Diasem

Diasem is a 9-note max-variety-3, generator-offset 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 is chiral, with two rotationally non-equivalent variants: right-handed diasem LMLSLMLSL and left-handed diasem LSLMLSLML; these are mirror images. 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 has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset a subminor third or a supermajor second. 21edo is the smallest edo to support a non-degenerate diasem.

Diasem 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.

"Diasem" is a name given by ks26 (though others have discussed the scale before her). 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.

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	├─────────┼─────┼─────────╫─────────┼─────┼─────────╫─────────┤

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 sizes in diasem

Interval class		Sizes	2.3.7 JI	21edo (L:m:s = 3:2:1)	31edo (L:m:s = 5:2:1)
1-step	small	s	64/63, 27.26¢	1\21, 57.14¢	1\31, 38.71¢
	medium	m	28/27, 62.96¢	2\21, 114.29¢	2\31, 77.42¢
	large	L	9/8, 203.91¢	3\21, 171.43¢	5\31, 193.55¢
2-step	small	L + s	8/7, 231.17¢	4\21, 228.57¢	6\31, 232.26¢
	medium	L + m	7/6, 266.87¢	5\21, 285.71¢	7\31, 270.97¢
	large	2L	81/64, 407.82¢	6\21, 342.86¢	10\31, 387.10¢
3-step	small	L + m + s	32/27, 294.14¢	6\21, 342.86¢	8\31, 309.68¢
	medium	2L + s	9/7, 435.08¢	7\21, 400.00¢	11\31, 425.81¢
	large	2L + m	21/16, 470.78¢	8\21, 457.14¢	12\31, 464.52¢
4-step	small	2L + m + s	4/3, 498.04¢	9\21, 514.29¢	13\31, 503.23¢
	medium	3L + s	81/56, 638.99¢	10\21, 571.43¢	16\31, 619.35¢
	large	3L + m	189/128, 674.69¢	11\21, 628.57¢	17\31, 658.06¢
5-step	small	2L + m + 2s	256/189, 525.31¢	10\21, 571.43¢	14\31, 541.94¢
	medium	2L + 2m + s	112/81, 561.01¢	11\21, 628.57¢	15\31, 580.65¢
	large	3L + m + s	3/2, 701.96¢	12\21, 685.71¢	18\31, 696.77¢
6-step	small	3L + m + 2s	32/21, 729.22¢	13\21, 742.86¢	19\31, 735.48¢
	medium	3L + 2m + s	14/9, 764.92¢	14\21, 800.00¢	20\31, 774.19¢
	large	4L + m + s	27/16, 905.87¢	15\21, 857.14¢	23\31, 890.32¢
7-step	small	3L + 2m + 2s	128/81, 792.18¢	15\21, 857.14¢	21\31, 812.90¢
	medium	4L + m + 2s	12/7, 933.13¢	16\21, 914.29¢	24\31, 929.03¢
	large	4L + 2m + s	7/4, 968.83¢	17\21, 971.43¢	25\31, 967.74¢
8-step	small	4L + 2m + 2s	16/9, 996.09¢	18\21, 1028.57¢	26\31, 1006.45¢
	medium	5L + m + 2s	54/28, 1137.04¢	19\21, 1085.71¢	29\31, 1122.58¢
	large	5L + 2m + s	63/32, 1172.74¢	20\21, 1142.86¢	30\31, 1161.29¢

The octave can be called the "perfect 9-step" in TAMNAMS.

Important properties

Any diasem scale has a fifth (large 5-step) between 4\9 (666.67¢) and 3\5 (720¢). The fifth is:

• > 4\7 if L > m + s
• = 4\7 if L = m + s
• < 4\7 if L < m + s

(This can be seen as follows: Let s' = m + s. Then the fifth generates the mos 5L 2s', which is either diatonic, 7edo or antidiatonic depending on the above conditions.)

Modes

Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. To minimize bias (e.g. towards certain tunings), this article names the modes after the semiquartal mode that the diasem mode is based on (the semiquartal modes are given in UDP notation).

