The Interval Matrix

DRAFT
This article introduces the concept of the interval matrix from a traditional music theory perspective, alongside a software tool designed to create and visualize these matrices. In this context, intervals refer to proportions or ratios between numbers.

The interval matrix is built from all possible representations of a set's values under an equivalence relation, using each element as a base, resulting in a numerical or geometrical table—a matrix—that represents this expansion.

These matrices are not initially intended for conventional matrix operations; the focus lies in the geometric structure that emerges from different sets and their elements' relationships.

Interval Matrix software. Prime numbers up to 19(set to periodic), with equivalence 1:2 (octave-space)
\(\mathbf{Ä}_{1:2}(P_{19})\)

For an infinite set, the matrix cannot be fully generated. However, if the set has a repeating pattern (period), a minimal generating set can be identified. The matrix is then built and completed using this minimal set, (n-by-n) as seen in a common musical tuning system (a set of pitches or rhythms used to create or perform music).

This period typically becomes the primary equivalence relation (equave) parameter in the set's function for constructing the matrix and analyzing the intervals within.

The matrix can be constructed for a finite set that isn't meant to repeat. For example, in music, this approach can be used to analyze notes on an instrument where there's no indication to continue calculating additional pitches. This method applies to any finite set. In a finite matrix, each row contains one element less than the previous row.

Set and matrix construction:

For analyzing a set \(S\) that is already normalized and within the desired range—such as in any pre-calculated musical tuning system—the set remains unaltered, and the matrix is built directly \(\mathbf{A}(S)\). The only required parameter is its periodicity: Is the given set a minimal generating set of an infinite set, or does it represent a fixed, finite number of elements?

Most examples here will use periodic matrices. To denote matrix periodicity or non-periodicity, we might use different notation, such as \(\mathbf{Ä}\) for periodic matrices and \(\mathbf{A}\) for non-periodic ones.

The generalization of the interval matrix construction allows us to relate different sets and reductions, enabling us to find congruences between systems. The reduction function (which corresponds to the chroma function when the space is the octave, 1:2) for a real matrix, where the set consists of any real numbers, operates as follows:

The absolute value of each element is taken, and the function then returns this value, reduced or remapped (if necessary) by an equivalence relation:

For a value \(s_x\) larger or smaller than the chosen equivalence relation \(r\), it is reduced to a new element \(\tilde{s}_x\) by applying the operation:

\(\tilde{s}_x = |s_x| \bmod 1:r\)

(This uses the mod symbol because it effectively returns the intervallic remainder. This process involves repeatedly multiplying or dividing \(s_x\) by \(r\) until \(s_x\) falls into \((1, r]\) space. This page has details about interval reduction.)

Since the matrix is defined by reinterpreting the set values with each element as the base, all rows inherently start with 1. Consequently, the reduction, or normalization, is consistently performed as \(\bmod 1:r\)

Optional: A constant \(\delta\) may be applied to each element of the set before performing the base change.(this in relevant for other uses explained in other article)

The reduction can be notated and performed for sets \(S_{1:r}\) without considering any matrix. It can also be used in constructing the matrix, \(\mathbf{A}_{1:r}(S)\), which implies both reduction and base shifting.

Example: If \(S\) = {1, 2, 3, 5}, then \(S_{1:2}\) ​= {1, 3/2, 5/4, 2}, and \(\mathbf{Ä}_{1:2}(S)\) would yield [{...},{...},{...},{...}]. (reduced and periodic)

Interval Matrix Definitions:

• Full Interval Matrix: \(\mathbf{A} = \mathbf{A}_{s_n}^{\delta}\)
  This matrix uses the last or largest element of its generating set as the equivalence relation.

• Local Interval Matrix: \(\mathbf{A}_{s_i}^{\delta}\)
  This matrix uses any element within the generating set as the equivalence relation, except for the largest one.

• External Interval Matrix: \(\mathbf{A}_{x}^{\delta}\)
  This matrix uses a value outside the generating set as the equivalence relation. 

A full interval matrix built from a periodic set is inherently a symmetric matrix.

A full or local interval matrix is not "useful" for isotropic sets (where the chosen period or relation is a member of the set). This leads to identical and overlapping shifts of the elements.

Musical Interpretation:
For example, the 12-tone equal temperament \(\text{12ed2}\) guitar is an interval matrix (incomplete) representing the infinite set generated by the constant \(2^{1/12}\). Each row is shifted by five elements from the previous row (except between \(\text{G}\) and \(\text{B}\), where the shift is four). The matrix is trivial for this set's intervallic analysis, as columns (frets) are always aligned regardless of the shift or element taken as base.

Interval matrices are tipically shifted by one element until they are complete.

Consider this group: \(\langle 2, 3 \mid 3^2 = 1 \rangle \). This represents a set of infinite fifths and octaves. One of its minimal generating sets is \(S\) = (1, 3/2, 2]. The resulting matrix \(\mathbf{Ä}_{1:2}(S)\) has only two rows:

(1,  3/2,  2]
(1,  4/3,  2]

Interval Matrix Accumulation: \(\text{Acc}(\mathbf{A}(S))\)

This is a new set with all the representations of the elements under the set equivalence relation, which unfiltered, might repeat values, helping to find prevalent proportions. Isotropic sets always have an accumulation identical to any of their matrix rows. (The accumulation is a vectorization or flattening of the matrix)

In this case, the infinite set generated by \(\langle 2, 3 \mid 3^2 = 1\rangle\) = { ..., 1/2, 2/3, 1, 3/2, 2, ...} has an interval accumulation (under the equivalence 1:2):  (1, 4/3, 3/2, 2].

The distinction between full, local and external interval accumulations reflects the matrix type.

For example, consider a local matrix \(\mathbf{A}_{1:2}(S)\) constructed from the set {1, 2, 3, 4} in octave space (with an equivalence relation of 1:2). The local accumulation would be:

\(\text{Acc}(\mathbf{A}_{1:2}(S))\) = {1, 4/3, 3/2, 2} (filtered, with non-repeated values)

To obtain the global or full accumulation, the space is set to the largest element in the set. Thus, the matrix built from the set {1, 2, 3, 4} under the equivalence relation 1:4 would yield:

\(\text{Acc}(\mathbf{A}(S))\) = {1, 4/3, 3/2, 2, 3, 8/3, 4} (filtered)

For larger and more complex sets, the accumulation also provides a method for finding a possible natural mode of the set, if any.

Let’s take the pentatonic \(\langle 2, 3 \mid 3^5 = 1 \rangle\)
a minimal generating set is { (1, 9/8, 81/64, 3/2, 27/16, 2/1] }, its full matrix (omitting 1):

{9/8,  81/64, 3/2,    27/16, 2/1}
{9/8,    4/3, 3/2,    27/16, 2/1}
{9/8,    4/3, 3/2,    16/9,  2/1} Natural Mode
{32/27,  4/3, 3/2,    16/9,  2/1}
{32/27,  4/3, 128/81, 16/9,  2/1}

The natural mode of any set is the particular representation that includes the most frequent values appearing after shifts; it is the most faithful or weighted representation of the set.

