“Mapping” has double meaning. A road atlas measures and records all the routes through a given territory. But we also call “mapping” the creative act of planning out a journey, using map information to choose between many possible routes. Composers use an array of processes to map out a musical journey. Designing a piece entails making a storytelling rhetoric, a pacing plan, and an architecture of interrelated components.
Each Map Lab in Mapping the Music Universe presents step-by-step recipes to compose simple pieces based on models of different musical genres. Each lab also includes an original sample piece following the Map Lab guidelines, illustrating one possible creative path and outcome.
Try your own experiment with any of these lab projects:
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.
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:
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.
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.
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 . . .
Some points of starlight are actually double stars or star clusters, as revealed through a sufficiently powerful telescope. Borrowing that term, one particular type of musical pitch constellation arising in the 20th century involves sounding adjacent scale steps together as a simultaneity.
CLUSTER — a constellation presented harmonically consisting of adjacent scale steps separated by small scalar intervals of one or two semitones.
If the intervals are one-semitone half-steps from the chromatic scale, the resulting harmony is intense, dark, dissonant. If the separating intervals are mostly whole steps, the quality can tend to be like bright glowing light.
It turns out that the diatonic scale is rich with cluster possibilities.
diatonic clusters
The names are borrowed from the Greek names of modes. One cluster array that can be readily found throughout the octatonic scale but not possible within a diatonic scale is the 1 2 1 array, since the diatonic scale has no one-semitone intervals that close to each other. One other cluster array not found in the diatonic scale is the intense, gritty 1 1 1 chromatic-scale array described above.
Harmonic constellations built from clusters can be still and radiant or animated by rhythmically active lines close together in pitch space. Here is a composed example that does both.
Photons excerpt
Symmetrical arrays
Dorian and Lydian clusters are symmetrical. Their scale-pattern arrays — 2 1 2 and 2 2 2, respectively — are the same when inverted or reversed. Many other interesting pitch constellations have this property.
Here is a sampling of other, taller symmetrical constellation arrays, reading the same from top to bottom as bottom to top:
many symmetrical arrays
Note that although each is a 4-pitch constellation, two of them (4 4 4 and 8 4 8) contain an octave and thus only three unique pitch classes.
The example below explores the first four arrays listed above. The stacked interval array of 4d is: 7 4 7. Note too that this constellation contains two 11-semitone intervals (+7+4 and +4+7) — in example 4d C up to B and G up to F#; and one very large interval of 18 semitones, C up to F# in the higher octave.
four symmetrical arrays
Three of the four could be analyzed in triadic harmony: 1c as an Ab Major-Major 7th chord in first inversion; 2c as an A minor-minor 7th in first inversion, or a C Major triad with jazz added 6; and 4c as a C Major 11th chord with 3rd and 9th missing! And 3c could be seen as a rearranged voicing of a segment of the “Circle of Fifths”! Obviously, I don’t recommend such contortions of traditional harmonic analysis to explain these beautiful, symmetrical constellations.
Each constellation’s successive-interval array, its stack, is symmetrical under mirror-inversion or palindromic — that is, its interval stack reads the same lowest-to-highest or highest-to-lowest: 7 1 7 and 7 4 7 are two of my favorite constellation stacks.
These interval stacks were chosen here for two of my particular interests. Each features the “Perfect-5th” 7-semitone interval at top or bottom, with a smaller interval in the middle. The Perfect 5th has a stable, rooted quality, but with two “roots” in the harmony, the overall stability of the sonority is compromised — complex yet balanced. It is like a “double star” in astronomy, to further pursue my constellation metaphor.
Acoustic quality
We started with clusters, sonorities that would be traditionally considered highly dissonant. To assess a constellation’s quality or sound character, we will transition from the concept of consonance and dissonance to an assessment of acoustical complexity.
When two pitches sound together, they make a harmonic interval, but also their distinct overtone series are interacting. In the purer sounding intervals, this interaction is mainly a closely compatible one, with some overtones matching. An example, take a Perfect 5th, C up to G. Their overtones are:
overtone match for a Perfect Fifth
The matching or interfering overtones make more of a difference with the lowest partials, as the higher overtones are fainter and fainter higher up in the series. So the c’ 4th partial of C interferes with the 5th partial b’ of the G overtones; but that interference is fainter than the lower g-to-g match. This gets scientifically and mathematically complex to calculate, as we will tackle in a while below.
For now, the ancient classification for counterpoint is accurate enough to adapt: Perfect Consonance, Imperfect Consonance, Dissonance . . . though I will distinguish between mild dissonance and strong dissonance.
