This article investigates the nature of "randomness" in music, specifically concerning pitch and tuning systems. I argue that true randomness is rarely achieved, even when using seemingly arbitrary number sources. Through examples and compositional analysis, I will demonstrate how traditional musical concepts continue to influence even the most unconventional pitch choices. The exploration includes the construction of diverse "random" scales, their integration within established music theory, and the impact of measurement challenges on our perception of randomness.
Defining Randomness?
Common responses to inquiries about randomness or aleatority include:
-"Unpredictability of future events."
-"Absence of discernible patterns or order."
-"Reliance on chance and luck."
-"Equal probability of all outcomes."
-"Chaos, lacking apparent causality."
-"Statistical independence of events."
-"Manifestation in natural phenomena (e.g., weather, genetic mutations, dice rolls)."
-"Fundamental aspect of quantum mechanics (e.g., particle behavior)."
These diverse responses reflect varying perspectives and levels of understanding. Individuals draw upon their experiences, education, and the specific context in which they encounter randomness. For example, a statistician might emphasize probability, while a physicist might invoke quantum mechanics.
In essence, randomness is a multifaceted concept intersecting diverse fields of knowledge and human experience. It challenges our notions of predictability and order, serving as both a source of complexity and a catalyst for creativity.
Defining Musical Tuning Systems
Just as the concept of music defies singular definition, so too do tuning systems. A traditional definition might be: a predefined set of pitches available for musical creation and performance.
Tuning systems are often defined by their generation process: a set of rules or algorithms that produce a finite set of pitches. For example, the Pythagorean scale, which yields 12 notes, involves one algorithm for note generation and another for application, the latter often constrained by the instrument's range. This application typically involves a period of repetition, most commonly the octave, also known as the interval of equivalence or "equave." The equave represents the most "informative" interval within the set. For instance, 12-tone equal temperament (12-EDO) offers multiple intervals of repetition, but the octave division is the most intuitive.
Some systems, like the harmonic series, may lack a defined period, as each successive "period" introduces additional notes (e.g., 1, 2, 3, 4, 5, 6, 7...).
Numerous generation processes exist, accompanied by a variety of justifications for their "validity." The standard 12-tone equal temperament, for example, has multiple origins and rationales. The Pythagorean concept of rational number metaphysics persists as a common explanation, despite the inherent "comma" (the misalignment of exponential sequences of 2 and 3). Modern theories often attribute the perceived "goodness" of the 12-tone scale to its approximation of rational intervals involving small prime numbers (3 and 5), though this is a fragile concept. A more robust explanation involves modern consonance models, which consider the complex timbre of sounds like the human voice or plucked strings. These sounds, rich in harmonics (integer multiples of the fundamental frequency), are analyzed through perceptual consonance models based on the beat effect, resulting in a dissonance curve. Applied to harmonic timbres, the minima of this curve align with the pitches of the 12-tone system.
This level of abstraction is crucial for isolating the principle and context of "randomness." By focusing on the "object" as a source of numbers or proportions, we can analyze it more effectively. While numbers may be sourced from various mediums and interpreted as random (even if they are not), the impact of precision and error becomes a key consideration. Furthermore, the extent to which a set of values can be "randomized" by a single defined rule warrants investigation.
Representing a set of numbers as proportions offers the advantage of base-independence. For example, when constructing a set based on the sizes of solar system objects, the specific unit of measurement (meters, inches, etc.) is irrelevant. The proportional relationship between objects, such as the moon's approximate quarter-size relative to Earth, remains constant. By normalizing to one value within the set, and because we will be creating periodic systems, any value as base renders the same set. (We will also create other types, non-periodic).
Examples of "Random" Scale GenerationThe following examples illustrate the creation of musical scales using "random" numbers derived from various sources. These examples demonstrate how even seemingly arbitrary number sources can generate musically coherent results, further supporting the argument that true randomness is elusive in this context.
