MAPPING MUSIC

Tag: music

  • Mapping Music 5. SCALES

    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:

    whole / whole / half / whole / whole / whole / half

    [octave repeats the cycle]

    Or in British terms:

    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 and 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.

    © 2026 – All Rights Reserved

    Thomas S. Clark

    TClarkArtMusic.com 

  • 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.”

    _____________

    © 2026 – All Rights Reserved

    Thomas S. Clark

    TClarkArtMusic.com

  • Mapping Music — PRELUDE

    “The heavenly motions are nothing

    but a continuous song for several voices,

    perceived not by the ear but by the intellect,

    music that sets landmarks

    in the immeasurable flow of time.”

    Galileo

    When we gaze at stars and planets, they appear as stationary points of light, fixed in place in what seems a random pattern across the entire night sky visible to our hemisphere. Time stands still.

    Throughout human time, humans have imagined that stars make picture patterns we name as constellations: fish, warriors, goddesses, animals. Only the persistent observers, such as astronomers, identify their nightly march across the sky, rising in the east and disappearing below the western horizon.

    Metaphor

    Musical sounds mark points in time, like stars. They form immediately into recognizable patterns we call chords, melodies, rhythms, memorable themes. They convey a sense of motion, time surging forward or slackening in our perception of their well choreographed parade.

    Astronomers observing and mapping (recording) the myriad points discovered that some of the stars are actually whole galaxies, with exotic forms of spirals and clouds. They observed through the color of the light that all these objects are racing away from us and each other in an expanding universe.

    Mapping music means cataloging many possible patterns, distinguishing their contrasts and commonalities. We will explore how to measure and compare the periodic rhythmic streams of musical events and their changing momentum. We will define and employ a simple but powerful math tool for cataloging and then creatively sculpting with all natures of harmony and melodic line in our 88-key chromatic universe. We will explore how master composers weave colorful fabrics and grand structures from skillfully crafted materials.

    Pursuing periodicity

    My music-mapping Periodicity Project began in 2021 as a comprehensive catalog of musical patterns and processes, meant to provide simple tools for understanding the complexities of modern music. It grew into this book, Mapping the Music Universe, written for anyone who is curious about how music works, especially in the 20th-21st-century modern and “post-modern” eras. For me as a composer, it is also an exploration of how some less traveled conceptual paths lead to interesting creative possibilities.

    In 1989 I co-authored a conceptually ground-breaking composition textbook with Larry Austin, Learning to Compose: Modes, Materials, and Models of Musical Invention. My next book, ARRAYS, was an aural skills workbook covering basic modal, tonal, and “post-tonal” music of the Renaissance through the Twentieth Century. Mapping the Music Universe draws in part on the ideas and approaches of both these now out-of-print publications.

    A common assumption within Western culture is that Science is all about observation, measurement, precision, and mathematical rigor . . . and Art is all about the “i” words: imagination, inspiration, intuition, improvisation. Science is Deductive, art is Creative. Our culture has begun to recognize the commonality of all these intellectual strengths, that the best Science can be creatively intuitive and great Art can be rigorous.

    Pioneer map makers

    As an educated musician and professional composer, I also have long been deeply interested in science, especially astronomy. Having read a great deal of general science writing, I am inspired particularly by ground-breaking pioneers who methodically and comprehensively mapped the possibilities of their particular field.

    Johann Joseph Fux — wrote Gradus ad Parnassum in 1725, codifying basic contrapuntal principles of Renaissance music.

    William Smith — a rural surveyor, in 1799 drew a colorful map of the subterranean rock strata of his county in English coal country, launching the modern science of geology.   

    Meriwether Lewis — kept extensive journals of the 1804-1806 Lewis and Clark Expedition, documenting and illustrating the discovered new world of the Northwest.  

    Dmitri Mendeleev — devised a “periodic table of the chemical elements,” published in 1869, providing a solid basis for modern chemistry through its graphic and organizational genius.

    Amédée Mouchez — launched an ambitious international star-mapping project (Carte du Ciel) in 1887 at the Paris Observatory.

    Henrietta Swan Leavitt — worked at the Harvard College Observatory as a “computer,” examining thousands of photographic plates from telescopes to measure and catalog the brightness of stars, identified 1777 variable stars.

