Blackdye

Blackdye is a 10-note AG scale with step pattern 5L 2M 3S (SLMLSLMLSL), equivalent to the 5L 5s mos with two of the small steps made larger and the other three made smaller. Unlike diasem, another diatonic-based scale with comma steps, blackdye is not chiral. It can be tuned as a 5-limit JI scale or a tempered version thereof, where L represents 10/9, M represents 16/15, and S represents 81/80. The scale can be generated by an alternating chain of 10/9 and 6/5. Blackdye can, of course, be used in 5-limit edos that do not temper out 81/80, such as 22edo and 34edo; however, there are more exotic tunings of the scale, such as 29edo's 4:3:1, 32edo's 4:3:2 and 5:2:1, and 36edo's 5:4:1.

Blackdye is a 10-note extension of the Zarlino pattern 3L 2m 2s, LmsLmLs.

"Blackdye" is a name given by ks26. The name is a pun on "blackwood" and "diatonic" since its step sizes are intermediate between that of diatonic (5L 2s) and blackwood[10] (5L 5s).

Intervals

The following is a table of blackdye 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 blackdye using the following expressions.

Interval sizes in blackdye

Interval class	Sizes	In 34edo (L:m:s = 5:3:1)
1-step	s m L	1\34, 35.29¢ 3\34, 105.88¢ 5\34, 176.47¢
2-step	L + s L + m	6\34, 211.77¢ 8\34, 282.35¢
3-step	L + 2s L + m + s 2L + s 2L + m	7\34, 247.06¢ 9\34, 317.65¢ 11\34, 388.24¢ 13\34, 458.82¢
4-step	2L + 2s 2L + m + s	12\34, 423.53¢ 14\34, 494.12¢
5-step	2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s	15\34, 529.412¢ 17\34, 600.00¢ 17\34, 600.00¢ 19\34, 670.59¢
6-step	3L + m + 2s 3L + 2m + s	20\34, 705.88¢ 22\34, 776.47¢
7-step	3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s	21\34, 741.18¢ 23\34, 811.77¢ 25\34, 882.35¢ 27\34, 952.94¢
8-step	4L + m + 3s 4L + 2m + 2s	26\34, 917.65¢ 28\34, 988.24¢
9-step	5L + 2m + s 5L + m + 2s 4L + 2m + 2s	29\34,1023.53¢ 31\34, 1094.12¢ 33\34, 1164.71¢

The octave in blackdye can be called the "perfect 10-step" in TAMNAMS.

Modes

The modes arranged in cyclic order: (Note: The mode names are based on the 5-limit JI interpretation; modes in a less JI-like tuning may differ greatly from what these names suggest.)

Pattern	Name
SLMLSLMLSL	Blackdye Bright Aeolian
LMLSLMLSLS	Blackdye Dark Aeolian
MLSLMLSLSL	Blackdye Locrian
LSLMLSLSLM	Blackdye Ionian
SLMLSLSLML	Blackdye Bright Dorian
LMLSLSLMLS	Blackdye Dark Dorian
MLSLSLMLSL	Blackdye Phrygian
LSLSLMLSLM	Blackdye Lydian
SLSLMLSLML	Blackdye Bright Mixo
LSLMLSLMLS	Blackdye Dark Mixo

Properties

1. The fifth (3L + M + 2S) of blackdye must be a diatonic fifth.
2. The fifth is:
   • sharper than 700¢, if L + 3S > 2M
   • equal to 700¢, if L + 3S = 2M
   • flatter than 700¢, if L + 3S < 2M

Tunings

Blackdye tunings

Tuning	L:M:S	Degrees of the mode SLMLSLMLSL
		1	2	3	4	5	6	7	8	9
5-Limit Interpretation		81/80	9/8	6/5	4/3	27/20	3/2	8/5	16/9	9/5
JI	8.481:5.195:1	21.506	203.910	315.641	498.045	519.551	701.955	813.686	996.090	1017.596
22edo	3:2:1	54.545	218.182	327.273	490.909	545.455	709.090	818.182	981.818	1036.364
27edo	4:2:1	44.444	222.222	311.111	488.889	533.333	711.111	800	977.778	1022.222
29edo	4:3:1	41.379	206.897	331.034	496.551	537.931	703.448	827.586	993.103	1034.483
32edo	4:3:2	75	225	337.5	487.5	562.5	712.5	825	975	1050
32edo	5:2:1	37.5	225	300	487.5	525	712.5	787.5	975	1012.5
34edo	5:3:1	35.294	211.765	317.647	494.117	529.412	705.882	811.764	988.235	1023.529
37edo	5:3:2	64.865	227.027	324.324	486.486	551.351	713.514	810.810	972.973	1037.838
36edo	5:4:1	33.333	200	333.333	500	533.333	700	833.333	1000	1033.333
39edo	5:4:2	61.538	215.385	338.462	492.308	553.846	707.692	830.769	984.615	1046.154
42edo	5:4:3	85.714	228.571	342.857	485.714	571.429	714.286	828.571	971.429	1057.143

Example

An example in Blackdye Bright Aeolian, SLMLSLMLSL

32edo (L:M:S = 5:2:1)

34edo (L:M:S = 5:3:1)

See also

• Diasem, a similar diatonic-like mos detempering but for 2.3.7