Interval Space Randomness

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 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, 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 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 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.

Interval Matrix for the Riemann 28 note Scale

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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, 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.

(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.

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 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 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 Intervention

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. 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 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" 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?

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.

This part of the study adopts a reverse approach, constructing musical scales based on the spectral locations of color attractors rather than imposing existing musical structures onto the light spectrum. These hues, identified as "best exemplars" in color science literature, exhibit notable individua internal consistency across studies. The derivation of scales from these data points reveals remarkably stable musical structures, distinct from the 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 reported in color science literature.

It is crucial to note that wavelengths, measured in nanometers, are part of a human-defined measurement system. The scales presented here are not constructed by directly mapping nanometers to frequencies (Hz). Instead, they are based 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:

The concept of a torsor within the context of hues and the spectral octave can be illustrated through an example.

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.

Auditory Examples:

The following auditory examples demonstrate the translation of unique hues into musical scales, revealing perceptual and structural parallels between light and sound.

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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 often surprisingly consonant with established musical practice. 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 non-uniform 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 notable.

[...]

This reinforces the principle that uniform distribution within the octave is a primary factor in creating musically usable scales.

Therefore, 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.

While the musical usability of these scales may be statistically probable, their origin in physical reality and human perception imbues them with additional meaning. 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 answer is affirmative. 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.

3D Fractal Xenharmonic Synth, Web App

 Virtual Virtual Instrument?

An experimental program for creating microtonal music.

Sound is generated from a 3D object, a fractal. Clicking on the object's surface produces a set of pitches based on the click point's coordinates (X, Y, Z, and distance to the surface). Each click remaps the keyboard with these new pitches.

The demo video showcases live performance with only drums added afterward.

Play using left and middle clicks, assigning sounds to four keys on two keysets (A, S, D, F and H, J, K, L).

The blinking lights indicate key presses (A, S, D...). The "chord compass" next to it simplifies chord prediction by referencing a 12-tone equal temperament framework.

Built with JavaScript and the browser's Audio API.

Note on microtonality: This program doesn't focus on specific fractal tunings. Musical patterns emerge from the object's geometry, but all fractals potentially contain the same intervals.

Core concept: Generating pitches from arbitrary object coordinates. The code includes primitive 3D shapes for experimentation.

How to play:

• Left and middle clicks trigger notes based on X, Y, Z, and D coordinates.
• Use keysets 1 (A, S, D, F) and 2 (H, J, K, L) to assign sounds.

Pitch generation: Pitch is derived from the click point's coordinates. In linear mode, X=1 corresponds to a pitch of 1 Hz.

SFINX - Xenharmonic Guitar Learning App

SFINX 
Stringed Fretted Instruments Notes Explorer

NEW VERSION IS COMING!

This free web-based app is designed to help you explore and experiment with xenharmonic (micro/macro-tonal) guitars.

Key Features:

• Generate scales and chord diagrams for custom tunings
• Play and find chord progressions on an interactive virtual guitar
• Utilize a built-in tuner and fretboard calculator

Basic Concepts:

• Xenharmonic Tuning: SFINX focuses on exploring tunings that go beyond the usual 12-tone equal temperament. It allows users to design custom tunings using any possible set definition.
• Virtual Guitar: The app provides an interactive virtual guitar interface to visualize and experiment with different tunings, scales, and chords.
• Preset Management: Users can save and load instrument and scale presets for quick access and experimentation.

Help:

Default presets include normal 12ed2 guitar and bass.
SFINX saves presets per browser, allowing you to import and export settings. Overwriting existing presets is possible, but remember to save the instrument preset after creating or modifying scales.

Tuning System(outdated)

• Strings and Frets: Define the physical parameters of your virtual instrument.
• Decimal Ratio List: The program accepts intervals in decimal ratio, comma-separated. The last ratio represents the octave equivalent.
• Equal Division: Divide the octave into a specified number of equal intervals.
• Tuning Pattern: Determine the open string note of each string relative to the previous one. This allows for various tuning configurations, including drop tunings.
  Each string you add, also adds a control on its left that sets the open string note in relation with its previous string (that is why the lowest string doesn't have one).
  The standard guitar tuning pattern is 5, 5, 5, 4, 5.
  A one step drop tuning pattern is: 7, 5, 5, 4, 5.
• Lowest Pitch: Set the fundamental frequency of the lowest open string.
  In hertz. Sets the lowest possible pitch, that is the lowest open string, the only control for setting pitch directly.
• Interval Rulers: Measure intervals and string lengths for in-depth analysis.
  - The pink ruler marks always the equave.(the last number in the interval list, or the interval used for the division in equal-division systems)
  - The fixed green 12ed2 ruler is useful to measure other tunings, since most musicians are familiar with the role of those tempered 12, is easy to relate new tunings using the ruler for quick comparison. (it can be moved)
  For deeper interval analysis and measurement beyond the basic tools provided, explore my other app, 'Interval Rulers.' It visualizes intervals as an 'abstract guitar' or interval matrix, enabling complex calculations like successive reductions and chroma finding. Unlike SFINX, which focuses on a fixed interval matrix inherent to the guitar, 'Interval Rulers' offers a more flexible and comprehensive approach to interval exploration.

Visualization and Interaction

• Highlights: Select specific notes on the fretboard to focus on particular scales or chords.
  Each interval on the list gets a control, or with the equal-division system, each division gets a control.(Note: with non-integer divisions, this gets truncated, floored. So with 11.66 divisions, it will count 11, this is non-sense, yes).
  Here is where you set the scale to be displayed on the fretboard.
  Each marked note, adds a control for arpeggiate a chord.
  
• Chord Pattern: Define the notes for each chord position.
  Each string added, adds a control for selecting a note for the chord-row.
• Colors: Customize the appearance of notes based on:
  Single: Plain color.
  Class: Notes get a unique color by its class.
  Row: Colorize by chord row, controlled by the chord pattern.
• Arpeggiator: Play notes sequentially in different patterns (up, down, converge, diverge).
• Speed: Arpeggio note separation speed in seconds.
• Synth Sound: Choose from available sounds for the virtual guitar.

Additional Controls

• Width, Length, Zoom: Adjust the visual representation of the fretboard without affecting the tuning or sound.
• Note Length: Control the duration of played notes.
• Volume: Adjust the overall volume of the virtual guitar.
• Calculator ruler: Displays a ruler (orange) with the relative size of the string when hovering a fret.
• Size: Sets the nut to bridge length for calculations. In units.

Limitations Regarding Fretboard Customization

The application presents certain inherent limitations regarding the direct creation of highly customized fretboard configurations, specifically:

1. Unique Individual Frets: It is not possible to directly add a single, unique fret at a specific position on only one string without incorporating that interval into the underlying pitch set definition for the entire instrument.
2. Fret-Skipping Patterns: Creating patterns where entire frets are omitted across all strings cannot be achieved through a direct fret-removal function.

These limitations are not arbitrary but stem from the application's core design philosophy, which prioritizes the generation and analysis of scales and patterns based on consistent, underlying pitch sets. Allowing arbitrary fret additions or removals would undermine this analytical framework. However, practical workarounds exist, grounded in established music theory principles. Understanding these limitations and their justifications clarifies the application's logic, which shares conceptual roots with historical approaches to instrument notation.

Historical Precedent and Theoretical Foundation

The methodology adopted finds a parallel in historical solutions developed during the Middle Ages for fretted instruments like the cittern. As instrument makers experimented with different fretting patterns (including microtonal variations, just intonation, and skipped frets), tablatures became inconsistent and difficult to interpret across instruments. A crucial development, particularly coinciding with the standardization towards 12-tone temperament, was to number frets according to their position within the complete theoretical chromatic scale, even if the physical fret was absent. (See Duodecimability)

For instance, if a string only possessed a single fret precisely at its midpoint (the octave), it was labeled "fret 12," not "fret 1." This indicated its position within the 12-tone system. This elegant solution not only standardized tablature but also facilitated theoretical understanding. It implicitly treated transposition algebraically (akin to coset shifts in modern terminology), ensuring that musical patterns like chord shapes remained consistent conceptually, even when physical frets varied. Attempting to transpose patterns based solely on naive fret counting on irregularly fretted instruments would quickly lead to inconsistencies.

