Just Intonation and Indian Aesthetic in Terry Riley’s The Harp of New Albion
Among major American minimalist composers, Terry Riley stands as a unique figure. Unlike his contemporaries with whom he is often associated - Philip Glass, Steve Reich, and perhaps John Adams - Riley’s work is notable for its improvisational element and also - since the late 1970s - for its employment of just intonation rather than equal temperament. Ironically, it is these unique aspects of Riley's music that have caused it to have been less studied; the music of his minimalist contemporaries Glass, Reich, and Adams has received more analytical attention. In addition, while Riley's extensive study of North Indian raga with the late Kirana singer Pandit Pran Nath has been frequently mentioned in the literature, the specific influence of Indian classical music on Riley’s compositional process and improvisational technique has not been addressed.
At the core of Riley’s music is the keyboard. The high-C pulse that holds together the composition for which he is best known, In C (1964), is played on a piano. In the late 1960s such extended cumulative tape-loop/delay works as A Rainbow in Curved Air and Poppy Nogood and the Phantom Band made extensive use of electronic organs (supplemented, in the case of Poppy Nogood, with soprano saxophone). Throughout the 1970s and early 1980s, Riley extensively performed works for multiple layers of electronic organs, such as Persian Surgery Dervishes (1971-72), performed solo through cumulative tape loops and delays. By the release of 1980’s Shri Camel album, Riley was using keyboards modified to play in just intonation, a result of his longtime friendship with La Monte Young and his studies with Pandit Pran Nath.
Then Riley returned to the piano. As he related in an interview, “Around 1980 I bought an old upright and started to play and develop music on piano again. Of course, I’d been aware of La Monte [Young]’s Well-Tuned Piano since ’64, but I’d also been playing both Indian music and electronic keyboards in just intonation. So I decided to tune the piano that way rather than in equal temperament.” Because of the way that the overtones of piano strings resonate sympathetically with other strings, working in just intonation turned out to be a powerful expressive tool for Riley: “I was able to give the music a different shape. The piano has a much greater scope of expressive possibilities than electronic instruments.”
Riley’s piano cycle The Harp of New Albion (1984) is therefore significant in the Riley canon; it not only remains his largest solo-piano work, but it “gives us an unusual insight into his improvisational process.” In this paper I will first describe the tuning used in The Harp of New Albion as a whole. I will then discuss specific techniques used in The Harp of New Albion that draw upon practices of Indian classical music such as tala, gamak, and jhala. This will be accomplished by a close study of two different performances (recorded within a few weeks of each other) of "The Magic Knot Waltz," one of the movements of The Harp of New Albion. Finally, I will analyze the recorded evidence to show which motives appear as constants in performance (which ideas make up the “seed” of the composition) and how those motives are developed through improvisation.
The Harp of New Albion is tuned to a five-limit chromatic scale (Figure 1) with C# as its tuning center. This pitch class was a tuning constant for Riley through much of his work from the mid-80s to mid-90s. In a 1986 interview for Keyboard magazine Riley explained:
For many years I played electronic organ and synthesizers. While I was playing the synthesizers, I got together with Krishna Bhatt, who’s a sitar master living in Berkeley. Krishna played in C#, so I started redoing all my pieces in C#. During that period I also started playing more and more piano. At that time I wrote the one piece that was the beginning of all the pieces I’ve done lately, called The Medicine Wheel. It was the first time I did this just tuning in C#, and I’ve kept the piano in that tuning to do both The Harp of New Albion and Salome Dances for Peace. They’re all done in the same tuning.