PERFECT CONSONANCE — intervals P8, P5 and P4 (12, 7 and 5 semitones) — “pure”
IMPERFECT CONSONANCE — intervals of major and minor 3rds and 6ths (3, 4, 8 or 9 semitones) — “triadic”
MILD DISSONANCE — intervals 2 semitones different in size from a unison, octave, or double octave (2, 10, 14, 22, 26 semitones) and the tritone (6 semitones)
We can use these distinctions to come up with an assessment of the general harmonic complexity of a constellation’s intervals.
PURE — containing no intervals except perfect consonances
SIMPLE — containing no mild or strong dissonances
MODERATE COMPLEXITY — containing at least one mild dissonance but no strong dissonance
STRONGLY COMPLEX — containing at least one strong dissonance
Examples of increasing complexity:
It is important to note that strongly complex does not mean “unpleasantly dissonant.” The Major-Major 7th chord in this category (C E G B) is quite a beautiful harmony. And the last two complex examples, quartal and quintal chords, are the sturdy mainstay of 20th-century American composers such as Copland.
In a recorded excerpt of Tyshawn Sorey’s Pulitzer Prize-winning Adagio (2023) for saxophone and orchestra, beautiful sonorities are quietly complex, tensely dissonant, dark and mysterious in their lyric unfolding. Like dark clouds, some morph to reveal brighter sounds, even simple triads. While there is no sense of any “chord progression,” there is a feeling of impending change in the air.
In notes on a recent composition, Frost Serenade, I described “changing tonal temperature.” Here is a deep dive into what that meant.
The metaphor of tonal color and temperature has to do with what we normally call consonance and dissonance in a chord or other harmonic entity. Centuries-old tradition classified musical pitch-intervals as pure, perfect consonances (“Perfect Fifth” and “Perfect Octave” for example); major or minor (exp. “Major Third” or “minor Sixth”); or problematic (“Augmented Fourth” and “diminished Fifth”). Some major and minor intervals (thirds and sixths) were considered imperfect consonances; the others (seconds and sevenths) were considered dissonant. Every music student learns these categories while studying 16th-century model counterpoint.
Using the color spectrum in temperature order:
harmonic color spectrum
Let’s convert the consonance/dissonance concept, going back to a pitch-interval’s acoustic complexity. Reviewing what was explained above: every musical tone has a fundamental pitch, plus faint overtones that give the sound its color. They are of fading intensity and felt (as color) more than actually heard as distinct pitches. Discovered by Pythagoras as partial vibrations in whole-number fractions, the overtones are always in a fixed interval ladder, rising from the fundamental: Up an octave, then a Perfect Fifth, then a Perfect Fourth, a Major third, minorthird, then to the eccentric seventh partial, which is out of tune by our scale-trained pitch perception (and shown a pale gray below), and on to the eighth partial, which is three octaves above the fundamental. (An octave is a multiply-by-2 operator, so partials 2, 4, 8, and 16 of the C overtone series are also the pitch-class C. Likewise, partials 3, 6, and 12 are all octave related.)
Two different fundamental pitches sounding together each bring into the acoustical mix their distinct overtones. The overtones from one either match (simple) or clash with (complex) overtones of the other. This is what makes the sonic complexity or perceived purity of the interval between two fundamental pitches. Using this relationship, we theorize that the higher we need to go to start finding matching overtones between the two pitches, the more complex is the interval. Following this logic, here is an overtone-match analysis of all harmonic intervals smaller than an octave. (We’ll show these horizontally to fit better what would otherwise be very tall slender graphics!) Each interval is shown from a fundamental pitch C up to a higher pitch.
PERFECT CONSONANCES
overtone match for perfect consonances
The rather pure Perfect Fifth interval between fundamental pitches, C up to G, matches overtones at G’s partial 2, a low level in the series, matching the C’s partial 3. The interval makes four such matches in this lowest-two-octaves span. The pitch match up of the G’s 2nd partial with the C’s 3rd partial (both are the same pitch, G) will be duplicated in all higher octaves, making this an acoustically simple interval. The two pitches’ overtones mostly match and don’t interfere with each other much.
IMPERFECT CONSONANCES
overtone match for imperfect consonances
The triadic consonant Major 3rd interval between fundamental pitches, C up to E, matches overtones at a somewhat higher level in the series, partial 4, and makes two matches in this lowest-two-octaves comparison.