Application Context: Timbre and Octave Equivalence

It is pertinent to note that while the application offers various synthesized guitar timbres, all are fundamentally harmonic. This guarantees the perceptual validity of the octave as the primary cycle of pitch repetition. Consequently, the application's chroma-based color-coding system is fixed to octave equivalence, reinforcing the focus on cyclical pitch structures. (For manipulation of more abstract pitch/timbre relationships, the generalized interval matrix tool should be utilized).

Workaround: Simulating Unique Fret Additions

Consider the scenario of adding a single extra fret to a standard 12-tone guitar – for example, a quarter-tone fret between the 3rd and 4th traditional frets on one string. This might represent a neutral third, a specific microtonal inflection, or serve another modal/contextual purpose.

Directly modeling this single fret addition is not supported. The required workaround involves redefining the instrument's fundamental pitch set to include the desired new interval. This means the interval will be replicated at octave-equivalent positions across the entire fretboard. While a single physical fret might be intended as ornamental, incorporating its pitch interval categorically into the underlying set maintains theoretical consistency.

This approach offers significant advantages:

• It preserves a coherent, cyclical pitch structure, which is easier to implement computationally.
• It results in a more intuitive playing and analysis experience, as patterns remain consistent.
• Chord progressions and scale analyses remain valid across the fretboard. Integrating the new note reveals its systematic relationship within the established structures.

To implement this, you would define the new pitch set using cents notation, incorporating the new interval. For instance, adding a quarter tone (50 cents) near the beginning of the scale: {50, 100, 200, 300, ... , 1200}. Tuning shifts can then be used to position this interval relative to the open string as desired.

Workaround: Simulating Fret-Skipping

Fret-skipping, common in constructing instruments based on subsets of larger microtonal systems (e.g., selecting specific pitches from 31-EDO), presents a different challenge. It's crucial to distinguish between:

• Removing specific notes from a system: This is how regular temperaments are often realized (e.g., a 7-note diatonic scale derived from 12-EDO). On a guitar, this might appear as skipped frets, but the same set of notes remains available across the entire fretboard, just spaced differently.
• Skipping entire frets across all strings: This creates a complex scenario where the availability of specific pitches becomes inconsistent across different positions and octaves.

While it is technically possible to pre-calculate the exact pitches resulting from a fret-skipping pattern and input them as a fixed, non-repeating scale into the application, this is highly impractical for analysis. It essentially defines a unique, non-standard instrument configuration, limiting the applicability of standard analytical tools and pattern recognition.

The recommended and more practical approach mirrors the historical solution: analyze the instrument based on the full, underlying pitch set (e.g., the complete 31-EDO scale if the physical instrument omits some 31-EDO frets). Although diagrams might show chords or scales utilizing frets physically absent on the specific instrument, this method allows:

• Understanding the underlying theoretical structure and harmony.
• Identifying alternative voicings or inversions of desired chords/patterns that are physically playable.
• Avoiding the creation of highly specific notation tied exclusively to one instrument's unique fretting.

Conclusion

The application's limitations on direct, arbitrary fret manipulation encourage users to work within consistent theoretical frameworks. By defining instruments based on complete, cyclical pitch sets (standard or custom), the application facilitates robust musical analysis, pattern recognition, and understanding of transposition and harmony, echoing effective principles developed historically for managing complex fretting systems. The focus remains on the underlying musical structure rather than idiosyncratic physical layouts.

Understanding Chroma, Pitch Class, and Equave in Scale Generation

Note: The next section of the article is slightly outdated, and I'm currently improving its clarity. The main focus is on distinguishing the types of equivalences in pre-selected pitch sets (perceptual, analytical, geometrical, and none).

To navigate the application's scale generation and analysis effectively, it's crucial to distinguish between three types of equivalence: Perceptual (Chroma), Analytical (Class-Based Equave), and Geometrical (Interval of Repetition).

1. Perceptual Equivalence (Chroma)

• Definition: The equivalence of pitches separated by intervals that are perceived by humans as functionally the "same note" at different registers.
• Context: For harmonic timbres (like those of the synthesized guitars), the primary chroma is the octave (1200 cents).

2. Analytical Equivalence (Class-Based Equave)

• Definition: An equivalence established by the chosen number of pitch classes. This defines the modular space within which musical elements (like scales and chords) are analyzed and manipulated. The Equave in this context refers to the interval spanning the chosen number of classes, often a musically significant interval that serves as the boundary of the analytical cycle.
• Relationship to Class: The Equave here is derived from the chosen number of classes. If you choose 12 classes for 12-EDO, the Equave of the class system is the octave. If you choose 13 classes for 13-EDT, the Equave of the class system is the tritave.
• Flexibility: The application allows you to set the number of classes (and thus the "Equave" of the analytical system) independently of the scale's inherent geometrical properties or perceptual chroma. This allows for diverse analytical perspectives.

3. Geometrical Equivalence (Interval of Repetition)

• Definition: The smallest pitch interval within the scale's structure such that transposition by this interval (or multiples of it) results in a repetition of the intervallic pattern or transpositional invariance of musical figures (like chords). This is an inherent property of the scale's construction.
• Examples:
• 12-EDO: The geometrical interval of repetition is the 100 cent step. Transposing any interval or chord by 100 cents (or multiples) within the 12-note octave will maintain the same intervallic structure relative to the steps of the scale. The octave (1200 cents) is the Equave of the 12-class system.
• 12-tone Pythagorean: The geometrical intervals between scale degrees are not uniform. While the octave is the perceptual chroma and also the interval after which the scale repeats in terms of its seven degrees, the exact intervallic relationships within each octave (and thus the sound of transposed chords) are not identical across different starting pitches (due to the nature of just intervals). In this case, the octave is the Equave (class-based) and also a period of repetition for the scale degrees, but not for the precise sonic character of transposed chords.
• 13-EDT: The geometrical interval of repetition is the single, uniform step size of approximately 146.3 cents. Transposing by this amount maintains the intervallic structure. The tritave is the Equave of the 13-class system.

In Summary:

• Chroma: Perceptual octave equivalence.
• Equave (Class-Based): The interval spanning the chosen number of pitch classes, defining the analytical cycle.
• Interval of Repetition (Geometrical): The smallest pitch interval that produces transpositional invariance of patterns within the scale's structure.

This distinction, especially between the class-based "Equave" (for analytical purposes) and the inherent "Interval of Repetition" (a structural property), is very important for understanding how scales behave and how transposition affects musical elements within them.

Photoshop Scripting +

kMaps - 32bit HDR 360 Sky Creator PS Plug-In

 kMaps - HDR 32bit Procedural 360° Sky Maps Latitude/Longitude Cylindrical Mapping

Important: This plug-in is free and open-source. It was created and tested with Photoshop 2020 (21.2.1) (with CEP 9) in English on Windows 10. Compatibility with other versions is unknown.

Update: I've confirmed compatibility with the latest Photoshop (24.3.0 with CEP 11), but it still requires the English version. Importantly, CEP is being deprecated, which may limit this plug-in's usefulness in the future due to advancements in AI image synthesis.

kMaps offers simple, perlin, and fractal noise-based clouds in a classic style.

As an unsigned plug-in, manual installation and enabling Photoshop's debug mode is required (see video for instructions).

Please note that the plug-in contains some bugs, primarily in the user interface.

Information:

• Groundless Images: Generated images lack a ground plane, allowing for versatile use. Each image can effectively serve two purposes.
• Resolution: Generating 16K images depends on GPU and driver capabilities. 8K is consistently reliable, and 4K is generally suitable.
• 32-bit Depth: The 32-bit output is computationally intensive. The plugin initially creates a 16-bit PSD, simulating different exposures, and then combines them into a 32-bit image.
• Preset Compatibility: Presets are optimized for default cloud settings. Modifying advanced noise options might require adjustments to other parameters. Experiment with Seed and Noise Evolution values before applying presets.
• Aspect Ratio: Images are generated in a 1:1 aspect ratio to accommodate engines like Unity, which often resize and pack textures in this format.

Installation (Windows 10):

1. Extract the /com.k.maps folder from the .rar file and place it in C:/Users/***/AppData/Roaming/Adobe/CEP/extensions/.
2. Install the included Actions as usual.
3. Enable Debug Mode in Photoshop (follow the video guide).

Simple Guide:

• Close all other files.
• Click "Render .PSD".
• Select a preset and click "Run".
• Optionally, click "Convert to HDR".
• Save the image.

