Music and Geometry - part I: pitch class

Introduction

In this series of 4 articles I’m going to introduce you to what for many will be the most exotic piece of music theory you’ve ever come across. Even if you think you’re not good at theory, you may want to try following along and – who knows – perhaps pick up an insight or two.

Music shows deep inner structure on multiple levels that appeals to mathematically inclined people. Math professor Guerino Mazzola has written a book called the “Topos of Music”. It’s safe to assume that only a handful of people in the world really understand the finer details of what it describes. You literally need a PhD in mathematics (in category theory to be more precise) to even begin making sense of it. Professor Mazzola argues that all this complexity is unavoidable because the music it models is just as complex.

Luckily also other books showing neat connections between music and mathematics have been written. And luckily these books are accessible to a wider audience. One of the more intriguing ones, in my humble opinion, is “A Geometry Of Music” by professor Dmitry Tymoczko of Princeton University. In these articles we will dip our toes in some of the very basics of his ideas. To be perfectly clear: I’m not affiliated to professor Tymoczko or the book in any way, and all awesome ideas are his, whereas all mistakes in the explanation are mine.

Pitch line

If you just consider notes in a chromatic scale, starting with the very lowest note you can imagine, and rising to the very highest note you can imagine, it’s not so difficult to depict these notes as points on a line. Although the picture below highlights only a few special notes on the line, actually all conceivable pitches, including microtonal ones or pitches so high they can only be heard by dogs, have their place on the pitch line which has no beginning and no end. It’s an infinite line. In geometry, such a line is considered a 1-dimensional space.


Some simple composition operations can be done by reasoning on a pitch line directly. Take transposition, e.g., and observe how it corresponds to sliding along the line. Or consider inversion of a melody and observe how it corresponds to mirroring pitches around some other pitch on the pitch line.

Pitch class

Yet this pitch line doesn’t model all existing intuitions about pitches. One intuition that is not visible on this line, is the feeling that notes one octave apart in some sense sound the same. Who doesn’t remember the lyrics to the Do-Re-Mi song from the sound of music? When reaching the end of the song, it’s made very clear that “this will lead us back to do”, as if all do’s (C notes) are equal in some sense.

A more difficult way to say the same thing is to say that in some contexts it makes sense to abstract away the exact pitch, and instead reason on pitch class. In other words we don’t speak of a concrete C4 or D3 note anymore, but of a more abstract concept of a “C” note, or a “D” note, with no indication of exactly which octave it belongs to.

The question arises what introducing this “pitch class” abstraction does to our pitch line. How does incorporating pitch class change the pitch line?

The solution is straightforward, but not necessarily trivial to come up with by yourself. To model the concept of pitch class, we need to transform our pitch line so that all differences between octaves disappear. This can be done by curling up the pitch line in three dimensions, so that all equivalent notes (e.g. all “C” notes) perfectly line up. After this is done, we will squash the curled line until it’s flat again. If you think deeply about it, the lining up and especially the squashing of equivalent pitches indeed is a kind of geometrical equivalent to replacing concrete notes with pitch classes. Just like with the pitch line, the pitch class circle has plenty of room to host all microtonal pitches you can come up with. In case you’ve been wondering: the pitch class circle as such has nothing to do with the so-called “circle of fifths”. Other than the fact that both are represented on a circle, there’s no special relation between the two circles.

Please watch the first video animation to better understand how replacing concrete pitch with pitch class has the effect of curling up the pitch line into a pitch class circle.

Before going on to our next topic, let’s sum up a few quick facts about the concept of pitch class and related subjects:

• “Pitch” indicates a specific point on a pitch line.  C4, for example, is a different pitch than C5.  The higher the number, the higher the sound.
• “Pitch class” indicates an entire group of pitches, related by their “octave equivalence”.  The pitch class of C, for example, includes C4, C5, C9 and so on.
• “Pitch class” includes notes with enharmonic spellings.  The pitch class of “C”, for example, would include such pitches as C3, B#5, D double flat 6, and so forth.

Pitch classes are “octave equivalent”, which means that pitches in different octaves are still in the same class.   Pitch classes are also “spelling equivalent” which means that notes spelled differently but sounding the same, which we call enharmonic spellings, are in the same pitch class.

