If a scale is transposed, the overlap of scale degrees between the transposed version and the original is defined like this:
Where g(S) is a transposition of a scale family. S may be any member of the scale family without affecting overlap. Balzano gives an example matrix of overlap between pairs of pentatonic scale family members:
| Member | 0 | 1 | 2 | 3 |
4 | 5 | 6 | 7 |
8 | 9 | 10 | 11 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 5 | 0 | 3 | 2 | 1 | 4 | 0 | 4 | 1 | 2 | 3 | 0 |
| 1 | 0 | 5 | 0 | 3 | 2 | 1 | 4 | 0 | 4 | 1 | 2 | 3 |
| 2 | 3 | 0 | 5 | 0 | 3 | 2 | 1 | 4 | 0 | 4 | 1 | 2 |
| 3 | 2 | 3 | 0 | 5 | 0 | 3 | 2 | 1 | 4 | 0 | 4 | 1 |
| 4 | 1 | 2 | 3 | 0 | 5 | 0 | 3 | 2 | 1 | 4 | 0 | 4 |
| 5 | 4 | 1 | 2 | 3 | 0 | 5 | 0 | 3 | 2 | 1 | 4 | 0 |
| 6 | 0 | 4 | 1 | 2 | 3 | 0 | 5 | 0 | 3 | 2 | 1 | 4 |
| 7 | 4 | 0 | 4 | 1 | 2 | 3 | 0 | 5 | 0 | 3 | 2 | 1 |
| 8 | 1 | 4 | 0 | 4 | 1 | 2 | 3 | 0 | 5 | 0 | 3 | 2 |
| 9 | 2 | 1 | 4 | 0 | 4 | 1 | 2 | 3 | 0 | 5 | 0 | 3 |
| 10 | 3 | 2 | 1 | 4 | 0 | 4 | 1 | 2 | 3 | 0 | 5 | 0 |
| 11 | 0 | 3 | 2 | 1 | 4 | 0 | 4 | 1 | 2 | 3 | 0 | 5 |
Balzano states that "'A' is too strong a property to be of much use for most scales." (p. 330) because it is too specific to the diatonic set. There are only 2 other 7-note scales that exhibit this property:
{0, 1, 2, 4, 6, 8, 10}, and {0, 1, 2, 3, 4, 5, 6}.
Balzano
Lerdahl's Constraints