Rhythmic intensity is an important factor in shaping musical form. A former research project “Density Functions in the Structure of Modern Music” in the 1970s sought to quantify it along with several other core aspects of structure at play in shaping large-scale form.
In the TIME chapters, we previously mentioned pace and showed how it can accelerate or decelerate in a line while tempo remains steady. (The Beethoven string quartet example Op. 135 illustrated that.) We have now also defined composite rhythm as an intersecting sum of rhythmic time points of lines, the layers of a textural fabric.
Density
In physical terms, density is a ratio comparing the amount of mass to the amount of space it takes up. Measuring time space, tempo (expressed in “M.M.” beats per minute) can convert a count of beats into a time-length in seconds:
DURATION (in seconds) — multiply BEATS times 60, then divide by TEMPO
Now we’re ready to measure the pace of a line for a bar or a whole phrase:
PACE (Notes Per Second) — number of notes divided by the duration of the stream
And then to quantify for a whole texture of rhythmic activity:
RHYTHMIC DENSITY (Attack-Points Per Second) — number of note-starting time-points in the composite rhythm of the whole texture divided by the duration of the stream
Let’s go back to the Webern Symphonie Op. 21. Though called a symphony, it has only two movements. The second movement is a theme and variations with coda, each exactly 11 bars long in two-four meter. Here’s the theme:
Op. 21, II — theme
Each variation, though 11 bars long like the theme, is in a different marked tempo. Each is distinguished by a contrasting degree of rhythmic density. And though the theme is a sparse (pointillistic) fabric, some variations are contrapuntally thick and intense.
Rhythmic density and what we might define as textural density (how many lines woven into what octave span) basically trace the same unfolding through the variations. The exception is Variation V. There they diverge, intensely active rhythms but only three textural elements in a diffuse pitch span of almost four octaves.
A graph of changing rhythmic density values in each variation highlights rhythmic density as the bolder line:
density graph of Op. 21 II
About broad form, this reveals that from the beginning, rhythmic density increases to a subordinate peak in Variation III and overall peak in Variation V, then variation by variation steps down to a coda that matches how we started with the sparse theme. In rhythmic density, the whole movement is an arch form, with Variation V the “climax.”
In the first “abstract sound mobile” of my 2024 work, FOLIO, it is easier to hear changing density as the changing thickness of clouds of sound, swelling and subsiding.
“Music of the Spheres”
Relativity
Modeling, the process of creating an overall design, can mean creating a new model or expanding the possibilities of an existing model. In Learning to Compose we identified and described three basic musical approaches:
NARRATIVE MODELING — Designing by telling a story, with characters, themes, gestures, suspense. What will happen next?
SPACIAL MODELING — Designing the size, shape, and texture of blocks or sections of material
TEMPORAL MODELING — Designing the flow and momentum of events in the passing of perceived time
Variation and contrast
Contrast is the essential complement to developmental continuity in musical material, driving musical momentum. Theme and variations form is a straightforward, traditional example of narrative modeling balancing contrast and continuity. Each variation preserves some basic element of structure such as harmonic progression (or in the Webern example, the tone row). Each variation presents a setting of that theme element in distinctly different orchestration, texture, mode, tempo, or rhythmic character.
The composer determines not just how and when to make a contrast, but how dramatic the contrast will be. Their fluctuations over time are the core of the composer’s instinctive variation skill. This is the impelling force that gives musical form a sense of going somewhere, of leading up to and flowing away from stable plateaus marking the structural pillars of large-scale form.
FLUCTUATION — Magnitude of contrast from one moment or event to the next
When analytically quantifying fluctuating data, the time scale of measurement matters. In avant-garde or experimental music, a stream of events may be high-contrast on the moment-to-moment scale but steady-state over broader time spans. Conversely and more traditionally, surface events may be continuous, while the bigger chunks of events, like one variation to the next, may pose more dramatic changes in parameters such as rhythmic density.
In typical Beethoven or Brahms variations, material within each variation is continuous, not at all fluctuant. The contrast comes altogether in the next variation.
That consideration plays out differently in Op. 21 II. There is the obvious contrast from one variation to the next; but within each variation, moment-to-moment surface continuity also fluctuates. Surface fluctuation in density factors occurs, especially from one 3-to-4-second “moment” to the next. (We can’t really call them phrases.)
For the Op. 21 II. Theme and Variations, we can now say something deeper about changing rhythmic density as the variations progress. From the Theme through the first two variations, rhythmic density increases gradually to Variation III. But then the fluctuation of rhythmic density spikes, dropping significantly for Variation IV, then suddenly increasing to its highest level in Variation V.
large-scale time form
It is not only Variation V’s greatest rhythmic intensity but also dramatically increased roller-coaster fluctuation, dropping then surging, that makes Variation V the climax of the movement.
Macro-structure
Though Webern may not have thought consciously about Schwankung (fluctuation), this is how composers manipulate momentum to make a climax and shape large-scale form. Likewise, approaching a final ending, not only do fluctuations typically diminish, but also rate of change subsides — the overall change factor levels out to zero. These are examples of temporal modeling.
The parameters of a musical event are numerous, a multidimensional matrix of at least six distinct, interacting qualities: each sound event’s loudness, resonance, timbre or sound color, duration, pitch (frequency), and time point of initiation. Imagine this as a six-dimensional space. In fact, physicists have imagined the structure of matter as exhibiting many more than six dimensions in string theory, M theory, etc.
Musical structure establishes the relativity of these parameters, though not exactly the way Einstein explained time, space, gravity, and energy with mathematical precision. Some structures such as the Schoenberg Farben example relate constellation harmony to sound color. Threnody relates rhythmic activity to fabrics of sound in a broad pitch space (spatial modeling). Counterpoint balances rhythmic relationships, metric placement of lines, and synchronicity with their intervallic relationships of consonance and dissonance. Ostinato music manipulates phase relationships.
And, as observed in Part I, temporal density, the rapidity of fluctuations and larger contrasts in these structures, propels our experience of the whole in time.
In Thinking in Numbers, Daniel Tammet wrote about a mathematical study of poetry,
“The best poems . . . combined in equal parts the predictability of meter with the novelty of unusual words. Too much meter made a poem banal; too much freewheeling . . . rendered it hard to follow. The delicate balance of convention and invention gives meaning to what we say.”
The essence of music’s large-scale temporal form is the relativity of overlapping, fluctuating musical structures in time, repeating, contrasting, interrupting, truncating, expanding, certainly recurring, or simply evolving. Designing a large-scale musical form combines temporal modeling, narrative modeling, and spatial modeling — a pacing plan, a storytelling rhetoric, an architecture of interrelated components.
–
–
Coda
sound mass . . . sound color . . . pitch constellations
ostinato repetition . . . changing density
evolving form . . . cosmic time
In Become Ocean (2013), John Luther Adams takes a deep dive into a serene sound sea, incorporating all of the elements and structures we have explored in our mapping journey.
John Luther Adams – Become Ocean (2013)
. . . and we have just begun gazing into
the vast space of color and complexity
in the Music Universe . . .
–
© 2026 – All Rights Reserved
Thomas S. Clark
–
Continue reading Mapping the Music Universe . . .
MapLab 1. Generate a Gymnopédie
–
MAPPING MUSIC 