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IN THE LINER NOTES to the 1993 recording of Lou Harrison's Fugue Percussion (1941), the composer relates how he and John Cage met in "a good pie shop" in San Francisco to work out the "is-tos and as-tos" of employing in rhythmic form the tonal relations found in a fugue. [1] Harrison had become interested in advanced rhythmic issues during the 1930s; in particular he was concerned with cross-rhythms, which he conceived of as "transferring rhythmic patterns into other lengths of time. "[2] He identified cross-rhythms with the overtone series, a concept borrowed from his one-time teacher, Henry Cowell. [3] The task that Harrison discussed with Cage over pie, then, was the application of this correspodence between pitch and rhythm to a musical structure with established pitch or tonal associations, namely the fugue.

This paper will begin with the source of Harrison's pitch/rhythm ideas, that is with a discussion of Cowell's rhythmic innovations. By analyzing the subject and answer statements in the exposition of Harrison's Fugue for Percussion, I will then discuss how Harrison adapted Cowell's ideas, and explain precisely what he is comparing when he abbreviates the phrase "this is to this, as that is to that." An examination of the middle section of the Fugue will follow, focusing on the ways in which Harrison creates rhythmic equivalents to traditional fugal techniques. This discussion will lead to an overview of the formal structure of the piece, and I will address aspects of pitch and instrumentation. I will close with remarks about the musical effectiveness of Harrison's Fugue for Percussion.


When the informally schooled Henry Cowell encountered his first music-theory textbook at the University of California in 1914, he discovered that the ratios formed by the lower partials of the overtone series corresponded to the ratios he had been using as cross-rhythms in his compositions. [4] Intrigued by this, he got together with a graduate student in physics to conduct an experiment to prove his hypothesis that there was a "demonstrable physical identity between rhythm and harmony." They tuned two simultaneous sirens in the relationship 3/2 and confirmed that the sirens sounded the interval of a perfect fifth. Then they slowed the sirens down, keeping the same 3/2 relationship, and discovered that they arrived at a rhythm of 3 against 2, "heard as gentle bumps but also visible in tiny puffs of air through the holes in the sirens." [5]

This experiment proved to Cowell that the ratios express a single physical relationship which can be heard as rhythm when slow and as pitch when fast. He went on to formalize the relationship in New Musical Resources, proposing a means by which vibration ratios of intervals derived from the overtone series can be translated into rhythmic ratios, thereby creating what Cowell called "parallel time-systems" (51). In his representation, the whole note is the basic durational unit, and the ratio of an acoustically pure interval becomes the ratio between simultaneous durational subdivisions of the whole note. For example, the interval of a perfect fifth, vibration ratio 3/2, translated into time (as Cowell expresses this process in New Musical Resources), produces a whole-note measure of three equal notes set over another measure of two equal notes, as shown in Example 1. [6]

Cowell also applied these principles compositionally. He wrote two polyphonic "rhythm-harmony" quartets, Quarter Romantic (1917) and Quartet Euphometric (1919), in which he used pitch ratios to generate aspects of the rhythmic organization. In Quartet Romantic the rhythmic content of the first movement is derived from the harmonic ratios of a precomposed harmonic theme (similar to a four-part chorale), based on the overtone series of [C.sub.2]. Each quarter-note duration of the harmonic theme generates one whole-note measure in the Quartet. Quartet Euphometric is also based on a precomposed harmonic theme, but in it the ratios are applied to simultaneous meters, not to durational subdivisions of the whole note.

An excerpt from the harmonic theme and the corresponding measures in the Quartet Romantic are reproduced in Example 2. [7] I have added annotations to the harmonic theme to indicate the ratios associated with each note, relative to the fundamental [C.sub.2]. So, for example, the E two octaves and a major third above the fundamental is the fifth overtone, vibration ratio 5/1, and corresponds to subdivision of the whole-note measure into five in measures 1-4 (Flute 2). [8] This quintuple subdivision sounds simultaneously with three other subdivisions of the whole note that are derived from the three notes that complete the opening chord of the harmonic theme. In the next chord of the harmonic theme the F one octave and a fourth above the fundamental is expressed by the fraction 8/3 (2/1 x 4/3). This produces the subdivision of the whole-note measure into 2 2/3rd beats in measures 5-6 (Viola). Again, the rhythmic subdivisions that occur simultaneously in the other instruments are generated by the three other pi tches comprising the F major chord in the harmonic theme.

