Yahoo Tuning Groups Ultimate Backup tuning-math Scala's ennealimmal notation

--- In tuning-math@yahoogroups.com,
"Gene Ward Smith" <gwsmith@s...> wrote:

>
> Manuel has installed an
> ennealimma notation for
> 441 which uses sybol pairs
> for the following: 32805/32768,
> 225/224, 2048/2025, 81/80,
> 64/63, 128/125, (64/63)^2,
> and 2187/2048. I'd be interested
> in comments; this system is
> pretty elaborate, but it works.

i snipped Gene's table, and here
is my own version, which
includes cents values:

symbol ..... ratio .......... ~cents

< , > .. 32805 / 32768 .... 1.953720788
........ 65625 / 65536 .... 2.349476659

s , $ .... 225 / 224 ...... 7.711522991
......... 1029 / 1024 ..... 8.432720273
........ 15625 / 15552 .... 8.107278862

v , ^ ... 2048 / 2025 .... 19.55256881

\ , / ..... 81 / 80 ...... 21.5062896
......... 875 / 864 ..... 21.90204547
......... 3125 / 3087 .... 21.18084819

L , 7 ..... 64 / 63 ...... 27.2640918
.......... 686 / 675 ..... 27.98528908

( , ) .... 128 / 125 ..... 41.05885841

[ , ] .... 4096 / 3969 ... 54.5281836 = (64/63)^2

b , # .... 2187 / 2048 .. 113.6850061

-monz
http://tonalsoft.com
microtonal music software

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...>
wrote:
> >
> > Gene,
> >
> > Wouldn't:
> > s z
> > make a better symbol pair than:
> > s $
>
> I think you'e right: "s" looks like an up symbol, and "z" the
> corresponding down one to me.

You may be getting stuck in a local optimum here which will prevent
you from finding a global one. That is, we found that the pairing of s
and z prevented us from making optimum use of _all_ the available
ASCII characters as up/down accidental pairs.

And I don't understand why you say one looks more up, or more down,
than the other since they are both symmetrical under rotation by 180
degrees.

George and I went through many iterations before settling on the
following down up ASCII pairs.

We first decided not to use any uppercase letters A-Z or digits 0-9 so
these could be reserved as nominals, octave numbers, interval numbers
in chords etc (so the pair L7 was rejected on those grounds). We also
reserved lowercase "x" as double-sharp even though there is no
single-character for double-flat. And of course we retained the
conventional pair b#. We also decided to keep the existing HEWM pairs
-+ <> [] and of course the other bracket pairs () and {}. Then we have
other pairings such as \/ v^ un wm dq which are obvious enough that no
one is likely to disagree with them.

The following were more difficult and sometimes we had to break up an
otherwise good pairing so we could give another character a partner
with any kind of sense to it at all.

Colon : might have been used instead of semicolon ; below but we
decided to retain it as a separator in chords.

We have the following pairs of special characters where one is clearly
set lower than the other:

,`
.'
;"
_=

Now we have two equipositioned pairs of special characters:

!| (the dot on the exclamation mark has to be the head of the arrow)
&% (the embedded slashes \/ determine which is down and which is up)

The remaining special characters have no obvious pairs with each other
and were paired instead with lowercase letters. Note that the letter
is always the down character and the special character is always up.
This agrees with the conventional pair b#. In approximate order of
decreasing obviousness:

o@
j?
s$
z~
i*

Now we have some more pairs of lowercase letters. For these, up or
down was determined by imagining balancing each character about its
horizontal midline (ignoring its usual placement relative to the
baseline and ignoring any serifs). If it is top-heavy (fgpyr) it is an
up character, otherwise it is a down character. In approximate order
of decreasing obviousness:

tf
ag
hp (would have been bp except b is already used for b#)
ky
cr

Of all the printable ASCII characters, that leaves only e and l which
don't look like a pair by any stretch of the imagination, and the
lowercase letter l shouldn't be used in any case since it could be
mistaken for the digit 1, uppercase letter I, vertical bar | or
right-square bracket ].

In the far from obvious pairings where we were really scraping the
bottom of the barrel, such as i* and cr (where e might have been
used), our choices were also informed by the shapes of the _real_
sagittal symbols that we wanted them to stand for. Otherwise we first
paired them all up and then considered how best to match them to
sagittal pairs.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Danny Wier" <dawiertx@s...>
wrote:
> > Thank Manuel for me for adding 441-tone notation. Is he planning
on adding
> > 612-tone anytime soon? I know that has to be a lot of work.
>
> Hi Danny,
>
> Actually, adding new sagittal ET notations shouldn't involve any
work
> for Manuel at all. You can implement them yourself by editing the
file
> sag_et.par.
>
> Try adding the following lines to the end of it and then let Manuel
> and I know if there are any problems when you SET NOTATION SA612.
Try
> the various combinations of SET SAGITTAL (PURE | MIXED | LONG |
SHORT)
> with EQUAL 612 and SHOW and the Chromatic clavier etc.
>
> d 612 358
> '| .|( |( '|( .)~| )~| ')~| .~~| ~~| ./| /| '/|
> .|) |) '|) .(| (| '(| .(|( (|( .//| //| '//|
> ./|) /|) '/|) .(/| (/| '(/| |\) '|\) .(|\ (|\ '(|\
> .)N( )N( ')N( ~N( '~N( .)N~ )N~ ')N~ .N) N) 'N)
> .N\ N\ 'N\ ~N) '~N) .~N\ ~N\ '~N\ ./N) /N) '/N)
> ./N\ /N\

Danny, be forewarned that this is a division for which we haven't yet
agreed upon a symbol set, so some of these symbol assignments are
tenative at best.

