MAPPING MUSIC

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  • MapLab 8. A Small Sonata

    Sonata is typically a multi-movement piece for solo piano or an instrument with piano. A shorter form with just three connected sections, the middle slower and quieter, can be called a sonatina.

    An inside look at how one was composed gives a guided tour in the form of a recipe to write your own sonata.

    1. 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.

    2. 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.

    3. 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.

    4. 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.)

    5. 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.

    6. 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.

    7. 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.

    8. 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.

    9. 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”:

    10. 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.

    11. 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.

    Sonatine

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    © 2026 – All Rights Reserved

    Thomas S. Clark

    Back to the beginning of . . .

    Mapping the Music Universe

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  • MapLab 7. Twelve-Tone Trichords in a Ternary Trio

    Schoenberg’s famous twelve-tone row technique, devised in the early 20th century, prescribed a compositional method to order choice of pitches in the dense forest of possibilities with 12 chromatic scale steps. Music composed in this manner was called “atonal,” but this Lab will create a distinctive individual tonality from a 12-tone series of four trichords.

    The sample piece, in a classic ternary form, will be scored for a chamber trio of three different instruments.

    The so-called atonal music was complex and mostly dark and dissonant. Can a technique like this also be used to methodically create coherent tonal characters of brighter, simpler sonority?

    1. Choose a model

    One of my favorite 12-tone pieces is Schoenberg’s Wind Quintet, Op. 26. Premiered in 1924 on Schoenberg’s fiftieth birthday, it was dedicated to Schoenberg’s young grandson Arnold. In four movements, it has a graceful neo-classical Viennese quality. It also is historically only the second piece ever (after his Suite for Piano, Op. 25) composed with Schoenberg’s new 12-tone row technique.

    The slow third movement, tempo marked “Etwas langsam,” displays a clear use of a 12-tone row in graceful counterpoint between the five instruments. The facile scoring of these instruments explores textures ranging from isolated solo to duos, trios, and even dense five-part counterpoint.

    2. Design a 12-tone series

    I imagine the typical way composers design a 12-tone row is by writing a line, choosing the interval from one pitch class to the next while not reusing any pitch class, until all 12 are included. That leads readily to a linear, contrapuntal approach.

    For our example in this lab, we’ll work instead with trichords, each a group of three pitch classes making a three-note array and set of three intervals. Here is such a row (A) and a slight variation of it (B):

    The A version also shows the three intervals in each trichord.

    Note that the 2 5 and 5 2 scale patterns are the same, representing a set that is inversionally symmetrical. (That is not true of 4 1 and its unique inversion, 1 4.) Its complete-octave array is 2 5 5. Think of it as a circle of the twelve semitones in an octave. Unwinding the circle through octaves looks like: 2 5 5 2 5 5 2 5 5 2 . . . etc. This is a palindrome, reading backwards the same as forwards.

    Here is a voicing into 3-voice chords:

    You can see (and hear) that the voiced-out trichords can make constellations predominantly of 5-semitone and 7-semitone intervals. This is what makes the sound of the progression coherent and somewhat more like Copland than Schoenberg in character.

    3. Twelve-tone rows

    Now we can delve into the linear potential of the series we’ve constructed. I’ll call my principal row P in keeping with standard theory, designating the transposition by the name of the starting pitch class. (Standard theory uses pitch class numbers — C = 0 — to designate the starting pitch.)

    Serial technique is just a box of 12-tone tools, which you can use in any way you wish. For the last row above, I shuffled the order of the trichords. The shuffle puts the trichords inside out, with the 1 4 array now at the beginning. This gives me more options when building counterpoint between two lines with each its own row form. Like rearranging four toy blocks, the shuffling still builds a 12-tone aggregate collection no matter the order of trichords or pitches within the trichords.

    4. Imitative lines

    As we observed earlier in Opus 26, a row is a natural material for building contrapuntal lines that are similar in interval shape. When these similar shapes are reinforced with similar or identical rhythmic patterns, the lines mimic each other in their back-and-forth conversation.

