Contributions to module-theoretic classification of musical motifs
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Classification, part 3 |
| No. | Representant | Volume | Subset classes |
|---|---|---|---|
| 0 | { (0,0), (1,0), (2,0) } | 0 | 1, 1, 2 |
| 1 | { (0,0), (1,0), (5,0) } | 0 | 1, 1, 4 |
| 2 | { (0,0), (1,0), (6,0) } | 0 | 1, 1, 6 |
| 3 | { (0,0), (1,0), (3,0) } | 0 | 1, 2, 3 |
| 4 | { (0,0), (1,0), (4,0) } | 0 | 1, 3, 4 |
| 5 | { (0,4), (0,2), (6,0) } | 0 | 2, 2, 2 |
| 6 | { (0,0), (2,0), (4,0) } | 0 | 2, 2, 4 |
| 7 | { (0,2), (0,0), (6,0) } | 0 | 2, 2, 6 |
| 8 | { (0,0), (2,0), (6,0) } | 0 | 2, 4, 6 |
| 9 | { (0,0), (3,0), (6,0) } | 0 | 3, 3, 6 |
| 10 | { (0,0), (4,0), (8,0) } | 0 | 4, 4, 4 |
| 11 | { (0,6), (0,0), (6,0) } | 0 | 6, 6, 6 |
| 12 | { (0,1), (0,0), (1,0) } | 1 | 1, 1, 1 |
| 13 | { (0,1), (0,0), (2,0) } | 2 | 1, 1, 2 |
| 14 | { (0,2), (0,1), (3,0) } | 3 | 1, 1, 1 |
| 15 | { (0,1), (0,0), (3,0) } | 3 | 1, 1, 3 |
| 16 | { (0,3), (0,0), (3,0) } | 3 | 3, 3, 3 |
| 17 | { (1,2), (0,0), (2,0) } | 4 | 1, 1, 2 |
| 18 | { (0,1), (0,0), (4,0) } | 4 | 1, 1, 4 |
| 19 | { (0,2), (0,0), (2,0) } | 4 | 2, 2, 2 |
| 20 | { (0,2), (0,0), (4,0) } | 4 | 2, 2, 4 |
| 21 | { (0,4), (0,0), (4,0) } | 4 | 4, 4, 4 |
| 22 | { (0,2), (3,1), (0,0) } | 6 | 1, 1, 2 |
| 23 | { (0,1), (0,0), (6,0) } | 6 | 1, 1, 6 |
| 24 | { (0,2), (0,0), (3,0) } | 6 | 1, 2, 3 |
| 25 | { (0,3), (0,0), (6,0) } | 6 | 3, 3, 6 |
The representants, of course, are not selected with respect to musical beauty.
Under a non-invertible transformation, the volume is multiplied with the determinant of the matrix. Note that non-invertible transformations can map two tones into one, so the result can also be a motif with 2 or even 1 element. On top of the hierarchy stands class 12. If we look at the shape of the example (the origin and the base of the module) it becomes obvious that every motif can be obtained from it: 2 arbitrary vectors v1, v2 defining a linear transformation whose base elements are mapped to v1 and v2. This will not hold for motifs with more than 3 elements. (The complete Hasse diagram for motifs with 3 elements is in Mazzola's book, but I don't have it available just now.)
Class 12 is also the largest of all classes.
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| © Hans Straub, 1999 | Previous: Classification, part 2 - Index - Next: Mathematics, part 1 |