# Blackdye

Blackdye (also denoted 3s in groundfault's aberrismic theory) is a 10-note, bipentatonic, generator-offset 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. 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:

1. LSLSLMLSLM (Lydian) > LSLMLSLSLM (Ionian) > LSLMLSLMLS (Grave Mixo) > LMLSLSLMLS (Grave Dorian) > LMLSLMLSLS (Grave Aeolian)
2. SLSLMLSLML (Acute Mixo) > SLMLSLSLML (Acute Dorian) > SLMLSLMLSL (Acute Aeolian) > MLSLSLMLSL (Phrygian) > MLSLMLSLSL (Locrian)

## Properties

1. Assuming positive step sizes, the fifth (3L + M + 2S) of blackdye must be a diatonic fifth.
2. 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
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

## Example

An example in Blackdye Acute Aeolian, SLMLSLMLSL (score)