    5. Ternary form

    Of all the classic form models, ternary is one of the simplest in concept and strongest to perceive. Opening material is contrasted with a middle section of different tempo, texture, register, rhythmic pace, etc., followed by a return to the opening material: A B A. (Several familiar classic model forms are ternary, including Sonata Allegro, Da Capo Aria, and Minuet and Trio.) Often the returning opening material of the third section is embellished or varied in some way while not clouding recognition that it is a recurrence of the opening.

    For Schoenberg’s Etwas langsam third movement of the Wind Quintet, the opening material consists of two-part counterpoint between horn, marked as the Hauptstimme preeminent line, and bassoon (Fg) marked as the Nebenstimme subordinate line.

    Opus 26 middle section is faster flowing, with much more rapid, complex rhythms in a denser four-part counterpoint:

    The third section brings back but alters the opening’s two lines, moved from horn and bassoon to oboe and bassoon, with the addition of a third contrapuntal voice in the clarinet.

    For our MapLab example, the chorale-like chordal setting of the trichords we’ve already seen will be the serene opening and ending material. Imitative contrapuntal setting of the row in a faster tempo and much faster pace make the contrasting middle. (Many classic three-part forms, by the way, are the converse, fast – slow – fast.)

    Opening:

    The opening material continues with a variation of the chords, transposing them and adding rhythmic interest with single 8th-notes each arpeggiating a pitch of the chord.

    Contrasting middle:

    Note: if you’re wondering how I pulled the lower line of dotted-half-note steps from the row, I didn’t. It is free counterpoint for a simple supporting “bass line.” Those pitches are drawn from the trichords expressed above them, as unison or octave “doubling” of single pitches from the upper lines.

    The recap first brings back the opening’s single-8th-note arpeggiations then proceeds back to the still chords of the very beginning.

    6. Scoring the trio

    For this lab, choose any three different instruments. They can be in the same instrument family, like a brass trio of trumpet, horn, and trombone. Or it can mix instruments from different families, like flute, horn, and cello. You can guess from the audio above, I choose a woodwind trio of flute, oboe, and Bb clarinet.

    Get to know your chosen instruments, not only their overall range but also the particular characteristics of sub-registers. For orchestral string instruments, it is good to know their open string tunings and recognize those strings’ different qualities; the lowest string of each instrument is thicker and produces a thicker, grittier timbre. For my chosen three, flute and clarinet have wonderfully rich, dark lower registers, while the lowest few pitches of the oboe’s range are squawky. Each of the three instruments have upper ranges that thin a bit in timbre but can be elegant or powerful.

    My scoring will emphasize use of those rich low registers of the flute and clarinet. Many chords will be voiced with oboe on top, flute in the middle, and clarinet lowest. Nonetheless, the score will show them in standard score order with flute at the top. Also, the clarinet is a Bb transposing instrument. That means in the score and player’s part, a written C will sound one whole step lower as a concert Bb. When I want, say, an F to sound, the clarinet part will show a written pitch one whole step higher, G.

    For a well-balanced chamber-music trio should balance how much time each instrument gets as the active, preeminent voice in the texture (which Schoenberg would call the Hauptstimme).

    7. Finished score

    Tempo and dynamics are essential to vividly portray where the music is going — launching, growing, swelling, subsiding, approaching a cadence or climax. Articulation marks will be particular to each instrument type. Rehearsal letters or measure numbers will be helpful in rehearsal.

    Since my finished example has a childlike playful curiosity, its title follows Schoenberg’s Opus 26 dedication to his grandson. Imagine a child in a sunny garden.

    For Little Arnold

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    Continue reading Mapping the Music Universe . . .

    MapLab 8. A Small Sonata

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  • MapLab 6. Paint a Landscape

    Many 19th cen. American landscape artists painted fascinating panoramic scenes, including Albert Bierstadt, who captured the grandeur of Western, mountainous landscapes.

    Much mid-20th-cen. large ensemble music “painted” with sound masses, animated and evolving instead of harmonically progressing — notably“Variations for Orchestra” (Bassett),“The Seventh Trumpet“ (Erb),“Aftertones of Infinity“ (Schwantner). My “Animated Landscapes” (1973) for orchestra was titled echoing Cage’s“Imaginary Landscapes.”

