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 be a positive integer, and a subset containing elements. Then every occurs as a difference between two elements .
Proof. Consider the translate . Since both and have size , they have an element in common. If , then for some , so .
Here are some applications to music theory:
Example 1 (scales containing every chromatic interval). Any scale consisting of at least out of the available chromatic notes contains every interval. Indeed, , 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 semitones) in our scale , we consider the scale and its transpose by a minor third. Because , 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 overlaps to get from down to . 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 major fifths in C major, since C major and G major overlap at 6 notes. Conversely, if an interval occurs many times in a key , that means that the transposition of by the interval has many notes in common with the old key . (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 and have size , 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 notes in a major or minor scale contains every diatonic interval. Indeed, modelling the scale as integers modulo , we observe that , so the lemma above shows that every diatonic interval occurs at least once.
For example, a seventh chord contains the notes¹ of the key. It contains a second between and , a third between and , a fourth between and , 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 -tone tuning system, then any scale consisting of at least of those notes contains every chromatic interval available in this tuning.
¹ A strange historical artefact of music is that chords start with instead of .