Dave, since you've used some less common symbols, ~~| and especially )
~| (with accents), it appears that you're trying to use 7-limit
symbols as much as possible (and only 11-limit when absolutely
necessary). Since 612 is perfectly good at the 11-limit, and since
you have to use (| and (|( anyway, why not simplify this by using the
11-symbols )|(, |\, /|\, and (|) as well?

Some time ago, when I was investigating the problem of which Sagittal
symbols (plus accents) could notate complex 7-limit ratios
(consistent with the application of those symbols to the 494 and 612
divisions), I was wishing that |~ could be more meaningful at the 7
limit, since it would be the only unaccented symbol needed between )|
( and /| and, besides, it's so easy to read. Now, after taking a
closer look, I find that 243:245 would be .|~., making |~.
32768:33075, a schisma higher (and much simpler than whatever ratio
you get for .~~| ). For 612, just drop the right accents, giving the
following:

d 612 358
'| .|( |( '|( )|( )|(' .|~ |~ ~~| ./| /| '/|
.|) |) '|) |\ (| '(| .(|( (|( .//| //| '//|
./|) /|) '/|) /|\ (/| '(/| |\) (|) .(|\ (|\ '(|\
.)N( )N( ')N( ~N( '~N( .)N~ )N~ /N .N) N) 'N)
.N\ N\ 'N\ ~N) /N~ '/N~ .//N //N ./N) /N) '/N)
./N\ /N\

There is also the possibility of replacing ~~| with '|~, but I don't
think that's necessary or desirable.

When it comes to some of these complicated applications, I think it's
rather nice to have a generous inventory of symbols from which to
select.

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>> d 612 358
>> '| .|( |( '|( .)~| )~| ')~| .~~| ~~| ./| /| '/|
>> .|) |) '|) .(| (| '(| .(|( (|( .//| //| '//|
>> ./|) /|) '/|) .(/| (/| '(/| |\) '|\) .(|\ (|\ '(|\
...
> Danny, be forewarned that this is a division for which we haven't yet
> agreed upon a symbol set, so some of these symbol assignments are
> tenative at best.

Yes. Sorry George. I should have explained to Danny that that's only
one possible sagittal notation for 612-ET. And we have not yet agreed
on any notations involving accented symbols or any ETs beyond 224. Not
because we have any great disagreements about it, but mostly because
we have little idea of what principles should guide the design of such
complex notations.

And we have other priorities such as helping software designers
implement the simpler sagittal notations and completing the mythology
to help explain them.

But Danny, we'd really appreciate the input of someone like yourself
who is actually using (or at least thinking deeply about using)
notations for tunings as complex as 612-ET.

> Dave, since you've used some less common symbols, ~~| and especially )
> ~| (with accents), it appears that you're trying to use 7-limit
> symbols as much as possible (and only 11-limit when absolutely
> necessary).

No, that wasn't the intention. You will find that about one third of
the symbols primarily represent ratios of 11.

My approach this time was inspired by Gene's approach to 441-ET, but
with nominals in a chain of best fifths and no more than one sagittal
accidental per note. Like Gene I aimed to have a minimum number of
components that would cover it, while notating the most common commas.
Or looking at it another way, I wanted as simple a pattern as possible
to the sequence so that there would be some hope of being able to
remember it or figure it out without having to look up a table every time.

> Since 612 is perfectly good at the 11-limit, and since
> you have to use (| and (|( anyway, why not simplify this by using the
> 11-symbols )|(, |\, /|\, and (|) as well?

Well these are certainly symbols for common commas, but at the time I
thought that they could not be used without increasing the total
number of root symbols required (the set of unaccented symbols
obtained by removing the accents from all symbols used). I wanted to
see what it looked like when this minimisation was carried out.

There may however be other solutions with a pattern at least as
regular as mine and with no more root symbols, which include some of
the symbols you suggest. I didn't explore it exhaustively. I will do
so below.

You say "simplify by using [more unaccented symbols]", but this is
only one kind of simplification. The kind of simplification I was
aiming for was that of a regular pattern to the construction of the
symbols, with accents.

Given that the 5-schisma left accent represents a single step of
612-ET the simplest pattern would be to have unaccented symbols for
only every multiple of 3 steps and notate the rest as a schisma up and
down from these. Unfortunately this is not possible, since even with
the full superset, as shown in figure 3 of the XH article, there are
no unaccented symbols for 12, 15 or 24 steps. And what's more, two of
the most common notational commas 80:81 and 63:64 are not at multiples
of 3 but are 11 and 14 steps respectively. These are so popular that
they _must_ be notated with their usual unaccented symbols.

So it ocurred to me that the next best thing would be to have
unaccented symbols at every third degree, but with a hiccup at every
11th degree. And it turned out that this was possible. i.e. unaccented
symbols at degrees
0, 3, 6, 9,11,14,17,20,22,25,28.
where the step sizes between unaccented symbols are
3, 3, 3, 2, 3, 3, 3, 2, 3, 3.
There's no need to look beyond the half apotome (degree 29) since
those symbols are determined by the standard complementation rules.