    1. Establish a model

    Albert Bierstadt: Passing Storm over the Sierra Nevadas (1870) – San Antonio Museum of Art

    Choosing one of Bierstadt’s finest works, you can see stark contrasts in brightness and in sense of motion between the mirror-smooth water and roiling clouds. Even the word “passing” in the title suggests change, a necessary ingredient of an analog musical landscape.

    2. Build big constellations

    The fixed, stationary nature of a landscape painting suggests a static harmonic approach, projecting and prolonging one constellation as an underlying sonority for the whole musical “canvas.”

    Here are two examples of a 12-tone constellation. The first was used as the underlying sonority of the pointillistic first movement of Anton Webern’s Symphonie Op. 21 (1929). Note that to accomplish the palindromic symmetry of the interval array, Eb is placed in two different octaves, thus 13 pitches total.

    Another example of placing all 12 pitch classes in particular octaves is from Witold Lutosławski’s Jeux vénitiens (1961):

    TC example

    A large register-spanning constellation need not be 12-tone. Building a sonority of very different (less dissonant) quality, let’s start with just octave C’s and their fifths, G. Pitches a whole step away from these tonal pillars are added. The result is 13 pitches only from the F major scale. Then their complement, a big 8-pitch constellation of similar array, is built from the pentatonic scale of the flats, the other “black keys” .

    3. Animations

    Different basic techniques for animating in time a large constellation of pitches . . .

    Arpeggio – one note at a time in any order, is a kind of stacking.

    Stacking – introduce sustained tones one or two at a time. The reverse, unstacking, works too.

    Swells – crescendo/diminuendo the whole sonority (think ocean waves in Debussy’s La Mer)

    Pointillism – a more complex texture of distinct sound points separated by registral and rhythmic spacing. Although they can be different sound colors, in this example they are all the same timbre:

    4. Develop a sound palette

    Larger ensembles can be good for musical landscapes, offering a varied palette of instrumental timbres. Like pigments, these sound colors group in hues: the

    • double reeds
    • other woodwinds
    • cylindrical brass (trumpet, trombone)
    • conical brass (horn, euphonium, tuba)
    • bowed strings
    • plucked strings (harp)
    • metal pitched percussion
    • wood percussion; drums; exotic gongs, etc.

    For synthesized sounds, beyond emulating these instrumental timbres, envelope and spectrum distinguish the sounds:

    • sharply accented or gradually emerging sound onset
    • bright metallic to dark hums
    • reverberation.

    For Passing Storm, I chose three “choirs” with distinct sound qualities:

    • Clanging percussion and plucky harp
    • Gentler emerging-onset sounds (Sibelius Pad 2 – warm) functioning like woodwinds
    • Brighter sustained sounds (Sibelius FX4 – atmosphere) in the orchestral role of bowed strings

    5. Compose the whole canvas

    In painting, “composition” refers to the main parts of an image and how they are arranged with each other in two-dimensional and the illusion of three-dimensional space. For music, we might call this the macro form. As with most landscape paintings, the musical textures may overlap in continuous sound. There may be sectional divisions, analogous to strong demarcation lines like a horizon. The sense of broad, distant scale of view will be best captured with extended textures of expansive rhythm and even with reverberation.

    While not trying to actually map the physical composition of a painting, we can get musical inspiration from considering the painting’s features of background, foreground, and highlights of strong visual focus. My example was coming together starting with distant swelling sonorities, which as they crescendo feel like they are emerging forward toward us.

    After deciding to name the piece Passing Storm after the Bierstadt painting, however, I realized I had no storm in the music, just gentle sprinkles. Thus was created a stronger sonic rendering of the sprinkles to provide a more aggressive introduction. The proceeding four minutes overlaps sound masses exhibiting the various animations in time we explored earlier. Dark and light in a painting are portrayed by loud and soft, timeless sustained sound vs. busy points of sound.

    Animated Landscape No. 4: Passing Storm

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    Continue reading Mapping the Music Universe . . .

