The Beginning

Unfolding the map

It is a cliche to observe that music is very mathematical, and millenia of observation and experiment have demonstrated the close connection between mathematics and the behavior of the physical world. This document investigates the idea of a 'physics of music' and constructs a metaphor between musical and physical processes and structures.

Immanuel Kant believed that we have an a priori concept of time and space, and that all our perceptions are prestructured into those categories. Kant believed, on the evidence of Newtonian theory and an absence of any existing alternative models, that time and space were uniform and noncontingent.

Einstein showed that time and space are not static and unchanging; they are dynamically intertwined with each other and the world-lines of observers. Extreme objects and conditions, like those near black holes or at the moment of the Big Bang, distort time and space in unimaginable ways that our mathematical tools have yet to decipher. The ultimate nature of time seems tied to a full theory of quantum gravity and a deeper understanding of cosmological entropy.

We know that music is a highly temporal art - does it have a spatial aspect? I assert that the principles and rules of musical scales, harmony, and form create musical analogs to physical structure and motion. Musical time, like physical time, is not a flat Euclidean expanse, it is affected by the content of the musical manifold just as physical time is affected by the matter energy structure of the gravitational manifold.

Signposts and pointers

Metaphors and models to initiate investigation

The relationship between dance and music is an anchor point for the mapping of music and motion through spacetime. A dancer's body traces out a complex set of arcs, controlled by the levers of the skeletal system and synchronized with a musical piece. The motions of the dancer's body map the music.

The sounds of the natural world create associations with musical characteristics. The birds of the sky make "high-pitched" sounds, the earth produces "low" rumbles. Our language for varying frequency is mapped onto physical space, and in a way that corresponds with experience.

We associate rhythm with motion; the cadence of walking and running are instantly recognizable by ear. When rhythmic values become shorter, we say the music has 'speeded up', like a runner quickening her pace.

These are connections that are allusory and suggestive; the purpose of this document is to investigate the possibility of a formal mathematical connection linking musical structure and fundamental physics.

Human time perception and rhythmic cycles

Days, Years, and Heartbeats: Cyclical rhythmic time

Our experience of time is based on the regularly recurring cyclical events that we use to measure and define it. Start out by forgetting about 'clock time' completely - pretend you have never seen a clock or a watch or heard anyone talk about time. Where does the concept of time begin?

The cycle of day and night is the predominating temporal rhythm in human and animal life. The cycle of the seasons over the course of a year is the other fundamental external time unit that defines our lives.In musical terms, a year is a single measure with 365 beats, one day per beat! On the smaller, personal scale, the pace of breathing and heartbeat create a steady rhythm of life processes.

The defining element of natural time units is that they are based on repetition, on cycles. We measure the time of day by a series of hours which repeate every single day. "Oh, its 7 AM AGAIN" — we say that the 'same' time has returned. This is a circular model of time, or perhaps I should say elliptical, because it has its origin in the elliptical orbits of the bodies of the solar system. This model of time has experiential reality, and it has a more complex structure than the naive idea of time as a uniform straight line.

Musical time is also cyclical. Music is usually constructed from repeating rhythmic units, often in the form of nearly symmetrical 'loops'. A series of 4 bar phrases exploring a theme can create a sense of 'rotational motion', as if the music was orbiting around a central idea. Large scale musical form makes use of substantial exact or parallel repetition.

Pitch space

Pitch, the fundamental dimension of musical space

Pitch refers to the frequency of a note - how "high" or "low" it is. Let's take a minute to study this omnipresent metaphor, which is so common in musical discourse that we forget that it is a metaphor at all.

High and low refer primarily to altitude, but they are analogized to magnitude also — a high number is one of larger magnitude, a high note has a greater frequency. Additionally, the experience of a practicing musician will often add another spatial element, in the form of an instruments keys or frets. The hands of a violinist move across the fretboard, mapping pitch space onto the neck of the instrument.

There is also a connection with the natural world. Birdsongs are high-pitched, as well as sung from high perches! Western music is full of both literal imitations of birdcalls, as well as a wealth of material that echos birdsong stylistically without direct quotation. It is a commonplace of musical description to describe certain types of melody as 'floating' or 'flying'.

