Blackdye
Blackdye is a 10-note AG, MV4 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 minor thirds and major thirds. 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 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 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 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¢ |
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
- The fifth (3L + M + 2S) of blackdye must be a diatonic fifth.
- (To see this: Consider the chain of fifths generated by the fifth. The diatonic chroma = 7 fifths - 4 octaves = 21L + 7M + 14S - (20L + 8M + 12S) = L - M + 2S. Since L - M >= 0, this must always be nonnegative, so the fifth must be >= 4\7. The fifth cannot be sharper than 3\5, since it generates the 2L' 3s' subset of blackdye with L' = ML and s' = SL.)
- 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 | 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 |
See also
- Diasem, a similar diatonic-like mos detempering but for 2.3.7