Analyzing post-tonal or atonal music is a far-cry from classical musical analysis of functional harmony. If you've studied classical theory, the familiar jargon (tonic chords, dominant chords, tonicization, modulation, etc.) is done away with. In its stead we are left with numbers, sets, and transformations; entities that are deeply rooted in mathematics. So before discussing anything about music, it's more than appropriate to familiarize ourselves with this math that will be in the background of all post-tonal analysis.
In particular we will try to understand modular arithmetic, a subject in abstract algebra, but only the sorts of statements as in the following:
If we understand these statements, then we can understand pitch classes.
Whenever I set text with a gray border to the left, like this, I'll either be summarizing some mathematical notation and what it means, or going deeper into some of the math previously introduced.
Chances are you know about Euclidean division. For example, $1\div 2$ is 1 divided by 2 (or 2 dividing 1) and represents symbolically the number of times 2 goes into 1 plus some remainder. What might not be as familiar are the two parts of Euclidean division taken separately: the quotient and the remainder.
Let $a$ and $b$ both be integers.
We can then express $a$ as the sum $a = qb + r$.
As an aside on notation, mathematicians are often lazy. The "let" statement above could be alternatively expressed "fix $a,b\in\Z$", where the symbol $\in$ means that $a$ and $b$ are variables with values taken from $\Z$, where $\Z=\{\ldots,-2,-1,0,1,2,\ldots\}$ is the set containing all integers.
The $\{\ldots\}$ notation is called set-builder notation, and is a way to group things, in this case integers. A set can be finite or infinite, for example: $\{0,1,2\}$ is a finite set, while $\{1,2,3,\ldots\}$ is an infinite set containing all of the positive integers, commonly denoted $\Z^+$ or $\N$. We can also define sets using rules, as for example again the positive integers can also be expressed as $\{d\in\Z:d>0\}$, meaning the set of any integer $d$ as long as $d$ is greater than zero.
Here's a few example of quotient-remainder theorem when $b=12$ for a few different values of $a$:
With the last of the examples above you might notice that there were written two different ways to express -5 as a sum of a multiple of 12 and a remainder. But these aren't the only ways, in-fact there is an infinite number of ways to express -5 using quotient remainder theorem since for any arbitrary quotient $q$ there is a unique remainder $r$ such that $a=qb+r$:
What we're getting at is that these remainders are all equal to -5 when we add some multiple of 12, and we call these remainders congruent/equivalent to one-another modulo 12.
Fix $a$, $b$ and $n$ all be integers where $a$ and $b$ are congruent modulo $n$. We express this congruence: $a\equiv b\bmod n.$
The $\equiv$ symbol means equivalence/congruence and is not the same as equality. For example, -5 and 7 are not equal to one-another, but they are equivalent/congruent to one-another modulo 12:
Also note that whenever we "fix" variables, they are completely new meanings to the variables, and no other variables are assumed to exist. In the definition of congruence, the variables $a$ and $b$ are unrelated to the $a$ and $b$ used in the definition of the quotient-remainder theorem.
Here are a couple examples of modular equivalence over the divisor $2$:
And here are a couple examples of modular equivalence over the divisor $12$:
In the example with $-5=(0)12+(-5)=(-1)12+(7)$ we saw that there is an infinite amount of remainders congruent to $7 \bmod 12$. Instead of listing them out (which might take a while, since there's an infinite number of them), we call this set of congruent integers the congruence class of 5 modulo 12, and denote it:
Fix an integer $a$, and another integer $n$ the divisor. We call $[a]_n$ the congruence class of $a$ modulo $n$, and denote it:
What is important to see though is that there is a finite number of congruence classes with respect to equivalence. For example, when 12 is our divisor, there's exactly 12 distinct congruence classes:
And while there are an infinite number of ways to denote congruence classes:
We stick with the canonical representations of them: $[a]_n$ where $0\le a < n$.
As a bonus, $\Z_n$ is the set of congruence classes of integers modulo $n$, which is a finite set, with respect to equivalence, of $n$ elements where every element is itself an infinite set:
Math is neat 🤯
With all this out of the way, now we can now see how atonal music theory and math relate.
If you are familiar with music, then you know that there are 12 distinct pitches, each with various octaves. For example, middle C lies at the center of a piano and is also the note called C4, and C3 and C5 lie one octave below and above C4 respectively. However several notes may be indistinct to one-another, that is they are enharmonic. For example the notes D-sharp 2 and E-flat 2 are enharmonic.
