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Composition II – Pitch Sets

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The study of set theory is one that is deep and often covered in music theory courses devoted to the analysis of 20th century music. A “nutshell” description of it is here. The typical approach to pitch set analysis involves considering pitches numerically (0=C, 1=C#, etc.), understanding the different pitch-class sets that are possible (there are 208!) considering three-note collections (trichords), four-note (tetrachords), five-note (pentachords), and so forth, with each set representing both the notes in “proper” order and under both ansposition and inversion, as is the common practice.

Why use set theory as a composer?

One point of set theory is to create a “language” for understanding music that is not based on triads as a fundamental feature of melody and harmony. Indeed, pitch sets are used like notes of a chord — they can be a framework for melodies, stacked as chords, embellished, and so forth. The order of pitches isn’t what is important (as it is with twelve-tone music).

For instance, music could be composed that features the pitch-class set 0,1,4 (it’s called “3-3,” since it is listed third on the standard table of pitch-class sets), transpositions of that set, and inversions of it:

3-3 is, in fact, a main feature of the second movement of Bartok’s String Quartet No. 2. We can most easily see this by looking at the intervals between notes of the piece; when motives are framed by a major third that is divided by a half-step on one side or the other, it is “3-3,” like in this music at rehearsal 1:

This passage also has a couple of extra notes – the G# in bar 19 helps expand 3-3 into more of a scale, as does the C# in bar 23. The C# in bar 23 also helps make another version of 3-3 (0,1,4 transposed up by a half step).

Another way that Bartók used this trichord is as main notes of the melody in rehearsal 4:

At rehearsal 7, this trichord is found alongside 3-4 (0,1,5 – imagine a perfect fourth with a half step-in the middle, like C-C#-F or C-E-F).

At bar 166, there is a very strong chord that combines 3-3 and 3-4 into a single unit.

As the music continues from this point, the cello performs a melody that also blends 3-3 and 3-4.

…and forth it continues.

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Blog Posts Composition & Orchestration Composition II supplements Rhythm Teaching Materials

Composition II – Rhythm and Meter

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Plenty of contemporary art music has ordinary rhythm. There are other approaches, however, that can be used to provide different sorts of rhythmic interest.

Music that seems to be free or only loosely coordinated

One example of this music is the first movement of Olivier Messaien’s Quartet for the End of Time, entitled Liturgy of Crystal. Listen to this music while following the score, noting how each instrument seems to move at its own pace. What aspects coordinate this music? Do instruments seem to align with the meter? Or to each other? Where do you perceive downbeats?

Some music provides extreme freedom in its interpretation, giving little information about how it is to be rhythmically interpreted. For instance, the beginning of Aleksandra Vrebalov’s My Desert, My Rose has performers coordinate by reading from the score (it’s here) until rehearsal letter T [here], at which point the music (more or less) settles into a consistent 7/8 meter.

Music with a propulsive beat

Much music of the 20th and 21st century has a strongly propulsive beat, not unlike some classical music of the past and certainly similar to much rock and jazz. Complex rhythmic patterns can be especially clear when they are accompanied by a persistent groove. The second movement of Béla Bartók’s String Quartet No. 2 is a great example of how small bars (in this case, 2/4) can be combined to make phrases of different lengths with a constant flow of 8th notes providing a rhythmic underpinning.

Music in unbalanced meters

While traditional music tends to be either in simple or compound time, music can also combine aspects of each (for instance in 5/8 meter, in which each bar has one simple beat and one compound, not necessarily in that order) or can shift from one to the other. Returning to Messaien’s Quartet for the End of Time, this approach can be seen in the sixth movement, called Dance of the Furies, for the Seven Trumpets. A tour-de-force entirely in octaves and unisons, notice how bars combine aspects of simple and compound meters (and more complicated rhythms, like beats with five 16th notes), and how Messaien did not even notate meter.

Music can also sound fresh and unpredictable in a consistent meter with an uncommon time signature (like Steve Reich’s Eight Lines in 5/4), or free and flexible in meters that change constantly (like Reich’s Proverb), or this part of Igor Stravinsky’s The Rite of Spring. Another terrific example is #140 (“Free Variations”) from Bartók’s Mikrokosmos.

Tempo fluctuation

Some composers, most notably Elliott Carter (1908-2012), explored the use of metric modulation to move from one tempo to another. Here’s a charming video of a drummer demonstrating the concept. Carter’s music is particularly complicated in this regard because the metric modulations are combined with an existing complexity of rhythm. For instance, in his solo guitar work Shard, a metric modulation in bar 5 uses one sixteenth note of the previous tempo as a single triplet unit in the next. In bar 17, the duration of five 16th notes becomes used as the new quarter note pulse. This continues…

One fantastic online resource is the Metric Modulation Calculator. It helps calculate tempi based on meter changes.