However, none of the eleven movements of The Harp of New Albion are in the key of C#, making it necessary to distinguish here between tuning center and tonal center. For each movement Riley chooses a tonal center in varying degrees of relationship to the C# tuning center, exploiting the intervallic differences that result from choosing different tonal centers and employing different modes based on those tonal centers (Figure 2). The fact that just intonation does not transpose equally to any key was, of course, historically one of the reasons for the adoption of equal temperament. Riley's music, on the other hand, celebrates the differences in quality that result from transposition. Figure 3 diagrams Riley’s tuning system by pitch-class dyads reflecting their layout on the piano keyboard. A full list of these intervals arranged by size, with their commonly accepted names in just-intonation terminology, is provided in Figure 4.
|(ratio)||(in cents from equal temperment)|
Figure 1: Tuning in The Harp of New Albion, compared with equal temperament.
|A:||Cadence of the Wind||(8:5 from C#)|
|A#:||The New Albion Chorale||(5:3 from C#)|
|The Orchestra of Tao|
|B:||Land's End||(16:9 from C#)|
|B#:||Ascending Whale Dreams||(15:8 from C#)|
|Circle of Wolves|
|D:||Riding the Westerleys||(16:15 from C#) DE F# G# AB C|
|Premonition Rag||DE F AB (C)|
|Return of the Ancestors||DE F# G# AB C(#)|
|The Magic Knot Waltz||DEb F# G# A C|
Figure 2: Tonal Centers in The Harp of New Albion.
The aesthetic behind Riley’s use of just intonation, "from which the particular consonances and dissonances determine the emerging energies that flow through both instrument and performer," is similar to the Indian concept of raga. For example, not only does Riley choose different tonal centers from movement to movement, much in the fashion of a romantic-era multi-movement work, but each of the four pieces that share D as their tonal center employ a different scale (Figure 2). For Riley, music in just intonation is a kind of yogic practice: “You know, the idea of yoga is union, union with God. And tuning means atonement, or trying to make two things one, right? So, just intonation has a lot to do with achieving the correct proportional balances of notes in order to create one.” He goes on to say in the same interview:
the effect of music is heightened by being in tune. Resonant vibration that is perfectly in tune has a very powerful effect. If it’s out of tune, the analogy would be like looking at an image that is out of focus. That can be interesting too, but when you bring it into focus you suddenly see details that you hadn’t seen before. What happens when a note is correctly tuned is that it has a detail and a landscape that is very vibrant.
Of course, the reverse axiom is also true; the effect of music is heightened in a different way by being “out of tune” (by equal tempered standards). A particularly striking example in The Harp of New Albion is the piece “Circle of Wolves.”
As a result of Riley’s just intonation tuning, three “fifths” are identifiable as so-called “wolf fifths” - 40/27 (D#-A# and E-B) and 1024/675 (B#-G). It was the “out of tune” quality of these intervals that led to certain keys historically being considered unsuitable for modulation. (Figures 3 and 4 show how strikingly different such intervals are from the “pure” just-intonation ratio of 3/2.) Of these “wolf fifths,” the most complex (therefore “out of tune”) ratio is 1024/675, heard in this tuning only between the pitch classes B# (C) and G. As a result, B-sharp - the tonal center of “Circle of Wolves” - is the key “most distant” from the C-sharp tuning center. The B-sharp/G dyad (or C-G) makes up the tonic and “fifth” scale degree of the pitch collection that characterizes “Circle of Wolves”; the other pitches in the collection are the two 40/27 fifths. The resulting sonority is as “out of tune” as possible within the tuning system, and Riley explores its unusual quality deliberately by making it the main sonority of his improvisation (Example 1). The title is a punning reference not only to the “wolf” quality of these fifths but to the classical “circle of fifths” that results from equal temperament.
Click to view Figure 3: Chart of interval sizes in The Harp of New Albion
Click to view Figure 4: Complete list of interval ratios in The Harp of New Albion (arranged smallest to largest), with names of intervals where applicable.
Example 1: "Circle of Wolves," reduction of pitch collection.
Riley and Indian Classical Music
Riley recalls first hearing the music of Ravi Shankar and Ali Akbar Khan “sometime in the early sixties” but was not interested in studying it at that time. It was not until 1970 that he began studying as a disciple of the North Indian kirana vocalist, Pandit Pran Nath, and making numerous trips to India to study with the Master. Riley appeared frequently in concert with the legendary singer as tamboura, tabla, and vocal accompanist until Pran Nath’s death in 1996.