How the Interval Matrix App Works

It accepts a list of numbers, treating them always as a minimal generating set(for now).

If the list/set is an already a reduced tuning system, the matrix is full and the equave(period, interval of equivalence) parameter should initially be set to match that of the set, typically the last and largest value. It does not adjust it automatically.

The matrix displays for each element in each row: the original value inserted, the reduced value(if it was reduced), a delta value(if it was displaced), and a rational approximation of the value.

The delta value comes from the delta parameter, usually 0. This value is added to every element in the original set before the rest of the calculations. This is useful for understanding how a minimal set, while maintaining its original absolute difference between members, shapes through this change.

For example, you can start with period/equave 1:2, and this set {1,2,3} reduces to {1, 3/2, 2}, but with delta = 3, it becomes {4, 5, 6}, and reduced, {1, 5/4, 3/2, (2)}.

Prime numbers up to 19. Delta = -1, octave-space.
\(\mathbf{Ä}_{1:2}^{-1}(P_{19})\)

The rational approximation has an adjustable tolerance value.

On top of the interval matrix, there is a configurable equal division ruler that helps with intervallic/ratio measures.

The chroma matrix has a fixed equivalence relation of an octave and, by default, starts at red. You can select whether the chromas displayed are absolute or relative to each row. When selecting relative, the full spectrum located in the bottom UI expands to display all the possible chroma shifts. (The full spectrum isn’t really "full"—you set a maximum space to occupy, with a logical maximum of the human hearing range.)

This last part is the most important when dealing with musical tuning systems; practical tuning systems have a simpler chroma matrix.

Unlike Scala files, the 1 must be inserted (remove it to understand what happens). You can, if you want, omit the equave in this list; it will be added (invisibly) from the equave parameter. However, it’s useful to keep it too, for example, when analyzing a non-octave tuning using an equave 3 (tritave). You can omit it, but if you want to inspect these intervals reduced to an octave, you might want to keep it and track it. So if when the set has an element equal to the equave, you will find two identical rows in the matrix.

Future Development

If you paid close attention to the code of this app and the SFINX app, you may have noticed that they use the same engine. That’s because, as I have pointed out, a guitar is essentially an interval matrix by string length.

My goal is to finally reunite both apps—SFINX was developed to aid in the graphic and diagram generation of scales for microtonal guitars, while the Interval Matrix was developed ideally for geometric analysis of sets and chromas.

(DRAFT)

Link to the apps:

jbcristian.github.io/xeneize/

Another Aural Temperament or Golden Harmonics

Phi (\(\phi\)) is defined as the arithmetic mean of \(1\) and \(\sqrt{5}\), similar to how the fifth \((3/2)\) is the arithmetic mean of the octave \((2/1)\). This system maps powers of \(\phi\) into \(\sqrt{5}\) space, making it the period. As an interval ,\(\sqrt{5}\) , represents a traditionally consonant (though irrational) ninth, situated between \(16/7\) and \(15/7\).

By stacking and folding (rotations on the circle in log-coordinates) four powers of \(\phi\) into \(\sqrt{5}\) space ( \(\{ (\sqrt{5})^n \times \phi^m \} \in [1,\sqrt{5})\) ), with \(n\) and \(m\) integers, the first five notes (zero index) include unison/identity and four effective powers, implying five quasi-equal divisions of \(\sqrt{5}\).

Similarly, using nine powers of \(\sqrt{\phi}\) approximates a 10-ed√5, which is also close to a 6-edΦ.(see note) 

This introduces a smaller step of about 139¢, forming a suitably sized leading tone, slightly smaller than that in the Bohlen-Pierce scale and 8-edo.

The small difference provides a good initial fifth, approximately \(1.489... = (\sqrt{\phi})^5 \times (\sqrt{5})^{-1}\) It is locally tonic, allowing for 3 or 4 consecutive notes within the "same scale." The next notes, guided by "consonance," align with a different tonal center scale out of phase with the previous one (sparse duodecimability). Additional "belonging" notes appear farther up, as this is an infinite chroma system that avoids near octaves in most practical ranges (nearest at 1253 cents, next at 2368, etc.), while other notes can create a distinct mode/s, as traditionally understood.

The tuning also provides a spanned but functional \(V_7 \rightarrow I\) chord progression (video/audio 1:10).

"Acoustic phi" is dissonant with harmonic timbres, so I avoided it in this guitar solo. It acts as a pivot for the separated tonics, creating a centerless sound.

Why \(\phi\) Acts Like a Generator of \(\sqrt{5}\)-space

The connection comes from expressing \(\phi\) in terms of equal divisions of \(\sqrt{5}\). 

Already at the first step, \(\phi\) itself is an excellent approximation to a rational power of \(\sqrt{5}\):

\(\phi = (1+\sqrt{5})/2 ​\approx (\sqrt{5})^{3/5}\).

Numerically,

\(\phi \approx 1.6180\), (\(\sqrt{5})^{3/5} \approx 1.6206\).

The error here is tiny (~ 833 vs 835 cents), small enough that, for all perceptual and practical purposes, \(\phi\) can be treated as though it were exactly a fractional division of \(\sqrt{5}\).

This means that stacking powers of \(\phi\) within \(\sqrt{5}\)-space is essentially the "same" as running an equal-step generator chain (just like stacking fifths in 12-EDO approximates octaves).

From this perspective: Four powers of φ fold neatly into \(\sqrt{5}\), yielding an effective 5-ED(√5) division. Using \(\sqrt{\phi}\) instead, you get 10-ED(√5) (≈ 139¢ steps), which is the scale i actually play with. (here we get the inexact acoustic phi at \((\sqrt{5})^{6/10}\) ).

edit: i just noticed some type errors in the table graphics,,, and on notation:

In the video, the tuning is described with generators in the form

\( (\sqrt{5})^n \times (\sqrt{\phi})^{m \bmod 10}\), with \(n, m \in \mathbb{Z}\)

 For \(m\), it doesn’t matter whether \(m \in \mathbb{Z}\) or \(m \in \mathbb{N_0}\), as modular reduction absorbs negative values.

For example,

\((−4 \bmod 10)=6\).

To avoid confusion, we can write \(m \in \{0,1,2,…,k−1\}\), so that m explicitly ranges within this set.

---

The Phi “Intervalizations”

In Western music theory, the constant phi has been interpreted as a musical interval in a couple of ways, with the most common being “acoustic phi” and “logarithmic phi.” Acoustic phi takes the normalized value of phi (about 1.618) and treats it as a frequency ratio, similar to how intervals like a fifth are 3/2 or 1.5. An interesting feature of this approach is that the resulting combination tones are also in golden proportion. Logarithmic phi, on the other hand, represents the golden ratio of the octave, calculated directly in cents as 1200 × 0.618 approx 741 cents.

Formulas and calculations vary among authors and theorists, and there are many different ways to interpret or "hear" the golden ratio.

