I thought this chart was interesting; from WIK, it shows the various 'interval projections' I have been discussing. These are the same ideas as Hanson's.

Pitch class multiplication modulo 12

When dealing with pitch class sets, multiplication modulo 12 is a common operation. Dealing with all twelve tones, or a tone row, there are only a few numbers which one may multiply a row by and still end up with a set of twelve distinct tones. Taking the prime or unaltered form as P0, multiplication is indicated by Mx, x being the multiplicator:

The following table lists all possible multiplications of a chromatic twelve-tone row:

M M × (0,1,2,3,4,5,6,7,8,9,10,11) mod 12

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 2 3 4 5 6 7 8 9 10 11

2 0 2 4 6 8 10 0 2 4 6 8 10

3 0 3 6 9 0 3 6 9 0 3 6 9

4 0 4 8 0 4 8 0 4 8 0 4 8

5 0 5 10 3 8 1 6 11 4 9 2 7

6 0 6 0 6 0 6 0 6 0 6 0 6

7 0 7 2 9 4 11 6 1 8 3 10 5

8 0 8 4 0 8 4 0 8 4 0 8 4

9 0 9 6 3 0 9 6 3 0 9 6 3

10 0 10 8 6 4 2 0 10 8 6 4 2

11 0 11 10 9 8 7 6 5 4 3 2 1

Note that only M1, M5, M7, and M11 give a one to one mapping (a complete set of 12 unique tones). This is because each of these numbers is relatively prime to 12. Also interesting is that the chromatic scale is mapped to the circle of fourths with M5, or fifths with M7, and more generally under M7 all even numbers stay the same while odd numbers are transposed by a tritone. This kind of multiplication is frequently combined with a transposition operation. It was first described in print by Herbert Eimert, under the terms "Quartverwandlung" (fourth transformation) and "Quintverwandlung" (fifth transformation) (Eimert 1950, 29–33), and has been used by the composers Milton Babbitt (Morris 1997, 238 & 242–43; Winham 1970, 65–66), Robert Morris (Morris 1997, 238–39 & 243), and Charles Wuorinen (Hibbard 1969, 157–58). This operation also accounts for certain harmonic transformations in jazz (Morris 1982, 153–54).

Look at this chart as being "stacks" of a certain interval, going left to right.

The top row "0" means nothing.

1X row (beginning with "1") is an ascending chromatic scale.

2X row is a whole-tone scale (stacking seconds, or "2").

3X row is a diminished scale

4X row is major thirds, or an augmented chord;

5X row is stacks of fourths (eventually giving all 12 notes)

6X row is the tritone, invertible with itself;

7X row is stacks of fifths (eventually giving all 12 notes)

8X is the inversion of row 4, also augmented

9X is the inversion of row 3, diminished

10X is the inversion of row 2, whole tone

11X is the inversion of row 1X, a descending chromatic scale