MAPPING MUSIC

Category: time points

  • Mapping Music 3. CHANGE

    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)

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    © 2026 – All Rights Reserved

    Thomas S. Clark

    TClarkArtMusic.com

  • Mapping Music 2. RHYTHM

    Rhythm is a stream of event durations. Often repetitive but potentially elastic, rhythm can be steady, as a simple march, or volatile, as a melody that hovers, trots, then suddenly starts running.

    Defining some basic terms:

    TIME POINT — a precise moment in time marking the beginning of a span of time extending until a comparable event marks the next time point

     TIME SPAN — the duration of time from one point to the next point marking a comparable event

     STREAM — a series of events formed by the elements of consecutive groups

     PERIODIC — a stream of events whose elements exhibit equivalent time spans

     PULSE — ungrouped periodic time marking in a speed of about 48 to 1,000 per minute (above 1000 per minute becomes pitch)

     METER — a nested hierarchy of periodic streams of of time points

    Nested means that two or more time-spans at a quicker level are synchronized with the next longer durational level. Two eighth-notes “fit” within the metric time-span of one quarter-note, for example. Meter creates the perception/expectation of events happening at periodic time points on more than one level of speed/duration.

    BEAT — pulse that constitutes the primary level of a meter’s hierarchy, the connector between both quicker subdivided and slower grouped levels

    PACE — general quickness / slowness of rhythmic activity in a line or whole fabric

    HYPERMEASURE — extension of metric hierarchy grouping measures, typically in two- or three-measure units

    PHRASE — grouping of molecules, shapes, motives, chord changes, etc. in a coherent stream, typically the length of the human breath

     PERIOD — grouping of phrases, typically concluded with a significant harmonic cadence and/or melodic sense of arrival

    Pulse and Beat

    Underneath the sense of a beat, pulse is primordial periodicity – clapping hands, stomping feet, banging rocks, running strides – features we inherit from our primitive musical ancestry and still use to organize our musical actions.

    Susie Ibarra – Sky Islands (2025)

    A rapid pulse can drive music with frenetic motion. Much of jazz relies on this power source. The tension between a fast, steady pulse vs. unpredictable accents (syncopations) and other turbulence generates energy and excitement.

    SYNCOPATION — an accent not synchronized with the beats, or a note length shifted from its regular metric starting point

    Listen to how a relentless, fast pulse drives this music.

    Julia Wolfe – Believing (2012)

    Rhythm makes meter, Meter drives rhythm

    Rhythm first generates meter by marking periodic time points at different levels of speed. The marking is just the moment of initiation of each note but also accents, chord changes, etc. at broader time levels. As a great example, let’s use a famous theme from a piece nicknamed for a planet:

    Jupiter Symphony theme

    This melody first establishes periodicity of half-notes but nothing shorter for a while. This could be any duple meter. Once we hear these first four equal notes, we tend to perceive them as two pairs, establishing the whole-note measure-level meter. We feel the time point that begins the third measure on two levels of periodicity even though no note happens to mark it. In this third measure, eighth-notes then divide those half-note time spans into four parts. In the middle of measure 3, a quarter-note fills in the missing level of meter. Finally, in the last beat of the measure, a burst of sixteenth-notes establishes the fourth, quickest periodicity of the nested hierarchy. This example extends the hierarchy to show 2-bar hypermeasures in a 4-bar phrase.

    Once the tune repeats, our sense of that nested hierarchy of speeds is in full cognitive play. While jumping from metric level to level, half-notes to eighth-notes to quarter-notes then 16th-notes, the melodic rhythm undergoes a compression of pace, moving from longer rhythmic values to quicker and quickest, while the Allegro tempo does not change.

    More definitions

     

    CYCLE — the duration of equivalent time spans in a periodic stream of events

     ELEMENT — a point and following time span of an event or group of events, relating to other consecutive elements to form a group at a broader (slower) level of time

    COMPRESSION — an element of a group or stream is a shorter span than the previous element

    Example: rhythm changing from half-notes to quarter-notes to eighth-notes, compression of pace.