Tips:

• The preview is a style guide, not a definitive representation.
• Ignore potential banding when zooming out. Gradients are accurate.
• To disable a layer or option without affecting presets, set its opacity to 0 instead of hiding it.
• The sun and moon may appear distorted depending on their position.
• The animation feature is experimental and prone to crashes. Use with caution.

Color Coding:

• Red: Do not modify these elements as they are controlled by the plugin.
• Blue and Green: These elements are editable.

Downloads:

kMaps-v2 Download < 184Kb
kMaps-v2n Download Photoshop 2023 < 184Kb
HDR examples Download < 476Mb
PNG examples Download < 3.5Gb

AE 3D Audio Spectrum (no plugins) +free templates

 

3D Wave Spectrum Audio Visualizer

After Effects CC Template Guide

Basic Usage:

1. Choose a template from the downloaded .rar file and open the Project (some may take longer to load). Locate the replaceMe.mp3 file in the project panel, right-click > Replace Footage, and select your audio file. Trim the composition length to match your audio; it is set to 10 minutes by default.
2. Render the project or customize the included camera and lights to your preference.

Modifying the Template:

The first layer, from top to bottom, called 3DSpectrumControls, contains all the controls to change and animate the aspect of the wave spectrum. The first option called 3DLayersPadding allows you to change the distance, in pixels, between the spectrum layers to create depth. A value of 0px makes it flat; it should be at least 1px. (Note: Each layer is flat on its own. If you place the camera from an absolute side view, the spectrum will not be visible.)

All other options are from the AE Wave Spectrum regular effect. Refer to the AE Manual for detailed explanations of each control.

• Start Point/EndPoint: The 2D coordinates (X, Y) in pixels defining the starting and ending points of the audio wave.
• Start Frequency/End Frequency: The frequency range (in Hertz) to be visualized, from 0 to 22050 for 44100Hz files.
• Frequency Bands: The number of subdivisions within the specified frequency range.
• Maximum Height: The spectrum's height relative to the audio amplitude.
• Audio Duration: The length of the audio sample used for spectrum analysis (in milliseconds). The default is 300ms. Higher values increase render time, lower values reduce animation smoothness. Keep this value between 200-500ms.
• Audio Offset: The time offset (in milliseconds) between spectrum layers. The default is 20ms. Controls the sampling time difference for each layer.

An Expression script (see below) automatically applies the audio offset to all spectrum layers. You only need to modify the Audio Offset value in the 3DSpectrumControls layer.

audioOffset = thisComp.layer("3DSpectrumControls").effect("AudioOffset")("Slider");
thisComp.layer(thisLayer, 1).effect("Audio Spectrum")(11) + audioOffset

The same happens with the 3DLayersPadding control.

padding = thisComp.layer("3DSpectrumControls").effect("3DLayersPadding")("Slider");
[ thisComp.layer(thisLayer, 1).transform.position[0], [ thisComp.layer(thisLayer, 1).transform.position[1] ], thisComp.layer(thisLayer, 1).transform.position[2]-padding ]

Both Audio Offset and Audio Duration affect the spectrum's refresh rate. The default values for each template typically work well for most audio files.

• Thickness: The thickness of the line (in pixels) that forms the audio wave.
• Softness: The softness of the line that forms the audio wave.
• Inside Color/Outside Color: The color of the line and its outline.
• Hue Interpolation: Interpolates the selected color based on the hue wheel (in degrees). A value of 360° displays the entire color wheel starting from the chosen color. The default is 35°, which interpolates from blue to cyan/green.
• Dynamic Hue Phase: Uses audio phase information to randomly change colors.
• Color Symmetry: Mirrors colors along the line, useful for hiding seams in round spectrum shapes (like template 04).
• Display (Digital/Lines/Dots): Changes the wave's appearance between these three options.
• Side (A/B/A&B): Displays the positive, negative, or both sides of the amplitude wave.

To use a custom shape for the spectrum, open the "3dWaveSpectrum_shape.aet" project. In the 3DspectrumLayer_1, you'll find the original AE Wave Spectrum effect, which allows you to use a mask or path to trace the wave. Duplicate this layer (Ctrl+D/Cmd+D) as many times as needed to create the desired spectrum depth.

About the Audio File

The AE Wave Spectrum effect used in this template represents the spectrum linearly in both frequency and amplitude, unlike most spectrum visualizers. This means that when displaying the entire spectrum (0 to 22,000 Hz), you'll see more activity on the left side (low frequencies) with higher peaks, and less activity on the right side (high frequencies), even with maximum height settings.

While selecting a smaller frequency range (e.g., 20 to 2000 Hz) can help, it might not be ideal as it eliminates higher frequencies and may not accurately represent the audio.

You can use After Effects' native audio effects, like the parametric equalizer, to adjust frequency levels. Create a pre-composition for better performance.

For more precise control, using an external audio editor like Audition is recommended. You can apply an equalizer to reduce frequencies below 800 Hz, use a multi-band compressor to target lower frequencies, and then a tube-modeled compressor, hard limiter, and declipper to even out volume levels. While this process might distort the audio, it's intended for spectrum visualization only. Use the original audio for the final sequence. Refer to the Audition documentation for detailed instructions on these effects.

About Performance

This template is computationally intensive, potentially taking up to 100 hours to render on an average PC. Performance decreases with more layers. To improve editing speed:

• Set preview resolution to "1/4" instead of "Full."
• Disable the three default lights during editing (re-enable for rendering).
• Avoid effects like Glow, Blur, background images, and footage during editing. Add these later in a new project.
• Use the Classic 3D Renderer with the lowest Shadow Resolution.
• Employ compressed audio formats like MP3 instead of uncompressed formats like WAV.
• Disable Depth of Field and Motion Blur.
• Reduce the frame rate from 24fps to 18fps and use frame blending after rendering for smoothness.

Rendering

Render the composition as a PNG sequence to preserve the alpha channel for adding background images or footage later in a new project.

To accelerate render times, consider using the After Effects command-line renderer (aerender.exe) to run multiple instances of the program simultaneously.

Exampe code:

aerender -project X:\examplePath\exampleProject.aep -comp "exampleComp" -RStemplate "Multi-Machine Settings" -OMtemplate "PNG Sequence" -output X:\examplePath\exampleSubPath\exampleOutput_[#####].png

Ensure the selected Render Settings template (-RStemplate) has the "Skip File if Already Exists" option enabled within After Effects. The pre-included "Multi-Machine Settings" template can be used.

Create an Output Module template (-OMtemplate) with your desired PNG settings or other sequence format.

This command-line rendering approach can be used on multiple computers with After Effects installed to accelerate the process.

For more details on multi-machine rendering, consult the After Effects manual.

This Plug-in is free and provided as-is without any support beyond this documentation.

Copyright laws apply. You can create unlimited commercial projects using it, but selling or distributing any of the templates is prohibited.

Tested exclusively in After Effects CC 2017.2 (Version 14.2.1.34).

DOWNLOAD

Project on Behance

The Spiral Harp

6-sided spiral harp, concept art

A spiral tuning system is a layout for string instruments based on any of the diverse configurations of a spiral polygonal chain, known as a spirangle, utilizing the segment's length as source for pitch, either as string length or frequency.

These systems are aperiodic (with exceptions) and possess an infinite range of possibilities. Among these configurations defined by their sides and segments, many prove musically practical, with potential for some to manifest as tangible instruments, such as spiral harps. (An instrument with a single wound string where pitch is linked solely to string length, and tension becomes relative.)

Each unique configuration unveils distinct chords and progressions, often showcasing geometric patterns.

This audio clip features a short melody played on a digitally modeled harp-like instrument tuned according to a six-sided spiral polygonal chain (as depicted in the concept art). The final, highest note corresponds to the shortest 'string' of the spiral.

Theory:

Each tuning can be mainly defined by the amount of sides and the margin, and can be named:

S6m1 - "S" for spiral, followed by the number of sides, "m" for margin; if its value is 1, it can be omitted (e.g., S6m1 = S6). [This tuning is of main interest.]

S5.5 - Five-and-a-half-sided spiral with margin 1 (omitted).

S1m1.05946 - One-sided spiral with a margin of the twelfth root of 2.

