Equivalence Groups

Equivalence Groups


Musical set theory groups objects at multiple levels of detail: from the lowest level, with tens of thousands of playable chords and scales, to the highest level, where every possible chord or scale can be grouped into around a hundred elements.

Modern musical set theory terminology allows us to remain exhaustive, yet to proceed without ambiguity, while capturing the richness inherent in centuries of western music theory. (This section is based generally on Callender, et al., 2008.)

Answering Q1 (How many chords & scales?)

Annoyingly, there is no way to count some type of object without knowing exactly what type of object we are counting. We will clarify the terminology used for the levels of detail discussed to group clock diagrams together in the previous section, where we found the lower bound answer, 115, to how many possible 3- to 9-note chord and scale objects there can be (Q1).

In this section we will start with that lower bound of 115 rows (of one to four set classes) and show how to get chords and scales that can be played on an instrument. We will go from black and white clocks back to colored clocks (unordered collections of notes), then on to named, playable chords.

The upper bound for playable chords and modes is still in the hundreds of thousands—so we now have the short answer to Q1, hundreds to hundreds of thousands—but in the next section, the number of useful (non-exhaustive) chords and scales will be reduced to tens of thousands.

Equivalence Relationships

If you can count, you can understand these simple equivalence relationships, which work for both chords and scales, treated uniformly as “musical objects.”

Here, equivalence relationship just means “when the equivalence rule is applied, we consider this group of objects to be one set, to have one identity, to be on the same team, so to speak.” A chord or scale may be part of many groups, depending on the level of detail being discussed.

Equivalence Label Small Steps Larger Steps
Octave Exact Match Chord Voicing
OC Match Mode Chord Inversion
OPC Inversion
(Permutation of)
Scale Chord
OPTC Transposition of Set Class (Prime Form)
OPTIC Involution Set Class (Forte Code)
OPTIC/K Complement Complementary Set Classes

This table lists the terminology used for labels in the interactive search pages, as well as conventional terminology for scale-like objects with many small steps and chord-like objects with fewer larger steps.

Continue reading below for examples of all of the equivalence rules listed in the table, where the following abbreviations are used:

OPTIC/K Equivalence

Row of Set Classes Table: just over a hundred possibilities

Each row of the Set Classes Table groups one or two Forte codes (one to four clock diagrams) since they have the same Lewin-Quinn fourier components.

Example OPTIC/K-Equivalent objects. a single row of the Set Classes table, 3-11/9-11, grouped by Complementarity and Involution. Top row: 3-11, (037), min and 3-11, (047), Maj. Bottom row: 9-11, (01245679t) and 9-11, (01235679t).

In this example, each black and white clock groups 12 colored clocks (pitch class sets) so this row encompasses 48 OPC-equivalent objects.

OPTIC Equivalence

Forte Code (accounting for Involution): over two hundred possibilities

Example OPTIC-Equivalent objects: a single Forte code, 3-11, grouping two black and white clock diagrams by Involution: 3-11, (037), min and 3-11, (047), Maj.

All 24 major and minor triads are treated as equivalent, and they have the same Lewin-Quinn fourier components and interval-content vector, and evenness.

OPTC Equivalence

Prime Form (ignoring Involution): over three hundred possibilities

Two different representations of one OPTC-Equivalent object: a single prime form or a single black and white clock diagram, grouping 12 colored clocks by transposition: 3-11, (047), Maj.

All transpositions of the major triad are considered a single object, no matter how many times each note appears, or in what order.

OPC Equivalence

INVERSION: around four thousand pitch class sets

This equivalence relation groups related scales and chords with different fingering or hand positions under shared theoretical terms (“C chord” or “E major scale”), which means all C chords are only “equivalent” in an abstract sense, but understanding this distinction reduces the amount of memorization significantly, with all “C chords” functioning in related ways for the purpose of analysis and composition.

Two different OPC-Equivalent Objects. Left: C Major triad in different inversions. Right: F Major triad in different inversions. Bottom: pitch class sets for C Maj and F Maj, respectively.

The two different Inversions of C Major on the left are treated as equivalent, and the two Inversions of F Major on the right are considered equivalent, but they are not equivalent to each other, since C Major uses C, E, and G, but F Major uses F, A, and C. They are different transpositions of prime form (047), the set class of all twelve major triads.

We saw in Clocks & Pitch Classes that 4,096 colored clock diagrams capture this distinction.

OC Equivalence

MATCH: tens of thousands of possibilities

OPC-Equivalent Objects. Left: Two voicings of C Major in Root Position are treated as OC-Equivalent (same root, same unordered set of note names). Right: Two voicings of C Major on the right are not OC-Equivalent to one another since the first one with an E in the root is in First Inversion.

Harmonious catalogues different chord inversions and modes of scales by pairing a colored clock diagram (pitch class set, an unordered collection of notes) with a root note, for example C Maj, C/G, D Dom 7, etc. This is a common way for chord charts or lead sheets (in Jazz fake books, or pop song write-ups with lyrics and chord names) to represent the harmonic structure of a piece, leaving the work of where to play the chord or how to voice it to the instrumentalist.

Converting an OPC-Equivalent object (e.g. our running example, the unordered set of notes C, E, G) into every possible mode or chord inversion is an involved process. Many chord bibles do this manually, listing ways to play given chords based on the author’s memory or personal experience. Some online chord resources use a cleverly-crafted algorithm to generate every possible way to play a given chord (for example, taking the note names and generating finger positions on the fretboard, even when these are sometimes unplayable).

Harmonious combines both approaches, starting with an exhaustive, heuristic-driven list generated by a computer, then manually filtering and curating it, leaving chords and scales playable and understandable by human beings with five-finger hands. After completing this process, looking up OC-equivalent ways to play a given chord in Harmonious is as simple as entering the notes of the chord into the interactive key search or interactive fret search and looking under the heading Match.

Octave Equivalence

EXACT MATCH: hundreds of thousands of possibilities

This is the level of detail at which music can actually be played. One cannot play the “note” C, only C in a specific register, e.g. C2, C4 (middle C) or C6. Similarly one cannot play a “C major triad” on a fretboard or keyboard; rather one can play a C major triad voiced in a particular inversion, in a particular register.

Octave-Equivalent Objects. Left: C Major chord in Root Position, played in a certain register. Right: The same chord shape played up an octave. Under Octave-Equivalence, these are treated as the same object, since they are played with the same fingering on these instruments.

For the keyboard and the guitar, moving the hand up by exactly one octave uses the exact same fingering, so this is the lowest level at which the theory bottoms out, reducing the number of interesting musical objects by a factor equal to the number of octaves available to the various instruments. For example, the seven-plus-octave piano has seven times fewer theoretical chord and scale voicings to learn, despite the piano’s extensive sonic range.

Reducing the Numbers and Focusing More

The next section will take a high-level view of easy-to-understand trends in the set classes and introduce a simple way of eliminating objects that are less useful (the concept of chromatic-cluster-containing versus cluster-free), while preserving all the chords and scales of jazz theory and reducing the number of OC-Equivalent chords and scales Harmonious will focus on from hundreds of thousands to tens of thousands.