The Turnip

"Music is the space between notes" - Claude Debussy.

A mathematical/computational investigation into the structure of ambiguity in Western harmony.

THIS IS A BIG WORK IN PROGRESS. THIS README IS A VERY ROUGH DRAFT. you've been warned ;)

Driving Curiosity

What chords and scales are the most versatile? By versatile, I mean what chords can be used in the most contexts? The more versatile a chord is, the more opportunities a composer has to use that chord. I might liken versatility in harmony to 'ambiguity' of feeling or resolution. But also, the more ambiguous a harmony is, the safer that chord is to use, because it's harder to juxtapose it with something so that the chords collectively (emergently) sound bad.
The less ambiguous a harmony is, the more the listener expects what will come next, and the more opportunities the composer has to break or follow the expectations of the listener.
Playing with expectations--going back and forth between satisfying and denying the listener's predictions--is a part of music I enjoy the most.

Ambiguity example: Chords derived from which key signatures?

Let's look at a simple example of ambiguity in harmony. Consider the C Major triad. This is composed of the 3 notes C, E, and G. This chord is written as [0, 4, 7], using pitch class notation.

To what musical key signature does this chord belong? Short answer: C Major, G Major, and F Major.

Not only does it belong to these three key signatures, but it also belongs to numerous additional scales, some jazzy, some exotic, some scales perhaps never deliberately used in recorded music.

Quick refresher on key signature: I use the term key signature and a diatonic major scale interchangeably, whereas a lot of musicians don't. If you're familiar with the concept "key signature of A minor", using my conventions 'A minor' is not a key signature, but rather it is equivalent to the key signature of C major. There are twelve unique key signatures just as there are 12 unique diatonic Major scales.
If you're familiar with modes, D dorian is equivalent to C Ionian or the C Major scale. C dorian is equivalent to Bb Ionian or the Bb Major scale.

What about modes? You actually don't have to worry about modes in this research project.
Modes are defined by where the root is. The root is the lowest note or where you "start" in a fixed collection of pitches, that is, which note is placed at the bottom.
This project is about unique collections, not where you start.
In this project, a C Major scale you can think of as all 7 modes at the same time, or superimposed.

So back to the question, which key signatures does C Major triad belong to? If you know all twelve key signatures if you learned the basics of the theory of musical harmony, it's not too hard to run through all of them and check whether the C Major triad can be constructed using the 7 pitches in each key signature.
You can of course find C Major Triad in C Major scale.
But you can also find it in G Major scale. And the F Major scale!

These three key signatures are neighbors on the circle of fifths.
Notes further away on the circle of fifths have fewer notes in common and so it is less likely that you'd find the same chord in two scales if those two scales are far apart from one another on the circle of fifths.

We can say that C Major triad is a member of G Major, C Major, and F Major scales.

A more mathematical way of saying member is subset. C Major triad is a subset of the G Major scale.

What is a chord?

A chord is a collection of pitches.
Chords generalize single notes, intervals, triads, and everything in between triads and scales.

Since a scale can have at most 12 notes corresponding to the chromatic scale

Definition of chord class

A chord class is a chord that is unique under inversion and transposition.

Assumptions of chord classes

• Inversion doesn't matter. G Bb D F is the same as Bb D F G, is the same as G D Bb F
• Repeating pitch classes doesn't matter. G Bb D F is the same as G G G G Bb D F G
• Transposition doesn't matter for uniqueness.
• Chords that are reflections of one another are not necessarily the same. Major and minor chords are not

Chord Enumeration

Chord Size	Number of unique chord classes	Example chord name and pitches
0	0	NA
1	1	The note C natural. [ 0 ]
2	6	Major third interval [ 0, 4 ]
3	19	Major third triad [ 0, 4, 7 ]
4	43	Dominant 7th chord [ 0, 4, 7, 10 ]
5	66	Pentatonic scale [ 0, 2, 4, 7, 9 ]
6	80	Wholetone scale [ 0, 2, 4, 6, 8, 10 ]
7	66	Major Scale [ 0, 2, 4, 5, 7, 9, 11 ]
8	43	Octatonic Scale [ 0, 1, 3, 4, 6, 7, 9, 10 ]
9	19	?
10	6	?
11	1	Chromatic scale minus 1 note [ 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
12	1	The chromatic scale [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]

For an excellent explanation of the mathematics behind the calculation of the number of unique chord classes, please refer to either "The symmetry of the equal temperament scale" by Athanassios Economou,"Enumeration in music Theory" by David Reiner or "Polya's Counting Theory" by Mollee Huisinga.

Dependencies

Python 3

References

• Economou, Athanassios. "The symmetry of the equal temperament scale." Mathematics and Design 98: Proceedings of the Second International Conference. 1998.
• Hook, Julian. "Why are there twenty-nine tetrachords? a tutorial on combinatorics and enumeration in music theory." Music Theory Online 13.4 (2007).
• Reiner, David L. "Enumeration in music theory." The American mathematical monthly 92.1 (1985): 51-54.