S7r2c1 - Seven-sided spiral with a margin of 1 (omitted), with an initial radius of 2, and constant increment c = 1. When omitted, spirals initial radius is 0, c = 1.

iS6m1 - Inverted six-sided spiral with a margin of 1.

The parameters affecting the resulting relative segment length progression are:

Amount of sides: from 0 to infinity.

Margin: usually 1 (to mimic spider-webs). This property can be (unnecessarily) employed to generate equal-division systems. For example, the angle is calculated with \( \frac {2\pi}{sides}\), so when sides are \(1\), \( \frac{1}{2}\), or \( \frac{1}{4}\), etc., it leaves the margin as the sole control for segment length increase. For instance, a one-sided spiral with a radius of approximately \( 1.05946 = \sqrt[12]{2} \) generates a 12 equal division system. From this perspective, equal-division systems can be seen as a subset of spirals.

Initial radius: usually 0 Using a different initial radius opens another dimension of progression; however, it seems to mostly affect the initial segments, and the rest of the spiral converges quickly with its version with radius 0.

Inversion: This parameter doesn't affect the progression but rather how the progression is treated, as string length or as frequency.

Spiral polygonal chains with different margins,

Spiral polygonal chains with different sides.

Construction:

Starting from the center, and considering the segment's length as string length, the first being the shortest, becomes the highest pitch so the tunings are defined inversely.
Since, in most cases, they are aperiodic, the system sizes are infinite, it will depend on how many notes one wants to calculate.
Most spiral settings cover the audible range with less than 300 segments.
For instance, a six-sided spiral harp with margin 1, comprised of 120 segments spans approximately five octaves.
The spiral can be of any size, a diameter, or scale property, while changing the length of the segments, won't alter their relative length.(if started at 0,0)
We assign a frequency to the first segment, e.g. 8000hz, and the rest of the notes are calculated from it.

Unwound spirals next to each other, firsts 10 segments. With margin 1. From 0.5 to 2 sides (150 spirals, in 0.01 step) First segment from each spiral is normalized to the same length. Segments are colored by octave, this means, every red is the same chroma. The 1-sided spiral has all its segments of the same length in this configuration with margin 1.

Algorithms for Segment Length Generation:

1- Euclidean distance between consecutive points on a spiral:

Given:

Radius: \(r= (z \times m)\times (m^n)\) where \(z\) is the constant size increment, \(m\) is margin and \(n\) is the point's index, starting at \(0\).

Angle \(a = \frac{2\pi}{s}\) where \(s\) is the amount of sides of the spiral

The x-coordinate and y-coordinate of a point on the spiral are calculated using:

\(x = r \times \cos(a)\)

\(y = r \times \sin(a)\)

The distance between two consecutive points on a spiral in Cartesian coordinates \((x_1,y_1)\) and \((x_2,y_2)\) is calculated using the Euclidean distance formula:

\(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)

2- Direct Polar Calculation: Instead of converting to Cartesian coordinates, we can calculate the distance directly in polar coordinates. Let \(r_n\) and \(r_{n+1}\) be the radii of two consecutive points, and \(a\) be the constant angle between them. The distance \(d\) can be calculated using the law of cosines:

\(d = \sqrt{r_n^2 + r_{n+1}^2 - 2r_n r_{n+1}\cos(a)}\)

Since \(r_n = (z \times m) \times m^n\) and \(r_{n+1} = (z \times m) \times m^{n+1}\), we can simplify this to:

\(d = (z \times m) \times m^n \sqrt{1 + m^2 - 2m\cos(a)}\)

Properties:

One significant characteristic that differs from most tunings is that each successive lower octave has more notes. At first sight, the different progressions don't seem to say much. It helps to analyze each tuning by looking at its full interval matrix, revealing that some strings have many more types of minor thirds, while others have more fifths. Some completely dodge certain harmonics, regardless of how many "strings" you add; some combinations just never happen.

The crucial factor is the number of sides in the spiral, as this directly determines the chords exposed in each row. Some configurations naturally lend themselves to more intuitive chord progressions. The initial segments of the progression are the most susceptible to alteration due to the inherent truncation of the spiral at its beginning. However, even with these truncated beginnings, distinct patterns still emerge. The remainder of the progression exhibits a high degree of similarity across most configurations. From a string length perspective (moving away from the center, where truncation errors diminish), the progression approximates an arithmetic series, seemingly increasing at a constant rate. However, a closer examination reveals a subtle, gradually increasing ratio.

Considerations for spiral tunings and possible harps:

For a real spiral, a logical number of sides starts at 3 (greater than 2, avoiding string overlap) and ends at some point depending on the expected range (e.g., 12 sides). However, a real harp beyond this will have too many or too short strings to be practical. Regarding the margin, the value is usually 1; going too far away from this eliminates the possibility of the spiral as an instrument, and so does this tuning inversion, the progression as frequency.

An open-source, virtual playable version is accessible link.

The new version is available as a interface in MIND.

-----

The concept art is a 3d model post processed with AI.

The app has one pending update....

draft

Tonal Constancy

Tonal Constancy and the Perceptual Forging of Pitch 

(draft, while im still writing this, never ending study, you can read it)

A fundamental paradox lies at the heart of music perception: we consistently identify tonal structures in acoustic environments where, by any mathematical measure, they should not exist. This raises foundational questions about the nature of musical categories. Are there finite notes, or an infinite spectrum of chromatic shades? Are their functions absolute, or are they defined purely by context and cognitive bias?

This essay posits a unifying principle to resolve this paradox: Tonal Constancy. It is the theory that the human brain, far from being a passive receiver of frequencies, is an active forger of meaning. Steeped in the patterns of culturally dominant musical systems, the mind projects this learned framework onto ambiguous acoustic signals, constructing familiar relationships from unfamiliar materials in an act of what might be called musical pareidolia.

What follows is both an argument and an exploration. It will establish a precise lexicon to move beyond conventional terminology, deconstruct the mechanisms of tonal reconstruction through vivid examples, and trace the origins of our perceptual biases to the intersection of physics, cognition, and culture. This work invites not only musicians and theorists, but any reader interested in the architecture of perception, to reconsider the very substance of what we hear when we listen to music.

Part I: The Phenomenon - A New Framework for Perception

    1: An Introduction to Tonal Constancy.

    2: From 'Xenharmonic' to 'Induodecimable': The Need for a Precise Lexicon.

    3: A Case Study in Perceptual Forcing: Tonal Reconstruction in 7-EDO.

Part II: The Mechanism - A Classification of Translatability

    4: The Four Layers of Duodecimability.

            Perceptual Forcing (e.g., 7-EDO)

            Relational Reference (e.g., Maqamat, 19-EDO)

            Structural Alienation (e.g., Bohlen-Pierce)

            Timbral Dissolution (e.g., Gamelan)

    5: The Spectrum of Familiarity: Microtonal 'Flavor' vs. New Functional Categories.

            A deeper dive into the distinction between Layers 1/2 and Layer 3.

    6: Anchor Density: A Model for Perceptual Alienation.

            "Anchor Density" spectrum explains the experiential difference between various induodecimable systems.

            Contrasts "High-Density" systems (11/13-EDO: slippery, locally familiar) with "Low-Density" systems (Bohlen-Pierce: truly alien, no footholds).

Part III: The Foundation - The Substrate of Hearing

    7: The Indispensable Cycle: Range, Resolution, and Perceptual Distance.

            "How real is the octave?"

    8: The Blueprint for Pitch: A Collision of Two Logics.

            Presents the theory that our pitch system arises from the conflict between two forces:

                The Logic of Symmetrical Partition (2^n binary division of the cycle).

                The Logic of Acoustic Resonance (The asymmetrical, "timbre-locked" force of the 3:2 fifth).

Part IV: The Context - History, Ontology, and Evidence

    9: The Riverbeds of Culture: Re-examining Pythagoras and the Great Convergence.

            What did Pythagoras really do?

            "Practice Before Theory" (lutes, frets) and the convergence of Chinese and Greek systems.

    10: Is Anything Fundamental? On Ontology

    11: Exhibit A: The Anti-Randomness Engine.

            The brain is a relentless ordering engine, and Tonal Constancy is its primary tool for forging                    meaning from the chaos of sound.

