--------------------------------------------------------------- Hexapentadic 17-note tuning, ~8:9:11:12:13 in 6 locations Tetrachordal lattice showing intervals and available genera And also 7 triads of 1:1:72:59-3:2 and 8 of 1:1-59:48-3:2 This tuning being a prime 59 version Of Erv Wilson's Rast-Bayyati matrix ---------------------------------------------------------------- Prompted by Kraig Grady's proposal and template for a lattice structure oriented to 4/3 tetrachords, I have attempted to devise a format which can show the intervals within each 4/3 fourth in a given tuning both as absolute ratios referenced to some 1/1 for the system as a whole, and as relative ratios to the lowest note of the given fourth or tetrachord space. In this lattice each such space appears as a square, with absolute tuning steps or ratios displayed on the left-hand border of the square, and relative tetrachord ratios within the square. Thus the lattice ascends vertically in 4/3 fourths, with the horizontal dimension showing neutral thirds at 72/59 (left to center) or 59/54 (center to right). These neutral third ratios are also noted on the lines connecting two notes forming one of these ratios. For each tetrachordal square, the note in the lower left corner is the local "1/1." For the squares in the left column, this note has a 72:59 third above it, while for squares in the right column it has a 59:48 third. Thus wherever there are squares in both columns, the three notes joined by the horizontal line defining the upper or lower border of these two squares form a 1:1-72:59-3:2 triad, for example: C* E G* 1/1-------- (72:59) ------ 72/59 ------ (59:48) ------- 3/2 Additionally, each square defines two neutral third triads, one at 118:144:177 (1:1-72:59-3:2), and the other at 48:59:72. These triads may be found by visualizing a diagonal bisecting a square from the lower left to the upper right corner into two triangles, each representing such a triad. For example: F* A C* 4/3 ------ (72:59) ----- 96/59 ---- (59:48) ----- 1/1 | , | , | | 118:144:177 , | 48:59:72 , | | , | , | | , | , | | , | , | | , 48:59:72 | , | | , | , 118:144:177 | C* E G* 1/1 ------- (72:59) ---- 72/59 ---- (59:48) ----- 3/2 While these neutral triads and related structures like an approximation of ~8:9:11:12:13 (in this 2-3-59 prime tuning, actually realized as 472:531:648:708:768) shape the tuning, it is intended above all for music influenced by Near Eastern styles such as the maqam and dastgah traditions which are above all melodic. At the same time, polyphonic textures drawing on these traditions may benefit from the variety of melodic and vertical intervals available. I should emphasize that the division of a 32:27 minor third into steps of 64:59:54 was mentioned by Ibn Sina in the early 11th century and made the basis of a recognized genus by Safi al-Din al-Urmawi in the 13th century (with Dr. Fazli Arslan giving this genus as 72:64:59:54, and the Scala archives, safi_diat2.scl, giving 64:59:54:48), which would fit respectively Ibn Sina's mode of Mustaqim or the Persian gushe of Shekaste in Mahur Dastgah; and a modern Persian Shur, Arab Bayyati, or Turkish Ushshaq, for example. The division of a 32:27 minor third with the smaller neutral second step (here 64:59) preceding the larger (here 59:54) is typical of these modalities, but stands in contrast to a modern Arab Rast, where the larger neutral second characteristically precedes the smaller, as also in al-Farabi's "mode of Zalzal," as Cris Forster terms it. Note that 64:59:54 division with the smaller neutral second first occurs in tetrachords in the left-hand column, while the 54:59:64 division with the larger neutral step first may be found in the right-hand column. This 17-note tuning is a constant structure, meaning that a 4/3 fourth always results from seven subtended tuning steps, simplifying this lattice format since each square includes the same number of steps. An important advantage of this format is that it shows all of the possible steps within a tetrachordal space, so that one may search for and identify desired modal patterns -- or possibly come upon new ones! Again, I warmly thank Kraig Grady for his suggestions leading to this lattice, while emphasizing that responsibility for any flaws, errors, or infelicities is entirely mine. C Eb* 1024/531 ----- 59:48 ------- 32/27 | | 108/59 81/64 | 16/9 59/48 | 27/16 14337/8192 | 96/59 9/8 | 729/472 2187/2048 | 3/2 531/512 | | | Eb* G Bb* 32/27 ----- 72:59 ------ 256/177 ----- 59:48 ------- 16/9 | | | 9/8 81/64 81/59 81/64 | 64/59 72/59 4/3 59/48 | 243/236 2187/1888 81/64 14337/8192 | 2/1 9/8 72/59 9/8 | 1024/531 64/59 32/27 59/54 | 108/59 243/236 9/8 531/512 | | | | Bb* D F* 16/9 ------ 72:59 ------- 64/59 ------ 59:48 -------- 4/3 | | | 27/16 81/64 243/236 81/64 | 96/59 72/59 2/1 59/48 | 729/472 2187/1888 1024/531 32/27 | 3/2 9/8 108/59 9/8 | 256/177 64/59 16/9 59/54 | 81/59 243/236 27/16 531/512 | | | | F* A C* 4/3 ------ 72:59 ------- 96/59 ------ 59:48 -------- 1/1 | | | 81/64 81/64 729/472 81/64 | 72/59 72/59 3/2 59/48 | 32/27 32/27 256/177 32/27 | 9/8 9/8 81/59 9/8 | 64/59 64/59 4/3 59/54 | 243/236 243/236 81/64 531/512 | | | | C* E G* 1/1 ------ 72:59 ------- 72/59 ------ 59:48 -------- 3/2 | | | 1024/531 2848/1593 32/27 944/729 | 108/59 72/59 9/8 59/48 | 16/9 32/27 64/59 32/27 | 27/16 9/8 243/236 9/8 | 96/59 64/59 2/1 59/54 | 729/472 243/236 1024/531 256/243 | | | | G* B D* 3/2 ------ 72:59 ------ 108/59 ------ 59:48--------- 9/8 | | | 256/177 2048/1593 16/9 944/729 | 81/59 72/59 27/16 59/48 | 4/3 32/27 96/59 32/27 | 81/64 9/8 729/472 9/8 | 72/59 64/59 3/2 59/54 | 32/27 256/243 256/177 256/243 | | | | D* F# A* 9/8 ------ 72:59 ------- 81/59 ------ 59:48 ------- 27/16 | | | 64/59 2048/1593 4/3 944/729 | 243/236 72/59 81/64 59/48 | 2/1 32/27 72/59 32/27 | 1024/531 16384/14337 32/27 7552/6561 | 108/59 64/59 9/8 59/54 | 16/9 256/243 64/59 256/243 | | | | A* C# E* 27/16 ----- 72:59 ------- 243/236 ---- 59:48 ------- 81/64 | | 96/59 2048/1593 2/1 944/729 729/472 72/59 1024/531 8192/6561 3/2 32/27 108/59 32/27 256/177 16384/14337 16/9 7552/6561 81/59 64/59 27/16 59/54 4/3 256/243 96/59 256/243 | | E* G#* 81/64 ----- 72:59 ------- 729/472 Margo Schulter 19 January 2011