    Lawrence Herbert — invented the Pantone system in 1956 to systematize color for printing ink and fabrics.

    Allen Forte — published an article in 1964 that launched musical set theory, defining, classifying and comparing all possible collections of “pitch classes” drawn from the equal-tempered 12-tone chromatic galaxy.

    The work and insights of the two on the list representing rigorous study of music, Fux and Forte, were part of my formal education in music and later an integral part of my teaching of composition and music theory.

    Maps

    Carte du Ciel was an ambitious second phase of an international star-mapping project initiated in 1887 by Paris Observatory director Amédée Mouchez.  A new photographic process revolutionizing the gathering of telescope images inspired the first phase, the Astrographic Catalogue of a dense, whole-sky array of star positions. Carte du Ciel, never completed after 70 years, used the Catalogue as a reference system for a complex survey of the vast field of even fainter images.

    Celebrating the grand metaphor relating astronomy to art music, here is my 8-minute computer-music sound sculpture. In the music, ghostly wisps of sound are punctuated by brighter bursts, clustered in a natural, not-quite randomly dispersed texture.

    _______________

    Looking ahead

    The blog-post chapters of Mapping the Music Universe will proceed in three broad phases, progressing logically from fundamental — time and periodicity — to pitch space, then to larger structures, texture and form. Within each phase, various topics are presented in a progressive order, but jumping in at any point is fine.

    Terms will sometimes be freshly coined. Graphic figures will include notated musical examples, tables, and graphic illustrations of patterns and their relationships. Big Ideas — Periodicity, Complexity, Symmetry, Relativity — will be explored using precise mathematical arrays as well as broad metaphors. Newly composed sample etudes will illustrate aurally.

    Along the way, “Map Labs” will present step-by-step recipes to compose simple pieces based on models of different compositional genres. Each Lab includes an original sample piece following the Map Lab guidelines, illustrating one possible creative outcome.

    Welcome! Join this creative journey of discovery . . .

    a composer’s expedition.

  • journal 13. Millennium

    Taiwan, 2001 —

    In my administrative career, 2001 was a high point, as interim dean of UNT College of Music, then the largest music school in the nation. Though I hired several professors, launched a magazine and Dean’s donor group, and headed up our part of a university capital campaign, I didn’t do any composing.

    Cho (left) and NTUA president (right)

    There was international travel, though. My Chinese-American colleague Gene Cho had established an exchange relationship with the National Taiwan University of the Arts, and he guided me to Taipei for the grand ceremony to sign the formal agreement.

    While there, we saw a traveling exhibit of the Qin Shi Huang Terracotta Army, took a train to visit Hsiuping University of Science and Technology in Changhua City, and went by car to the northern tip of the island of Formosa. In a cold mist on the rocky shore, we gazed out at the infinite expanse of the Pacific Ocean.

    California composer Robert Erickson wrote that the stimulus for his music “is usually some noise or some non-music sound composing the environment in which I live, its sounds, its ambience.” In 1968 he composed Pacific Sirens (ocean sounds) involving taped sounds gathered from the environment with acoustic instruments.

    LISTEN ›

    Pacific Sirens

    GreyWing Ensemble

    Global warming

    I was commissioned by North Carolina State University’s Arts Now Series, directed by Dr. Rodney Waschka II, for an artistic contribution to The Ericka Fairchild Symposium on Climate Change. “The Fourth Angel” refers to one of the “seven last plagues” as they were called in the King James Version of the Bible. In the NRSV translation, Revelation 16:8 reads:

    “The fourth angel poured his bowl on the sun, and it was allowed to scorch people with fire; they were scorched by the fierce heat.”

    The other six angels and their bowls wrought plagues of painful sores, bloody seas, bloody rivers, darkness, a dried up Euphrates, and finally the seventh angel’s loud voice pronounced, “It is done!”

    Standing in the middle of the sequence, the prophecy of the fourth angel is a dramatic metaphor for global warming.

    The Fourth Angel

    Clark 2006 (TC-77)

    Though there are some literal sound references, the angel is portrayed more broadly as a metaphor for the forces of nature. Rather than capturing actual samples of nature sounds, the computer-generated sounds are all synthesized, musical objects constructed employing a now-common computing technique called grain-table synthesis. (The choice of machine synthesis over nature sampling suggests a particular belief about the causes of global warming.) These synthetic sound images form a broad range of simple and complex musical rhythms and textures evocative of the natural world:

    • sunlight reflected off water and ice
    • glaciers calving and cascading into the ocean
    • solar radiation
    • night sounds.

