User:Xenoindex/MOS Production Rules

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MOSes have a recursive structure so they have rules to produce them.

I'm not really sure how to explain what I've done here with words so I'll just show an example worked out by hand.


Lets figure out the MOS Production Rules for the 4L3s MOS scale LLsLsLs.

We begin by chunking also known as dividing the scale into segments called chunks that each contain exactly one s on the right-most end.

Start chunking from the brightest mode to avoid having to wrap around.


LLsLsLs chunks into the three chunks LLs, Ls, and Ls.

The largest chunk LLs will be replaced with L and the smallest chunk Ls will be replaced with s.

Note the length of the largest chunk LLs is 3, add it to the beginning of a list: [3]


LLs -> L

Ls -> s

LLsLsLs becomes Lss


Lss chunks into two chunks Ls and s.

The largest chunk Ls will be replaced with L and the smallest chunk s will be replaced with s.

Note the length of the largest chunk is 2, add it to the beginning of the list. [2, 3]


Ls -> L

s -> s

Lss becomes Ls


Ls chunks into only one chunk so the process is complete.

Note the length of the final chunk is 2, add it to the beginning of the list. [2, 2, 3]

[2, 2, 3] are the production rules for this MOS (LLsLsLs).

In a less compact form the production rules [2, 2, 3] would be something like:


axiom: Ls

iteration 1 rules: (L -> Ls), (s -> s)

iteration 2 rules: (L -> LLs), (s -> Ls)


But that isn't as nice and compact as a simple list of numbers.


Now to use the production rules [2, 2, 3] to generate the MOS.


The first element from the rules [2, 2, 3] is 2 which is the axiom.

Start with a chunk size of 2: Ls


The next element from the rules [2, 2, 3] is 2 which is the size of chunk to replace each L by.

2 means each L is to be replaced with a chunk of size 2 or the replacement rule: L -> Ls

The replacement rule for s is always one smaller than the one for L. (This feels like a discovery)

2-1 is 1 so that means each s is to be replaced with a chunk of size 1 or the replacement rule: s -> s


L -> Ls

s -> s

Ls becomes Lss


The next and final element from the rules [2, 2, 3] is 3 which is the size of chunk to replace each L by.

3 means each L is to be replaced with a chunk of size 3 or the replacement rule: L -> LLs

3-1 is 2 so that means each s is to be replaced with a chunk of size 2 or the replacement rule: s -> Ls


L -> LLs

s -> Ls

Lss becomes LLsLsLs


The end of the production rules has been reached so the process is complete.

The MOS scale of the MOS production rules [2, 2, 3] is: LLsLsLs


Below is a table of the production rules of all MOSes up to 19 notes.

Red rows are Multi-MOSes.


