Partch lattice in 72-tET notation

For those who are still pondering the useful-/uselessness
of 72-tET vs 144-tET notation, this post is an attempt to
illustrate the 'consistency' of 72-tET. For a definition, see:

http://www.ixpres.com/interval/dict/consiste.htm

and follow the links to the Erlich and Ozzard-Low papers
for more info on consistency.

One of the reasons 72-tET notation works so well for approximating
just-intonation is that its set of 'accidentals' give deviations
from 12-tET that are remarkably close to the deviations of 12-tET
itself from the 'basic' prime ratios:

('~diff' gives the comparison in approximate cents, 'deg' means
degree of the note in that particular ET, and the single hyphens
are minus signs with values also in approximate cents)

--- JI ---- ---- 12tET ------ ----- 72tET --------
ratio ~cents deg cents JI-12tET deg ~cents 12-72tET ~diff

3/2 702.0 7 700 2.0 42 700.0 0 2.0
4/3 498.0 5 500 - 2.0 30 500.0 0 2.0
5/4 386.3 4 400 -13.7 23 383.3 -16.7 3.0
8/5 813.7 8 800 13.7 49 816.7 16.7 3.0
7/4 968.8 10 1000 -31.2 58 966.7 -33.3 2.2
8/7 231.2 2 200 31.2 14 233.3 33.3 2.2
11/8 551.3 6 600 -48.7 33 550.0 -50.0 1.3
16/11 648.7 6 600 48.7 39 650.0 50.0 1.3
13/8 840.5 8 800 40.5 50 833.3 33.3 7.2

It's easy to see how the deviation increases more dramatically
at the 13-limit.

But because the deviation is so small for the primary 3-, 5-,
7-, and 11-limit ratios, 72-tET is able to represent quite
extensive systems which use these ratios.

I made a lattice diagram of Partch's 43-tone scale represented
in 144-tET back in April 1999:

http://www.egroups.com/message/tuning/2398

The inconsistencies of 144-tET are plain to see (where the
accidental symbol in a chain of 3/2 changes, for instance).

I thought it would be a good idea to redraw the Partch lattice
in 72-tET to show how consistent *it* is.

I've tried to follow my own lattice formula, which is different
from the usual 'triangular' lattices posted here, because I
think it shows the 4 dimensions better ... for an explanation
see:

http://www.ixpres.com/interval/monzo/lattices/lattices.htm

This looks just a bit different from the lattices on my website
because of the limitations of ASCII. Here are the basic metrics:
(here, the caret ^ means 'raised to the power')

11^1
\
\
\
\
\
\
\ 3^1
\ /
5^1 _ \ /
'-._\ /
7^-1------------ n^0 --------------7^1
/ \'-._
/ \ ' 5^-1
/ \
3^-1 \
\
\
\
\
\
\
11^-1

The ASCII accidentals I use for 72-tET are:

sharp flat cents

+ - 1/12-tone 16.2/3
> < 1/6-tone 33.1/3
^ v 1/4-tone 50
# b 1/2-tone 100

And (drum roll, please...) here's the Partch lattice:

HARRY PARTCH 11-LIMIT 43-TONE SYSTEM

in Monzo ASCII 72-tET notation

G^
/ \ G+
/ \ /
/ \ E /
D#- ------------ C^ \ / '-._ /
\ / \'-._ \ / ' C+
\ / \ A< \ / /
\ / B>\----\--\------ A /
\ F^ / \ \ \ / \'-._ /
\ / \ / \ F#- \ / \ F+
\ / \/ \/ '-\_ \ / \ /
\ / E>\------/\----\- D -------/------ C<
\Bb^ / \ / \ \/ \'-._ /\ / '-._
C#+ --\\-------\- B- \ /\ \ Bb+ -----/----- Ab-
/ '-._\\/ \/ \'-._\ / \ \ / \ / /
/ A> -----/\--\--- G ---\-----------F< /
/ / \ / \ \ 1/1'-._\ /\ /\\'-._ /
F#+ ----------\- E- \ \/ \ Eb+ -----/--\\--- Db-
'-._ / \/ '-._ \ /\ \ / \ / Ev\
D> -----/\-----' C ---------------Bb< / \
/ \ / \'-\_ /\ /\ / \
A- \ / \ \Ab+ \ / \ / \
/ '-._ \ / \ \ \ / Av \
/ ' F ---------\-----Eb< / \
/ / \ F> \ / \
D- / \ '-._\ / \
/ '-._ / \ Dv ------------- B+
/ ' Bb \ /
/ \ /
G- \ /
Gv

The largest error for this notation from any ratio in the system
is ~3.9 cents for 9/8 and 16/9. Notice that, unlike the 144-tET
lattice, all chains of 3/2s have the same accidentals.

-monz

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html