    Extending the metaphor, sounds echo and swirl in sound space, just as do the dynamic, powerful weather systems that shape our global climate.

    Other angels

    Thus pieces about angels began with The Fourth Angel. Portraying imagery from Revelation, the seven angels in chapter 16 inflict suffering upon humanity. Angels of Bright Splendor evokes an equally awesome but more hopeful experience of our life-giving sun.

    In Zuni origin mythology, thunder sounded, and The People climbed from darkness, emerging into the daylight world. Seeing the Sun (Awonawilona, they cried, not used to such intense light. Where their tears fell, sunflowers sprang from the earth.

    Angels of Bright Splendor

    Spirits

    Heavenly light, voiced musically with metamorphic chord clusters, became an iconic sound in a famous 1968 movie. György Ligeti describes the technique for his 1966 piece for 16-part mixed choir, Lux Aeterna (“eternal light”):

    “The complex polyphony of the individual parts, embodied in a harmonic-musical flow in which the harmonies do not change suddenly, but merge into one another; one clearly discernible interval combination is gradually blurred, and from this cloudiness it is possible to discern a new interval combination taking shape.”

    LISTEN ›

    Lux Aeterna

    A Cappella Amsterdam

    Angels in most world religions and mythologies seem to serve one of two functions: wielding controlling power over the physical world or over human affairs; or making spiritual announcements to humans. The next piece in the angels series, scored for antiphonal double SATB choirs, brass, and strings, gives voice to the unseen voices of angels and other spirits. The choir pronounces the names of Native American and Hebrew spirits representing the power and beauty of nature – wind, moonlight, rainbows.

    •   Gǎoh – chief wind spirit (Iroquois)
    •   Yaogah – bear spirit of the north wind (Iroquois)
    •   Neoga – fawn spirit of the south wind (Iroquois)
    •   Oyandone – moose spirit of the East Wind (Iroquois)
    •   Amitolane – rainbow spirit (Zuni)
    •   Nokomis – daughter of the moon (Algonquin)
    •   Gabriel – archangel of justice, annunciation (Hebrew)
    •   Maris stella – star of the sea (Latin)

    Unseen Voices

    Clark 2018 (TC-94)

    Messengers of peace and assurance . . . and hope for the future of this millenium?

    ___________

  • journal 12. Zweite Wiener Schule

    Vienna, 1992 —

    The so-called “Second Viennese School” consisted of influential master composer Arnold Schoenberg and his protegés, Alban Berg and Anton Webern in early 20th-century Vienna. They pioneered a compositional approach described succinctly by Wikipedia as “totally chromatic expressionism without a firm tonal centre, often referred to as atonality; and later, Schoenberg’s serial twelve-tone technique.”

    When I began studying composition at Michigan in 1968, I quickly became immersed in exploration of pitch structure and broader tonality freed from the long-traditional restrictive limits of tonality: diatonic major and minor keys and their chromatic extensions, triadic sonorities and tonal centers. The complexity of this new musical realm is not truly “atonal” but rather an opening to a universe of fascinating, colorful possibility.

    Three pieces of the early 20th century, which I studied deeply in the 1970s and later used extensively in my teaching of modern music, were each masterful explorations of musical sound color.

    • Claude Debussy’s La Mer (1905), an iconic tone poem of Impressionistic musical painting, was discussed in Journal 1. Musique Française.
    • Anton Webern’s Symphony, Op. 21 (1928) is maybe the briefest piece ever titled as a symphony, a succinct, two-movement work whose first movement is a delicate gem of pointillistic color and complex 12-tone harmony.
    • Arnold Schoenberg’s Five Pieces for Orchestra, Op. 16 (1909); the third piece is a gentle study of orchestral sound color titled “Sommermorgen an einem See (Farben)” — (Summer Morning by a Lake: Colors”.

    After fifty years, these works are embedded more deeply than ever in my musical consciousness.

    It was only in 1992, on a side trip by bus from Brno, that I visited Vienna, the great musical city of Haydn, Beethoven, and Mozart in his last years. Mozart’s grave, not in the main cemetery but on the edge of the city, was hard to find but emotionally powerful to visit.