Scala file:

! rphi9-rfive.scl

!
9 powers of square root of phi, mapped into square root of five space. rational approximations with gcd tolerance .000001
10
!
2983/2768
1597/1364
491/386
451/329
1741/1169
987/610
1109/636
646/341
743/361
2207/987

group theory notation https://xcjb.blogspot.com/2024/08/pythagorean-scale-z12z-z.html

Pythagorean Scale ≅ Z/12Z ⊕ Z

If you’re familiar with some Group Theory but not much about music theory, especially tuning theory and ancient tuning methods, this article will help you see how they connect. On the flip side, if you know tuning theory, the Pythagorean scale, and common interval manipulations but nothing about Group Theory, this article will guide you in understanding their relationships.

Pythagorean Scale and Group Theory

The Pythagorean Scale and similar tuning systems across cultures exhibit a clear group-theoretic structure,(for example forming finitely generated abelian groups).


I. The Pythagorean Scale

The Pythagorean Scale is one of the most well-known tuning systems from antiquity and continues to influence Western music theory. While similar intervals and generative methods are found in other cultures (1), the scale remains a fundamental example within the broader Pythagorean framework of number and harmony. Musicians beginning their study of tuning theory often learn about the Pythagorean Scale as a precursor to modern equal temperament (12EDO). (However, this characterization is not entirely accurate (2)).

A multitude of tuning systems has existed since antiquity, and in modern times, many more have emerged due to the ease of implementation and experimentation with synthesizers and computers. While some contemporary tuning systems employ sophisticated mathematical concepts, group theory is frequently applied to both tuning definitions and musical analysis. Despite the prevalence of textbooks linking music to algebraic structures, the Pythagorean Scale itself has not been explicitly identified as an instance of group-theoretic structure in either musicological or mathematical literature.(i havent found any direct mention, maybe there is, most music theory sits at the silo-obscure)

Ancient theorists did not conceptualize musical intervals as elements of an algebraic group. Instead, they developed practical tuning methods that implicitly embody group-theoretic principles, driven by the acoustical properties of intervals and human perceptual preferences. Still, it is accurate to describe the Pythagorean Scale (and its cross-cultural analogs) as one of the oldest implicit examples of a finitely generated abelian group (FGAG). This retroactive classification underscores the universality of mathematical patterns in music, even when the underlying theory remained undiscovered for millennia.

This study shows that the group structure is inherent to the algorithm used to construct the scale, as reflected in modern interpretations found in numerous music theory textbooks and historical references (e.g., Boethius, Ptolemy, Guido d'Arezzo, Vincenzo Galilei).

It is important to acknowledge that this analysis presents a specific perspective on the Pythagorean Scale, focusing on its algorithmic structure. Historically, the scale has been interpreted through various lenses, including harmonic theory, philosophical considerations, and perceptual studies. This article does not seek to invalidate those interpretations but rather to provide a complementary perspective rooted in group theory. The focus remains on the mathematical properties of the algorithm itself, independent of any particular musical application or aesthetic judgment.

Some may argue that labeling ancient tuning systems with modern algebraic terminology is anachronistic without explicit recognition of group axioms. However, in mathematics, it is common practice to retroactively classify structures once their properties are understood. For example, ancient symmetries are now described using group theory.

The following sections will review the historical context, examine the algorithmic generation of the scale, and formalize it using group theory, revealing a direct correspondence.


II. Historical Context & Algorithmic Generation

Many tuning systems share a common foundation, historically referred to as "chaining/stacking and reducing/folding" or its linguistic equivalents (e.g., "encadenamiento y cancelación" in Spanish). This method, exemplified in the Pythagorean tuning system, involves repeatedly adding intervals (specifically, perfect fifths) and reducing the results by octaves (a 1:2 ratio). This principle finds parallels in ancient Mesopotamian and Chinese musical systems, suggesting a universal approach to generating scales and temperaments.

The Chinese sanfen sunyi system, also known as the shí’èr lǜ (十二律) or "twelve-pitch" system, documented in texts such as the Lüshi Chunqiu and the Huainanzi, involves successively raising a pitch by a perfect fifth and then lowering it by an octave. This process closely resembles the "chaining/stacking and reducing/folding" method and results in a twelve-tone scale identical to the Pythagorean system. This historical evidence suggests that the concept of generating scales through interval manipulation was present in ancient Chinese musical thought, even if not formalized in group-theoretic terms.

Similarly, recent translations of cuneiform tablets from ancient Mesopotamia (3) reveal sophisticated tuning practices. These tablets describe step-by-step scale generation and document modal relationships as cyclic permutations of interval sequences. This implicit understanding of group-like structures highlights the mathematical depth of early musical systems.


The Algorithm

The Pythagorean tuning algorithm is introduced here in its most common interpretation. While historically (or folklorically) Pythagoras is said to have derived the scale from a monochord, bells, or even hammers (4), the fundamental method remains consistent regardless of the starting point. The arithmetic operations are adjusted accordingly for either string-length or frequency-based interpretations. This study adopts the frequency-based interpretation, as modern music theory represents tuning systems as sets of frequency multiples and provides clear mathematical notation for these operations.

The algorithm can be understood as follows:

1. Establish octave equivalence: Pitches at twice the frequency (or half the string length) are perceived as equivalent, forming a cyclic structure with the ratio 1:2.
2. Generate new pitches using the perfect fifth (3/2): This interval is derived from the third harmonic (3/1), reduced to the octave range.
3. Stack fifths and fold back into the octave: Iteratively applying the fifth and reducing by octaves when necessary.

For simplicity, examples use the Pythagorean pentatonic scale, corresponding to the first five notes obtained from the method.


Pythagorean Pentatonic Scale Construction:

- Initial notes: {1/1 (Unison), 2/1 (Octave)}

- Generate the first fifth: 1/1 * 3 = 3/1 → Reduced to 3/2

- Compute another: (3/2) * 3 = 9/2 → Reduced to 9/8

- Compute next: (9/8) * 3 = 27/8 → Reduced to 27/16

- Continue iterating…

Stopping at five iterations for the pentatonic, the resulting scale in ascending order is:

{ 1, 9/8, 81/64, 3/2, 27/16, 2/1 }

\(2^0 \times 3^{0}\) \(2^{-3} \times 3^{ 2}\) \(2^{-6} \times 3^{4}\) \(2^{-1} \times 3^{1 }\) \(2^{-4} \times 3^{3 }\) \(2^1 \times 3^{5 }\)
1 9/8 81/64 3/2 27/16 2/1

This set embodies the distinct elements of the pentatonic group, excluding octave duplicates.

(Note: The scale is often shifted using a different base, such as 9/8, yielding {1, 9/8, 4/3, 3/2, 16/9, 2/1}. Due to its cyclical nature, the starting point is relative.)

Since the octave serves as a period, the generated set is duplicated to extend the scale across an instrument’s range.

Some may argue that historical theorists, such as Guido d'Arezzo, worked with a fixed number of pitches without explicitly considering infinite extension. However, as musical practice expanded, scales were extended using the underlying infinite representation inherent in the algorithm.