MOS len MOS Production Rules
2 Notes
1L 1s 1 [ 2 ]
3 Notes
1L 2s 2 [ 2, 2 ]
2L 1s 1 [ 3 ]
4 Notes
1L 3s 3 [ 2, 2, 2 ]
2L 2s 1 [ 2 ]
3L 1s 1 [ 4 ]
5 Notes
1L 4s 4 [ 2, 2, 2, 2 ]
2L 3s 2 [ 3, 2 ]
3L 2s 2 [ 2, 3 ]
4L 1s 1 [ 5 ]
6 Notes
1L 5s 5 [ 2, 2, 2, 2, 2 ]
2L 4s 2 [ 2, 2 ]
3L 3s 1 [ 2 ]
4L 2s 1 [ 3 ]
5L 1s 1 [ 6 ]
7 Notes
1L 6s 6 [ 2, 2, 2, 2, 2, 2 ]
2L 5s 3 [ 3, 2, 2 ]
3L 4s 2 [ 4, 2 ]
4L 3s 3 [ 2, 2, 3 ]
5L 2s 2 [ 2, 4 ]
6L 1s 1 [ 7 ]
8 Notes
1L 7s 7 [ 2, 2, 2, 2, 2, 2, 2 ]
2L 6s 3 [ 2, 2, 2 ]
3L 5s 3 [ 2, 3, 2 ]
4L 4s 1 [ 2 ]
5L 3s 2 [ 3, 3 ]
6L 2s 1 [ 4 ]
7L 1s 1 [ 8 ]
9 Notes
1L 8s 8 [ 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 7s 4 [ 3, 2, 2, 2 ]
3L 6s 2 [ 2, 2 ]
4L 5s 2 [ 5, 2 ]
5L 4s 4 [ 2, 2, 2, 3 ]
6L 3s 1 [ 3 ]
7L 2s 2 [ 2, 5 ]
8L 1s 1 [ 9 ]
10 Notes
1L 9s 9 [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 8s 4 [ 2, 2, 2, 2 ]
3L 7s 3 [ 4, 2, 2 ]
4L 6s 2 [ 3, 2 ]
5L 5s 1 [ 2 ]
6L 4s 2 [ 2, 3 ]
7L 3s 3 [ 2, 2, 4 ]
8L 2s 1 [ 5 ]
9L 1s 1 [ 10 ]
11 Notes
1L 10s 10 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 9s 5 [ 3, 2, 2, 2, 2 ]
3L 8s 4 [ 2, 3, 2, 2 ]
4L 7s 4 [ 2, 2, 3, 2 ]
5L 6s 2 [ 6, 2 ]
6L 5s 5 [ 2, 2, 2, 2, 3 ]
7L 4s 2 [ 4, 3 ]
8L 3s 2 [ 3, 4 ]
9L 2s 2 [ 2, 6 ]
10L 1s 1 [ 11 ]
12 Notes
1L 11s 11 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 10s 5 [ 2, 2, 2, 2, 2 ]
3L 9s 3 [ 2, 2, 2 ]
4L 8s 2 [ 2, 2 ]
5L 7s 3 [ 2, 4, 2 ]
6L 6s 1 [ 2 ]
7L 5s 3 [ 3, 2, 3 ]
8L 4s 1 [ 3 ]
9L 3s 1 [ 4 ]
10L 2s 1 [ 6 ]
11L 1s 1 [ 12 ]
13 Notes
1L 12s 12 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 11s 6 [ 3, 2, 2, 2, 2, 2 ]
3L 10s 4 [ 4, 2, 2, 2 ]
4L 9s 3 [ 5, 2, 2 ]
5L 8s 3 [ 3, 3, 2 ]
6L 7s 2 [ 7, 2 ]
7L 6s 6 [ 2, 2, 2, 2, 2, 3 ]
8L 5s 3 [ 2, 3, 3 ]
9L 4s 4 [ 2, 2, 2, 4 ]
10L 3s 3 [ 2, 2, 5 ]
11L 2s 2 [ 2, 7 ]
12L 1s 1 [ 13 ]
14 Notes
1L 13s 13 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 12s 6 [ 2, 2, 2, 2, 2, 2 ]
3L 11s 5 [ 2, 3, 2, 2, 2 ]
4L 10s 3 [ 3, 2, 2 ]
5L 9s 5 [ 2, 2, 2, 3, 2 ]
6L 8s 2 [ 4, 2 ]
7L 7s 1 [ 2 ]
8L 6s 3 [ 2, 2, 3 ]
9L 5s 2 [ 5, 3 ]
10L 4s 2 [ 2, 4 ]
11L 3s 2 [ 3, 5 ]
12L 2s 1 [ 7 ]