Additive rhythm is a name given to the technique of using flexible lengths for musical fragments, constantly changing the lengths of a particular gesture. This can be reflected in changes of time signatures, or with a consistent time signature (as in Jacob her Veldhuis’ Caterpillar – here is one page of the score).

Polyrhythms

The use of two simultaneous rhythmic “paces” (“polyrhythm“) can provide great rhythmic interest. This can happen through the use of tuplets, like in this segment of the second movement of Charles Ives’ Trio (layering triplets in the cello and the pianists’ right hand against sextuplets and septuplets in the other parts). It can also occur in the layering of different-length patterns that share a beat-division, such as Adam Silverman’s Pounding Fists; notice in this score, starting in bar 5, how a raucous texture is created with repeated ostinati of different lengths (the triangle pattern repeats every two beat, the bell tree every four; the glockenspiel ostinato lasts five eighth-notes; the two vibraphones loop every three eighth-notes; the two marimbas every five sixteenth-notes).

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Composition II – Minimalism

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What is minimalist music?

Reduced from https://courses.lumenlearning.com/musicapp-medieval-modern/chapter/minimalist-music/

Minimal music is an aesthetic, a style, or a technique of music that originated in the New York Downtown scene of the 1960s and represents a new approach to the activity of listening to music by focusing on the internal processes of the music, which lack goals or motion toward those goals. Prominent features of the technique include consonant harmony, steady pulse (if not immobile drones), stasis or gradual transformation, and often reiteration of musical phrases or smaller units such as figures, motifs, and cells. It may include features such as additive process and phase shifting which leads to what has been termed phase music. Minimal compositions that rely heavily on process techniques that follow strict rules are usually described using the term process music.

Composer and music critic Tom Johnson wrote “The idea of minimalism is much larger than many people realize. It includes, by definition, any music that works with limited or minimal materials: pieces that use only a few notes, pieces that use only a few words of text, or pieces written for very limited instruments, such as antique cymbals, bicycle wheels, or whiskey glasses. It includes pieces that sustain one basic electronic rumble for a long time. It includes pieces made exclusively from recordings of rivers and streams. It includes pieces that move in endless circles. It includes pieces that set up an unmoving wall of saxophone sound. It includes pieces that take a very long time to move gradually from one kind of music to another kind. It includes pieces that permit all possible pitches, as long as they fall between C and D. It includes pieces that slow the tempo down to two or three notes per minute.”

What techniques are associated with minimalist music?

  • Some minimalist music features the sound of a drone (or pulsing drone), such as La Monte Young’s “The Well-Tuned Piano.” Music may gradually “evolve” by repeating notes or fragments for spans of time, as in Julius Eastman’s “Gay Guerrilla” [score | audio].”
  • Minimalist music tends to have a rhythmic characteristic of stillness or slowness (such as in David Lang’s “The Passing Measures“) or else it is motoric and incessant (like in John Adams’ Shaker Loops).
  • Ostinato and repetition of a melodic cell (a few pitches, either steady or rhythmic) is often found in place of a melody, such as in Philip Glass’ Two Pages.
    • Terry Riley’s seminal piece In C (the score is here) features a drone note “C” and an indefinite number of players improvising by moving from one small cell to another at an intuitive pace.
    • These cells often “evolve” through the use of a conceptual technique, like additive or subtractive process – this may be at the beginning or end of a melody (as in Rzewski’s Coming Together – here’s an analysis that explains it) or by filling in gaps of a pattern until it is complete (as in Reich’s Drumming – notice [in this score fragment] how each repeated bar of music adds one note at a time until the pattern is full).
    • Sometimes one or more bars are repeated for definite or indefinite numbers of times before progressing to a new cell (like in Two Pages, or in Steve Reich’s Nagoya Marimbas.
    • Permutation is the changing of a pattern through methodical changes, such as in David Lang’s Cheating, Lying, Stealing, which achieves rhythmic variety by repeating a pattern while inserting tiny pauses between each note of the pattern [read about it here].
    • Classic counterpoint techniques are also sometimes used. In Arvo Pärt’s Fratres, an additive process is used to create an eight-chord melody (it builds from the outside-in, performing chords 1-2–7-8, then 1-2-3-6-7-8, then 1-2-3-4-5-6-7-8). This pattern is then repeated in inversion. Those two patterns are then repeated in transposition again and again until the piece is complete.
  • Layering is an important compositional element in minimalism, both in how layers may be added or subtracted abruptly (as in Michael Gordon’s Trance) or through fading in and out (as in Reich’s Music for 18 Musicians).
    • One special technique of minimalism is the use of extremely close canon, such as in Reich’s Clapping Music (notice how the same pattern is performed by both groups of clappers, but one layer is shifted by an eighth note bit-by-bit) or in Louis Andriessen’s Hout, in which four players perform the same quick line, one behind the other at a distance of one note.
    • When two identical lines move in a canon that has no discrete steps (one layer “floats” away from the other, either by being at a slightly different speed, or by the temporary speeding or slowing of one part, a technique called phasing occurs, as in Steve Reich’s Piano Phase. 
  • Harmony can be chromatic, but it is usually triadic or pan-diatonic (using all the notes of a key as the basic harmony), such as in Arvo Pärt’s Cantus in Memoriam Benjamin Britten. This piece also uses the medieval process of mensuration canon, in which each melodic layer moves at half the rate of the line above it. A composer’s use of harmony is often one of the characteristics that makes them distinctive, such as Glass’ use of triads or Reich’s use of extended jazz chords.
  • Dynamics in minimalist music are typically steady (loud throughout, or quiet throughout), or in a single trend (quiet to loud over the entire span of time, or the opposite).
  • Instrumentation of minimalist music varies from one composition to another, but certain trends have emerged. Early minimalist music tended to be written for their composer-led ensembles, so Steve Reich’s works often were written for percussion and pianos; Philip Glass’ ensemble blended keyboards, voices and saxophones. Ambient music like Brian Eno’s Music for Airportsis often associated with electronic media. One common feature in much minimalist music is homogeneity of timbre, resulting in single-family ensembles like Reich’s New York Counterpoint (for multi-tracked clarinets) or Different Trains (for four string quartets with pre-recorded sounds)

