MAPPING MUSIC

Tag: rock

  • Mapping Music 10. COUNTERPOINT

    Two lines woven into a shared time stream — counterpoint — can be relatively more or less independent. How similar or diverse are their rhythmic patterns (congruent or diverse)? How often do their note-initiating time points “line up” (synchronous or independent)?

    In an example of congruent, matching rhythmic material, the upper line’s rhythm is echoed in the trailing lower line in the first five bars below. But the lines are rhythmically independent, sharing only one time point, the downbeat of bar 4. This echo process is known as . . .

    CANON — leading line is echoed after some delay by one or more answering lines of identical rhythmic values and melodic shape (possibly transposed)

    For more on canons, go to BOOK OF CANONS, 14 short 3-part canonic studies.

    example of two-voice counterpoint

    Bars 6-11 show diverse rhythms (the upper line in mostly shorter durations than the lower), and not in canon but synchronized at most of their time points.

    Rhythmic alignment

    Johann Joseph Fux established a theoretical construct for pedagogical purposes in which contrapuntal lines in a 16th-century style progressed from congruent, synchronous rhythms (“First Species”) to one line twice the pace of the other (“Second Species”), and so on. Only in Fourth Species was the relationship reversed, back to matching, congruent rhythmic values but in studied alternation avoiding synchrony.

    COMPOSITE RHYTHM — stream of durations between time points marked by an attack of a note in one or more lines of the fabric

    Here is a graphic identification of the composite rhythm of each contrapuntal phrase above.

    composite rhythm

    You can see in the first example that there are 7 notes in the upper line and the same 7 rhythmic values in the lower line. But the composite rhythm shows 12 durational values, due to the non-synchrony of the lines. In the second example, the upper line has 9 notes, but the lower line’s 5 notes all align with them. The “sum” of the two lines is a composite rhythm of only 9 durational values, identical to the upper line.

    Contrapuntal intervals (in number of semitones) are identified between the staves. The time points of the composite rhythm, moments when both lines are starting a note, are contrapuntally accented and emphasize the contrapuntal intervals (boldface) formed at those points. The consistency — in this example the contrapuntally accented intervals of 7, 8, 2 (and 2+octave), and 5 (and 5+octave).

     

    CONTRAPUNTAL ACCENT — prominence of contrapuntal intervals formed by notes starting together on a time-point

    Refraction

    This term refers to the metaphor of light going through a prism or drop of water, revealing a spectrum of colors. In that sense, a musical refraction might refer to a line presented by instruments of changing sound color. (See Klangfarbenmelodie below.) But let’s apply the refraction concept to pitches in a line of consistent color.

    Refraction can also be a simple way to make two lines out of one, splitting up its notes into two lines shared by alternation or some other less strict pattern. The pitch assigned to one line can be sustained to make a companion pitch to the pitch or pitches that come next in the other line. In this way, the vertical intervals can be strategically controlled to generate a coherent contrapuntal harmonic flow.

    To demonstrate, here is the opening theme to Jupiter Rising:

    Jupiter Rising theme

    Now splitting this violin line into two violin parts:

    Jupiter theme refracted

    Identifying the contrapuntal intervals (by number of semitones) that are formed reveals a preference for contrapuntal intervals of 2, 4, and 5 semitones.

    Some might say this is not real counterpoint, but the total rhythmic independence of the lines argues for that distinction. Mandelbrot, pioneer of fractal mathematics, described fractional spatial dimensions. Maybe we can call our refraction one-and-a-half voice counterpoint.

    Canon

    Repeating the definition of this ancient form of Rumpelstiltskin magic, spinning complex counterpoint out of a single melodic line:

    CANON — leading line is echoed after some delay by one or more answering lines of identical rhythmic values and melodic shape (possibly transposed)

    For a collection of 21st-century examples, 14 studies in 3-voice canon, go to BOOK OF CANONS.

    Now let’s look closely at a more famous canon, in four parts scored for seven different instruments. Here is a contrapuntal example of canonic threads expressed through changing instrumental colors, the opening of the first movement of Webern’s Symphonie Op. 21:

    Webern Symphony opening

    Instead of showing each instrument’s part, I have rearranged the score so that each staff line strings together the successive pitches of a 12-tone row:

    • On the top staff, A F# G Ab played by horn; E F B Bb played by clarinet; then D by cello, continuing past this excerpt to complete the 12-tone row with C# C Eb.
    • The second staff answers in canon one bar later, starting on F plucked by harp and proceeding with a mirror inversion of the lead-line row: F Ab G F# Bb A Eb E C C# D B.
    • The third staff is also an inversion of the row starting on A.
    • The fourth staff, entering last, is a transposition of the original lead-line row starting on C#.

