Topic: 7-limit planar lattice via fractional monzos: attn Monz

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From: Gene Ward Smith (2005-04-08)
Subject: 7-limit planar lattice via fractional monzos: attn Monz

This 7-limit 2401/2400 lattice can be approached via Joe's favorite
fractional monzos, and from that I think he could figure out how to do
planar 7-limit lattice diagrams pretty easily. If we represent a 7 by
(2400)^(1/4), we are representing it by |5/4 1/4 1/2>. My lattice
business is simply what you get if you use this for 7s, and stick the
result inside the plane containing the 5-limit lattice: the pitch
class for 3 is [1,0], for 5 is [0,1], and for 7 is [1/4,1/2].

From this, given any 5-limit Fokker block, you can find the
corresponding 7-limit object by adding all the 7-limit lattice
elements which fall in the range of the block. A Fokker block obtained
from 25/24 and 81/80 is 1, 10/9, 6/5, 4/3, 3/2, 5/3, 9/5. If I use the
same range on the 7-limit planar lattice, I end up with the following
scale of 28 notes, which as expected makes use of 2401/2400
approximations to obtain extra 7-limit harmony.

! bigblok.scl
Bigblok
28
!
49/48
21/20
15/14
49/45
10/9
8/7
7/6
6/5
49/40
9/7
21/16
4/3
49/36
7/5
10/7
72/49
3/2
32/21
14/9
49/30
5/3
12/7
7/4
9/5
90/49
28/15
40/21
2