Part I: The Phenomenon - A New Framework for Perception

1: An Introduction to Tonal Constancy

Tonal Constancy refers to the brain’s tendency to project learned tonal structures onto acoustically unfamiliar or ambiguous data. It explains how we perceive familiar relationships—like semitones, tonics, or functional resolutions—even when listening to mathematically alien or statistically random pitch materials. This theory proposes that the mind is not a passive receiver of sound, but an active constructor of order: a kind of musical pareidolia, shaped by culture, cognition, and the deep architecture of hearing itself.

The term is proposed in analogy to color constancy in vision: the phenomenon by which colors remain identifiable despite changes in lighting. In the auditory domain, pitch constancy works similarly. If we slightly stretch or shrink the intervals between pitches—analogous to altering the illumination of a color palette—we still recognize the underlying “shape” of the set. These transformations, when small enough, are heard not as deviation but as expression. Even in more extreme conditions, such as microtonal systems that differ radically from 12-tone equal temperament, listeners often reconstruct familiar tonal relationships through the motion of pitch trajectories and context-dependent inference.

For such reconstruction to be possible, the brain requires a stable framework. This reveals a deeper process: the perceptual categorization of cycles. Just as we naturally group spatial patterns into shapes or rhythms into meters, we instinctively categorize frequencies into closed, circular structures—with the octave being the most dominant. This circular mapping is not merely cultural convenience. It is perceptually anchored, strongly supported by the harmonic series (and the missing fundamental phenomenon), which offers the octave as an early perceptual prior.

This theory thus distinguishes between three intertwined concepts:

    -Range — the full span of frequencies audible to a human.
    -Resolution — the minimum perceivable difference between two pitches (JND).
    -Cycle — the perceptual tactic of wrapping pitch into repeating units to create a structure.

While pitch range and resolution describe physical thresholds, the cycle is an inferential structure: a strategy used by the brain to generate meaning from undifferentiated acoustic space.

This framework helps explain phenomena such as:

    -Why microtonal scales like 7-EDO can still evoke a sense of “semitone” or “dominant” when heard in motion.
    -Why random or uniformly spaced tones often produce surprisingly familiar tonal structures.
    -Why even trained musicians, when exposed to unfamiliar scales (e.g. 13-EDO), often fail to notice that they’ve left the 12-tone universe—so long as timbre and motion remain within certain thresholds.

In other words: most pitch systems are statistically doomed to sound tonal—unless they are carefully designed to evade the listener’s learned priors and pattern-matching instincts.

Tonal Constancy, then, is not just an artifact of musical tradition—it may be a deep feature of auditory perception itself. This essay aims to define the concept rigorously, construct a classification system for types of translatability into tonal space, and explore its implications for theory, composition, and cognition.

2: From 'Xenharmonic' to 'Induodecimable': The Need for a Precise Lexicon.

The term xenharmonic was originally coined to describe musical materials that lie outside the framework of Western 12-tone equal temperament (12-EDO). Implicit in its early usage was a sense of perceptual alienness—sounds that could not be reconciled with conventional tonal expectations, or even approximated within the familiar scalar structures of Western music.

Over time, however, the term’s scope has broadened. Today, xenharmonic may be applied to any music using non-standard tunings, alternate instruments, or unfamiliar timbres. As its use has expanded, its precision has diminished. A piece might now be labeled xenharmonic even if it maps closely onto 12-EDO, or if it retains gestures that remain tonally functional within familiar paradigms. In this diluted form, the term no longer guarantees that the music is truly untranslatable into the 12-tone system.

To address this ambiguity, I propose a more semantically precise term: induodecimable—from Latin roots meaning not reducible to twelve. It describes musical structures, scales, or timbres that cannot be effectively translated into 12-EDO without a perceptual or functional loss. Unlike xenharmonic, this term emphasizes irreducibility, not just unfamiliarity. Moreover, its morphology is cross-linguistically stable (e.g., induodecimable reads identically in English and Spanish), and it admits extensions for greater specificity—such as indiatonizable, referring to pitch content incompatible with diatonic function.

It is important to note that this property is not binary. Whether a given musical structure is duodecimable—that is, whether it is approximable by 12-EDO—is ultimately a perceptual judgment. Some cases are obvious: inharmonic spectra perceived as noise, or microtonal systems with step sizes that fall far outside typical pitch categories. But others lie in a gray zone where perceptual context, cultural exposure, and learned listening habits strongly shape what we "hear."

This gray zone is precisely where Tonal Constancy becomes critical. Even when a melodic or harmonic structure defies analytical reduction to diatonic scales, listeners often project familiar tonal frameworks onto it—constructing implied functions, modes, or centers through context and inference. The ability to “make tonal sense” of unfamiliar material is not evidence of universality in the structure itself, but of the perceptual elasticity of the listener.

As this study will show, the diatonic scale functions as a tonal attractor—a kind of perceptual sink into which ambiguous or approximate materials are pulled. The 12-tone system serves as its most stable host, offering a resolution of the scale’s unequal steps into evenly spaced units that align (imperfectly, but reliably) with physical redundancies in the harmonic series.

The question then arises: why twelve? Why does this particular internal subdivision serve as the dominant attractor, as opposed to systems based on ten, fifteen, or nineteen tones? Why does duodecimability feel like a perceptual threshold?

The answer is not solely historical. Nor is it purely acoustic. Instead, we find ourselves in the deeper terrain of categorical emergence: how perceptual systems construct stable reference frames from continuous data. Just as we learn to divide the color spectrum into culturally specific “basic colors,” so too do we divide the pitch continuum into categories that are both learned and constrained—by cognition, biology, and acoustics.

This chapter lays the foundation for a more precise taxonomy of perceptual translatability in music. The aim is not only to explore how Tonal Constancy works, but to examine the deeper question: where do musical categories come from at all? At what point do categories cease to form, or become unstable? And when do they dissolve into pure context-dependence—when Tonal Constancy, in effect, runs out?

3: Tonal Reconstruction in 7-EDO and the Elasticity of Pitch Meaning

A particularly vivid demonstration of Tonal Constancy arises within the 7-tone equal division of the octave (7-EDO)—a tuning system in which every interval spans approximately 171 cents. Unlike the diatonic system in 12-EDO, which features varying step sizes (whole tones and semitones), 7-EDO contains no built-in hierarchy. Its scale degrees are evenly spaced, devoid of structural bias. And yet, when shaped melodically, listeners often report perceiving familiar tonal centers, harmonic functions, and modal implications.

This raises a crucial question:

How does a scale composed entirely of uniform steps give rise to perceived modes, cadences, and tonal direction?

The answer lies in trajectory, rhythm, and expectation. When notes are arranged in a musically structured sequence—when certain tones fall on emphasized beats, when contours mimic known harmonic arcs—the perceptual system imposes meaning where none exists acoustically. That is tonal constancy at work: the brain reconstructs pitch relationships that aren't literally present in the signal.

 
The Case of 7-EDO: Structure Without Hierarchy

Let us consider a concrete example. The full 7-EDO scale, in cents, is:

[0, 171, 342, 513, 684, 855, 1026]


[missing]

 
Audio Evidence and Cognitive Confirmation

Audio examples accompanying this section allow the reader to directly confirm this phenomenon. In the absence of harmonic overtones (e.g., using pure sine waves), the effect still holds. The listener perceives functional pitch relationships despite the absence of spectral cues.

While the evidence here is intersubjective rather than quantitative, it aligns with existing empirical findings in psychoacoustics and pitch perception. Later chapters will explore more controlled studies that provide data-driven support for this auditory elasticity.
Trajectory, Momentum, and Torsor Space

These examples demonstrate another key mechanism: pitch momentum. As a melody unfolds, successive steps form a trajectory—a kind of auditory vector through pitch space. Even when the underlying tuning lacks functional landmarks, the shape of the melodic path builds expectation and invites closure. This perceptual momentum allows the brain to infer tonality where none exists acoustically.

We might say that musical space behaves more like a torsor than a vector space: there is no fixed origin—no absolute zero point—only relationships. A pitch acquires meaning not from its frequency but from its relational chroma: its position within a trajectory, its placement in rhythm, and its alignment with culturally learned tonal archetypes.

 
When Does Meaning Collapse?

This leads to deeper questions that this study will return to again and again:

How much can an interval be stretched or displaced before it ceases to evoke its expected function?
At what point does a “minor third” cease to be heard as such?
How much cognitive reinterpretation is too much before tonal constancy fails?