The relation of pitch-space to time is crucial for the metaphor we are pursuing. At the base, physical level, pitch corresponds to time — more rapid frequency of vibration = higher pitch. When we hear a melody as moving 'up and down' we are translating a temporal characteristic (frequency of vibration) into a spatial one. Here, time becomes space.

Time scales and phase transitions

Phase transitions between different musical time scales

All musical materials are ways of structuring time. The smallest temporal divisions are the high frequency overtones that provide timbre and 'color' to sounds. At a larger scale, about 40-5000 Hz, are the majority of the commonly used base pitches in western music - the standard notes of the bass and treble clefs.

Rhythm represents levels of temporal division below the audible range. Common rhythmic values range from about 16 Hz (32nd notes at 120 quarters/min) to 0.5 Hz (whole notes at 120 quarters/min). Rhythmic values are grouped together in metric systems such as 4/4, which are in turn organize in terms of phrases. In a standard classical-era work such as a Mozart sonata outer movement, the progression of balanced 8 measure periods creates a 'structural rhythm' of about 4 structural units per minute (assuming 4/4 time, 120 qbpm).

Each structural level is constructed via regular rhythmic division in simple ratios. Musical scales are composed of simple frequency ratios, musical rhythms are again expressed in simple 2:1 and 3:1 units (quarters, eighths, triplets, etc), and common meters and phrase structures are again composed of symmetrical groupings of units, usually by 2:1 ratios — 4 beats per measure, 4 measures per phrase, 2 units of 4 measures a balanced dominant-tonic period.

As arithmetic temporal division is the structural mechanism at each level, the differences between pitch, rhythm, and form are an example of a phase transition in the underlying medium. Rhythm is 'frozen' pitch, possessed of a characteristic scale of a different order of magnitude. These different structural levels can be mapped onto different spatial dimensions. The 'one-dimensional' temporal manifold acquires a fractal dimension as it is subjected to recursive 'folding' on different scales.

A metric for musical space

Measuring musical spacetime

What is the structure of the musical manifold? This is not an esoteric question; in fact, answering it is usually the first step in formal musical education.

The base material of the musical manifold is the frequency/pitch space. The musical scale and the graphical staves that represent it establish a metric upon the surface. The most interesting feature to note is that the tonal metric is logarithmic in relation to the measurement in Hz.

In other words, the musical staff maps the powers-of-two progression of the octaves as a linear progresion; as measured in Hz, an octave in the bass is a smaller interval than an octave in the treble, but they each have the same 2:1 ratio.

In the same way that the Minkowski metric captures many of the properties of Einsteinian space-time, the musical staves express the dynamics of the auditory manifold. The musical scale is a kind of conformal mapping of pitch space.

The Quantum Nature of Musical Materials

Quantization of musical elements in scales and rhythms

The basic nature of sound is smooth and continuous, and not inherently differentiated by anything other than our audible range. As a siren sweeps up and down, it travels through each point of the frequency spectrum. By contrast, a musical scale is composed of distinct steps.

The transition between scale steps is similar to the behavior of a particle such as an electron. The available energy states of an atomically bound electron are quantized as multiples of the fundamental charge. The frequency values available to a musical note are quantized as set ratios of the fundamental tuning pitch.

Musical rhythm is also quantized. The possible durations of a sound are an undifferentiated continuum of length-values, but our musical systems quantize this space into set ratios: the familiar whole note, half note, quarter note system where the rhythmic values are again simple ratios of the given base tempo unit.

This is a foundational element of the mapping between music and physics: the quantum nature of the fundamental,"microscopic" components. A single note is analogous to a small scale element of the universe such as a particle, atom, or molecule, and is characterized by a set of quantized relationships with the given tuning pitch (A=440) and rhythmic value (60 quarter notes/minutes). Of course, the 'quantum' in QED is connected to more than simple quantization. There are additional connections in the form of an 'uncertainty principle' for the significance of a given note.

The Uncertainty principle of note interpretation

Schrodinger's chord: the Uncertainty principle

The physical pitch and duration of a given note may be well-defined, but an uncertainty principle comes into play in the interpretation of a note's musical function. In the musical equivalent of an 'isolated quantum system' we find an analogous role for unpredictability.

