MAPPING MUSIC

Category: music structure

  • 9. Mapping

    Leelanau, 1983 —

    My last summer working at what was then called the National Music Camp in Interlochen, Michigan was 1983. We spent as much time off as possible on the nearby shore of Lake Michigan. Three spots on the western edge of the Leelanau peninsula were favorite magical places. Otter Creek played out into a sandy delta at the beach, perfect for a picnic. Good Harbor Bay was an excellent shore for finding gray Petoskey stones, revealing fascinating hexagonal-shaped fossils when wet. Farther north, the Great Sleeping Bear Sand Dunes rise majestically hundreds of feet above the water’s edge.

    Béla Viktor János Bartók’s monumental 1937 work, Music for Strings, Percussion and Celeste, begins with a mysterious, meandering line played by subdued violas. It sounds to me like walking at the water’s curving edge on a fog-shrouded beach. The line becomes the subject of a gigantic fugue, building to a powerful climax. In my imagination, we reach the sheer cliff of a massive bluff at the end of a Lake Michigan bay.

    Music for Strings, Percussion and Celeste

    I. Andante Tranquilo

    Chicago Symphony

    LISTEN > YouTube

    Shores

    Of course, Bartók never saw Lake Michigan. But shorelines are a fascinating kind of fractal patterns in nature.

    In 1980, Larry Austin received a commission from the Canadian Broadcasting System and KPFA for an experimental radiophonic work. For the premiere broadcast, the performers were in three different Canadian cities, synchronized by electronic signals! The mind-boggling result was a piece consisting of

    “a massively contrapuntal texture, with many instruments playing continuous, independent lines, all in different, independent tempos. The contours of each contrapuntal part were determined using maps of Canadian coastlines.”

    [Clark — Larry Austin: Life and Works of an Experimental composer. Borik Press, 2012, p. 40]

    Canadian Coastlines

    I.C.M.C. 1981, Denton Texas

    LISTEN › YouTube

    Glacially-etched shorelines also inspired sonic imagery for a series of my pieces culminating in PENINSULA. Mappings of the natural contours of the Leelanau Peninsula provided richly varied patterns as basic coordinate numbers for sculpting sound patterns. The piano explores some of the endless possibilities for articulating a spectrum of sonorities. A surrounding environment of synthetic sounds was made by digitally analyzing timbral qualities of acoustic instruments, mostly with percussive articulations (metaphorically the rocky shore). The timbres were modified and resynthesized into a pointillistic sound texture. The density of the sound events rises and falls in waves according to changing values derived from the basic mappings. Larger confluences of waves are located in time by map points of special significance on the graph.

    The coexistence of piano sonorities and synthetic sounds is a metaphorical meeting of seascape and landscape, both animated in time.

    PENINSULA

    Clark 1984 (TC-50) Borik Press

    Clifton Matthews, piano, Winston-Salem NC, Feb. 2007

    There were many other groundbreaking pieces by my late friend and collaborator, Larry Austin. The first, Improvisations for Orchestra and Jazz Soloists, brought him to national prominence in 1964 with highly publicized broadcast performances by Bernstein and the New York Philharmonic.

    As Austin moved into computer music, he began exploring compositional algorithms using mathematical models such as fractals.

    Some of Charles Ives’ sketches for his monumental, never completed Universe Symphony were tracings of the outlines of rock formations. Austin studied deeply this Ives work starting in 1974 and eventually completed a version of Universe Symphony for expanded orchestras in 1993. In Austin’s own work beginning in 1976, mapping contours of mountain ridges and star constellations yielded musical patterns for First Fantasy on Ives’ Universe Symphony, Maroon Bells, and *Stars.

    Constellations

    Always interested in astronomy, I tried plotting star constellations on two-dimensional matrix graphs. The coordinates of each star in a constellation could be interpreted as time-point and pitch information, resulting in a complex arpeggiated group of notes. More intriguing was the capability to rotate the map, resulting in many possible variants that stretch or compress the rhythm and chord structure.

