual dating of the score, which, we believe, corresponds to 1974 in the Thai Solar Calendar. Harrison used a number of different calendars, in accord with his interest in different musical traditions). 

As an experiment, we can rewrite this short free style score in “strict style,” with its accompanying absolute ratios (to some fixed “tonic”—in this case the opening E). Bold ratios under each note are the Free Style intervals from the score, and the smaller ratios beneath them are the accumulated “absolute ratio” to 1/1 (E). [You can download a PDF of the notated example from this link: /Portals/2/arions_leap_recopied.4.pdf ]

Harrison’s Free Style ratios already involve some large numbers (128/75, 96/75), and note how quickly the ratios become complex—640/351, 234/225, and so forth: absurdly so! (Harrison: “From the Latin, ‘surd’ = Deaf. … The intervals of equal temperament are surds. Our word ‘absurd’ carries the suggestion ‘from deafness!’”). But also note how Harrison controls them: the smaller, more complex intervals give way to simple just ratios at the end of phrases — the first phrase, for example, goes up a perfect to a just A, and then up another fourth to a just D, yielding the Pythagorean minor seventh. Even though the piece seems to be wandering out of control when we notate it in term of absolute pitch, it’s actually very tightly constructed from a harmonic point of view.

Why is this so important? With the advent of easily available and instantly tunable electronic keyboards and synthesizers in the ‘80s, this idea of “tuning on the fly” or “real-time adaptive tuning” (what Harold Wagge nicely dubbed “the intelligent keyboard”) goes from being a nearly impossible performance feat to an entirely feasible musical resource. For over thirty years, composers have been working with computer software and synthesis hardware that decides how to tune on its own and is not limited to any fixed scale. Harrison’s work presaged this, and in fact, was the inspiration to a number of composers who knew his work. His vision, first implemented in strings, voices, and harps, has become one of the most fertile ideas in contemporary computer-aided music.

Gamelan Tunings
Harrison never stopped being fascinated by modes and instrument building. Later on, when he and Colvig began building gamelan (note: the plural of gamelan is gamelan), they designed just intonation modes for the instruments inspired by the tunings of Central Javanese slendro/pelog pairs. Broadly speaking, slendro is a pentatonic mode with nearly-equidistant steps, whereas pelog’s seven steps alternate between large and small in size. Interestingly, the two modes/tunings have one note in common, called the tumbuk.

One of the Harrison’s most interesting tunings was the one he designed for the large aluminum “double” gamelan Si Darius/Si Madeleine, named after Darius and Madeleine Milhaud. Harrison held the Milhaud Chair of Composition at Mills College, an institution with which the French composer was affiliated from 1940 to 1971. The tuning of this gamelan is anchored to A=440Hz, which is pitch 6 in both pelog and slendro — the tumbuk. Each mode has its own distinct tuning rationale. The two tunings are shown in the figure by the ratios between adjacent notes (which is how Harrison often thought of them), but the “absolute” ratios are easily derived from those successive intervals.

Gamelan Tunings:
Si Darius and Si Madeleine, Mills College, 1981

(Adjacent intervals, ratios, and cents values of intervals)
slendro (Darius)
5-6 (8:7)    6-1 (7:6)    1-2 (8:7)     2-3 (8:7)    3-5 (7:6)
231¢          267¢           204¢           231¢          267¢

pelog (Madeleine)
1-2 (13:12)    2-3 (14:13)    3-4 (17:14)    4-5(18:17)    5-6(19:18)
139¢              128¢                 336¢              99¢              94¢
6-7 (21:19)    7-1 (8:7)
173¢                 231¢

Si Darius’ slendro is a “septimal” pentatonic, meaning that the interval ratios involve the prime seven. The “seconds” in this tuning (1-2, 2-3, 5-6) are the characteristic wide septimal major second (8:7, or 231¢), an interval which is suggested or approximated in many Central Javanese gamelan. Because the seconds are wide, the septimal minor thirds are narrow, and approximate the equally characteristic 7:6 (267¢) interval (a kind of “blues” third). The “cadential chords” (seleh, in Javanese gamelan) are tuned as pure fifths in Harrison and Colvig’s tuning, unlike what would be the case in Javanese tunings. Thus the intervals between 5 and 2, and 6 and 3, are “pure” P5ths. The octaves in this tuning are also “pure” (2:1), whereas in Javanese gamelan they are generally stretched between 10–20¢. Aside from Harrison’s preference for non-beating intervals, one factor to keep in mind was that the gamelan he built were aluminum, while Javanese gamelans are typically bronze (or iron). As such the spectra of the American gamelan tend to be more harmonic than the inharmonic Javanese counterparts; the former material invites the tuning of “pure” octaves, whereas the wide partials of the latter lead to octave stretching.

Si Madeleine’s pelog is also interesting, featuring the notion of superparticular ratios (or epimores), of which Harrison and Colvig were especially fond. These are intervals in which the numerator is one greater than the denominator; when small integers are involved, our ears tend to perceive them as consonant. Si Madeleine’s unusual and inventive pelog is built with superparticular ratios from the 12th to the 24th harmonic. Note the unusual quality of this pelog (and of most pelogs), that certain single intervals are bigger than the combined width of other pairs of adjacent intervals, for example 336¢ is bigger than 99¢ + 94¢. Scales with this property, that of being “unevenly spaced” and which are relative uncommon, are called improper (or non-coherent).

Get In Tune
Whether exploring centuries-old tuning solutions or forging new forward-looking paths, Harrison wrote music that is intellectually rigorous, beautifully accessible and ultimately satisfying from almost any perspective. Check out our essential listening list for a sampler of Harrison’s rich, diverse, and mesmerizing sound-worlds. His do-it-yourself attitude can and should be contagious—construct your own monochord, learn how to tune a piano, or explore the deeper settings of your DAW or digital keyboard. Musicians owe it to ourselves to hear and play these tunings, and new ones of our own, first-hand; we may find ourselves on an unexpected and rewarding journey.

References (for further reading and listening)
Lou Harrison’s Music Primer, C.F. Peters, 1971
Composing a World: Lou Harrison, Musical Wayfarer, Leta Miller and Fredric Lieberman, University of Illinois Press, 1998
“Paratactical Tuning: An Agenda for the Future Use of Computers in Experimental Intonation,” Larry Polansky, Computer Music Journal, 11(1).
“The Intelligent Keyboard,” Harold Wagge, 1/1, The Journal of Just Intonation, Vol. 1, No. 4. 1985
A Lou Harrison Reader, Soundings Press, ed. Peter Garland, 1987
A Phrase for Arion’s Leap, Tellus Cassette Magazine #14, 1986
Harrison: Piano Concerto, Suite for Violin, Piano, and Small Orchestra, NWR 366-2, 1988
Harrison: In Retrospect, NWR 80666, 2007
Harrison: Complete Harpsichord Works; Music for Tack Piano & Fortepiano in Historic and Experimental Tunings, New Albion Records 11724, 2002.
Lukas Foss and the Brooklyn Philharmonic: At The Tomb Of Charles Ives / Divertimento / Party Pieces / Academic Graffiti. Gramavision 7006, 1983.
American Festival of Microtonal Music: Chamber. Pitch (journal) 2002, 2003, 2004

Concerto for Piano with Javanese Gamelan


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