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A General Approach to Enharmonicism

Caleb Scott Alons

arXiv:2609.10603v1math.HOcs.SD

TL;DR

The paper asks whether enharmonicism can be generalized as abstract musical syntax rather than derived from pitch and acoustic constructions. It formalizes arbitrary enharmonic systems and argues that reflection and maximal evenness help assess their musical practicality, while offering a mathematical rationale for preferring the standard system.

  • Problem

    The paper addresses how to generalize enharmonic theory without presupposing the standard seven diatones or acoustic preferences.

    Source: §1.1 · p. 2

  • Method

    The authors formalize enharmonic equivalence and related structures for arbitrary diatonic cardinalities as a syntactic theory independent of acoustic considerations.

    Source: §1.1 · p. 2, §4.1 · p. 7

  • Results

    The standard enharmonic system has the reflection property, and the authors argue its preference can be motivated mathematically; the 19-tone example is not reflectable.

    Source: §4.5 · p. 12, §4.7 · pp. 14–15

  • Takeaways & Limitations

    Reflection and maximal evenness are proposed as criteria for assessing whether arbitrary enharmonic systems are musically practical.

    Source: §4.7 · pp. 14, 16

    The theory does not relate enharmonic labels to pitches in cents or prescribe their use in a tuning system.

    Source: §4.3 · p. 9

Abstract

from arXiv · show

We propose a general approach to enharmonicism within syntactic music theory; that is, we formalize several abstract structures of enharmonicism without any acoustic considerations. The structures of concern to this paper are constructed from diatonics, chromatics, and enharmonics, and we seek to present a general enharmonic theory that builds upon Hook's enharmonic theory published in "Enharmonic Systems: A Theory of Key Signatures, Enharmonic Equivalence and Diatonicism" (2007). We also propose that the two criteria of reflection and maximal evenness can be used to determine whether an arbitrary enharmonic system is musically practical. Moreover, we argue that preferential treatment of the standard enharmonic system (SES) can be motivated from an abstract mathematical perspective divorced from pitch consideration and conventional acoustic constructions of SES.

1 Context and Motivation

The paper situates enharmonicism in a history shaped largely by acoustic concerns, while distinguishing abstract musical syntax from sound. It develops a general syntactic approach to enharmonicism that does not presuppose a standard number of diatones or acoustic preferences.

1.1 Historical Considerations

  • Historical enharmonic systems often addressed acoustic tuning, from Vicentino’s 36-key Archicembalo to later 19-pitch systems and Mersenne’s work on temperament.The cited examples include Costeley, Salinas, and Mersenne.Source: §1.1 · p. 1
  • After standard equal temperament became widely accepted, accidentals took on a more syntactic role, though some later composers pursued expanded systems grounded in just intonation.Partch and Johnston derived expanded pitch vocabularies from rational frequency ratios.Source: §1.1 · p. 1
  • The authors develop a general syntactic theory of enharmonicism from an arbitrary number of diatones, avoiding presupposed acoustic preferences.They build on Hook’s syntactic enharmonic theory.Source: §1.1 · p. 2

1.2 Acoustics and Syntactics

  • Syntactics concerns abstract formal structures without inherent reference to pitch, whereas acoustics concerns relationships between sound waves and their physical properties.The authors note that music theory has recognized this distinction implicitly and explicitly.Source: §1.2 · p. 2
  • Music theory has often derived notational syntax from acoustic phenomena, a history the authors say has encouraged acoustics and syntactics to be conflated.Examples include just intonation, concords, the overtone series, and temperaments; syntax also serves practical preservation and reproduction.Source: §1.2 · p. 2
  • The authors present Hook’s enharmonic theory as an example of treating musical syntax as an abstract study and suggest general enharmonic theory may reveal reasons for standard enharmonicism’s prevalence.The paper frames music as a combination of mathematics (syntactics) and sound (acoustics).Source: §1.2 · p. 2

2 Diatonic Theory

The paper generalizes diatonic theory by treating diatones as finite indexed sets and defining transpositions, interval-based orderings, and rotation classes. It then relates diatonic systems to modular arithmetic and formalizes reflection as a structural property.