The two "rhythm-harmony" quartets, while innovative and structurally fascinating, are almost unplayable, except by providing each player with a set of headphones and a prepared click tape. This is especially true of the Quartet Romantic with its intricate web of metrically unrelated simultaneous subdivisions of the whole note. In fact, Cowell himself described the Quartets as being "obviously unperformable by any known human agency" and said that he thought of them as "purely fanciful." [9]


Harrison borrowed Cowell's pitch-derived rhythmic concepts, but modified them so that he could produce cross-rhythms that were indeed playable. While Cowell's system expresses the interval-derived ratios in simultaneous subdivisions of the whole note, Harrison is not restricted by the whole-note duration and uses the ratios successively. He takes a passage comprising notes with varying durations and applies an operation to each member to produce a second passage, in which the interval-derived ratio is expressed in the durational relationship between each note of that passage and its corresponding member in the original passage.

Let us examine Harrison's Fugue for Percussion as an example of his use of rhythmic ratios. Harrison divides his Fugue into the three main sections of the traditional fugue (Example 3) and maintains their conventional functional roles. The exposition, measures 1--34, [10] presents the subject four times. The middle section, measures 34--99, develops subject material and consists of an episode, measures 34--45, followed by a group of subject entries, measures 45--75, followed by a second episode, measures 76--84, and a second set of subject entries, measures 88--97. Short transitions connect episode 2 to the second group of subject entries (measures 85--87) and the end of the last middle section subject statement to the start of the recapitulation (measures 98--99). Finally the recapitulation, measures 99--132, restates the material of the exposition.

Harrison's Fugue is written for four percussion parts, and is scored for fourteen percussion instruments, as shown in Example 3. The four parts are analogous to the voices of a four-voiced fugue, except that, despite the implication of a single identity implied by the word, each "voice" has more than one timbral form. Instrumental changes in voices frequently coincide with (or follow shortly after) the start of new formal sections or subsections, helping to identify them as such.

In the exposition, Voice 1 (flexatone) sounds a rhythmic pedal (discussed in relation to Example 11), and is thus excluded from the exposition of the fugal subject. Since Harrison still wishes to present the subject as if in four different voices, after Voices 2 (metalaphone) and 3 (meditation bells) each state the subject, he has Voice 4 present two subject entries, first articulated by the triangles and then by the bell coils.

The subject, first stated in measures 3--6 and comprising twenty-one notes of varying durations (Example 4a), is followed immediately by a second subject statement in measures 7--12 (Example 4b). The notes in parentheses in Example 4b are used by Harrison to indicate the length of the gruppetto. [11]

Each note in the second subject statement is 1.5 times as long as the corresponding note in the opening statement, creating a ratio of 3/2 between the duration of each note in the second subject statement and the duration of the corresponding note in the original statement, the interval ratio of the acoustically pure fifth. Harrison, however, has inverted the ratio, so that the rhythms produced are analogous to notes a fifth below instead of a fifth above (the typical subject-answer relationship). His inversion of the ratio appears to be inadvertent, and, since he is consistent in his application, in this essay I will follow along with him in his oversight by indicating the relationship "above" in quotation marks. The second subject statement in Harrison's Fugue thus corresponds to the traditional answer at the fifth "above" in a tonal fugue. Transposition occurs through durational augmentation.

After the opening "tonic" subject and "dominant" answer, the exposition of Harrison's Fugue continues in a similar fashion, alternating between subject statements in the "tonic" and the "dominant." The third subject statement occurs in measures 13-20, with each note being twice as long as the corresponding note in the original subject, i.e. in the ratio 2/1 to the original and equivalent to an octave "above" the original subject. This "tonic" statement is then answered by the fourth and final subject statement "at the dominant" in measures 22-33. Proportional durations between the fourth statement and the original indicate the relationship 3/1, making the final statement equivalent to a perfect twelfth "above" the original subject.

