Blackville

Blackville is a rank-3 dipentatonic generator-offset and step-nested scale with step signature and mapping 2L 5m 3s = (10/9, 16/15, 135/128). It may be generated as a generator offset scale by two chains of 3/2, offset by 16/15, or by the variable generator sequences GS(15/8, 8/5) or GS(16/15, 45/32); or as a step-nested scale by adding 16/15 above the bottom or below the top of every step of the Pythagorean pentatonic scale, which is generated by 3/2.

Tempering L = s leads to Blackwood[10], and tempering out s leads to Mavila[7]. Since tempering s = m leads to the pentachordal decatonic of Srutal/diaschsimic temperament, we may name Blackville's modes through extrapolating the modes of the pentachordal decatonic. The first edo to represent Blackville without redundancy is 17edo using the 17c val, where in the standard major mode (smLmsmLmsm) it is 2131213121, i.e., 2L 5m 3s = (3, 1, 2). It may also be tuned without redundancy to 19edo as 1232123212 with 2L 5m 3s = (3, 2, 1), 26edo as 1343134313 with 2L 5m 3s = (4, 3, 1), 27edo as 3242324232 with 2L 5m 3s = (4, 2, 3), 29edo as 2343234323 with 2L 5m 3s = (4, 3, 2), 31edo as 2353235323 with 2L 5m 3s = (5, 3, 2), and 41edo as 3464346434 with 2L 5m 3s = (6, 4, 3), for example.

Blackville may be used for harmonic tonality by way of it's 3-step triads, which include 5-limit major and minor triads (3 of each, along with wolf major, minor, and augmented triads).

Blackville is a superset of Nicetone / Zarlino, the "just" major scale 9/8 5/4 4/3 3/2 5/3 15/8 2/1, and of it's mirror inverse, the "just" minor scale 9/8 6/5 4/3 3/2 8/5 9/5 2/1 (available in the standard major and standard minor modes respectively as 2112121).

Mode number	Mode in JI	Step pattern	Pentachordal Decatonic	Pent. Dec. Mode name	Blackwood[10]	UDP	Mode height
-5	16/15 9/8 6/5 81/64 27/20 3/2 8/5 27/16 9/5 2/1	msmsmLmsmL	sssssLsssL	Dark minor	sLsLsLsLsL	0|1 (5)	-.0745
-4	135/128 9/8 1215/1024 81/64 45/32 3/2 405/256 27/16 15/8 2/1	smsmLmsmLm	ssssLsssLs	Alternate minor	LsLsLsLsLs	1|0 (5)	-.0592
-3	16/15 9/8 6/5 4/3 64/45 3/2 8/5 27/16 9/5 2/1	msmLmsmsmL	sssLsssssL	Standard minor	sLsLsLsLsL	0|1 (5)	-.0411
-2	135/128 9/8 5/4 4/3 45/32 3/2 405/256 27/16 15/8 2/1	smLmsmsmLm	ssLsssssLs	Dark major	LsLsLsLsLs	1|0 (5)	-.0258
-1	16/15 9/8 6/5 4/3 64/45 3/2 8/5 16/9 256/135 2/1	msmLmsmLms	sssLsssLss	Bright minor	sLsLsLsLsL	0|1 (5)	-.0077
1	135/128 9/8 5/4 4/3 45/32 3/2 5/3 16/9 15/8 2/1	smLmsmLmsm	ssLsssLsss	Standard major	LsLsLsLsLs	1|0 (5)	.0077
2	16/15 32/27 512/405 4/3 64/45 3/2 8/5 16/9 256/135 2/1	mLmsmsmLms	sLsssssLss	Alternate major	sLsLsLsLsL	0|1 (5)	.0258
3	10/9 32/27 5/4 4/3 45/32 3/2 5/3 16/9 15/8 2/1	LmsmsmLmsm	LsssssLsss	Bright major	LsLsLsLsLs	1|0 (5)	.0411
4	16/15 32/27 512/405 4/3 64/45 128/81 2048/1215 16/9 256/135 2/1	mLmsmLmsms	sLsssLssss	Dark Augmented	sLsLsLsLsL	0|1 (5)	.0592
5	10/9 32/27 5/4 4/3 40/27 128/81 5/3 16/9 15/8 2/1	LmsmLmsmsm	LsssLsssss	Bright Augmented	LsLsLsLsLs	1|0 (5)	.0745