The Music Toolkit introduces the physics and math behind musical scales, and covers some basic theory that will help you write a chart-topping hit.

Frequency and Octave Equivalence

Intro to the makeascale browser instrument, setting a fundamental frequency, and octave equivalence.

You can open and play with the makeascale browser instrument here.

The Third Harmonic

Different ways to represent oscillation and octave equivalence. Adding a new note derived from the third harmonic to our musical system.

This video uses the sine waves observable, which you can open and play with here.

Adding Harmonics Together

The harmonic series, and changes in tone quality when harmonics are added together.

The Ratio 3-to-2

Playing with a two-note musical system, adding a third note, and playing in the space of a single octave. The differences between "frequency," "pitch," and "note."

Expanding to five, then seven notes

Playing with a five-note (pentatonic) musical system, and using a seven-note system to play along with a song.

A few songs you can play along with using this seven-note system:

Tonal Center and Modes

Ordering the notes of a seven-note musical system, listening to the musical effects, ordering the same notes in different ways. Here, we are playing along with Black or White by Michael Jackson.

Building from the tonal center

Constructing the same seven-note system starting from a different fundamental frequency.

Expanding to twelve notes

Completing a twelve note musical system and coming full circle... almost.

Equal Temperament

Dividing the octave into twelve equal parts, what's gained, what's lost.

Comparing Pure and Equal Temperaments

Listening to the sum and difference between pitches in different tuning systems.

Degrees and Intervals

Naming notes relative to the tonal center, and measuring distances between notes.

Mathematics of Intervals

Representing intervals more abstractly within in a Mod 7 system.

Chords and Triads

Triads constructed using tertian, quartal, and quintal harmony. Common triads in a mode.