    MapLab 7. Twelve-Tone Trichords in a Ternary Trio

  • MapLab 4. Model a Metamorphosis

    As with MapLab 3, this will be multi-layer counterpoint utilizing canon in a homogenious texture. Now it will be entirely a repetitive ostinato texture — flowing, periodic rhythmic activity building a continuous texture of repeated arpeggios or melodic motives. Commonly called “minimalism,” its texture and overall rhythmic character are maximally dense.

    Multiple layers generate complex phase relationships between contrapuntal voices, with patterns of differing length repeating and changing at different times in the four layers.

    Layers of texture will change at different times to a new pattern, overlapping each other. Thus overall change of harmony unfolds gradually and continuously instead of at definite time points of harmonic rhythm, building a metamorphic form (instead of a traditional episodic sequence of chords, phrases, and sections).

    1. Choose a model

    The classic granddaddy of this whole genre is Terry Riley’s monumental 60-to-90-minute improvisatory piece, In C. My own 1984 homage to that classic, EFFULGENCE, models with Riley’s many innovative techniques.

    2. Select a source scale

    While any scale can work, those most commonly used are diatonic scales. In the TC example, we’ll go with the same as In C, a C-major/A-minor no-sharps-or-flats key signature. (We’ll see later, however, that a motive can be transposed into another diatonic scale and key signature.)

    3. Make motives

    First, design two or three motives, basic shapes of 3 to 7 pitches from the source scale.

    TC example:

    Motive R gets extended by the addition of two pitches, F and E. The last example shows motive T’s shape shifted to a different level of the diatonic scale (what Sibelius calls a diatonic transposition). A motive can also be truncated to as few as two notes:

    4. Plan a stream of motive variants

    Motive patterns can and should vary in length, especially when rhythmic values are mostly all 8th-notes, providing a changing landscape of rhythmic vitality. In the TC example, however, most patterns are 5 8th-notes long. Since 5 is a prime number, and set in a 3 4 meter, the overlaps of these 5-patterns in the competing lines fulfills that energetic complexity of rhythmic fabric.

    TC example

    For the pitch motives, a process of adding or abandoning pitches to make the next pattern creates the metamorphic unfolding process that is the true magic of this lab. In the TC example below, this add/abandon process is color coded:

    • GREEN for newly added pitches
    • BLUE for pitches appearing in a different octave than in the previous pattern
    • PURPLE for pitches that will appear next in a different octave
    • RED for pitches that will be abandoned in the next pattern

    You can see that by letter K the original C-major diatonic is modulating to a new diatonic, Bb major. These two keys have in common 5 pitch classes, and the patterns capitalize on the F, G, and C common tones to connect smoothly. (Riley’s In C also modulates, eventually adding F# and Bb in much the same way Bach inflects the C-major tonality toward the end of his famous C Major Prelude that launches Book I of the Well-Tempered Klavier.)

    Here is the lead voice of the ostinato canon:

    You can see that the number of repetitions of a pattern and the overall duration of its presence in the texture vary throughout. Patterns E, F, K, and P run for five full measures in the lead line alone (plus delayed answers in the whole texture), while the simple transitional pattern N runs for only five beats in the lead line.

    5. Spin the canonic counterpoint

    The time delays of canonic answer should be chosen not to match the length of the typical pattern. Otherwise, the answers would lock into fixed duplications of each other, making a rigid, uninteresting periodicity. Each new motive-pattern entry is highlighted below with a new dynamic marking. Here is a sample excerpt starting around pattern H:

    The answers all enter at unison or octave, with timings determined by a mostly trial-and-error method as follows:

    • PP – 9 beats later at unison, then 9 more beats down an octave
    • P – same
    • MF – almost same, but shortened last answer comes one 8th-note early
    • MP – (for a 3-8th-note pattern) 5 8th-notes later then 7 8th-notes after that
    • PP – top voice leads, answers are 2 beats later then 3 8th-notes after that

    This last is what we described in MapLab 3 as a stretto, answers coming in with very short time delay.

    6. Interrupt with an interlude

    As with In C, the ostinato texture can blast through from beginning to end in a continuous monolithic stream. Another form scheme, which I will invoke in the TC example, breaks the stream with an interrupting interlude before a coda to come. Of course, it’s another canon, a stretto of cascading downward dotted quarter-notes.

