JUST TUNING, akj.scl by Gene Ward Smith (aka pipedum_12i.scl) Each quadrangle shows a complete 12:14:18:21 (1:1-7:6-3:2-7:4) from the note in lower left corner of the quadrangle (0-267-702-969 cents) with the horizontal dimension representing 3:2 fifths and the vertical dimension 7:6 minor thirds. 63 765 267 28/27 - 14/9 --- 7/6 / / / / / / 702 204 906 16/9 --- 4/3 --- 1/1 --- 3/2 --- 9/8 --- 27/16 996 498 0 / / / / / / 9/7 --- 27/14 - 81/56 435 1137 639 Here is a TEMPERED VERSION in Zest-24 system, where the same note may appear more than once in a similar lattice showing approximations of 12:14:18:21, since some of these approximations represent septimal ratios using notes from the same chain of tempered fifths. An asterisk (*) indicates a note in the upper circle of fifths in the tuning, raised by about 50.28 cents. Note that in order to realize all fifths of the just version with intervals close to 3:2 while using the best septimal approximations, the note 7/6 (267 cents) in this version sometimes maps to either Db (275 cents) or C* (254 cents). Thus 13 steps of the tempered system are used to represent Gene Ward Smith's 12-note just tuning. Numbers within each quad show sizes in cents of the outer minor seventh, lower and upper minor thirds, and middle major third -- respectively 969 cents (7:4), 267 cents (7:6), and 435 cents (9:7) in the just version. 50 758 254|275 983 492 Bb* ------ F* ---- C*|Db ---- Ab ------ Eb / 975 / 963 / 983 / 983 / / 267 267 / 267 254 / 274 274 / 274 287 / / 441 / 441 / 434 / 421 / Db ----- Ab ------ Eb ------ Bb ------ F ------- C ------- G 274 983 492 0 708 ----- 204 ----- 900 / 958 / 958 / / 262 262 / 262 262 / / 434 / 434 / D* ------ A* ------ E* 446 1142 637 An interesting exercise is to seek out a just system sharing this lattice shape, for example: 63 765 267 969 471 28/27 - 14/9 --- 7/6 --- 7/4 -- 21/16 / / / / / / / / / / 32/27 --- 16/9 --- 4/3 --- 1/1 --- 3/2 --- 9/8 -- 27/16 294 996 498 0 702 --- 204 -- 906 / / / / / / 9/7 -- 27/14 - 81/56 435 1137 639 In this just version, we have only a single step for 7/6 (as in the original akj.scl), but separate steps for 7/6 and 32/27, and also for 21/16 and 4/3 -- pairs represented in the tempered version by the single steps Db (274 cents) and Eb (492 cents). Thus starting with a 12-note just tuning by Gene Ward Smith, we arrive at a 13-note tempered version -- which, in turn, has a lattice shape suggesting a 15-note just tuning. * * * As a postscript, one might note that a 15-note extension of the tempered version has the attractions of making the chain of fifths from the upper 12-note circle of Zest-24 continuous by adding G*, and also letting Db serve as the lowest note (lower left corner on the lattice) of a complete 12:14:18:21 approximation. Thus we would have: 542 50 758 254|275 983 492 Eb* ----- Bb* ------ F* ---- C*|Db ---- Ab ------ Eb / 975 / 975 / 963 / 983 / 983 / / 267 267 / 267 267 / 267 254 / 274 274 / 274 287 / / 441 / 441 / 441 / 434 / 421 / Db ------ Ab ------ Eb ------ Bb ------ F ------- C ------- G 274 983 492 0 ------ 708 ----- 204 ----- 900 / 958 / 958 / 958 / / 250 262 / 262 262 / 262 262 / / 446 / 434 / 434 / G* ------ D* ------ A* ------ E* 950 446 1142 637 An equivalent lattice shape would be provided by this 17-note just tuning: 561 63 765 267 969 471 112/81 - 28/27 - 14/9 --- 7/6 --- 7/4 -- 21/16 / / / / / / / / / / / / 32/27 - 16/9 --- 4/3 --- 1/1 -- 3/2 --- 9/8 -- 27/16 294 996 498 0 ---- 702 --- 204 -- 906 / / / / / / / / 12/7 --- 9/7 -- 27/14 - 81/56 933 435 1137 639 Finally, I might comment that in these tempered sets growing out of a focus on Gene Ward Smith's akj.scl, a four-voice sonority with minor third, fifth, and minor seventh above the lowest voice is taken to have a "septimal" quality suggesting 12:14:18:21 if the outer minor seventh has a size of 958, 963, 975, or 983 cents, thus keeping within some 14 cents of a just 7:4. Minor thirds in these rather diversely shaded sonorities have a range of 250-287 cents, with major thirds at 421-446 cents. Margo Schulter 8 January 2007