Building Arpeggio Pt. 2: Music Theory for Programmers

I’m building a domain-specific language called Arpeggio that compiles code into music. In Part 1, I outlined the basic language design and syntax. Here, I’m going to do a very shallow dive into music theory from a programmer’s perspective, focusing on the terms and concepts needed to build the music backend that powers Arpeggio.

Disclaimer: Music theory is a huge field of study that’s filled with complexity and ambiguity, which I’m going to vastly oversimplify down to the basic math and patterns that I can understand. Apologies to any real musicians who stumble onto this.

The Terminology

Songs are fundamentally composed of notes, which are just names assigned to frequencies: A4 is a signal oscillating at 440Hz¹. There are 12 other distinct notes², each separated by one semitone. Move 12 semitones up from a note and you double its frequency³, getting the same note in the next octave (A5 is 880Hz).

Not all notes sound good together⁴, which is why modes exist. A mode describes a set of semitone intervals between notes that are frequently played together. The most common modes in western music are Ionian and Aeolian, better know as the major and minor⁵ scales, which are called diatonic and contain 7 distinct notes each. The intervals of a mode can be played from any starting note, or tonic. Combining a tonic with a mode, like A major, defines a key.

Calculating Frequency

Putting all that together, we can get a very simple representation of musical keys in Python with:

Note = float
Interval = int
Mode = list[Interval]

class Key:
    tonic: Note
    mode: Mode

class AMajor(Key):
    tonic = 440.0
    mode = [2, 2, 1, 2, 2, 2, 1]

We can calculate the frequency of any other note in that key by counting how many semitones are needed to reach the desired interval. The 4th interval in the key of A major is 5 semitones (2 + 2 + 1) above the 440 Hz tonic. Each semitone is 1/12th of an octave, so our new note is:

$$ 440 \text{ Hz} \cdot 2^{\frac{5}{12}} = 587.33 \text{ Hz} $$

In Python:

class Note:
    def __init__(self, frequency: float):
        self.frequency = frequency

    def __add__(self, semitones: int) -> Note:
        """Return the note n semitones above this note."""
        return Note(self.frequency * 2 ** (semitones / 12))

Calculating Chords

A chord is a collection of notes played together. The most common chord in western music is a triad, which is three notes at the root, third, and fifth interval. Triads can be played at any interval in the mode by calculating intervals relative to the new root, usually written in Roman numerals. For example, a IV chord in the key of A major has a root at the 4th interval and other notes at the 7th and (wrapping around to the next octave) 2nd intervals.

Rhythm

In addition to the frequency, we also need to know how long to play each note for. This depends on the note length, measured in fractions of a whole note, and the tempo and time signature of the song. In a song with 60 beats-per-minute (BPM) in 4/4 time⁶, a single quarter note lasts for 1 second.

from fractions import Fraction

class Duration(Fraction):
    def to_millis(cls, bpm: int, beats_per_measure: int = 4) -> float:
        """Return the number of milliseconds this note duration lasts."""
        return float((60_000 * float(self) / bpm) * beats_per_measure)

QuarterNote = Duration(1, 4)

assert QuarterNote.to_millis(bpm=60) == 1000

Wrapping Up

And that’s it! We can define a song with a key, use modes and intervals to find semitones, and use semitones and beats to calculate notes and chords. All that’s left is turning frequencies and durations into audio that we can hear, which you can read about in Part 3!