Diasem
Diasem is a max-variety-3 scale with step pattern 5L 2M 2s, equivalent to the semiquartal (5L 4s) mos with two of the small steps made larger and the other two made smaller. Diasem has two rotationally non-equivalent variants, right-hand diasem LMLSLMLSL and left-hand diasem LSLMLSLML; these are mirror images. It can be tuned as a 2.3.7 subgroup JI scale or a tempered version thereof. The fact that the small step of diatonic is made smaller results in 26edo and 31edo diasem having better melodic properties than the respective diatonic scales. The scale can be generated by an alternating chain of subminor thirds and supermajor seconds. The name "diasem" is a portmanteau of "diatonic" and "semiquartal" (or "semaphore") since its step sizes are intermediate between that of diatonic (5L 2s) and semiquartal (5L 4s); it is also a pun based on the diesis, which appears as the small step in the scale in the 31edo and 36edo tunings.
| Name | Structure | Step Sizes | Graphical Representation |
|---|---|---|---|
| Semiquartal | 5L4s | 10\62, 3\62 | ├─────────┼──┼─────────┼──┼─────────┼──┼─────────┼──┼─────────┤ |
| Diasem | 5L2m2s | 10\62, 4\62, 2\62 | ├─────────┼───┼─────────┼─┼─────────┼───┼─────────┼─┼─────────┤ |
| Diatonic | 5L2s | 10\62, 6\62 | ├─────────┼─────┼─────────╫─────────┼─────┼─────────╫─────────┤ |
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.
Tunings
| Tuning | L:m:s | Good Just Approximations | other comments | Degrees of the mode LMLSLMLSL | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||
| 9/8 | 7/6 | 21/16 | 4/3 | 3/2 | 14/9 | 7/4 | 16/9 | ||||
| JI | 7.479:2.309:1 | Just 7/6, 8/7, and 3/2 | 203.910 | 266.871 | 470.781 | 498.045 | 701.955 | 764.916 | 968.826 | 996.090 | |
| 21edo | 3:2:1 | 23/16 and 39/32 | 171.429 | 285.714 | 457.143 | 514.286 | 685.714 | 800 | 971.429 | 1028.571 | |
| 26edo | 4:2:1 | 14/11, 8/7 and 11/8 | 184.615 | 276.923 | 461.538 | 507.692 | 692.308 | 784.615 | 969.231 | 1015.385 | |
| 28edo | 4:3:1 | Pental thirds | 171.429 | 300 | 471.429 | 514.286 | 685.714 | 814.286 | 985.714 | 1028.571 | |
| 30edo | 4:3:2 | cross between Mavila and Semaphore | 160 | 280 | 440 | 520 | 680 | 800 | 960 | 1040 | |
| 31edo | 5:2:1 | Pental thirds and 7/5 | 193.548 | 270.968 | 464.516 | 503.226 | 696.774 | 774.194 | 967.742 | 1006.452 | |
| 33edo | 5:3:1 | Septimal and Neogothic thirds and 10/9 | 181.818 | 290.909 | 472.727 | 509.091 | 690.909 | 763.636 | 981.818 | 1018.182 | |
| 35edo | 5:3:2
5:4:1 |
Uses 21/16 as inconsistent 4/3 | 171.429 | 274.286
308.571 |
445.714
480 |
514.286 | 685.714 | 788.571
822.857 |
960
994.286 |
1028.571 | |
| 36edo | 6:2:1 | Septimal thirds and 3/2 | 200.000 | 266.667 | 466.667 | 500.000 | 700.000 | 766.667 | 966.667 | 1000.000 | |
| 37edo | 5:4:2 | 35/32 | cross between Mavila and Semaphore | 162.162 | 291.892 | 454.054 | 518.919 | 681.081 | 810.811 | 972.973 | 1037.838 |
| 38edo | 6:3:1 | 189.474 | [[1]] | 473.684 | 505.263 | 694.737 | 789.474 | 978.947 | 1010.526 | ||
| 39edo | 5:4:3 | cross between Mavila and Semaphore | 153.846 | 276.923 | 430.769 | 523.077 | 676.923 | 800 | 953.846 | 1046.154 | |
| 40edo | 6:3:2
6:4:1 |
Uses 21/16 as inconsistent 4/3 | 180 | 270
300 |
450
480 |
510 | 690 | 780
810 |
960
990 |
1020 | |
| 41edo | 7:2:1 | 204.878 | 263.415 | 468.293 | 497.561 | 702.439 | 760.976 | 965.854 | 995.122 | ||