It becomes evident that the algorithm simultaneously generates the group and selects a subset.

A more concise representation of the algorithm considers the exponential sequence {3^0, 3^1, 3^2, ...}, reduced modulo 1:2, and ordered by size. \(r_x = a^x \times b^{y_x} \in [1, b)\).


III. Group-Theoretic Formulation

Defining the Generators

As the algorithm implies, every pitch in the Pythagorean Scale, whether the full 12-tone system or the pentatonic subset or any k-cycle, can be expressed as products of powers of its fundamental generators: the octave (2) and the perfect fifth (3) modulo the cycle.

These two harmonics serve distinct roles:
- The octave (2/1) functions as a free generator, unrestricted in its powers.
- The fifth (3/2) is constrained by a modular cycle in the pentatonic case, specifically, a 5-cycle.

Thus, each pitch in the pentatonic scale can be represented as a product of powers of these generators. Using standard group notation:\[
\text{Pentatonic} = \langle 2, 3 \,|\, 3^5 \equiv 1 \rangle
\] where any pitch \( p \) can be written as:
\[
p = 2^n \cdot 3^{m \bmod 5}, \quad \text{with } n, \; m \in \mathbb{Z}
\] This notation aligns with standard finitely generated abelian group (FGAG) representations, analogous to:
\[
G = \langle a, b \,|\, b^k = 1 \rangle.
\]

Group Properties

The structure of the Pythagorean scale follows naturally from the algorithmic process of stacking fifths and reducing by octaves:

- Commutativity: Since multiplication in the frequency domain is commutative, the group operations inherit this property.
- Identity: The unison (1/1) acts as the identity element, represented as \( 2^0 \cdot 3^0 = 1 \).
- Inverses: The group inherently contains inverse elements due to the modular restriction.
- Closure: Any two pitches \( p_1 = 2^{n_1} \times 3^{k_1 \bmod 5} \) and \( p_2 = 2^{n_2} \times 3^{k_2 \bmod 5} \) multiply as:
\[
p_1 \cdot p_2 = 2^{n_1 + n_2} \times 3^{(k_1 + k_2) \bmod 5}
\]Since exponents of 3 are taken modulo 5, results remain within the defined group, ensuring closure.

Structural Clarification

The Pythagorean scale, and its cyclic subsets like the pentatonic, are not built from arbitrary powers of 2 and 3. Instead, each pitch class is of the form: \(p = 2^n \cdot 3^{m \bmod k}, \quad \text{with } n \in \mathbb{Z},\; m \in \mathbb{N_0},\; k \in \mathbb{N}\).

This definition differs crucially from the unrestricted "3-limit tuning group" \(\langle 2, 3 \rangle \subset \mathbb{Q}^+\), where both exponents range freely over \(\mathbb{Z}\), and the resulting structure is infinitely generated and not bounded within an octave.

Here, the modulo operation on the exponent of \(3\) constrains it to a cyclic subgroup of order \(k\), making the set of pitch classes isomorphic to: \(\mathbb{Z}/k\mathbb{Z} \oplus \mathbb{Z}\), which is a finitely generated abelian group: a product of a finite cyclic group (mod-k fifths) and the infinite cyclic group generated by octave shifts.

The operation remains standard multiplication in \(\mathbb{Q}^+\),
But the set is closed under modular identification of one of the generators, resulting in a well-structured group.

Notes:
1. The 3-Limit is Dense: The set \(P = \{ 2^n \times 3^m \,|\, n, m \in \mathbb{Z} \}\) under standard multiplication is a group (isomorphic to \(\mathbb{Z} \oplus \mathbb{Z}\)), but it represents all possible intervals generated solely by octaves and perfect fifths/fourths. It's dense within the positive rationals and doesn't represent a discrete scale with a repeating structure.
2. The \(\mod k\) Creates the Scale Structure: The crucial step in defining a specific Pythagorean scale (like the 12-tone or 5-tone) is imposing the cyclic identification \(3^k \sim 1\) (modulo octaves). This is what limits the distinct pitch classes derived from the \(3\) generator to \(k\) consecutive possibilities.



IV. Cultural Analogs

While the FGAG structure has been demonstrated for the Pythagorean scale, other historical tuning systems require careful consideration. For example, the Chinese temperament has a rich and multifaceted history. While many musicologists equate it with the Pythagorean system, some disagree. Given its nuances, a rigorous classification of its group structure needs a separate study.

The Sanfen Sunyi method (三分损益法, c. 500 BCE), which constructs scales by alternating multiplication by \(3/2\) and division by \(3\) (equivalent to multiplying by \(2/3\)), followed by octave reduction. This process explicitly generates a cyclic subgroup of \( \mathbb{Q}^+/\langle 2 \rangle \), aligning with FGAG structures.

A more challenging case is the Mesopotamian tuning system, dating back to 2500 BCE. Cuneiform tablets describe tuning procedures that cyclically permute intervals, akin to generating cosets in a quotient group. While less explicit than the Pythagorean or Chinese systems, this suggests an intuitive grasp of modular arithmetic and group-like structures.

The key controversy lies in interpretation: these tablets do not explicitly reference octaves, fifths, or the numbers 2 and 3. Instead, reconstructions rely on geometric depictions of tuning procedures for the lyre.


Algorithmic Basis of the Structure

The group structure arises naturally from the algorithm rather than any inherent musical qualities. The selection of generators and modular constraints is parametric rather than fundamental. For instance, in this video [link], the scale demonstrated is constructed using the same framework but employs different generators. Instead of the octave and fifth, it uses the golden ratio (phi) and the square root of 5 as the period. \[ \langle \sqrt{5}, \sqrt{\phi} \,|\, \sqrt{\phi}^{10} \equiv 1 \rangle \]
This insight has direct applications in modern music theory, which already incorporates algebraic methods.

Note: different theoretical schools often introduce overlapping terminology. Some branches of xenharmonic music theory, for example, employ group-like concepts but might not fully embrace the existing mathematical framework. A frequent disclaimer that "this group is not a group in the mathematical sense" only adds unnecessary complexity. In reality, both set theory and group theory already provide comprehensive tools for analyzing musical structures.

Not all tuning systems can be fully described as groups, clearly

While this study focuses on well-structured cases, many historical systems do not rely on the same principles and may be better understood as sets rather than other algebraic groups. However, group theory remains a powerful tool for analyzing ancient musical structures, and many lesser-studied tuning systems may reveal even deeper mathematical properties.

Revisiting ancient mathematical and musical traditions continues to enrich both fields, with potential applications in modern tuning theory, digital synthesis, and mathematical musicology.

draft


V. 

This is not merely a retroactive classification; rather, it underscores the universal and enduring nature of these structures across musical traditions.

This analysis reveals that the algorithmic basis of these scales not only defines their musical properties but also implicitly encodes a method for approximating logarithms, as explored in [this] study. This suggests that early music theory, across diverse traditions, may represent a proto-group-theoretic framework with unexpected computational capabilities.