Planetary Data and the "Music of the Spheres"The concept of the "music of the spheres," associating celestial bodies with musical harmony, has resonated across cultures, from ancient Greece to pre-Columbian America. While some specific examples of simple harmonic ratios exist in celestial mechanics (e.g., orbital resonances), many planetary properties do not readily translate into easily recognizable musical intervals. This section explores the creation of musical systems based on planetary data, examining whether these seemingly arbitrary values can generate musically meaningful results.
Scales were constructed using data from NASA (2018), specifically:
- Average surface temperature
- Orbital period
- Planet size (including the Sun)
Pitch generation employed octave equivalence. For example, in the planet size (diameter) scale, values were normalized relative to Earth (Earth = 1). The Sun's diameter, for instance, is approximately 109 times Earth's. These normalized values were then octave-folded into the range of 1 to 2 (representing Earth to "2 Earths") and then duplicated to cover the audible or instrumental range. This process was repeated for the other planetary properties.
The resulting music reveals that these seemingly arbitrary values can generate surprisingly stable chords and progressions, sometimes even exhibiting a clear tonal center. Most remarkably, the scale derived from planetary sizes contains a fully functional pentatonic blues scale, inspiring the track title "The Astrocaster Blues." This suggests a deeper, perhaps unexpected, connection between planetary proportions and familiar musical structures.
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Video.01 Description: "Astrocaster Blues"
This video showcases the planetary diameter data used to calculate the pitches for "Astrocaster Blues." It displays the numerical values and visualizes the scale construction process.
A prominent feature of the video is a pitch dial, displaying a single octave for each instrument (piano, guitar, and bass). This allows viewers to clearly see the interactions of chords and the intervallic relationships within the scale as the music is played.
The most striking aspect of this scale is the presence of a fully functional pentatonic blues scale. This led to the lighthearted observation that the search for extraterrestrial life might be best focused on solar systems with a high potential for blues musicians. The irony is that, within this planetary-diameter-derived blues scale, Earth itself is assigned the "bluesy" microtonal inflections. While the pitches are normalized relative to Earth's diameter, the tonal center of the music gravitates towards Saturn.
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Video.02 Description: "The Dance of Entropy"
"The Dance of Entropy," based on planetary temperatures, is a waltz-like composition with a distinctly 18th-century European neoclassical orchestral vibe. The addition of a bandoneon, however, infuses the piece with a touch of tango, creating a unique blend of styles.
This video provides a visual representation of the scale constructed using planetary temperatures and the resulting musical composition, "The Dance of Entropy." It clearly displays the planetary temperature values used in the scale's construction, along with a step-by-step visualization of the process itself.
A key feature of the video is a "planet grid-keyboard" that illuminates the notes as they are played. This allows the viewer to directly observe the consonant relationships within the scale and identify the tonal center of the music.
It's important to note that the order of the planets on the grid-keyboard does not correspond to their spectral order within the solar system. The octave folding process used to create the scale results in a different arrangement of pitches. As with other planet-based scales in this work, Earth is used as the base for normalization. However, because these systems are periodic, the choice of base is inconsequential; the resulting musical relationships remain consistent regardless of which planet is used as the reference point. The scales are not geocentric in any meaningful sense.
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"Soles Mortem," using orbital periods, produced the least consonant of the three scales. While it still contains numerous usable chords (as demonstrated in the audio example), identifying a stable tonal center within traditional musical frameworks proved challenging.
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Order or Chance?The question arises: is it even possible to encounter a scale derived from planetary data that is truly unusable? The consistent emergence of musical coherence from these seemingly "random" sources suggests that, even within the vastness of space and the complexity of celestial mechanics, underlying order and harmonic potential may be far more prevalent than one might initially assume.
Clearly, these planetary data do not conform to the strict definition of randomness—"all outcomes have equal chance"—as the number and sizes of planets within a solar system are inherently constrained. The question remains: is it randomness, or simply a form of order we have yet to fully comprehend?
Furthermore, the near-uniform distribution of these values within the octave raises another crucial point. Why is it so difficult to find a set of "random" numbers that, after octave folding, reveals only near-octave values or a cluster of notes that lack any tonal meaning? Is this simply a matter of chance?