The aspects of music we focus on, harmony, melody, form, and rhythm, are all emergent, 'classical' entities — they exist only as a result of the aggregation of multiple musical events. A note in isolation has only pitch and duration, we cannot identify its location within a scale or value within a rhythmic system.

Once we are dealing with a system of several notes, we enter the realm of multivalenced possibilities, and 'collapsing wavefunctions'. A music piece can open with a two-note interval, a 'bare fifth', for instance between C and G. This suggests the key of C, but we do not know the mode. The next note to enter is E natural, and now the harmonic wavefunction collapses to the state of C major.

After a few permutations of these notes, the state evolves further; the pitch Bb is introduced, and suddenly the C Major becomes a dominant-seventh harmony. All the pitches change simultaneously to F, A, and C, and now the system has evolved a 'classical history', and we can place all of these notes within a system of relations: the key of F major, beginning from the dominant and moving to the tonic. However, just as in quantum mechanics, we can see an alternate, less probable possible evolution. Suppose the second set of pitches was not F, A, C, but rather D, F, A? This is also a plausible 'classical history', but it describes a deceptive cadence onto a minor chord.

An overview of the mapping

Overview of modern physics and its mapping to musical processes

Modern physics uses several distinct sets of mathematical tools to analyze the behavior of the universe at various scales. On the microscale, quantum field theory (the standard model of particle physics) describes the fundamental structure and interactions of the building blocks of the universe. In music, this corresponds to our fundamental systems of scales, tunings, rhythms, and meters. Both are built of quantized elements organized by symmetry groups.

The macroscale of cosmological structure is described by General Relativity, which governs systems of size ranging from the universe as a whole to gravitationally bound structures like the solar system. In musical terms, this corresponds to the large scale tonality and form of musical pieces. The harmonic relationships between sections in a piece of music are structured by the musical content in the same way the shape of the spacetime manifold is determined by its matter/energy content. There are musical analogs of the tensorial structures of the GR field equations.

Physics also uses a wide variety of tools to analyze 'intermediate' structures - in other words, the physics of the human scale. These include the traditional tools of Newtonian mechanics, the vast field of condensed matter physics, and, of particular interest to us, the laws of thermodynamics and entropy. This the physics that we have the most direct physical experience of, and it corresponds to the dynamics of music as we hear it - the principles of melodic construction, rhythmic continuity, and local harmonic progression. The relation we will be focused on the most is that between consonance, dissonance, entropy, and the arrow of time.

The structure of the mappng:

The building blocks of music, a note as existing within a given scale and meter, are governed by principles akin to those of quantum field theory.

The large scale structure of music, the principles of tonal harmony and form, are governed by principles akin to those of general relativity.

The content of music as we hear it, a progression of melodies with particular rhythms and harmonies, is governed by principles akin to those of newtonian mechanics and the laws of thermodynamics.

Where to from here?

This is a fragmentary sketch of an attempt to reconstruct and continue an ambitious project mapping the structure of classical music to the entirety of physics. The claim to be made is that the overall structure of reality is manifested on the cosmological scale, in abstract mathematics, in human emotion, and in human artistic creation - and that this overall structure has the same shape everywhere. Where was I going next?

I know that in fact all of this was part of writing BACKGROUND to an investigation to be done with statistical analysis of the tendencies of acknowledged musical masterpieces, and then finding a way to show that there were measurable mathematical correlations between the structural principles of classical music and the laws of physics.

Consonance, dissonance, and entropy and the arrow of time was to be the central study, with the claim that the reason classical music is structured by harmonic tension and release is that this corresponds to the principle of entropy - evolution of the state of the system, but with a big twist. Instead of showing entropy increasing linearly, musical dynamics use increase of entropy - dissonance - to increase the emotional tension, and then resolve this by moving to consonance - lower entropy. Moving to lower entropy on the local scale is possible by dumping waste heat into an external thermodynamic sink so this is not non-physical.

I know there is a lot more even than this and some of it is still rolling around in my brain back behind the Plan 9 obsession. I still play piano and make music of course, but the attempt to show that reality is structured in the same way both in terms of cosmology and human artistic creation seemed to be more than I could reasonably handle. These somewhat fragmentary notes may provide a signpost to someone else with an interest in these topics.

Ben Kidwell


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