Interval Space Randomness

This section explores the generation of "random" scales (tuning systems, pitch sets), examining formal definitions of randomness and their manifestation in pitch selection. Surprisingly, generic mathematical disorder does not appear to erode available tonality. Although the generated scales consistently fall outside standard 12edo and Just Intonation tolerance windows across all rotations—suggesting they should sound wildly microtonal or unusable—audio examples demonstrate otherwise. The created music can sound remarkably conventional, revealing that compositional structure can guide listeners' categorical expectations to assimilate unfamiliar acoustic material. While these scales generally cannot support all musical styles, randomness alone fails to prevent local coherent structures from forming stable, larger compositions. Building on these findings, the work expands into the concept of "tonal constancy"

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This article examines the nature of “randomness” in music, focusing on how arbitrary pitch data interacts with listener tonal expectations. One might assume that generating pitches from non-musical sources—such as planetary data, mathematical attractors, zeta zeros, terrain elevations, or light wavelengths—would yield unstable, incoherent, or musically unusable results. Yet the examples presented here demonstrate the opposite: even highly irregular pitch sets with no internal design principles and no relation to 12-EDO frequently allow for the production of recognizable chords, tonal centers, and harmonic progressions.

This paradox arises primarily because “randomness” lacks a unified definition across disciplines. It may refer to unpredictability, statistical independence, or the simple absence of intentional structure. For the scope of this study, our narrow operational definition treats randomness as the absence of deliberate musical scale design. The scales used here were not crafted to reflect any tuning tradition or harmonic logic; some were merely constrained to a usable pitch range via octave folding, while others underwent no folding at all.

Despite this lack of musical intent, the statistical density and uniformity of the generated values still give rise to stable tonal functions. This suggests that tonal order does not strictly require intentional pitch selection. Instead, tonality can emerge through the listener’s perceptual framework, which actively maps unfamiliar acoustic input onto high-probability categories derived from diatonic experience.

This article frames these findings within established music theory, examining how measurement, folding methods, and categorical perceptual biases shape our experience of “randomness.” Ultimately, the accompanying audio examples serve as primary evidence: they reveal that even when structural intention is removed, tonal possibilities persist. The challenge of random tunings, therefore, is far from straightforward.

Nature rarely produces "white noise".

To argue that arbitrary pitch sets can support coherent music is not to claim that traditional scale design is an arbitrary illusion. Historical tuning systems—whether hyper-diatonic Just Intonation or equal temperaments—are optimized configurations that maximize acoustic resonance, timbral alignment, and intervallic symmetry. What these experiments reveal is not that traditional scales are unnecessary, but that the human capacity for tonal organization is far more resilient than traditional theory assumes. Tonal Constancy does not require ideal acoustic conditions; it requires only a continuous pitch space and strong compositional syntax to build stable perceptual worlds.

Music theory isn't a delusion, but it is an over-determined description of an under-determined signal.

Defining Randomness

Common responses to inquiries regarding randomness or aleatoricism typically encompass several distinct angles:

Unpredictability: Uncertainty regarding future events or outcomes.
Absence of Pattern: A lack of discernible structural order.
Equiprobability: Uniform probability distribution across all possible states.
Causality Breakdown: Stochastic processes lacking apparent deterministic origin.
Process vs. Phenomenon: Statistical independence in data processing, or physical manifestations in natural systems (e.g., genetic mutation, quantum mechanical state collapse).

These responses highlight how randomness is fundamentally context-dependent. A statistician prioritizes probability distributions; a physicist invokes non-deterministic mechanics.

In music, however, the situation is fundamentally different. An inference engine like human tonal perception does not process raw, uninterpreted data. Tonal randomness is relative and observer-dependent; consequently, standard mathematical disorder does not automatically cause the perceptual system to fail.

Informational Randomness vs. Categorical Complexity

To understand this discrepancy, we must separate informational (external) randomness from categorical (internal) complexity:

Informational Complexity: Kolmogorov (or algorithmic) complexity offers a near-Platonic definition of observer-independent randomness: a sequence is random if it is incompressible, meaning its shortest description is the sequence itself.

Categorical Complexity: A tuning system can possess maximal algorithmic complexity yet remain functionally coherent to a listener. Consider a frequency ratio infinitely close to a perfect fifth \(\approx 1.5\). Its decimal expansion may form an incompressible, algorithmically random string of digits. Yet to a tonal observer, its categorical complexity is minimal—it resolves cleanly into the perceptual category of a fifth. The informational "randomness" is merely noise in the decimal representation, not an intrinsic property of the pitch relationship.

