MAPPING MUSIC

Tag: Schoenberg

  • Mapping Music 11. TEXTURE

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

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

    Sound color

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

    instrumental color rainbow

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

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

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

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

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

    Augusta Read Thomas – Dance Mobile (2021)

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

    Symmetry

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

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

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

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

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

    Webern 13-pitch constellation

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

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

    Op. 21 color map

    Pointillism

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

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

    Andromeda sound color map

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

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

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

    Here is the whole colorfully pointillistic 3-minute study:

    Sound Mass

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

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

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

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

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

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

    Schoenberg Farben scoring

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

    Farben constellations

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

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

    framing dissonances

    Foreground / background

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

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

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

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

    Hauptstimme handoffs

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

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

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

    Julia Wolfe – Anthracite Fields I: Foundations (2015)  

    Galaxy groups

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

    Laniakea score excerpt

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

    Laniakea score abstracted

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    Thomas S. Clark

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    Continue reading Mapping the Music Universe:

    12. FORM

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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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    Thomas S. Clark

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    9. LINE

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  • Mapping Music 4. TUNING

    “To understand the Universe,

    you must understand the language in which it’s written,

    the language of Mathematics.”

    — Stephen Hawking

    Galileo revolutionized astronomy, in part by using a new tool: the telescope.

    Schoenberg revolutionized harmony by evolving an existing concept, the chromatic scale, into a new tool: the 12-tone scale, and devised a new compositional tool of the 12-tone row.

    Allen Forte took Schoenberg’s ideas to another level of abstraction: defining Pitch Class and applying basic math to the 12-tone universe.

    Chrisman focused on the interval essence of pitch patterns: defining the “successive interval array.”

    I am merely another explorer using their maps but choosing my own creative path. In doing so, I will define some of my own terms, while adapting and clarifying some established terms that fit what I’m thinking and expressing.

    From Tuning to Tonality

    We think of traditional common-practice Tonality of the 17th through 19th centuries being synonymous with the major and minor scales. But there’s more to traditional common-practice Tonality than just the scale. Here are the four basic factors that determine any tonal design:

    SOURCE SCALEHARMONIC TYPETONAL CENTER
    ancient modeperfect intervalsfixed by mode
    Major / minortriadmodulatory shifting
    extended chromaticextended triadpolytonal centers
    exotic / syntheticnon-triadestablished contextually
    12-tonediversenone

    tonal design factors

    As you can see, there is much to explore: scales, modes, intervals, consonance . . .

    Tuning

    Taking the overtone series and partial vibrations as a natural acoustical model, Pythagoras identified pitch intervals as simple integer ratios of lengths of a vibrating string. The same ratios describe frequency ratios.

    fundamental pitch C and overtones

    For example, what we call a Perfect Fifth, the interval of the Third Partial to the Second Partial of a natural overtone series, is a 3:2 ratio. Such natural tuning is always employed by orchestras, bands, and a cappella choirs.

    • Octave = 2:1
    • Perfect 5th = 3:2
    • Perfect 4th = 4:3
    • Major 3rd = 5:4
    • Minor 3rd = 6:5
    • Major 6th = 5:3
    • Minor 6th = 8:5
    • Major 2nd = 9:8

    This approach requires, however, that intonation be constantly adjusted as the key changes or tonal context shifts. For a keyboard that can’t make those adjustments, the fixed tuning devised in the 18th century, called Equal Temperament, compromises the Perfect Fifth, shrinking it from a 1.5 ratio to 1.498307 so that it and all other intervals are very slightly but equally mis-tuned in every possible key or tonal context. The ratio for a semitone is derived mathematically from the 12th root of 2: 1.059643094. That ratio, multiplied by itself 12 times, results in 2.000, the ratio of the octave.

    comparing tuning systems

    While “chromatic” historically meant extending a key with accidentals — temporary extra sharps or flats — now we refer to the 12-half-step scale as the chromatic scale. Two pitch names for the same piano key — C-sharp or D-flat — are said to be enharmonic and considered equivalent, almost interchangeable.

    Equal Temperament became the basis for the 20th-century system of 12 equal semitones per octave, the basis not only for all keyboard instruments but also for harmonic theory in the post-tonal world of 12-tone music. We should not forget, however, that choirs, orchestras and bands still use the purer natural tuning, even with music that has no key signature.

    Other tuning systems

    Long before equal temperament, the Chinese culture developed several systems. A fascinating history is described in Gene Jinsiong Cho’s monograph, LU-LU: A study of Its Historical, Acoustical and Symbolic Signification (Caves Books, Ltd., Taipei, 1989). Cho (a music theory professor colleague at the University of North Texas) explains the LU system from the Chin Dynasty, which extended beyond 12 increments in an octave as far as to the arcane realm of Jing fang’s sixty LU series.

    In the West and into the 20th century, two American composers experimented with microtonal tunings splitting the octave into finer increments than our 12 semitones.

    Working with American Lou Harrison, California composer Harry Partch (1901-1974) devised his own tuning system with 43 increments, described in Genesis of a Music (1947). The system necessitated invention of specialized percussion and string instruments to precisely intone the sounds, which felt exotic both in tuning and sound quality.

    Harry Partch – Castor & Pollux (1952)

    University of Illinois professor Ben Johnston (1926-2019) wrote music for standard orchestral string instruments using the ancient just intonations of Pythagorus. This involved specifying pitches microtonally slightly higher or lower than the equal-tempered standard pitch classes – a notational challenge of pitch-adjustment symbols.  

    Ben Johnston – String Quartet No. 7 (1984)

    In the 21st century, Japanese composer norokusi has produced a broad catalog of microtonal music, apparently using a 17-increment division of the octave.

    norokusi – Piano Sonata n.718 (2018) 17EDO/TET

    Such complex systems as described above never became mainstream. The vast bulk of 20th-century and now 21st-century music is based on the equal-tempered 12-increment system found on a well-tuned piano, with subtle adjustments by orchestral strings, wind bands and a cappella choirs to momentarily purify some sonorities.

    © 2026 – All Rights Reserved

    Thomas S. Clark

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    Continue reading Mapping the Music Universe:

    5. SCALES

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