    7. Ending an ostinato stream

    Several considerations . . .

    First, since you’ve built a canon with staggered entrances, the last notes will be staggered as well. To make any kind of cadential closure, however, you’ll want to have them stop at the same time, right? That is accomplished simply by truncating the answering lines and/or adding repetitions of the final pattern in the lead voice.

    Think about the lead line and its answers leading to a point of harmonic stability and finality — somewhere that feels like tonic home base.

    More repetitions help slow and stop the harmonic momentum.

    In the TC example, an ostinato coda after interruption settles into and prolongs what will sound like a dominant chord in C major, then crash lands on a tonic C-major stinger.

    8. Title and listen

    The picturesque metaphor of a babbling creek made me reminisce about a favorite adventure on days off from working at the National Music Camp in Interlochen, Michigan back in the ’70s and early ’80s. We would canoe down the Platte River to its end flowing into Platte Bay on Lake Michigan. There was also a nearby spot where tiny Otter Creek trickled out onto a more secluded sandy Lake Michigan beach offering northward a spectacular view of Empire Bluff.

    Otter Creek

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    Continue reading Mapping the Music Universe . . .

    MapLab 5. Spin a solo

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  • 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. 

    –

    –

    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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    MapLab 1. Generate a Gymnopédie

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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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    Continue reading Mapping the Music Universe:

    11. TEXTURE

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  • Mapping Music 8. TONALITY

    In traditional tonal music, or for a composer’s personal design, there are four main factors defining a tonal language: source scale (covered in Mapping Music 5); harmonic type; horizontal (voicing) connection; and tonal center, a basic concept for Common-Practice tonal music.

    A diatonic major or minor scale and harmonic structures built from it define a key and “tonic” home-base tonal center. (In the ancient modal music of the monophonic Gregorian chant it was called the “finalis,” as it was the expected final arrival destination of an extended melody.) Triads taken from the scale build a scaffold of harmonies, featuring the dominant chord (scale degrees 5, 7, 2, and sometimes 4) with its scale-degree 7 “leading tone” propelling a progression to resolve back to the tonic chord (scale degrees 1, 3, 5).

    In 20th-century music, some composers (notably Bartók) began to define tonal center contextually rather than by scale-and-key, writing melodic patterns and counterpoint that branched out from and converged back to a core base (but not necessarily bass) pitch. Twelve-tone music, derived from the full chromatic scale, would seem to be avoiding any tonal center, but some composers still built textures whose lines and counterpoint would emphasize one focal pitch-class.

    A matrix of choices

    In forging a tonal language, the composer develops preferences in each of these factors. Choices from each factor column can be mixed in a variety of ways. The composer designs by delving into more specific patterns, especially for the source scale (possibly, say, a six-note pitch-class set) and the harmonic type, establishing a preference for certain harmonic intervals (such as my favoritism for 7-semitone Perfect 5ths and 11-semitone Major 7ths).

    There are, of course, thousands if not millions of possible combinations of all these factors, a universe of tonal possibilities for the individual composer and a particular piece.

    Next, let’s dive more deeply into harmonic types and the factor of horizontal connections between successive harmonies.

    Constellation streams

    A stream of successive constellations, which we might nickname a “constream,” would traditionally be called a chord progression. In the following example, all stacks are 10 semitones tall; no common tones in the transposition choices.

    no common-tone connections

    In the next example, stacks of differing heights, with constellations that reduce to three different scale patterns: scale array 5 2, then 2 3, back to 5 2, then 4 1, and finally 2 5, inversion of 5 2.

    common-tone connection

    Now a longer, more mixed succession of interval stacks of constellations belonging to these same three scale patterns (2 5 or 5 2; 1 4 or 4 1; and 2 3).

    extended constreams

    Back to my constellation friends of Mapping Music 6, we can make some constreams with them.

    diatonic and chromatic successions of symmetrical constellations

    An intriguing example from the literature of great early modern music, an interlude near the beginning of Stravinsky’s L’Histoire du Soldat:

    L’Histoire du Soldat excerpt

    This passage is intriguing in many ways. It looks like counterpoint between two woodwind instruments in high register. But both lines are quite simple and don’t seem to go anywhere. (In our GALAXIES: Structure chapter, we’ll discuss these questions of texture and counterpoint.) Introducing it here raises the question of harmony, of constellations and their arrays, though the passage doesn’t look at all chordal. Here is an array analysis of the constellations formed in the first through fourth bars then jumping to bar 10 and, finally, bar 14.

