Maximum variety/MOS summing
Scales with three step sizes are also derivable by summing MOS scales together. All 5-note and 7-note scales derived this way will have max variety 3.
| 2L 3s | 3L 2s | 4L 1s | |
|---|---|---|---|
| 1L 4s | 4 2 3 2 2
1L 1m 3s |
4 3 3 2 2
1L 2m 2s |
4 3 3 3 2
1L 3m 1s |
| 2L 3s | 2L 3s | 4 3 3 3 2
1L 3m 1s |
4 3 4 3 2
2L 2m 1s |
| 3L 2s | 3L 2s | 4 4 3 4 2
3L 1m 1s |
| 2L 5s | 3L 4s | 4L 3s | 5L 2s | 6L 1s | |
|---|---|---|---|---|---|
| 1L 6s | 4 2 2 3 2 2 2
1L 1m 5s |
4 2 2 3 2 3 2
1L 2m 4s |
4 3 2 3 2 3 2
1L 3m 3s |
4 3 2 3 3 3 2
1L 4m 2s |
4 3 3 3 3 3 2
1L 5m 1s |
| 2L 5s | 2L 5s | 4 2 3 3 3 2 2 | 4 3 2 4 3 3 2
2L 3m 2s |
4 3 2 4 3 3 2
2L 3m 2s |
4 3 3 4 3 3 2
2L 4m 1s |
| 3L 4s | 3L 4s | 4 3 2 3 2 3 2 | 4 3 2 4 3 4 2
3L 2m 2s, Nicetone |
4 3 3 4 3 4 2
3L 3m 1s | |
| 4L 3s | 4L 3s | 4 4 2 4 3 4 2
4L 2m 1s |
4 4 3 4 2 4 2
4L 2m 1s, Mohajira | ||
| 5L 2s | 5L 2s | 4 4 3 4 4 4 2
5L 1m 1s |
However, if a scale derived this way has any other number of notes, its three step sizes will not always produce max variety 3.
| 2L 4s | 3L 3s | 4L 2s | 5L 1s | |
|---|---|---|---|---|
| 1L 5s | 4 2 2 3 2 2
1L 1m 4s |
4 2 3 2 3 2
1L 2m 3s fails, max variety 4 |
4 3 2 3 2 2
1L 2m 3s fails, max variety 4 |
4 3 3 3 3 2
1L 4m 1s |
| 2L 4s | 2L 4s | 4 2 3 3 3 2
1L 3m 2s fails, max variety 4 |
4 3 2 4 3 2
2L 2m 2s |
4 3 3 4 3 2
2L 3m 1s fails, max variety 4 |
| 3L 3s | 3L 3s | 4 3 3 4 3 2
2L 3m 1s fails, max variety 4 |
4 3 4 3 4 2
3L 2m 1s fails, max variety 4 | |
| 4L 2s | 4L 2s | 4 4 3 4 4 2
4L 1m 1s |
Since Indonesian gamelan music stems from a tradition with 9-note scales converging from max variety 7 to equal, sums of 9-note MOS make for great compromise gamelan tunings when they fail to produce max variety 3.
| 2L 7s | 3L 6s | 4L 5s | 5L 4s | 6L 3s | 7L 2s | 8L 1s | |
|---|---|---|---|---|---|---|---|
| 1L 8s | 4 2 2 2 3 2 2 2 2
1L 1m 7s |
4 2 2 3 2 2 3 2 2
1L 2m 6s fails, max variety 4 |
4 2 2 3 2 3 2 3 2
1L 3m 5s fails, max variety 4 |
4 3 2 3 2 3 2 3 2
1L 4m 4s |
4 4 2 3 2 2 3 2 2
2L 2m 5s |
4 3 3 2 3 3 3 3 2
1L 6m 2s fails, max variety 4 |
4 3 3 3 3 3 3 3 2
1L 7m 1s |
| 2L 7s | 2L 7s | 4 2 2 3 3 2 3 2 2
1L 3m 5s fails, max variety 4 |
4 2 2 3 3 3 2 3 2
1L 4m 4s |
4 3 2 2 4 3 2 2 2
2L 2m 5s |
4 3 2 3 3 2 3 3 2
1L 5m 3s fails, max variety 4 |
4 3 3 2 4 3 3 3 2
2L 5m 2s |
4 3 3 4 4 3 3 3 2
1L 6m 2s fails, max variety 4 |
| 3L 6s | 3L 6s | 4 2 2 4 3 2 3 3 2
2L 3m 4s fails, max variety 4 |
4 3 2 4 2 3 3 3 2
2L 5m 2s |
4 3 2 4 3 2 4 3 2
3L 3m 3s |
4 3 3 3 3 3 3 4 2
2L 6m 1s fails, max variety 4 |
4 3 3 4 3 3 4 3 2
1L 5m 3s fails, max variety 4 | |
| 4L 5s | 4L 5s | 4 3 2 4 2 4 2 4 2
4L 1m 4s |
4 3 2 3 3 2 3 4 2
2L 4m 3s fails, max variety 4 |
4 4 3 3 3 4 3 4 2
3L 5m 1s fails, max variety 4 |
4 3 3 4 3 4 3 4 2
4L 4m 1s | ||
| 5L 4s | 5L 4s | 4 4 2 4 3 3 3 4 4 2
5L 2m 2s, diasem |
4 4 3 3 3 4 3 4 2
4L 4m 1s |
4 4 3 4 3 4 3 4 2
5L 3m 1s fails, max variety 4 | |||
| 6L 3s | 6L 3s | 4 4 3 4 3 3 4 4 2
5L 3m 1s fails, max variety 4 |
4 4 3 4 4 3 4 4 2
6L 2m 1s fails, max variety 4 | ||||
| 7L 2s | 7L 2s | 4 4 4 3 4 4 4 4 2
7L 1m 1s |
As the 6-note and 9-note addition tables prove, the requirement of having the same number of steps of different sizes makes max variety 3 scales with three step sizes rare.