Diasem
Diasem is a max-variety-3 MODMOS that is equivalent to semaphore[9] with two of the small steps made larger and the other two made smaller. This results in near-just septimal intervals and better melodic properties than the meantone scales of 26edo and 31edo, which both support it. The scale can be generated by an alternating chain of subminor thirds and supermajor seconds. The name "diasem" is a portmanteau of "diatonic" and "semaphore" since it the way it tempers the 64/63 is intermediate between superpyth and semaphore; it is also a pun based on the diesis, a defining step size in the scale.
| Name | Structure | Step Sizes | Graphical Representation |
|---|---|---|---|
| Semaphore | 5L4s | 10\62, 3\62 | ├─────────┼──┼─────────┼──┼─────────┼──┼─────────┼──┼─────────┤ |
| Diasem | 5L2m2s | 10\62, 4\62, 2\62 | ├─────────┼───┼─────────┼─┼─────────┼───┼─────────┼─┼─────────┤ |
| Meantone | 5L2s | 10\62, 6\62 | ├─────────┼─────┼─────────╫─────────┼─────┼─────────╫─────────┤ |
| Tuning | L:m:s | Good Just Approximations | Degrees | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26edo | 4:2:1 | Neogothic thirds and 8/7 | 0.000 | 184.615 | 276.923 | 461.538 | 507.692 | 692.308 | 784.615 | 969.231 | 1015.385 | 1200.000 |
| 31edo | 5:2:1 | Pental thirds and 7/5 | 0.000 | 193.548 | 270.968 | 464.516 | 503.226 | 696.774 | 774.194 | 967.742 | 1006.452 | 1200.000 |
| 36edo | 6:2:1 | Septimal thirds and 3/2 | 0.000 | 200.000 | 266.667 | 466.667 | 500.000 | 700.000 | 766.667 | 966.667 | 1000.000 | 1200.000 |
| JI | 7.479:2.309:1 | Just 7/6, 8/7, and 3/2 | 0.000 | 203.910 | 266.871 | 470.781 | 498.045 | 701.955 | 764.916 | 968.826 | 996.090 | 1200.000 |
| 1/1 | 9/8 | 7/6 | 21/16 | 4/3 | 3/2 | 14/9 | 7/4 | 16/9 | 2/1 |
Like superpyth, diasem is great for diatonic melodies in the 2.3.7 subgroup; however, it does not temper 64/63, adding two diesis-sized steps to what would normally be a diatonic scale. Not tempering 64/63 is actually quite useful, because it's the difference between only two 4/3 and a 7/4, so the error is spread over just two perfect fourths, unlike the syntonic comma where the error is spread out over four perfect fifths. As a result, the results of tempering out 81/80 are not as bad, because each fifth only needs to be bent by about half as much to achieve the same optimization for the 5-limit. So in the case of 2.3.7, it may actually be worth it to accept the addition of small step sizes in order to improve tuning accuracy. Another advantage of detempering the septimal comma is that it allows one to use both 9/8 and 8/7, as well as 21/16 and 4/3, in the same scale. Semaphore in a sense does the opposite of what superpyth does, exaggerating 64/63 to the point that 21/16 is no longer recognizable, and the small steps of diasem become equal to the medium steps.