Notating the 13 limit in Sagittal

One way we might go about doing this is as follows: we first note the
index of {3/2, 2187/2048} in arrow is 88, which means one out of
every 88 13-limit intervals can be expressed with only fifths and
sharp/flats. Choosing some metric on the exponents of 64/63 and
81/80, we choose the closest lattice point to the interval in the
lattice defined by 3/2 and 2187/2048. Now we use symbols for 1,2,3
64/63 and 1,2,3,4 81/80 to get from the lattice point to the
interval, and that will be something in Sagittal, and least if we are
willing to add symbols for 40/39 and 22/21.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> One way we might go about doing this is as follows: we first note
the
> index of {3/2, 2187/2048} in arrow is 88, which means one out of
> every 88 13-limit intervals can be expressed with only fifths and
> sharp/flats. Choosing some metric on the exponents of 64/63 and
> 81/80, we choose the closest lattice point to the interval in the
> lattice defined by 3/2 and 2187/2048. Now we use symbols for 1,2,3
> 64/63 and 1,2,3,4 81/80 to get from the lattice point to the
> interval, and that will be something in Sagittal, and least if we
are
> willing to add symbols for 40/39 and 22/21.

There is already a separate symbol for 39:40 in extreme-precision
Sagittal:
|~)

And if there is already a symbol for any rational interval <68.57c,
there will also already be one for its apotome-complement. Since
there is already a symbol for 45056:45927:
(|
then there is also already a symbol for 21:22:
)||~

Now what was it that you were trying to do (that I haven't read about
yet)?

--George

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

> Now what was it that you were trying to do (that I haven't read
about
> yet)?

I start by pointing out that up to equivalence by schisminas, the
largest of which is 1716/1715 at 1 cent, and the other three of which
are already in use by you and Dave, you can express every 13 limit
interval in terms of the two most basic of the Sagittal commas, 81/80
and 64/63. This is by using the "arrow" linear temperament with that
basis. Since 81/80 gives a straight flag and 64/63 gives a curved
flag, you ought to be able to expeditiously notate the 13-limit (and
the arrow temperaments 46, 224, 270 and 494) by suitable Sagittal
symbols.

This is a finite problem--one out of every 22 13-limit intervals is
notatable already just by octaves and fifths. Hence you need only a
finite collection of symbols, which should be concoctable from
straight and curved flags, to fully notate the 13-limit in a
perspicuous manner, consistent with arrow and with the schisminas you
already have introduced. The 81/80-64/63 representation of 3/2 is [4,
23] and of 2 is [6, 40]; you can make a sort of Fokker block out of
these, and ask for symbols for all 22 of the pairs [a, b] defined by
the parallogram [0,0], [4,23], [6,40], [10,63], which we can
translate to [-5, -31], [-1,-8], [1,9], [5, 32]. Then every 13-limit
note becomes so many octaves, so many fifths, and one of these 22
symbols (11 symbol pairs will do it), which are combinations of 81/80
and 64/63.

Arrow is really an obvious choice if we want to consider the 13-
limit, and let higher limits do their best to fit in. I ran a search
for the very best 13-limit ets up to 10000, and obtained the
following:

224 .780361
270 .732188
494 .688526
1506 .761715
6079 .663004

The second number is a 13-limit consistent logflat badness figure,
which I bounded by 0.8. If you take any two of 224, 270 or 494
together, you get arrow, and arrow happens to have 81/80 and 64/63 as
a generator pair, which is perfect for Sagittal.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> > Now what was it that you were trying to do (that I haven't read
about
> > yet)?
>
> I start by pointing out that ...
>
> The second number is a 13-limit consistent logflat badness figure,
> which I bounded by 0.8. If you take any two of 224, 270 or 494
> together, you get arrow, and arrow happens to have 81/80 and 64/63
as
> a generator pair, which is perfect for Sagittal.