Here is an example of a piece which is built entirely on one pitch class, until the final few measures.

Musical key and mood

Key determines mood? Really?!

It is often argued that with the advent of equal temperament all differences between musical keys disappeared, i.e. C major doesn't sound fundamentally different from Eb major. Yet composers still attribute certain moods to certain musical keys. In part this is probably determined by tradition, and in part it may be influenced by the mechanics of playing an instrument: playing a piece in C major on a piano (using only white keys) is quite a different experience from playing that same piece in C# major (using many black and some white keys) because of the differences between physical location of black and white keys on a piano. Also, bowed instruments will often play an F# differently than a Gb, despite both being the same note on a piano.

So what are these traditional moods associated to certain keys?

This list is based on a list made up by Christian Schubart in his book Ideen Zu Einer Ästhetik Der Tonkunst...

Key	Mode	Mood
C	major	completely pure key; speaks of innocence, simplicity, naivety, child's talk
Db	major	key to bring out unusual feelings; can smile but not laugh; can grimace but not cry
D	major	key of triumph, hallelujah, war-cries and victory-rejoicing
Eb	major	key of love, devotion, intimate conversation with God
E	major	noisy shouts of love, laughing pleasure and not-yet-complete full delight
F	major	key of complaisance and calmness
Gb	major	key of triumph over difficulty, sigh of relief after difficulaties have been overcome
G	major	key that is rustic, idyllic and lyrical, calm and satisfied passion, any gentle and peaceful emotion of the heart
Ab	major	key of the grave, death, putrefaction, judgement, eternity
A	major	declaration of innocent love, satisfaction with one's state of affairs, hope of seeing one's beloved one again when departing, youthful cheerness and trust in God
Bb	major	cheerful love, clear conscience, hope, aspiration for a better world
B	major	strongly colored, announcing wild passions, anger, rage, jealousy, fury, despair and every emotion of the heart
C	minor	declaration of love with lament of unhappy love, sighing of the lovesick soul
C#	minor	penitential lamentation, intimate conversation with God, sighs of disappointed friendship and love
D	minor	melancholy, womanliness
Eb	minor	feelings of anxiety, the soul's deepest distress, brooding despair, blackest depression, most gloomy condition of the soul
E	minor	naive declaration of love, lament without grumbling, sighs accompanied by a few tears, desires to resolve into the pure happiness of C major
F	minor	deep depression, funeral lament, groans of misery and longing for the grave
F#	minor	a gloomy key, resentment and discontent, it languishes for the calm of A major or happiness of D major
G	minor	discontent, uneasiness, worry about a failed scheme, bad-tempered gnashing of teeth, dislike
G#	minor	grumbling, struggling with difficulty, heart squeezed until it suffocates
A	minor	pious womanliness, tenderness of character
Bb	minor	mocking God and the world, discontented with itself and everything, preparation of suicide
B	minor	key of patience, calm awaiting one's fate, submission to divine dispensation, mild lament without breaking out into offensive murmuring or whimpering,

Tutorial on my technique for writing a canon

I promised to write a tutorial on how to write a canon like the one I posted in my previous post Canon in 5, and I did.

The tutorial is 12 pages long, but most of that space is taken up by white-space and examples. You will probably get the most out of it if you already know how to read music, although strictly speaking it is not a requirement to already benefit from the material in the tutorial.

You can get the tutorial in .pdf format by clicking this link. You may have trouble downloading it with some versions of internet explorer. In that case use chrome or firefox instead.

You can listen to the result of the tutorial here:

Drop me a note if you've been able to use the techniques explained in there.

Modes

Using modes in composition

I have some fascination with composing based on modes. It can make music sound so refreshing. For the longest time i've been very confused about modes. Here's an attempt at clarifying some aspects related to modes. The explanation assumes you are already familiar with key signatures in major and minor scales (i.e. number of sharps and flats required to make a key like D major sound like D major).

