------------------------------------------------------
Lattice for Pegasus-20
With some notes appearing twice
------------------------------------------------------
A complete genus musicum of 2^m 3^9 59^1
------------------------------------------------------
This Euler-Fokker genus musicum is formed by two chains
each with nine 3:2 fifths (3^9), these chains being
situated at a distance of 59:32 (59^1), with the interval
of periodicity or repetition being a 2:1 octave (2^m).
Five of the 20 notes in this genus appear twice in the
lattice, as approximate factors of 7 (diagonal) and
neutral or Zalzalian intervals of 59 (vertical), and
are bracketed when appearing in the latter dimension.
Horizontal = 3:2
Diagonal = ~7:6 (4779:4096)
Vertical = 59:48 or ~16/13
Eb* Bb* F* C* [G*] [D*] [A*] [E*] [B*]
(~128/117) (~64/39) (~16/13) (~24/13) (~112/81) (~28/27) (~14/9) (~7/6) (~7/4)
59/54 ---- 59/36 ---- 59/48 ---- 59/32 --- 177/128 --- 531/512 -- 1593/1024 -- 4779/4096 -- 14337/8192
| | | | | | | | |
G* | D* | A* | E* | B* | F#* | | | |
~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 ----- 81/64 ----- 243/128 ---- 729/512
F C G D A E B F# C# G#
Ratios of 59 differ from corresponding ratios of 7 by 14337:14336 (0.121 cents)
Ratios of 59 differ from corresponding ratios of 13 by 768:767 (2.256 cents)
While precise 59-based values are shown at the top portion
of the diagram along with approximate 7-based or 13-based
values, one ratio appearing only in approximate septimal form
in the diagonal or "depth" dimension is the narrow fourth
almost identical to the indicated 21/16 (470.781 cents), but
more precisely at 43011/32768 (470.902 cents). The 21:16, or
here this close equivalent, is a basic interval of Erv
Wilson's 1-3-7-9 hexany, represented by any two adjacent
slanted quadrilaterals or parallelograms, so that there are
four such hexanies included in this genus musicum.
Margo Schulter
13 January 2011