Blackdye: Difference between revisions
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"Blackdye" is a name given by [[User:ks26|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]]). | "Blackdye" is a name given by [[User:ks26|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]]). | ||
== Properties == | == Properties == | ||
The fifth (3L + M + 2S) of blackdye must be a diatonic fifth. | # 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|2L' 3s']] subset of blackdye with L' = ML and s' = SL.) | |||
(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|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 == | == Tunings == | ||
Revision as of 08:07, 14 August 2021
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 minor thirds and major thirds. Blackdye can be used in 5-limit edos that do not temper out 81/80, such as 22edo and 34edo.
"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).
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 detempering but for 2.3.7