Model-dependent definitions of randomness—such as Martin-Löf randomness, where a sequence is deemed random if it passes every computable statistical test—provide a far better starting point for musical intervals. Under this framework, a pitch sequence is random within a tonal system if and only if it fails the "tonal test."

Furthermore, a scale or tuning system is not merely a static sequence of numbers; it defines the available state space for melodic and harmonic progressions. Listeners do not perceive a tuning system simultaneously. When a pitch space is sampled sequentially over time, a flexible perceptual system with strong priors can actively reconstruct tonal categories by exploiting local consonances and expectation-building structures.     

Order, Predictability, and Improbability

This perceptual mapping illuminates the frequent conflation of order, predictability, and probability:

Order vs. Probability: A sequence of 100 consecutive zeros exhibits low Kolmogorov complexity (highly ordered), yet in a fair binary process, it represents an extremely improbable event. Order does not negate randomness if a sequence persistently defies probabilistic expectations.

Specialized Predictability: Western music is generally highly predictable due to structural regularities, phrasing, and tonal hierarchies. However, this does not make music probable in an absolute statistical sense. Music is predictable only because human cognitive predictors are extraordinarily specialized.

Observer-Dependent Perception: To a "simple" observer lacking low-complexity tonal priors, a Bach fugue is indistinguishable from white noise because the observer lacks the specific compression algorithm (tonality) to parse the pattern.

Predictability is local; improbability is global. Music can thus be understood as a form of coordinated improbability.

Constructing the "Unusable" Scale

How, then, does one construct an genuinely "unusable" pitch collection—a scale that actively resists assimilation into familiar tonal categories?

If tonal recognition were purely statistical—reducible to static tolerance windows around standard 12-EDO pitch classes—expectation-driven pitch flexibility would remain unexplained. In real listening conditions, pitch deviations can exceed 100 cents, and such discrepancies accumulate without collapsing the active tonal center.

Early music cognition research (e.g., Roger Shepard, Diana Deutsch, Carol Krumhansl, Ernst Terhardt) demonstrated that pitch perception is neither passive nor fixed. Musical hearing is dynamically shaped by context, expectation, and learned tonal inference.

Consequently, even when a tuning system fails every conventional tolerance test against standard pitch categories—regardless of interval rotation or acoustic dissonance—compositional strategy can still impose tonal order. Through cadential syntax and structural expectation, highly irregular pitch sets can be rendered functionally diatonic.

These findings challenge conventional formalizations of musical randomness and highlight a frequent misconception in microtonal theory: while some tuning systems function as hyper-diatonic refinements optimized for acoustic resonance, others represent fundamental reorganizations of tonal logic—offering not "better fifths," but entirely distinct structural frameworks.

Ultimately, randomness in pitch selection is far from a trivial problem. These experiments demonstrate that convoluted or non-musical pitch-generation procedures are readily absorbed by human tonal perception through compositional context. The ease with which coherent music emerges from radical, non-standard pitch structures underscores the remarkable resilience of tonal organization.

Operational Bounds of the Test: What Makes a "Fair" Random Scale?

The moment we define what constitutes a "fair" random set for musical human hearing, the premise unravels in a fascinating way: the "bias" isn't a cheat code but a prerequisite for auditory perception to exist at all.

If we hand an observer one frequency, 10,000 frequencies squeezed within 10 cents, or steps that jump 8 octaves at a time, we haven't created a "more pure" random scale. We've simply exited the operational bounds of human pitch-space cognition.

framing the boundary conditions:

To test whether arbitrary data can support tonality, the pitch collection must meet basic cognitive and physical constraints:

Audible Bounding: Pitches must land within the human hearing/instrumental spectrum (roughly 20 Hz to 20 kHz, weighted toward the mid-range) 

Granularity Threshold: Interval steps cannot be smaller than the Difference Limen for frequency (otherwise it collapses into continuous noise/unresolvable cluster) nor so sparse that no melodic step-wise motion is possible. 

Continuity (The "Stair" Condition): The set must form a reasonably consistent "staircase" across the spectrum—avoiding giant blank abysses or hyper-dense clumps.

Once a generator satisfies these basic perceptual criteria, it creates a viable state space. 