The Average Tuning System: Scala Archive Statistics

The Average Tuning System (ATS) represents a set of pitches derived from a descriptive statistical analysis of the Scala Archive, a renowned database/collection of global tunings. The primary goal of this work was to identify common structural elements and tendencies across diverse historical and contemporary tuning practices.

A core aspect of the analysis involved interpreting the data under the assumption that most tuning systems within the archive function as periodic pitch sets. To comprehensively assess the interval content, an interval matrix expansion was performed for each tuning file. Why? : cyclic permutation and base changes inherent in periodic sets mean that the initially presented sequence of intervals (the "key") may not fully reveal the system's most prominent interval relationships. Matrix expansion calculates all possible intervals generated within the set, providing a complete picture independent of the starting note. This process revealed that while some systems appear distinct initially, expansion shows they are permutations of the same underlying structure (torsors), often highlighting strong internal interval preferences (like the perfect fifth) not explicit in the original file listing.

Despite the potential for the matrix expansion to alter perceived interval prominence, key statistical findings remained consistent whether analyzing the initial keys directly or the fully expanded interval matrices. The average (mean), median, and mode for the number of notes per system, as well as the ranking of the most frequent intervals (top 10), showed strong convergence in both scenarios, indicating powerful underlying tendencies within the archive.

The analysis navigated inherent challenges related to data representation, including precision issues arising from converting between fractional ratios and cent values, the limitations of decimal representations for logarithmic pitch data, and the effects of necessary truncation and clustering. While acknowledging these potential sources of error (and noting that analysis on a logarithmic scale would be optimal), the fundamental trends in interval popularity and system size proved robust even when accounting for these factors.

Therefore, while not an exhaustive approach, this analysis confidently identifies key features:
Dominant Equave: The octave (2/1) serves as the interval of equivalence in over 95% of the analyzed systems.
Common System Size: The statistical average size was 17 notes, with 12 being the median and clear mode. For the ATS, this was refined to 14 notes, considering practical application constraints (guitar fretting).
Most Prominent Intervals: The analysis yielded a set of the 14 most frequent intervals, forming the basis of the ATS:

 
{16/15, 10/9, 7/6, 6/5, 5/4, 4/3, √2, 3/2, 8/5, 5/3, 12/7, 9/5, 15/8, 2/1}



Further details of the analysis, including specific statistical distributions, graphical representations, data processing considerations, and comparisons, are presented in the following draft analysis section. Further research, potentially involving more sophisticated correlation analyses, remain possible.

DRAFT: 

This tuning system is a simple descriptive statistical representation of the scala archive, a renowned curated database of global tunings, seeking common ground and practical use among diverse world tunings.

Interval    Traditional Western Name
16/15       minor diatonic semitone
10/9        minor whole tone
7/6         septimal minor third
6/5         minor third
5/4         major third
4/3         perfect fourth
√2
3/2         perfect fifth
8/5         minor sixth
5/3         major sixth
12/7        septimal major sixth
9/5         just minor seventh
15/8        classic major seventh
2/1         octave

Statistics and tuning construction:

Out of the 5,176 files, the range of system sizes extends from 2 to 579. The average system size is 17, with a median of 12. The mode is also 12, appearing 1,546 times, followed by 7-note size tunings with 715 occurrences. This signifies a diverse collection, albeit with a notable concentration of systems hovering around the 12-note mark.

Top 5 Sizes

Size  Occurrences
12    1546
7     715
5     231
19    218
8     206

While some files span multiple octaves or include non-reduced intervals below the unison, these instances are relatively rare. Most are periodic tunings in alignment with the octave, the archive's most common interval. (Note: rather than relatively rare, some are intentionally wrong, since scala file definition specifies the omission of the 1, and conclude with the equave, implementations may totally ignore those values)

In a direct analysis of the files, the first key from each tuning, totaling 87,558 notes, reveals the octave as the most common, appearing with its exact representation in 4,481 total files and with close variations in practically all tunings.

The perfect fifth emerges as the second most popular interval, succeeded by the perfect fourth and major third.

Distribution of intervals. The two graphics depict identical data. The first graphic displays both vertical and horizontal axes on a linear scale, while the second utilizes a logarithmic scale for the vertical axis. This logarithmic scale highlights intervals that occur only once, significantly beyond the octave, as well as those appearing below a value of 1.

Top 5 Intervals

Interval  Name              Occurrences
2/1       octave            4481
3/2       perfect fifth     2001
4/3       perfect fourth    1743
5/4       major third       1290
9/8       major whole tone  1095


Assuming all tunings are periodic, cyclical pitch sets, the octave is identified as the interval of equivalence in 4,379 tuning files. The next most common equave is the twelfth, with only 93 files.

When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.

The two graphics present distinct datasets. The first graphic represents the scan of the initial key in each file, while the second illustrates the scan subsequent to computing all matrices. Both graphics showcase the top 17 intervals, which exhibit remarkable similarity. Each graph encompasses a single octave, with both vertical and horizontal axes set to a logarithmic scale.

(Why is it important to calculate the interval matrix and added tones to determine the most common intervals?

Take this periodic tuning, for example: 16/15 6/5 8/5 9/5 2/1.

If you're not very familiar with intervals, simply seeing the initial key doesn't tell you anything. However, upon computing the matrix for this 5-note periodic tuning, it reveals 14 unique intervals. Among these, the most common intervals are the fifth (3/2), the fourth (4/3), the major whole tone (9/8), and the Pythagorean minor seventh (16/9) , all of which aren't explicitly mentioned in the "first" key.)

There are precision issues affecting interval categorization, resulting from the conversion of fractions and cents, the dual languages of scala files, to a common decimal representation. This inherits machine number problems. When calculating the complete matrix of equal division systems, where a size of any given number should imply the same diversity, the precision nuances in floating-point arithmetic may lead to some being counted as different.

Another problem arises in categorizing cent tunings. Some files may refer to the same note, but due to differences in the amount of digits in their definitions, no program will consider them equal. (701.955 != 701.95)

You can attempt to correct this by equally limiting the number of digits, which would effectively reduce the number of individual distinct intervals. However, since truncation occurs in their decimal format, an uneven definition loss of musical notes is observed due to their original distribution, which is nonlinear (without repetitions).

The graph represents the tuning space horizontally and accumulates identical exact repetitions vertically.

Both graphics portray identical data, but the second one illustrates the data after truncation (with a maximum error of approximately 0.2 cents). Both visuals display the top 17 intervals, which remained consistent even after truncation. This reduction resulted in 242,538 unique intervals being compressed to just 9,997. The logarithmic view in the graphic also highlights the uneven definition loss of musical notes post-truncation, which was executed on the decimal data.

Progressively truncating the notes in this way, doesn't significantly alter popularity, even a 2-cent error proved insufficient to dislodge any peak prominence.

Additionally, the graph experiences intrinsic truncation due to its fixed resolution, significantly lower than the data range. Consequently, different notes are depicted on the same pixel, this is used to add a third dimension to the graph, highlighting note concentrations, which are always very close to some of the already favored intervals. For example, the perfect fifth has a concentration of notes next to it, hinting at systems like 12-tone equal temperament, where the fifth is 700 cents. However, without altering the graphical scale, these clusters won't even be apparent.

