MAPPING MUSIC

Category: music structure

  • MapLab 3. Construct a Canon

    Canon is a venerable, centuries-old compositional device, building counterpoint between a melodic line and one or more delayed and possibly transposed echoes of itself. Like a magic trick, it makes a strongly cohesive contrapuntal texture of rhythmically independent lines that are like clones of each other. Canon is more intense than a fugue, which formalizes the echo cloning technique, interspersed with free counterpoint.

    1. Study historical models

    There are many great models to study. Many 16th-century composers (notably Josquin and di Lasso) wrote canonic choral mass movements. Known more for his fugues, the great 18th-century contrapuntal master, Bach, also wrote several intriguing canons in his late work The Musical Offering. No more elegant model exists than the first movement of Anton Webern’s Symphonie Op. 21 (1928), in which four voices are spun out by successions of instruments each in turn differently coloring two to four notes of the same 12-tone line.

    Like every fine magic trick, there are several basic techniques we can learn to construct a canon. I’ll cover three, which I will call Zigzag technique, Trial-and-error technique, Rhythmic alternation, and Stretto echo.

    In 1610, Venetian composer Diruta wrote Il Transilvano analyzed Renaissance polyphonic style by codifying five species of rhythmic relationships between contrapuntal lines. Johann Joseph Fux, in his monumental 1725 pedagogy, Gradus ad Parnassum, explicated 16th-century counterpoint using these rhythmic species, of which the following are of special importance for us in this lab:

    • FIRST Species – note against note
    • FOURTH Species – lines alternating, seldom moving simultaneously
    • FIFTH Species – a mixture of rhythmic values in all lines

    2. Zigzag

    My name for it says it simply, like laying bricks one at a time but staggered to overlap.

    • Compose a few notes of the lead line. (In the example below, it is just three notes in two measures.)
    • ZIG: Establish a time delay. (in the example, one measure of two half-note beats). Duplicate the first notes (rhythm and melodic interval shape) in the following line, starting on a chosen pitch that makes the kind of vertical contrapuntal interval you desire to emphasize.
    • ZAG: Select new notes for the lead line that overlap with the ZIG notes, again making your desired vertical contrapuntal intervals. These ZAG notes need not match one-to-one the rhythms of the ZIG notes, providing the opportunity if desired to establish a Fifth-species rhythmic mixture.
    • The notes of this ZAG now ZIG into the following line, preserving the same transpositional level you established in the first ZIG.
    • Keep going as long as you wish or have stamina for. When ready to cadence, arrive at a longer note of stable pitch-sense in the lead line.

    The canonic material you just contructed can be reused transposed. Just be sure you transpose all lines together by the same transpositional interval.

    In the example below, my seven zigzag-composed measures are transposed down one semitone.

    Starting on Eb might be useful to follow the first statement of the material, which ended on D in the lead (lower) line. Or I could transpose the whole thing up 8 semitones to start on C, eliding with the middle C (bass clef) that ended the following line.

    Adding the third part enables this stair-step sequential transposition of the two-voice canon to go on and on . . .

    3. Trial and error

    Let’s try a different technique to add a canonic answer, one that is facilitated by notation software such as Finale or Sibelius. This way involves

    • copy the whole lead line, not just a head motive
    • choose a time delay or maintain one already established. Paste into the new answering voice the lead line
    • Playback the synthesized audio to test aurally for contrapuntal viability.
    • If it sounds bad, analyze the vertical intervals to discover why.
    • Make a strategic choice of a transposition of the pasted-in answer, then test it aurally.
    • Keep trying different transpositions until you find one you really like.

    For traditional diatonic tonal subjects, common transpositional choices are: unison; octave; Perfect 5th (7 semitones); Perfect 4th (5 semitones).

    In the following examples, I show in the first system a trial of an added third voice in the middle, starting on E (alto clef) transposed an octave up from the lead. For the second system, I tried adding a third voice on top, transposed up a Major 9th (14 semitones) from the lead’s start on Eb to start on F (treble clef).

    Horrible, yes? Why? What vertical contrapuntal invervals are the sour ones to your ear?

    I’ll jump to a better trial that succeeds in both places.

    In this successful trial, the first system’s added middle voice transposes from the lead’s E up 13 semitones (minor 9th), and later in the second system the added upper voice transposes up also 13 semitones from the lead’s Eb to a second answer starting on E. The minor 9th is unusual, unorthodox, chromatic, not a solution we might predict . . . but it works!

    4. Rhythmic alternation

    This will be like Fux’s Fourth Species. The lead subject is best with some long note values, leaving ample time for answering voices to present pitches when it is not moving. Transpositional choices for entering answers become fixed as predominant vertical intervals throughout the canon. In this example, the first answer chooses down 11 semitones plus an octave, and the second answer enters up 7 semitones (Perfect 5th) from the first answer, which is down a Major 10th (14 semitones) from the lead line. Thus vertical (harmonic) intervals of 11, 7, and 14 semitones end up projecting harmonies based on the 7 4 array: G up 7 to D up 4 to F#, which is up 11 from G. That sets the harmonic character of pitch constellations throughout the canon.

    5. Stretto echo

    Stretto is the term used in fugue structure for when an answer to the subject happens before the subject is finished, sometimes with a delay as short as only one or two beats. For a canon, this offers an interesting strategy for choosing pitches to shape a subject that makes its own arpeggiated harmony as it goes. The answers at unison (not transposed) are literally echoes. Even with answers octave-transposed, the effect is a multivoice arpeggiation. The fascinating wrinkle, however, is that the “chord” being arpeggiated is constantly evolving, dropping one pitch and adding one new at each note of the lead line.