     EXPANSION — opposite of compression, elements of a group or stream are longer spans than previous elements

     ACCELERATION — consistent successive small compressions of beat or pulse

    PROPORTION — relationship of time spans expressed as a ratio, reduced to smallest-possible integers (whole numbers)

     RHYTHMIC GROUP — consecutive related elements, with a point of initiation and accumulated durational span

     RHYTHMIC RANGE — ratio of longest duration to the shortest

    In the Mozart Jupiter example above, the rhythmic range is 1:1 conformity for the first two bars, then 4:1 with three different note values in the third and fourth bars.)

    RHYTHMIC VARIETY — number of different note values in a stream of notes

    In the Mozart Jupiter example above, the rhythmic variety is 4 (halfs, quarters, eighths, sixteenths).

    Stress and accent

    Classic poetry classifies each syllable grouping (a “foot”) in a line of poetry according to which syllable in the grouping is stressed or longer length (agogic stress). (The last, a “reversibrach,” is my addition to complete the set of possibilities for musical purposes.)

    Metric “feet” in poetry

    Rhythmic molecules, groupings of two or more notes, can be similarly characterized, though with more stress possibilities:

    • Strength (accent) stress
    • Length (agogic) stress
    • Metric stress (strong beat vs. weak beat; on the beat vs. off-beat)

    A molecule can have one stress pattern in accent contradicting a different stress pattern in length or metric placement. Another familiar Mozart example: the opening themes of the first movement of Mozart’s Symphony No. 40 in G Minor. The first theme is fast (allegro) but steady (low elasticity).

    Symphony No.40 1st theme

    Analyzed as three rhythmic molecules:

    Three quick, predictable anapests, in which the longer, metrically-accented note is precisely the same length as the pair of shorter notes that lead to it. It also shows a narrow rhythmic range, 2:1, and little rhythmic variety, with only two rhythmic values, the eighth-note and quarter-note.

    Now the contrasting second theme, a soaring oboe line, highly elastic in rhythm.

    Symphony No.40 2nd theme

    This phrase launches with a long trochee, beginning-stressed in both length and metric placement. This rhythm uses four different note values, the longest of which is 12 times the length of the shortest. The first note sounds stretched, like an elastic band that is then released after the third note, unleashing the quick notes that scamper to the last.

    This is just a glimpse on the micro-level of rhythmic contrasts and a temporal elasticity that propels the exciting roller-coaster allegro opening of this great symphony. All our perceptual and gestalt faculties are engaged in a grand game of play with time.

    Molecules

    Here are the opening notes of three famous 20th-century unaccompanied flute pieces, by Debussy, Varese, and Berio, respectively.

    Each uses a three-pitch motive that, when analyzed as a pitch-class set, is a segment of the chromatic scale.

    • Syrinx: A Bb B = +1 +1 semitones chromatic scale pattern
    • Density 21.5: E F F# = +1 +1 semitones, same scale pattern
    • Sequenza: G G# A = same +1 +1 chromatic scale pattern (G displaced by an octave)

    Shown above in their chronological order of writing, it is likely that one influenced the next, and it the next in a chain of evolving variation. While this shared pitch-class-set characteristic is the usual basis for comparison, it is also interesting to compare the rhythmic molecules of their generating motives.

    Syrinx starts with a clear front-stressed dactyl, repeated then echoed in bar 2.

    The Density 21.5 motive is more complicated. Its opening three notes, from a short/long durational standpoint, is an end-stressed anapest. But the first note, though short, has the metric accent, being the only note of the three written on a beat. That first note is also emphasized by the tenuto mark. Those accent factors point toward a front-stressed dactyl like Syrinx. The next three notes starting with the C# are a more ambiguous stress shape.

    The opening three notes of Sequenza have no clear metric or dynamic accent difference; they are all strong. But by duration, the third note is “longer” in effect in the time stream (agogic stress), as the silent time after it, before the next note comes along,is longer. Also the third pitch, G, is much higher, giving it a registral or contour accent. This 3-note molecule is an end-stressed anapest.

    In all three pieces, however, the sense of simple repetition of matching poetic feet is not established or maintained. It is more productive to understand throughout each piece how rhythmic range and variety expand and contract and pace intensifies or subsides.