Another sequence that equally satisfies the above criteria would be
0, 3, 6, 8,11,14,17,19,22,25,28
3, 3, 2, 3, 3, 3, 2, 3, 3, 3

This gives:
d 612 358
'| .|( |( '|( .)~| )~| .|~ |~ '|~ ./| /| '/|
.|) |) '|) .(| (| ./|~ /|~ '/|~ .//| //| '//|
./|) /|) '/|) .(/| (/| '(/| |\) '|\) .(|\ (|\ '(|\

Note that the layout I'm using is intended to make the pattern plain,
in that same-accent symbols are lined up underneath each other, at
least up to the half-apotome. In this case one could even replace '(/|
on the bottom line with .|\) and continue the pattern past the
half-apotome.

Here's the only other such sequence (maximally-even and minimally
populated while including 0, 11 and 14):
0, 3, 5, 8,11,14,16,19,22,25,27
3, 2, 3, 3, 3, 2, 3, 3, 3, 2

This gives:
d 612 358
'| .|( |( '|( )|( ')|( .|~ |~ '|~ ./| /| '/|
.|) |) '|) |\ '|\ ./|~ /|~ '/|~ .//| //| '//|
./|) /|) '/|) /|\ '/|\ '(/| .(|) (|) .(|\ (|\ '(|\

This does get rid of those rarely used )~| and ~~| symbols and limits
us to the standard set plus (/|. The only one that needs an 8-bit
short ASCII symbol is /|~ . So this is my favourite SA612 notation so
far. It differs from your latest, George, in only a few places.

Notice that the above has the property that whenever a right barb /|
can be added to a symbol to increase it by 11 steps (within our
constraint of no more than two flags) it actually does so. Except that
/| + |( = |) and so we do not have any symbol /|( .

> d 612 358
> '| .|( |( '|( )|( ')|( .|~ |~ ~~| ./| /| '/|
> .|) |) '|) |\ (| '(| .(|( (|( .//| //| '//|
> ./|) /|) '/|) /|\ (/| '(/| |\) (|) .(|\ (|\ '(|\
...
> There is also the possibility of replacing ~~| with '|~, but I don't
> think that's necessary or desirable.

I do.

> When it comes to some of these complicated applications, I think it's
> rather nice to have a generous inventory of symbols from which to
> select.

I agree. Although it turns out we don't need them in this particular case.

Here's the short ASCII for my latest 612 proposal. The numbers are
degrees of 612-ET.

dn up
---------
1 . '
2 'c .r
3 c r
4 .c 'r
5 i *
6 'i .*
7 .z '~
8 z ~
9 'z .~
10 '\ ./
11 \ /
12 .\ '/
13 't .f
14 t f
15 .t 'f
16 k y
17 .k 'y
18 'ø .ð
19 ø ð 248 lowercase o slash, 240 lowercase eth (iceland)
20 .ø 'ð
21 '_ .=
22 _ =
23 ._ '=
24 'u .n
25 u n
26 .u '^
27 v ^
28 .v '^
29 .a 'g
30 'o .@
31 o @
32 'w .m
33 w m
34 .w 'm

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

It used to be that we aimed to notate 494-ET with no accented
characters. But that's no longer possible and no longer seems important.

The 5-schisma is one step of 494-ET. I'm interested in what we get if
we apply the principles I outlined in the 612-ET proposal. Namely, the
distribution of unaccented symbols should be maximally-even and
minimally-populated within the contraint of repeating at the 5-comma
and including the 7-comma.

In 494 the 5-comma is 9 steps and the 7-comma is 11 steps.

This means we must have
/| |)
0 2 ... 9 11 ... 18 20

If we also include the 11-medium-diesis at 22 steps then we have:
/| |) /|\
0 2 4 ... 9 11 13 ... 18 20 22

So for maximal evenness we can have either
/| |) /|\
0 2 4 6 9 11 13 15 18 20 22
or
/| |) /|\
0 2 4 7 9 11 13 16 18 20 22

Both are possible, but the latter uses the more common symbols. It is
unfortunate that it involves a lateral confusability between ~| and |~

|( )~| |~ /| |) |\ ~|) //| /|) /|\
0 2 5 7 9 11 13 15 18 20 22

(|(
|( ~| |~ /| |) |\ /|~ //| /|) /|\
0 2 4 7 9 11 13 16 18 20 22

441-ET is similar. We have
5-comma = 8 steps
7-comma = 10 steps
11-m-diesis = 20 steps

If we want all three of these commas to be unaccented and want to
repeat at the 5-comma then the maximally-even minimally-populated
sequence is unique and completely even.
~~|
~|( (| (|(
|( )~| |~ /| |) |\ /|~ //| /|) /|\
0 2 4 6 8 10 12 14 16 18 20

-- Dave Keenan

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> It used to be that we aimed to notate 494-ET with no accented
> characters. But that's no longer possible and no longer seems
important.

Yes, because there were too many symbol cores to remember. But I
think you've gone too far in the opposite direction with what you're
now proposing, if you're intending these to be the "standard" or
default 494 and 612 symbol sets -- too many symbols with accents, too
many less-common symbol cores, and not enough commonality between the
two divisions.

The property of evenness is interesting, but I don't see that it
offers any clear advantages to insist that it be employed in a
standard symbol set.