Gene, I intended that to be a rhetorical question -- but thanks for
the explanation, anyway! :-)

Best,

--George

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I start by pointing out that up to equivalence by schisminas, the
> largest of which is 1716/1715 at 1 cent, and the other three of
which
> are already in use by you and Dave, you can express every 13 limit
> interval in terms of the two most basic of the Sagittal commas,
81/80
> and 64/63. This is by using the "arrow" linear temperament with
that
> basis. Since 81/80 gives a straight flag and 64/63 gives a curved
> flag, you ought to be able to expeditiously notate the 13-limit
(and
> the arrow temperaments 46, 224, 270 and 494) by suitable Sagittal
> symbols.

You certainly can. Except that we made a decision not to have more
than two flags per symbol as it would lead to symbols which were
typographically too complex to be distinct on the staff, and indeed
we found that we didn't _need_ to have more. So some of the symbols
required for the 13-limit don't consist only of straight and convex
flags (barbs and arcs) but instead use convex or wavy flags (scrolls
or boathooks) via other (23-limit) schisminas.

The schisminas indicated as vanishing in table 1 in the XH paper do
not necessarily vanish in the Olympian (extreme precision) JI
notation which involves added 5-schisma accents (which is the only
way we sort-of allow 3 flags per symbol). Olympian is not yet
settled and it would be good if you could take a look at the current
proposal and see if it can be improved. After I get it together and
put it up on the Sagittal site. I'll let you know.

Generally only schisminas smaller than about 0.4 c will continue to
vanish at the Olympian level.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> The schisminas indicated as vanishing in table 1 in the XH paper do
> not necessarily vanish in the Olympian (extreme precision) JI
> notation which involves added 5-schisma accents (which is the only
> way we sort-of allow 3 flags per symbol). Olympian is not yet
> settled and it would be good if you could take a look at the
current
> proposal and see if it can be improved. After I get it together and
> put it up on the Sagittal site. I'll let you know.

You could kick the ball off by telling me which intervals you
tentitively propose to make equivalent.

> Generally only schisminas smaller than about 0.4 c will continue to
> vanish at the Olympian level.

Any other rules? Do we want to go to the 31 limit, or where?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > The schisminas indicated as vanishing in table 1 in the XH paper
do
> > not necessarily vanish in the Olympian (extreme precision) JI
> > notation which involves added 5-schisma accents (which is the
only
> > way we sort-of allow 3 flags per symbol). Olympian is not yet
> > settled and it would be good if you could take a look at the
> current
> > proposal and see if it can be improved. After I get it together
and
> > put it up on the Sagittal site. I'll let you know.
>
> You could kick the ball off by telling me which intervals you
> tentitively propose to make equivalent.

It looks like 1171-ET with some of the gaps filled in. That's all I
can give you at short notice.

> > Generally only schisminas smaller than about 0.4 c will continue
to
> > vanish at the Olympian level.
>
> Any other rules? Do we want to go to the 31 limit, or where?

We found we only needed to go to the 23-limit for primary roles of
the symbols and then we could get the 47-limit (insane I know) as
sufficiently good approximations.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> It looks like 1171-ET with some of the gaps filled in. That's all I
> can give you at short notice.

Why 1171? A decent 23-limit system, but at first blush, notable
mostly as a 5-limit et. 1578 or 2460 seem a more obvious route to the
23-limit; the latter is divisible by 12 in case anyone cares about
that.

Putting the two together gives the following 23-limit linear
temperament:

[<6 13 16 14 13 35 20 41 38|, <0 27 16 -22 -60 99 -35 120 84|]

81 symbol pairs would suffice to get you an octave-fifth-symbol
representation of the 23-limit to extreme accuracy.

If we put 1171 with 1578, we get

[<1 85 35 -551 -6 8 -548 -203 63|, <0 97 38 -644 -11 5 -642 -241 68|]

Now 49 symbol pairs will give you octave-fifth-symbol.

I'll take a closer look at 1171, and post what I find on tuning-math.