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    © 2026 – All Rights Reserved

    Thomas S. Clark

    TClarkArtMusic.com 

  • Mapping Music 1. TIME

    “If you want to find the secrets of the universe,

    think in terms of energy, frequency, and vibration.”

    (Nikola Tesla)

    We start with time. Everything in music involves time, is of time, sound events occurring in our perceived flow of time.

    Sound itself is periodic vibration, a repetition of compression waves of energy in air (or water). Repetition of an event or series of events establishes a frequency of repetition and the period or cycle length, the elapsed time duration from each event’s starting time point (moment) to the starting point (moment) of its repetition.

    We perceive the frequency of air-compression waves as pitch if they are faster than 20 per second and slower than about 4,000. Frequency is typically measured in cycles per second, called Hertz. Non-periodic waves faster than about 20 Hz are perceived as noise. Events or time cycles slower than 20 Hz are perceived as pulses, tempo, rhythm, phrase structure, etc. At these slower sub-sonic event speeds, it is more convenient to identify the duration of the cycle, its period, than the frequency.

    Periodicity, this repetitive aspect of sound events in time, gives us a dimension to map all the possibilities, from extremely fast to almost frozen slowness, and from simple, highly regular repetitions to a very complex succession of variants.

    the periodic time/sound universe

    In this illustration, the Y-axis is speed/frequency (slowest at bottom, fastest on top), the X-axis is regularity of repetition (perfectly regular at left, randomly sporadic time spans at right). The blocks have sharp rectangular edges; if I were a better artist, the boundaries between descriptive categories would actually be curving and very blurred. Though the graph shows firm straight lines separating pitch and noise, there is actually a fuzzy, curving grayscale continuum from pure, simple pitch through complex, colorful pitched timbres to noise.

    Defining time

    What is time and how does it work in our lives and in the rhythms that are the fundamental “substance” of music? I say substance metaphorically, because time does not exist as any physical matter. It is a perceptual construct, a complex quilt stitched out of human experience.

    Discover magazine ran an article in June of 2007 titled, “Time May Not Exist”.

    “Efforts to understand time below the Planck scale have led to an exceedingly strange juncture in physics. The problem, in brief, is that time may not exist at the most fundamental level of physical reality. If so, then what is time? And why is it so obviously and tyrannically omnipresent in our own experience? ‘The meaning of time has become terribly problematic in contemporary physics,’ says Simon Saunders, a philosopher of physics at the University of Oxford. ‘The situation is so uncomfortable that by far the best thing to do is declare oneself an agnostic.’”

    The mysteries of time were explored as early as sixteen hundred years ago by the great Saint Augustine of Hippo, in Book XI of his deeply philosophical work, Confessions.

    “. . . What is time? Who can give that a brief or easy answer? Who can even form a conception of it to be put in words? Yet what do we mention more often or familiarly in our conversation than time? We must therefore know what we are talking about when we refer to it, or when we hear someone else doing so. But what, exactly, is that? [Book XI, Section 17]

    Nicholas Stratas’ thought-provoking article in the July 2007 issue of Wake County Physician, “Time – Continuous Yet Bidimensional” asserts that most of us have a firm concept of Past, Present, and Future. But defining them is challenging, and sorting out how these constructs interact in our consciousness even more so. Michael Spitzer, in The Musical Human (Bloomsbury Publishing, 2021), wrote:

    “Musical time is a window into time consciousness in general. We listen to music in the moment, sitting in the saddle of an ever-shifting Now, as the past whizzes by to become memory, and the present anticipates what is just around the corner. Music’s present tense is really a bundle of memories and anticipations . . .”

    Many years ago, I first read an article translated from Die Reihe, written by a preeminent avant-garde experimental composer, Karlheinz Stockhausen. “Structure and Experiential Time” described Stockhausen’s view that time does not flow uniformly through the experience of a serious musical composition. It ebbs and surges as the composer shapes not just the tempo but the flow of information in the form of repeated or new musical events, simple or complex musical structures.

    “When we hear a piece of music, processes of alteration follow each other at varying speeds; we have now more time to grasp alterations, now less.”

