Blackdye

Blackdye (also denoted 3s in groundfault's aberrismic theory) is a 10-note, bipentatonic, generator-offset scale with step signature 5L 2m 3s (step pattern 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. Under this interpretation, the scale has two chains of 3/2 generators with an offset of 10/9. Blackdye can, of course, be used in 5-limit equal temperaments that do not temper out 81/80, such as 22edo and 34edo; however, there are more exotic tunings of the scale, such as 32edo's 5:2:1 and 37edo's 6:2:1, which have neogothic thirds and naiadics instead of 5-limit thirds and Pythagorean thirds, and 29edo's 4:3:1 and 36edo's 5:4:1, which have submajor and supraminor thirds instead of 5-limit thirds. One can even interpret diasem as a 9-note subset of blackdye with a negative s step.

Blackdye is a 10-note extension of the nicetone scale 3L 2M 2s, LMsLMLs.

Blackdye, a name given by groundfault, 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	5-limit JI	32edo (L:m:s = 5:2:1)	34edo (L:m:s = 5:3:1)
1-steps	s m L	81/80, 21.51¢ 16/15, 111.73¢ 10/9, 182.40¢	1\32, 37.50¢ 2\32, 75.00¢ 5\32, 187.50¢	1\34, 35.29¢ 3\34, 105.88¢ 5\34, 176.47¢
2-steps	L + s L + m	9/8, 203.91¢ 32/27, 294.14¢	6\32, 225.00¢ 7\32, 262.50¢	6\34, 211.77¢ 8\34, 282.35¢
3-steps	L + 2s L + m + s 2L + s 2L + m	729/640‚ 225.42¢ 6/5, 315.64¢ 5/4, 386.31¢ 320/243, 476.54¢	7\32, 262.50¢ 8\32, 300.00¢ 11\32, 412.50¢ 12\32, 450.00¢	7\34, 247.06¢ 9\34, 317.65¢ 11\34, 388.24¢ 13\34, 458.82¢
4-steps	2L + 2s 2L + m + s	81/64, 407.82¢ 4/3, 498.04¢	12\32, 450.00¢ 13\32, 487.50¢	12\34, 423.53¢ 14\34, 494.12¢
5-steps	2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s	27/20, 519.55¢ 64/45, 609.78¢ 45/32, 590.22¢ 40/27, 680.45¢	14\32, 525.00¢ 15\32, 562.50¢ 17\32, 637.50¢ 18\32, 675.00¢	15\34, 529.41¢ 17\34, 600.00¢ 17\34, 600.00¢ 19\34, 670.59¢
6-steps	3L + m + 2s 3L + 2m + s	3/2, 701.96¢ 128/81, 792.18¢	19\32, 712.50¢ 20\32, 750.00¢	20\34, 705.88¢ 22\34, 776.47¢
7-steps	3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s	243/160, 723.46¢ 8/5, 813.69¢ 5/3, 884.36¢ 1280/729, 974.58¢	20\32, 750.00¢ 21\32, 787.50¢ 24\32, 900.00¢ 25\32, 937.50¢	21\34, 741.18¢ 23\34, 811.77¢ 25\34, 882.35¢ 27\34, 952.94¢
8-steps	4L + m + 3s 4L + 2m + 2s	27/16, 905.87¢ 16/9, 996.09¢	25\32, 937.50¢ 26\32, 975.00¢	26\34, 917.65¢ 28\34, 988.24¢
9-steps	5L + 2m + s 5L + m + 2s 4L + 2m + 2s	9/5, 1017.60¢ 15/8, 1088.27¢ 160/81, 1178.49¢	27\32, 1012.50¢ 30\32, 1125.00¢ 31\32, 1162.50¢	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

Cyclic order

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 Acute Aeolian
LmLsLmLsLs	Blackdye Grave Aeolian
mLsLmLsLsL	Blackdye Locrian
LsLmLsLsLm	Blackdye Ionian
sLmLsLsLmL	Blackdye Acute Dorian
LmLsLsLmLs	Blackdye Grave Dorian
mLsLsLmLsL	Blackdye Phrygian
LsLsLmLsLm	Blackdye Lydian
sLsLmLsLmL	Blackdye Acute Mixo
LsLmLsLmLs	Blackdye Grave Mixo

By generator chain

Because blackdye has two chains of 5 fifth generators, it has two chains of 5 modes each separated by one generator:

Lydian > Ionian > Grave Mixo > Grave Dorian > Grave Aeolian
Acute Mixo > Acute Dorian > Acute Aeolian > Phrygian > Locrian

Properties

Assuming positive step sizes, the fifth (3L + m + 2s) of blackdye must be a diatonic fifth.
Assuming positive step sizes, 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
Tuning	L:m:s	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	800.000	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	75.000	225.000	337.500	487.500	562.500	712.500	825.000	975.000	1050.000
32edo	5:2:1	37.500	225.000	300.000	487.500	525.000	712.500	787.500	975.000	1012.500
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	200.000	333.333	500.000	533.333	700.000	833.333	1000.000	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
39edo	5:4:2	61.538	215.385	338.462	492.308	553.846	707.692	830.769	984.615	1046.154
37edo	6:2:1	32.432	227.027	291.892	486.486	518.919	713.514	778.378	972.973	1005.405
39edo	6:3:1	30.769	215.385	307.692	492.308	523.077	707.692	800.000	984.615	1015.384
42edo	6:3:2	57.143	228.571	314.286	485.714	542.857	714.286	800.000	971.429	1028.571
41edo	6:4:1	29.268	204.878	321.951	497.561	526.829	702.439	819.512	995.122	1024.390
47edo	6:4:3	76.596	229.787	331.915	485.106	561.702	702.439	714.894	970.213	1046.809
42edo	7:2:1	28.571	228.571	285.714	485.714	514.286	714.286	771.429	971.429	1000.000
44edo	7:3:1	27.273	218.182	300.000	490.909	518.182	709.091	790.901	981.818	1009.091
46edo	7:4:1	26.087	208.696	313.043	495.652	521.739	704.348	808.696	991.304	1017.391
53edo	8:5:1	22.642	203.773	316.981	498.113	520.755	701.887	815.094	996.226	1018.868