It's unlikely. The consistent emergence of usable musical scales from diverse "random" sources, including planetary data, suggests that there's more at play than mere coincidence. The constraints of human hearing, the inherent properties of number relationships within a bounded space (like the octave), and perhaps even deeper, undiscovered principles of harmonic organization seem to guide the formation of musical structures, even when starting from seemingly arbitrary data. The "randomness" we perceive may be, in fact, a manifestation of underlying order that we are only beginning to understand. The universe, it seems, may be inherently musical.
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The Riemann Zeta FunctionThe Riemann Zeta function, a complex-valued function with deep connections to number theory, was used to generate a set of pitches for musical composition. The imaginary parts of the Zeta function's zeros, while not truly random, exhibit statistical properties that make them a suitable source of seemingly random numbers. Unlike previous examples that focused on selecting a small set of values and applying a fixed interval of equivalence (like the octave), this approach directly utilized 28 consecutive imaginary parts of the Zeta function's zeros. These values were interpreted as cents and directly applied to control synthesizer pitches.
Octaves, in traditional periodic tuning systems, provide confinement for the total pitch availability. Knowing that any interval present in one period is found in the next (up or down, depending on the instrument's range) allows for predictability, manageability, and perceptual substitution of pitches. The scale constructed with Riemann Zeta function values doesn't inherently contain octaves. Any octaves, or approximations thereof, that appear, do so by chance, as do other consonant intervals. This absence of a pre-defined octave is a key element of the experiment, challenging conventional notions of consonance and scale construction.
The deliberate omission of a defined equave or period of repetition makes the results particularly intriguing. As demonstrated in the audio example below, the resulting music, while using an unconventional scale, sounds surprisingly "normal." It rarely registers as overtly microtonal. Instead, it possesses a rock-folk-like quality, suggesting the use of unusual but not entirely foreign scales. It certainly does not sound atonal or xenharmonic. Clear, recognizable chord progressions emerge readily, and consonance is not compromised across a wide range of timbr
(Zeta function 28 notes music)
The timbres, all synthesized, contribute to a rich sonic spectrum. The distinct harmonic characteristics of the synthesized guitar and strings create clear timbral differentiation. Some instruments handle otherwise dissonant intervals more gracefully than others. The guitar, with its inherently harmonic timbre, serves as a kind of consonance "stress test." If an interval sounds good on the guitar, it generally passes a basic consonance check, even if that consonance is subjective. Essentially, if it sounds good on the guitar, it's likely to be perceived as consonant.
The resulting musical texture underscores the central point: these values, derived from a complex mathematical function, do not sound as "random" as one might initially expect. The emergence of familiar musical elements, even without a pre-defined tonal framework, suggests an underlying order and reinforces the idea that true randomness is elusive in musical contexts.
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| Interval Matrix for the Riemann 28 note Scale |
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The Question of RandomnessThe observation that simple octave folding of seemingly random values consistently leads to relatively uniform interval distributions, often guaranteeing a degree of tonality, raises fundamental questions about the nature of randomness itself. It forces us to reconsider not only the "randomness" of our chosen values but also the "randomness" of our conventionally accepted methods for scale construction.
A previous study [link] analyzing the Scala Archive (a vast library of over 5000 world tunings) revealed a striking phenomenon. Generating random pitch sequences using even simple pseudo-random number generators (like those built into web browsers) often resulted in scales that closely approximated (within ±10 cents) or even perfectly matched existing scales in the archive. These archived scales, of course, have established origins, structures, and "mathematical justifications."
This finding is significant. It suggests that what we perceive as "random" in the context of pitch spacing may be less random than we think. The inherent properties of number relationships, coupled with the constraints of human hearing, appear to guide the formation of musical scales, even when starting from seemingly arbitrary data. This raises a profound question: what, then, is true randomness, at least within the realm of musical pitch and human perception?
The Challenges of Measurement and Their Impact on "Randomness"
As previously discussed, some "random" number sources, such as the output of random functions, were used directly without imposing a specific period or equave. For instance, one scale was derived directly from the zeros of the zeta function, with values interpreted as cents.