2.1 Diatonic Sets

  • A diatonic set is a finite indexed set of diatones, with its number of elements called its diatonic cardinality.The standard set D7 contains C, D, E, F, G, A, and B.Source: §2.1 · p. 2

2.2 Diatonic Transpositions and Prime Diatonic Orderings

  • Diatonic transposition shifts a diatone’s index modulo the set’s cardinality, while interval length and prime diatonic orderings generalize familiar circles such as the circle of fifths.An interval length p generates an ordering when p − 1 is coprime with the cardinality n.Source: §2.2 · pp. 2–3

2.3 Diatonic Systems

  • A diatonic space combines an ordering by a circle of pths with a diatonic transposition, and its rotations form a rotation class.The standard diatonic system groups the rotation classes for D7; for example, the fifths orderings shown are rotations of one another.Source: §2.3 · p. 4
  • Diatonic spaces and rotation classes correspond to Zn, while diatonic systems correspond to its multiplicative group of units U(Zn).The standard seconds and sevenths classes are reflections of one another because their corresponding units in Z7 are additive inverses.Source: §2.3 · pp. 4–5
  • The reflection property requires every rotation class in a diatonic system to be reflectable, and the standard diatonic system satisfies it.A class’s reflection is defined by the corresponding interval lengths having additive-inverse units in Zn.Source: §2.3 · p. 5

3 Chromatic Theory

The paper builds chromatic structures by combining diatones with modifiers, then defines how chromatics can be transposed, arranged, and compared within ordered chromatic sets. This formalization treats chromaticism as a syntactic interaction between diatonic structure and modification.

3.1 Chromatic Sets

  • A chromatic set pairs each diatone with a modifier, where modifiers correspond to integers and natural chromatics remain distinct from diatones.The modifier map is an isomorphism between Z and the modifier set.Source: §3.1 · pp. 5–6

3.2 Chromatic Transpositions and Prime Chromatic Orderings

  • Chromatic transposition changes both the diatonic component and modifier, while a modified interval joins chromatics whose modifiers differ.The modifier difference names the interval modifier, such as a sharp-sharp third or flat eleventh.Source: §3.2 · p. 6
  • A prime chromatic ordering arranges a chromatic set using a diatonic space, yielding lines of pths such as the standard line of fifths.The standard ordered chromatic set corresponds to the central row of the Tonnetz.Source: §3.2 · p. 6
  • The authors frame chromaticism as interaction between diatonics and modifiers, with diatonic-space structure informing ordered chromatic sets and enabling syntactic complexity used by enharmonic equivalence.This is presented as a non-acoustic motivation for chromatic theory.Source: §3.2 · p. 7

4.1 Preliminaries

The section develops enharmonic equivalence by treating enharmonics as equivalence classes of chromatics. It uses the standard piano keyboard to motivate a general assignment scheme.

4.1 Preliminaries

  • Enharmonics are equivalence classes of chromatics, obtained by defining a relation that partitions a chromatic set.The section formalizes this relation after introducing the familiar piano-key example.Source: §4.1 · p. 7
  • The standard keyboard has 12 keys—seven white and five black—and black-key placement between white keys underlies the definition of equivalence.The white keys correspond to the seven diatonics in D7.Source: §4.1 · p. 7, Figure 3 · p. 7
  • The tuple E = (2, 2, 1, 2, 2, 2, 1) encodes conventional equivalence, with positive integer entries whose sum, 12, gives the keyboard’s total keys.Each entry specifies the number of steps between successive white keys; the number of intervening black keys is one less.Source: §4.1 · pp. 7–8
  • The function A_E uses the consecutive sums of E to computationally assign conventional enharmonic equivalence.For example, the accumulated offsets relate chromatics built on one diatonic to naturals built on subsequent diatonics.Source: §4.1 · p. 8

4.2 Enharmonic Assignments

An enharmonic assignment is built from a tuple of positive integers and has a weight equal to the tuple’s sum. The standard assignment is one example; other tuples define more general systems.

4.2 Enharmonic Assignments

  • An enharmonic assignment A_E is defined from an n-tuple E of positive integers indexed by Z_n, and its weight is the sum of E’s entries.The standard enharmonic assignment uses E = (2, 2, 1, 2, 2, 2, 1).Source: §4.2 · p. 8
  • The standard enharmonic assignment is the assignment associated with E = (2, 2, 1, 2, 2, 2, 1).Source: §4.2 · p. 8
  • The tuple E10 = (4, 2, 7, 1, 1, 4, 2, 6, 5, 9) has weight 41, corresponding to a keyboard with 41 keys, and illustrates the definition’s generality beyond standard assumptions.That keyboard would contain 10 white keys and 31 black keys arranged according to E10.Source: §4.2 · p. 8

4.3 Enharmonic Sets

The section defines enharmonic equivalence through modular arithmetic and identifies enharmonics as the resulting equivalence classes. It specifies the standard set while keeping the theory syntactic rather than acoustic.