    Even tempo, a supposedly steady clock in most music, ebbs and flows. Computer music composers in synthesizing musical sounds have found that a mechanistically rigid clock tempo sounds artificial. Human musicians are constantly flexing tempo in subtle ways to convey almost subliminally where the music is “going” (another metaphor, that of travel through space).

    Saint Augustine recognizes the slippery challenge of measuring time:

    “ . . . We observe the different ways times lapse, and compare them, and call some longer and some shorter. . . . It is passing time we measure, as we experience it. . . . Time can only be measured as it passes. Once past, it is no longer there to be measured.” [Book XI, Section 21]

    “We measure time as it passes . . . . But how can we measure the present, when it has no extent of its own? . . . Time must be measured in something with extent . . . But in what extended thing do we measure time as it passes?” [Book XI, Section 27]

    “So time is measured, my mind, in you. Raise no clamor against me—I mean against yourself—out of your jostling reactions. I measure time in you . . . because I measure the reactions that things caused in you by their passage, reactions that remain when the things that occasioned them have passed on. . . . Time has to be these reactions for me to be able to measure it.” [Book XI, Section 36]

    Time perception

    Pulling all this together, I’d like to suggest several things about time in classical music.

    • Time is perceptual.
    • Time is multidimensional.
    • Time is elastic.
    • Time is experienced in complex ways as the fundamental basis of music’s richness.

    In LEARNING TO COMPOSE, co-author Larry Austin and I begin the chapter titled “Time Streams” with a quote from a philosopher, and then express in our own words the fundamental nature of time.

    “ ‘Music makes time audible and its form and continuity sensible.’
    —Suzanne Langer

    Music exists in time. Time exists as we sense it, articulated on many levels by changing and cyclically recurring events.

    As beautiful, colorful and essential as sound is in making music, musical sounds are the means to an end, building blocks for events that primarily mark articulations of time.

    We sometimes like to think of music as having two fundamental dimensions, like a graph. The horizontal dimension is the parameter of time. The vertical dimension is the parameter of pitch. But pitch is actually a temporal phenomenon – the frequency (periodic change over time) of sound waves. How amazing are the human ear and human mind to perceive waves of air coming at us a thousand times a second or much faster and distinguish the small differences that make a pitch “in tune” (or not) and the even subtler differences that identify an oboe instead of a violin producing that pitch. All of this from a perception simply of periodic rates in time!

    Stockhausen pointed out that in mentally processing all of these sonic distinctions, we are forced to pay more attention to changes in their qualities, combinations, and “spacing” in time. These are his “alterations”.

    “The greater the temporal density of unexpected alterations . . . the more time we need to grasp events, and the less time we have for reflection, the quicker time passes; the lower the effective density of alteration (not reduced by recollection or the fact that the alterations coincide with our expectation), the less time the senses need to react, so the greater intervals of experiential time lie between the processes, and the slower time passes.”

    The concepts of expectation and information help make some sense of things. “Information” is perceptual data that is similar to what you just heard or logically confirms what you were expecting next. “Entropy” is the opposite perception – surprise, contrast, noticeable change. In musical listening, though we don’t do so consciously, we are constantly “computing,” assessing, retaining, and predicting.

    Saint Augustine connects Past, Present, and Future with memory, experience, and expectation:

    “What should be clear and obvious by now is that we cannot properly say that the future or the past exist, or that there are three times, past, present, and future. Perhaps we can say that there are three tenses, but that they are the present of the past, the present of the present, and the present of the future. This would correspond, in some sense, with a triad I find in the soul and nowhere else, where the past is present to memory, the present is present to observation, and the future is present to anticipation.” [Confessions, Book XI, Section 26]

    And to make matters more complicated, it is not at all a linear process. Let’s take a metaphor. I can’t resist one that Einstein was very fond of in his thought experiments.

    As listeners, we’d like to imagine ourselves as a train riding on tracks through time, a train that keeps moving forward and doesn’t back up. The clickety-clack of our wheels is a steady tempo measuring time. We only remember back to the tracks the locomotive has passed but still lie under the wheels of our caboose at the end. And we only look ahead a little bit, as the tree-bordered tracks curve, preventing a longer straight view.

