Harmonic rhythm is the pace at which chords change in common-practice tonal music. Often in songs or simpler instrumental music, the harmony changes periodically, like once every measure or every half-note or every beat. Even when the rate of chord change is this uniform, it often accelerates approaching a cadence at the end of a phrase or other sectional unit. Calculus suggests that there can be a change in the rate of change, a second-order differential. Beethoven offers something like this in his very late work, the String Quartet No. 16 Opus 135. Here is an excerpt from the Allegretto first movement:
String Quartet Op. 135 Allegretto, mm. 25-48
An analytic sketch of the harmonic-root-foundation bass line reveals that F major gives way in the first four bars to a tonicization of the dominant, C major, starting with its dominant, G major:
The rate of chord change starts as every 4 beats, eventually quickening toward the end of the excerpt to a different chord every 8th-note — an eight-fold quickening of harmonic pace! As you listen again, notice if you feel this intensifying compression of events.
Going deeper into the tonal groupings of these harmonies, the starting key of F Major gives way to various tonicizations of G, then C. Here is a reductive sketch showing the durations of these tonicizations.
Op.135 excerpt harmonic reduction
Since this section is 24 measures long, it could have been composed as three equal 8-measure periods. Instead, the middle-ground tonal rhythm is surprisingly non-periodic, an irregular durational stream consisting of 8, 10, 10, 4, 7, 1, 1, 5, and 2 quarter-notes.
Beethoven was beyond eccentric at this late point in his career; Op. 135 was the last work he ever completed. Yet the elasticity of harmonic rhythm found in it is a hallmark of his earlier styles as well.
Beyond meter
Arising in the middle of the 20th century, highly complex, elastic rhythms began to be composed, in which every durational value was different and notes or events do not group into periodic measures or phrases. An example composed in 1971 is an elegy that makes a conscious effort to avoid articulating periodic beats or falling into groups of notes of periodic duration.
Meter signatures are present only for notational purposes and change four times in the passage. Only four of the 22 notes fall “on the beat” and only three of those articulate a downbeat.
Since the note values are so slightly or drastically different, we can measure each duration from the start of a note to the start of the next note as a multiple of fine “time particles” each one-twelfth of a quarter-note. The durational stream is blatantly non-periodic: 30, 12, 44, 8, 14, 6, 21, 9, 31, etc. The rhythmic range of the first four measures is higher than 7, rhythmic variety at 9. The next three measures have a higher rhythmic range of more than 11 and rhythmic variety of 8 (due to the 9-particle dotted 8th-notes that occur five times).
Beyond a mathematical comparison, a time graph mapping the durations reveals to the eye no periodicity, no perceived meter or regular conforming rhythmic pattern.
Elegy rhythm graphed
The rhythm floats above or beyond meter or pulse in a dreamlike, elastic stream. [From Night Songs (1971)]
Free time
Defeating the notated meter in this way, by avoiding beats and periodic, conforming note values, was developed to free a stream of events from periodic pulse, thus freeing the listener’s sense of time flow – free time itself. The logical next step, developed concurrently in mid-20th century, was to remove meter entirely as even a notational necessity. Just like the time graphs we have been using to visualize timing of events, a horizontal, proportional scale (such as one half-inch equals one second of time) enables the horizontal placement and spacing of notes on a staff to suggest visually subtly different durations, both of sustained sounds and the time spacing from one event to another.
Spatial notation
Spatial notation — non-metric representation of time by proportional horizontal spacing of notes
After “Elegy,” the first movement of the unaccompanied trombone piece Night Songs, the third movement, “Somniloquy,” was originally notated in this manner – what came to be known as “spatial time.”
Somniloquy notated spatially
In his one partially preserved manuscript, On Time, the Greek philosopher Heraclitus wrote about “the unity of opposites” and “flux,” meaning change. “It is not possible to step into the same river twice.” He also imagined that the cosmos is shaped as an enormous vortex of fire.
That image ignited musical sparks in my imagination for the third movement of my early solo piano work, Geography of the Chronosphere (1975), subtitled “Heraclitean Vortex.”
The score, in non-metric spatial notation, articulates explosive bursts of notes separated by irregular spans of reverberation.
Heraclitean Vortex excerpt
An analytic graph of loudness shows these bursts occurring at unpredictable time intervals, in moments (not so much phrases) of varying length, from 3 to 11 seconds.
time graph of 11 moments in Heraclitean Vortex excerpt
Prime time
Meter, as a periodic grouping of beats, almost always involves groups of two, three, or multiples of these factors. We call them duple meters if the groups are multiples of two, triple meter if multiples of three. Likewise, subdivisions of beats are usually subdivided into twos, threes, or multiples. Sixteenth-notes divide by two to the fourth power.
A prime number is defined as having no integer (whole number) factors other than one and itself. In metric structure, prime numbers, with no sub-grouping factors of two or three, are more complex – 5 8 or 7 4 time for example. A musical stream that avoids metric regularity can be built with the interaction of prime number series. When repeated periodic streams of note values equivalent to 5, 7, 11 or 13 smaller time values (such as eighth-notes or sixteenth-notes) interact in time, layers of rhythm will seldom strike notes together to make a contrapuntal accent that feels like a downbeat.
Here is a map illustrating this potential for non-metric independence:
repeating prime numbers interact
The bottom row of numbers shows rhythmic values of the composite rhythm, time points marked by an attack of a sound in one strand. If the streams start together as shown, they don’t all come together again until after 5,005 time-units. If each time-unit were a sixteenth-note duration, that would be after 312 four-four measures!
This is the hidden rhythmic scheme for Night Sky, layers of pitched sounds that don’t synchronize into any meter or composite periodicity. Though not regular and certainly not metric with a pulse, time points are not at all random. Listening to it while not looking at the score’s notational details, pay attention to the way in which the sounds mark points in the flow of time – as stars mark light points in the night sky.
Night Sky score
A direct photographic rendering of the middle system of the score illustrates the non-metric, asynchronous timing of note events in a broad texture of sounds.
Night Sky score abstracted
Do stars make spatial patterns? Of course, that’s what our fanciful constellation names are all about. But are those patterns regular, metric, periodic, symmetrical? No – that is part of their magic, a magic that can be metaphorically translated into floating musical time.
Beyond Time
From the classical tradition of Beethoven’s accelerating harmonic rhythm, we jump finally to the very modern stretching of time itself. Einstein explained gravity as the stretching of “Space/Time.” From composers such as Cage and Feldman in the ‘50s, we experience isolated events, moments of sound separated by extended pause. No pulse drives the clockwork of time; it stretches immeasurably into contemplation. Listen.
Lei Liang, My Windows (2007)
MAPPING MUSIC 