Other sources, like planetary properties (e.g., sizes interpreted as Hertz values), required a different approach. Scales were constructed from a fixed set of values (e.g., ten planetary sizes). However, to expand these sets to a wider pitch range, a rule for extending the values was necessary. While the planetary sizes themselves might be considered a source of "randomness," the choice of how to extend their range (e.g., by octave transposition) could introduce additional bias. Simply using octaves based on Earth's values, while seemingly logical, doesn't constitute a purely "random" approach. Compositions were created based on various planetary properties, like size, temperature, and rotation, including their octave transpositions.
A more complex challenge arose with data from sources like mountain heights. Using the "fourteeners" (the tallest Himalayan mountains), the very definition of height became problematic. The influence of sea level, for example, significantly affects perceived height. The same mountain, measured from different baselines, will have different proportional heights relative to other mountains, even though the absolute difference in height remains constant. This necessitated the introduction of both an equave and a sea-level reference point. Music based on these mountain heights has also been composed.
Images Description:
This image visually represents the process of constructing the Himalayan tuning scale. The fourteen highest peaks in the Himalayas (the "fourteeners"), a list derived from mountaineering tradition and historical convention rather than strict geological definition, are depicted as individual rectangles, arranged horizontally to illustrate their relative heights above the 8000-meter mark (indicated by the red baseline). It's important to note that these peaks are geographically dispersed across the Himalayan range; their proximity in the image serves only to convey their height relationships. The inherent ambiguity in defining a "peak" (as opposed to a sub-peak or shoulder of a larger mountain mass) is also acknowledged, highlighting the challenges in establishing a definitive list.
Below the height representation, the image demonstrates the octave duplication process used to extend the scale. While the typical method involves octave folding the initial set of mountain heights into a single octave and then duplicating that octave, the image illustrates an alternative, but mathematically equivalent, approach: multiplying and dividing the original heights by powers of 2 to generate octaves, and then selecting the results that fall within the audible range. The duplicated heights are marked on the image, and some may visually align with other mountain silhouettes in the background. These background mountains are purely illustrative; the fourteeners (shown in white) are the primary focus of the scale construction.
The resulting music evokes the epic and adventurous spirit often associated with the Himalayas, reflecting the grandeur and challenge of these mountains.
The choice of sea level appeared to be a more significant factor in altering the proportional relationships within the scale than the choice of octave. Small changes in sea level resulted in drastically different proportions between mountain heights (whether analyzed in octave space or any other proportional space). While all combinations of the 14 mountain heights were theoretically possible, certain proportions became more probable than others due to the influence of the chosen sea level.
Illustrative Examples: Normalization and Equivalence
To illustrate the impact of these measurement choices, consider a simplified example. Let's start with a set S = {2, 800, 1040}. We can normalize this set by choosing a base value (e.g., 2) and dividing all elements by that base: S(base:2) = {2/2, 800/2, 1040/2} = {1, 400, 520}.
Next, we can create a new, reduced set by applying an equivalence relation, such as the 1:2 octave relationship. We find representatives of 400 and 520 within the octave range (1 to 2).
400 / 2^8 = 400 / 256 = 25/16
520 / 2^9 = 520 / 512 = 65/64
The minimal generating set in octave space becomes {1, 65/64, 25/16, 2}. This set preserves the original proportions of S within the octave.
With unambiguous measurements (like planetary sizes), this process works well. However, with context-dependent measurements (like mountain heights), changing the sea level alters the proportions, rendering the normalized set no longer representative of the original relationships. For example, adding a constant delta of 150 to each element of S results in a completely different generating set.
The Challenge of Precise Proportional Calculations
Calculating precise proportions within a given musical space presents a significant challenge, particularly when dealing with numbers spanning vastly different orders of magnitude or when measurement precision is limited.