    Though this setup can work with other rhythmic “species” of lines, it is particularly interesting in the note-against-note conforming rhythms of “First Species.”

    Here is how it can work, using the canon above as a straightforward example.

    This analysis sounds as a rather nice progression of arpeggiated chords and simple flute line! The important point, though, is that this progression did not come first. It was built by the canonic subject line as each new pitch was chosen to make a certain array with the previous two pitches in an ongoing, evolving flow. Magic!

    6. Spin a piece

    For our example, we’ll follow the order of the example techniques:

    • two-voice zigzag canon
    • add a third voice by trial and error
    • stretto echo of the same subject
    • Rhythmically spacious subject allowing non-synchronous timing of answers
    • Recapitulation of the stretto echo canon

    The result is a fuller working out of No. 10 of the 14 specimens in my Book of Canons:

    Black Canyon

    The title comes from my photographic memories of the Black Canyon of the Gunnison River, named for the ever-present shadows the narrow canyon’s steep, sheer, tall rock walls cast on the river flowing far below. The sheer cliffs of the Black Canyon are metamorphic Precambrian gneiss and schist, streaked with thin, brighter-colored layers of pegmatite. These streaks sketched on the darker rock look like maps of ancient contrapuntal lines.

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    Continue reading Mapping the Music Universe . . .

    MapLab 4. Model a metamorphosis

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  • MapLab 2. Sketch a Song

    Many great art-song models . . .

    Schubert‘s famous lied, Erlkönig — a dramatic setting of Goethe’s poem with a hammering piano ostinato as the running horse’s hooves, it uses tonal changes and vocal tessitura to draw distinctions between four dramatic voices.

    Or Charles Ives‘ “The Cage” — utilizes whole-tone scales and “quartal chords” to depict the restless pacing of a leopard in its cage.

    2. Find simple lyrics

    A short poem or single stanza that evokes colorful or dramatic images — or write your own. Limit the total number of syllables so that the vocal line isn’t forced to be too “note-busy” just to cover each syllable. This leaves room for some syllables to have more than one pitch, a melisma that extends the duration of an important syllable’s vowel with beautiful melodic curves.

    TC example

    Speaking of curves, a recent visit to the shores of Michigan’s Leelanau Peninsula inspired me to write a poem:

    Yin Yang

    Peninsula upon peninsula upon grand peninsula,
    Lee upon Leelanau upon Lower.
    Cove from bay from great lake,
    Suttons Bay off Grand Traverse Bay off Lake Michigan.

    Land curves in myriad shore shapes,
    Reaching out to blue water.
    Fresh wind weds the land and water,
    Sun warms bright sails and sailor.

    That is a total of 76 syllables. Though it does not rhyme, there is a simple poetic structure. Each stanza has two 2-line sentences. In both sentences of the first stanza, the first line describes a general recursive process, then the following line particularizes that with geographic names. The second stanza follows this same two 2-line sentences pattern. Land touches water in the first 2-line sentence. The last sentence, like a traditional sonnet-ending couplet, introduces the melding elements of wind and sun.

    3. Design tonal material

    For this lab, let’s start with a scale pattern, something different than a major or minor scale. Let’s limit it to a pattern of no more than 6 pitches in an octave.

    TC example

    I am choosing a six-note pattern, array 2 2 2 1 2, that is actually a truncated Lydian mode scale:

    It also has a similarity to a whole-tone scale, with three consecutive array intervals of 2 semitones (the “whole-tones”). Both the Lydian and whole-tone characters are exotic sounding, conducive to the Impressionistic landscape painting quality I want.

    It also interests me from the remarkable standpoint that its complement, the six other pitch-classes of a 12-tone scale not included, make an incomplete Dorian scale pattern whose array, 2 1 2 2 2, is just the reverse/inverse of the incomplete Lydian. Cool!

    4. Make a melodic theme or motive

    Think about shape: a line can step through the scale pattern or skip or leap to non-adjacent tones of the scale. A line can go straight up or down (like an arpeggio), or it can turn (change direction) occasionally or frequently, or even incessantly (making a stationary oscillation).

    TC example

    This melodic shape has three turns in direction, on the F then on D then on B. (Only the A is not a turning point.) It also uses five different melodic interval sizes, each only once. The mirror inversion has these same features, here starting on G#.

    5. Construct prototype constellations

    Drawing pitches from the chosen scale, establish preferred harmonic interval arrays.

    TC example

    Mine emphasize the intervals 2, 5, 7, and 10, setting a harmonic character

    5. Build the song’s form

    The large-scale form of a song will usually be prescribed by the nature of the lyrics, such as the stanza structure of a poem. The music’s sectional form may use changes in tonality, tempo, or rhythmic character to parallel changes of tone or image in the lyrics.

    TC example

    Instead of marking sections or stanzas by tonality, I will choose to differentiate with tempo and rhythmic fabric. Bright introductory chords are sustained for different prime numbers of 8th-notes — 7 then 5 then 3 then 7.

    The land sentence will be set in continuous quarter-notes. Pitches are again drawn from our primary Lydian-and-Dorian scale patterns but with varying orders and octave placements.

    In a faster tempo and pace, water will be set in continuous flowing 8th-notes.

    The second stanza will transition from the 8th-note flow to slower, more mixed rhythms and, finally, back to an echo of the static chords from the beginning.

