Tag: symmetry

  • 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

    © 2026 – All Rights Reserved

    Thomas S. Clark

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  • Mapping Music — PRELUDE

    The heavenly motions are nothing

    but a continuous song for several voices,

    perceived not by the ear but by the intellect,

    music that sets landmarks

    in the immeasurable flow of time.”

    Galileo

    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.

    Metaphor

    Musical sounds mark points in time, like stars. They form immediately into recognizable patterns we call chords, melodies, rhythms, memorable themes. They convey a sense of motion, time surging forward or slackening in our perception of their well choreographed parade.

    Astronomers observing and mapping (recording) the myriad points discovered that some of the stars are actually whole galaxies, with exotic forms of spirals and clouds. They observed through the color of the light that all these objects are racing away from us and each other in an expanding universe.

    Mapping music means cataloging many possible patterns, distinguishing their contrasts and commonalities. We will explore how to measure and compare the periodic rhythmic streams of musical events and their changing momentum. We will define and employ a simple but powerful math tool for cataloging and then creatively sculpting with all natures of harmony and melodic line in our 88-key chromatic universe. We will explore how master composers weave colorful fabrics and grand structures from skillfully crafted materials.

    Pursuing periodicity

    My music-mapping Periodicity Project began in 2021 as a comprehensive catalog of musical patterns and processes, meant to provide simple tools for understanding the complexities of modern music. It grew into this book, Mapping the Music Universe, written for anyone who is curious about how music works, especially in the 20th-21st-century modern and “post-modern” eras. For me as a composer, it is also an exploration of how some less traveled conceptual paths lead to interesting creative possibilities.

    In 1989 I co-authored a conceptually ground-breaking composition textbook with Larry Austin, Learning to Compose: Modes, Materials, and Models of Musical Invention. My next book, ARRAYS, was an aural skills workbook covering basic modal, tonal, and “post-tonal” music of the Renaissance through the Twentieth Century. Mapping the Music Universe draws in part on the ideas and approaches of both these now out-of-print publications.

    A common assumption within Western culture is that Science is all about observation, measurement, precision, and mathematical rigor . . . and Art is all about the “i” words: imagination, inspiration, intuition, improvisation. Science is Deductive, art is Creative. Our culture has begun to recognize the commonality of all these intellectual strengths, that the best Science can be creatively intuitive and great Art can be rigorous.

    Pioneer map makers

    As an educated musician and professional composer, I also have long been deeply interested in science, especially astronomy. Having read a great deal of general science writing, I am inspired particularly by ground-breaking pioneers who methodically and comprehensively mapped the possibilities of their particular field.

    Johann Joseph Fux — wrote Gradus ad Parnassum in 1725, codifying basic contrapuntal principles of Renaissance music.

    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.   

    Meriwether Lewis — kept extensive journals of the 1804-1806 Lewis and Clark Expedition, documenting and illustrating the discovered new world of the Northwest.  

    Dmitri Mendeleev — devised a “periodic table of the chemical elements,” published in 1869, providing a solid basis for modern chemistry through its graphic and organizational genius.

    Amédée Mouchez — launched an ambitious international star-mapping project (Carte du Ciel) in 1887 at the Paris Observatory.

    Henrietta Swan Leavitt — worked at the Harvard College Observatory as a “computer,” examining thousands of photographic plates from telescopes to measure and catalog the brightness of stars, identified 1777 variable stars.

    Lawrence Herbert — invented the Pantone system in 1956 to systematize color for printing ink and fabrics.

    Allen Forte — published an article in 1964 that launched musical set theory, defining, classifying and comparing all possible collections of “pitch classes” drawn from the equal-tempered 12-tone chromatic galaxy.

    The work and insights of the two on the list representing rigorous study of music, Fux and Forte, were part of my formal education in music and later an integral part of my teaching of composition and music theory.

    Maps

    Carte du Ciel was an ambitious second phase of an international star-mapping project initiated in 1887 by Paris Observatory director Amédée Mouchez.  A new photographic process revolutionizing the gathering of telescope images inspired the first phase, the Astrographic Catalogue of a dense, whole-sky array of star positions. Carte du Ciel, never completed after 70 years, used the Catalogue as a reference system for a complex survey of the vast field of even fainter images.

    Celebrating the grand metaphor relating astronomy to art music, here is my 8-minute computer-music sound sculpture. In the music, ghostly wisps of sound are punctuated by brighter bursts, clustered in a natural, not-quite randomly dispersed texture.

    _______________

    Looking ahead

    The blog-post chapters of Mapping the Music Universe will proceed in three broad phases, progressing logically from fundamental — time and periodicity — to pitch space, then to larger structures, texture and form. Within each phase, various topics are presented in a progressive order, but jumping in at any point is fine.

    Terms will sometimes be freshly coined. Graphic figures will include notated musical examples, tables, and graphic illustrations of patterns and their relationships. Big Ideas — Periodicity, Complexity, Symmetry, Relativity — will be explored using precise mathematical arrays as well as broad metaphors. Newly composed sample etudes will illustrate aurally.

    Along the way, “Map Labs” will present step-by-step recipes to compose simple pieces based on models of different compositional genres. Each Lab includes an original sample piece following the Map Lab guidelines, illustrating one possible creative outcome.

    Welcome! Join this creative journey of discovery . . .

    a composer’s expedition.

    © 2026 – All Rights Reserved

    Thomas S. Clark

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