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Parallelogram-shaped (parallelogram substring with full first and last rows) [[MOS substitution|MOS-substitution]] scales for scale sizes from 6 to 36. Only axbycz for which [a, b, c] is a partition of the scale size are shown.
Parallelogram-shaped (parallelogram substring with full first and last rows) scales for scale sizes from 6 to 36. Only axbycz for which a > b > c > 0 are shown.


It is conjectured that fully parallelogram scales are always MOS-substitution. This has been checked for sizes <= 12.
Ternary parallelogram scales are always MOS substitution. (Proof forthcoming)
== 6 note scales ==
== 6 notes ==
* xxyxxz row_count: 2 full_row_len: 3, lattice basis: gx: [2, 0, 1], gy: [3, 1, 1]
* xyxyxz row_count: 2 full_row_len: 3, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
== 8 note scales ==
* xxyxxz row_count: 2 full_row_len: 3, lattice basis: gx: [3, 1, 3], gy: [4, 2, 3]
* xyxyxyxz row_count: 2 full_row_len: 4, lattice basis: gx: [1, 0, 0], gy: [3, 2, 1]
== 8 notes ==
* xxxyxxxz row_count: 2 full_row_len: 4, lattice basis: gx: [3, 0, 1], gy: [5, 1, 1]
* xyxzyxyz row_count: 2 full_row_len: 4, lattice basis: gx: [4, 0, 2], gy: [4, 1, 2]
== 9 note scales ==
* xyxyxyxz row_count: 2 full_row_len: 4, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
* xxyxxyxxz row_count: 3 full_row_len: 3, lattice basis: gx: [5, 2, 1], gy: [4, 1, 1]
* xxxyxxxz row_count: 2 full_row_len: 4, lattice basis: gx: [2, 1, 3], gy: [4, 2, 3]
== 10 note scales ==
== 9 notes ==
* xyxzyxzxyz row_count: 2 full_row_len: 5, lattice basis: gx: [2, 2, 1], gy: [1, 1, 1]
* xxyzxyxzy row_count: 3 full_row_len: 3, lattice basis: gx: [1, 1, 1], gy: [2, 1, 1]
* xyxyxyxyxz row_count: 5 full_row_len: 2, lattice basis: gx: [4, 3, 1], gy: [1, 0, 0]
* xxyxxyxxz row_count: 3 full_row_len: 3, lattice basis: gx: [3, 1, 3], gy: [4, 2, 3]
* xxxxyxxxxz row_count: 2 full_row_len: 5, lattice basis: gx: [4, 0, 1], gy: [7, 1, 1]
== 10 notes ==
== 12 note scales ==
* xyxzyxzxyz row_count: 2 full_row_len: 5, lattice basis: gx: [3, 1, 1], gy: [1, 1, 1]
* xyxyxyxyxyxz row_count: 6 full_row_len: 2, lattice basis: gx: [5, 4, 1], gy: [1, 0, 0]
* xyxyxzxyxz row_count: 2 full_row_len: 5, lattice basis: gx: [4, 2, 3], gy: [3, 1, 2]
* xxyxxyxxyxxz row_count: 4 full_row_len: 3, lattice basis: gx: [6, 2, 1], gy: [7, 3, 1]
* xyxyxyxyxz row_count: 2 full_row_len: 5, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
* xxxyxxxyxxxz row_count: 4 full_row_len: 3, lattice basis: gx: [8, 2, 1], gy: [6, 1, 1]
* xxxxyxxxxz row_count: 2 full_row_len: 5, lattice basis: gx: [1, 1, 3], gy: [4, 2, 3]
* xxxxxyxxxxxz row_count: 2 full_row_len: 6, lattice basis: gx: [5, 0, 1], gy: [9, 1, 1]
