Notation for n*12-ETs (was: A common notation for JI and ETs)

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>> > One thing that I thought should be taken into consideration is
>that,
>> > where appropriate, ET's that are subsets of others should make
>use of
>> > a subset of symbols of the larger ET. This would especially be
>> > advisable for ET's under 100 that are multiples of 12 -- if you
>learn
>> > 48-ET, you have already learned half of 96-ET.

Below, I notate all the most important multiples of 12-ET consistently with
each other, using symbols that correspond to 17-limit commas as described
previously in the "A common notation for JI and ETs" thread.

The real symbols that correspond to the ASCII versions below (and a lot
more), can be seen in
/tuning-math/files/secor/notation/Symbols3.bmp

But first some observations about apotome complements.

There is often a conflict between wanting to
(a) make ||\ the complement of /| or make /|| the complement of |\, and (b)
make the second half-apotome follow the same pattern of flags as the first.

There is another problem with making the second half apotomes agree between
n*12-ET and every second step of 2n*12-ET, even when there's perfect
agreement in the first half apotome. This problem occurs between 36 and 72,
but 24, 48 and 96 are ok.

In general, complement symbols are a pain in the posterior, and I'll leave
it for you to wrestle with them. I'm starting to think that the only way to
make them work is to make the second half-apotome the mirror image of the
first, (with the addition of a second shaft to each symbol).

So here's the first half-apotome of the most important n*12-ETs notated in
such a way that the same number of cents always has the same symbol. |) is
always the 7 comma and (| is always the 11'-7 comma. In all but 60, 72 and
132 they are also the 13-5 and 13'-(11-5) commas respectively.

1 2 3 4 5
Cents -> 012345678901234567890123456789012345678901234567890
12 (6 4 3 2):
24 (8): /|\
36 (18): |)
48 (16): ~|) /|\
60 (15, 20, 30): /| |\
72: /| |) /|\
84 (42): /| |) /|)
96 (32): /| ~|) |\ /|\
108 (54): /| //| |) /|)

132 (44 66): ~|( (| |) |\ /|)
Cents -> 012345678901234567890123456789012345678901234567890
1 2 3 4 5

If you really don't want //| as 2deg108 (and hence 1deg54) then you could
use /|( which isn't correct for 108-ET, but is correct for every 2nd
degree of 216-ET. //| is correct for both.

120-ET is omitted because it does not serve to notate any lower ETs (as
shown in parenthesis for those above). Also, it is not possible to notate
it consistently with 60-ET while respecting the same comma values for the
symbols. The 20 cent accidental is the problem. In 120-ET it must be (|,
which is zero steps in 60-ET. In 60-ET it must be /| which is 1 step (10
cents) in 120-ET.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com