    6. Shape vocal melodies to lyrics

    For singing, multi-syllable words should be divided the way a singer would sustain the vowel before ending the syllable with the consonant initiating the next syllable.

    Vocal range should be considered and the pitch space used limited to the likely capabilities of the kind of singer you’re writing for. The higher tessitura (portion of the range) might be reserved to effect a climax if appropriate to the lyrics.

    In determining rhythmic values for the melodic vocal pitches, it is important to recognize the accent pattern of the words, giving accented syllables a musical accent, either by:

    • metric — placing them on a beat or strong beat
    • agogic — sustaining them for longer duration
    • contour — placing the accented syllables on pitch high or low arrival points
    • combination of any of these emphases

    TC example

    Trying to limit the vocal range required to sing this simple song, Yin Yang extends from middle C to the D an octave and a step higher . . . except saving an Eb yet one semitone higher for the dramatic last note on the last word.

    Pe-nin-su-la u-pon pe-nin-su-la u-pon grand pe-nin-su-la,
    Lee u-pon Lee-la-nau u-pon Lo-wer.
    Cove from bay from great lake,
    Sut-tons Bay off Grand Tra-verse Bay off Lake Mi-chi-gan.

    Land curves in my-ri-ad shore shapes,
    Rea-ching out to blue wa-ter.
    Fresh wind weds the land and wa-ter,
    Sun warms bright sails and sai-lor.

    Notice how the incidence of consecutive stressed syllables increases toward the end.

    7. Fit the melodic and accompanying lines together

    Melodic pitches can be drawn from the underlying chord. Or they can represent “non-harmonic tones” forming a dissonance against some pitch of the harmony.

    TC example

    My Peninsula melodic pitches are taken from the underlying chord.

    Since the piano presents the chord as a moving line, vocal pitches often are a simultaneous with the same piano pitch, as in “Fresh” and “weds” above. Melodic tones can also occur not at the same time as the matching harmonic pitch, but instead make a contrapuntal (vertical) interval between the two parts. Under each new vocal pitch below, I’ve indicated the contrapuntal interval it forms with the differing piano pitch of that moment.

    You can see a contrapuntal interval consistency between the voice and piano, even as their rhythmic streams contrast.

    8. Assemble the song

    Now it’s time to put everything together. A traditional approach will include a piano-only introduction and at least one interlude without the voice.

    Normally I suggest listening to a whole piece without watching a score. Since my synthesized rendering here cannot pronounce the words in the synthetic voice, however, I suggest watching below to get the feel of the lyrics that, after all, drive the whole song.

    Yin Yang

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    Continue reading Mapping the Music Universe . . .

    MapLab 3. Construct a canon

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  • MapLab 1. Generate a Gymnopédie

    For this first mapping lab, a basic experimental process is outlined step-by-step and demonstrated with examples from a sample composition. Once you’ve studied the example piece, you can start over and craft your own experiment using the same open steps. General instructions leave you free to openly consider and choose from many musical possibilities.

    1. Choose a model

    Trois Gymnopédies (1888) by Erik Satie

    Simple in harmony, meter, melody, texture, repetitive form.

    2. Design a theme

    Start with a pair of 4-note constellations of considerable interest due to their symmetrical interval stacks and “perfect fifth” 7-semitone interval separated by a smaller interval. (See “Symmetrical interval arrays.”)

    We’ve made two chords, both with the same identical interval stack.

    3. Choose a meter and rhythm/tempo character.

    A prime-number meter (such as the 7 4 meter used for the Finale of Stravinsky’s Firebird) can have a more “timeless” quality, due to its lack of layers of nested pulse between beats and bars. The prime number of beats prevents them from grouping into regular sub-measure groupings.

    To follow through further on the floating feel of lacking groupings, let’s stretch the timings a bit between arpeggios.

    4. Add a line and sound color to the texture

    I call this technique extraction or refraction, pulling selected tones of a complex line into a separate voice:

    5. Make variations

    Arpeggios with refracted color line:

    Pull the 8th-note arpeggios into a continuous stream:

    Canon at the octave:

    Rhythmic augmentation, without then with the refracted color line:

    Mirror inversion of augmentation, canon:

    6. Assemble the large-scale form

    The theme and each variation end with a clear cadence, a sustained final note and pause in rhythmic activity . . . except Variation 3, the continuous 8th notes. It morphs into a transition that both interrupts the 8th-note flow and slows the tempo, preparing for calmer, much less dense quarter-note variation:

    The variation process is serial, each one progressing from the previous idea, rather than “starting over” each time. Thus the overall unfolding form feels evolutionary rather than episodic. Then a kind of recap does start over with a return to the opening idea, making a rather traditional coda ending,

    6. Title

    This musical sketch, like most of my pieces, was composed without a title or guiding image. The compositional process began with the basic challenge to make a small piece out of simple, limited material. The adopted model was Satie’s radically sparse, (one could even say) minimalist style in his Trois Gymnopédies for piano (1888), Its title may have been taken from a French poem by J. P. Contamine de Latour — the poem ends with the word gymnopédie:

    Oblique et coupant l’ombre un torrent éclatant
    Ruisselait en flots d’or sur la dalle polie
    Où les atomes d’ambre au feu se miroitant
    Mêlaient leur sarabande à la gymnopédie

    Slanting and shadow-cutting a bursting stream
    Trickled in gusts of gold on the shiny flagstone
    Where the amber atoms in the fire gleaming
    Mingled their sarabande with the gymnopaedia.