The four subject entries in the exposition introduce Harrison's method of transforming the ratios of acoustically pure intervals into rhythmic ratios. Furthermore, the durational relationships between successive subject statements mimic the alternation of tonic subjects and dominant answers in a tonal fugue. The body of Harrison's "is-to, as-to" phrase can now be fleshed out: the dotted half-note is to the half-note as the dominant is to the tonic. Tonal relationships have been transformed into rhythmic relationships.

Like a traditional fugue, the middle section consists of alternating episodes, in which subject material is developed, and groups of subject entries. Unlike the traditional fugue, however, Harrison does not explore new tonal areas in his subject entries, and all of the subject statements in the middle section of the Fugue repeat those of the exposition. There are two groups of subject entries, measures 45--75 and measures 88--97. The first of these two groups comprises four subject statements in which "tonic" and "dominant" forms alternate in exactly the same way as they did in the exposition (with changes in instrumentation); this could be seen as a counterexpostion. [12]

The second group of subject entries, occurring at the end of the middle section, comprises two subject statements in which the durations are presented in retrograde. A statement of the original "tonic" subject in retrograde (measures 88--91) is immediately followed by a retrograde statement of the original "dominant answer" in measures 92-97. In Example 5 the retrograde "tonic" subject starting in measure 88 is reproduced together with the original "tonic" subject from the exposition for comparison.

While the use of the retrograded subject is a technique found in traditional tonal fugues, albeit rarely, and one that is easily translated from pitch to duration, Harrison does not use other common traditional techniques, like the exploration of new "keys" in the subject entries of the middle section. In other words, he does not extend his method of expressing pitch transposition as durational augmentation to new levels of transposition. It seems likely that this is because the rhythmic patterns, already challenging to perform in the "tonic" and "dominant" versions of the subject, would become unplayable at other transpositions.

In the episodes, however, Harrison finds rhythmic equivalents to other compositional procedures traditionally found in fugal episodes, namely subject development, sequences, and the establishment of pedal points. The primary method for developing subject material in the first episode (measures 34--45) involves fragmentation, together with the same kind of transposition by durational augmentation that is used for the subject statements.

Starting in measure 34 there is a series of imitative entries in which each of three voices presents a fragment of the subject at a different transpositional level. The basic unit that is imitated is derived from the first ten notes of the original subject. Example 6 presents both the first ten notes of the original subject (the source), and the fragment as it is presented in measures 34--35 (the basic unit of imitation).

The fragment comprises two parts. The first part, presented in measure 34, consists of the first five notes of the fugue subject in their original "tonic" form. The second part of the fragment, presented in measure 35, is the quintuplet gesture with each note twice the duration it was in the original, i.e. still a "tonic" form, but the equivalent of an octave "higher." In this way transposition "at the octave" occurs within the subject fragment that forms the basic unit of imitation.

This fragment is then imitated and transposed at two different levels by two other voices in succession. In measures 36-38 the fragment is imitated by a second voice, transposed to the "dominant." As can be seen in Example 7, notes belonging to the first part of the fragment have 1.5 times the duration of the corresponding notes in the original subject, equivalent to transposition "up" a perfect fifth. Notes corresponding to the original quintuplet gesture are three times as long as those in the original, thus maintaining the interior transpositional relationship of the fragment (2/1).

The fragment is imitated by a third voice in measures 39-43, reproduced in Example 8. This time the subject fragment is transferred to a new transpositional level, that of the "supertonic." [13] Since the interval ratio of an acoustically pure ninth is 9/4 = 2.25, the first part of the fragment is presented in durations 2.25, times the length of the opening notes of the original subject, and the quintuplet gesture in durations 4.5 times the original.