    Cygnus
    Cygnus rotated 90º
    Orion
    Orion rotated 90º

    The first compositional product of this design work, LIGHTFORMS 1 – Constellations (TC-65), scored for piano, was published by Borik Press in 1992. Naming these patterns, pitch-time chord arpeggios, as constellations became a breakthrough concept

    In my book, Mapping the Music Universe, I cite a remarkable pioneer of cartography. “William Smith, a rural surveyor, in 1799 drew a colorful map of the subterranean rock strata of his county in English coal country, launching the modern science of geology.”  The map was extraordinary not only as a scientific breakthrough, but also visually by his hand coloring each huge copy.

    As digital synthesizers came along, sound making with computers offered more calculated control of the timbral (tone color) spectrum. My astronomical metaphor continued with a 1993 piece, using the then state-of-the-art Synclavier II digital synthesizer to “color” the constellation patterns of LIGHTFORMS 1. Reflecting the varied colors of stars, I built color families of sound, distinguishing unique frequency-modulation ratios for each group.

    LIGHTFORMS 2: StarSpectra

    Clark 1993 (TC-68)

    In 1887, French astronomer Amédée Mouchez launched an ambitious international star-mapping project (Carte du Ciel) at the Paris Observatory. It was never finished, until now the challenge has been taken up by the new Vera C. Rubin Observatory (formerly the Large Synoptic Survey Telescope) in Chile. It is conducting the Legacy Survey of Space and Time, repeated astronomical surveys of the entire southern sky.

    From wandering forest paths to trekking scenic shorelines, my life has always been full of ambient exploration. Mapping has become my grand metaphor for exploring musical territory, culminating in the book, Mapping the Music Universe. It begins:

    “The heavenly motions are nothing
    but a continuous song for several voices,
    perceived not by the ear but by the intellect,
    a figured music that sets landmarks
    in the immeasurable flow of time.”

    — Galileo Galilei

    “When we gaze at stars and planets, they appear as stationary points of light, fixed in place in what seems a random pattern across the entire night sky visible to our hemisphere. Time stands still.

    “Throughout human time, humans have imagined that stars make picture patterns we name as constellations: fish, warriors, goddesses, animals. Only the persistent observers, such as astronomers, identify their nightly march across the sky, rising in the east and disappearing below the western horizon.”

    In Mapping the Music Universe, a studied journey through musical time, pitch, and structure, many composed examples took on characters of named constellations, galaxies, and galaxy clusters. They coalesced into 12 etudes, collected here as “a continuous song.”

    Mapping the Cosmos

    Clark 2021 (TC-114)

    Listen, imagining a 24-hour 360º rotation of our earthbound telescope, viewing the entire cosmos in 24 minutes.

    _____________

  • A Small Sonata

    A sonata is typically a multi-movement piece for solo piano or for an instrument with piano. A shorter form with just three connected sections, the middle slower and quieter, can be called a sonatina. Here is an inside look at how one was composed, step by step. Like the MapLabs in Mapping the Music Universe, this guided tour is in the form of a recipe you can follow to write your own sonata.

    Choose a model

    I started formal composition study in 1968, first with composer Eugene Kurtz, based in Paris but filling in that semester at the University of Michigan. A proponent of modern French music, his compositional models included Debussy and Ravel. He assigned me to immerse myself in deep study of their music, in particular Ravel’s 1905 work, SONATINE.

    I met Beth, a flower lover, in Interlochen in 1975. She had been a promising flute student at Aspen, but was then embarking on a journalism career specializing in horticultural writing.

    The Ravel study came back to me later in my career, as I began to adopt its lush, bright harmonic language and a gentle French Impressionist quality. My SONATINE for Beth (2025) brings together the Ravel study, the flute sound, and (in my video version on YouTube) even the flower motif.

    Start with a generating idea

    The impelling theme can be a melody, a rhythmic pattern, a special kind of chord, or a non-musical image such as a painting or poem.

    Sonatine for Beth is spun entirely from a single harmonic progression, seven chords, each stacking one Perfect 5th interval above another.