Such boundaries are neither fixed nor purely acoustic. They are context-dependent, modulated by familiarity, cultural exposure, and moment-to-moment listening conditions.

We might ask:
Have we learned categories so deeply that we now impose them reflexively? Have we trained ourselves into hearing functions that are not there? And if so, can we ever unlearn them?

These questions will be addressed further in upcoming chapters, where we explore tonal attractors, cultural priors, and the emergence of categorization itself.

In summary, tonal constancy in 7-EDO reveals just how much pitch meaning is a construct—a dynamic synthesis of trajectory, rhythm, relational placement, and cognitive inference. The auditory system is not a passive receptor of interval sizes. It is an active interpreter, constantly reconstructing tonal function from even the most alien input.

Where conventional theory sees symmetry and ambiguity, perception hears hierarchy and direction. And in this, we find the essence of tonal constancy—not as a property of sound, but as a property of mind.

Audio Examples

The following audio examples present melodies in both 7-EDO and their 12-tone equivalents. Despite the equal step sizes in 7-EDO, the melodies evoke a sense of familiar tonal functions. For example, the second step in 7-EDO, a neutral third at 342 cents (approximately halfway between a major and minor third), is often perceived as having either a "major" or "minor" quality depending on the surrounding musical context, such as the implied harmony or the melodic contour. This effect, which persists even with pure sine waves (thereby eliminating harmonic artifacts), demonstrates tonal constancy: the listener's brain interprets the neutral intervals within a tonal framework, resolving them into functionally familiar pitches. When the same progression is rendered in 12-tone equal temperament, the listener/performer naturally resolves each step into the "correct" functional pitch to satisfy the implied cadence.

Ξ Example A - 7edo

Ξ Example A - 12edo

Ξ Example B - 7edo

Ξ Example B - 12edo

(Image.1) This geometric visualization compares 7-EDO with the diatonic scale in 12-tone equal temperament on a logarithmic scale. Transposition of the 7-EDO structure yields identical intervallic relationships, whereas transposition of the diatonic scale reveals the seven familiar modes of 12-tone music.

4: The Four Layers of Duodecimability

Why do we even pre-select pitches to begin with?

This question leads us directly to the core assumption embedded in most musical thinking: that pitch exists within some kind of lattice, and that we can—or must—choose a finite subset of it. Whether this subset is the Western 12-tone equal temperament (12-EDO), or any other system, the act of pre-selection implies a structure. And the moment we impose a structure, we are already translating sound into something else—a symbolic language, a tool of control, a perceptual grid.

But what exactly does that structure point toward?

This chapter introduces the idea of duodecimability—a classification of how different tuning systems interact with, defer to, or break from the gravitational field of the 12-tone system. This is not a claim about ontological “truth” in tuning, but rather a pragmatic framework: a way to talk about tonal constancy using known musical concepts, while acknowledging that these concepts are culturally situated and not universal.

We must walk the tightrope here: 12-EDO is not natural, but it has become our natural. Thus, while future chapters will dismantle the idea of 12-tone supremacy more explicitly, this chapter works within it, to define different degrees of translatability—from systems that are subtly coercible into 12-EDO interpretation, to those that fundamentally resist it, even at the level of acoustic substrate.

This framework is not merely theoretical—it offers a way for microtonal composers and instrument designers to better understand where they sit in relation to tradition. It also helps reveal why some alternative systems feel like “flavors,” while others feel like new species.
The Four Layers of Duodecimability

A tuning system’s “duodecimability” refers to the degree to which its pitches, functions, or perceptual structures can be interpreted—or misinterpreted—through the lens of the 12-tone system.

Each layer below marks a progressively deeper departure from 12-EDO as both perceptual default and theoretical grammar:

 
Layer 1: Perceptual Forcing
("Duodecimability by Momentum")

Definition:
A system that is mathematically unrelated to 12-EDO, but is perceptually coerced into a 12-tone framework by the listener’s tonal expectations.

Mechanism:
Tonal constancy combined with melodic trajectory. The listener’s brain fills in “missing” functions based on contour, rhythm, and cultural conditioning.

Example:
7-EDO. Despite its equal-step structure (~171 cents per step), melodies played in 7-EDO can imply tonic-dominant relationships, cadential closure, and even modal coloration. When three consecutive 171-cent steps are interpreted as [0, 100, 300] in 12-EDO, listeners hear something resembling a major third. The actual intervals don’t match—but the function does. The translation happens inside the brain, not the score. This is tonal constancy at its most extreme: imposed, not earned.

 
Layer 2: Relational Reference
("Duodecimability by Analogy")

Definition:
A system that contains more than 12 pitches, but still describes itself in terms derived from the 12-tone world—thus retaining it as a conceptual anchor, even if not a physical one.

Mechanism:
Category refinement. These systems acknowledge and orbit around 12-EDO concepts like major/minor, fifths, thirds, etc., often subdividing them or redefining their boundaries.

Examples:

Arabic Maqamat: While using microtonal intervals, the maqam system still frames its identity in terms of scalar steps and modal gravity. The intervals are described with reference to their 12-EDO cousins: neutral seconds, sub-minor thirds, etc.

19-EDO, 22-EDO: These temperaments extend the lattice but rarely abandon diatonic thinking. We still hear “tonics,” “leading tones,” and “dominants,” even if their sizes shift. The systems are enriched, not alienated.

The language used in pedagogy and notation often betrays this link: if you’re still calling something a “flattened third,” you’re still playing the 12-tone game—just with extra tiles.

 
Layer 3: Structural Alienation
("True Induodecimability with Harmonic Timbre")

Definition:
A tuning system whose internal logic—how it generates scales, harmonies, and motion—has no connection to diatonic or 12-tone principles.

Mechanism:
Constructed from different mathematical or harmonic seeds. Even when using harmonic instruments (e.g., with overtone series), the structural rules cannot be mapped onto 12-EDO categories.

Examples:

The Bohlen-Pierce Scale: Built on the tritave (3:1) instead of the octave (2:1), and divided into 13 steps. Its chords are derived from odd harmonics (3, 5, 7...), and its “fifths” and “thirds” have no analogues in 12-EDO. It creates functional progressions—just not those functions.

10-EDO: This system, while octave-repeating, produces arpeggios whose internal logic is highly ambiguous. A sequence may contain chords that, individually, can be squeezed into 12-EDO, but collectively shift too much—they demand multiple conflicting translations. The system resists stable mapping.

Certain Just Intonation Lattices: Especially those incorporating the 11th or 13th harmonics. These systems often sound “smooth” or “consonant” when played with harmonic timbres, but their intervallic logic has no fixed counterpart in 12-EDO. Trying to approximate them with standard pitches is like translating poetry using only rhymes—it misses the meaning entirely.

This is the tipping point: these systems can sound beautiful, but their logic is alien. You can enjoy them, but you can't name them with your old words.

 
Layer 4: Timbral Dissolution
("Induodecimability by Substrate")

Definition:
A musical system whose inharmonic timbres prevent any pitch structure from aligning with 12-EDO attractors—rendering not only tuning, but pitch itself, unstable or secondary.

Mechanism:
The overtone series is no longer harmonic. With no integer multiples, the usual anchors—octave, fifth, third—do not emerge naturally in perception. The substrate itself erodes tonal identity.

Example:
Gamelan music. The metallophones used in Balinese and Javanese ensembles produce inharmonic spectra. Their tuning systems (e.g., slendro, pelog) are not “approximations” of 12-EDO—they are entirely separate epistemologies. A tone's pitch is defined by its timbral fingerprint, not by harmonic ratios. Even the concept of an interval can dissolve into a cloud of color and resonance.

Attempting to analyze this using Western theory is like applying Latin grammar to birdcalls: the medium does not support the metaphor. These musics are not microtonal—they are extratonal.

 
Closing Reflection

These four layers don’t represent value judgments. They describe degrees of translatability, not superiority or purity. Each layer tells us more about how listeners—trained and untrained—perceive, categorize, and force-fit sound into symbolic boxes.

They also suggest that duodecimability is not a binary. It is a gradient, and perhaps a contested one: where you place a system may depend not only on its design, but on your listening history, your training, and your linguistic tools.

We have learned to hear with twelve ears. But what happens when we encounter a sound that won’t sit still on any of those notes?