If you are familiar with music theory, then you also know that a C-major chord is called a C-major chord regardless of what octave of C it is built upon. Nevertheless octaves of the same pitch are still distinct; C4 and C5 are different notes on the keyboard or on a music staff, just as 0 is not equal to 12.
On the other hand, different octaves of C are equivalent to one-another in that they are all C.
A pitch class is a collection of pitches related by an integer multiple of an octave and/or enharmonic equivalence.
With this definition we can draw a relation between music and modular arithmetic by associating each pitch class with a congruence class over the integers modulus 12. But since, as far as music is concerned, we will always be working over the integers modulus 12. Thus we can simplify notation going forward by writing $0$ instead of $[0]_{12}$, $1$ instead of $[1]_{12}$, etc. (values of $\Z_{12}$), with the express understanding that these values, as well as the value resulting from any operation on these values, is equivalent to some canonical representation. We may also mangle notation by using the equality operator $=$ in place of the equivalence operator $\equiv$.
| Pitch class | Congruence class/value | Pitch class | Congruence class/value |
|---|---|---|---|
| $\text C$ | $[0]_{12} \Rightarrow 0$ | $\text F\sharp/\text G\flat$ | $[6]_{12} \Rightarrow 6$ |
| $\text C\sharp/\text D\flat$ | $[1]_{12} \Rightarrow 1$ | $\text G$ | $[7]_{12} \Rightarrow 7$ |
| $\text D$ | $[2]_{12} \Rightarrow 2$ | $\text G\sharp/\text A\flat$ | $[8]_{12} \Rightarrow 8$ |
| $\text D\sharp/\text E\flat$ | $[3]_{12} \Rightarrow 3$ | $\text A$ | $[9]_{12} \Rightarrow 9$ |
| $\text E$ | $[4]_{12} \Rightarrow 4$ | $\text A\sharp/\text B\flat$ | $[10]_{12} \Rightarrow 10$ |
| $\text F$ | $[5]_{12} \Rightarrow 5$ | $\text B$ | $[11]_{12} \Rightarrow 11$ |
Here are a few examples of values, either on their own or resulting from an operation, and their corresponding pitch classes:
| Value | Pitch class |
|---|---|
| $1=1$ | $\text C\sharp$ |
| $12=0$ (since $12\equiv 0\bmod 12$) | $\text C$ |
| $13=1$ (since $13\equiv 1\bmod 12$) | $\text C\sharp$ |
| $1+1=2$ | $\text D$ |
| $1+10=11$ | $\text B$ |
| $1+11=0$ (since $12\equiv 0 \bmod 12$) | $\text C$ |
| $3-8=7$ (since $-5\equiv 7 \bmod 12$) | $\text G$ |
A map $f:A\mapsto B$ is a function $f$ such that for every $a\in A$ there is a unique $f(a)\in B$, and the sets $B$ and $A$ are called the image and preimage (or range and domain) respectively. The names map and function are largely used interchangeably.
Fix $P$ the set of musical pitch classes. Let $g:P\mapsto\Z$ be the map from the pitch classes to congruence classes over the integers modulus 12, which we define:
It is worth pointing out that this map is bijective.
A map $f:A\mapsto B$ is called injective, or "one-to-one", if whenever $f(a)=f(b)$ also $a=b$. In other words, every element in the preimage is mapped to a unique value in the image.
A map $f:A\mapsto B$ is called surjective, or "onto", if for every $b\in B$ there is an element $a\in A$ such that $f(a)=b$. In other words, every element in the image is mapped to by $f$, from at least one element in the preimage.
A map is called bijective if it is both injective and surjective (a bijective map is also called a bijection). A function $f$ admits an inverse function $f^{-1}$ if and only if it is bijective.
Since $g$ is a bijection, we can map freely between the pitch classes and the congruence classes. In fact this map is an isomorphism between the two sets, but I'll leave it at that and skip the necessary definitions and explanations for some other time.
Pitch classes will allow us to study music in a mathematical way, where we will quantify distances between notes (intervals), and look at relations between set of notes in terms of operations, i.e. transposition and inversion.
Intervals are nothing new to anyone learned in music theory. However, instead of identifying them by names like major third, perfect fourth, or minor sixth, we identify them by the value of the semitone distance between pitches.
| Interval name | Value |
|---|---|
| Unison | 0 |
| Minor second/augmented unison | 1 |
| Major second | 2 |
| Minor third | 3 |
| Major third | 4 |
| Perfect fourth | 5 |
| Augmented fourth/diminished fifth | 6 |
| Perfect fifth | 7 |
| Minor sixth | 8 |
| Major sixth | 9 |
| Minor seventh | 10 |
| Major seventh | 11 |
| Octave | 12 |