Scales containing every interval

This is a maths/music crossover post, inspired by fidgeting around with diatonic chords containing no thirds. The general lemma is the following (see also the examples below):

Lemma. Let n be a positive integer, and I \subseteq \mathbf Z/n\mathbf Z a subset containing k > \frac{n}{2} elements. Then every a \in \mathbf Z/n\mathbf Z occurs as a difference x - y between two elements x, y \in I.

Proof. Consider the translate I + a = \{x + a\ |\ x \in I\}. Since both I and I + a have size k > \frac{n}{2}, they have an element in common. If x \in I \cap (I + a), then x = y+a for some y \in I, so a = x - y. \qedsymbol

Here are some applications to music theory:

Example 1 (scales containing every chromatic interval). Any scale consisting of at least 7 out of the 12 available chromatic notes contains every interval. Indeed, 7 > \frac{12}{2}, so the lemma shows that every difference between two elements of the scale occurs.

The above proof in this case can be rephrased as follows: if we want to construct a minor third (which is 3 semitones) in our scale S, we consider the scale S and its transpose S + 3 by a minor third. Because 7 + 7 = 14 > 12, there must be an overlap somewhere, corresponding to an interval of a minor third in our scale.

In fact, this shows that our scale must contain two minor thirds, since you need at least 2 overlaps to get from 14 down to 12. For example, the C major scale contains two minor seconds (B to C and E to F), at least two major thirds (C to E and G to B), and two tritones (B to F and F to B).

The closer the original key is to its transpose, the more overlaps there are between them. For example, there are 6 major fifths in C major, since C major and G major overlap at 6 notes. Conversely, if an interval a occurs many times in a key S, that means that the transposition S + a of S by the interval a has many notes in common with the old key S. (Exercise: make precise the relationship between intervals occurring ‘many times’ and transpositions having ‘many notes in common’.)

We see that this argument is insensitive to enharmonic equivalence: it does not distinguish between a diminished fifth and an augmented fourth. Similarly, a harmonic minor scale contains both a minor third and an augmented second, which this argument does not distinguish.

Remark. We note that the result is sharp: the whole-tone scales 2\mathbf Z/12\mathbf Z and (2\mathbf Z + 1)/12\mathbf Z have size 6 = \frac{12}{2}, but only contain the even intervals (major second, major third, tritone, minor sixth, and minor seventh).

Example 2 (harmonies containing every diatonic interval). Any cluster of 4 notes in a major or minor scale contains every diatonic interval. Indeed, modelling the scale as integers modulo 7, we observe that 4 > \frac{7}{2}, so the lemma above shows that every diatonic interval occurs at least once.

For example, a seventh chord contains the notes¹ \{1,3,5,7\} of the key. It contains a second between 7 and 1, a third between 1 and 3, a fourth between 5 and 1, etcetera.

Thus, the largest harmony avoiding all (major or minor) thirds is a triad. In fact, it’s pretty easy to see that such a harmony must be a diatonic transposition of the sus4 (or sus2, which is an inversion) harmony. But these chords may contain a tritone, like the chord B-F-G in C major.

Example 3. If you work with your favourite 19-tone tuning system, then any scale consisting of at least 10 of those notes contains every chromatic interval available in this tuning.

¹ A strange historical artefact of music is that chords start with 1 instead of 0.