In this section of the first episode then, Harrison creates a series of imitative entries using a fragment from the fugue subject. Development takes place in pitch-derived rhythmic forms in two ways: there is a "tonal" relationship within each fragment (the "octave" relationship between the first and second parts of the fragment), and another "tonal" relationship between the imitative fragments where "transposition" is through the "circle of fifths" from the "tonic" to the "dominant" and, finally, to the "supertonic."

The use of sequence is of course common in fugal episodes, and Harrison specifically mentions sequence as one of the tonal elements for which he worked on finding a rhythmic equivalent in his Fugue for Percussion. [14] To create an equivalent, Harrison replaces the melodic unit with a rhythmic unit, which is then repeated in the same voice in different metric positions within the constant 2/2 meter. In other words, he uses metric shifting rather than durational augmentation to create a rhythmic equivalent to pitch transposition in sequential passages.

Harrison uses this form of rhythmic sequence twice in episode 1 to accompany the imitative subject fragments discussed above. The two instances occur concurrently starting in measure 39, as shown in Example 9. Although the basic rhythmic unit of each sequence is different in length, they are coordinated so that both sequences complete a metric cycle simultaneously after three measures.

Another tonal feature often used in fugues is the pedal point. Harrison employs a simple rhythmic equivalent in the middle section of his Fugue, by having the bass drum present an extended drum roll. In this way a "single note" is sustained through measures 42-47, and a rhythmic pedal point is created over which other rhythmic patterns continue. This bass pedal functions in part as a link between two sections, joining the end of episode 1 to the beginning of the first group of middle section subject entries (measure 45). The passage is reproduced in Example 10.

A different kind of pedal point is presented by the flexatone in the exposition and recapitulation, as can be seen in Example 11. In this case the instrument produces a continuously fluctuating sound that creates a background over which the other instruments present their rhythmically differentiated subject statements. The flexatone's continuousness and lack of attack points help to identify it as a rhythmic pedal.


As was discussed earlier, the exposition of the Fugue presents the subject four times, alternating "tonic" and "dominant" entries. In the diagram in

Example 12 shading is used to differentiate "tonic" and "dominant" statements, with horizontal waves indicating "tonic" subject entries, and vertical waves indicating "dominant answers." Furthermore, each subject statement is labeled according to how it relates durationally to the original subject.

There are no countersubjects in the Fugue, and each voice states different material to accompany the subject statements in the exposition. However, Harrison repeats almost all of the original material to accompany the four subject statements that make up the first set of subject entries in the middle section, reinforcing the idea of a counterexposition. He makes only two, relatively minor changes: the flexatone pedal is replaced by material in the maracas in measures 45-59, and new material is added in measures 64-75 to replace rests. Harrison restates the original accompanying material again in the recapitulation.

The recapitulation presents a retrograde of the order in which subjects, "answers," and various accompanying materials occur in the exposition. Furthermore, the content of each rhythmic unit (subject, material accompanying a subject entry, etc.) occurs in retrograde, a technique that was first explored in the middle section where subjects were presented in durational retrograde in the second group of subject entries (Example 5).

This technique of retrograding large sections of original material is also applied to the episodic material of the middle section, where the second episode is an almost complete retrograde of episode 1. The content of the first episode (measures 34-45) is slightly truncated--measure 34 and measures 42 (middle) to 45 are omitted--and otherwise presented in exact retrograde in measures 76-84. As with the recapitulation, not only is the order in which voices enter with specific material retrograded, but the durations within each voice are also presented in reverse order.

As a result of Harrison's large-scale use of retrograde, the standard, three-part fugal design that he adopts in his Fugue for Percussion is expressed through a palindromic plan. The center of the palindrome is the first group of subject entries in the middle section. The only anomaly in this otherwise strict palindromic structure is the passage from measure 85 to measure 99, which includes the second set of subject entries (measures 88-97) and the two short transitions that lead to and from it. [15] The palindromic design of Harrison's Fugue is diagrammed in Example 13, with similar shading connecting retrograde-related sections.