    My title will adopt the English translation of one selected metaphor: Amber Atoms in the Fire Gleaming.

    7. The finished piece

    In keeping with the Satie models, this study generates entirely from one modern harmonic constellation, arpeggiated repeatedly in a gentle, almost imperceptible meter, then growing colorful “amber” sustained highlight sounds. Eventually the arpeggios begin to spin and swirl in a layered, kaleidoscopic texture that is “minimalist” in the 20th-century usage as the description for repetitive ostinato music.

    8. Test sample

    Listen without looking at a score, the best way to first sample created art:

    Amber Atoms in the Fire Gleaming

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    Continue reading Mapping the Music Universe . . .

    MapLab 2. Sketch a Song

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  • MapLabs — Modeling Music

    “Mapping” has double meaning. A road atlas measures and records all the routes through a given territory. But we also call “mapping” the creative act of planning out a journey, using map information to choose between many possible routes. Composers use an array of processes to map out a musical journey. Designing a piece entails making a storytelling rhetoric, a pacing plan, and an architecture of interrelated components. 

    Each Map Lab in Mapping the Music Universe presents step-by-step recipes to compose simple pieces based on models of different musical genres. Each lab also includes an original sample piece following the Map Lab guidelines, illustrating one possible creative path and outcome.

    Try your own experiment with any of these lab projects:

    MapLab 1. Generate a Gymnopédie

    MapLab 2. Sketch a Song

    MapLab 3. Construct a Canon

    MapLab 4. Model a Metamorphosis

    MapLab 5. Spin a Solo

    MapLab 6. Paint a Landscape

    MapLab 7. Twelve-Tonal Trichords in a Ternary Trio

    MapLab 8. A Small Sonata

  • Mapping Music 12. FORM

    Rhythmic intensity is an important factor in shaping musical form. A former research project “Density Functions in the Structure of Modern Music” in the 1970s sought to quantify it along with several other core aspects of structure at play in shaping large-scale form.

    In the TIME chapters, we previously mentioned pace and showed how it can accelerate or decelerate in a line while tempo remains steady. (The Beethoven string quartet example Op. 135 illustrated that.) We have now also defined composite rhythm as an intersecting sum of rhythmic time points of lines, the layers of a textural fabric.

    Density

    In physical terms, density is a ratio comparing the amount of mass to the amount of space it takes up. Measuring time space, tempo (expressed in “M.M.” beats per minute) can convert a count of beats into a time-length in seconds:

    DURATION (in seconds) — multiply BEATS times 60, then divide by TEMPO

    Now we’re ready to measure the pace of a line for a bar or a whole phrase:

    PACE (Notes Per Second) — number of notes divided by the duration of the stream

    And then to quantify for a whole texture of rhythmic activity:

    RHYTHMIC DENSITY (Attack-Points Per Second) — number of note-starting time-points in the composite rhythm of the whole texture divided by the duration of the stream

    Let’s go back to the Webern Symphonie Op. 21. Though called a symphony, it has only two movements. The second movement is a theme and variations with coda, each exactly 11 bars long in two-four meter. Here’s the theme:

    Op. 21, II — theme

    Each variation, though 11 bars long like the theme, is in a different marked tempo. Each is distinguished by a contrasting degree of rhythmic density. And though the theme is a sparse (pointillistic) fabric, some variations are contrapuntally thick and intense.

    Rhythmic density and what we might define as textural density (how many lines woven into what octave span) basically trace the same unfolding through the variations. The exception is Variation V. There they diverge, intensely active rhythms but only three textural elements in a diffuse pitch span of almost four octaves.

    A graph of changing rhythmic density values in each variation highlights rhythmic density as the bolder line:

    density graph of Op. 21 II

    About broad form, this reveals that from the beginning, rhythmic density increases to a subordinate peak in Variation III and overall peak in Variation V, then variation by variation steps down to a coda that matches how we started with the sparse theme. In rhythmic density, the whole movement is an arch form, with Variation V the “climax.”

    In the first “abstract sound mobile” of my 2024 work, FOLIO, it is easier to hear changing density as the changing thickness of clouds of sound, swelling and subsiding.

    “Music of the Spheres”

    Relativity

    Modeling, the process of creating an overall design, can mean creating a new model or expanding the possibilities of an existing model. In Learning to Compose we identified and described three basic musical approaches:

    NARRATIVE MODELING — Designing by telling a story, with characters, themes, gestures, suspense. What will happen next?

    SPACIAL MODELING — Designing the size, shape, and texture of blocks or sections of material

    TEMPORAL MODELING — Designing the flow and momentum of events in the passing of perceived time

    Variation and contrast

    Contrast is the essential complement to developmental continuity in musical material, driving musical momentum. Theme and variations form is a straightforward, traditional example of narrative modeling balancing contrast and continuity. Each variation preserves some basic element of structure such as harmonic progression (or in the Webern example, the tone row). Each variation presents a setting of that theme element in distinctly different orchestration, texture, mode, tempo, or rhythmic character.

    The composer determines not just how and when to make a contrast, but how dramatic the contrast will be. Their fluctuations over time are the core of the composer’s instinctive variation skill. This is the impelling force that gives musical form a sense of going somewhere, of leading up to and flowing away from stable plateaus marking the structural pillars of large-scale form.