The punchline? Almost any arbitrary dataset that satisfies these basic operational constraints will yield usable tonal structures, it requires more than noise for a pitch set with these restrictions to escapesdiatonic logic, see Tonal Constancy

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 (12ed2) offers multiple intervals of repetition, but the octave division is the most intuitive. (for example, \(6\text{ed}\sqrt{2}\) 6 divisions of the square root of two, is the same)

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...).(after every new octave, more notes are between, no translation symmetry)

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). Canon theories, like Just Intonation, often attribute the perceived "goodness" of the 12-tone scale to its approximation of rational intervals involving small prime numbers, though this was more belief than proof (note that i'm a fan of Just Intonation systems, but as music theory framework). A more robust explanation involves modern consonance models, which consider the complex timbre of sounds like the human voice or plucked strings. sounds , are analyzed through perceptual consonance models based on the beat effect, resulting in a dissonance curve. Applied to harmonic timbres (overtones are integer multiples of the fundamental frequency), the minima of this curve align with some of the pitches of the 12-tone system and/or J.I. intervals.(see note on consonance)

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 is a puzzle

Representing a set of numbers as periodic 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).

Note on Consonance

Physicists and mathematicians sometimes joke that "music is solved" upon encountering modern consonance models—particularly William Sethares’ framework for sensory dissonance. The joke highlights a fundamental misunderstanding: while these models illuminate important acoustic mechanisms, they are far from an exhaustive account of musical perception.

Traditionally, Western music theory prioritized small integer frequency ratios (2:1,3:2,4:3) as the objective foundation of consonance. This perspective was historically reinforced by string physics, where overtone series naturally exhibit simple integer relationships. Yet as a universal model of musical organization, this premise was never fully established; it persists primarily as an intuitive theoretical framework rather than a complete perceptual law.

Sethares’ model—extending the foundational 1965 work of Plomp and Levelt on critical bandwidths and beating—demonstrates how a sound's spectral timbre can be algorithmically tailored to minimize sensory dissonance across specific intervals. 

However, minimizing dissonance is rarely the sole objective of composition; musicians frequently seek the opposite effect for expressive tension. "Music is solved" only if one reduces music to a singular optimization problem: arranging fundamentals and overtones so that acoustic beating disappears.

Critical perceptual dimensions remain that sensory dissonance models leave unaddressed:

Melodic Context vs. Harmonic Beating: Sensory dissonance models focus heavily on simultaneous, sustained tones. In sequential, melodic contexts, perceived "out-of-tuneness" is often a function of category mismatch or unexpected pitch trajectories rather than physical acoustic interference.

Pitch Cyclicity and the Equave: For a harmonic timbre, the octave (2:1) is far more than a point of local dissonance minimization—it acts as an identity shift and perceptual equivalence class. While one can synthesize an inharmonic timbre whose dissonance minimum lands on a non-octave interval (e.g., 3:1 or 2.1:1), this acoustic shift does not automatically replace the deep cognitive machinery of octave equivalence. (See Spectral Congruence.).

To isolate the cognitive capacity for tonal organization from timbral assistance, the experiments presented here deliberately avoid dynamic timbre matching. All musical examples utilize familiar, standard instrument timbres—such as pianos, acoustic strings, and plucked guitars—forcing the listener's pitch inference system to process the raw, un-adapted interval structures.

The Probability of Order

If we temporarily suspend the purely musical goals of this study, we are left with a fundamental statistical question: What is the probability that a set of unrelated, arbitrary values—when normalized to an octave framework via logarithmic modulo reduction \(\log_2 n \bmod 1\) (or not)—will yield a sufficiently uniform distribution to support a functional scale?

Furthermore, what are the odds that such a distribution will not only achieve spatial evenness, but actively enable traditional tonal utility?

As the results show, the probability is higher than traditional microtonal theory assumes.

This suggests that functional tonality is not a fragile, hyper-specific artifact of careful mathematical design 12-EDO or pure rational ratios), but rather a statistical inevitability of uniform distribution within a bounded state space

Once an arbitrarily generated pitch set achieves a critical threshold of density and evenness across the pitch-class cycle, two complementary forces take over:

1.Statistical Density: A uniformly distributed pitch space guarantees that for almost any desired tonal trajectory or step-size interval, a "near-enough" pitch candidate exists within the set.