In the next chapter, we turn from classification to emergence: how tonal categories form in the first place, and what kinds of mental scaffolding make tonal constancy—and its resistance—possible.

5: The Spectrum of Familiarity — Microtonal Flavor vs. Functional Break

Music doesn’t become "otherworldly" just because it uses strange intervals. Not all microtonality is revolutionary. In many cases, it is ornamental, expressive, a kind of seasoning—a flavor layered atop an underlying structure that is still resolutely tonal.

The difference between microtonal flavor and functional departure is a spectrum, not a binary. But it’s crucial—because it defines whether a piece of music is interpretable, translatable, or cognitively disorienting. And that distinction hinges on tonal constancy: whether the listener can still rely on familiar perceptual anchors—tonic, cadence, resolution—even as the tuning system mutates around them.

 
Historic Flavors: Chopin and Meantone Coloration

A famous example: Chopin referred to D minor as the “saddest” key.

At first glance, this seems metaphysical or poetic. But in fact, there was a physical reason. During his time, many pianos were tuned in meantone temperament, a system optimized for certain intervals using simple integer ratios. While 12-tone equal temperament (12-EDO) was theoretically known—and even in use—it was still rare for instruments to be tuned to it precisely. Ear-based tuning methods favored rational approximations. Algorithmically, meantone was simply more practical before electronic tuners.

The result: each key had a unique color, a subtle deviation in interval sizes that made D minor sound distinctly different from, say, B minor. These were microtonal inflections, not fundamental departures. The harmonic framework remained diatonic. What changed was the flavor profile of each key.

 
Modern Examples of Flavor: Bends, Blues, and Maqamat

Today, the idea persists in many styles:

Blues music bends between notes of the pentatonic and chromatic scales, sliding into pitches that don't "exist" in 12-EDO notation. These expressive bends act as stylistic inflections, not harmonic challenges. The tonic remains the tonic. 

 

Arabic Maqamat and Persian Dastgah systems incorporate quarter-tones and nuanced scalar steps, often creating pitches "between the keys." Yet these systems still rely on cadential logic and tonal gravitation. The microtones serve as ornaments, bridges, flavors. They rarely seek to dissolve the entire structure—they aim to enrich it.

In both cases, duodecimability remains possible, even if imperfect. A skilled listener can still find the center of gravity. These are flavored tonalities, not alternative logics.

 
Functional Break: When Tonal Constancy Fails

What happens, though, when the system no longer submits to interpretation?

Below is an example from 8-EDO—a tuning system that divides the octave into eight equal steps (150 cents each). It contains two maximally symmetric diminished scales, and enough pitch density to form chords and melodies. However, the logic of this system is non-diatonic by design.

Try to map its harmonic progressions to 12-EDO, and tonal constancy breaks. No amount of perceptual coercion or melodic expectation can fully translate its motion. The listener doesn’t "mishear" it as tonal—they simply hear it as strange.

Why?

Because 8-EDO sits out of phase with 12-EDO. There are no simple ratios shared between their step sizes. Their intervals don’t approximate one another; they contradict each other. This is the threshold at which duodecimability fails entirely. Translation is not fuzzy—it is impossible.

 
Octave Retention vs. Structural Alienation

Interestingly, 8-EDO still uses the octave as a repeating unit. This gives it a slight advantage in group performance and instrument design: parts can be transposed, ranges can be shared.

Compare that to the very similar scale, Bohlen-Pierce (13-ED3)—a system that replaces the octave (2:1) with the tritave (3:1). While rich in harmonic possibilities (especially with odd harmonics), it loses the universal reference point that the octave provides. The result: true structural alienation, especially in chordal writing. Melodies still function, but harmonies drift into perceptual limbo. An approximate 1.96 ratio, close to the octave, exists—but it is harmonically incoherent with traditional instruments.

This is why 8-EDO, though less famous, can feel more playable. Its symmetrical design makes it excellent for exploring alien harmonic functions while maintaining just enough structure for ensemble use.

 
The Takeaway: The Diatonic Ghost is Hard to Kill

Even in highly divided systems like 19, 22, or 31-EDO—often used for their greater consonance or intonation precision—diatonic templates resurface. Musicians use them to better approximate known categories, not to invent new ones. In fact, the higher the division, the more tempting it becomes to overfit microtonal pitch space to traditional harmonic roles.

By contrast, systems like 8-EDO or 10-EDO—low-subdivision tunings that avoid rational alignment with 12-EDO—offer fewer handholds. Their symmetry, spacing, and internal logic prevent easy mapping. They don't flavor tonal music—they replace it.

These systems are functionally distinct, and their progressions defy tonal constancy. This is the boundary line: where the mind stops hearing “altered chords” and starts hearing new grammar.

 
Closing Note

The difference between flavor and functional break is not merely theoretical. It defines whether music can still operate within a shared perceptual vocabulary, or whether it demands the invention of a new one.

In the chapters ahead, we’ll explore this boundary more formally: how tonal categories form, and what kinds of cognitive attractors allow—or prevent—the perception of coherence when pitch structures drift too far.

Or put more provocatively: when does a microtone become a mutiny?

6: Anchor Density — Diatonic Memory and the Illusion of Familiarity

Tonal constancy does not act on systems uniformly. Some tunings invite diatonic reinterpretation easily. Others resist it entirely. The key difference is not simply the number of steps per octave or the presence of harmonic intervals, but what we call Anchor Density—the frequency and distribution of elements that are close enough to recognizable tonal functions that the brain tries to interpret them as such.

This is not a binary switch. Induodecimability, as introduced earlier, isn’t “on” or “off.” It’s a gradient of perceptual traction—the ease with which the listener's cognitive machinery can hallucinate tonal relevance from non-diatonic material.

 
Anchors: The Seeds of Tonal Illusion

We define an anchor as a moment—a pitch, a dyad, a short progression—that approximates a recognizable function within the 12-EDO system. These are not structural absolutes; they are perceptual affordances. A chord that sounds like a major triad—even if it's technically off—is an anchor. A cadence that feels like resolution is an anchor, regardless of its tuning origin.

The brain uses these moments to project a familiar tonal grid over unfamiliar territory. It’s a form of perceptual compression, and it’s why even radically mistuned systems can feel “not quite right” instead of “completely alien.”

 
Case Study: The Diatonic Categorization Experiment

A striking example of this phenomenon appears in Jason Yust’s paper, "Diatonic Categorization in the Perception of Melodies." In the study, ~30 participants—primarily musicians or audio professionals—were asked to categorize melodies played in an unusual subscale of 13-EDO: a seven-note scale with step intervals [0, 2, 4, 8, 10, 12,13].

Despite the scale’s deep structural departure from 12-EDO, participants consistently used diatonic terms to describe what they heard—“major third,” “perfect fourth,” “leading tone,” etc. Not a single subject identified the system as non-12-EDO. Even with highly trained ears, the internal tonal schema overrode the signal.

This was not 13-EDO as a novel tuning. This was 12-EDO imposed on a foreign substrate.

One participant with absolute pitch was excluded from the results for a fascinating reason: they did detect the detuning, but failed to map it to any tonal framework—an inability to “force” the signal back into the familiar. This suggests that tonal constancy relies more on relational familiarity than on fixed pitch memory.

 
The Anchor Density Spectrum

Let’s break down two critical points on the anchor density gradient.

 
1. High-Density Anchors: Partial Duodecimability

Systems: 11-EDO, 13-EDO, high-fidelity Just Intonation subsets
Perceptual Experience: Slippery, ambiguous, “almost tonal”
Mechanism: Local phrases strongly resemble 12-EDO intervals, triggering familiar categories

These systems produce local illusions of tonality. You might hear a chord that “feels” like a major triad—your brain engages tonal constancy, and you momentarily experience a key center. But when the next chord arrives, the illusion fails. The logic collapses. The system can't sustain a consistent diatonic mapping over time.

We call this the Shifting Grid Problem: tonal constancy can win a battle, but it loses the structural war. The mind would have to rebuild the entire tonal scaffold for each new event—a computation it can't maintain over time.

This explains why many listeners describe such music as “drifting,” “haunting,” or “unstable.” It’s not unfamiliar in total—it’s familiar in fragments, and that inconsistency is deeply disorienting.