    The Perfect 5ths in the two hands are separated by one or more octaves, highlighting this strong interval as a characteristic sound for the piece.

    Now some basic tools to develop and vary a generating theme.

    Transposition

    The whole five-chord progression can be transposed. The harmony is heard plainly in a middle section as ten block chords. The last five chords are a transposition of the first five, up three semitones, starting on the bass pitch Eb instead of C.

    Sequence is successive statements of a pattern transposed by a consistent interval.

    Here is another transposition of the whole ten-chord sequence:

    This harmonic material generates melodic lines and many arpeggio patterns, in successive variations of changing register, intensity, and rhythmic pace. Let’s go through the compositional unfolding of this thematic idea.

    Extract a melody and bass

    Since the starting idea is simply a chord progression, we can select individual tones from each chord for a melody. The most obvious selection is the highest pitch of each chord, even if it is not in a soprano singing range.

    At letter A the melody is given a slightly independent rhythm to help set it off from the chords, in addition to the different sound color of the flute. Also, the lower chord tones are articulated one at a time, making a bass line also rhythmically distinct, faster than the half-note chords. (The Bb in the bass line’s first bar is a passing tone, not a chord tone.)

    Add arpeggios

    An arpeggio is any pattern articulating chord tones one at a time. Usually in order lowest to highest or back down, the individual chord tones can be articulated in any order. At letter A shown above, we already saw the left hand articulate its chord tones one at a time. In the introduction, the right hand is partially broken up into arpeggios.

    In the next variation below, right-hand treble chord tones and still some bass chord tones are arpeggiated. Now all three lines (flute, right hand, left hand) have distinct rhythmic patterns, though congruent with each other in the established 4 4 meter.

    Next, the flute arpeggiates chord tones in eighth-notes, with the left hand simplified to quarter-notes of two pitches from each chord.

    Rhythmic variations

    Variation D simplifies the flute melody to just two half-note chord tones per bar.

    The two hands reunite rhythmically to place some chords after the downbeat and between flute notes.

    Counterpoint

    The original term, contrapunctus, translates “point against point” — two or more independent lines interacting in time.

    A more active rhythm for the flute line leaves time gaps that can be filled in by another line. The right hand selects chord tones to make a similarly playful rhythmic line that mostly alternates and sometimes lines up with the flute rhythm.

    The harmonic progression is still there but just hinted at by the chord tones selected for these interacting lines.

    Variation F continues this back-and-forth rhythmic interaction of the flute and piano right hand, now adding back in the left-hand chord-tone pairs with a simple rhythm for a supporting third contrapuntal line.

    Texture

    Having reached a complex level of three rhythmically interacting, independent contrapuntal lines, a nice contrast will be to simplify. Variation G reduces to a lower-register flute line and only a much simplified skeletal supporting line above it in the right hand.

    Then the texture begins to revert rhythmically to a simpler alignment of all chord tones.

    This paves the way back to a simple piano texture revealing the fundamental thematic chord progression.

    Shape a time form

    What is the plan for the whole? How will the various versions of the generating idea unfold in the larger time span of the whole piece?

    The quiet letter I variation is the apex of an arch form . . .

    • starting with simple
    • building up more rhythmic and textural complexity
    • reaching a stable plateau
    • subsiding back to what started it all.

    That sets up a recapitulation of the whole process, building up textural complexity again, first with the high two-part counterpoint:

    Then with three voices:

    Flute line “calming down”:

    Coda

    A good essay ends with a conclusion or a summary restatement of the thesis.

    Our musical coda summarizes with a last return to the beginning. The chords are back to their very low and very high registers. The flute makes a small melodic arch, ascending to the pitch B, then climbing down gently to its lowest possible pitch, C.

    Fine

    A final edit and audit are mandatory. In the case of our example, listening revealed that the beginning needed a piano introduction with some rhythmic vitality. Some sections were also reordered to improve the flow. Thus, the piece will not begin with a plain statement of the progression, and there will be a somewhat different order of other events.