> The 5-schisma is one step of 494-ET. I'm interested in what we get
if
> we apply the principles I outlined in the 612-ET proposal. Namely,
the
> distribution of unaccented symbols should be maximally-even and
> minimally-populated within the contraint of repeating at the 5-comma
> and including the 7-comma.
>
> In 494 the 5-comma is 9 steps and the 7-comma is 11 steps.
>
> This means we must have
> /| |)
> 0 2 ... 9 11 ... 18 20
>
> If we also include the 11-medium-diesis at 22 steps then we have:
> /| |) /|\
> 0 2 4 ... 9 11 13 ... 18 20 22
>
> So for maximal evenness we can have either
> /| |) /|\
> 0 2 4 6 9 11 13 15 18 20 22
> or
> /| |) /|\
> 0 2 4 7 9 11 13 16 18 20 22
>
> Both are possible, but the latter uses the more common symbols. It
is
> unfortunate that it involves a lateral confusability between ~| and
|~
>
> |( )~| |~ /| |) |\ ~|) //| /|) /|\
> 0 2 5 7 9 11 13 15 18 20 22
>
> (|(
> |( ~| |~ /| |) |\ /|~ //| /|) /|\
> 0 2 4 7 9 11 13 16 18 20 22

The second one could also be either of the following, taking the )|
flag as 2deg:

|( )|( |~ /| |) |\ /|~ //| /|) /|\
|( )|( |~ /| |) |\ (|( //| /|) /|\
0 2 4 7 9 11 13 16 18 20 22

But what's the reason for using /|~ instead of the more common (|(,
other than that it makes a matching symbol sequence in pure
Sagittal? Is it that for 15deg we would like to represent 48:49
and/or 49:50, and ./|~ comes much closer than .(|(? If so, we can do
still do much better than that (see below).

I would think that anyone notating anything as complicated as 494 or
612 would (or should) already know the athenian symbol set. For 612
I would therefore want to see an 11-limit symbol set that uses as
many of the athenian-level core symbols as possible (taking |~ as
32768:33075):

612: '| .|( |( '|( )|( ')|( .|~ |~ '|~ ./| /| '/| .|)
|) '|) |\ (|
'(| .(|( (|( .//| //| '//| ./|) /|) '/|) /|\
(/| '(/| .|\) |\) (|) .(|\ (|\ '(|\

I would much prefer (/| to '/|\, because it represents a simpler
ratio and is one less accented symbol to deal with.

For 494, I would think that it would make a lot of sense to use a
subset of the above:

494a: '| |( '|( )|( ')|( .|~ |~ ./| /| '/| |) '|) |\ (|
'(| (|( .//| //| ./|) /|) '/|) /|\ '(/| .|\) (|) .(|\
(|\ '(|\

Where there were choices to be made between accented symbols, I chose
the one that would correspond to a less complex ratio, e.g., ./|) in
preference to '//| . However, I chose '(| over .(|( because it comes
much closer to notating both 48:49 and 49:50, which we would expect
to see frequently in other important temperaments.

I also used '(/| instead of '/|\ because it corresponds to a much
simpler ratio (392:405 vs. 10935:11264) at a lower prime limit (7 vs.
11); this symbol is useful for notating the generator for the
temperament family that Gene called "hemififth," and we can also
expect to see this one in other temperaments.

In case we wanted to make this more athenian-core, we can always use
the 17-comma symbol:

494b: '| |( '|( )|( ')|( ~|( '~|( ./| /| '/| |) '|) |\
(|
'(| (|( .//| //| ./|) /|) '/|) /|\ '(/| .|\) (|) .(|\
(|\ '(|\

and this would make it completely athenian-core:

494c: '| |( '|( )|( ')|( ~|( '~|( ./| /| '/| |) '|) |\
(|
'(| (|( .//| //| ./|) /|) '/|) /|\ '/|\ .(|) (|) .(|\
(|\ '(|\

> 441-ET is similar. We have
> 5-comma = 8 steps
> 7-comma = 10 steps
> 11-m-diesis = 20 steps
>
> If we want all three of these commas to be unaccented and want to
> repeat at the 5-comma then the maximally-even minimally-populated
> sequence is unique and completely even.
> ~~|
> ~|( (| (|(
> |( )~| |~ /| |) |\ /|~ //| /|) /|\
> 0 2 4 6 8 10 12 14 16 18 20

Here's how I would do it:

441a: '| |( '|( ')|( .|~ |~ ./| /| '/| |) '|) (|
'(| (|( .//| //| ./|) /|) '/|) /|\ '(/| (|) .(|\
(|\ '(|\

441b: '| |( )|( ')|( .|~ |~ ./| /| '/| |) '|) (|
'(| (|( .//| //| ./|) /|) '/|) /|\ '(/| (|) .(|\
(|\ '(|\

I wasn't at all happy with )~|, and I thought that ')|( as 2816:2835
would be much better. I was reluctant to replace '|( with )|(, since
a 7-limit ratio, 224:225, seems to be more suited to this division.
For accidentals to a 9-nominal (7-limit) ennealimal notation, I would
probably use:

441a: '| .~| ~| '~| .|~ |~ ./| /| '/| |) '|) .~|)
~|) '~|) .//| //| ./|) /|) '/|) (/| '(/| |\) .(|\
(|\ '(|\

Something I consider more important than evenness is how many
ennealimmal generators are involved with each symbol:

deg symbol generators ratio
-------------------------------------------
0: |//| +0G 1:1
3: ~| -8G 224:225
5: .|~ +3G 243:245, 125:126
6: |~ -16G 32768:33075
8: /| -5G 80:81
10: |) +6G 63:64
13: ~|) -2G 48:49, 49:50
16: //| -10G 6400:6561
18: /|) +1G 35:36
20: (/| +12G 3969:4096
21: .(/| -7G 392:405
23: .(|\ +4G 27:28
24: (|\ -15G 8192:8505

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Yes, because there were too many symbol cores to remember. But I
> think you've gone too far in the opposite direction with what you're
> now proposing, if you're intending these to be the "standard" or
> default 494 and 612 symbol sets -- too many symbols with accents, too
> many less-common symbol cores, and not enough commonality between the
> two divisions.