== 12 notes ==
== 14 note scales ==
* xyxyxzyxyxyz row_count: 2 full_row_len: 6, lattice basis: gx: [3, -1, 2], gy: [3, 0, 2]
* xyxzyxzyxyzxyz row_count: 7 full_row_len: 2, lattice basis: gx: [4, 4, 3], gy: [2, 3, 2]
* xyxyxyxyxyxz row_count: 2 full_row_len: 6, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
* xyxyxzxyxzxyxz row_count: 7 full_row_len: 2, lattice basis: gx: [2, 1, 1], gy: [6, 4, 3]
* xxyxxyxxyxxz row_count: 4 full_row_len: 3, lattice basis: gx: [3, 1, 3], gy: [4, 2, 3]
* xyxyxyxyxyxyxz row_count: 7 full_row_len: 2, lattice basis: gx: [6, 5, 1], gy: [1, 0, 0]
* xxxyxxxyxxxz row_count: 3 full_row_len: 4, lattice basis: gx: [2, 1, 3], gy: [4, 2, 3]
* xxxxxyxxxxxz row_count: 2 full_row_len: 6, lattice basis: gx: [0, 1, 3], gy: [4, 2, 3]
== 14 notes ==
* xyxzyxzyxyzxyz row_count: 2 full_row_len: 7, lattice basis: gx: [3, -1, 1], gy: [4, 1, 2]
* xyxyxzxyxzxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [4, 2, 3], gy: [3, 1, 2]
* xyxyxyxzxyxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [4, 2, 3], gy: [2, 0, 2]
* xyxyxyxyxyxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [1, 0, 0], gy: [4, 1, 3]
* xxyxzxyxxzxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [4, 2, 1], gy: [3, 1, 1]
* xxyxzxyxxzxyxz row_count: 2 full_row_len: 7, lattice basis: gx: [4, 2, 1], gy: [3, 1, 1]
* xxxxxxyxxxxxxz row_count: 2 full_row_len: 7, lattice basis: gx: [6, 0, 1], gy: [11, 1, 1]
* xxxxxxyxxxxxxz row_count: 2 full_row_len: 7, lattice basis: gx: [-1, 1, 3], gy: [4, 2, 3]
== 15 note scales ==
== 15 notes ==
* xxyxyxxzxyxxyxz row_count: 3 full_row_len: 5, lattice basis: gx: [6, 3, 1], gy: [4, 2, 1]
* xxyzxyzxyxzyxzy row_count: 3 full_row_len: 5, lattice basis: gx: [3, 2, 2], gy: [1, 1, 1]
* xxyxxyxxzxxyxxz row_count: 3 full_row_len: 5, lattice basis: gx: [1, 0, 0], gy: [6, 2, 1]
* xxyxzxyxxzyxxyz row_count: 3 full_row_len: 5, lattice basis: gx: [3, 1, 1], gy: [2, 1, 1]
* xxyxxyxxyxxyxxz row_count: 5 full_row_len: 3, lattice basis: gx: [8, 3, 1], gy: [9, 4, 1]
* xxyxxyzxyxxyxzy row_count: 3 full_row_len: 5, lattice basis: gx: [4, 2, 1], gy: [3, 2, 1]
* xxxxyxxxxyxxxxz row_count: 5 full_row_len: 3, lattice basis: gx: [11, 2, 1], gy: [8, 1, 1]
* xxyxyxxzxyxxyxz row_count: 3 full_row_len: 5, lattice basis: gx: [2, 1, 2], gy: [4, 2, 1]
== 16 note scales ==
* xxyxxyxxzxxyxxz row_count: 3 full_row_len: 5, lattice basis: gx: [1, 0, 0], gy: [1, 1, 2]
* xyxzyxzyxzxyzxyz row_count: 8 full_row_len: 2, lattice
* xxyxxyxxyxxyxxz row_count: 5 full_row_len: 3, lattice basis: gx: [3, 1, 3], gy: [4, 2, 3]
* xxxxyxxxxyxxxxz row_count: 3 full_row_len: 5, lattice basis: gx: [1, 1,
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