Lattice for Zeta-24 tuning:

        3:2    3:2  176:117   3:2    3:2   182:121 176:117  3:2    3:2    3:2    3:2
     Gp --- Dp --- Ap ... Eb* --- Bb* --- F* .... C* ... G* --- D* --- A* --- E* --- B*
    13/9   13/12  13/8   11/9    11/6    11/8   91/88   14/9   7/6    7/4   21/16  63/32
   636.6   138.6  840.5  347.4  1049.4  551.3    58.0  764.9  266.9  968.8  470.7 1172.7

      176:117  3:2    3:2     3:2    3:2   176:117 182:121  3:2  176:117  3:2    3:2
     Eb ... Bb ---- F ---- C ---- G ----- D ..... A .... E ---- B .... F# --- C# --- G#
   13/11   16/9    4/3    1/1    3/2     9/8    22/13  14/11  21/11  56/39  14/13  21/13
   289.2   996.1  498.0   0.0   702.0   203.9   910.8  417.5 1119.5  626.3  128.3  830.3

In this notation, the Persian koron (ASCII "p") is used for the three notes at the lower
end of the upper chain, Gp (13/9), Dp (13/12), and Ap (13/8), classical intervals from
the tunings of Ibn Sina (980-1037) approximately 1/3-tone lower than the Zeta-24 steps
G (3/2), D (9/8), and A (22/13).


     Another perspective on chains of fifths, showing G#-Eb* connection between chains:

        3:2    3:2  176:117   3:2    3:2   182:121 176:117  3:2    3:2    3:2    3:2
     Gp --- Dp --- Ap ... Eb* --- Bb* --- F* .... C* ... G* --- D* --- A* ---- E* --- B*
    13/9   13/12  13/8   11/9    11/6    11/8   91/88   14/9   7/6    7/4    21/16  63/32
   636.6   138.6  840.5  347.4  1049.4  551.3    58.0  764.9  266.9  968.8   470.7 1172.7
                           \
                            \______________________________ 286:189 (717.2) ______________________________        
                                                                                                          \
                            176:117  3:2    3:2     3:2    3:2  176:117 182:121  3:2   176:117  3:2    3:2 \
                           Eb ... Bb ---- F ----- C ---- G ---- D .... A ..... E ---- B .... F# --- C# --- G#
                         13/11   16/9    4/3     1/1    3/2    9/8   22/13   14/11  21/11  56/39  14/13  21/13
                         289.2   996.1  498.0    0.0   702.0  203.9  910.8   417.5 1119.5  626.3  128.3  830.3



     Zalzalian lattice: A web of fifths and middle thirds 
     (Dudon's entrelacs or interlacing, or Wilson's Rast/Bayyati matrix, with 5 sizes of generating thirds)

             3:2       3:2   176:117    3:2      3:2   182:121   176:117    3:2     3:2       3:2      3:2
       636.6     138.6    840.5    347.4   1049.4   551.3    58.0      764.9   266.9    968.8     470.7   1172.7
       13/9      13/12    13/8     11/9     11/6     11/8    91/88     14/9     7/6      7/4      21/16   63/32
        Gp -------Dp ----- Ap ..... Eb* ---- Bb* ---- F* ..... C* ..... G* ----- D* ----- A* ----- E* ---- B*
        /\        /\       /\       /\       /\       /\       /\       /\       /\       /\       /\      /
       /  \      /  \     /  \     /  \     /  \     /  \     /  \     /  \     /  \     /  \     /  \    /
      3    5    1    5   1    5   3    4   3    4   3    5   2    5   3    4   3    5   1    5   1    5  1
     /      \  /      \ /      \ /      \ /      \ /      \ /      \ /      \ /      \ /      \ /      \/
   Eb ...... Bb ------ F ------ C ------ G ------ D ...... A ...... E ------ B ..... F# ----- C# ----- G#
 13/11      16/9      4/3      1/1      3/2      9/8     22/13    14/11    21/11   56/39     14/13   21/13
 289.2     996.1     498.0     0.0     702.0    203.9    910.8    417.5   1119.5   626.3     128.3   830.3
     176:117     3:2       3:2     3:2       3:2   176:117   182:121    3:2    176:117   3:2      3:2

Key to Zalzalian thirds

1 = 39/32 (342.483 cents)
2 = 1183/968 (347.245 cents)
3 = 11/9 (347.408 cents)
4 = 27/22 (354.547 cents)
5 = 16/13 (359.472 cents)


Margo Schulter
mschulter@calweb.com
30 December 2013