The nexus between the two divisions is of course 494&612, which is
kwazy. In the 5-limit it is defined by the kwazy comma of |-53 10 16>,
about 4/7 of a cent in size. This leads to a 5-limit temperament with
a half-octave period and a generator which is a schisma larger than
9/7, and hence is a "quasi" 9/7. The 5-limit Graham complexity of 26
is high, as expected for a microtemperament, but much less than the
7-limit or 11-limit Graham complexity of 162. In kwazy, you get a lot
of essentially just 5-limit harmony, and the "quasi" 7-limit, before
eventually the 11-limit turns up.

> I would think that anyone notating anything as complicated as 494 or
> 612 would (or should) already know the athenian symbol set. For 612
> I would therefore want to see an 11-limit symbol set that uses as
> many of the athenian-level core symbols as possible (taking |~ as
> 32768:33075):
>
> 612: '| .|( |( '|( )|( ')|( .|~ |~ '|~ ./| /| '/| .|)
> |) '|) |\ (|
> '(| .(|( (|( .//| //| '//| ./|) /|) '/|) /|\
> (/| '(/| .|\) |\) (|) .(|\ (|\ '(|\

The trouble with this, of course, is that in ascii characters they are
very complicated.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> Yes, because there were too many symbol cores to remember.

Sorry I forgot we already agreed on the word "cores" for what I called
"roots". "Cores" it is.

> But I
> think you've gone too far in the opposite direction with what you're
> now proposing, if you're intending these to be the "standard" or
> default 494 and 612 symbol sets

Yes I am. But I prefer to think of them as the "example" sets in the
case of these complex ETs. This should become clearer below.

> -- too many symbols with accents, too
> many less-common symbol cores, and not enough commonality between the
> two divisions.

Accented symbols are only complex if you insist on knowing what comma
the whole symbol primarily represents. If you are using an ET, you
don't necessarily care. It may be good enough to know that the accent
itself represents a fixed number of steps of the ET, e.g. one, and if
you're a JI type, that it represents a 5-schisma alteration.

Why is comonality between 494 and 612 more important than say 494 and
224. Many such pairings represent good (multi-)linear temperaments and
only one such pairing can be honored in the example notations.

Note that an example notation for 612 or 494 ET is a different thing
from a standard notation for JI that has resolution comparable to 612
or 494-ET.

I think it's futile to try to come up with a single best notation for
such complex ETs. Their very complexity means that people can approach
them from so many different directions. JI or not. Linear temperament
or not. And if linear temperament then which one.

And yet it is important that there should be at least one example
given on the sagittal website and implemented in Scala and FTS etc.
just so people know it is possible to notate such large ETs in
sagittal and don't just go off and start inventing more incompatible
systems. Once they know it is possible they can develop their own
sagittal notation for it, that matches their perspective.

And I am concerned to minimise the "freak-out factor". Anyone seing a
notation for such a complex ET for the first time is going to be
freaked out by the complexity. People with little knowledge of
sagittal, let alone the athenian set, will be looking at these things.
I am concerned to minimise this factor by having the example notation
be as regular as possible.

That is, for the examples I'd like a notation that is "compressible",
i.e. can be described in a few words or a short table with a short set
of symbol construction rules. Think of this as a public-relations
requirement if you like.

When people learn more symbol-comma relationships, and if they come
from a JI perspective, they will automatically want to substitute some
of the more common unaccented athenian symbols for some of the
accented ones in the regularised "example" notation. And that will be
just fine.

> The property of evenness is interesting, but I don't see that it
> offers any clear advantages to insist that it be employed in a
> standard symbol set.

I hope I have explained that above.
> > |( )~| |~ /| |) |\ ~|) //| /|) /|\
> > 0 2 5 7 9 11 13 15 18 20 22
> >
> > (|(
> > |( ~| |~ /| |) |\ /|~ //| /|) /|\
> > 0 2 4 7 9 11 13 16 18 20 22
>
> The second one could also be either of the following, taking the )|
> flag as 2deg:
>
> |( )|( |~ /| |) |\ /|~ //| /|) /|\
> |( )|( |~ /| |) |\ (|( //| /|) /|\
> 0 2 4 7 9 11 13 16 18 20 22
>
> But what's the reason for using /|~ instead of the more common (|(,
> other than that it makes a matching symbol sequence in pure
> Sagittal? Is it that for 15deg we would like to represent 48:49
> and/or 49:50, and ./|~ comes much closer than .(|(? If so, we can do
> still do much better than that (see below).

Look again. I didn't choose. I showed both (one above the other).
Another reason for choosing /|~ however is that both /| and |~ are
already in the sequence, so there's less to remember and more that can
be figured out as a pattern, which is the same principle I am pushing
to its extreme here with minimal core-count and maximal evenness.