A typical explanation about modes is something as follows. First you start with a C major scale

c d e f g a b c

Now you play the same notes, but you start on the note "d"

d e f g a b c d

and you have created a dorian mode. Similarly, starting on the note "e": "e f g a b c d e" results in a phrygian mode, starting on "f": "f g a b c d e f" results in a lydian mode, starting on "g": "g a b c d e f g" results in a mixolydian mode, starting on "a": "a b c d e f g a" results in a aeolian mode, and starting on the "b": "b c d e f g a b" creates a locrian mode.

If you're like me, immediately a few questions arise (which are never answered by most basic tutorials)

1. Is "d e f g a b c d" is a dorian mode of the C major scale? or of the D major scale?
2. Do different modes have a specific sound to them? some mood or character?
3. How do you quickly construct a dorian mode (or any other mode) of - say - the E major scale without memorizing the notes for each possible combination of (scale, mode)?
4. What difference does it make if you play "c d e f g a b c" or "d e f g a b c d", it's all the same notes anyway?

Is "d e f g a b c d" a dorian mode of the c major or of d major ?

The rule is simple: if it starts on a "d" it's derived from a d-scale

What different modes exist? Do they have a specific sound to them? some mood or character?

Some modes lend themselves more naturally to achieving specific moods in your music, but you can by no means generalize. Some very sad and melancholic music has been written in major keys (I'm thinking e.g. of Scriabin's prelude op 17, no 6, or the Plainte by Caix d'Hervelois) Despite all those over-generalizations, the following list seems to work well (these are all modes derived from a major scale; you could derive even more modes by starting from other scales, say a harmonic or melodic minor scale, too):

• c ionian "c d e f g a b c": happy music. This is also called the c major key.
• c dorian "c d es f g a bes c": irish folk-like
• c phrygian "c des es f g aes bes c": spanish sounding
• c lydian "c d e fis g a b c": happy, playful, somewhat comical effect, attention raising (think: the simpsons theme)
• c mixolydian "c d e f g a bes c": uniting pleasure and sadness; creating a effect of yearning for something or someone
• c aeolian "c d es f g aes bes c": sad music. This is also called the c natural minor key.
• c locrian "c des es f ges aes bes c": (this is not often used as it sounds weird to most western ears)

Some nice effects can be achieved e.g. if a melody is written in one mode, and later on is echoed in a different mode.

How do you quickly construct the dorian mode of - say - the E major scale ?

This may not work for you, but it works for me. It requires that you don't have to think about key signatures for major keys (i.e. you do know those by heart), and you also don't have to think twice about the names of the simplest modes built on the major keys (i.e. ionian "c d e f g a b c", dorian "d e f g a b c d", phrygian "e f g a b c d e", lydian "f g a b c d e f", mixolydian "g a b c d e f g", aeolian "a b c d e f g a", locrian "b c d e f g a b")

• First I remember that the simplest dorian mode is "d e f g a b c d"
• Then I remember that D major really needs two sharps (fis and cis) to sound like D major.
• From those two memories I can quickly construct the following rule: "the dorian mode is like the major scale, but with two less sharps (or two extra flats, or one less sharp and one extra flat)." This rule can be applied to any scale.
• Example: E major requires four sharps. If I want to have the dorian mode I just need to drop two of those sharps, so I end up with "e fis g a b cis d e". Second example: now I want the E lydian mode. Simplest lydian mode is "f g a b c d e f". Compared to F major, this has one less flat (bes became b), or equivalently: one extra sharp. E lydian therefore requires 5 sharps instead of 4, i.e. fis, cis, gis, dis, ais.

What difference does it make if you play c ionian "c d e f g a b c" or d dorian "d e f g a b c d", it's all the same notes anyway?

• One difference is in which notes you emphasize on the strong beats (typically first, third, fifth notes of the mode) also the notes you chose to return to at the end of a musical fragment (often something like the first note of the mode).
• If you're writing a melody in a mode, you will want to use the notes that make it sound different from a major scale more prominently to emphasize that you're not working in a major/minor scale. So a melody in G mixolydian should feature the natural f, and a melody in d dorian should feature the natural f and the natural c. It's exactly those altered notes (compared to the major scale) that lend your melody its extra qualities/mood.