    That’s way too simple, a two-dimensional time frame in which we either recall a little of what we just heard or maybe guess a little what might happen next. As Meyers, Stockhausen, Spitzer, and Dr. Stratas all observe, in keen listening to music our minds are filled with memories of not just the previous measure or phrase, but the very beginning of the piece, its theme or launching impetus (Grundgestalt as Schoenberg named it) and, in a more diffuse sense, all that has “happened” up to the present moment. The present moment is not one single phenomenon in time either. Melody, countermelody, bass line, chordal texture, and punctuating sounds are simultaneously tracing distinct paths, each with its own pace through time. At the same time, we are constantly expecting what’s coming, or at least “feeling” where the music might be going. And, as if that weren’t complicated enough, we are busy reevaluating what we just heard in relation to what we had been expecting. Saint Augustine describes it more succinctly:

    “Only in the mind can this [the experience of time] be accomplished, because of three activities there—the acts of anticipating, of observing, and of remembering.” [Book XI, Section 37]

    None of this is conscious, but in describing it in concrete terms, we recognize the dizzying multidimensionality, time arrows pointing in all directions and curling back on themselves. This is what I believe constitutes deep listening, “getting lost in the music”.

    Just one more idea – elasticity. Stockhausen recognizes that in music the sense of time passing changes, stretches or compresses, depending on how much “alteration” is being encountered. This is why music can seem “steady” or “surging ahead” or dissipating and almost “frozen”. It is not at all the tempo that causes this, but rather the rate of change, sharp contrast or subtle evolution, in the harmonies, the melodic character, or the rhythm.

    A rhythmic playfulness in modern music stretches our sense of timing. Tempos change, are interrupted, break down, tumble into avalanches, come to rest. Time itself stretches and becomes the titled thematic element in pieces such as Time Cycles (1960) by Lukas Foss. Here is another example titled about time, written at the starting gun of the new millennium.

    Fred Lerdahl – Time After Time (2000)

    Awe

    In his book When (Riverhead Books, 2018) Daniel H. Pink writes,

    “I used to believe that timing was everything. Now I believe that timing is everything. . . . The experience of awe changes our perception of time. When we experience awe, time slows down. It expands. We feel like we have more of it. And that sensation lifts our well-being.”

    He quotes researchers Rudd, Vohs, and Aaker in Psychological Science 23 No. 10 (2012):

    “Experiences of awe bring people into the present moment, and being in the present moment underlies awe’s capacity to adjust time perception.”

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    © 2026 – All Rights Reserved

    Thomas S. Clark

    TClarkArtMusic.com

  • journal 8. Zeitmasse

    Interlochen, 1976 —

    The German word Zeit means “time.” While masse sounds like “masses,” it actually tranlates “measures.” The two together make the musical term “tempos”. Karlheinz Stockhausen composed his “Time Measures” in 1956, N. 5 Zeitmasse for five woodwinds. Time is unmetered, rapidly and unpredictably ebbing and flowing in a dense texture of highly stretchy, elastic rhythms. Instead of predictably placed bricks, musical notes are dancing sparkles or sustained starlight.

    Nr. 5 ZEITMASSE

    London Sinfonietta

    LISTEN › YouTube

    Compare that time sense with a work of mine composed 67 years later.

    DARK MATTER

    Clark 2023 (TC-133)

    Lucas Foss was among many composing works that were all about time. The opening song of his 1960 work,Time Cycle, for soprano, piano, clarinet, cello, and percussion, sets a poem by W. H. Auden, “We’re Late.” It begins:

    “Clocks cannot tell our time of day
    For what event to pray
    Because we have no time, because
    We have no time until
    We know what time we fill,
    Why time is other than time was.”

    TIME CYCLE – “We’re Late”

    Judith Kelloch, soprano

    LISTEN › YouTube

    Refracting time

    I met Foss in 1973 or ’75 when he guest conducted at the National Music Camp in Interlochen, Michigan, where I worked as assistant to the director of the university-level program (my first adminstrative gig). Back in Ann Arbor in the fall of 1975, I began writing my doctoral dissertation, a large three-movement score for symphony orchestra. Following the direction of my previous orchestral work, Animated Landscapes: Nocturne (1971), ILLUMINATIONS built sound masses animated in widely varying tempi, meters, and even unmetered free sprays of notes.