4.3 Enharmonic Sets

  • Two chromatics are enharmonically equivalent by E exactly when their accidental offsets plus their assignment values are congruent modulo the assignment’s weight N.The resulting relation is an equivalence relation, and its classes define the enharmonics in an enharmonic set.Source: §4.3 · p. 9
  • A key is an enharmonic, a keyboard is an enharmonic set, and white and black keys are distinguished by whether an enharmonic contains a natural chromatic.Source: §4.3 · p. 9
  • The standard enharmonic set contains 12 enharmonics, with slash-paired names denoting interchangeable labels for the same enharmonic.The paper expresses these elements using π-notation.Source: §4.3 · p. 9, Figure 5 · p. 9
  • The definitions do not assume equal temperament or specify pitch relationships, so equal key counts do not establish equal acoustic intervals or transposability.The paper focuses on syntactic structures and does not assign pitches in cents to enharmonic labels.Source: §4.3 · p. 9
  • The notation distinguishes membership of a chromatic in an enharmonic from equivalence between two chromatics; context resolves symbols that may name either diatonics or enharmonics.For example, E♯ ∈ F denotes membership, whereas E♯ ≈E F♮ denotes equivalence.Source: §4.3 · pp. 9–10

4.4 Major Scales and Enharmonic Intervals

The section extends transposition, scale, and interval terminology to arbitrary enharmonic sets. It defines how scales are ordered and how intervals receive quality classifications.

4.4 Major Scales and Enharmonic Intervals

  • Enharmonic transposition τ_x is a diatonic transposition by x within an enharmonic set.Source: §4.4 · p. 10
  • Whole and fractional steps generalize from the largest entry w of E, while a transposition by 1 is a chromatic step.In the standard set, τ_2 is a whole step and τ_1 is both a half step and a chromatic step.Source: §4.4 · p. 10
  • A π_k major scale orders the transpositions of its starting enharmonic by the assignment values; the standard assignment reproduces the familiar whole-whole-half-whole-whole-whole-half sequence.The A major example in the standard set is A, B, C♯, D, E, F♯, G♯.Source: §4.4 · p. 10
  • A pth is perfect when its assignment value is not N − 1 and is coprime with N; other non-unison pths are major unless classified through augmentation or diminution.In the standard set, fourths and fifths are the only perfect intervals; seconds, thirds, sixths, and sevenths are major.Source: §4.4 · pp. 10–11
  • Interval naming is not unique, and the paper informally favors the minimal amount of augmentation or diminution without claiming unique classifications.Diminution is bounded at the perfect unison in the paper’s treatment.Source: §4.4 · p. 11

4.5 Prime Enharmonic Orderings and Enharmonic Systems

The paper constructs enharmonic orderings and systems from perfect intervals, then defines reflection among their rotation classes. In the standard enharmonic system, the two rotation classes are reflectable.

4.5 Prime Enharmonic Orderings and Enharmonic Systems

  • A prime enharmonic ordering is generated from a perfect interval, and every PEO is a prime diatonic ordering, though not every PDO is a PEO.The PEO generator choices are narrower because the interval must be perfect.Source: §4.5 · p. 11
  • The standard PEO orders the enharmonics as C, G, D, A, E, B, F♯/G♭, C♯/D♭, G♯/A♭, D♯/E♭, A♯/B♭, F.The text gives interchangeable labels rather than privileging one enharmonic spelling, since key signatures are not formalized.Source: §4.5 · p. 12
  • An enharmonic space is a diatonic space whose PDO is a PEO, while an enharmonic system collects qualifying diatonic systems of an enharmonic set.The system includes those p for which AE(p − 1) is not N − 1 and is coprime with N.Source: §4.5 · p. 12
  • The standard enharmonic system is the enharmonic system of p, and its rotation classes are the standard rotation classes.Reflection pairs classes whose assignment values sum to N.Source: §4.5 · p. 12
  • The standard enharmonic system has two rotation classes, and each is reflectable, making it a reflection-practical system.Figure 6 represents enharmonic spaces for the two standard rotation classes.Source: §4.5 · p. 12, Figure 6 · p. 12

4.6 Practical Chromatics

The section defines practical chromatics for arbitrary enharmonic sets using bounds on modifier values. It contrasts those formal bounds with the rarity of high modifiers in musical practice.