If you mean why didn't I use (|( instead of /|~ in my 612 proposal,
it's because (|( is 20 steps whereas /|~ is the 19 steps that I needed.

> I would think that anyone notating anything as complicated as 494 or
> 612 would (or should) already know the athenian symbol set.

Yes. But that doesn't mean they have any idea how the athenian symbols
end up spaced out in 494 or 612. They are highly irregular and
therefore very hard to remember. I think we should make more use of
regular patterns of spacing, and flag arithmetic. And of course
respect the odd consistency limit of the division.

> I would much prefer (/| to '/|\, because it represents a simpler
> ratio and is one less accented symbol to deal with.

I suppose that's OK since I have an accented version of this symbol
anyway.

>
> For 494, I would think that it would make a lot of sense to use a
> subset of the above:
>
> 494a: '| |( '|( )|( ')|( .|~ |~ ./| /| '/| |) '|) |\ (|
> '(| (|( .//| //| ./|) /|) '/|) /|\ '(/| .|\) (|) .(|\
> (|\ '(|\

But in 494-ET the primary comma for Poseidon's trident )|( [the
7:11-kleisma 891:896] is not 4 steps but 5.

-- Dave Keenan

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> > But Danny, we'd really appreciate the input of someone like
yourself
> > who is actually using (or at least thinking deeply about using)
> > notations for tunings as complex as 612-ET.
>
> My recommendation is that the notation be aimed at the most
> characteristic prime limit and temperament of the et. That would
mean,
> for 224, the 11 and especially the 13 limit, and 13-limit octoid.
For
> 612 it could be the 5, 7 or 11 limit; going with the highest limit
> (another good principle, I think) says to notate it in
> hemiennealimmal, but ennealimmal would also make sense. 270 again is
> strong in multiple limits: 7, 11, and 13. More than one notation
makes
> sense, ennealimmal, hemiennealimmal, and the 270&494 temperament all
> being live possibilities, with others such as 80&270 being
interesting
> also.

Dave & I are presently trying to establish standard symbol sets for 7
nominals in a chain of (best) fifths for the better ET's. This is
the approach that will be most easily understood by virtually all
musicians, many of whom would freak out at the idea of learning new
nominals. Nevertheless, we're very much in favor of developing such
notations, because they're going to be very attractive to those with
a more adventurous spirit, such as yourself.

One issue we're going to be wrestling with is how closely the
selection of the set of accidentals for a particular ET should
coincide when different sets of nominals are used. Deciding these
things can be very time-consuming, and someone who has had some
experience actually composing music in the temperament would probably
have additional insight regarding what would be desirable. Thus,
some of our present decisions are at best tentative.

> > My approach this time was inspired by Gene's approach to 441-ET,
but
> > with nominals in a chain of best fifths and no more than one
sagittal
> > accidental per note.
>
> What did you think of my proposal of taking a chain of eight fifths
> (so nine notes in the chain) as a starting point for ennealimmal? If
> you could notate 171 by such a system, you could get things such as
> 441 simply by tacking on or taking off the appropriate number of
schismas.

Unfortunately, a set of nine tones in a chain of fifths isn't a
constant structure, hence wouldn't be suitable as nominals for a
notation. It looks like we'll have to stick with nominals separated
by 1/9-octave.

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > Yes, because there were too many symbol cores to remember.
>
> Sorry I forgot we already agreed on the word "cores" for what I
called
> "roots". "Cores" it is.
>
> > But I
> > think you've gone too far in the opposite direction with what
you're
> > now proposing, if you're intending these to be the "standard" or
> > default 494 and 612 symbol sets
>
> Yes I am. But I prefer to think of them as the "example" sets in the
> case of these complex ETs. This should become clearer below.
>
> > -- too many symbols with accents, too
> > many less-common symbol cores, and not enough commonality between
the
> > two divisions.
>
> Accented symbols are only complex if you insist on knowing what
comma
> the whole symbol primarily represents. If you are using an ET, you
> don't necessarily care. It may be good enough to know that the
accent
> itself represents a fixed number of steps of the ET, e.g. one, and
if
> you're a JI type, that it represents a 5-schisma alteration.
>
> Why is comonality between 494 and 612 more important than say 494
and
> 224.

It isn't.

> Many such pairings represent good (multi-)linear temperaments and
> only one such pairing can be honored in the example notations.

By commonality I mean that:

1) ET's should have in common as many accidentals as possible; and
2) They should have as many commonly-used accidentals (i.e., those
with athenian cores) as possible.

> Note that an example notation for 612 or 494 ET is a different thing
> from a standard notation for JI that has resolution comparable to
612
> or 494-ET.

Yes.

> I think it's futile to try to come up with a single best notation
for
> such complex ETs. Their very complexity means that people can
approach
> them from so many different directions. JI or not. Linear
temperament
> or not. And if linear temperament then which one.
>
> And yet it is important that there should be at least one example
> given on the sagittal website and implemented in Scala and FTS etc.
> just so people know it is possible to notate such large ETs in
> sagittal and don't just go off and start inventing more incompatible
> systems.

Yes, of course.

> Once they know it is possible they can develop their own
> sagittal notation for it, that matches their perspective.

Meaning a modified symbol set, or a different set of nominals and
accidentals. I doubt that very many would acquire the expertise to
do that properly without the aid of detailed reference manual. (So
little time, so many things ...)