 
2. Low-Density Anchors: Global Induodecimability

Systems: Bohlen-Pierce, 8-EDO, some dissonant Just Intonation networks
Perceptual Experience: Profound alienation, unfamiliarity, or unclassifiability
Mechanism: Few (or zero) interval categories resemble 12-EDO constructs

Here, tonal constancy doesn't just fail intermittently—it fails completely. There are no islands of recognition. The harmonic series is differently parsed, the scale is divided in unfamiliar ratios, and resolution itself might not even be meaningful.

These are truly induodecimable systems. Even approximate mappings to 12-EDO don’t make sense. The brain has to make a choice: either accept this new musical logic on its own terms, or reject it as noise.

 
Implications: When Hallucination Meets Constraint

Anchor density reveals a deep cognitive tradeoff. Tonal constancy can only operate within a limited domain of error. Systems like 19-EDO or 22-EDO can stretch that domain without breaking it; systems like 8-EDO snap it in half.

Flavor becomes function when too many anchors accumulate. Microtonality starts as expression—but once enough structural anchors reappear, the listener’s mind imposes full tonality onto the scale.

Neural Mechanisms and Predictive Models of Tonal Constancy

Neuroimaging and electrophysiological studies reveal that specific regions of the auditory cortex selectively respond to structured sound patterns such as speech, melody, and harmonic sequences, but remain relatively inactive during exposure to unstructured noise. Notably, areas such as the planum temporale, located posterior to the primary auditory cortex, appear to engage dynamically when pitch structures exhibit internal regularities, even when those regularities are statistically subtle or culturally learned.

These regions are not merely passively decoding incoming sound—they participate in an active predictive process. The brain constructs internal models of melodic or harmonic progression, and generates expectations for future events. When a pitch contour unfolds predictably, it minimizes error between the expected and actual input, triggering dopaminergic reward responses in associated circuits such as the nucleus accumbens. These reward-linked responses, observed even in anticipation of musical climaxes, suggest that successful pattern prediction is inherently satisfying, reinforcing the learned tonal templates over time.

Such findings align well with a Bayesian perspective: the brain updates internal priors based on the statistical structure of the sound environment, forming what we might call tonal basins—perceptual attractors that stabilize around culturally salient pitch configurations. These basins guide pitch interpretation even when the physical signal is ambiguous, distorted, or derived from non-standard tunings (e.g., 7-EDO or inharmonic timbres). The result is tonal constancy, rooted in statistical expectation, contextual prediction, and hierarchical sensory processing.

Given these parallels, we may consider whether concepts from statistical mechanics or nonlinear dynamical systems could be adapted to describe perceptual behavior in pitch space. Several potential analogies emerge:

Neural/Perceptual Concept   || Physical/Mathematical Analog:

Tonal template / basin      || Potential well in an energy landscape
Contextual pitch trajectory || Particle path with momentum
Dopaminergic reinforcement  || Entropy minimization with energy input

Such analogies are speculative, but not without foundation. Perception is inherently path-dependent, sensitive to both immediate context and long-term learning. The momentum of pitch trajectories, as previously discussed, may correspond to cumulative Bayesian updating or even to forms of inertial processing, where prior motion in tonal space biases future interpretations.

Tonal constancy may not yet be a widely codified term in music cognition, but the phenomena it describes lie at the heart of musical experience. From the brain's predictive machinery to the statistical learning of pitch spaces, it offers a compelling bridge between acoustic structure, cultural form, and neural function. As we continue to unravel the relationship between sound, meaning, and expectation, tonal constancy may prove to be not just a perceptual curiosity, but a fundamental principle in how humans find coherence—and beauty—in the musical world.

The Perceptual Construction of the Octave: Cyclicity, Distance, and the Substrate of Tonal Constancy

This raises a foundational question regarding the limits of perceptual structuring: could the brain impose a cyclical framework, such as an octave, within a narrow frequency band like that between 100 Hz and 101 Hz?

This question cuts to the root of how we perceive pitch structures. The octave, usually defined as the interval between a frequency and its double, is often treated as a given in music and psychoacoustics. But what is an octave, perceptually? Is it a fixed truth, or a learned, contingent structure imposed by cognitive and biological processing?

What Is the Octave?

Historically, the octave inherits its importance from multiple overlapping domains:

Physically, it manifests as half the length of a string or air column.
Musically, it frames the diatonic scale and tonal structures.
Mathematically, it represents a 2:1 frequency ratio—elegantly simple, multiplicative, recursive.

It’s tempting to accept the octave as self-evident, especially under Pythagorean or harmonic frameworks. However, perceptual mechanisms introduce complexities that challenge this physical simplicity. In the brain, the octave is not merely heard—it's constructed.

Tonal Constancy Requires a Cycle

In our 12-tone system, it's the octave that closes the loop. Without a perceptual cycle, there’s no frame of reference; no concept of “return”; no meaningfully bounded space in which melodies can live and relate.

But crucially, this cyclicity is not just physical. The pitch continuum (20 Hz to 20 kHz) is logarithmic, but our perception of distance across it is nonlinear and mediated by internal frameworks like tonal templates. The perceptual salience of an interval depends on its place within a cycle—not just on raw frequency ratios or ranges.

JND, Range, and the Myth of Resolution

To illustrate this distinction, consider two hypothetical listeners. Listener A possesses a standard auditory range (20–20,000 Hz) and a just noticeable difference (JND) of 10 cents. Listener B, in contrast, has a severely restricted range (10–11 Hz) but a proportionally finer JND, resulting in the same total number of 1,200 discriminable pitch steps. While both listeners can technically differentiate an equal number of unique pitches, their perceptual experiences of the pitch world are fundamentally dissimilar. The topology of their perceived pitch space differs radically; one spans ten octaves, while the other encompasses less than a semitone.

Even if their pitch resolution is equalized, the topology of pitch space is not. One spans 10 octaves; the other spans less than one semitone. The experience of distance—of musical identity, direction, and hierarchy—is rooted in cyclical frameworks, not in raw resolution.

This demonstrates that perceptual distance is not simply a product of range or JND. It emerges relative to cycles, and is shaped by shared statistical and structural expectations—like those derived from the harmonic series.

Cycles Are the Substrate of Hearing

So what gives rise to perceptual constants like pitch equivalence, or intervals like the octave?

The answer is statistical redundancy in the harmonic series. When you double a frequency, all overtones align perfectly—no new information is introduced. Whether simultaneous (consonance) or sequential (temporal prediction), the brain notices this and treats such transformations as perceptual "returns."

Even with pure sine waves—devoid of harmonic cues—the octave retains salience. This suggests that the octave is a learned statistical attractor, reinforced by both acoustic regularities and internal models.

Importantly, if you stretch this cycle—say, to 1100 or 1300 cents—and still use recognizable melodies within it, perception holds. The melodic identity persists, even as the physical cycle distorts. This shows that tonal constancy is momentum-based: the mind keeps cycling where it's been taught to.

Toward a Definition of Perceptual Cyclicity

Let’s propose a basic abstraction:

Perceptual cyclicity: cyclical sameness in continuous one-dimensional perceptual space, where constant motion in one direction encounters regularly spaced, similar percepts.

This is distinct from raw discriminability. The cycle introduces relational structure: every point within it has a maximum antipode (e.g., 600 cents from a reference tone in a 1200-cent cycle), and recursive midpoint divisions (50%, 25%, etc.) define internal geometry.

Even with degraded range or resolution, this internal geometry remains. That’s what makes tonal constancy constant.

Timbre and the Breaking of Cycles

An interesting experiment is to consider whether this system could be artificially manipulated. Suppose we design timbres where "sameness" is defined by a 2.5:1 ratio. Could we train a new perceptual cycle?

Maybe. But statistical coherence breaks down. Combination tones, missing fundamentals, and overtone interactions all push the brain back toward the harmonic series. You can stretch the cycle, but not endlessly. Eventually, you lose pitch identity—and with it, the cycle.

This suggests the octave is not a physical given but rather a 'habit' of perception..., anchored in physics but shaped by cognition. Without a cycle, there is no tonal identity—only pitch. Tonal constancy, diatonizability, and musical distance all require some cycle to act as a reference structure.

And that raises the real question:

Could the brain construct a "cycle" from 100 Hz to 101 Hz?

While the successful construction of such a cycle may be impossible, the crucial insight is the brain's persistent attempt to impose cyclical structure on acoustic phenomena.