    L’Histoire du Soldat constellations

    Now you can see and hear more clearly the role played by array interval of 7 semitones (“Perfect 5th” as in above examples) and also 5, and 2 semitones in the harmonic continuity of the passage. (Also note 7 + 7 = 14; 5 + 2 = 7; 5 +5 = 10; 2 + 12 = 14; etc.)

    To illustrate that this is not all just theoretical, here is a simple etude composed using exactly the constellations and successions explored in Examples 12 and 17. It took only about an hour to compose this minute and a half in Sibelius. The title: the constellation Pleiades (“Seven Sisters”) is a tight cluster of 7 stars tagging along in the winter sky with Taurus as the Zodiac sails westward every night.

    12-tone sets

    Let’s keep going. How about designing a succession of three four-pitch constellations, so that all 12 pitch classes of the chromatic scale are included but none repeated? (Traditional terminology calls such a set a 12-tone aggregate.)

    three sets make a row

    Constellations a) and c) are different “chord voicing” of the same scale pattern, 2 4 2 . Both scale patterns and all three interval stacks are symmetrical. And they all contain two 6-semitone “tritones,” giving the whole succession the tritone’s quality of ambiguity and the character of the succession a feeling of mystery.

    Altering arrays

    Similarity of interval patterns can build coherence in a stream of constellations. Beyond functional common-practice harmony, this is a kind of process that composers of the 20th century and today can use to create a “new tonality”.

    Possible operations to transform an interval array into a closely related array:

    OPEN — Expand an interval by an octave, adding 12 semitones

    FUSE — Join two adjacent intervals to make a larger interval, the sum of their sizes

    DELETE — Remove an interval, shortening the stack’s height

    SUBDIVIDE — Insert a pitch to divide an interval into two smaller intervals, whose sum equals the original interval

    PROPOGATE — Append or insert an interval of a size already present into the stack

    INVERT — Reverse the registrar order of the stack — turn it upside down

    alteration examples

    There are operations that more significantly alter the character of the interval array.

    REDISTRIBUTE — Fuse two adjacent intervals into one larger interval then re-subdivide it into two different smaller intervals

    SHRINK / STRETCH — Alter one interval size by other than an octave, leaving others unchanged

    COMPRESS / EXPAND — Alter all intervals in the stack by adding or subtracting each by the same number of semitones, or multiplying each by a constant

    These alterations are listed in order, from the mildest alteration producing a similar array (redistribution) to the most dramatic producing a substantially different array, compression or expansion of the whole array (preserving little from the original but its symmetry). Here is an example employing these altering transformations.

    more alterations, with common-tone connections

    The other element of coherence in this example is the many common-tone connections between one chord and the next, establishing a slow-moving stability. Another example of the same interval stacks, same succession of alterations, but choosing transpositional level of each constellation to create as many 1-semitone voicing connections as possible (10 such voicing connections in the following example) makes the con stream’s sense of progressive change stronger.

    more alterations, with semitone connections

    Finally, another example etude, using this last constream . . .

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    9. LINE

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  • Mapping Music 3. CHANGE

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

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

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

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

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

    Op.135 excerpt harmonic reduction

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

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

    Beyond meter

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

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

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

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

    Elegy rhythm graphed

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

    Free time

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

    Spatial notation

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

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

    Somniloquy notated spatially

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

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

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

    Heraclitean Vortex excerpt

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

    time graph of 11 moments in Heraclitean Vortex excerpt

    Prime time

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

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

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

    repeating prime numbers interact

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

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

    Night Sky score

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

    Night Sky score abstracted

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

    Beyond Time

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

    Lei Liang, My Windows (2007)

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  • Mapping Music 2. RHYTHM

    Rhythm is a stream of event durations. Often repetitive but potentially elastic, rhythm can be steady, as a simple march, or volatile, as a melody that hovers, trots, then suddenly starts running.

