Diasem
Diasem is a 9-note max-variety-3, AG 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-handed diasem LMLSLMLSL and left-handed diasem LSLMLSLML; these are mirror images. It can be tuned as a 2.3.7 subgroup JI scale or a tempered version thereof, where L represents 9/8, M represents 28/27, and S represents 64/63. 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. 21edo is the smallest edo to non-trivially support diasem.
"Diasem" is a name given by ks26. The name 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 | ├─────────┼─────┼─────────╫─────────┼─────┼─────────╫─────────┤ |
Intervals
The following is a table of diasem intervals and their abstract sizes in terms of L, m and s. Given concrete sizes of L, m and s in edo steps or cents, you can compute the concrete size of any interval in diasem using the following expressions.
| Interval class | Size of large k-step | Size of medium k-step | Size of small k-step |
|---|---|---|---|
| 1-step | L | m | s |
| 2-step | 2L | L + m | L + s |
| 3-step | 2L + m | 2L + s | L + m + s |
| 4-step | 3L + m | 3L + s | 2L + m + s |
| 5-step | 3L + m + s | 2L + 2m + s | 3L + m + 2s |
| 6-step | 4L + m + s | 3L + 2m + s | 3L + m + 2s |
| 7-step | 3L + 2m + 2s | 4L + 2m + s | 4L + m + 2s |
| 8-step | 5L + 2m + s | 5L + m + 2s | 4L + 2m + 2s |
Interval sizes in one example tuning:
| Interval class | Size of large k-step | Size of medium k-step | Size of small k-step |
|---|---|---|---|
| 1-step | 5\31, 193.55¢ | 2\31, 77.42¢ | 1\31, 38.71¢ |
| 2-step | 10\31, 387.10¢ | 7\31, 270.97¢ | 6\31, 232.26¢ |
| 3-step | 12\31, 464.52¢ | 11\31, 425.81¢ | 8\31, 309.68¢ |
| 4-step | 17\31, 658.06¢ | 16\31, 619.35¢ | 13\31, 503.23¢ |
| 5-step | 18\31, 696.77¢ | 15\31, 580.65¢ | 14\31, 541.94¢ |
| 6-step | 23\31, 890.32¢ | 20\31, 774.19¢ | 19\31, 735.48¢ |
| 7-step | 25\31, 967.74¢ | 24\31, 929.03¢ | 21\31, 812.90¢ |
| 8-step | 30\31, 1161.29¢ | 29\31, 1122.58¢ | 26\31, 1006.45¢ |
Modes
Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. Modes with LLS are left-handed; modes with LLM are right-handed. Fourteen of these modes are versions of diatonic modes which come in LH and RH pairs; four are less diatonic-like.
Here, "brightened" (resp. "darkened") means that the diatonic scale is overall sharpened (resp. flattened) perfect 5th (resp. the perfect 4th) is missing. These may sound somewhat more like oneirotonic modes.
The modes arranged in cyclic order:
| Left-handed modes | Right-handed modes |
|---|---|
| LSLMLSLML LH Diasem Mixo |
LMLSLMLSL RH Diasem Aeolian |
| SLMLSLMLL LH Diasem Brightened Aeolian |
MLSLMLSLL RH Diasem Locrian |
| LMLSLMLLS LH Diasem Aeolian |
LSLMLSLLM RH Diasem Ionian |
| MLSLMLLSL LH Diasem Locrian |
SLMLSLLML RH Diasem Brightened Dorian |
| LSLMLLSLM LH Diasem Ionian |
LMLSLLMLS RH Diasem Dorian |
| SLMLLSLML LH Diasem Dorian |
MLSLLMLSL RH Diasem Phrygian |
| LMLLSLMLS LH Diasem Darkened Dorian |
LSLLMLSLM RH Diasem Lydian |
| MLLSLMLSL LH Diasem Phrygian |
SLLMLSLML RH Diasem Mixo |
| LLSLMLSLM LH Diasem Lydian |
LLMLSLMLS RH Diasem Darkened Mixo |
In JI and similar tunings
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. On the other hand, 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 JI approximations | other comments | Degrees of the mode LMLSLMLSL | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||
| 2.3.7 subgroup interpretation | 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 | 5/4 and 13/7 | 171.429 | 300 | 471.429 | 514.286 | 685.714 | 814.286 | 985.714 | 1028.571 | |
| 30edo | 4:3:2 | 13/8 | superdiatonic fifth | 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 | 9/7, 13/11 and 10/9 | 181.818 | 290.909 | 472.727 | 509.091 | 690.909 | 763.636 | 981.818 | 1018.182 | |