2.Perceptual Assimilation: The human cognitive system does not require zero-error acoustic alignment to establish a tonal hierarchy. Instead, strong compositional syntax (rhythmic grouping, metric placement, voice leading) acts as an error-correcting filter, pulling the densely available pitches into active, functional tonal categories.

Under this view, tonality is not an elusive needle in a haystack of noise; it is an emergent property that arises naturally whenever a pitch collection provides sufficient density for cognitive category attraction.

[In pure math and statistics, this phenomenon is intimately tied to Equidistribution Theory (such as Weyl’s Equidistribution Theorem for fractional parts like \(\log_2 n \pmod 1\) and Maximum Entropy Distributions. When you fold arbitrary continuous datasets mod 1, they very quickly smear into a remarkably uniform, continuous probability distribution.]

The Perceptual Stress Test

Traditional microtonal analysis relies on deviance metrics—calculating how far a given pitch set strays from an idealized reference point, such as Just Intonation ratios or 12-EDO pitch classes. However, this approach assumes a static, passive listener. This study posits that pitch perception is fundamentally path-dependent: the functional meaning of a pitch is governed by its sequential trajectory, voice-leading syntax, and local metric context, rather than its isolated frequency value.

Consequently, static mathematical proximity is a poor predictor of whether a scale can support coherent music. A static deviance chart cannot account for Tonal Constancy—the real-time, active cognitive mechanism that reconciles acoustic discrepancies against strong perceptual priors.

To isolate and stress-test this mechanism: We generate pitch collections from non-musical or stochastic data streams—such as planetary orbital parameters, mathematical attractors, or raw noise distributions—and subject them to compositional constraints.

The core of this test is straightforward: Take an arbitrary, un-designed collection of frequencies and construct a piece of music that functions normally within a recognizable tonal grammar.

The accompanying audio examples serve as functional proofs. They demonstrate that "musicality" and "tonality" are not intrinsic, immutable properties hidden inside specific frequency ratios. Rather, tonality is an emergent property created when compositional syntax actively guides human cognitive inference to impose structural order onto chaotic data.

Examples of "Random" Scale Generation

The 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. The frequency and pitch positions are flexible enough for the brain’s “error correction” to fill in predictions, and many of these scales unintentionally resemble “maximally even” sets, the mathematical basis of diatonic scales (our perception of key and tonality is probabilistic. Temperley, D. 2007. Music and Probability).

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 (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. The scale derived from planetary sizes allows a fully functional pentatonic blues scale, inspiring the track title "The Astrocaster Blues."

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Planetary Diameter

Table.01: Solar System Object Size - Planetary Diameter, Tuning Construction

	Sun	Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune
Index(i)	0	1	2	3	4	5	6	7	8
Size(Base:Earth)	109.2	0.3829	0.9499	1	0.5320	10.97	9.14	3.981	3.865
Octave-Space fold: \( P_i \longleftarrow P_i \times 2^n, \, n \in \mathbb{Z} \Rightarrow P_i \in (1,2]\)	1.70625	1.5316	1.8998	1	1.064	1.37125	1.1425	1.9905	1.9325
New Index(j)	5	4	6	0	1	3	2	8	7

Interval Matrix for the Diameter Tuning
The set and all its rotations against the 12edo ruler (log scale) practically looks like noise,
the last row has the matrix accumulation (the interval density function).

 

Video.01 Description: "Astrocaster Blues"

This video showcases the planetary diameter data used to calculate the pitches for "Astrocaster Blues."

A main feature of the video is a pitch dial, displaying a single octave for each instrument (piano, guitar, and bass). This allows clearly see the interactions of chords and the intervallic relationships within the scale as the music is played.(and how off-12edo the scale is, yet music is normal)

The scale allows a fully functional pentatonic blues scale (near enough, this led to the 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.

Average Surface Temperature

Table.02: Solar System Planets Average Surface Temperature, Tuning Construction

	Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune
Index(i)	0	1	2	3	4	5	6	7
Temp(K)	452	726	285	230	120	88	59	48
Base:Earth \( P_i \longleftarrow  P_i / P_2\)	1.5859	2.5473	1	0.8070	0.4210	0.3087	0.2070	0.1684
\( P_i \longleftarrow P_i \times 2^n, \, n \in \mathbb{Z} \Rightarrow P_i \in (1,2]\)	1.5859	1.2736	1	1.6140	1.6842	1.2350	1.6560	1.3473
New Index(j)	4	2	0	5	7	1	6	3

Interval Matrix for the Planet Temperature Tuning

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. (+ a bandoneon)

This video provides a visual representation of the scale constructed using planetary temperatures and the resulting musical composition, "The Dance of Entropy."