    Repetition

    Any musical element can be repeated — a note, an arpeggio, a measure, a phrase, a whole section of a form, as in the baroque rounded-binary model or the exposition of a classical sonata-allegro form. When a melodic motive or molecule is continuously repeated many times, it is called an ostinato, usually forming a background to some changing line or evolving stream of events. We can analyze two critical factors:

    CYCLE — duration length of a repeating pattern

     PHASE — time point at the start of a cyclic repetition

    Some 20th-century composers, especially Americans, started to bring background patterns or structures into the foreground, as primary objects rather than accompaniments. The incessant repetition of an ostinato, often a chord arpeggio, became the basis for simple structures. With a relentless pulse at its rhythmic core, most ostinato music generates simple highly congruent rhythmic lines in simple or no counterpoint.

    Classic works by composer Philip Glass, such as the ‘70s pieces Music in Twelve Parts, are continual repetition of chord arpeggios, with the chord changing gradually and subtly over many repetitions. This has two effects: making a very slow harmonic change rhythm and time flow under an animated surface; and creating a broad time form that is monolithic and metamorphic, rather than a more traditional multi-section recurrence form.

    John Adams brought this relentlessly repetitive approach to appealing prominence in symphonic music. His Fearful Symmetries (1988) has a pulsing persistence reminiscent of the great Stravinsky ballets, such as Le Sacre du Printemps (1913).

    John Adams – Fearful Symmetries (1988)

    Steve Reich continued this energetic vein of repetitive rhythmic construction into the 21st century with works such as Double Sextet (2008).

    Steve Reich – Double Sextet (2008)

    Despite its sometimes lush fabric of harmony and animated rhythmic activity, persistent-repetition music has unfortunately been labeled “Minimalist,” often having no melody, no sense of harmonic progression or tonal modulation, no themes, no sectional cadences and divisions, and no discernable large-scale recurrence form. (A music more truly described as Minimalist can be found in the more radical works of John Cage, with sparse sounds — or no prescribed sounds at all — in a time-space of mostly “silence.”)

    Phasing

    Back to ostinato — what about more than one ostinato layered into a more complex texture? Even if the ostinato patterns are of the same length, it is possible for their repetitions at different times to not synchronize but overlap. We would say their repetitions are out of phase.

    Using Webern’s canon technique to place identical lines out of phase:

    Milky Way score excerpt

    The Milky Way is our own barred spiral galaxy. The musical fabric is adapted closely from Buckingham Fountain, the third movement of my Chicago Sketches for flute choir.

    There is also the potential for each ostinato pattern to have its own cycle length of repetition. And if the lines repeat different cycle lengths, their phase, the start of another repetition, cannot always align in synchrony. This can be described as multi-cycle/multi-phase ostinato music, pioneered among others by American composer Terry Riley.

    Inspired by tape loops continuously replaying recorded sequences of sounds, in 1968 Riley produced a massive (45- to 90-minute length) multi-phase ostinato work, In C. Becoming iconic, it has been recorded commercially more than 36 times and performed by countless new music ensembles, finding its improvisatory freedom and large flexible instrumentation attractive. (A 2006 performance at the Walt Disney Concert Hall featured 124 musicians.) It consists of 53 ordered patterns of specified, notated rhythm and pitch, to be continually repeated against a steady eight-note pulse. The patterns range in length from only 4 eighth-notes to extended phrases sprawling across a part’s entire manuscript line (without bar lines). Thus the variety of repetition cycle lengths is enormous. And because each musician chooses when to start and how many times to repeat each pattern, multiple phases are also guaranteed.

    Rather than analyze this iconic piece, I will show and explore a piece of mine inspired by In C, originally composed in 1984. It employs the canon technique and differing-length patterns to create the constant overlapping of patterns out of phase with other lines, This makes it difficult to express all the patterns in one common meter signature. Riley’s solution, and mine, is to use no meter signature, with all lines (parts) aligning only with a constant eighth-note pulse.

    Effulgence improv score

    Before we dive into its structure, let’s listen to its beginning.

