Music and Mathematics: On Symmetry

Monday, February 11th, 2019

Introduction

What role does symmetry play in aesthetics? Is symmetrical art more beautiful than asymmetrical art? Is music thatcontains symmetries (whatever that means), more sonorous? Or is it through the balance and juxtaposition of thesymmetrical and the asymmetrical that beautiful art is made?

This is something that Hermann Weyl explored in his book, Symmetry. There, he defined symmetry as follows:

... symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort ofconcordance of several parts by which they integrate into a whole. Beauty is bound up with symmetry.

Discussion

What are some examples of symmetry intersecting with common conceptions of the beautiful?

What are some ways in which symmetry serves a functional purpose?

Ways of describing symmetry

Consider patterns/designs that are repetitive in one direction - these are called frieze patterns.

These kinds of symmetries can be found in all sorts of artistic, architectural, and musical works. Play around with thisweb app to see some visual examples: https://eschersket.ch/

Of course, there are tons of visual examples, but how about some musical examples?

Musical examples of symmetry

One of the reasons that Friezes are good examples of symmetry to consider is that we can transfer some of these ideasto melodic symmetry. While comfort with musical notation isn't necessary for understanding these basic music-theoreticconcepts or identifying musical symmetry, it can be visually useful. Take a look at the following examples and see if youcan identify the kinds of symmetry being used.

These visual explorations of symmetry are a great start, but to dig even deeper, we can call upon the field of mathematicsthat explores symmetry: abstract algebra.

Abstract Algebra

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Music theory is very "structural." That is to say, it is very pattern heavy. For example, we saw that major and minor chordsfollow a particular pattern:

While all of mathematics could be said to be about structure and pattern, the field of abstract algebra, which is the studyof algebraic structures, is particularly well suited to exploring the relationships among a given set of objects and theoperations that can be performed on them. That said, abstract algebra can provide a great framework for analyzing musicand abstracting the relationships found in modern (Western) music theory to uncover other possible musics.

Let's start with a basic introduction to Abstract Algebra.

Group Theory

Let's revisit the earlier question: what is symmetry?

One very precise way to define symmetry is this: A symmetry is an undetectable motion; an object is symmetric if it has symmetries.

To get us thinking symmetrically let's do the following exercises.

1. Catalog all the symmetries of a (nonsquare) rectangular card. Get a card and look at it. Turn it about. Mark its parts asyou need. Write out your observations and conclusions.

You should have found four symmetries total:

: no motion: 180 degree rotation around the -axis: 180 degree rotation around the -axis: 180 degree rotation around the -axis

Let's consider combinations of those operations by filling out a multiplication table. In each cell, write down the result ofdoing the two corresponding motions in sequence.

2. List the symmetries of an equilateral triangular plate and work out the multiplication table for the symmetries.

Definition of a Group: A group consists of a nonempty set together with a binary operation

Satisfying the following conditions

1. Closure: , the result .2. Associativity: 3. Identity: such that 4. Inverses: such that

What are some examples of groups, and some examples of non-groups?

The above are all examples of infinite groups, while the symmetries we discussed earlier were examples of finite groups.

Much of abstract algebra is concerned with categorizing different types of groups. For example, one important categoryis that of cyclic groups.

We say that a group is cyclic with generator provided that every element of can be expressed as (or ) for some .

What examples can you think of?

For those less familiar, let's quickly review modular arithmetic.

We'll say that - or in plain English "a is congruent to b mod n if and only if n divides b-a."

Some examples:

Consider the integers modulo 4. How many distinct integers do we get mod 4? Do the integers mod 4 form a groupunder some operation?

They do! All integers can be put into one of four equivalence classes: [0], [1], [2], [3] - i.e. those integers congruent to 0mod 4, those integers congruent to 1 mod 4, those integers congruent to 2 mod 4, those integers congruent to 3 mod 4.Since , once we get back to 4, we're really back to [0]. Sometimes modular arithmetic is referred to as clockwork

arithmetic.

The integers modulo 4, written form a group under addition mod 4. How do we check? Well, we need to confirm thatunder addition mod 4, the following conditions hold. Make a multiplication table to check!

