Diasem

Diasem is a max-variety-3 scale with step pattern 5L 2M 2s (with two rotationally non-equivalent variants LMLSLMLSL or LSLMLSLML; these are mirror images) that is equivalent to the semiquartal (5L 4s) mos with two of the small steps made larger and the other two made smaller. It can be tuned as a 7-limit (specifically 2.3.7 subgroup) JI scale or a tempered version thereof. This results in better melodic properties than the diatonic 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 "semiquartal" 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.

Comparison with semiquartal and diatonic in 62edo

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

Common Diasem Tunings

Tuning	L:m:s	Good Just Approximations	other comments	Degrees
				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 and 8/7		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	771.429	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	284.2105	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	954.5455 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	286.9565 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	783.6735 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	268.9655 290.394 311.084	435.222 455.911 476.601	517.98	682.02	786.946 807.6355 828.325	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	145.4545	272.727	418.182	527.273	672.727	800	945.4545	1054.5455

Links

• Play JI diasem - Sevish Scale Workshop
• Play 26edo diasem - Sevish Scale Workshop