DISSONANCES
overtone match for dissonances
The dissonant minor 7th interval between fundamental pitches, C up to Bb, matches overtones makes only one match in this lowest-two-octaves comparison, at partial 5. That means its harmonic quality is more complex, with most of the lower overtones interfering, not matching. Not a strong dissonance, but more complex than the others.
By contrast, with the more complex Major Seventh interval (ex. C up to B), you have to go all the way up three octaves to the B’s 8th partial (matching the C’s 15th partial!) to find an overtone that matches and doesn’t conflict/interfere. The Major 7th interval can be considered much more complex at a rating of 8 than a Perfect 5th at rating 2.
The most complex interval analyzed, the minor 2nd, clashes all the way up until the 15th partial.
A colorful summary depiction of this Pythagorean analysis of harmonic intervals looks rather like a modern-day sound mixing board.
overtone matching for 13 intervals
Summarizing the analysis with a complexity rating number for each interval:
interval complexity ratings
Now we can add up the ratings of each interval in a chord and take an average complexity quotient. And we can think of complex as darker than simple, or we can invoke the color spectrum. In digital photo imaging, we use a temperature metaphor, seeing red as warmest (infrared heat) down through orange, yellow, green, down to blue, the coolest. The “hottest,” most complex harmonic interval is the minor 2nd. The “coolest,” purest (other than the octave) is the Perfect 5th.
The intervals in the following example are shown in semitones. Each chord has four pitch classes and six intervals between them. The Blue chord has an average complexity rating of 3.8. Green chord is slightly more complex, at 4.3. Yellow, which includes the more complex 11-semitone Major 7ths, rates 5.5. And Orange, with the only minor 2nd 1-semitone hot dissonance, is warmest at 6.2. Try to hear the differences. (No attempt here to demonstrate the red-hot complexity of 10 or higher for a cluster chord!)
examples of four temperatures
The following demonstration phrase uses hose four chord types to build a progression of tonal temperature colors. Again, as you listen, try to feel the temperature warm up then cool back down.
Pursuing our grand space metaphor, here is an important new term:
CONSTELLATION — a group of pitches occurring in a perceived relationship, either vertical (a chord simultaneity), horizontal (a segment of a melodic line), or diagonal, a combined collection of pitches from various lines sounding in temporal proximity.
This is intentionally a broadly inclusive concept. Larry Austin and I first coined the term in our 1989 book, Learning to Compose. A constellation can be any number of pitches, but those of three to six pitches are most manageable to analyze, categorize, and manipulate.
In Mapping Music 5. SCALES, we explored pitch classes (all the D’s in any octave, for example). For now, let’s not go there. A constellation can be very tall, spanning even five octaves, or very narrow, as in three or four close-together pitches well within one octave. (As a chord, we might call these a “cluster.”)
Common names for types of pitch grouping, “sonority,” “chord,” “harmony,” “melodic motive,” “arpeggio,” or “chord voicing” will all be considered manifestations of a pitch constellation.
Jennifer Higdon’s 2007 work, Percussion Concerto, driven by rhythmic vitality, romps through a dazzling variety of pitch constellations. Most are more complex sonorities consisting of 4 different pitches, drawn from diatonic scales but extending beyond the basic triads of the scale’s traditional harmony.
Jennifer Higdon – Percussion Concerto (2007)
Interval arrays
NOTE: In place of traditional interval names, which literally don’t add up, we will consistently measure every interval by how many chromatic semitones (half-steps) it spans.
When pitches of a constellation are considered out of time, like a chord, and rearranged from lowest to highest, we can study their harmonic structure. The stack of intervals makes a successive interval array of semitones from lowest to next, on up to the top.
For example, the following line of four pitches, in order E – B – C – D, rearranged lowest to highest, yields C – D – B – E. Its interval stack = 2 9 5. (Going back to 5. SCALES, the four pitch classes can be derived from the set / diatonic scale-pattern 1 2 2.)
sample constellation 2 9 5
This constellation’s particular pitch-pattern shape shows a stack of successive intervals from lowest to highest: 2 9 5.
INTERVAL ARRAY — stack of intervals that identifies the constellation’s particular intervallic shape in vertical pitch space, listing the successive, additive upward intervals from lowest to highest pitch
Note: I tend to use “interval stack” and “successive upward interval array” interchangeably. If we wanted an acronym, how about Successive Upward Interval Series Stack — SUISS? No, maybe Vertical Interval Array — VIA? But vertical is not quite right, as the pitches might occur in musical context diagonally in 2-D pitch-time space and only be vertical when theoretically aligned as a chord stack. So let’s stick with interval array — and since conventional music theory doesn’t use the word for anything else, let’s just call it an ARRAY.