In spite of this lengthy tutelage, Riley is careful to distinguish his work from raga in the traditional sense of the word:
The purest form of music that I’ve ever heard is raga. I wouldn’t try to create anything different if I were singing a concert of raga, for instance. I would try to sing it exactly as I’ve been taught to sing it. But the other work, the work that is outside of raga, that’s definitely a different category in my mind. I don’t feel like that’s doing a disservice to raga, because it’s not raga.
The Belgian minimalist composer Wim Mertens similarly observes, “one cannot find any real influence [of study with Pandit Pran Nath], since he [Riley] came to Indian music more by thinking through his own musical ideas.” Nevertheless, certain techniques of raga have their counterparts in Riley’s improvisational practice. For example, each raga in North Indian music belongs to one of ten parental scales called thats. Each that has a definite set of seven scale degrees or swaras. While Indian theorists usually acknowledge the existence of 22 srutis - musical tones - within a saptaka (octave), the srutis are nevertheless grouped into seven swaras for each that - and hence for each raga. Individual ragas may omit a particular swara, but the same swara would not be found in two different forms (F and F#, for example, in Western terms). Swarup lists the intervals between the suddha (“pure”) swaras in Indian tuning as, in ascending order: 9/8, 10/9, 16/15, 9/8, 9/8, 10/9, 16/15. The European equivalents, he claims, are 9/8, 10/9, 16/15, 9/8, 10/9, 9/8, 16/15. This corresponds to Riley’s C# major scale.
The scale of “Magic Knot Waltz” is made up of six pitches: D, Eb, F#, G#, A, and C; in Indian terminology this would correspond to Sa, komal (“soft” or “flat”) Ri, Ga, tivra (raised) Ma, Pa, and komal Ni (the sixth swara, Dha, is silent). The combination of komal ni (D) with tivra ma (G#) does not occur in North Indian ragas (unless borrowed from the south), because the combination is not acknowledged among the thats. Riley’s scale, therefore, is not one of the traditional North Indian ragas. His performances of “Magic Knot Waltz” are, however, consistent with raga practice in their virtual exclusion of all pitches foreign to the scale. Also, Ma is the only interval that is raised in North Indian theory, so Riley’s use of G# is consistent with tradition in that regard.
Another parallel with North Indian classical practice can be found in Riley’s application of tala, the rhythmic cycles that are characteristic of a raga performance. The “live” performance, recorded in 1986 but not released until 1992 as part of The Padova Concert, fits quite well into a thirteen-beat tala, the left-hand 3:2 ostinato also analogous to the repeating lahara in Indian performance (Example 2). A lahara is a short melodic ostinato that is usually used in accompanying a tabla, particularly in tabla solos. It is often used as the equivalent of a metronome when a student is working on any fixed composition.
Example 2: "Magic Knot Waltz," the labara as used in the Padova performance, 20 January 1986.
At one point in the Padova performance Riley comes back in, after a cadential flourish, on the “wrong” starting note of his lahara (C4 rather than D4) - an eighth note early, as it were (the “wrong” note entry is the first note of the left-hand pattern in Example 3). He corrects this “error” in mid-cycle by removing the Eb3 from his final ascent (toward the end of system 2 of Example 3) and instead moving directly to F#3, ensuring that the cycle will again begin on D4. This correction “in midstream” suggests that Riley was aware of the need to make a correction earlier in the cycle; in fact, he shifts to a repetitive right-hand pattern (a moment of improvisational “autopilot”) immediately after the left-hand correction, perhaps in order to concentrate on the lahara and ensure that the change would bring about the desired realignment. In summary, this correction in the midst of an improvisational flourish is likely evidence of “tala thinking” in Riley’s structural plan.