    The surface rhythmic relationship of overlapping patterns is simple, all conforming to a common eighth-note pulse, as in Riley’s In C. The differing bar lengths, however, produce different periodicities, different repetition cycles. Patterns of 2, 4, 6 or 8 eighth-notes relate to each other to establish a common quarter-note based meter, a feel of 2/4, 3/4 or 4/4 meter. But the patterns of a prime number of eighth-notes, 3, 5 or 7, oppose the sense of a quarter-note beat.

    The prime numbers mean also that the repetition cycles will rarely synchronize, creating a more complex, floating or flying fluidity of motion. Three against four is fairly simple, as with Patterns 6 and 7. Repetition of primes seven against five, as in Patterns 19 and 20, make a much more complex composite, taking some 35 eighth-note pulses to return to a synchronous starting point.

    multi-phase combinations

    To control the interaction between successive patterns that will overlap in canonic lines, each pattern’s pitch content must work with the pitches of patterns before and after it. By “work” means that the collective, cumulative constellation should be of an intervallic character, an array, that conforms with the overall harmonic character desired.

     Assuming a performance spread of three patterns, here is a sample analysis of the middle, Patterns 16 through 21, showing the three-pattern collective constellation. Each pattern intersects with common pitches of its neighbor patterns, adding pitches to the sonority that will eventually disappear.

    intersecting pitch collections

    This is the mechanics of a metamorphic harmonic process that gives multi-phase ostinato music its graceful evolving form.

    Now let’s listen to the complete composition from 1984 (revised 1994), one of my personal favorites.

    Effulgence

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    © 2026 – All Rights Reserved

    Thomas S. Clark

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    Continue reading Mapping the Music Universe:

    11. TEXTURE

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  • Mapping Music 3. CHANGE

    Harmonic rhythm is the pace at which chords change in common-practice tonal music. Often in songs or simpler instrumental music, the harmony changes periodically, like once every measure or every half-note or every beat. Even when the rate of chord change is this uniform, it often accelerates approaching a cadence at the end of a phrase or other sectional unit. Calculus suggests that there can be a change in the rate of change, a second-order differential. Beethoven offers something like this in his very late work, the String Quartet No. 16 Opus 135. Here is an excerpt from the Allegretto first movement:

    String Quartet Op. 135 Allegretto, mm. 25-48

    An analytic sketch of the harmonic-root-foundation bass line reveals that F major gives way in the first four bars to a tonicization of the dominant, C major, starting with its dominant, G major:

    The rate of chord change starts as every 4 beats, eventually quickening toward the end of the excerpt to a different chord every 8th-note — an eight-fold quickening of harmonic pace! As you listen again, notice if you feel this intensifying compression of events.

    Going deeper into the tonal groupings of these harmonies, the starting key of F Major gives way to various tonicizations of G, then C. Here is a reductive sketch showing the durations of these tonicizations.

    Op.135 excerpt harmonic reduction

    Since this section is 24 measures long, it could have been composed as three equal 8-measure periods. Instead, the middle-ground tonal rhythm is surprisingly non-periodic, an irregular durational stream consisting of 8, 10, 10, 4, 7, 1, 1, 5, and 2 quarter-notes.

    Beethoven was beyond eccentric at this late point in his career; Op. 135 was the last work he ever completed. Yet the elasticity of harmonic rhythm found in it is a hallmark of his earlier styles as well.

    Beyond meter

    Arising in the middle of the 20th century, highly complex, elastic rhythms began to be composed, in which every durational value was different and notes or events do not group into periodic measures or phrases. An example composed in 1971 is an elegy that makes a conscious effort to avoid articulating periodic beats or falling into groups of notes of periodic duration.

    Meter signatures are present only for notational purposes and change four times in the passage. Only four of the 22 notes fall “on the beat” and only three of those articulate a downbeat.

    Since the note values are so slightly or drastically different, we can measure each duration from the start of a note to the start of the next note as a multiple of fine “time particles” each one-twelfth of a quarter-note. The durational stream is blatantly non-periodic: 30, 12, 44, 8, 14, 6, 21, 9, 31, etc. The rhythmic range of the first four measures is higher than 7, rhythmic variety at 9. The next three measures have a higher rhythmic range of more than 11 and rhythmic variety of 8 (due to the 9-particle dotted 8th-notes that occur five times).

