Blackdye
Blackdye is a 10-note, dipentatonic, 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 edos 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" is a name given by ground fault. 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 class | Sizes | 5-limit JI | 32edo (L:m:s = 5:2:1) | 34edo (L:m:s = 5:3:1) |
|---|---|---|---|---|
| 1-step | 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-step | 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-step | 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-step | 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-step | 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-step | 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-step | 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-step | 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-step | 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 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 |
By generator chain
Because blackdye has two chains of 5 fifth generators, it has two chains of 5 modes each separated by one generator:
- LSLSLMLSLM (Lydian) > LSLMLSLSLM (Ionian) > LSLMLSLMLS (Dark Mixo) > LMLSLSLMLS (Dark Dorian) > LMLSLMLSLS (Dark Aeolian)
- SLSLMLSLML (Bright Mixo) > SLMLSLSLML (Bright Dorian) > SLMLSLMLSL (Bright Aeolian) > MLSLSLMLSL (Phrygian) > MLSLMLSLSL (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
| 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 Bright Aeolian, SLMLSLMLSL (score)
22edo (L:M:S = 3:2:1)
27edo (L:M:S = 4:2:1)
29edo (L:M:S = 4:3:1)
32edo (L:M:S = 5:2:1)
34edo (L:M:S = 5:3:1)
36edo (L:M:S = 5:4:1)