    Farben

    “Farben” is an early Schoenberg piece that is all about instrumental sound color and exotic harmonic color. The chords are not triads but rather atonally “dissonant” sonorities that place the instrumental colors in close, glowing pitch-interval proximity.

    LISTEN ›

    Five Pieces for Orch – III (Farben)

    Chicago Symphony on YouTube

    My recent piece, Farben, pays special homage to Schoenberg’s masterpiece, layering kaleidoscopic wind-instrument colors to build massive, morphing constellations, echoing Webern’s hidden chord-color symmetry.

    FARBEN

    Clark 2025 (TC-149)

    I have long admired and been influenced by the music of Anton Webern. Known historically as a member of the Second Viennese School with Alban Berg and mentor Arnold Schoenberg, the three were pioneers of so-called atonal music and 12-tone-row serial harmonic organization. I find the term “atonal” misleading and negative, as their 12-tone processes achieved new “12-tone tonalities” — not simply a rejection of traditional tonal harmony but also striving to create new and more complex tonalities.

    What I admire about Webern’s mostly-quiet instrumental miniatures (his Symphonie Op. 21 has only two sparsely-scored movements) is the delicate, crystalline quality of his pitch constellations; and their gently lyric, precious setting into transparent, pointillistic textures, pearl-strings of separate, delicate instrumental colors (called Klangfarbenmelodie). The first movement is built on one enormous, static, 13-pitch chord containing all 12 pitch classes of the chromatic universe in a symmetrical interval pattern, a palindrome interval pattern, the same top to bottom as bottom to top.

    Todesfall in Mittersill

    Webern’s mentor, Schoenberg, as a Jew was compelled to emigrate to the U.S. in 1933 before it was too late. Webern, not Jewish, stayed in Vienna, where he was born, suffered through and survived World War II, only to be fatally shot by a U.S. Army soldier during the Allied occupation of Austria in 1945. My homage to this beautiful musical mind tries to capture his music’s “lyrical, poetic concision” (Wikipedia).

    WEBERN ELEGY

    Clark 2024 (TC-115)

    Neue Tonalität

    My compositional excursions in 12-tone tonality traverse many of my compositions. One that sums it up well, if not succinctly, is VIENNESE SKETCHES. A set of “Twelve Miniatures in Twelve Tones,” parts I through IV are adapted from Webern Elegy , and V through XII from MapLab7 – For Little Arnold from my book, Mapping the Music Universe.

    Not intended to portray the historical European city, VIENNESE SKETCHES instead explores various textures and tonalities using the musical techniques of the Second Viennese School. My goal was to create a complex counterpoint of sound constellations that is less dissonant and more sonorous — my sense of a new tonality.

    VIENNESE SKETCHES

    Clark 2023 (TC-131)

    ___________

  • journal 11. Moravská Hudba

    Brno, 1991 —

    I first visited Czechoslovakia in 1991 to perform at the Brno International Music Festival. How this opportunity came about is a story in itself. My colleague Tom Sovik at the University of North Texas joined a group promoting the City of Dallas as a sister city with Brno, the second largest Czech city and capital of the Moravian province, where he had done his doctoral musicology research. At his suggestion, I wrote a short piece as a gift to Brno. Its mayor turned over the gift score to the secretariat of Brno’s International Music Festival, a distinguished Moravian composer Arnošt Parsch. He invited me to come to the festival and conduct my music. The result was an October 1991 performance in Brno’s New Town Hall of two of my works, ANTIPHONS (1989) and CANZONA, for combined woodwind and brass quintets, which I conducted.

    rehearsing with Czech ensemble

    Parsch invited me back in 1992 for the 27th Brno International Festival’s Experimental Music Exposition V. I presented my LIGHTFORMS 2: StarSpectra multimedia computer music and played trombone in an experimental multimedia piece by my friend, Rodney Waschka. I had performed the same program early that fall at the Festival Internacional Alfonso Reyes in Monterrey, Mexico.

    PTACí

    While in Brno for the 1991 festival, I met choreographer Hana Smičkova, who invited me to compose a work for her Mimi Fortunae Dance Theater, which rehearsed in the ancient Spilberk Castle. I began studying the great 20th-century Moravian composer Leoš Janáček’s music as background for the ballet’s composition.