The “studio” Celestial Harmonies performance, recorded at the Academy of Music in Munich, is actually edited together from two live performances on January 3 and 4, 1986 (made a week before the Padova recording). Despite Riley’s assurances that “all the takes here were recorded as I performed them with only a few minor cuts and all the breakthroughs and unplanned moments intact,” the impact that such edits make on the final product is nevertheless difficult to ascertain. Possibly because of their assembled nature, the cyclic aspect of the Munich “Magic Knot Waltz” is irregular - ostinato lengths tend to fit into 12- or 13-beat cycles, with cadenza-like interjections of varying lengths.
The Munich performance, however, contains a striking moment that is entirely absent from the more cyclical Padova recording: an alternation of a hoedown-like theme with a section that is strikingly similar to Indian jhala. As David Courtney defines jhala:
Jhala is undoubtedly the most characteristic of the instrumental styles. Indian stringed instruments are noted by [sic] a few special purpose drone strings called chikari. These strings are never fretted but are struck whenever the tonic needs to be emphasized (i.e., Sa and Pa). The jhala is a fast paced alternation of main melody string and chikari. This lends itself to interesting permutations of both rag and tal simultaneously.
The construction of the piano, of course, does not allow for special drone strings. Riley’s jhala passages, however, always emphasize A3 and C4 (Pa and komal Ni). Although there is not a literal “rapid alternation” of the drone notes with a melody, Riley does pit the drones against an oscillating pair of dyads (see Example 4). In the first two of the three jhala sections, these dyads have attack points between the attack points of the drone stream, possibly allowing for an “alternating” interpretation.
Click to view Example 3: "Magic Knot Waltz," Padova recording, "error" in tala with Riley's compensation [2:50].
Example 4:“Magic Knot Waltz,” a jhala section from the München recording [4:36 ff].
Another aspect of Indian classical music, all the more striking in Riley’s case because it is found in Indian vocal (rather than instrumental) style, is a technique of ornamentation called gamak. Peter Manuel describes gamak as “a technique in which every note in a passage is approached from its lower neighbor,” and notes that the practice has “crossed over” from Indian classical music to popular genres such as film music. Riley’s improvised passages in “Magic Knot Waltz,” especially the long rhapsodic lines that come at cadential points, employ the same “lower neighbor” ascending-step contour cited by Manuel as essential to gamak. One example of this technique in Riley’s improvisational style is found in Example 5.
Example 5:“Magic Knot Waltz,” example of gamak developing initial turn of the theme [from München recording, 3:43-3:53].
This vocal orientation in Riley’s music (incorporating a technique such as gamak in an instrumental context, for example) explains the fundamentally melodic character of Riley’s music. Like his minimalist counterpart Philip Glass, Riley downplays Western harmony in favor of developing other parameters. But whereas Glass, in his early work, chose to develop the rhythmic element, Riley’s music is more melodically focused. This emphasis on melodic development perhaps derives from his vocal study with Pandit Pran Nath, although it may also result from his saxophone improvisation technique on pieces such as Poppy NoGood and the Phantom Band (1969), which predate his formal study of Indian classical music. Today, however, Riley clearly attributes this interest in melody to non-Western sources:
Melodic complexity is inversely proportional to the amount of harmonic or polyphonic movement or density of parts. What you have in Eastern or Renaissance music or Gregorian Chant is very ornate melody, but if anything starts to compete with that, it cancels out the effects of the melody. So in the West we opted for the drama, the harmonic and polyphonic complexity. That’s great, but we’re a little bit stuck at this point.
Motives in The Harp of New Albion
More important to me than the Minimalist theme in my work is the interrelationship of motives. Both In C and the later works have a really strong developmental quality, a lot of variation and permutation of motives. This isn’t theoretical; it’s the way I hear.