The History of Minimalism

Early/conceptual minimalism
A simple process is allowed to progress with little intervention from the composer. There is often little more than one or two musical layers in combination, or a single layer in (often haphazard) counterpoint with a copy of itself.

Progressive minimalism
Characteristics of early-period minimalism are employed in combination with more intuitive approaches to composition or mutiple techniques in combination. Static or expansive textures and vivid colors are favored over process-based conceptualization. 

More listening

  • Steve Reich “Music for 18 Musicians” (1974-76) [video]
  • Philip Glass “Einstein on the Beach” (1976) [YouTube audio playlist]
  • John Adams “Shaker Loops” (1983) [video]
  • Pauline Oliveros “A Woman Sees How The World Goes With No Eyes” (1989) [YouTube audio]
  • Michael Torke “Four Proverbs” (1993) [YouTube audio playlist]
  • David Lang “Cheating, Lying, Stealing (1993) [video] [analysis]
  • Michael Gordon “Trance” (1995) [YouTube audio playlist]
  • Anna Thorvaldsdottir “Aeriality” (2011) [video]
  • Nico Muhly “Drones” (2012) [audio]

How Does One Create a Minimalist Composition?

  • Choose amazing timbres
    • Great instruments
    • Effects
    • Long-tones
    • Flurried sounds
    • Interlocking patterns
    • Amplification & effects
    • Similar timbres (8 guitars, 6 voices, etc.) or contrasting (rock band, orchestra, etc.)
    • Spoken word (choral spoken word?)
    • Overwhelming sonic combinations, or spare sounds?
    • Fading in/out (of chords, tones, layers, etc.)
  • Choose a stylistic approach or process(es)
    • Audible process
      • Additive/subtractive linear process
        • Notes/chords/sections/phrases before/between/after melody notes
      • Additive/subtractive textural process
      • Additive/subtractive harmonic process (notes expanding registrally from a small core, etc.)
      • Phasing; canon (unison? at expanding/contracting intervals?)
      • Key “rotation” through repetitions that change one note at a time
    • Steady-state
      • Polyrhythmic layers
      • Other types of unpredictable variety
  • Choose a rhythmic approach
    • Are lengths of phrases consistent, or do they vary with the process?
    • Are repetitive patterns symmetrical, or in meters such as 5/4 or 7/8? Or expanding and contracting?
    • Is the music pulse-oriented or free?
  • Choose a notational approach that suits the concept or procedure
    • fully/strictly notated
    • strict notation with loose repetitions
    • musical “cells” with repetitions or improvisations
    • set of instructions
  • Choose a duration and adapt musical process to accommodate it
    • Infinite music (cyclical)
    • Finite music
      • Indivisible: completes one cycle of a process
      • Sectional: different processes occur successively, or same process is repeated with new material
      • Rounded: opening material returns after digression