Both graphics represent the analysis of the initial keys, displaying the same dataset. However, the first graphic features a vertical logarithmic scale, while the second employs a linear scale. Presented as a heat map, red areas denote note concentrations (which are not visible in the linear view), while blue indicates fewer notes.

The generated systems employing the 17 most frequent intervals, are symmetric in both cases, reflecting a mirror image via the square root of 2. They comprise half superparticular intervals and half their reduced inversions, the perfect fourth and fifth, major third and minor sixth, minor third and major sixth, etc.

Nonetheless, some of these intervals are very small in practice, which poses minimal concerns for keyboard or synthesizer configurations but imposes constraints upon the guitar's limited space, among other factors that make it less suitable for very precise tunings; and 17 was just the average system size.

The final generated system consists of 13 notes, or 14 when including the square root of 2. This selection exhibits near-complete coverage of the tuning space. Graphically, their common-tone aggregate resembles the added tones for the entire collection, which is interesting. The intervals that were left out from the average 17 due to their proximity haven't disappeared entirely; they remain popular, even surpassing those included, although the major whole-tone was removed from the main key, it still exists in some of the others.

The first image corresponds to the analysis of the full archive's interval matrix, showcasing the 17 most popular intervals. The second image depicts the same graphic process, computing the interval matrix and accumulating the repetitions vertically, but on the newly generated tuning system. The general contour of both is similar, this type of tuning analysis typically provides the fingerprint for a tuning. This means the 14-note system generates a similar fingerprint to the entire database of 2.5 million notes.

The system does not match any of the existing files.

Analysis using subsets of the archive half or a third selected randomly still yielded the same most frequent intervals. However, for a more accurate representation of an average world tuning system, it's essential to curate the data better. This would involve handpicking the most well-known tunings that are or were actually in use, rather than relying on the full Scala archive, which contains numerous modern tunings seldom used.

Composition with the average system

Improvisation with the average system

TODO: Additional statistics:

The first ~500 most frequent intervals comprise just, rational, and integer ratio intervals before cent-defined intervals like the octave at 1200 cents appear.

How to:

The program developed for this analysis is open-source and available at [LINK]. It's designed for straightforward usage, simply load any .scl file or files, and it will promptly conduct and showcase statistics on them. The analysis comes in two modes: 'direct' examines files as they are, focusing on the first key, while 'full' generates interval matrices for all files. Notably, the 'full' analysis uses a fixed equave of 2:1, a setting implemented after discovering that 95% of the database concludes with a 2:1 equave. This equave parameter can be adjusted within the code for further exploration and customization.

Interval Space Randomness

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

draft

1.

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

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

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

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

Nature rarely produces "white noise".

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

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

Defining Randomness

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

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

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

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

Informational Randomness vs. Categorical Complexity

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

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

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

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

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

Order, Predictability, and Improbability

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

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

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

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

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

Constructing the "Unusable" Scale

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

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

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

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

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

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

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

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

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

framing the boundary conditions:

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

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

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

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

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

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

Defining Musical Tuning Systems

Just as the concept of music defies singular definition, so too do tuning systems.  A traditional definition might be: a predefined set of pitches available for musical creation and performance.

Tuning systems are often defined by their generation process: a set of rules or algorithms that produce a finite set of pitches. For example, the Pythagorean scale, which yields 12 notes, involves one algorithm for note generation and another for application, the latter often constrained by the instrument's range.  This application typically involves a period of repetition, most commonly the octave, also known as the interval of equivalence or "equave." The equave represents the most "informative" interval within the set. For instance, 12-tone equal temperament (12ed2) offers multiple intervals of repetition, but the octave division is the most intuitive. (for example, \(6\text{ed}\sqrt{2}\) 6 divisions of the square root of two, is the same)

Some systems, like the harmonic series, may lack a defined period, as each successive "period" introduces additional notes (e.g., 1, 2, 3, 4, 5, 6, 7...).(after every new octave, more notes are between, no translation symmetry)

Numerous generation processes exist, accompanied by a variety of justifications for their "validity."  The standard 12-tone equal temperament, for example, has multiple origins and rationales. The Pythagorean concept of rational number metaphysics persists as a common explanation, despite the inherent "comma" (the misalignment of exponential sequences of 2 and 3). Canon theories, like Just Intonation, often attribute the perceived "goodness" of the 12-tone scale to its approximation of rational intervals involving small prime numbers, though this was more belief than proof (note that i'm a fan of Just Intonation systems, but as music theory framework). A more robust explanation involves modern consonance models, which consider the complex timbre of sounds like the human voice or plucked strings. sounds , are analyzed through perceptual consonance models based on the beat effect, resulting in a dissonance curve. Applied to harmonic timbres (overtones are integer multiples of the fundamental frequency), the minima of this curve align with some of the pitches of the 12-tone system and/or J.I. intervals.(see note on consonance)

This level of abstraction is crucial for isolating the principle and context of "randomness."  By focusing on the "object" as a source of numbers or proportions, we can analyze it more effectively. While numbers may be sourced from various mediums and interpreted as random (even if they are not), the impact of precision and error becomes a key consideration. Furthermore, the extent to which a set of values can be "randomized" by a single defined rule is a puzzle

Representing a set of numbers as periodic proportions offers the advantage of base-independence. For example, when constructing a set based on the sizes of solar system objects, the specific unit of measurement (meters, inches, etc.) is irrelevant. The proportional relationship between objects, such as the moon's approximate quarter-size relative to Earth, remains constant. By normalizing to one value within the set, and because we will be creating periodic systems, any value as base renders the same set. (We will also create other types, non-periodic).

Note on Consonance

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

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

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

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

Critical perceptual dimensions remain that sensory dissonance models leave unaddressed:

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

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

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

The Probability of Order

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

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

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

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

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

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

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

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

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

The Perceptual Stress Test

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

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

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

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

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

Examples of "Random" Scale Generation

The following examples illustrate the creation of musical scales using "random" numbers derived from various sources. These examples demonstrate how even seemingly arbitrary number sources can generate musically coherent results. The frequency and pitch positions are flexible enough for the brain’s “error correction” to fill in predictions, and many of these scales unintentionally resemble “maximally even” sets, the mathematical basis of diatonic scales (our perception of key and tonality is probabilistic. Temperley, D. 2007. Music and Probability).

Planetary Data and the "Music of the Spheres"

The concept of the "music of the spheres," associating celestial bodies with musical harmony, has resonated across cultures, from ancient Greece to pre-Columbian America. While some specific examples of simple harmonic ratios exist in celestial mechanics (orbital resonances), many planetary properties do not readily translate into easily recognizable musical intervals. This section explores the creation of musical systems based on planetary data, examining whether these seemingly arbitrary values can generate musically meaningful results.