Most of Riley’s keyboard music through the 1970s was improvised to particular raga-like scale patterns. (It was not until his “G Song,” composed in 1980 for the Kronos Quartet, that Riley returned to fully notated composition.) Riley’s involvement in improvisation goes back to childhood:
One of my cousins played really well by ear, so I would listen to him and try to play what he played. He would make up copies of pieces like Tchaikovsky’s Piano Concerto. You know, he’d start out with the theme and then improvise on it. I thought that was really neat, so I tried to do that kind of stuff, too.
In fact, Riley was first attracted to Indian classical music through its improvisational aspects: “I’d always been interested in improvised music, and here was an improvised music that had such precision that it was a classical music, too. That idea, I thought, was quite a stunning one: to be able to develop improvisation to the degree that it sounded like it was all composed.”
The goal of tightly structured improvisation is evident throughout The Harp of New Albion. Much of the work is improvised, but improvised passages are found side-by-side with composed ideas. A performance of a movement from The Harp of New Albion can perhaps be compared to a jazz improvisation, in which the theme or “head” is followed by solos over the chord structure before concluding with a return to the “head.” The interchange between “composed” and “improvised,” however, is much more fluid in Riley’s music; John Schaefer describes the piece as having a “spiral form.” As Riley explains it, “Something spins off a little motif¼and gets larger and more arpeggiated, more embroidered. It cycles back to a certain note, but it’s very irregular; it takes a circuitous route.”
Examples 6 and 7 illustrate what is meant by “spiral form.” In comparing the two recorded performances of the movement “Magic Knot Waltz,” one finds certain consistencies between the two performances that are perhaps “composed” in a loose fashion. Example 6 depicts various transformations of the “Magic Knot Waltz” theme, as heard in the opening two minutes of the released Munich performance. These versions of the theme share an E-flat/C turn around D6 at the outset and culminate with some variation on an ornamented D-C figure (these thematic statements are labeled on Example 6). Beyond these generalizations, however, there is considerable variation, with very little direct repetition from one thematic statement to another. The E-flat/C turn that begins the theme also becomes the catalyst for the gamak-like cadential flourishes that occur numerous times in both recorded performances (see Example 5). The gentle arc that closes the theme also undergoes development. Example 7 shows how this truncated fragment is varied in three different ways (labeled A, B, and C).
The Padova performance, recorded barely a week later, is noticeably different in its treatment of thematic materials. The jhala sections described earlier are absent, the cadential flourishes much more extensive in range, particularly in the low register (as low as A1), and the revelation of the theme much more tentative. In fact, the first hints of the theme in the Padova performance are oblique references to the D-C-C or D-C-A motive with an upper pedal tone that closes the concluding arc of the theme. Example 7 provides instances of this three-note or three-chord figure in the München performance (see, for example, the last three notes of the first system, the three notes immediately preceding statement B, the last three notes immediately preceding statement C, and the last three notes of the example). Riley’s Padova performance introduces this figure as slowly shifting dyads over the ostinato (see Example 8). When fragments of the theme do appear, after nearly a minute, they appear with interjections of the earlier shifting dyads. The theme is not completely stated until [2:12], nearly halfway through the performance.
Click to view Example 6: “Magic Knot Waltz,” first four thematic appearances, München recording [0:42-1:33].
Example 7: “Magic Knot Waltz,” three different thematic truncations [labeled A., B., and C.], München recording [2:07-2:23].
Click to view Example 8: “Magic Knot Waltz,” Padova recording, gradual revelation of thematic material [0:24-1:08].
The Harp of New Albion as a whole is an important transitional piece among Riley’s compositions, crystallizing the keyboard-oriented improvisations of the previous fifteen years while pointing toward his more formally notated works for acoustic ensembles. Riley’s compositions prior to the release of The Harp of New Albion tended to involve electronic keyboards, in which “tape loops and studio processing often melted Riley’s organ and synthesizer lines into a larger, more liquid context.” The acoustic medium, therefore, allows the listener to focus more on the musical line, and on ideas as they emerge, rather than to be seduced by textures and ambience. In addition, this piece differs from some of Riley’s later recordings - such as Chanting the Light of Foresight, written for the ROVA Saxophone Quartet - in that we hear Riley as his own interpreter. It is this piece’s clarity of texture, combined with its unmediated performance, that makes it faithful to his philosophical and aesthetic aims.