13L 1s 1 [ 14 ]
15 Notes
1L 14s 14 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 13s 7 [ 3, 2, 2, 2, 2, 2, 2 ]
3L 12s 4 [ 2, 2, 2, 2 ]
4L 11s 5 [ 2, 2, 3, 2, 2 ]
5L 10s 2 [ 2, 2 ]
6L 9s 2 [ 3, 2 ]
7L 8s 2 [ 8, 2 ]
8L 7s 7 [ 2, 2, 2, 2, 2, 2, 3 ]
9L 6s 2 [ 2, 3 ]
10L 5s 1 [ 3 ]
11L 4s 2 [ 4, 4 ]
12L 3s 1 [ 5 ]
13L 2s 2 [ 2, 8 ]
14L 1s 1 [ 15 ]
16 Notes
1L 15s 15 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 14s 7 [ 2, 2, 2, 2, 2, 2, 2 ]
3L 13s 5 [ 4, 2, 2, 2, 2 ]
4L 12s 3 [ 2, 2, 2 ]
5L 11s 3 [ 6, 2, 2 ]
6L 10s 3 [ 2, 3, 2 ]
7L 9s 3 [ 2, 5, 2 ]
8L 8s 1 [ 2 ]
9L 7s 4 [ 3, 2, 2, 3 ]
10L 6s 2 [ 3, 3 ]
11L 5s 5 [ 2, 2, 2, 2, 4 ]
12L 4s 1 [ 4 ]
13L 3s 3 [ 2, 2, 6 ]
14L 2s 1 [ 8 ]
15L 1s 1 [ 16 ]
17 Notes
1L 16s 16 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 15s 8 [ 3, 2, 2, 2, 2, 2, 2, 2 ]
3L 14s 6 [ 2, 3, 2, 2, 2, 2 ]
4L 13s 4 [ 5, 2, 2, 2 ]
5L 12s 4 [ 2, 4, 2, 2 ]
6L 11s 6 [ 2, 2, 2, 2, 3, 2 ]
7L 10s 4 [ 2, 2, 4, 2 ]
8L 9s 2 [ 9, 2 ]
9L 8s 8 [ 2, 2, 2, 2, 2, 2, 2, 3 ]
10L 7s 3 [ 4, 2, 3 ]
11L 6s 2 [ 6, 3 ]
12L 5s 3 [ 3, 2, 4 ]
13L 4s 4 [ 2, 2, 2, 5 ]
14L 3s 2 [ 3, 6 ]
15L 2s 2 [ 2, 9 ]
16L 1s 1 [ 17 ]
18 Notes
1L 17s 17 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 16s 8 [ 2, 2, 2, 2, 2, 2, 2, 2 ]
3L 15s 5 [ 2, 2, 2, 2, 2 ]
4L 14s 4 [ 3, 2, 2, 2 ]
5L 13s 4 [ 3, 3, 2, 2 ]
6L 12s 2 [ 2, 2 ]
7L 11s 3 [ 4, 3, 2 ]
8L 10s 2 [ 5, 2 ]
9L 9s 1 [ 2 ]
10L 8s 4 [ 2, 2, 2, 3 ]
11L 7s 4 [ 2, 2, 3, 3 ]
12L 6s 1 [ 3 ]
13L 5s 3 [ 2, 3, 4 ]
14L 4s 2 [ 2, 5 ]
15L 3s 1 [ 6 ]
16L 2s 1 [ 9 ]
17L 1s 1 [ 18 ]
19 Notes
1L 18s 18 [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
2L 17s 9 [ 3, 2, 2, 2, 2, 2, 2, 2, 2 ]
3L 16s 6 [ 4, 2, 2, 2, 2, 2 ]
4L 15s 6 [ 2, 2, 3, 2, 2, 2 ]
5L 14s 6 [ 2, 2, 2, 3, 2, 2 ]
6L 13s 3 [ 7, 2, 2 ]
7L 12s 4 [ 3, 2, 3, 2 ]
8L 11s 3 [ 3, 4, 2 ]
9L 10s 2 [ 10, 2 ]
10L 9s 9 [ 2, 2, 2, 2, 2, 2, 2, 2, 3 ]
11L 8s 4 [ 2, 3, 2, 3 ]
12L 7s 3 [ 2, 4, 3 ]
13L 6s 6 [ 2, 2, 2, 2, 2, 4 ]
14L 5s 2 [ 5, 4 ]
15L 4s 2 [ 4, 5 ]
16L 3s 3 [ 2, 2, 7 ]
17L 2s 2 [ 2, 10 ]
18L 1s 1 [ 19 ]
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