    FLUCTUATION — Magnitude of contrast from one moment or event to the next

    When analytically quantifying fluctuating data, the time scale of measurement matters. In avant-garde or experimental music, a stream of events may be high-contrast on the moment-to-moment scale but steady-state over broader time spans. Conversely and more traditionally, surface events may be continuous, while the bigger chunks of events, like one variation to the next, may pose more dramatic changes in parameters such as rhythmic density.

    In typical Beethoven or Brahms variations, material within each variation is continuous, not at all fluctuant. The contrast comes altogether in the next variation.

    That consideration plays out differently in Op. 21 II. There is the obvious contrast from one variation to the next; but within each variation, moment-to-moment surface continuity also fluctuates. Surface fluctuation in density factors occurs, especially from one 3-to-4-second “moment” to the next. (We can’t really call them phrases.)

    For the Op. 21 II. Theme and Variations, we can now say something deeper about changing rhythmic density as the variations progress. From the Theme through the first two variations, rhythmic density increases gradually to Variation III. But then the fluctuation of rhythmic density spikes, dropping significantly for Variation IV, then suddenly increasing to its highest level in Variation V.

    large-scale time form

    It is not only Variation V’s greatest rhythmic intensity but also dramatically increased roller-coaster fluctuation, dropping then surging, that makes Variation V the climax of the movement. 

    Macro-structure

    Though Webern may not have thought consciously about Schwankung (fluctuation), this is how composers manipulate momentum to make a climax and shape large-scale form. Likewise, approaching a final ending, not only do fluctuations typically diminish, but also rate of change subsides — the overall change factor levels out to zero. These are examples of temporal modeling.

    The parameters of a musical event are numerous, a multidimensional matrix of at least six distinct, interacting qualities: each sound event’s loudness, resonance, timbre or sound color, duration, pitch (frequency), and time point of initiation. Imagine this as a six-dimensional space. In fact, physicists have imagined the structure of matter as exhibiting many more than six dimensions in string theory, M theory, etc.

    Musical structure establishes the relativity of these parameters, though not exactly the way Einstein explained time, space, gravity, and energy with mathematical precision. Some structures such as the Schoenberg Farben example relate constellation harmony to sound color. Threnody relates rhythmic activity to fabrics of sound in a broad pitch space (spatial modeling). Counterpoint balances rhythmic relationships, metric placement of lines, and synchronicity with their intervallic relationships of consonance and dissonance. Ostinato music manipulates phase relationships.

    And, as observed in Part I, temporal density, the rapidity of fluctuations and larger contrasts in these structures, propels our experience of the whole in time.

    In Thinking in Numbers, Daniel Tammet wrote about a mathematical study of poetry,

    “The best poems . . . combined in equal parts the predictability of meter with the novelty of unusual words. Too much meter made a poem banal; too much freewheeling . . . rendered it hard to follow. The delicate balance of convention and invention gives meaning to what we say.”

    The essence of music’s large-scale temporal form is the relativity of overlapping, fluctuating musical structures in time, repeating, contrasting, interrupting, truncating, expanding, certainly recurring, or simply evolving. Designing a large-scale musical form combines temporal modeling, narrative modeling, and spatial modeling — a pacing plan, a storytelling rhetoric, an architecture of interrelated components. 

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    Coda

    sound mass . . . sound color . . . pitch constellations

    ostinato repetition . . . changing density

    evolving form . . . cosmic time

    In Become Ocean (2013), John Luther Adams takes a deep dive into a serene sound sea, incorporating all of the elements and structures we have explored in our mapping journey.

    John Luther Adams – Become Ocean (2013)

    . . . and we have just begun gazing into

    the vast space of color and complexity

    in the Music Universe . . .

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    © 2026 – All Rights Reserved

    Thomas S. Clark

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    Continue reading Mapping the Music Universe . . .

    MapLab 1. Generate a Gymnopédie

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    TClarkArtMusic.com

  • Mapping Music 11. TEXTURE

    Imagine a piece of music exploring texture in time, made of single sounds and sonorities occurring one at a time in sustained resonance. Then imagine the points of sound are separated by rests, silence. As the texture drifts in and out of a resonant cloud, the sound events remain unconnected. Suddenly, their pace explodes into a torrent of notes. That describes the following powerful piece by my UNT colleague, Joseph Klein.

    Joseph Klein – Pathways IV: Rhymes & Spirals (2024)

    Sound color

    Our next music map shows a simple color-coding graphic system for classifying most musical timbres, informally the tone quality of sounds. The map intuitively chooses colors of the rainbow. While the color spectrum orders the frequencies of light (another manifestation of periodicity), our sound-color classifying map does not imply any ordered quantification of timbral complexity.

    instrumental color rainbow

    Though we think first of an orchestra for a rainbow of color, chamber music can incorporate a variety of instrumental colors, each produced in vivid isolation by one instrument, standing out or changeably mixed with other colors.

    Augusta Read Thomas wrote Dance Mobile in 2021, scored for 13 instruments: Woodwind quartet (Flute, Oboe, Clarinet, Bassoon); Trombone; String quintet: (2 Violins, Viola, Cello, Contrabass); Piano; 2 Percussion (vibraphone/metal, marimba/wood, drums).

    The piece starts with a single pitch, blending several colors that swell in intensity. Then ensues a kaleidoscopic dance of at least seven distinct color combinations, of two basic types:

    Sustained sounds – strings; high woodwinds; lone brass of the trombone

    Sparks – pizzicato strings; ringing metal sounds; drum strokes; staccato piano

    Augusta Read Thomas – Dance Mobile (2021)

    Though the piece is dedicated “in memoriam Oliver Knussen,” the memory is a joyous dance of color.