When asked by an interviewer whether “being in tune really does have a profound meaning,” Riley’s response was “Yes, that’s Pandit Pran Nath’s main philosophy. It’s called surma, being in tune. That recognition, that appreciation of the subtle frequencies, is an insight into music [that] we need¼. Being in tune, putting total being and concentration in each note, living through each moment in music as a divine link in the ecstatic experience: These things are important to me now.”
One can find throughout The Harp of New Albion eloquent evidence of these aesthetic concerns. The importance Riley places on “being in tune,” for instance, manifests itself in the careful intricacies of Riley’s tuning system, in which the arrangement of semitone sizes is virtually symmetrical with the exception of F-sharp/G and G/G-sharp (see Figure 3). The ramifications of just intonation, in which tuning is not homogenized by temperament but instead results in differing shades of consonance or dissonance, lead to a listening mode that is conducive to appreciating such nuances. For example, in the “Ascending Whale Dreams” movement, Riley employs a whole-tone scale over a B-sharp tonal center (which, in just intonation, actually features three different types of “whole step” - see Figure 3). This scale dictates the pitch material until nearly halfway into Riley’s eight-and-a-half minute recorded performance, when he abruptly shifts to the other whole-tone collection at [3:54]. The effect of this change is dramatic, in a way that could not be possible using equal tempezrament.
In just intonation, allowing for pitch-class permutation, more than two whole-tone scales are possible, but in “Ascending Whale Dreams” Riley limits himself to two. This self-imposed limitation, and Riley’s extensive development of such limited materials (also discussed earlier in reference to “Circle of Wolves”), is what Riley means by “putting total being and concentration in each note.” According to Wim Mertens, “Riley examines only a small amount of material intensively, and only when the material is exhausted does he add something new to it.” Although Mertens was chiefly referring to earlier keyboard works, such as Persian Surgery Dervishes (1972) or the Keyboard Studies (1964), one can certainly hear evidence of Riley’s exhaustive concentration on the smallest of materials in some movements of The Harp of New Albion. While listening to the “Riding the Westerleys” movement, for example, the listener becomes aware of how Riley’s subtle pedal technique and choice of notes that reinforce the overtones of the D tonal center create a subtly shifting wash of sympathetic resonance. This allows for a listening experience that shifts from foreground to background and back again.
“Living through each moment in music as a divine link in the ecstatic experience” not only refers to Riley’s improvisational style but also his raga-inspired aesthetics. “The Magic Knot Waltz,” among other movements, offers an excellent realization of these aims; “Magic Knot Waltz” is a particularly fine example for study since one can compare the two recorded versions. One is struck not only by Riley’s considerable dexterity but by the fecundity of his ideas.
Nevertheless, Riley’s emphasis on the “moment” in performance means that to fully grasp his art in an analytical fashion, one must go to the recorded artifact. This means that Riley’s music may indeed have more in common with John Coltrane or Ravi Shankar than it does with that of composers such as Philip Glass or Steve Reich. It is also for this reason that Riley’s music has not enjoyed the same scholarly attention that the music of Reich or Glass has. Much of Riley’s music, therefore, remains largely unexplored by music analysts; but careful study of the recorded performances of his music may well lead to a new aural-based theory, in which Western analytical technique is linked with centuries-old traditions.
Canright, David. 1985. “On Piano Retuning.” Downloaded from the World Wide Web on
November 29, 2000 at http://www.mbay.net/~anne/david/piano/index.htm
Capwell, Charles. 2000. Personal e-mail communication, 12 December 2000.