4.6 Practical Chromatics

  • Modifiers beyond double sharps and flats are rarely used in practice, and higher modifiers have essentially no practical role in standard composition and performance.The text notes only isolated theoretical possibilities for quadruple modifiers in Ben Johnston’s alternate notation context.Source: §4.6 · pp. 12–13
  • Although each enharmonic has infinitely many chromatics, only a much smaller subset is used to convey music in composition and performance.The paper cites triple-sharp and triple-flat score examples, while noting that such modifiers are exceedingly rare.Source: §4.6 · pp. 12–13
  • A chromatic is practical for an enharmonic set when its modifier m lies between −e_(i−1) and e_i; values outside that range are impractical.The criterion applies to arbitrary enharmonic sets and provides a principled way to restrict useful chromatics.Source: §4.6 · p. 13
  • For the standard enharmonic set, practical E chromatics are E♭, E♮, E♯, and E3, while E♭5 and E5 are impractical.The criterion is applied across diatonic components C through B to identify practical chromatics.Source: §4.6 · p. 13

4.7 Nonstandard Enharmonic Systems

The paper applies its general syntactic theory to nonstandard enharmonic systems and argues that reflectability and maximal evenness help assess their musical usefulness. These examples also support a mathematical rationale for preferring the standard enharmonic system (SES).

4.7 Nonstandard Enharmonic Systems

  • The authors present SES as a central construction uniting the acoustic and syntactic branches of music theory.They argue that a generalized account makes the intuition behind a syntactic preference for SES visible.Source: §4.7 · p. 14
  • Examples show the theory accommodates nonstandard systems, including 19-tone p19 and an arbitrarily constructed 11-enharmonic PE.The authors present p19 as a known system and PE as an example demonstrating the theory’s generality.Source: §4.7 · pp. 14, 16
  • The authors suggest p19 has useful structure if reflectability signals practicality, since ep19^5 is reflectable.This argument is made independently of pitch considerations.Source: §4.7 · p. 16
  • PE is irregular and fails RP; its example motivates maximal evenness as a second criterion for assessing enharmonic-system usefulness.The authors ultimately propose reflectability and maximal evenness as basic properties of musically viable systems.Source: §4.7 · pp. 16–17

4.8 Maximal Evenness

The paper generalizes maximal evenness by requiring spectrum sizes across an enharmonic set to differ by at most one, relative to the smallest element of E. Under this definition, SES and p19 are maximally even, but only SES is both RP and ME among the three systems considered.

4.8 Maximal Evenness

  • A diatonic’s spectrum is the subset of enharmonics generated by transposing πAE(i) through ei steps; each spectrum is unique and has size ei.The paper uses these spectra to define maximal evenness.Source: §4.8 · p. 17
  • An enharmonic set is maximally even when every spectrum size lies between emin and emin + 1, inclusive, where emin is the smallest element of E.An enharmonic system is ME exactly when its enharmonic set is ME.Source: §4.8 · p. 18
  • The paper’s generalized criterion extends Clough and Douthett’s standard-system condition that each spectrum size is 1 or 2.The broader definition bounds spectrum-size differences by one for all diatonics.Source: §4.8 · pp. 17–18
  • SES and p19 are ME, while PE is not; among these three systems, only SES is both RP and ME.For PE, the spectrum of K has size 4, exceeding the stated bound of 1 to 2.Source: §4.8 · p. 18

5 Concluding Remarks

The paper recommends RP-ME enharmonic systems for musical application and analysis, arguing that its abstract theory supports preferring SES. It leaves efficient identification, uniqueness, and construction of RP-ME systems for future work.

5 Concluding Remarks

  • The authors recommend RP-ME enharmonic systems for musical application and analysis and argue that an abstract perspective supports preferring SES.The paper frames this preference as divorced from pitch considerations.Source: §5 · p. 18
  • Open questions include efficient RP-ME identification, uniqueness, and whether n-tuple formulas can guarantee RP-ME systems for a given diatonic set.The author says these questions will be explored in a follow-up paper.Source: §5 · p. 18
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