    Symmetry

    In the exposition of Webern’s Symphony, Op. 21, we saw that each contrapuntal line duplicates the exact rhythm of the lead line, with each entrance one bar later — a classic canon. But each contrapuntal line presents a different succession of instrumental colors:

    Horn . . . . . . . . . Clar. . . . Cello
          . . . Harp . . . Cello pizz. . . . Cello arco . . . Violin . . . Harp . . . Horn . . . Harp
          . . . . . . Horn . . . . . . . . . Bass Clar. . . . Viola
           . . . . . . . . . Harp . . . Viola pizz . . . Viola arco . . . Violin . . . Harp . . . Horn . . . Harp

    The German term for this is so elegant, we’ll use it here:

    KLANGFARBENMELODIE — melodic or contrapuntal line expressed by a string of changing tone colors

    Webern placed each pitch in every line in a particular fixed octave, except Eb that appears in two different octaves. This makes a striking, symmetrical 13-pitch constellation with a palindromic array, the same array going down as going up.

    Webern 13-pitch constellation

    Not only was he obsessed with symmetry in this piece, but this constellation’s symmetry also proves that he was thinking specifically about the chord voicing in what I have identified in successive interval array form.

    We can use this constellation as a Y-axis for a graph mapping the timbres as they appear in the various parts in canonic lines in pitch space for the first 9 bars. This farben color map looks like one of the later geometric paintings of Piet Mondrian.

    Op. 21 color map

    Pointillism

    Though we often share musical terms and concepts with visual art, we sometimes mean different things by the same term. In painting, a technique developed in the Impressionist style period of the late 19th century that became known as pointillism. The most famous example is Georges Seurat’s “A Sunday Afternoon on the Island of La Grande Jatte” at the Chicago Art Institute. Instead of sweeping brush strokes and palette-blended colors, it used small separate spots of subtly varied colors to make a texture that, when viewed from a distance, seems to merge into a color cloud, giving the impression of animated light.

    Musical pointillism, unlike painting, separates sounds in time and pitch space, not to blend them into a texture so much as to highlight the different qualities of each unique sound event. Webern was a pioneer of musical pointillism in works such as Op. 21. Let’s graph the first 10 bars of this fabric using our timbre color-coding (BLUE = wind, ORANGE = percussion, VIOLET = plucked string) on a broadly distinguished 6-octave pitch range. We get something as colorful as a Mondrian painting!

    Andromeda sound color map

    As a musical fabric, isolation — using the vast available range of pitch and the empty time of rests and silence — is a fitting analog for the vast, mostly empty space of a galaxy. Let’s use it for a demonstration etude.

    Andromeda is the nearest large galaxy, 2.5 million light-years from our own Milky Way galaxy. Our sound color demonstration study uses every sound quality on our sound color spectrum except red. Here is a score of the first 10 bars.

    Notice that the green woodwind notes are doubled with a synthesized vocal-type sound. Yellow brass notes are punctuated by orange metallic percussion attacks. Likewise, blue string notes are articulated by the plucked string sounds of harp.

    Here is the whole colorfully pointillistic 3-minute study:

    Sound Mass

    At a time when electronic music was emerging in the 1950s, new instrumental resources were also developing a new style that was all about animating massive layers of sound.

    German experimentalist Karlheinz Stockhausen composed two early, influential sound mass works, Gruppen (1957) for three orchestras, and Carré (1960) for four orchestras and four choirs. The scores were huge, dense, 12-tone, and monolithic in form.

    A 2002 piece by John Adams, On the Transmigration of Souls, harkens back to a mid-century masterpiece of the Avant Garde. In 1961, Polish composer Krzysztof Penderecki wrote a piece for a massive score of 52 string instruments. Conceived as an abstract, freeform, dense massing of animated and intense musical fabrics, it represents a pioneer in the genre of sound mass music, winning the UNESCO Prize that year. Only after it was heard in performance, he said, “I was struck by the emotional charge of the work … I searched for associations and decided to dedicate it to the Hiroshima victims” — thus the title, Tren Ofiarom Hiroszimy (translated Threnody for the Victims of Hiroshima).

    As a young composer in the ‘70s, I reflected this approach in some pieces titled Animated Landscapes. (The title was inspired by John Cage’s famous Imaginary Landscapes no. 4 for 12 radios.) Beyond referring to the painting genre of landscapes, the title sets the imagination for solid, continuous textures like viewing the shapes of a mountain range, but set into rhythmic motion. (This approach became prevalent in ensemble music, especially of Midwestern composers such as Donald Erb.)

    Considerably predating the music mentioned above, Schoenberg’s Fünf Orchesterstücke, Op. 16 (1909), was originally scored for a large orchestra of 37 parts. It is not thought of as sound mass music, as its five movements each have Expressionist or Impressionist titles: “Vorgefühle” (“Premonitions”); “Vergangenes” (“The Past”); “Farben” (“Summer Morning by a Lake”); ”Peripetie” (“Peripeteia”); “Das obligate Rezitativ”(“The Obligato Recitative”). The third movement, Farben, is of special interest not only for its exquisite mixed-palette painting of orchestral timbres, but also for its thick though delicate fabric of sustained sounds. At the start, nothing moves, the subtle shimmer of instrumental colors fading in and out of a continuous fabric of delicate, faint sounds. (A sound mass can be delicate, not necessarily “massive.”)

