# 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