    ILLUMINATIONS – Three Refractions of Time

    Clark 1976 (TC-33)

    1. PROJECTION (future)

    2. REFLECTION (past)

    3. EMANATION (present)

    Receiving a Bicentennial commission from the Federation of Women’s Clubs, it was premiered in 1976 by the World Youth Symphony at Interlochen.

    with conductor and World Youth Sym.
    my prof Leslie Bassett

    Chronos

    Avant garde music in the middle of the 20th century was exploring an exploding range of new tonal possibilities — from by-tonaility to all-chromatic scales, from massive unresolved dissonance to pitchless noise. At the same time, composers were intensely interested in stretching rhythmic possibilities to the extremes of aperiodic time perception.

    The term trope in medieval music was an elaboration inserted into a liturgical passage. In the 20th century, Hauer used it to name an unordered collection of six different pitches (called an unordered hexachord), half of an equal-tempered twelve-tone set. Italian Niccolo Castiglioni’s TROPI (1959) uses this pitch organizing technique, but is more about blocks of contrasting rhythmic texture separated by extended silences. Moment by moment, these blocks succeed each other, recur, combine, and dissapate — a kaleidoscope of sound in time.

    Castiglioni – TROPI

    Gruppo “Musica Insieme” di Cremona

    LISTEN › YouTube

    American George Crumb composed ground-breaking works for piano, beginning with Five Pieces for Piano in 1962. With imaginative, free-wheeling non-metric notation, he conjured a menagerie of sound sprites dancing through an ever-changing timescape.

    Crumb – Five Pieces for Piano

    Thomas Little, piano

    LISTEN › YouTube

    In 1973, he went deeper with the first of four volumes of innovative piano music, Makrokosmos I – Twelve Fantasy Pieces after the Zodiac. Here is Nr. 12:

    Makrokosmos I – Spiral Galaxy

    Scott Sherman, piano

    LISTEN › YouTube

    In 1975 at Rackham Auditorium on the University of Michigan campus, I heard a live performance of this monumental work by David Burge, for whom the piece was composed. It had a profound impact. My own dive into this sound-in-time cosmos had only four character studies:

    1. precession of the equinoxes
    2. Stonehenge at dawn
    3. Heraclitean vortex
    4. lunar litany

    Geography of the Chronosphere

    Clark 1975 (TC-32)

    Max Lifchitz, piano

    Before Time

    Stephen Hawking, the great theoretical physicist and cosmologist, is famous for solving in 1974 the mind-boggling mathematics of black holes and what became known as their Hawking Radiation. He also wrote a fascinating book, A Brief History of Time. Now, after Hawking’s death, his last collaborator, Thomas Hertog, has published On the Origins of Time explaining Hawking’s theory of how Time itself began at the Hot Big Bang birth of the universe. The idea, in grossly simplified geometry, is that Space and Time were united as one primordial sphere that dramatically split apart at the Big Bang’s initial hyperinflation into expanding Space and progressing Time. Before that moment, there was no time, no before.

    The musical challenge: how to express utter timelessness before the explosion; and how to build a sound space that sits still then explodes. You’ll hear an initial sound space of just one pitch, G, which at first quivers in color but without perceivable rhythm. While standing still, the sound space expands by octaves and eventually explodes with a fuller spectrum of chromatic pitch color.

    HAWKING’S TIME

    Clark 2023 (TC-133)

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  • Book of Canons

    My compositional fascination with musical canons began in the early 1970s with study (at the University of Michigan) of Ockeghem’s 15th-century polyphony, the 10 canons in Bach’s 18th-century The Musical Offering, and Webern’s 20th-century Symphonie Op.21. As a young professor in the 1980s teaching 16th-century counterpoint at what was then North Texas State University (now UNT), I used canon as a challenging contrapuntal writing assignment. In 1985, a wind ensemble piece, Parallel Horizons (Homage to Schoenberg), was my first formal composition constructed by canon. In Dark Matter, other contrapuntal writing surrounds an extended canon. Now canon pervades much of my 21st-century writing, a challenging yet stimulating and gratifying approach to texture and continuity of material.