The "planet grid-keyboard" illuminates the notes as they are played. This allows to directly observe the categorical 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, a permutation on planets order. 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.

Orbital Period

Table.02: Solar System Planets Orbital Period, Scale|Set|Tuning Construction

	Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune
Index(i)	0	1	2	3	4	5	6	7
Earth Days	87.96	224.68	365.65	686.98	4331.6	10832.3	30799	60190
Base:Earth \(P_i \longleftarrow P_i / P_2\)	0.2405	0.6144	1	1.8787	11.8463	29.6247	84.2308	164.6109
\( P_i \longleftarrow P_i \times 2^n, \, n \in \mathbb{Z} \Rightarrow P_i \in (1,2]\)	1.9244	1.2289	1	1.8787	1.4807	1.8415	1.3161	1.2860
New Index(j)	7	1	0	6	4	5	3	2

"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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The Riemann Zeta Function

The 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 ~30 consecutive imaginary parts of the Zeta function's zeros. These values were interpreted as frequencies 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.

The deliberate omission of a defined equave or period of repetition makes the results rarer. the resulting music, while using an unconventional scale, sounds surprisingly "normal", maybe 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.

(Zeta function 28 notes music)

The distinct harmonic characteristics of the synthesized guitar and strings (all synthesized) 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.

Interval Matrix for the Riemann 28-note Scale
In music and tuning theory, there are various types of interval matrices. The one shown here is designed for geometric visualization. The set, or tuning, is displayed horizontally, using a logarithmic scale, and each new row shifts the base and rearranges the elements. Since this tuning isn’t periodic, each row removes one element, unlike periodic tuning matrices where elements rotate. The red ruler at the top indicates 12edo steps of 100 cents, with a ±15 cent tolerance, aiding in visualizing how closely the intervals align with 12edo across all permutations.

The idea that the Riemann Zeta zeros sound “normal” is actually backed by quantum chaos theory. Their distribution is thought to mirror the statistics of energy levels in heavy atomic nuclei, described by the Gaussian Unitary Ensemble (GUE). According to Montgomery’s Pair Correlation Conjecture, the zeros aren’t random, they repel each other. This repulsion keeps them from clustering too closely, naturally creating a well-spaced musical scale. (Spectral Rigidity)

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The Question of Randomness

The observation that simple octave folding of seemingly random values consistently leads to relatively uniform interval distributions, often guaranteeing a degree of tonality, is paradoxical. 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 an interesting 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."

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.

Other sources, like planetary properties (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 (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. 

A different 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.

(Himalayan 14 periodic notes music)

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.

Absolute Vs Relative Proportions

The choice of sea level is a more significant factor in altering the proportional relationships within the scale than the choice of period, (e.g. octave). Small changes in sea level results 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 are 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). ( \( S_i \times 2^n \in (1,2]\, \vert \, n \in \mathbb{Z}\) )

\(400 / 2^8 = 400 / 256 = 25/16\)

\(520 / 2^9 = 520 / 512 = 65/64\)

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 \((1, 2]\).  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 \((1, 2]\) requires multiplying by a power of \(2\).  In this specific example, \(B \times 2^n\) 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: \(B \times 2^n = 1.25\), 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 Structure

This 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 composition.

While 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.

Therefore, while randomness can play a role in pitch selection, its impact on musical composition is less direct and less compelling. The "randomness" of the initial pitch set, in a sense, becomes a canvas upon which human musicality is expressed.

Fitting "Random" Scales within Established Frameworks

Earlier, 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" tonally. 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.

EXTRA:

Color Attractor Spectral Location and Wavelength-Derived Musical Scales

Historically, attempts have been made to establish connections between the musical and visual domains. Isaac Newton famously associated the colors of the rainbow with musical notes. Despite the prevalence of equal temperaments, such as the 12-tone system, during his era, Newton's pitch calculations were rooted in Pythagorean metaphysics and rational harmony. However, the challenge of consistently aligning scales, intervals, and light wavelengths with musical octaves prevented the development of a definitive model.