    Here is a score of the first page, showing sounding concert pitches for all instruments.

    Schoenberg Farben scoring

    Each measure presents one constellation, recolored with different instruments in the second half of the measure. For the first three bars, the constellation does not change, and then only subtly in the next five bars, maintaining the constant C pedal point in the low strings.

    Farben constellations

    The bass clarinet’s F3 in bar 7 is considered an ornamental non-harmonic pitch. While you can see many recurring smaller constellations imbedded within these changing large constellations, such as 5 5, 3 5 and its inversion 5 3 (which are triads), and some transformations of smaller constituent constellations: 8 3 redistributed to 9 2, 4 7 shrinking to 4 5 (another triad), and 3 4 (also a triad) shrinking to 2 4.

    Though there are many triads embedded in the constellations, the overall quality of the sonorities is complex, as the triads are framed within critical dissonances:

    framing dissonances

    Foreground / background

    Most landscape paintings, distant textures of forest, mountains, sky, waves on the sea, or clouds, have some sharp focal point. Often on the horizon (in itself a focusing anchor of the visual display), it may be a barn, a setting sun, a boat, a farmer and dog. If we consider proportion and symmetry in a visual composition, the focal point is best not dead center. A more interesting balance, according to expert photographers, follows the Rule of Thirds, placed one-third from the left or right, one third from the top, or both. Two-thirds is a ratio of 0.667. The Greeks famously defined the Golden Ratio, an ideal ratio dividing a whole length or height into two parts such that the ratio of the smaller part to the larger is the same as the ratio of the larger part to the whole. The ratio is 1.618:1, the solution to the equation: x2 – x – 1 = 0; a 62% and 38% division.

    In a simple traditional musical texture, an accompanying harmonic texture is designed as a background for the focal element of a melody. Sound masses may lack such focus, like the forest or sea waves. When there is to be perceived a standout element of the texture, Schoenberg called this focal element of the musical fabric the Hauptstimme. Though that might translate “highest voice,” the melody or other focal events are not necessary to be higher in the pitch range of the fabric than other elements. But there must be some isolation or distinction setting them off from background in at least one of the parameters mentioned above. The Hauptstimme focal line or textural element can be:

    • in a pitch range isolated from background
    • a color isolated as a single timbre, not a mixed diffusion of background colors
    • slower or faster than background
    • more rhythmically elastic, varied than background
    • not synchronized with background
    • loudest line (the most obvious)

    Schoenberg devised a special symbol for the focal Hauptstimme line of a fabric, a boldface stylized capital H, which you see marking the bass clarinet entrance in bar 7 of the Farben example. Here is how that principal Hauptstimme line continues, a Klangfarbenmelodie of changing color, from bass clarinet to clarinet with trombone to three solo contrabasses.

    Hauptstimme handoffs

    Notice the aggressive rhythmic motive, each time stepping down 2 semitones; and the  7 7 7 quintal-chord constellations in the contrabasses. (The rhythmically aligned clarinet and trombone are separated by 14 semitones, 7 + 7.)

    Beyond color isolation, Learning to Compose makes a distinction for a timbre mixed with itself or other colors spread over some pitch register (“diffuse”) or reinforcing itself in a narrow, confined pitch space (“concentrated”). While Farben’sbackground is diffuse, its Hauptstimme color is isolated in the low pitch register of the bass clarinet and then also concentrated with the three solo contrabasses.

    In the first movement of Anthracite Fields (2015) by Julia Wolfe, the bass clarinet emerges as a focal sound by its loudness and singularity of pitch in a cloud mass of softer sound. Then aggressively loud clusters suddenly interrupt the steady-state background, yielding eventually to repetitive sung chords and floating vocal duets. The sound fabric maintains a three-dimensional depth of contrasting intensities.

    Julia Wolfe – Anthracite Fields I: Foundations (2015)  

    Galaxy groups

    Our sample etude composition for sound mass is a thick score of 10 wind parts and harp, with a fabric the opposite of pointillism: everything sustains and overlaps. There are basically no pauses or holes in the continuous 2-minute sound fabric. Its title, Laniakea, is the name of the supercluster of galaxies that includes the Milky Way.

    Laniakea score excerpt

    Having shown the score with all its notational details, to better illustrate the main point of the example, sound mass, here is a graphic rendering of that actual second system of notes. We can reveal its pointillism by increasing the contrast in a negative image of light on dark. That makes the attack beginning of each sound show up but not the staff lines or sustained resonances . . . a fanciful art image of Laniakea, a vast empty part of the universe dotted with millions of galaxies.

    Laniakea score abstracted

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    12. FORM

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  • 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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    11. TEXTURE

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  • Mapping Music 9. LINE

    Think about levels of structure scientists study in our universe.

    They dive deep into atomic structure, below electrons spinning around a nucleus of protons and neutrons, discovering subatomic particles like the meson and boson. At the other extreme, they gather 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.

    “We are slowed down sound and light waves,

    a walking bundle of frequencies tuned into the cosmos.”

    — Albert Einstein

    Think about how artists build structures that establish a style  . . .

    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.

    Structure and Relativity

    Shrinking our metaphor from the vastness of the universe down to the physical immediacy of cloth . . .

    A woven fabric has a longitudinal warp and a perpendicular crossing weft. Part II explored the vertical-pitch “weft” of harmonic design. Now we return to the “warp” in music, longitudinal time streams of events. In keeping with our standard conception of time as horizontal and pitch as vertical, let’s name each longitudinal “warp” element:

    LINE — an element of a musical fabric consisting of a conforming stream in time of similar events (notes, pitches, colors, drum sounds, etc.)