Courtney, David. 1992. “New Approaches to Tabla Instruction.” Percussive Notes 30, 4:
______________, n.d. “Jhala.” Downloaded from the World Wide Web on January 19, 2001
Duckworth, William. 1995. Talking Music: Conversations with John Cage, Philip Glass,
Laurie Anderson, and Five Generations of American Experimental Composers. New York:
Gann, Kyle. 1998. “Anatomy of an Octave.” Downloaded from the World Wide Web on
June 8, 2001 at http://home.earthlink.net/~kgann/Octave.html.
Ghosh, Nikhil. 1968. Fundamentals of Raga and Tala with a New System of Notation.
Bombay: Popular Prakashan.
Glass, Philip. 1987. Music by Philip Glass. Edited by Robert T. Jones. New York: Harper
Mancini, Patricia and Joseph. 1986. “On Just Intonation and the Spiritual Source of
Music: Terry Riley.” Keyboard 12, 7: 52-65.
Manuel, Peter. 1988. Popular Musics of the Non-Western World. New York: Oxford
Mertens, Wim. 1983. American Minimal Music. Trans. J. Hautekiet. London: Kahn &
Partch, Harry. 1974. Genesis of a Music. 2nd ed. New York: Da Capo Press.
Riley, Terry. 1986. The Harp of New Albion, liner notes. Celestial Harmonies CD CEL
Schaefer, John. 1987. New Sounds: A Listener’s Guide to New Music. New York: Harper
Strickland, Edward. 1991. American Composers: Dialogues on Contemporary Music.
Bloomington, IN: Indiana University Press.
Swarup, Rai Bahadur Bishan. 1933. Theory of Indian Music. Maithan, Agra, India: Swarup
 See Edward Strickland, American Composers: Dialogues on Contemporary Music (Bloomington, IN: Indiana University Press,1991); Patricia and Joseph Mancini, “On Just Intonation and the Spiritual Source of Music: Terry Riley,” Keyboard 12, 7 (July 1986): 52-65; and Wim Mertens, American Minimal Music, trans. J. Hautekiet (London: Kahn & Averill 1983), 35-45.
 Strickland, 121.
 Ibid., 122.
 Mancini, 65.
 The term “five-limit” as applied to tuning means that all ratios of the chromatic scale are derived from the first five harmonics of the harmonic series. A five-limit tuning allows for pure pure major and minor thirds (5:4 and 6:5) - see David Canright, “On Piano Retuning,” downloaded from the World Wide Web on November 29, 2000 at http://www.mbay.net/~anne/david/piano/index.htm. A thorough discussion of the five-limit can be found in Harry Partch, Genesis of a Music (New York: Da Capo Press, 1974), 109-119.
 Recorded evidence of Riley’s collaboration with Krishna Bhatt can be heard on the soundtrack to Alain Tanner’s film No Man’s Land (Planisphere compact disc PL 1267, 1985; out of print).
 Mancini, 59.
 Sources: Pitch and Interval: Terry Riley. The Harp of New Albion, liner notes (Celestial Harmonies CD CEL 018/19 1986). Deviation from equal temperament: David Canright, "On Piano Retuning," http://www.mbay.net/-anne/david/piano/index.htm.
 The names of pitch classes provided in Figure 3 (e.g., B# rather than C natural) are Riley’s, reflecting a chromatic (all-sharp) scale above the instrument’s tuning center of C#. In the musical transcriptions from “Magic Knot Waltz” accompanying this article, however, I have chosen to use enharmonic equivalents that more accurately reflect the scale based on the tonal center for “Magic Knot Waltz.”