The constellation above also contains a “Major 7th” 11-semitone interval (+2+9=11), C up to B; a 14-semitone Major 9th, D up to E; and one very large interval of 16 semitones, C up to E in the next higher octave.
sample constellation 2 9 5
Below is a sample etude made with just this one 4-pitch constellation and its transpositions (bars 4-6 two semitones down), all with the same interval stack, 2 9 5, or its upside-down inversion, 5 9 2 (bars 12-14 bass clef).
Pisces etude
The etude is based on this 2 9 5 array. Bars 11 through 14 in the right hand are a constellation with a slightly altered array: ascending F# G# E A = interval stack 2 8 5, transformed from 2 9 5 by shrinking the middle interval of the stack by one semitone. Why? Sticking with 2 9 5 would have made e# or f and then b-flat on the top, not such great counterpoint against the b-natural in the lower line. And why not? The minor-9th interval A up to B-flat, 13 semitones, is a particularly gritty, unpleasant dissonance.
One example with pitch classes would be [F B E], in which F up to B is 6 semitones, B up to E is 5 semitones, and F up to E is 11 semitones. This example with all three pitch classes drawn from a C major scale illustrates that [6 5] is correctly shown in white as a diatonic pattern, despite the fact that it is not commonly used as a harmony in common-practice tonal music other than as a Mahler-style suspension.
In the table below, each column groups stacks of the same height – each stack also forms a larger interval (not shown) that is the sum of the adjacency intervals shown. For example, reading bottom up, the stack 6 5 also forms an 11-semitone interval, the stack’s total height. All 3-pitch-class interval stacks:
3-pitch interval-stack arrays
It may be helpful to see example pitches on a staff illustrating all these possibilities. Each line below shows a family, one Forte set class: first Forte’s “best normal order” with example pitches, then their chord voicings with stacked-interval sizes; then the set’s inverse, if there is a unique one.
Here the color shadings denote special degrees of interval complexity: RED = sharply dissonant; ORANGE and YELLOW = mildly dissonant; GREEN = minor and major triads; BLUE = quartal/quintal chords of P4 and P5 intervals.
3-pitch arrays, families 1-6
3-pitch arrays, families 7-12
As with scale-pattern maps, these maps and their notated lists represent the entire chromatic universe of possible constellations within a two-octave range. Each could be expanded by adding an octave to any stacked interval. And of course, each can become a line, a chord, or a temporal proximity of pitches in a texture.
What is a scale? Its essence is an interval pattern, selecting which pitches out of the entire chromatic possibilities become scale steps. Successive interval arrays are a vivid way to describe its pattern:
SCALE PATTERN — periodic interval pattern that cycles through each octave, defining which pitch-classes from the 12 possibilities are degrees of the scale
In that sense, it is a theoretical circle, starting over in each octave — or more imaginatively, a spiral. Let’s visualize the natural-note white keys on the keyboard, a prime example of the ubiquitous diatonic scale, as a circle.
diatonic scale circle
Now an unlooped visualization as stair steps, rungs on a spiral ladder:
diatonic scale cycling through three octaves
Anyone familiar with the white and black keys of a piano will recognize this pattern!
Chroma
Almost all scales in both Western music and other art-music traditions are built on the framework of octave equivalence, the close affinity of two pitches that are one or more octaves apart. We give them the same pitch name – all called “C” or “F#” for example. This makes the circular nature of a scale, that its pitch names and the intervals between them start over at the octave and repeat.
We also have the feature on an equal-tempered piano that one black key produces a pitch with two possible names depending on the scale in which they appear. For example, the D# seventh scale degree in an E Major scale is the same piano key as an Eb, the fourth scale degree in a Bb Major scale. The two pitch names are said to be “enharmonic.”
When a melodic line in an all-white-key C major scale introduces an F# for color or to temporarily alter the interval terrain, we call it a chromatic tone, after the Greek word for color, chroma. Now we have a comprehensive scale of all possible pitches. Going further, theorist Allen Forte defined a way to reduce all the pitches in an entire eight-octave chromatic pitch space into just twelve categories:
PITCH CLASS — a set of all pitches that are octave and/or enharmonically related
He gave them pitch-class numbers 0 through 11.
chromatic scale
In the advent of computer systems to produce, edit, and analyze musical sound, a sound’s identified pitch class is termed its chroma.