    PTACí (“Birds”) was premiered in Brno in 1993 by the Moravian Chamber Orchestra, which I conducted. The ballet, choreographed by Smičkova, was performed by Mimi Fortunae in historic Mahunovo Divadlo, the first building in Europe to be equipped by Thomas Edison with electric lights.

    During these years, Parsch and I became composer friends. Our visits to each other always included long walks in nature and deep discussions of music, art, and culture. In 1991 I had visited the northern Moravian mountain village of Hukvaldy, the summer home of Janáček. He loved nature walks and studied bird songs.

    Hukvaldy Sketches was first a concert suite of PTACI, my set of modern musical impressions of old Moravia, in the ancient heart of Eastern Europe. Scored for a chamber quartet, it was premiered February 6, 2018, at Texas State University Performing Arts Center, by Ian Davidson (oboe), Vanguel Tangarov (clarinet), Ames Asbell (viola), and Kari Klier (marimba).

    The final transformation of this work was a re-scoring of Hukvaldy Sketches for the original PTACI orchestration. Its five scenes:

    Hrad – morning climb to the castle ruins

    Ptáci – watching Leoš’s birds

    Vody – forest streams and shadows

    Bystroušky – mouflons and other mountain wildlife

    Podzim – autumn sunset

    PTACí / Hukvaldy Sketches

    Clark 1993/2016 (TC-69/80)

    Morava

    In my intense study of Janácek, I reveled in the expressive depth of his uniquely modern Moravian music. His powerful String Quartet No. 2 and his collection of gentle piano music, Po zarostlém chodníčku, affected me deeply.

    In one of my Brno performances, Parsch’s Czech colleagues commented on my music’s affinity to modern Moravian musical style. I was informally dubbed an honorary Moravian Composer, a distinction I proudly took as a high honor of their acceptance. Since then, I have written many pieces with Czech imagery:

    Two of these are vocal music that include some Czech lyrics. The treble choir piece A NEW LIDICE begins with “We build a new village, while a just world watches. Stavíme novou vesnici. Spravedlivý svêt bude sledovat.” Children (including my daughter Alison) sang a short phrase in Czech in MORAVIAN MOUNTAIN SONGS, written for the Woodrow Wilson Elementary School Choir in Denton, Texas.

    Sinfonietta

    with Parsch at the spring outside Brno

    Leos Janácek composed his great concert work, Sinfonietta, in 1926 for the Sokol Gymnastic Festival in Prague. Janáček said it was intended to express “contemporary free man, his spiritual beauty and joy, his strength, courage and determination to fight for victory.” It is what I call musical sketches of his home city, Brno, the largest city in the Moravian east of what was then Czechoslovakia.

    I visited Brno several times starting in 1991 to perform my music at its International Music Festival.

    LISTEN ›

    Janácek Sinfonietta

    UNT Symphony Orch. on YouTube

    The festival traditionally ends with a performance of Sinfonietta by the Brno Philharmonic in Janácek Divadlo (theatre). In 1993 my ballet, PTACI, was premiered at historic Mahunovo Divadlo, across a plaza from Janácek Divadlo.

    Though I could have continued my “Sketches” series with a “Brno Sketches,” instead a 2024 work is a set of more abstract variations partly based on and quoting themes from Sinfonietta (in the tradition of Brahms’ Variations on a Theme of Haydn).

    • Variation 1 “Canon” engages that ancient musical technique, evoking Brno’s medieval history.
    • Variation 2 “Overtones” explores two harmonic series, C and Bb, painted over each other in layers of color, with hints of fanfare emerging through the clouds.
    • Variation 3 “Constellations” is a kaleidoscopic succession of large sonorities built on stone-sturdy Perfect Fifth intervals brightened by jazz-like added tones.
    • Variation 4 “Fanfare” is an ostinato pattern-music fantasia on Sinfonietta‘s grand fanfare themes.

    Brno Variations

    Clark 2024 (TC-138)

    ___________

  • journal 7. Carte du Ciel

    U.Mich. Electronic Music Studio, 1975 —

    Mapping the stars —

    My 2024 book, Mapping the Music Universe, begins with recognition of historic, world-changing pioneers in science and the arts. It includes Carte du Ciel (“Map of the Heavens”), an ambitious second phase of an international star-mapping project initiated in 1887 by Paris Observatory director Amédée Mouchez.  A new photographic process revolutionizing the gathering of telescope images inspired the first phase, the Astrographic Catalogue of a dense, whole-sky array of star positions. Carte du Ciel, never completed after 70 years, used the Catalogue as a reference system for a complex survey of the vast field of even fainter images.