The modes arranged in cyclic order:

Left-handed modes	Right-handed modes
LSLMLSLML LH Diasem 4|4 LH Diasem Mixo	LMLSLMLSL RH Diasem 4|4 RH Diasem Aeolian
SLMLSLMLL LH Diasem 0|8 LH Diasem Brightened Aeolian	MLSLMLSLL RH Diasem 0|8 RH Diasem Locrian
LMLSLMLLS LH Diasem 5|3 LH Diasem Aeolian (*)	LSLMLSLLM RH Diasem 5|3 RH Diasem Ionian
MLSLMLLSL LH Diasem 1|7 LH Diasem Locrian	SLMLSLLML RH Diasem 1|7 RH Diasem Bright Dorian
LSLMLLSLM LH Diasem 6|2 LH Diasem Ionian	LMLSLLMLS RH Diasem 6|2 RH Diasem Dark Dorian
SLMLLSLML LH Diasem 2|6 LH Diasem Bright Dorian	MLSLLMLSL RH Diasem 2|6 RH Diasem Phrygian
LMLLSLMLS LH Diasem 7|1 LH Diasem Dark Dorian	LSLLMLSLM RH Diasem 7|1 RH Diasem Lydian
MLLSLMLSL LH Diasem 3|5 LH Diasem Phrygian	SLLMLSLML RH Diasem 3|5 RH Diasem Mixo (*)
LLSLMLSLM LH Diasem 8|0 LH Diasem Lydian	LLMLSLMLS RH Diasem 8|0 RH Diasem Darkened Mixo

Notes:

• Note that there are two Dorians for each chirality, Bright and Dark Dorian.
• (*) We take these as the default LH Aeolian and RH Mixolydian because they can be dechiralized (see below). The adjectives "brightened" and "darkened" are modifications that flatten the default modes by a comma step; these modes have no dechiralized equivalents.

Ambi modes

By taking the pitches of both chiralities corresponding to each diatonic mode, we obtain the achiral scale LSCSLMLSLM, where c = L − s, termed ambi-diasem (after ambidextrous). This is a stretched version of the blackdye scale with the same step ratio (with the L in the "SLS" of blackdye made smaller). There are 8 ambi modes that result from changing an occurrence of SL and LS in it to SCS (on the same root as the original chiral modes).

Ambi modes
LSCSLMLSLM Ambi Diasem Lydian
LSLMLSCSLM Ambi Diasem Ionian
SCSLMLSLML Ambi Diasem Mixo
SLMLSCSLML Ambi Diasem Bright Dorian
LMLSCSLMLS Ambi Diasem Dark Dorian
LMLSLMLSCS Ambi Diasem Aeolian
MLSCSLMLSL Ambi Diasem Phrygian
MLSLMLSCSL Ambi Diasem Locrian
CSLMLSLMLS
SLMLSLMLSC

The last two modes are exceptional because they begin and end in a C and thus do not result from a union-taking that preserves the tonic.

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

Diasem 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	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 7|1 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

Extensions

The diasem scale extends to a 14-note AG scale: LmLsLmLsL and LsLmLsLmL both extend to the scale mcmcmsmcmcmsmc (c = L − m), with two 7-note mosses generated by the diasem's fifths separated by m. This scale is not chiral. This scale extends diasem like how blackdye is a 10-note non-chiral AG extension of the Zarlino scale's AG pattern 3L 2m 2s, LmsLmLs. The 14-note extension is one of:

• 5L 7m 2s (if m < c < L)
• a 2-step modmos of 12L 2s (if c = m)
• 7L 5m 2s (if s < c < m)
• 7L 7s (if c = s)
• 7L 2m 5s (if c < s).

5L 7m 2s must have a diatonic fifth, since L > 2m > m + s. The 31edo tuning (c = 3\31, m = 2\31, s = 1\31) of the scale is ideal for the 81/80-tempering 2.3.5.7 interpretation.

Another extension is scscsmscscsmsc, with c = L - s (5L 2m 7s if c > m). 31edo diasem yields 5L 2m 7s with step ratio 4:2:1.

Both these tunings, 5L 7m 2s and 5L 2m 7s, have been named crossdye ("crossed eyes" referring to the two copies of 5L 2s diatonic + "blackdye", courtesy of cellularAutomaton). 5L 7m 2s could be called chromatic crossdye and 5L 2m 7s could be called dietic crossdye.

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