Of the fourteen instruments in Harrison's Fugue, all but one, the fiexatone (or saw), is capable of producing distinct rhythmic patterns and can thus participate in the rhythmic fugue. The melodic capabilities of the ensemble instruments, however, are limited, with only six instruments or instrument groups being able to produce different tones. The metalaphone has the greatest melodic potential of seven tones; [16] the cowbells, meditation bells, and brakedrums are grouped to present five different tones; the three triangles produce three tones, and a pair of gongs generate two; the bell coils are each capable of producing two tones. Furthermore, the box and washtub can each produce two different unpitched tones, the former by being played on the top or side, the latter by being struck in the center or near the edge.

The melodic capabilities of the instruments give rise to questions about how important pitch is in the fugue and whether the subject has a melodic profile. While Harrison does assign the instrument with the widest melodic range, the metalaphone, the task of introducing the fugal subject in measure 3, melodic aspects are clearly subordinate to rhythmic ones. Subsequent subject statements occur m instruments with minimal melodic capabilities, including the two-toned bell coils (measure 22) and gongs (measure 55), as well as the bass drum (measure 64), which is not capable of any melodic differentiation. Even when Harrison does state the subject in instruments capable of producing five tones, the contour of the original metalaphone statement is only loosely maintained, and he is not consistent in the way he treats different five-toned statements melodically. This is illustrated in Example 14 where three subject statements are reproduced: the original metalaphone statement from measures 3-6, the "answer" in the meditation bells in measures 7-12, and the middle section subject entry in the cowbells, measures 45-48.

A comparison of these three subject statements reveals Harrison's varied treatment of the melodic aspects of the fugue subject. Although both the meditation bells and the cowbells can produce five tones, Harrison uses only four of the five possible tones in the subject statement in the meditation bells, thus limiting its melodic capabilities further. In addition, some figures do not correspond melodically to the original. For example the original eighth-note figure from measure 5 is quite different in the meditation bells statement, being presented with an inverted contour in measure 10. The cowbells subject statement adheres more closely to the original metalaphone statement in terms of contour; in comparison to the meditation bells statement, however, it is noticeably dissimilar melodically, in part because it does use all five tones at its disposal. From this flexible approach to contour, one can conclude that Harrison's primary concern lies in the domain of rhythm, and that pitch aspects are secondary.


In summary, this paper has examined the "is-tos and the as-tos," the ways in which Harrison transposed traditional tonal elements of the fugue into rhythmic form to produce a rhythmic fugue. We have seen how Harrison draws on Cowell's durational innovations based on the overtone series to create rhythmic equivalents for tonal relationships that then form the basis of a traditional fugal exposition with alternating "tonic" and "dominant" subject entries. In the middle section of the fugue, Harrison creates rhythmic equivalents for other tonal features frequently found in fugues, including imitative motives derived from the subject, melodic sequences, and pedal points. An examination of the overall form reveals a palindromic structure projected onto the traditional fugal structure, created by large sections of music occurring in retrograde.

It seems curious that after having carefully worked out some of the ways in which tonal relations could be transformed into rhythmic forms, Harrison should choose to simply repeat a third of his musical material in retrograde. Possibly he tired of working out the intricacies of the "is-tos and as-tos." Or perhaps Harrison realized that, from an aural perspective, the extensive use of retrograde is irrelevant. Given the rhythmic complexity of the original material, the retrogrades are almost impossible to recognize as such, and the listener is quite unaware of these procedures.

What the listener is aware of are musical processes like climax, cadence, and structural delineation, which are carefully articulated through instrumentation, contour, texture, and dynamics. Consider, for example, the transition from the end of the second group of subject entries in the middle section to the start of the recapitulation, reproduced in Example 15. Voice 3 (washtub) concludes its retrograded subject statement in measure 97 with a diminuendo. It then rests until its entry on meditation bells in measure 99, heralding the start of the recapitulation. The texture gradually thins as Voice 3 nears the end of its retrograde subject, Voice 4 dropping out in measure 95 with a diminuendo to pp, and Voice 1 falling silent in measure 96. Voice 2 continues across Voice 3's measure of rest, in a descending arc, growing ever softer until its final note melts into the accented meditation bells attack in measure 99.