Here i construct musical scales based on the spectral locations of color attractors rather than imposing existing musical structures onto the light spectrum. These "unique-hues", identified  in color science literature, exhibit notable individual internal consistency across studies. The derivation of scales from these data points reveals remarkably stable musical structures, distinct from the ideal rational intervals sought by Newton, yet no less compelling.

This section presents short musical examples based on tuning systems derived from the wavelengths of color attractors("unique-hues") reported in color science literature.

note that wavelengths, measured in nanometers, are part of a human-defined measurement system. The scales presented here are constructed on the proportional relationships between color attractors, abstracting away from specific unit systems.

For the creation of these musical scales, wavelengths are considered proportionally relative to a base color and adapted for practical implementation on specific instruments. For example, a synthesizer may map a central tone to 261 Hz (middle C), with subsequent scale values expressed as frequency multiples to establish a periodic system. Within this framework, the perceptual spectrum functions as a torsor, where relative relationships are of primary importance.

Torsor (in the context of color): A torsor describes a set lacking a distinguished origin or zero point, yet possessing a well-defined notion of relative position or displacement. In the context of color, the set of all possible hues constitutes a torsor. The difference between two hues can be defined (e.g., "this hue is 30 degrees clockwise from that hue"), but there is no absolute "zero hue." In this context, the hues form a torsor relative to the scales (nm, Hz, cents, mocts, etc.), meaning that the relationships between hues are preserved regardless of the measurement units employed.

Mathematical Process Summary:

While color science typically employs wavelength measurements (nm) within the electromagnetic spectrum, music utilizes audio frequencies (Hz). These quantities are inversely related. Analogous to musical frequency ratio calculation from string lengths (or wavelengths), where the specific frequency value is less important than the ratio itself (assuming constant string tension), the precise terahertz values or photon energy are not directly employed here. Wavelength units (nm) are sufficient for determining proportional frequencies, calculated as inverses of the wavelengths. For example, the frequency ratio from "red" (700 nm) to "cyan" (495 nm) is calculated as follows:

Red (base): 700/700 = 1

Cyan frequency ratio: 1 × (700/495) ≈ 1.414

In the generated scales, ratios are calculated relative to red. However, given the cyclical nature of the system, the choice of base color is arbitrary; the proportional intervals remain invariant regardless of which color is chosen as the root or unison. This invariance exemplifies the torsor nature of hues.

The position, wavelength, and corresponding musical note assigned to "magenta" are derived from the observed complementary relationships. Specifically, the frequency ratio assigned to magenta is the frequency ratio of green multiplied by √2. This methodology accounts for individual variations in the spectral octave range (e.g., 370–740 nm, 405–810 nm), which are dependent on the location of the green attractor. While the graphics presented here utilize a constant 375–750 nm range for illustrative purposes, this choice reflects the torsor nature of hues.

Examples of Unique Hue-Based Scales:

• Modern Trichromat Research: This scale utilizes median unique hue data from contemporary color vision studies on normal trichromats.
• Tetrachromat Data: This scale is derived from studies on individuals with genetic predispositions to a fourth photopigment.

Examples:

  

Color Spectrum Log-Scale 375-750nm \((\lambda, 2\lambda]\) Magenta bridges the gap

Musical Properties of Hue-Derived Scales and the Role of Uniform Distribution

If strikingly unusual or exotic microtonal sonorities are anticipated from these hue-derived scales, their relative conventionality may be surprising. While subtle microtonal inflections may be perceptible to trained listeners, the overall impression is normal. As previously mentioned, not only the tritone is frequently approximated by frequency ratios derived from hue data, but also other stable musical intervals, such as the major third and perfect fifth, emerge from various color combinations. The resulting scales exhibit major and minor chords, and each scale features varying degrees of consonance with other traditional intervallic relationships, corresponding to intervals such as sixths and sevenths. However, bad news for Newton, a single diatonic scale is not derived from a single root; multiple intervals are present, but their distribution prevents direct transposition of chords derived from one color to another. The fact that these scales exhibit musical usability with common timbres, as demonstrated by the piano example in Audio:Trichromats01, is just rare.

[...]

This reinforces the principle that uniform distribution is a primary factor in creating musically usable scales. the relative conventionality of the hue-derived scales is not entirely unexpected. The color attractors themselves are well-distributed across the "color octave," naturally facilitating traditional tonal and modal usage.(see color-coding chapter in Sfinx manual)

While the musical usability of these scales may be statistically probable. These are not merely arbitrary numerical values; they are rooted in the fundamental properties of light and its perception.