1. Closure: , the result .2. Associativity:

3. Identity: such that 4. Inverses: such that

Examples

1. Consider the nonzero integers modulo 5 under multiplication: . Is this a group? What if we take the nonzerointegers mod 4 instead?

2. Work out the full multiplication table for the set of permutations of three objects, where "multiplication" really meanscomposition. Confirm that this is a group.

Going back to symmetries, a dihedral group is the group of symmetries of a regular polygon, which includes reflectionsand rotations. Let's consider a Dihedral group of order . How many elements does this group have? Consider someexamples?

Alright...so how does this all tie into music? Thoughts?

Musical Actions of the Dihedral Groups

The dihedral group of order 24 is the group of symmetries of a regular 12-sided polygon. Algebraically, the dihedral groupof order 24 is generated by two elements, and , such that

Let's take some time to make sense of these properties in terms of symmetries. The operation , is rotation by andthe operation is reflection about an axis.

First musical action

The first musical action of the dihedral group of order 24 we consider arises via the familiar compositional techniques oftransposition and inversion. A transposition moves a sequence of pitches up or down. When singers decide to sing asong in a higher register, for example, they do this by transposing the melody. An inversion, on the other hand, reflects amelody about a fixed axis, just as the face of a clock can be reflected about the 0-6 axis. Often, musical inversion turnsupward melodic motions into downward melodic motions.

Let's do the following.

1. Define these operations explicitly.

1. Rotation: transposition by one note2. Reflection

2. Show that the set of twelve notes under either operation forms a group.

C C# D D# E F F# G G# A A# B C

0 1 2 3 4 5 6 7 8 9 10 11 0

Second musical action

The second action of the dihedral group of order 24 that we explore has only come to the attention of music theorists inthe past two decades. It's not that no one made these music moves before, but rather that they were not formalized. Itsorigins lie in the , , and operations of the 19th-century music theorist Hugo Riemann. The parallel operation maps a major triad to its parallel minor and vice versa. The leading tone exchange operation takes a major triad to theminor triad obtained by lowering only the root note by a semitone. The operation raises the fifth note of a minor triadby a semitone. The relative operation R maps a major triad to its relative minor, and vice versa. For example,

=

Pitch Classes and Integers Modulo 12

As we discussed above, the octave (a doubling of pitch) functions like modular arithmetic. If we let the note C = [0] (anarbitrary choice), then we get the following mapping:

So, of course, for any note , . Some examples:

so F + 8 = C#

We can add lots of notes for color to major (or minor) chords. Like a CMaj9 chord is formed like this:

.

Construct and then play a couple major 11 chords: .

Defining Transposition and Inversion

Verify that these operations act on the 12 notes of the chromatic scale like rotation and reflection. I.e.

Draw a "clock" of the twelve notes to see how these operations play out visually

We can apply these transformations to individual notes, to chords, or to melodies. In particular, the inversion operation isa good way to create interesting melodies.

The PLR-Group

We notice that in the T/I-Group, we can start with any major triad, and using only those two transformations, get all 24major and minor triads. Another way of navigating the major and minor triads is through the PLR-group, initiated byDavid Lewin. As we'll find, the PLR-group has a beautiful geometric depiction called the Tonnetz.

Consider the three functions defined as follows:

Take some time to play with these operations on major and minor triads. As you do so, look at triads on your "clock"and see if there are any symmetries.

Theorem: The -group is generated by and and is dihedral of order 24.

Recall: The dihedral group of order 24 is the group of symmetries of a regular 12-sided polygon. Algebraically, thedihedral group of order 24 is generated by two elements, and , such that

Proof: Closure should be clear. Now let's consider a sequence of triads starting on C-major and then alternately applying and .

This tells us that the 24 bijections (reversible operations)

are distinct and that we get at least 24 elements in our -group.

Can we define in terms of and ? (Look at the sequence we generated above.)

Now, set and . Then, and . Lastly, confirm that .

Fascinatingly, the progression we looked at above (in our proof) can be found in Beethoven's 9th (2nd movement,measures 143-176).

The Tonnetz

The PLR actions can be visualized as movements about a Tonnetz. And so, Beethoven's 9th (or that particular segment ofit) can be viewed as a path traced along a Tonnetz.

https://codedot.github.io/tonnetz/