| 35edo | 5:3:2 | 171.429 | 274.286 | 445.714 | 514.286 | 685.714 | 788.571 | 960 | 1028.571 | ||
| 35edo | 5:4:1 | 171.429 | 308.571 | 480 | 514.286 | 685.714 | 822.857 | 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 | superdiatonic fifth | 162.162 | 291.892 | 454.054 | 518.919 | 681.081 | 810.811 | 972.973 | 1037.838 |
| 38edo | 6:3:1 | 189.474 | 284.2105 | 473.684 | 505.263 | 694.737 | 789.474 | 978.947 | 1010.526 | ||
| 39edo | 5:4:3 | superdiatonic fifth | 153.846 | 276.923 | 430.769 | 523.077 | 676.923 | 800 | 953.846 | 1046.154 | |
| 40edo | 6:3:2 | 180 | 270 | 450 | 510 | 690 | 780 | 960 | 1020 | ||
| 40edo | 6:4:1 | 180 | 300 | 480 | 510 | 690 | 810 | 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 | 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 | 11/10 (and 9/7) | superdiatonic fifth | 163.636 | 272.727 | 436.364 | 518.182 | 681.818 | 790.909 | 954.5455 | 1036.364 |
| 44edo | 6:5:2 | 11/10 (and 9/7) | superdiatonic fifth | 163.636 | 300 | 463.636 | 518.182 | 681.818 | 818.182 | 981.818 | 1036.364 |
| 45edo | 7:3:2 | 186.667 | 266.667 | 453.333 | 506.667 | 693.333 | 773.333 | 960 | 1013.333 | ||
| 45edo | 7:4:1 | 186.667 | 293.333 | 480 | 506.667 | 693.333 | 800 | 986.667 | 1013.333 | ||
| 46edo | 6:5:3 | Neogothic thirds | superdiatonic fifth | 156.522 | 286.9565 | 443.478 | 521.739 | 678.231 | 808.696 | 965.218 | 1043.418 |
| 46edo | 8:2:1 | Neogothic thirds | gentle fifth | 208.696 | 260.87 | 469.565 | 495.652 | 704.348 | 756.522 | 965.218 | 991.314 |
| 47edo | 7:4:2 | 178.723 | 280.851 | 459.578 | 510.638 | 689.362 | 791.489 | 970.212 | 1021.277 | ||
| 47edo | 7:5:1 | 178.723 | 306.383 | 485.106 | 510.638 | 689.362 | 817.021 | 995.744 | 1021.277 | ||
| 48edo | 6:5:4 | superdiatonic fifth | 150 | 275 | 425 | 525 | 675 | 800 | 950 | 1050 | |
| 48edo | 8:3:1 | superdiatonic fifth | 200 | 275 | 475 | 500 | 700 | 775 | 975 | 1000 | |
| 49edo | 7:4:3 | 171.429 | 269.388 | 440.817 | 514.286 | 685.714 | 783.6735 | 955.102 | 1028.571 | ||
| 49edo | 7:5:2 | 171.429 | 293.878 | 465.756 | 514.286 | 685.714 | 808.163 | 979.592 | 1028.571 | ||
| 49edo | 7:6:1 | 171.429 | 318.367 | 489.796 | 514.286 | 685.714 | 832.653 | 1004.082 | 1028.571 | ||
| 50edo | 8:3:2 | 192 | 264 | 456 | 504 | 696 | 768 | 960 | 1008 | ||
| 50edo | 8:4:1 | 192 | 288 | 480 | 504 | 696 | 792 | 984 | 1008 | ||
| 51edo | 7:5:3 | superdiatonic fifth | 164.706 | 282.353 | 447.059 | 517.647 | 682.353 | 800 | 964.706 | 1035.294 | |
| 51edo | 7:6:2 | superdiatonic fifth | 164.706 | 305.882 | 470.588 | 517.647 | 682.353 | 823.529 | 988.235 | 1035.294 | |
| 52edo | 8:5:1 | 184.615 | 300 | 484.615 | 507.692 | 692.308 | 807.692 | 992.308 | 1015.385 | ||
| 53edo | 7:5:4 | 27/20 | superdiatonic fifth | 158.491 | 271.698 | 429.189
452.831 |
520.755 | 679.245 | 792.453 | 950.944 | 1041.509 |
| 53edo | 7:6:3 | 27/20 | superdiatonic fifth | 158.491 | 294.34 | 452.831 | 520.755 | 679.245 | 815.094 | 973.585 | 1041.509 |
Tuning examples
|
An example in the RH Diasem Lydian mode LSLLMLSLM. (score) 14edo, L:M:S = 2:1:1 (degenerate; this is basic semiquartal) 16edo, L:M:S = 2:2:1 (degenerate; this is basic superdiatonic) 19edo, L:M:S = 3:1:1 (degenerate; this is hard semiquartal) 21edo, L:M:S = 3:2:1 23edo, L:M:S = 3:2:2 (degenerate; this is soft semiquartal) 24edo, L:M:S = 4:1:1 (degenerate; this is superhard semiquartal) 26edo, L:M:S = 4:2:1 28edo, L:M:S = 4:3:1 31edo, L:M:S = 5:2:1 33edo, L:M:S = 5:3:1 35edo, L:M:S = 5:4:1 35edo, L:M:S = 5:3:2 36edo, L:M:S = 6:2:1 38edo, L:M:S = 6:3:1 41edo, L:M:S = 7:2:1 |
See also
- Blackdye, a similar diatonic detempering but for 2.3.5
Links
- Play JI diasem - Sevish Scale Workshop
- Play 26edo diasem - Sevish Scale Workshop