Let's first illustrate a scenario where this problem is less pronounced. Consider creating a scale based on the sizes of two planets. Assume their sizes are A = 200 and B = 1200 (in some arbitrary unit). The first step is to normalize the values by choosing a base. Using A as the base, we get {A = 200/200, B = 1200/200} = {A = 1, B = 6}. Next, we define our musical space, in this case, the octave (a 1:2 ratio). A remains at 1 (the unison). B must be scaled to fit within the octave (1 to 2]. We divide B by 2^2 (4) to get 6/4 = 3/2 = 1.5, representing a perfect fifth. Our generating set is {1, 1.5}. We can extend this scale by repeatedly multiplying by 2 (within the instrument's range).
In this scenario, small variations in the initial measurements have minimal impact on the final proportions. For example, if the measurements were slightly off (e.g., {201, 1205} or {199.32142, 1200.0000001}), the resulting proportions within the octave remain practically the same. The generating set might become {1, 1.50003}, but this tiny difference is negligible in musical terms. Planet B is still perceived as roughly a fifth above planet A.
The Problem of Scale and Precision
The problem becomes much more acute when dealing with values that span a vast range, such as particle energies, which can range from giga-electronvolts (GeV) to electronvolts (eV). Measurements at these scales often have varying degrees of precision. Consider a simplified example (not a real-world physics case) to illustrate the issue. Suppose we have two particle energies: A = 1000 and B = 0.09155... Normalizing to A gives us {1, 0.0009155...}. Scaling B to fit within the octave (1 to 2] requires multiplying by a power of 2. In this specific example, B * 2^x happens to equal 1.5, a perfect fifth.
So far, so good. But what if the measurement of B was slightly different due to limitations in precision? Let's say B = 0.0781 (a seemingly small difference). Now, when we scale B to fit within the octave, we get a different result: B * 2^n = 1.25, a major fourth. A tiny change in the initial value of B has resulted in a significant change in the musical interval.
Therefore, unless we have extremely precise values for particle energies (which span an even wider range than our simplified example), we cannot reliably claim consistent proportional relationships within a musical space. While we can say that "planet B is a fifth of planet A" with reasonable certainty, saying that "an electron is a major fourth of a muon" based on imprecise energy values would be misleading. The inherent uncertainty in the measurements prevents us from establishing such precise musical relationships.
Nature of Musical StructureThis exploration, while seemingly trivializing historical efforts in scale creation, is not intended to diminish their significance. Rather, it builds upon the observation that the Scala Archive contains over 5000 documented tuning systems, raising the question: does everything sound good? My approach of constructing scales from random sources is primarily for inspiration. The resulting scales often either already exist within the archive or possess inherent musicality that can be further enhanced with appropriate timbral choices.
This leads to a fundamental question: what are the underlying principles that govern musical structure? Octave equivalence, periodicity (though not always present, as seen in the Riemann scale example), and a tendency towards uniformity across the hearing range, with a bias toward octave-based relationships (even when the octave is not pure), appear to be recurring themes.
But what about the diatonic and pentatonic scales? The major chord itself? Are these truly foundational elements of music, or are they primarily learned constructs shaped by cultural context and familiarity? Is there anything objectively special about the function of tonal modes or chord inversions? Or are they simply products of neurological evolution favoring harmonic timbral preferences, coupled with cultural development?
The fact that the diatonic scale appears in diverse and geographically separated cultures suggests that it may not be purely a learned construct. It likely has a deeper connection to timbre perception, as recognized by modern music theory.
The Role of Human InterventionWhile the selection of pitches in the scales described above was often based on "random" sources, it's crucial to emphasize that the composition of the music was not. A human mind, with its inherent perceptual biases and musical understanding, ultimately shaped the final musical output. The composer, working within the constraints and possibilities presented by the "random" scale, makes choices about melody, harmony, rhythm, and form. This human element is essential for transforming a set of pitches into a meaningful musical experience.
This distinction is important. I've also explored algorithmic and generative music for years, both by studying the work of other artists and by developing my own algorithms. For example, this app [link] allows users to create music based on cellular automata (like Conway's Game of Life), translating cell states and coordinates into musical parameters.