    Defining some basic terms:

    TIME POINT — a precise moment in time marking the beginning of a span of time extending until a comparable event marks the next time point

     TIME SPAN — the duration of time from one point to the next point marking a comparable event

     STREAM — a series of events formed by the elements of consecutive groups

     PERIODIC — a stream of events whose elements exhibit equivalent time spans

     PULSE — ungrouped periodic time marking in a speed of about 48 to 1,000 per minute (above 1000 per minute becomes pitch)

     METER — a nested hierarchy of periodic streams of of time points

    Nested means that two or more time-spans at a quicker level are synchronized with the next longer durational level. Two eighth-notes “fit” within the metric time-span of one quarter-note, for example. Meter creates the perception/expectation of events happening at periodic time points on more than one level of speed/duration.

    BEAT — pulse that constitutes the primary level of a meter’s hierarchy, the connector between both quicker subdivided and slower grouped levels

    PACE — general quickness / slowness of rhythmic activity in a line or whole fabric

    HYPERMEASURE — extension of metric hierarchy grouping measures, typically in two- or three-measure units

    PHRASE — grouping of molecules, shapes, motives, chord changes, etc. in a coherent stream, typically the length of the human breath

     PERIOD — grouping of phrases, typically concluded with a significant harmonic cadence and/or melodic sense of arrival

    Pulse and Beat

    Underneath the sense of a beat, pulse is primordial periodicity – clapping hands, stomping feet, banging rocks, running strides – features we inherit from our primitive musical ancestry and still use to organize our musical actions.

    Susie Ibarra – Sky Islands (2025)

    A rapid pulse can drive music with frenetic motion. Much of jazz relies on this power source. The tension between a fast, steady pulse vs. unpredictable accents (syncopations) and other turbulence generates energy and excitement.

    SYNCOPATION — an accent not synchronized with the beats, or a note length shifted from its regular metric starting point

    Listen to how a relentless, fast pulse drives this music.

    Julia Wolfe – Believing (2012)

    Rhythm makes meter, Meter drives rhythm

    Rhythm first generates meter by marking periodic time points at different levels of speed. The marking is just the moment of initiation of each note but also accents, chord changes, etc. at broader time levels. As a great example, let’s use a famous theme from a piece nicknamed for a planet:

    Jupiter Symphony theme

    This melody first establishes periodicity of half-notes but nothing shorter for a while. This could be any duple meter. Once we hear these first four equal notes, we tend to perceive them as two pairs, establishing the whole-note measure-level meter. We feel the time point that begins the third measure on two levels of periodicity even though no note happens to mark it. In this third measure, eighth-notes then divide those half-note time spans into four parts. In the middle of measure 3, a quarter-note fills in the missing level of meter. Finally, in the last beat of the measure, a burst of sixteenth-notes establishes the fourth, quickest periodicity of the nested hierarchy. This example extends the hierarchy to show 2-bar hypermeasures in a 4-bar phrase.

    Once the tune repeats, our sense of that nested hierarchy of speeds is in full cognitive play. While jumping from metric level to level, half-notes to eighth-notes to quarter-notes then 16th-notes, the melodic rhythm undergoes a compression of pace, moving from longer rhythmic values to quicker and quickest, while the Allegro tempo does not change.

    More definitions

     

    CYCLE — the duration of equivalent time spans in a periodic stream of events

     ELEMENT — a point and following time span of an event or group of events, relating to other consecutive elements to form a group at a broader (slower) level of time

    COMPRESSION — an element of a group or stream is a shorter span than the previous element

    Example: rhythm changing from half-notes to quarter-notes to eighth-notes, compression of pace.

     EXPANSION — opposite of compression, elements of a group or stream are longer spans than previous elements

     ACCELERATION — consistent successive small compressions of beat or pulse

    PROPORTION — relationship of time spans expressed as a ratio, reduced to smallest-possible integers (whole numbers)

     RHYTHMIC GROUP — consecutive related elements, with a point of initiation and accumulated durational span

     RHYTHMIC RANGE — ratio of longest duration to the shortest

    In the Mozart Jupiter example above, the rhythmic range is 1:1 conformity for the first two bars, then 4:1 with three different note values in the third and fourth bars.)

