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About the Piece

On the Edges goes by in six sections contrasting in pulse and energy. The first section is a toccata with two-handed arpeggios gradually turning into chords that get thicker and thicker before returning to the opening arpeggio motive. A dialogue follows where slow-moving chords are in counterpoint with quicker, arpeggiated figures derived from the toccata’s opening figure. Then there’s a quiet lyrical reflection – again based on the opening motive – eliding into a cadenza with arpeggiated figures that increase in range and intensity before dissolving into a trill.

The next section turns the toccata’s opening motive into the subject of a four-part invention, the first in a series of three linked by thematic rotation: the counter-subject of one invention becomes the subject for the next. This cycle completes as the subject of the first invention becomes the counter-subject of the last, and ends as the three subjects are combined, segueing into an extended return of the opening toccata.

About the Musical Materials

On the Edges uses the symmetrical, or octatonic, scale (e.g., C#-D#-E F#-G-A-Bb-C). If played as a scale, it sounds a lot like the familiar minor scale – you can build it by taking the first four notes of the minor scale and then transposing them by a tritone.

On the Edges treats these notes more as a collection than a scale, exploiting the relationships between complementary groups of notes, that is, groups that taken together contain all the notes in the scale without repetitions. This is similar to much twelve-tone music, which is based on relationships among complementary parts of the chromatic scale.

In the symmetrical scale, the most symmetrical complementary relationships occur between complementary four-note collections (tetrachords) that are themselves symmetrical. There are three of these symmetrical tetrachords, including the “diminished seventh chord” familiar from tonal music, and they form the basis of the melody/harmony of On the Edges. The most angular of these (C-C#-F#-G) is expressed as a motive in the first measures of the piece and is used thematically throughout.

The most symmetrical tetrachord, the “diminished seventh,” is the generator of the symmetrical scale since it can combine with either of the other two transpositions of itself to form the scale. This means that there are three different collections of notes that form a symmetrical scale, which taken together exhaust the chromatic scale with each note occurring exactly twice. The larger scale structure of On the Edges exploits this: each of the first three sections uses a different collection, while the last three each use all three, and the switching from collection to collection grows closer to the foreground in each successive section.

About the Rhythm

While On the Edges’ pitch structure is based on symmetries, the rhythmic structure is asymmetrical. With each new section comes a metrical change; yet all meters in On the Edges use seven subdivisions to the measure, grouped 3+4 and 4+3. When the
unit note value is very short the effect is like a measured rubato; when it’s slightly longer what you hear is each measure divided into two lopsided beats (somewhat like dancing with a broken leg); and when it’s long the groupings sound like measured changes in tempo.

Listen to the entire piece here.

"...an almost ragtime sensibility to Bachian themes and suggested an otherworldly common ground between those distant influences."     --Allan Kozinn, The New York Times


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