Scales were constructed using data from NASA (2018), specifically:

• Average surface temperature
• Orbital period
• Planet size (including the Sun)

Pitch generation employed octave equivalence. For example, in the planet size (diameter) scale, values were normalized relative to Earth (Earth = 1). The Sun's diameter, for instance, is approximately 109 times Earth's. These normalized values were then octave-folded into the range of 1 to 2 (representing Earth to "2 Earths") and then duplicated to cover the audible or instrumental range. This process was repeated for the other planetary properties.

The resulting music reveals that these seemingly arbitrary values can generate surprisingly stable chords and progressions, sometimes even exhibiting a clear tonal center. The scale derived from planetary sizes allows a fully functional pentatonic blues scale, inspiring the track title "The Astrocaster Blues."

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Planetary Diameter

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

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

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

 

Video.01 Description: "Astrocaster Blues"

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

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

The scale allows a fully functional pentatonic blues scale (near enough, this led to the observation that the search for extraterrestrial life might be best focused on solar systems with a high potential for blues musicians.)  The irony is that, within this planetary-diameter-derived blues scale, Earth itself is assigned the "bluesy" microtonal inflections!. While the pitches are normalized relative to Earth's diameter, the tonal center of the music gravitates towards Saturn.

Average Surface Temperature

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

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

Interval Matrix for the Planet Temperature Tuning

Video.02 Description: "The Dance of Entropy"

"The Dance of Entropy," based on planetary temperatures, is a waltz-like composition with a distinctly 18th-century European neoclassical orchestral vibe. (+ a bandoneon)

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

The "planet grid-keyboard" illuminates the notes as they are played. This allows to directly observe the categorical relationships within the scale and identify the tonal center of the music.

It's important to note that the order of the planets on the grid-keyboard does not correspond to their spectral order within the solar system. The octave folding process used to create the scale results in a different arrangement of pitches, a permutation on planets order. As with other planet-based scales in this work, Earth is used as the base for normalization. However, because these systems are periodic, the choice of base is inconsequential; the resulting musical relationships remain consistent regardless of which planet is used as the reference point. The scales are not geocentric in any meaningful sense.

Orbital Period

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

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

"Soles Mortem," using orbital periods, produced the least consonant of the three scales. While it still contains numerous usable chords (as demonstrated in the audio example), identifying a stable tonal center within traditional musical frameworks proved challenging.

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The Riemann Zeta Function

The Riemann Zeta function, a complex-valued function with deep connections to number theory, was used to generate a set of pitches for musical composition. The imaginary parts of the Zeta function's zeros, while not truly random, exhibit statistical properties that make them a suitable source of seemingly random numbers. Unlike previous examples that focused on selecting a small set of values and applying a fixed interval of equivalence (like the octave), this approach directly utilized ~30 consecutive imaginary parts of the Zeta function's zeros. These values were interpreted as frequencies and directly applied to control synthesizer pitches.

Octaves, in traditional periodic tuning systems, provide confinement for the total pitch availability. Knowing that any interval present in one period is found in the next (up or down, depending on the instrument's range) allows for predictability, manageability, and perceptual substitution of pitches. The scale constructed with Riemann Zeta function values doesn't inherently contain octaves. Any octaves, or approximations thereof, that appear, do so by chance, as do other consonant intervals. This absence of a pre-defined octave is a key element.

The deliberate omission of a defined equave or period of repetition makes the results rarer. the resulting music, while using an unconventional scale, sounds surprisingly "normal", maybe suggesting the use of unusual but not entirely foreign scales. It certainly does not sound atonal or xenharmonic. Clear, recognizable chord progressions emerge readily, and consonance is not compromised.

(Zeta function 28 notes music)

The distinct harmonic characteristics of the synthesized guitar and strings (all synthesized) create clear timbral differentiation. Some instruments handle otherwise dissonant intervals more gracefully than others. The guitar, with its inherently harmonic timbre, serves as a kind of consonance "stress test." If an interval sounds good on the guitar, it generally passes a basic consonance check, even if that consonance is subjective. Essentially, if it sounds good on the guitar, it's likely to be perceived as consonant.

The resulting musical texture underscores the central point: these values, derived from a complex mathematical function, do not sound as "random" as one might initially expect.

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

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

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The Question of Randomness

The observation that simple octave folding of seemingly random values consistently leads to relatively uniform interval distributions, often guaranteeing a degree of tonality, is paradoxical. It forces us to reconsider not only the "randomness" of our chosen values but also the "randomness" of our conventionally accepted methods for scale construction.

A previous study [link] analyzing the Scala Archive (a vast library of over 5000 world tunings) revealed an interesting phenomenon. Generating random pitch sequences using even simple pseudo-random number generators (like those built into web browsers) often resulted in scales that closely approximated (within ±10 cents) or even perfectly matched existing scales in the archive. These archived scales, of course, have established origins, structures, and "mathematical justifications."

The Challenges of Measurement and Their Impact on "Randomness"

As previously discussed, some "random" number sources, such as the output of random functions, were used directly without imposing a specific period or equave. For instance, one scale was derived directly from the zeros of the zeta function.

Other sources, like planetary properties (sizes interpreted as Hertz values), required a different approach.  Scales were constructed from a fixed set of values (e.g., ten planetary sizes). However, to expand these sets to a wider pitch range, a rule for extending the values was necessary. While the planetary sizes themselves might be considered a source of "randomness," the choice of how to extend their range (by octave transposition) could introduce additional bias. Simply using octaves based on Earth's values, while seemingly logical, doesn't constitute a purely "random" approach. 

A different challenge arose with data from sources like mountain heights. Using the "fourteeners" (the tallest Himalayan mountains), the very definition of height became problematic. The influence of sea level, for example, significantly affects perceived height. The same mountain, measured from different baselines, will have different proportional heights relative to other mountains, even though the absolute difference in height remains constant.This necessitated the introduction of both an "equave" and a sea-level reference point. Music based on these mountain heights has also been composed.

(Himalayan 14 periodic notes music)

Images Description:

This image visually represents the process of constructing the Himalayan tuning scale. The fourteen highest peaks in the Himalayas (the "fourteeners"), a list derived from mountaineering tradition and historical convention rather than strict geological definition, are depicted as individual rectangles, arranged horizontally to illustrate their relative heights above the 8000-meter mark (indicated by the red baseline). It's important to note that these peaks are geographically dispersed across the Himalayan range; their proximity in the image serves only to convey their height relationships. The inherent ambiguity in defining a "peak" (as opposed to a sub-peak or shoulder of a larger mountain mass) is also acknowledged, highlighting the challenges in establishing a definitive list.

Below the height representation, the image demonstrates the octave duplication process used to extend the scale. While the typical method involves octave folding the initial set of mountain heights into a single octave and then duplicating that octave, the image illustrates an alternative, but mathematically equivalent, approach: multiplying and dividing the original heights by powers of 2 to generate octaves, and then selecting the results that fall within the audible range. The duplicated heights are marked on the image, and some may visually align with other mountain silhouettes in the background. These background mountains are purely illustrative; the fourteeners (shown in white) are the primary focus of the scale construction.