    RHYTHMIC VARIETY — number of different note values in a stream of notes

    In the Mozart Jupiter example above, the rhythmic variety is 4 (halfs, quarters, eighths, sixteenths).

    Stress and accent

    Classic poetry classifies each syllable grouping (a “foot”) in a line of poetry according to which syllable in the grouping is stressed or longer length (agogic stress). (The last, a “reversibrach,” is my addition to complete the set of possibilities for musical purposes.)

    Metric “feet” in poetry

    Rhythmic molecules, groupings of two or more notes, can be similarly characterized, though with more stress possibilities:

    • Strength (accent) stress
    • Length (agogic) stress
    • Metric stress (strong beat vs. weak beat; on the beat vs. off-beat)

    A molecule can have one stress pattern in accent contradicting a different stress pattern in length or metric placement. Another familiar Mozart example: the opening themes of the first movement of Mozart’s Symphony No. 40 in G Minor. The first theme is fast (allegro) but steady (low elasticity).

    Symphony No.40 1st theme

    Analyzed as three rhythmic molecules:

    Three quick, predictable anapests, in which the longer, metrically-accented note is precisely the same length as the pair of shorter notes that lead to it. It also shows a narrow rhythmic range, 2:1, and little rhythmic variety, with only two rhythmic values, the eighth-note and quarter-note.

    Now the contrasting second theme, a soaring oboe line, highly elastic in rhythm.

    Symphony No.40 2nd theme

    This phrase launches with a long trochee, beginning-stressed in both length and metric placement. This rhythm uses four different note values, the longest of which is 12 times the length of the shortest. The first note sounds stretched, like an elastic band that is then released after the third note, unleashing the quick notes that scamper to the last.

    This is just a glimpse on the micro-level of rhythmic contrasts and a temporal elasticity that propels the exciting roller-coaster allegro opening of this great symphony. All our perceptual and gestalt faculties are engaged in a grand game of play with time.

    Molecules

    Here are the opening notes of three famous 20th-century unaccompanied flute pieces, by Debussy, Varese, and Berio, respectively.

    Each uses a three-pitch motive that, when analyzed as a pitch-class set, is a segment of the chromatic scale.

    • Syrinx: A Bb B = +1 +1 semitones chromatic scale pattern
    • Density 21.5: E F F# = +1 +1 semitones, same scale pattern
    • Sequenza: G G# A = same +1 +1 chromatic scale pattern (G displaced by an octave)

    Shown above in their chronological order of writing, it is likely that one influenced the next, and it the next in a chain of evolving variation. While this shared pitch-class-set characteristic is the usual basis for comparison, it is also interesting to compare the rhythmic molecules of their generating motives.

    Syrinx starts with a clear front-stressed dactyl, repeated then echoed in bar 2.

    The Density 21.5 motive is more complicated. Its opening three notes, from a short/long durational standpoint, is an end-stressed anapest. But the first note, though short, has the metric accent, being the only note of the three written on a beat. That first note is also emphasized by the tenuto mark. Those accent factors point toward a front-stressed dactyl like Syrinx. The next three notes starting with the C# are a more ambiguous stress shape.

    The opening three notes of Sequenza have no clear metric or dynamic accent difference; they are all strong. But by duration, the third note is “longer” in effect in the time stream (agogic stress), as the silent time after it, before the next note comes along,is longer. Also the third pitch, G, is much higher, giving it a registral or contour accent. This 3-note molecule is an end-stressed anapest.

    In all three pieces, however, the sense of simple repetition of matching poetic feet is not established or maintained. It is more productive to understand throughout each piece how rhythmic range and variety expand and contract and pace intensifies or subsides.

    © 2026 – All Rights Reserved

    Thomas S. Clark

    –

    Continue reading Mapping the Music Universe:

    3. CHANGE

    _____________

    TClarkArtMusic.com

  • 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.