The fact that The Harp of New Albion was conceived for piano—the historical embodiment of equal temperament—requires some rethinking of assumptions about music in just intonation in general. For example, the sizes of particular interval classes (e.g., the “major second”) do indeed vary in Riley’s just intonation system, whereas all identical interval classes in equal temperament are of identical size. However, it does not follow that pitch classes considered to be enharmonically equivalent in equal temperament must differ in just intonation, at least when notation for the piano is considered. For example, the dyad A-B# in The Harp of New Albion will of necessity be the same as A-C, because B# and C correspond to the same piano key. Because Riley assumes octave equivalence in his tuning structure, there will still be only twelve pitch classes to the octave available. (This makes his approach different from that of composers such as Harry Partch, whose chromelodeon was tuned to the exigencies of his forty-three-tone system - with the result that playing two different “G” keys in different octaves would yield two different pitches. Had Riley composed this piece for a medium that allows more flexibility of pitch - a string quartet, for example - his tuning system might have indeed allowed for intervals that were enharmonic in equal temperament but not equivalent in just intonation.)
I nevertheless adopt Partch’s convention of placing interval names in quotation marks, to show that Riley’s intervals are not equivalent to their traditional equal-temperament counterparts.
 Terry Riley, The Harp of New Albion, liner notes. Celestial Harmonies compact disc CEL 018/19, 1986.
 William Duckworth, Talking Music: Conversations with John Cage, Philip Glass, Laurie Anderson, and Five Generations of American Experimental Composers (New York: Schirmer Books, 1995), 283.
 Although the written interval B#-G is technically a “diminished sixth,” Riley is clearly using the “B#” in this movement not as a leading tone to C# but as its own tonal center. This results in the aural impression that once is actually hearing a “fifth” between C (enharmonically equivalent with B# on the piano keyboard - see note 8) and G.
 Ibid., 279.
 Ibid., 284 [italics added].
 Mertens, 44.
 Rai Bahadur Bishan Swarup, Theory of Indian Music (Maithan, Agra, India: Swarup Brothers, 1933), 19. The “European scale” cited by Swarup is nevertheless in just intonation rather than equal temperament.
 Charles Capwell, personal e-mail communication, 12 December 2000. A specialist in the music of India, Dr. Capwell is Associate Professor of musicology at the University of Illinois at Urbana-Champaign. I am grateful to him for pointing out this observation.
 I am of course excluding “wrong notes” resulting from finger slips in performance—of which there are remarkably few. Riley’s Munich performance of “Magic Knot Waltz” (released on Celestial Harmonies CD CEL 018/019, 1986) does contain one left-hand E-natural3 (replacing the E-flat) at [3:04] and one left-hand E-natural5 at [3:41]. These seem to be more deliberately placed, as they both occur immediately before cadenza-like flourishes that interrupt the ostinato sections.
 See Ghosh 1968, 43-48 for a full but succinct discussion of North Indian tuning hierarchy, including srutis, swaras and thats.
 David Courtney, “New Approaches to Tabla Instruction.” Percussive Notes 30, 4 (1992): 28.
 Riley, The Harp of New Albion, liner notes.
 David Courtney, “Jhala.” Downloaded from the World Wide Web on January 19, 2001 at http://chandrakantha.com/articles/indian_music/jhala.html.
 Peter Manuel, Popular Musics of the Non-Western World (New York: Oxford University Press, 1988), 184.
Like Riley, Glass found his inspiration in Indian classical music, specifically that of Ravi Shankar. See for example his discussion in Glass 1987: 16-18.
 In John Schaefer’s book New Sounds, Riley is quoted as saying that the roots of his experiments with melodic improvisation lie in North African and Middle Eastern music. “When I was living in Spain¼I used to listen to the radio a lot, and I’d get the stations from Tangiers and the cities right across the Mediterranean. That’s the first music that really sank in. Those maqams, the Middle Eastern scales, have always attracted me. Even though I’m a student of Indian classical music and that’s my main love as far as ethnic music goes, when I write my own music it tends to have a Middle Eastern flavor. There’s always been a kind of dream world for me there.” (John Schaefer, New Sounds: A Listener’s Guide to New Music [New York: Harper and Row, 1987], 75.)
 Mancini, 57.
 Strickland, 123.
 Duckworth, 272.
 Ibid., 279.
 Schaefer, 76.
 Mancini, 53.
 Ibid., 65.
 Mertens, 44.