Absolute Vs Relative Proportions

The choice of sea level is a more significant factor in altering the proportional relationships within the scale than the choice of period, (e.g. octave). Small changes in sea level results in drastically different proportions between mountain heights (whether analyzed in octave space or any other proportional space). While all combinations of the 14 mountain heights are theoretically possible, certain proportions became more probable than others due to the influence of the chosen sea level.

Illustrative Examples: Normalization and Equivalence

To illustrate the impact of these measurement choices, consider a simplified example.  Let's start with a set \(S = \{2, 800, 1040\}\).  We can normalize this set by choosing a base value (e.g., \(2\)) and dividing all elements by that base: \(S_{base:2} = \{2/2, 800/2, 1040/2\} = \{1, 400, 520\} \).

Next, we can create a new, reduced set by applying an equivalence relation, such as the \(1:2\) octave relationship.  We find representatives of \(400\) and \(520\) within the octave range (1 to 2). ( \( S_i \times 2^n \in (1,2]\, \vert \, n \in \mathbb{Z}\) )

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

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

The minimal generating set in octave space becomes \(\{1, 65/64, 25/16, 2\}\).  This set preserves the original proportions of S within the octave.

With unambiguous measurements (like planetary sizes), this process works well.  However, with context-dependent measurements (like mountain heights), changing the sea level alters the proportions, rendering the normalized set no longer representative of the original relationships.  For example, adding a constant delta of 150 to each element of S results in a completely different generating set.

The Challenge of Precise Proportional Calculations

Calculating precise proportions within a given musical space presents a significant challenge, particularly when dealing with numbers spanning vastly different orders of magnitude or when measurement precision is limited.

Let's first illustrate a scenario where this problem is less pronounced.  Consider creating a scale based on the sizes of two planets.  Assume their sizes are \(A = 200\) and \(B = 1200\) (in some arbitrary unit).  The first step is to normalize the values by choosing a base.  Using \(A\) as the base, we get \(\{A = 200/200, B = 1200/200\} = \{A = 1, B = 6\}\).  Next, we define our musical space, in this case, the octave (a \(1:2\) ratio).  \(A\) remains at \(1\) (the unison).  \(B\) must be scaled to fit within the octave \((1, 2]\).  We divide \(B\) by \(2^2 =4\) to get \(6/4 = 3/2 = 1.5\), representing a perfect fifth.  Our generating set is \(\{1, 1.5\}\).  We can extend this scale by repeatedly multiplying by \(2\) (within the instrument's range).

In this scenario, small variations in the initial measurements have minimal impact on the final proportions.  For example, if the measurements were slightly off (e.g., \(\{201, 1205\}\) or \(\{199.32142, 1200.0000001\}\)), the resulting proportions within the octave remain practically the same.  The generating set might become \(\{1, 1.50003\}\), but this tiny difference is negligible in musical terms.  Planet \(B\) is still perceived as roughly a fifth above planet \(A\).

The Problem of Scale and Precision

The problem becomes much more acute when dealing with values that span a vast range, such as particle energies, which can range from giga-electronvolts (GeV) to electronvolts (eV).  Measurements at these scales often have varying degrees of precision.  Consider a simplified example (not a real-world physics case) to illustrate the issue.  Suppose we have two particle energies: \(A = 1000\) and \(B = 0.09155\)... Normalizing to \(A\) gives us \(\{1, 0.0009155\}\).  Scaling \(B\) to fit within the octave \((1, 2]\) requires multiplying by a power of \(2\).  In this specific example, \(B \times 2^n\) happens to equal \(1.5\), a perfect fifth.

So far, so good.  But what if the measurement of \(B\) was slightly different due to limitations in precision?  Let's say \(B = 0.0781\) (a seemingly small difference). Now, when we scale \(B\) to fit within the octave, we get a different result: \(B \times 2^n = 1.25\), a major fourth. A tiny change in the initial value of \(B\) has resulted in a significant change in the musical interval.

Therefore, unless we have extremely precise values for particle energies (which span an even wider range than our simplified example), we cannot reliably claim consistent proportional relationships within a musical space.  While we can say that "planet \(B\) is a fifth of planet \(A\)" with reasonable certainty, saying that "an electron is a major fourth of a muon" based on imprecise energy values would be misleading.  The inherent uncertainty in the measurements prevents us from establishing such precise musical relationships.

Nature of Musical Structure

This exploration, while seemingly trivializing historical efforts in scale creation, is not intended to diminish their significance. Rather, it builds upon the observation that the Scala Archive contains over 5000 documented tuning systems, raising the question: does everything sound good? My approach of constructing scales from random sources is primarily for inspiration. The resulting scales often either already exist within the archive or possess inherent musicality that can be further enhanced with appropriate composition.

While the selection of pitches in the scales described above was often based on "random" sources, it's crucial to emphasize that the composition of the music was not. A human mind, with its inherent perceptual biases and musical understanding, ultimately shaped the final musical output. The composer, working within the constraints and possibilities presented by the "random" scale, makes choices about melody, harmony, rhythm, and form.

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

Fitting "Random" Scales within Established Frameworks

Earlier, I mentioned that these "random" scales could be understood within the context of established music theory. While the preceding examples demonstrated this through the creation of musically coherent pieces, the underlying framework deserves further explanation. The sheer existence of the Scala Archive, with its thousands of diverse tuning systems, provides compelling evidence that, in a broad sense, "anything works" tonally. However, we can be more specific about how these "random" scales relate to established theoretical frameworks.

Modern music theorists have explored dividing the octave into an increasingly large number of intervals, often with the goal of cataloging and analyzing scales that more closely approximate specific intervals of interest, such as "perfect" fifths. However, the limits of human pitch perception must be considered. The just noticeable difference (JND) for pitch, averaging around 10 cents in the central hearing range, means that many of these highly refined scales contain distinctions that are imperceptible to the human ear. What, then, is the practical purpose of constructing scales with hundreds or even thousands of divisions per octave if these microtonal nuances are not perceivable? Such explorations are, of course, valuable from a theoretical standpoint, but their direct relevance to musical practice is less clear.

This framework of highly granular octave divisions, however, provides a context for understanding how our "random" scales can be "fitted" into established musical thinking. Any of these randomly generated tuning systems can be considered a subset of a highly divided equal temperament (e.g., 100-EDO or even less). For example, analysis of the Riemann Zeta function scale using an interval matrix reveals that numerous 12-EDO approximations (within ±15 cents) are present at various transpositions.

EXTRA:

Color Attractor Spectral Location and Wavelength-Derived Musical Scales

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

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

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

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

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

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

Mathematical Process Summary:

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

Red (base): 700/700 = 1

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

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

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

Examples of Unique Hue-Based Scales:

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

Examples:

  

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

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

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

[...]

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

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

About the "Spectral Octave":

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

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

A1. Color Wheel Construction

Addressing Color Space Transformations and Limitations

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

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

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

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

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

Physically-Based, Octave-Modeled, Logarithmic Hue Wheel

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

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

Key Graphics and Steps:

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

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

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

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

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

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

Meta:

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

Further Reading:

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