We know that in the 12-tone equal temperament musical system, each octave is equally divided into 12 semitones. In a major scale, for example the C major scale, the notes are C, D, E, F, G, A, and B, and let C' to be the C note in the next octave. The intervals between these notes and the first note C are 0, 2, 4, 5, 7, 9, 11, 12 semitones. Similarly, we can list the intervals in a table in terms of semitones:
| Note | C | D | E | F | G | A | B | C' | D' | E' | F' | G' | A' | B' | C'' |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| C | 0 | 2 | 4 | 5 | 7 | 9 | 11 | 12 | 14 | 16 | 17 | 19 | 21 | 23 | 24 |
| D | -2 | 0 | 2 | 3 | 5 | 7 | 9 | 10 | 12 | 14 | 15 | 17 | 19 | 21 | 22 |
| E | -4 | -2 | 0 | 1 | 3 | 5 | 7 | 8 | 10 | 12 | 13 | 15 | 17 | 19 | 20 |
| F | -5 | -3 | -1 | 0 | 2 | 4 | 6 | 7 | 9 | 11 | 12 | 14 | 16 | 18 | 19 |
| G | -7 | -5 | -3 | -2 | 0 | 2 | 4 | 5 | 7 | 9 | 10 | 12 | 14 | 16 | 17 |
| A | -9 | -7 | -5 | -4 | -2 | 0 | 2 | 3 | 5 | 7 | 8 | 10 | 12 | 14 | 15 |
| B | -11 | -9 | -7 | -6 | -4 | -2 | 0 | 1 | 3 | 5 | 6 | 8 | 10 | 12 | 13 |
| C' | -12 | -10 | -8 | -7 | -5 | -3 | -1 | 0 | 2 | 4 | 5 | 7 | 9 | 11 | 12 |
| D' | -14 | -12 | -10 | -9 | -7 | -5 | -3 | -2 | 0 | 2 | 3 | 5 | 7 | 9 | 10 |
| E' | -16 | -14 | -12 | -11 | -9 | -7 | -5 | -4 | -2 | 0 | 1 | 3 | 5 | 7 | 8 |
| F' | -17 | -15 | -13 | -12 | -10 | -8 | -6 | -5 | -3 | -1 | 0 | 2 | 4 | 6 | 7 |
| G' | -19 | -17 | -15 | -14 | -12 | -10 | -8 | -7 | -5 | -3 | -2 | 0 | 2 | 4 | 5 |
| A' | -21 | -19 | -17 | -16 | -14 | -12 | -10 | -9 | -7 | -5 | -4 | -2 | 0 | 2 | 3 |
| B' | -23 | -21 | -19 | -18 | -16 | -14 | -12 | -11 | -9 | -7 | -6 | -4 | -2 | 0 | 1 |
| C'' | -24 | -22 | -20 | -19 | -17 | -15 | -13 | -12 | -10 | -8 | -7 | -5 | -3 | -1 | 0 |
Since the scale is periodic, in this way, all integer intervals can be found in this scale.
In contrast, the C major pentatonic scale (composed of notes CDEGA) does not satisfy this requirement. For example, we cannot find two notes that are 1 semitone apart.
| Note | C | D | E | G | A | C' | D' | E' | G' | A' | C'' |
|---|---|---|---|---|---|---|---|---|---|---|---|
| C | 0 | 2 | 4 | 7 | 9 | 12 | 14 | 16 | 19 | 21 | 24 |
| D | -2 | 0 | 2 | 5 | 7 | 10 | 12 | 14 | 17 | 19 | 22 |
| E | -4 | -2 | 0 | 3 | 5 | 8 | 10 | 12 | 15 | 17 | 20 |
| G | -7 | -5 | -3 | 0 | 2 | 5 | 7 | 9 | 12 | 14 | 17 |
| A | -9 | -7 | -5 | -2 | 0 | 3 | 5 | 7 | 10 | 12 | 15 |
| C' | -12 | -10 | -8 | -5 | -3 | 0 | 2 | 4 | 7 | 9 | 12 |
| D' | -14 | -12 | -10 | -7 | -5 | -2 | 0 | 2 | 5 | 7 | 10 |
| E' | -16 | -14 | -12 | -9 | -7 | -4 | -2 | 0 | 3 | 5 | 8 |
| G' | -19 | -17 | -15 | -12 | -10 | -7 | -5 | -3 | 0 | 2 | 5 |
| A' | -21 | -19 | -17 | -14 | -12 | -9 | -7 | -5 | -2 | 0 | 3 |
| C'' | -24 | -22 | -20 | -17 | -15 | -12 | -10 | -8 | -5 | -3 | 0 |
But if we replace E with D# and construct this uncommon scale, then all integer semitones can be found.
| Note | C | D | D# | G | A | C' | D' | D#' | G' | A' | C'' |
|---|---|---|---|---|---|---|---|---|---|---|---|
| C | 0 | 2 | 3 | 7 | 9 | 12 | 14 | 15 | 19 | 21 | 24 |
| D | -2 | 0 | 1 | 5 | 7 | 10 | 12 | 13 | 17 | 19 | 22 |
| D# | -3 | -1 | 0 | 4 | 6 | 9 | 11 | 12 | 16 | 18 | 21 |
| G | -7 | -5 | -4 | 0 | 2 | 5 | 7 | 8 | 12 | 14 | 17 |
| A | -9 | -7 | -6 | -2 | 0 | 3 | 5 | 6 | 10 | 12 | 15 |
| C' | -12 | -10 | -9 | -5 | -3 | 0 | 2 | 3 | 7 | 9 | 12 |
| D' | -14 | -12 | -11 | -7 | -5 | -2 | 0 | 1 | 5 | 7 | 10 |
| D#' | -15 | -13 | -12 | -8 | -6 | -3 | -1 | 0 | 4 | 6 | 9 |
| G' | -19 | -17 | -16 | -12 | -10 | -7 | -5 | -4 | 0 | 2 | 5 |
| A' | -21 | -19 | -18 | -14 | -12 | -9 | -7 | -6 | -2 | 0 | 3 |
| C'' | -24 | -22 | -21 | -17 | -15 | -12 | -10 | -9 | -5 | -3 | 0 |
Therefore, we only need no more than 5 notes to construct a scale in the 12-tone equal temperament musical system to find all integer intervals.
Now, in the more general case of the N-tone equal temperament musical systems, what is the minimum number of notes M to construct a scale such that any integer intervals can be found?
Additionally, what is the maximum number of notes L for each N to construct a scale such that at least one integer interval cannot be found in this scale?