> And I am concerned to minimise the "freak-out factor". Anyone seing
a
> notation for such a complex ET for the first time is going to be
> freaked out by the complexity. People with little knowledge of
> sagittal, let alone the athenian set, will be looking at these
things.
> I am concerned to minimise this factor by having the example
notation
> be as regular as possible.
>
> That is, for the examples I'd like a notation that
is "compressible",
> i.e. can be described in a few words or a short table with a short
set
> of symbol construction rules. Think of this as a public-relations
> requirement if you like.
>
> When people learn more symbol-comma relationships, and if they come
> from a JI perspective, they will automatically want to substitute
some
> of the more common unaccented athenian symbols for some of the
> accented ones in the regularised "example" notation. And that will
be
> just fine.

I'm a bit concerned about introducing too many non-athenian cores in
these example notations. People have already been freaked out by the
total number of symbols we have, and if we have to introduce too many
symbol cores beyond the athenian set *overall* in order to have fewer
symbols *within* a given ET, it's going to be self-defeating.

> > The property of evenness is interesting, but I don't see that it
> > offers any clear advantages to insist that it be employed in a
> > standard symbol set.
>
> I hope I have explained that above.
> > > |( )~| |~ /| |) |\ ~|) //| /|) /|\
> > > 0 2 5 7 9 11 13 15 18 20 22
> > >
> > > (|(
> > > |( ~| |~ /| |) |\ /|~ //| /|) /|\
> > > 0 2 4 7 9 11 13 16 18 20 22
> >
> > The second one could also be either of the following, taking the )
|
> > flag as 2deg:
> >
> > |( )|( |~ /| |) |\ /|~ //| /|) /|\
> > |( )|( |~ /| |) |\ (|( //| /|) /|\
> > 0 2 4 7 9 11 13 16 18 20 22
> >
> > But what's the reason for using /|~ instead of the more common (|
(,
> > other than that it makes a matching symbol sequence in pure
> > Sagittal? Is it that for 15deg we would like to represent 48:49
> > and/or 49:50, and ./|~ comes much closer than .(|(? If so, we
can do
> > still do much better than that (see below).
>
> Look again. I didn't choose. I showed both (one above the other).

Okay, sorry. I've been very pressed for time lately and didn't look
carefully enough.

> Another reason for choosing /|~ however is that both /| and |~ are
> already in the sequence, so there's less to remember and more that
can
> be figured out as a pattern, which is the same principle I am
pushing
> to its extreme here with minimal core-count and maximal evenness.
>
> If you mean why didn't I use (|( instead of /|~ in my 612 proposal,
> it's because (|( is 20 steps whereas /|~ is the 19 steps that I
needed.

I didn't mean that, but it's just as well that you brought it up,
because it's an excellent example of using an unusual symbol /|~ to
achieve distributional evenness when an athenian-level symbol (within
a suitable odd limit) could have been used instead.

> > I would think that anyone notating anything as complicated as 494
or
> > 612 would (or should) already know the athenian symbol set.
>
> Yes. But that doesn't mean they have any idea how the athenian
symbols
> end up spaced out in 494 or 612. They are highly irregular and
> therefore very hard to remember. I think we should make more use of
> regular patterns of spacing, and flag arithmetic.

Which comes at the expensive of having to know more symbols overall
(and perhaps wondering what they really mean, besides degrees of an
ET) -- time to freak out!!! 8<O

> And of course
> respect the odd consistency limit of the division.

Of course.

> > I would much prefer (/| to '/|\, because it represents a simpler
> > ratio and is one less accented symbol to deal with.
>
> I suppose that's OK since I have an accented version of this symbol
> anyway.
>
> >
> > For 494, I would think that it would make a lot of sense to use a
> > subset of the above:
> >
> > 494a: '| |( '|( )|( ')|( .|~ |~ ./| /| '/| |) '|)
|\ (|
> > '(| (|( .//| //| ./|) /|) '/|) /|\ '(/| .|\) (|) .
(|\
> > (|\ '(|\
>
> But in 494-ET the primary comma for Poseidon's trident )|( [the
> 7:11-kleisma 891:896] is not 4 steps but 5.

How did you get that? 7:11k (891:896) is:
7L (27:28) minus 11M (32:33) or 26 - 22 = 4 deg.
The flags normally add up to 3deg, but )| isn't being used as a 19-
schisma anywhere, so both )| and |( can be 2deg494.

--George

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> By commonality I mean that:
>
> 1) ET's should have in common as many accidentals as possible; and
> 2) They should have as many commonly-used accidentals (i.e., those
> with athenian cores) as possible.

Yes those are good principles, but I'd allow the trojan symbols |~ and
/|~ in that as well, i.e. all the mythological symbols excluding )/| .
I'd also allow the non-mythological ~|) and (/| and ~| . I also don't
have a problem with including maybe one other non-mythological such as
)~| in an accented ET if it makes the sequence easier to remember and
does not introduce any lateral confusability.

Maybe we should try to agree on a ranking of all single-shaft
unaccented symbols less than a half-apotome, according to decreasing
commonness of use. This might be the basis of a simple algorithm for
deciding which symbol to apply to each degree of an ET.

I suggest it should start something like this:

/|\ /| |) /|) |( |\ //| (|( )|( (| ~|( ~|) (/| ~|
|-------spartan --------|
|----------------athenian------------------------|

/|~ )~| |~ ...