    Now listen to the whole 6-minute parade of variations on a single chord progression.

  • Book of Canons

    My compositional fascination with musical canons began in the early 1970s with study (at the University of Michigan) of Ockeghem’s 15th-century polyphony, the 10 canons in Bach’s 18th-century The Musical Offering, and Webern’s 20th-century Symphonie Op.21. As a young professor in the 1980s teaching 16th-century counterpoint at what was then North Texas State University (now UNT), I used canon as a challenging contrapuntal writing assignment. In 1985, a wind ensemble piece, Parallel Horizons (Homage to Schoenberg), was my first formal composition constructed by canon. In Dark Matter, other contrapuntal writing surrounds an extended canon. Now canon pervades much of my 21st-century writing, a challenging yet stimulating and gratifying approach to texture and continuity of material.

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

    CANON
    A 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 my BOOK OF CANONS in the appendices. For pedagogical demonstration purposes, the subject of each is shown, with indications for when and at what pitch level each answer will occur.

    Read more at Mapping the Music Universe: COUNTERPOINT.

  • Mapping the Music Universe – preview

    Mapping the Music Universe is written for the literate musician, including college music students through music scholars, and anyone who is intellectually curious about how music works, especially in the 20th-21st-century modern and “post-modern” eras.

    Purpose

    The mapping project is a comprehensive catalog of patterns and processes, meant to provide simple tools for understanding modern music. This is not a theoretical treatise but a practical guide for all educated or educating musicians and the intellectually curious, requiring only basic music literacy. For me as a composer, it is also an exploration of how some of the less travelled conceptual paths lead to interesting creative possibilities. Sample composed etudes will give examples to connect the abstractions back to our musical imaginations.

    In 1989 I co-authored a composition textbook with Larry Austin, Learning to Compose: Modes, Materials, and Models of Musical Invention. We felt it was conceptually ground-breaking. My next book, ARRAYS, was an aural skills workbook covering basic modal, tonal, and “post-tonal” music of the Renaissance through the Twentieth Century. They were intended as college-level textbooks, ARRAYS basic and Learning to Compose quite advanced — maybe too advanced to succeed as a textbook, turning out to be more of a technical monograph than a basic guide. Mapping the Music Universe draws in part on the ideas and approaches of both these now out-of-print publications.

    The three parts progress logically from fundamental — time and periodicity — to pitch space, then to larger structures — texture and form. They can be read in this sequence or separately in any order. Likewise, within each part, the various topics are presented in a progressive order, but jumping in at any point is not to be discouraged. As terms are defined, they are set off to the right. Figures include musical examples, sample etude compositions, tables, and graphic illustrations of patterns and their relationships.

    Read full Introduction . . .

    Mapping the Music Universe by Thomas S. Clark . . . CONTENTS

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  • GALAXIES: Musical Structure and Relativity

    Pursuing a grand cosmic metaphor, think about the levels of structure scientists study in our physical universe. They have dived deep below the atom’s structure of electrons spinning around a nucleus of protons and electrons to discover subatomic particles like the meson and boson. On the other extreme of scale, they have gathered observations to speculate about the shape of the entire expanding universe. We understand the structure of our planet, of our solar system, and our Milky Way galaxy.

    Texture

    Painting engages techniques to create texture, rising to broad descriptions of style that actually describe structure: impressionism, cubism, pointillism. Musically, macro-structure is thought of as texture and form. Texture has been treated in broad descriptive categories: monody, homophony, polyphony, counterpoint, and more recently, sound mass, each focusing on the number of distinct parts, voices, or layers and how they interrelate. At the risk of invoking too many different metaphors, I like to think of the musical texture as a fabric.

    Other topics

    • Counterpoint
    • Rhythmic alignment
    • Canon
    • Farben
    • Symmetry
    • Pointillism
    • Repetition
    • Multi-phase ostinato
    • Sound mass
    • Hauptstimme
    • Density
    • Relativity

    To read more, request a password from tc24@txstate.edu

    Mapping the Music Universe by Thomas S. Clark . . . CONTENTS