Introduction

A pitch-class collection is simply a set of pitch classes, in no particular order.  There are a number of different pitch-class collections, some more common than others in common-practice music.

In twelve-tone equal temperament, the chromatic collection is a set of every one of the twelve available pitch classes—play every key, black and white, between any C up to B (e.g. C4 to B4) and you have played the chromatic pitch class collection.

Other pitch class collections in common-practice, therefore, are subsets of the chromatic collection.  One of the most important for common-practice music is the diatonic collection.

The diatonic collection

The diatonic collection is a set of seven pitch classes.  One example of this collection is found in the white-key notes of the piano keyboard, or the seven notes that can be represented by a letter from A to G.  We can represent this as an unordered set (set X in Example 00).

One thing we can do with the diatonic collection that we cannot do with the chromatic collection is to transpose it.  For example, we could take the set X above and either raise every note by a semitone (A →  B♭, B → C, etc., see set Y) or lower every note by a tone and a semitone (A → F♯, B → G♯, etc., see set Z).

If you look at all these representations of the diatonic collection, one thing that is consistent is that each letter name is represented exactly once .  This is a distinctive feature of the diatonic collection, no matter how it is transposed.

Scales

A scale is an ordered collection of pitch classes.  This means that they are placed in ascending order from an arbitrarily selected starting pitch class.  Therefore, an ordered diatonic collection is a diatonic scale.  An order chromatic collection is a chromatic scale.

If we return to the white-key notes of the piano keyboard, and start the diatonic collection on A, then we get a scale starting on A, and ascending through B, C, D, E, F and G.  By convention, we usually enclose such a collection with a repetition of the initial pitch class, thus: A–B–C–D–E–F–G–A.  While this represents a total of eight notes, it is still a collection of seven pitch classes (A is represented twice).

We can rotate this collection, by starting it on different pitch classes: B–C–D–E–F–G–A–B, C–D–E–F–G–A–B–C, etc., until we return to starting on A.  Each of these rotations creates a different series of tones and semitones between each pair of adjacent notes.  This is shown in Example 00