    In my 2019 computer music of that title, ghostly wisps of sound are punctuated by brighter bursts, clustered in a natural, not-quite randomly dispersed texture.

    CARTE DU CIEL

    Clark 2019 (TC-98)

    Space sounds

    A pioneering work of early electronic music made a huge impact on my imagination when I first heard it on FM radio in the 1960s. Karlheinz Stockhausen made Kontakte (Nr. 12 in the composer’s catalogue of works) in 1958–60 at the Westdeutscher Rundfunk electronic-music studio in Cologne with assistance from Gottfried Michael Koenig. It originated as a tape piece for four-channel loudspeaker reproduction. The title refers to “contacts between various forms of spatial movement” of the sounds coming from four different directions.

    Stockhausen – KONTAKTE

    Deutsche Grammophon

    LISTEN › YouTube

    American composer Morton Subotnick’s Silver Apples of the Moon was released by Nonesuch Records in 1967. The title comes from a Yeats poem, “The Song of Wandering Aengus”. It was made with a Buchla 100 analog synthesizer, which Subotnick helped develop, a common practice of early electronic music pioneers to build their own tools.

    Part I is a calm exploration of tone quality. Part II generates rapid machine sequences of sounds.

    Silver Apples of the Moon

    Nonesuch Records

    LISTEN › YouTube

    Exigencies

    My works of analog electronic music were composed at the University of Michigan Electronic Music Studio in Ann Arbor starting in 1975. The studio, on an upper floor behind the stage and organ pipes of historic Hill Auditorium, was assembled by Michigan composition professor George Balch Wilson in 1962.

    Patterned after the Columbia-Princeton Electronic Music Center, the studio included reel-to-reel half-inch tape decks running at 15 or 30 inches per second, a mixing board and patch bay, an early model of the famous Moog Synthesizer, other tone generators, and a large wooden coffin containing a heavy metal plate to create electronic reverberation.

    Wilson’s first tape piece is an excellent sample of the analog studio’s sound and capability in expert hands.

    Wilson – EXIGENCIES

    Equilibrium records

    LISTEN › YouTube

    My first large work of analog electronic music, Celestial Ceremonies combines otherworldly sounds made with this now antiquated equipment at Wilson’s U.Mich. Electronic Music Studio. (You may hear a resemblance to the sounds of EXIGENCIES.) Updating my work in 2017 with digital enhancements, I also separated out a suite of four sound sketches with subtitles.

    Celestial Ceremonies

    Clark 1976 (TC-33)

    Dark Energy
    Black Hole
    Gravitation
    Luminescence

    Kraken

    For a sample of my current use of digital synthesis technology, we go back to La Mer. Diving into what has been described as our other unexplored frontier, here is a fantasy sketch of the deep sea on the blue planet.

    Mar Profundo

    Clark 2025 (TC-156)

    ______________

  • MapLab: A Small Sonata

    A sonata is typically a multi-movement piece for solo piano or for an instrument with piano. A shorter form with just three connected sections, the middle slower and quieter, can be called a sonatina. Here is an inside look at how one was composed, step by step. Like the MapLabs in Mapping the Music Universe, this guided tour is in the form of a recipe you can follow to write your own sonata.

    Choose a model

    I started formal composition study in 1968, first with composer Eugene Kurtz, based in Paris but filling in that semester at the University of Michigan. A proponent of modern French music, his compositional models included Debussy and Ravel. He assigned me to immerse myself in deep study of their music, in particular Ravel’s 1905 work, SONATINE.

    I met Beth, a flower lover, in Interlochen in 1975. She had been a promising flute student at Aspen, but was then embarking on a journalism career specializing in horticultural writing.

    The Ravel study came back to me later in my career, as I began to adopt its lush, bright harmonic language and a gentle French Impressionist quality. My SONATINE for Beth (2025) brings together the Ravel study, the flute sound, and (in my video version on YouTube) even the flower motif.