    The definition of this ancient form of Rumpelstiltskin magic, spinning complex counterpoint out of a single melodic line:

    CANON
    A 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 my BOOK OF CANONS in the appendices. For pedagogical demonstration purposes, the subject of each is shown, with indications for when and at what pitch level each answer will occur.

    Read more at Mapping the Music Universe: COUNTERPOINT.

  • Timing of Awe

    In his book When (Riverhead Books, 2018) Daniel H. Pink writes, “I believe that timing is everything. . . . The experience of awe changes our perception of time. When we experience awe, time slows down. It expands. We feel like we have more of it. And that sensation lifts our well-being.” He quotes researchers Rudd, Vohs, and Aaker in Psychological Science 23 No. 10 (2012): “Experiences of awe bring people into the present moment, and being in the present moment underlies awe’s capacity to adjust time perception.”

    Read more about time at Time perception

  • Mapping the Music Universe – preview

    Mapping the Music Universe is written for the literate musician, including college music students through music scholars, and anyone who is intellectually curious about how music works, especially in the 20th-21st-century modern and “post-modern” eras.

    Purpose

    The mapping project is a comprehensive catalog of patterns and processes, meant to provide simple tools for understanding modern music. This is not a theoretical treatise but a practical guide for all educated or educating musicians and the intellectually curious, requiring only basic music literacy. For me as a composer, it is also an exploration of how some of the less travelled conceptual paths lead to interesting creative possibilities. Sample composed etudes will give examples to connect the abstractions back to our musical imaginations.

    In 1989 I co-authored a composition textbook with Larry Austin, Learning to Compose: Modes, Materials, and Models of Musical Invention. We felt it was conceptually ground-breaking. My next book, ARRAYS, was an aural skills workbook covering basic modal, tonal, and “post-tonal” music of the Renaissance through the Twentieth Century. They were intended as college-level textbooks, ARRAYS basic and Learning to Compose quite advanced — maybe too advanced to succeed as a textbook, turning out to be more of a technical monograph than a basic guide. Mapping the Music Universe draws in part on the ideas and approaches of both these now out-of-print publications.

    The three parts progress logically from fundamental — time and periodicity — to pitch space, then to larger structures — texture and form. They can be read in this sequence or separately in any order. Likewise, within each part, the various topics are presented in a progressive order, but jumping in at any point is not to be discouraged. As terms are defined, they are set off to the right. Figures include musical examples, sample etude compositions, tables, and graphic illustrations of patterns and their relationships.

    Read full Introduction . . .

    Mapping the Music Universe by Thomas S. Clark . . . CONTENTS

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  • STARS: Time points, periodicity, perception

    We start with time. Everything in music involves time, is of time, sound events occurring in our perceived flow of time. Sound itself is periodic, a repetition of compression waves in air (or water).

    Repetition of an event or series of events, establishing a frequency of repetition and the period or cycle length of the elapsed time duration from each event’s starting time point (moment) to the starting point (moment) of its repetition.

    We perceive the frequency of the waves as pitch if they are faster than 20 per second and slower than about 4,000. Frequency is typically measured in cycles per second, called Hertz.

    Non-periodic waves faster than about 20 Hz are perceived as noise. Events or time cycles slower than 20 Hz are perceived as pulses, tempo, rhythm, phrase structure, etc. At these slower sub-sonic event speeds, it is more convenient to identify the duration of the cycle, its period, than the frequency.

    Periodicity, this repetitive aspect of sound events in time, gives us a dimension to map all the possibilities, from extremely fast to almost frozen slowness, and from simple, highly regular repetitions to a very complex succession of variants.

    Topics

    • Defining time
    • Time perception
    • Periodicity
    • Meter
    • Definitions
    • Stress and accent
    • Rhythmic molecules
    • Harmonic rhythm
    • Beyond meter
    • Prime numbers

    To read more, request a password from tc24@txstate.edu

    Mapping the Music Universe by Thomas S. Clark . . . CONTENTS