------------------------------------------------------ Zest 24: Zarlino Extraordinaire Spectrum Temperament A septimal (2-3-7) lattice and JI equivalents ------------------------------------------------------ One way of viewing Zest 24 is as a tempered equivalent of a JI lattice for ratios of primes 2-3-7, with 3:2 fifths shown in the horizontal dimension and 7:6 minor thirds in the vertical dimension. This scheme can account for all 24 notes in the tuning system, and also provides an opportunity to see how intervals shade from more or less close approximations of septimal ratios into other regions of the continuum. In the following diagram, each quadrangle shows an approximation of a 12:14:18:21 tetrad or quad, with "septimal" tetrads taken to be those with a minor seventh at 958, 962, 975, or 983 cents. The "1/1" of the tuning is taken to be F on the lower keyboard, and other pitches are specified in cents in relation to this note. An asterisk (*) shows a note on the upper keyboard, raised by an enharmonic diesis in Zarlino's regular 2/7-comma meantone temperament, about 50.28 cents. The numbers within each quad show the sizes in cents of the minor seventh, the two minor thirds, and the major third. Gb* Db* Ab* Eb* 121 ------ 817 ------ 325 ----- 1033 / 983 / 983 / 983 / / 287 275 / 275 275 / 275 287 / / 421 / 434 / 421 / Ab* Eb* Bb* F* C* 325 ----- 1033 ----- 54 ------- 50 ------ 746 / 962 / 975 / 975 / 962 / / 254 267 / 267 267 / 267 267 / 267 254 / / 441 / 441 / 441 / 441 / Gb Db Ab Eb Bb 71 ----- 766 ----- 275 ------ 983 ----- 492 / 983 / 983 / 983 / / 287 275 / 275 275 / 275 287 / / 421 / 434 / 421 / Eb Bb F C G D A E B 983 ----- 492 ------ 0 ------ 696 ------ 192 ---- 887 ----- 383 ---- 1079 ----- 575 / 958 / 958 / 958 / 958 / 958 / 958 / 958 / / 250 262 / 262 262 / 262 262 / 262 262 / 262 262 / 262 262 / 262 250 / / 446 / 434 / 434 / 434 / 434 / 434 / 446 / Gx ------- Dx ------ Ax ------ Ex ------ Bx ----- F#x ------ C#x ----- G#x 242 937 434 1129 625 121 817 325 This diagram consists of four "rows" of tetrads. The lowest row has tetrads with a 958-cent minor seventh formed from a regular meantone minor seventh (1008 cents) less a 50-cent diesis. The next row is formed within notes at the "remote" portion of the 12-note circle on the lower keyboard, where large 708-cent fifths generate some near-septimal intervals including 983-cent minor sevenths. The next row has tetrads with minor sevenths at 962 and 975 cents formed by adding a diesis to a large major sixth; these sevenths offer the best approximations of 7:4. The highest row has tetrads with 983-cent minor sevenths formed in the remote portion of the 12-note circle on the upper keyboard. In all there are 17 tetrads at 16 distinct "locations," since two alternative forms within our "septimal" range are available above Eb: either Eb-Gb-Bb-Db found within the lower keyboard (0-287-708-983 cents), or Eb-F*-Bb-C* (0-267-708-963 cents). These two sonorities nicely demonstrate the shadings and often "mixed" qualities of these sonorities, since the lower 287-cent minor third of the first tetrad is close to 33:28 (284 cents) or 13:11 (289 cents), while the upper 254-cent third of the latter is near 22:19 (also a rounded 254 cents) in the "interseptimal" region intervening between 8:7 (231 cents) and 7:6 (267 cents). Indeed, one could quite reasonably argue that sonorities with 958-cent minor sevenths should generally be considered interseptimal rather than septimal. Here this size is regarded as "septimal" in part because it is the regular augmented augmented sixth of Zarlino's regular 2/7-comma meantone, and is often used in this modification of his temperament as an equivalent for 7:4. This free equivalence might better fit a neo-medieval or other style where the focus is on ratios of 2-3-7-9 than a context where a near-just rendition of 4:5:6:7 is sought, for example. Two of the seven tetrads with 958-cent sevenths also include 250-cent minor thirds and 446-cent major thirds, notably diverging from the just ratios of 7:6 (267 cents) and 9:7 (435 cents) in an interseptimal direction, and further underscoring the fluidity of the continuum. Likewise, in a neo-medieval style, sonorities here regarded as "septimal" which include 287-cent minor thirds and 421-cent major thirds can be seen as shading toward the region of ratios such as 13:11 (289 cents) or 14:11 (418 cents), a kind of shading which is routine in 17-note circulating temperaments, for example. It is interesting to compare this tempered array of 24 notes with a lattice of the 31 just intonation pitches forming a similar pattern of tetrads, here all at a just 12:14:18:21. Again, the horizontal dimension shows 3:2 fifths and the vertical, 7:6 minor thirds. Two dots indicate a tetrad (3/2-7/4-9/8-21/16) which in Zest 24 moves outside our "septimal" range with a 996-cent minor seventh (virtually a just 16:9), but in this JI version would, of course, form a pure 12:14:18:21. 99 801 303 1005 343/324-343/216-343/288-343/192 | | | | | | | | 330 1032 534 36 738 98/81 - 49/27 - 49/36 - 49/48 - 49/32 | | | | | | | | | | 63 765 267 969 471 28/27 - 14/9 --- 7/6 --- 7/4 -- 21/16 | | | | . | | | | . 16/9 --- 4/3 --- 1/1 --- 3/2 --- 9/8 -- 27/16 -- 81/64 -- 243/128 -- 729/512 996 498 0 702 204 906 408 1110 612 | | | | | | | | | | | | | | | | 8/7 --- 12/7 -- 9/7 -- 27/14 - 81/56 - 243/224 - 729/448 - 2187/1792 231 931 435 1137 639 141 843 345 Here are Scala files for Zest 24 and this possible 31-note JI interpretation. ! zest24.scl ! Zarlino Extraordinaire Spectrum Temperament (two circles at ~50.28c apart) 24 ! 50.27584 25/24 120.94826 191.62069 241.89653 287.43104 337.70688 383.24139 433.51722 504.18965 554.46549 574.86208 625.13792 695.81035 746.08619 779.05173 829.32757 887.43104 937.70688 995.81035 1046.08619 1079.05173 48/25 2/1 ! zarte24-ji31_3-7.scl ! JI version of zarte24a.scl as lattice of 3/2, 7/6 31 ! 49/48 28/27 343/324 243/224 9/8 8/7 7/6 343/288 98/81 2187/1792 81/64 9/7 21/16 4/3 49/36 729/512 81/56 3/2 49/32 14/9 343/216 729/448 27/16 12/7 7/4 16/9 343/192 49/27 243/128 27/14 2/1 Most appreciatively, Margo Schulter mschulter@calweb.com