    Start with a generating idea

    The impelling theme can be a melody, a rhythmic pattern, a special kind of chord, or a non-musical image such as a painting or poem.

    Sonatine for Beth is spun entirely from a single harmonic progression, seven chords, each stacking one Perfect 5th interval above another.

    The Perfect 5ths in the two hands are separated by one or more octaves, highlighting this strong interval as a characteristic sound for the piece.

    Now some basic tools to develop and vary a generating theme.

    Transposition

    The whole five-chord progression can be transposed. The harmony is heard plainly in a middle section as ten block chords. The last five chords are a transposition of the first five, up three semitones, starting on the bass pitch Eb instead of C.

    Sequence is successive statements of a pattern transposed by a consistent interval.

    Here is another transposition of the whole ten-chord sequence:

    This harmonic material generates melodic lines and many arpeggio patterns, in successive variations of changing register, intensity, and rhythmic pace. Let’s go through the compositional unfolding of this thematic idea.

    Extract a melody and bass

    Since the starting idea is simply a chord progression, we can select individual tones from each chord for a melody. The most obvious selection is the highest pitch of each chord, even if it is not in a soprano singing range.

    At letter A the melody is given a slightly independent rhythm to help set it off from the chords, in addition to the different sound color of the flute. Also, the lower chord tones are articulated one at a time, making a bass line also rhythmically distinct, faster than the half-note chords. (The Bb in the bass line’s first bar is a passing tone, not a chord tone.)

    Add arpeggios

    An arpeggio is any pattern articulating chord tones one at a time. Usually in order lowest to highest or back down, the individual chord tones can be articulated in any order. At letter A shown above, we already saw the left hand articulate its chord tones one at a time. In the introduction, the right hand is partially broken up into arpeggios.

    In the next variation below, right-hand treble chord tones and still some bass chord tones are arpeggiated. Now all three lines (flute, right hand, left hand) have distinct rhythmic patterns, though congruent with each other in the established 4 4 meter.

    Next, the flute arpeggiates chord tones in eighth-notes, with the left hand simplified to quarter-notes of two pitches from each chord.

    Rhythmic variations

    Variation D simplifies the flute melody to just two half-note chord tones per bar.

    The two hands reunite rhythmically to place some chords after the downbeat and between flute notes.

    Counterpoint

    The original term, contrapunctus, translates “point against point” — two or more independent lines interacting in time.

    A more active rhythm for the flute line leaves time gaps that can be filled in by another line. The right hand selects chord tones to make a similarly playful rhythmic line that mostly alternates and sometimes lines up with the flute rhythm.

    The harmonic progression is still there but just hinted at by the chord tones selected for these interacting lines.

    Variation F continues this back-and-forth rhythmic interaction of the flute and piano right hand, now adding back in the left-hand chord-tone pairs with a simple rhythm for a supporting third contrapuntal line.

    Texture

    Having reached a complex level of three rhythmically interacting, independent contrapuntal lines, a nice contrast will be to simplify. Variation G reduces to a lower-register flute line and only a much simplified skeletal supporting line above it in the right hand.

    Then the texture begins to revert rhythmically to a simpler alignment of all chord tones.

    This paves the way back to a simple piano texture revealing the fundamental thematic chord progression.

    Shape a time form

    What is the plan for the whole? How will the various versions of the generating idea unfold in the larger time span of the whole piece?

    The quiet letter I variation is the apex of an arch form . . .

    • starting with simple
    • building up more rhythmic and textural complexity
    • reaching a stable plateau
    • subsiding back to what started it all.

    That sets up a recapitulation of the whole process, building up textural complexity again, first with the high two-part counterpoint:

    Then with three voices:

    Flute line “calming down”:

    Coda

    A good essay ends with a conclusion or a summary restatement of the thesis.

    Our musical coda summarizes with a last return to the beginning. The chords are back to their very low and very high registers. The flute makes a small melodic arch, ascending to the pitch B, then climbing down gently to its lowest possible pitch, C.

    Fine

    A final edit and audit are mandatory. In the case of our example, listening revealed that the beginning needed a piano introduction with some rhythmic vitality. Some sections were also reordered to improve the flow. Thus, the piece will not begin with a plain statement of the progression, and there will be a somewhat different order of other events.

    Now listen to the whole 6-minute parade of variations on a single chord progression.