While these explorations have yielded fascinating results, I've yet to discover a simple or complex algorithm that consistently generates music that sounds convincingly human. Many algorithms can create interesting sonic textures and patterns, but they often lack the expressive nuances, the sense of narrative, and the subtle irregularities that characterize human-created music. The only algorithms that seem to approach this level of "human-likeness" are those based on artificial intelligence and machine learning. These AI systems, however, operate on a fundamentally different principle than simple, rule-based algorithms. They learn patterns from vast datasets of existing music and then generate new music based on these learned patterns, rather than following a pre-defined set of instructions.
Therefore, while randomness can play a role in pitch selection, its impact on musical composition is less direct and less compelling. Human intervention, with all its subjective biases and creative insights, remains a crucial ingredient in the creation of music that resonates with human listeners. The "randomness" of the initial pitch set, in a sense, becomes a canvas upon which human musicality is expressed.
Fitting "Random" Scales within Established FrameworksEarlier, I mentioned that these "random" scales could be understood within the context of established music theory. While the preceding examples demonstrated this through the creation of musically coherent pieces, the underlying framework deserves further explanation. The sheer existence of the Scala Archive, with its thousands of diverse tuning systems, provides compelling evidence that, in a broad sense, "anything works" musically. However, we can be more specific about how these "random" scales relate to established theoretical frameworks.
Modern music theorists have explored dividing the octave into an increasingly large number of intervals, often with the goal of cataloging and analyzing scales that more closely approximate specific intervals of interest, such as perfect fifths. However, the limits of human pitch perception must be considered. The just noticeable difference (JND) for pitch, averaging around 10 cents in the central hearing range, means that many of these highly refined scales contain distinctions that are imperceptible to the human ear. What, then, is the practical purpose of constructing scales with hundreds or even thousands of divisions per octave if these microtonal nuances are not perceivable? Such explorations are, of course, valuable from a theoretical standpoint, but their direct relevance to musical practice is less clear.
This framework of highly granular octave divisions, however, provides a context for understanding how our "random" scales can be "fitted" into established musical thinking. Any of these randomly generated tuning systems can be considered a subset of a highly divided equal temperament (e.g., 100-EDO or even less). For example, analysis of the Riemann Zeta function scale using an interval matrix reveals that numerous 12-EDO approximations (within ±15 cents) are present at various transpositions. This demonstrates that even seemingly arbitrary number sources can generate scales that, upon closer examination, relate to familiar tonal structures.
This brings us back to the central question: what constitutes "random numbers" in this context? If even "random" data can be mapped onto existing musical frameworks, are we truly dealing with randomness, or simply exploring different facets of a deeper, underlying order?
The Chroma Theory: Connecting Music and Color
This work on "random" scales and the exploration of musical universals led to the development of the Chroma Theory, an in-depth study of the relationship between color and musical notes—another seemingly universal association. The first version of this ongoing study is available here: [link].
The study, which delves into afterimages, color constancy, and stereoscopic color mixing, concludes that color perception and categorization are indeed related to musical chroma and pitch categorization, revealing shared perceptual limits and ambiguities. Just as we might ask, "Is this interval a sub-minor third or a super-major second?", and resolve the ambiguity through musical context, we also ask, "Is this a yellowish-orange or an orangish-yellow?", with the context of visual perception providing the answer.
The construction of musical scales using data from hue-wavelength matchings yielded stable musical structures, as one might expect given the "randomness" argument. However, a key difference emerges: visual chroma is a "stand-alone" property, while musical chroma is inherently a two-note (interval) relationship. I propose that our tonal and harmonic preferences are fundamentally shaped by a shared perceptual mechanism involving color perception and its categorization. The fact that we use the same words (chroma, tone, brightness, color/timbre) across languages to describe both auditory and visual experiences suggests that our intuitive understanding of these phenomena is not far off.
The Chroma Theory study, which develops a color model to predict various perceptual phenomena, provides numerous examples supporting this connection. Newton wasn't wrong: the rainbow is an harp.