    Now we can go back to “monody, homophony, polyphony” and at least identify how many lines are in a musical fabric, from one (monody) to many (polyphony). But to distinguish between homophony with its matching, rhythmically aligned lines from polyphony with its more diverse set of lines of different nature, we must distinguish different types of lines to determine the extent to which the lines of a polyphonic fabric “match.”

    There are limitless number of combinations of characters for a line and thus an infinite number of fabrics possible. We will stick to six parameters and simple observational characterizations for each parameter. Since we are swimming in the painting and weaving metaphors, we will color-code these six parameters. Each parameter will be distinguished with just two binary descriptors, a simpler or purer character or a more intense or complex character in that parameter.

    distinguishing parameters

    Since there are 7 parameters and two possible descriptors for each parameter, the total number of permutations is 2 to the 7th power = 128 possible combinations. That means, however, if there are two lines in the fabric, the number of possible combinations rises to 16,384 — plenty of choice for creative composing. And with 4 lines, the number of possible combinations explodes to more than 268 billion!

    More simply, with these defined characteristics we can redefine “homophony” to mean more than one line that match characteristics, and typically are in rhythmic alignment (synchronized). Indeed, most musical fabrics involve quite a bit of similarity between multiple lines. In a typical traditional “melody-and-accompaniment” fabric, there are only three distinct lines, melody, bass line, and chords, even if the chords are actually in two or more matching instrumental or vocal parts.

    The following example is taken from the Allegretto movement of Beethoven’s String Quartet Op. 135.

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

    In the first two bars of this example from Op. 135, there are actually only two lines in the fabric, the melody (1st violin) and repeated chord tones (the other three instruments aligned in 16th-notes) — common homophony.

     Op. 135 violin vs. other lines

    Though there is no dynamic marking for the 1st violin, it will be played as a prominent line, what Schoenberg would have called the Hauptstimme. By the third bar of the second system (11th bar of the example), there are three lines, violins / viola / cello, and by the next bar, briefly, all four instruments have distinct fabric threads. By the end of the excerpt, all parts have joined in homophonic unity.

    Melodic shape

    Melodic connotes a singable tune of primary focus; here it is meant simply as any line of successive single pitches. In the general descriptors of texture, we referred to smooth and angular shape. Let’s be more precise. First, there is the general size of melodic intervals. As music practice moved from Medieval/Renaissance through 18th-Century styles, smaller intervals, steps and small skips predominated. 19th-Century styles introduced a greater proportion of larger “leaping” intervals, 6ths, 7ths, 9ths. And those large, disjunct intervals became the norm for much 20th-Cenury music.

    Another important melodic shape factor is directional.

    TURN — a melodic note is approached in one direction (up or down) and left in the opposite direction

    Some turns are trivial and do not complicate melodic shape, such as trills and back-and-forth oscillations.

    turns in Elegy line

    The first phrase, starting on Eb, goes up to E then down to D — turning on the middle note, E. The next two phrases are increasingly complex in shape.

    Elegy 2nd and 3rd phrases

    The phrase starting on the lower B rises to G# then turns down on that G# to A, then back up from A, and finally back down, turning on Bb. There are three turning points, G#, A, and Bb, in a phrase of only six pitches and five melodic intervals. Combined with the fact that each melodic interval is a different size (9 s.t., 3, 8, 1, 3) except the last (reusing the downward 3 semitone interval), this is a rather complex, angular shape.

    The third phrase, starting on the higher B, is even more complex in angular shape: turns on every pitch except the C# — that is five turns in just 7 melodic intervals between 8 pitches.

    A side note of analytic math:

    • Number of pitches (#P) minus 1 = number of melodic intervals
    • Number of melodic intervals minus 1 = number of “opportunities” for the line to turn (#P – 2)
    • Shape complexity = #T / (#P – 2) ranging from zero to 1

    Pitch recurrence

    RETRACING — melodic line returning within a phrase to the same pitch (in the same octave) as previously sounded in the phrase

    Distinguished from a pitch being repeated (immediately), a retracing is a recurrence after other intervening pitches. It contributes to structural stability in the phase, a sense of staying in one place. Conversely, when retracings are avoided in the shape of the line, the sense is more of progressing, even of wandering, as in the Elegy example above. (Use of a 12-tone row to construct a line is a way to methodically avoid retracing any pitch until all 12 pitch classes have been introduced.)

    Back to Bartók — two orchestral lines studied in the Pitch chapter. The first example (from the opening fugue of Music for Strings, Percussion and Celeste) is scored for viola, but here I show it in bass clef for those a bit challenged by alto clef.

    fugue subject

    Mapping the line on a time/pitch graph for analysis, the first phrase avoids any retracing. The next three phrases make only one retracing each: back to Bb in the second phrase (highlighted in blue); back to C# in the third (in red); retracing back to C in the fourth (in green). (There are fainter retracings back to the previous phrase in each not shown.)

    The second is a low string line from the opening of Concerto for Orchestra.

    Concerto for Orchestra retracings

    In this example, both phrases are built with two retracings, C# and F# in the first phrase, F# and B in the second phrase.

    In this manner, retracing of pitches builds the support structure for the architecture of many lines.

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    10. COUNTERPOINT

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  • journal 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.

    _____________

  • MapLab: 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.