Synesthesia – some people, such as the composer Scriabin, actually see a color when they hear a pitch or a tonal key. In his variant of synesthesia, C is red, G is orange, D yellow, and A green. Scriabin’s Promethius: The Poem of Fire (1910) includes a part for “clavier à lumières,” a color organ that emitted light of what he deemed the appropriate color for a pitch instead of sound.
Scale prototypes
When we describe a scale, we name the pitches in order within an octave. Better yet, we name the successive intervals going up within the octave. The classic description of the ubiquitous diatonic scale, in whole-steps or half-steps, in its major mode starting on the tonic pitch, is:
tone / tone / semitone / tone / tone / tone / semitone
In the chromatic 12-tone universe, that scale pattern measuring the intervals in semitone sizes would be:
2 2 1 2 2 2 1
That is what I would call a scale pattern . . . a Successive Upward Interval Sequence in Semitones (SUISS!). But let’s call it a scale pattern array, working exactly like the arrays describing constellations.
Now we can particularize our scale pattern definition to apply to any smaller set of pitch classes, even if they don’t look like a scale:
SCALE ARRAY — successive interval array describing the pitches of a constellation condensed by octave equivalence to their most compact pitch-class-equivalent arrangement within an octave, ordered lowest-to-highest (Forte’s “normal order”)
In this sense, the array of a smaller set or scale fragment is just like a scale pattern.
Successive Interval array is a versatile tool that can apply to any pitch collection, to a linear, scalar pitch pattern as well as to a vertical chord sonority or even an arpeggiated diagonal collection of pitches I call a constellation.
Modes
Most of our familiar scales are actually a different mode of the same 7-note diatonic scale, with a different starting and ending point called a tonic establishing the mode.
scale modes
Scale patterns / set classes
We can describe a set of pitches as an octave-compressed abstraction of 3 or 4 pitches as a lowest-to-highest ordering of pitch classes. It doesn’t produce anything like the 7 or so notes per octave we’re used to thinking of as a scale, as those shown above. It is conceptually powerful, nonetheless, to call the successive interval array of this compressed abstraction a scale pattern, even though it’s a scale fragment with no name. Its name can simply be the successive interval array, such as 2 4 2, the array describing a symmetrical pitch-class set called the French Augmented Sixth chord.
[Theoretical aside] In establishing set theory, Forte described these compact arrangements by naming the pitch-classes in order using a mod-12 number system shown above, C=0, C#/Db=1, D=2, etc. He identified twelve 3-note classes, including upside-down inversions reversing the scale pattern, as members of the same class. (Lewin kept these inversions separate, defining instead nineteen 3-note set classes. We’ll use Forte’s; the set classes as generalities are not as crucial to composing as to theoretical analysis.) Forte used cumbersome descriptions employing pitch-class numbers and “normal order.” In the Journal of Music Theory 15 (1971), Richard Chrisman defined and proposed successive interval arrays as a better, more revealing way to characterize the commonality of a family of pitch-class sets that are all related by transposition and/or inversion.
Relating to Forte’s concept of a set class, any set grouping three pitch-classes can be analyzed as an interval array or partial scale pattern.
scale patterns of all 3-pitch-class sets
Sets forming triads (or seventh chords below) are highlighted in BLUE; those that are atonal (cannot be found in a diatonic scale) are highlighted in GOLD.
While the number of possible interval arrays for constellations of four pitches is enormous — even if limited to interval stack sizes less than two octaves, there are more than 12,000 possibilities — we can use this scale-pattern abstraction tool to categorize them into forty-three 4-pitch-class families.
scale patterns of all 4-pitch-class sets
The blue-highlighted scale patterns have common triadic chord names:
1 4 3 = “Major Major 7th chord” (in any chord inversion)
3 2 3 = “minor minor 7th chord” (in any chord inversion)
3 3 2 = “dominant 7th chord” (in any chord inversion)
3 3 3 = “fully diminished 7th chord”
The scale pattern 2 4 2 is an interesting symmetrical, non-diatonic pattern called a “French augmented 6th chord”.
Vocabulary
These maps collecting 62 scale-patterns summarize all possible constellations of 3 or 4 unique pitches, our total harmonic vocabulary in the chromatic universe.
you must understandthe 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 SCALE
HARMONIC TYPE
TONAL CENTER
ancient mode
perfect intervals
fixed by mode
Major / minor
triad
modulatory shifting
extended chromatic
extended triad
polytonal centers
exotic / synthetic
non-triad
established contextually
12-tone
diverse
none
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.
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.
MAPPING MUSIC 