Examples like this clearly show that Harrison's concern was not restricted to the mechanics of creating a rhythmic system, and that he was equally concerned with musical processes. His approach to rhythmic complexity, ultimately, is elegantly successful. Unlike Cowell, Harrison manages to maintain a high level of complexity while still writing music that is playable by human performers. Moreover, despite the contrived way in which the system, the "is-tos and as-tos," is worked out, and despite the fact that it is difficult to hear the piece as a fugue per se, his Fugue for Percussion works on an aesthetic level, providing the listener with a satisfying and stimulating musical experience.


BRENDA RAVENSCROFT teaches theory at Queen's University in Kingston, Ontario, where she is Associate Professor of Music.


The author thanks Peter Schubert for his advice on earlier drafts of this article.

(1.) Lou Harrison, The Perilous Chapel, The San Francisco Contemporary Music Players, New Albion Records, Inc., NA055CD, p. 4.

(2.) Ibid.

(3.) Cowell's rhythmic theories are explained in New Musical Resources (New York: Knopf, 1930; reprint, with notes and an essay by David Nicholls, Cambridge: Cambridge University Press, 1996). Cage, like Harrison, had studied with Cowell in the early 1930s; his familiarity with Cowell's rhythmic innovations, together with his own compositional imagination, must have made him a useful collaborator for Harrison.

(4.) Arthur Foote and Walter R. Spalding, Modern Harmony in Its Theory and Practice (Boston: The Arthur P. Schmidt Co., 1905).

(5.) This paragraph is based on an account of an anecdote related by Henry Cowell in the Preface to Quartet Romantic (New York: C. F. Peters Corporation, 1974).

(6.) Cowell also proposed a new system of rhythmic notation whereby noteheads of various shapes (triangles, squares, etc.) would represent different subdivisions of the whole-note, thereby eliminating the need for brackets with numbers to denote irregular subdivisions of the beat. Harrison, however, does not use this system in his Fugue, preferring the traditional notation of durational subdivisions (like those used in this example).

(7.) Cowell uses his notational system in the Quartet, but also indicates subdivisions conventionally.

(8.) The pitch content is not related at all to the harmonic theme, the piece being described by Cowell as freely "atonal."

(9.) Preface to Quartet Romantic.

(10.) Many of the sectional breaks in the Fugue are not as clear-cut as is suggested by these measure numbers, and formal divisions are frequently elided (see number 15).

(11.) The published score is handwritten by the composer, and he appears to have made a few notational errors. For example, in measures 10-11 of the second subject statement he omits a tie across the bar. This is clearly needed for the subject to be consistent with the original, and I have added it in Example 4b. In later subject statements otherwise identical to this one, measures 49-54 and measures 121-126, the tie is present (although a different tie is omitted in measure 122).

(12.) Another difference between these entries and those of the exposition (besides instrumentation), is that the fourth statement is presented in two voices simultaneously, starting in measure 64.

(13.) This "supertonic" fragment does, for the only time in the piece, extend the tonal analogy beyond the "tonic" and "dominant." It is worth noting that the music that does this is fragmentary and brief, and therefore more flexible than the subject.

(14.) Harrison, liner notes to The Perilous Chapel, p. 4.

(15.) Two of the voices, Voices 1 and 2, have palindromic instrumental profiles, as would be expected from the overall palindromic form of the Fugue. However, as can be seen in Example 3, in Voices 3 and 4 the symmetrical instrumental design is interrupted by the introduction of new instruments, the washtub and cymbals respectively, in measure 83 when they enter early with transitional material the leads to the second group of subject entries. This is an example of the formal elision that was mentioned earlier. Although the retrograded material of the second episode continues in the claves until the end of measure 84, at which point the transition to the second set of subject entries begins, the washtub and cymbal, after a measure of rest, enter in measure 83 with their transitional material.

(16.) Harrison suggests the metalaphone part could be played on a celeste, restricted to seven selected tones.

[Table omitted]

[Graph omitted]
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