| 42edo | 6:5:1 | Uses 21/16 as inconsistent 4/3 | 171.429 | 314.286 | 485.714 | 514.286 | 685.714 | 828.571 | 1000 | 1028.571 | |
| 43edo | 7:3:1 | 195.349 | 279.07 | 474.419 | 502.326 | 697.674 | 781.395 | 976.744 | 1004.651 | ||
| 44edo | 6:4:3
6:5:2 |
11/10 (and 9/7) | cross between Mavila and Semaphore | 163.636 | 272.727
300 |
436.364
463.636 |
518.182 | 681.818 | 790.909
818.182 |
[[2]]
981.818 |
1036.364 |
| 45edo | 7:3:2
7:4:1 |
186.667 | 266.667
293.333 |
453.333
480 |
506.667 | 693.333 | 773.333
800 |
960
986.667 |
1013.333 | ||
| 46edo | 6:5:3
8:2:1 |
Neogothic thirds | cross between Mavila and Semaphore
Gentle fifth |
156.522
208.696 |
[[3]]
260.87 |
443.478
469.565 |
521.739
495.652 |
678.231
704.348 |
808.696
756.522 |
965.218 | 1043.418
991.314 |
| 47edo | 7:4:2
7:5:1 |
Uses 21/16 as inconsistent 4/3 | 178.723 | 280.851
306.383 |
459.578
485.106 |
510.638 | 689.362 | 791.489
817.021 |
970.212
995.744 |
1021.27h | |
| 48edo | 6:5:4
8:3:1 |
cross between Mavila and Semaphore | 150
200 |
275 | 425
475 |
525
500 |
675
700 |
800
775 |
950
975 |
1050
1000 | |
| 49edo | 7:4:3
7:5:2 7:6:1 |
Uses 21/16 as inconsistent 4/3 | 171.429 | 269.388
293.878 318.367 |
440.817
465.756 489.796 |
514.286 | 685.714 | [[4]]
808.163 832.653 |
955.102
979.592 1004.082 |
1028.571 | |
| 50edo | 8:3:2
8:4:1 |
192 | 264
288 |
456
480 |
504 | 696 | 768
792 |
960
984 |
1008 | ||
| 51edo | 7:5:3
7:6:2 |
cross between Mavila and Semaphore | 164.706 | 282.353
305.882 |
447.059
470.588 |
517.647 | 682.353 | 800
823.529 |
964.706
988.235 |
1035.294 | |
| 52edo | 8:5:1 | Uses 21/16 as inconsistent 4/3 | 184.615 | 300 | 484.615 | 507.692 | 692.308 | 807.692 | 992.308 | 1015.385 | |
| 53edo | 7:5:4
7:6:3 |
27/20 | cross between Mavila and Semaphore | 158.491 | 271.698
294.34 |
429.189
452.831 |
520.755 | 679.245 | 792.453
815.094 |
950.944
973.585 |
1041.509 |
| 54edo | 8:4:3
8:5:2 8:6:1 |
Septimal thirds
Neogothic thirds |
Uses 21/16 as inconsistent 4/3 | 177.778 | 266.667
288.889 311.111 |
444.444
466.667 488.889 |
511.111 | 688.889 | 777.778
800 822.222 |
955.556
977.778 1000 |
1022.222 |
| 55edo | 7:6:4 | cross between Mavila and Semaphore | 152.727 | 283.636 | 436.364 | 523.636 | 676.364 | 807.273 | 960 | 1047.273 | |
| 56edo | 8:5:3
8:7:1 |
Golden tuning
Uses 21/16 as inconsistent 4/3 |
171.429 | 278.571
321.429 |
450
492.857 |
514.286 | 685.714 | 792.857
814.286 |
964.286
985.714 |
1028.571 | |
| 57edo | 7:6:5 | cross between Mavila and Semaphore | 147.368 | 273.684 | 421.053 | 526.316 | 673.684 | 800 | 947.368 | 1052.684 | |
| 58edo | 8:5:4
8:6:3 8:7:2 |
(Septimal and) Neogothic thirds | cross between Mavila and Semaphore | 165.517 | [[5]]
290.394 311.084 |
435.222
455.911 476.601 |
517.98 | 682.02 | 786.946
[[6]] 827.586 |
952.463
973.153 993.842 |
1034.483 |
| 60edo | 8:7:3 | cross between Mavila and Semaphore | 160 | 300 | 460 | 520 | 680 | 820 | 980 | 1040 | |
| 62edo | 8:7:4 | Neogothic thirds | cross between Mavila and Semaphore | 154.839 | 290.323 | 445.161 | 522.581 | 677.419 | 812.903 | 967.742 | 1045.161 |
| 64edo | 8:7:5 | cross between Mavila and Semaphore | 150 | 281.25 | 431.25 | 525 | 675 | 806.25 | 956.25 | 1050 | |
| 66edo | 8:7:6 | Neogothic thirds | cross between Mavila and Semaphore | [[7]] | 272.727 | 418.182 | 527.273 | 672.727 | 800 | [[8]] | 1054.5455 |
Links
- Play JI diasem - Sevish Scale Workshop
- Play 26edo diasem - Sevish Scale Workshop