About the "Spectral Octave":

If the visible spectrum spanned a significantly different range either much smaller (e.g., 400–430 nm) or spanning multiple "octaves" (e.g., 400–3500 nm) the relationship between color and chroma would become less compelling. The fact that colors exist within a single spectral octave... strengthens the perceptual analogy.

This limited range also addresses the question of whether sufficient color distinctions exist to represent functional harmonies.. The fine distinctions made in color perception are analogous to the subtle distinctions made in musical intervals. Just as musicians may debate whether an interval is a "super major second" or a "sub minor third," distinctions are made between colors such as "yellowish orange" and "orangish yellow." This shared phenomenon highlights the fine granularity of both auditory and visual perception. (Goldstone, R. L., & Hendrickson, A. T. (2010). Categorical perception.)

A1. Color Wheel Construction

Addressing Color Space Transformations and Limitations

The construction of the color wheel presented requires careful consideration of color space transformations and the inherent limitations of representing the visible spectrum within the RGB color space. Converting a specific wavelength to RGB values involves several factors that can influence the final color representation:

• CIE XYZ Model Version: Different versions of the CIE XYZ color space (e.g., 1931, 1964, 2012) have slightly different color matching functions, leading to variations in the resulting XYZ coordinates for a given wavelength.
• Illuminant: The choice of standard illuminant (e.g., D65, A, C) affects the white point of the color space and, consequently, the mapping of wavelengths to XYZ coordinates.
• Gamma Correction: Gamma correction is a non-linear transformation applied to RGB values to account for the non-linear response of display devices. Different gamma values will result in different RGB representations for the same XYZ coordinates.

Consequently, obtaining a specific RGB value like (0, 255, 255) for cyan from a wavelength requires careful selection of the CIE XYZ model, illuminant, and gamma. Furthermore, achieving fully saturated RGB values for all spectral hues is often impossible. If a median render of the spectrum with equal power distribution is used, for example, the perceived saturation of red tends to decrease at longer wavelengths, making it difficult to accurately represent individual "best red" values at wavelengths like 710 nm.

It is crucial to emphasize that the wheels presented here is primarily concerned with the hue/chroma dimension of color, not with precise representations of luminance or gamma. The goal is to accurately represent the relative positions of hues within the spectrum and their complementary relationships, rather than to create a photometrically accurate rendering of the spectrum.

Therefore the final color attractor representations in the wheel are ultimately based on standard RGB values, chosen to represent the perceived hue as accurately as possible within the limitations of the RGB color space. The choice of RGB values for the attractors is done with a focus on maximizing saturation and perceptual distinctiveness, with the understanding that this might not perfectly align with a strict radiometric conversion. (a pseudo-color)

Physically-Based, Octave-Modeled, Logarithmic Hue Wheel

The following sequence of graphics illustrates the construction of the hue wheel, it demonstrates how and which region is assigned to non-spectral magenta.

While electromagnetic waves are often described in terms of wavelengths in color science, music theory typically focuses on frequency ratios.

Key Graphics and Steps:

1. Linear Spectrum (200–1600 THz): A linear representation of the electromagnetic spectrum, highlighting the visible range (~400–750 THz) and including black regions beyond visible light for reference.

2. Octave Doubling: Frequencies are repeated at \(2^k\) generating three "rainbows" separated by black gaps. Non-spectral magenta, which does not exist in the physical spectrum, will be placed within these gaps.

3. Logarithmic Scale: The three rainbows are equalized in size by compressing the scale logarithmically.

4. Magenta Addition: A normal distribution curve fills the gaps, smoothly blending red and blue without altering their intensity.

5. Hue Wheel: A single rainbow from magenta to magenta forms a continuous logarithmic hue wheel.

Color Spectrum Log-Scale 375-750nm \((\lambda, 2\lambda]\) Magenta bridges the gap

Meta:

None of the sources used are “random.” They’re all structured, but in hidden, non-musical ways; some of which are precisely the kinds of distributions that, when mapped onto a circle (log-frequency), naturally produce near-intervals and almost-scales.

Further Reading:

Plomp, R., & Levelt, W. J. M. (1965). Tonal consonance and critical bandwidth.
Temperley, D. (2007). Music and Probability.
Hermann, T., et al. (2011). The Sonification Handbook.
Purves, D. (2017). Music as Biology.
Mazzola, G. (2002). The Topos of Music.