If I list the symbols in order of decreasing popularity of the primary
commas shown below, according to my formula that takes into account
the Scala archive ocurrences of the ratios they notate and their
distance along the chain of fifths, I get the following ranking.

/| |) //| /|) /|\ |( ~|) (|( ~|( )|( (| (/| ~| )|
5 7 25 35 11 5:7 49 5:11 17 7:11 7:11 49 17 19

/|~ ~|\ )~| |\ )|) )/| )|~ |~ ~~| (|~ |~) |~\
7:13 23 11:35 55 7:19 5:19 19 23 11:25 11:19 ? ?

Notice that )~| as the 11:35-kleisma (11-prime-limit) comes out as
slightly more popular than the 55-comma represented by |\ and it
introduces no lateral confusability.

> > Once they know it is possible they can develop their own
> > sagittal notation for it, that matches their perspective.
>
> Meaning a modified symbol set, or a different set of nominals and
> accidentals. I doubt that very many would acquire the expertise to
> do that properly without the aid of detailed reference manual. (So
> little time, so many things ...)

No, not different nominals, just departing from the even sequence of
the example notation, by substituting a few unaccented athenians for
ratios that are meaningful to them in place of some accented symbols.

> I'm a bit concerned about introducing too many non-athenian cores in
> these example notations. People have already been freaked out by the
> total number of symbols we have, and if we have to introduce too many
> symbol cores beyond the athenian set *overall* in order to have fewer
> symbols *within* a given ET, it's going to be self-defeating.

Yes. You're quite right. But maybe we can do it by adding only

~|) (/| ~| /|~ )~| |~ (in order of choice).

> > I think we should make more use of
> > regular patterns of spacing, and flag arithmetic.
>
> Which comes at the expensive of having to know more symbols overall
> (and perhaps wondering what they really mean, besides degrees of an
> ET) -- time to freak out!!! 8<O

What do you mean by "having to know more symbols overall"? What
evidence is there that someone who uses say 612-ET will also want to
use several other divisions of similar size?

Anyway, after having explored maximal evenness, let me back off a
little and say merely that
(a) I don't think we should ever have two unaccented symbols next to
each other in notations that use accents, if we can avoid it by using
symbols from athenian plus ~|) (/| ~| /|~ )~|

And I think we should aim for maximal evenness and minimal core-count
within a notation that includes /| |) and /|\ .

> > > For 494, I would think that it would make a lot of sense to use a
> > > subset of the above:
> > >
> > > 494a: '| |( '|( )|( ')|( .|~ |~ ./| /| '/| |) '|)
> |\ (|
> > > '(| (|( .//| //| ./|) /|) '/|) /|\ '(/| .|\) (|) .
> (|\
> > > (|\ '(|\
> >
> > But in 494-ET the primary comma for Poseidon's trident )|( [the
> > 7:11-kleisma 891:896] is not 4 steps but 5.
>
> How did you get that? 7:11k (891:896) is:
> 7L (27:28) minus 11M (32:33) or 26 - 22 = 4 deg.
> The flags normally add up to 3deg, but )| isn't being used as a 19-
> schisma anywhere, so both )| and |( can be 2deg494.

I goofed. I was looking at 612, not 494. Sorry. Yes, we should use )|(
for 4deg494. I also goofed on the first of my 494 sequences that
supposedly repeated at the 5-comma. It should have been

|( )|( ~|( /| |) |\ ~|) //| /|) /|\
0 2 4 6 9 11 13 15 18 20 22
2 2 2 3 2 2 2 3 2 2
| |

~~| /|~
|( )|( |~ /| |) |\ (|( //| /|) /|\
0 2 4 7 9 11 13 16 18 20 22
2 2 3 2 2 2 3 2 2 2
| |

Here are some that repeat at the 7-comma.
~~|
|( )|( |~ /| |) |\ ~|) //| /|) /|\
0 2 4 7 9 11 13 15 18 20 22
2 2 3 2 2 2 2 3 2 2
| |

/|~
|( )~| |~ /| |) |\ (|( //| /|) /|\
0 2 5 7 9 11 13 16 18 20 22
2 3 2 2 2 2 3 2 2 2
| |

Here are some less regular ones, only using athenian symbols, but
still having no unaccented symbols next to each other.

|( )|( ~|( /| |) |\ (|( //| /|) /|\ (|)
0 2 4 6 9 11 13 16 18 20 22 25
2 2 2 3 2 2 3 2 2 2 3

|( )|( ~|( /| |) (| (|( //| /|) /|\
0 2 4 6 9 11 14 16 18 20 22
2 2 2 3 2 3 2 2 2 2

So, based on the second-last one above, I'm proposing the purely
athenian-cored:

494b: '| |( '|( )|( ')|( ~|( '~|( ./| /| '/| |)
.|\ |\ '|\ .(|( (|( .//| //| ./|) /|) ./|\
/|\ '/|\ .(|) (|) '(|) (|\ '(|\

I believe this also has the advantage of matching flag sequences in
the two half-apotomes in the pure notation. This is a result of the
fact that the pattern of steps between unaccented symbols from natural
to /|\ has reflective symmetry about its midpoint. Perhaps that's a
more useful requirement generally, than repeating at the 5-comma or
7-comma. I should go back and try that on 612-ET.

-- Dave Keenan

Scala's ennealimmal notation

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Danny Wier <dawiertx@sbcglobal.net>

🔗monz <monz@tonalsoft.com>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗George D. Secor <gdsecor@yahoo.com>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>