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.
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.
-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.
-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.
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?
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?
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
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.
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