    Beyond a mathematical comparison, a time graph mapping the durations reveals to the eye no periodicity, no perceived meter or regular conforming rhythmic pattern.

    Elegy rhythm graphed

    The rhythm floats above or beyond meter or pulse in a dreamlike, elastic stream. [From Night Songs (1971)]

    Free time

    Defeating the notated meter in this way, by avoiding beats and periodic, conforming note values, was developed to free a stream of events from periodic pulse, thus freeing the listener’s sense of time flow – free time itself. The logical next step, developed concurrently in mid-20th century, was to remove meter entirely as even a notational necessity. Just like the time graphs we have been using to visualize timing of events, a horizontal, proportional scale (such as one half-inch equals one second of time) enables the horizontal placement and spacing of notes on a staff to suggest visually subtly different durations, both of sustained sounds and the time spacing from one event to another.

    Spatial notation

    Spatial notation — non-metric representation of time by proportional horizontal spacing of notes

    After “Elegy,” the first movement of the unaccompanied trombone piece Night Songs, the third movement, “Somniloquy,” was originally notated in this manner – what came to be known as “spatial time.”

    Somniloquy notated spatially

    In his one partially preserved manuscript, On Time, the Greek philosopher Heraclitus wrote about “the unity of opposites” and “flux,” meaning change. “It is not possible to step into the same river twice.” He also imagined that the cosmos is shaped as an enormous vortex of fire.

    That image ignited musical sparks in my imagination for the third movement of my early solo piano work, Geography of the Chronosphere (1975), subtitled “Heraclitean Vortex.”

    The score, in non-metric spatial notation, articulates explosive bursts of notes separated by irregular spans of reverberation.

    Heraclitean Vortex excerpt

    An analytic graph of loudness shows these bursts occurring at unpredictable time intervals, in moments (not so much phrases) of varying length, from 3 to 11 seconds.

    time graph of 11 moments in Heraclitean Vortex excerpt

    Prime time

    Meter, as a periodic grouping of beats, almost always involves groups of two, three, or multiples of these factors. We call them duple meters if the groups are multiples of two, triple meter if multiples of three. Likewise, subdivisions of beats are usually subdivided into twos, threes, or multiples. Sixteenth-notes divide by two to the fourth power.

    A prime number is defined as having no integer (whole number) factors other than one and itself. In metric structure, prime numbers, with no sub-grouping factors of two or three, are more complex – 5 8 or 7 4 time for example. A musical stream that avoids metric regularity can be built with the interaction of prime number series. When repeated periodic streams of note values equivalent to 5, 7, 11 or 13 smaller time values (such as eighth-notes or sixteenth-notes) interact in time, layers of rhythm will seldom strike notes together to make a contrapuntal accent that feels like a downbeat.

    Here is a map illustrating this potential for non-metric independence:

    repeating prime numbers interact

    The bottom row of numbers shows rhythmic values of the composite rhythm, time points marked by an attack of a sound in one strand. If the streams start together as shown, they don’t all come together again until after 5,005 time-units. If each time-unit were a sixteenth-note duration, that would be after 312 four-four measures!

    This is the hidden rhythmic scheme for Night Sky, layers of pitched sounds that don’t synchronize into any meter or composite periodicity. Though not regular and certainly not metric with a pulse, time points are not at all random. Listening to it while not looking at the score’s notational details, pay attention to the way in which the sounds mark points in the flow of time – as stars mark light points in the night sky.

    Night Sky score

    A direct photographic rendering of the middle system of the score illustrates the non-metric, asynchronous timing of note events in a broad texture of sounds. 

    Night Sky score abstracted

    Do stars make spatial patterns? Of course, that’s what our fanciful constellation names are all about. But are those patterns regular, metric, periodic, symmetrical? No – that is part of their magic, a magic that can be metaphorically translated into floating musical time. 

    Beyond Time

    From the classical tradition of Beethoven’s accelerating harmonic rhythm, we jump finally to the very modern stretching of time itself. Einstein explained gravity as the stretching of “Space/Time.” From composers such as Cage and Feldman in the ‘50s, we experience isolated events, moments of sound separated by extended pause. No pulse drives the clockwork of time; it stretches immeasurably into contemplation. Listen.

    Lei Liang, My Windows (2007)

    © 2026 – All Rights Reserved

    Thomas S. Clark

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    Continue reading Mapping the Music Universe:

    4. TUNING

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    TClarkArtMusic.com