Atomic temperament as a basis for a valhalla-level notation

Atomic temperament, in 2460-et, has a period of exactly 100 cents and
a generator of size one step of 615-equal, a type of schisma. The
lowest Graham complexity for the 23-limit is given by

[<12 19 28 34 42 43 47 50 52|, <0 1 -7 -16 -25 72 105 50 117|]

so this I propose as the official 23-limit atomic mapping. 2460 is a
strong temperament up to the 27-odd-limit, and so makes a good basis
for defining atomic. Since 205 schismas make up 400 cents in this
tuning, we can obtain alternate mappings by raising or lowering the
generator steps by 205. We want -25 schismas for 11 at one end, and
+50 schismas for 19 at the other, which sticks us with this mapping if
we want minimal Graham complexity. 50+88 is 138, less than 117+25=142,
so we might bring 117 down to -88; however if we do that then 105+88
is 193, whereas bringing it down to -100 leaves us with 100+88=188,
and clearly leaving 117 as it is means we have to leave 72 and 105
alone also.

An atomic notation would, I presume, have twelve nominals dividing the
octave into twelve parts exactly. Other symbols would be defined in
terms of schismas and semitones. One curious feature would be that in
the 5-limit, with the atom as a comma, it would be a very, very, very,
very precise notation for the 5-limit. One question I have is how, and
how easily, 2460 and atomic could be notated in Sagittal.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Atomic temperament, in 2460-et, has a period of exactly 100 cents
and
> a generator of size one step of 615-equal, a type of schisma. The
> lowest Graham complexity for the 23-limit is given by
>
> [<12 19 28 34 42 43 47 50 52|, <0 1 -7 -16 -25 72 105 50 117|]
>
> so this I propose as the official 23-limit atomic mapping. 2460 is
a
> strong temperament up to the 27-odd-limit, and so makes a good
basis
> for defining atomic. Since 205 schismas make up 400 cents in this
> tuning, we can obtain alternate mappings by raising or lowering the
> generator steps by 205. We want -25 schismas for 11 at one end, and
> +50 schismas for 19 at the other, which sticks us with this
mapping if
> we want minimal Graham complexity. 50+88 is 138, less than
117+25=142,
> so we might bring 117 down to -88; however if we do that then
105+88
> is 193, whereas bringing it down to -100 leaves us with 100+88=188,
> and clearly leaving 117 as it is means we have to leave 72 and 105
> alone also.
>
> An atomic notation would, I presume, have twelve nominals dividing
the
> octave into twelve parts exactly. Other symbols would be defined in
> terms of schismas and semitones. One curious feature would be that
in
> the 5-limit, with the atom as a comma, it would be a very, very,
very,
> very precise notation for the 5-limit. One question I have is how,
and
> how easily, 2460 and atomic could be notated in Sagittal.

Well it certainly can't notate every step of 2460-ET. But it should
be able to notate the distributionally even (DE or "MOS") scales
having the following numbers of steps to the octave: 612 and 624,
and it would go close to doing 1236.

If you really had to do 2460-ET you'd have to invent a new kind of
accent mark that is defined as some comma (specifically a schismina)
somewhere in the 0.3 to 1 cent range, preferably around 0.6 or 0.7
of a cent, that doesn't vanish in the 12-fold schismas (atomic)
temperament.

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

> If you really had to do 2460-ET you'd have to invent a new kind of
> accent mark that is defined as some comma (specifically a schismina)
> somewhere in the 0.3 to 1 cent range, preferably around 0.6 or 0.7
> of a cent, that doesn't vanish in the 12-fold schismas (atomic)
> temperament.

I don't know that I really have to do it, but it seems worthwhile to
notate if you are on a quest for ultra notations, which I gathered
from what George said you were. We already have our nominals in a very
familiar form, say Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#. It would
be possible to find 23 limit intervals corresponding to n schismas for
various values of n, and then see how Sagittal would propose to notate
those intervals. This would, it seems to me, give a good start towards
notating atomic. Of course another approach would be to notate the
5-limit and then use the result for the 23-limit.

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

It would
> be possible to find 23 limit intervals corresponding to n schismas for
> various values of n, and then see how Sagittal would propose to notate
> those intervals. This would, it seems to me, give a good start towards
> notating atomic. Of course another approach would be to notate the
> 5-limit and then use the result for the 23-limit.

Here are some 11-limit intervals corresponding to certain quantities
of schismas in atomic. If we had symbols for these, it would be a start.

1: {32805/32768, 703125/702464, 5632/5625}
2: {441/440, 1375/1372, 6250/6237}
3: {3136/3125, 5120/5103}
4: {225/224}
5: {176/175, 896/891}
7: {4000/3969, 126/125}
9: {99/98, 100/99}
10: {2048/2025}
11: {81/80, 3125/3087, 245/242}
14: {64/63}
16: {56/55}
18: {50/49}
20: {45/44}
21: {128/125}
25: {36/35}
29: {405/392}
32: {648/625}
39: {256/245}
43: {360/343}

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> Here are some 11-limit intervals corresponding to certain
quantities
> of schismas in atomic. If we had symbols for these, it would be a
start.

Yes. We can supply a symbol for every multiple of the 5-schisma.

I'll put the ASCII longhand version of the symbol before the ratio
below that is it's definition or primary role. Any with an asterisk
after the ratio is still open for discussion about whether that
should be the symbol's defining ratio or not.

There was only one for which the symbol of the right size doesn't
currently have one of the commas you listed, as its primary role,
and that's the two schismas symbol .|(, for which we have 385/384 as
its primary role.

> 1: '| 32805/32768, 703125/702464, 5632/5625
> 2: 441/440, 1375/1372, 6250/6237, .|( 385/384*
> 3: 3136/3125, |( 5120/5103
> 4: '|( 225/224
> 5: .)~| 176/175*, )|( 896/891
> 7: 4000/3969, ')~| 126/125*
> 9: .)|~ 99/98*, ~~| 100/99*
> 10: ./| 2048/2025
> 11: /| 81/80, 3125/3087, 245/242
> 14: |) 64/63
> 16: .(| 56/55
> 18: '(| 50/49*
> 20: (|( 45/44
> 21: .//| 128/125
> 25: /|) 36/35
> 29: '(/| 405/392
> 32: '(|) 648/625
> 39: ||~ 256/245*
> 43: '/||360/343*

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

> There was only one for which the symbol of the right size doesn't
> currently have one of the commas you listed, as its primary role,
> and that's the two schismas symbol .|(, for which we have 385/384 as
> its primary role.

This suggests there may be a pretty good fit to atomic. 385/384 is
mapped by atomic to one semitone, down 49 schismas, so it isn't what
we'd want to use for 2 schismas; but a 441/440 symbol fits nicely.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> > There was only one for which the symbol of the right size
doesn't
> > currently have one of the commas you listed, as its primary
role,
> > and that's the two schismas symbol .|(, for which we have
385/384 as
> > its primary role.
>
> This suggests there may be a pretty good fit to atomic. 385/384 is
> mapped by atomic to one semitone, down 49 schismas, so it isn't
what
> we'd want to use for 2 schismas; but a 441/440 symbol fits nicely.

OK. Well you saw that 385/384 isn't set in stone for this symbol.
441/440 was second in line. That tentative decision was based on the
count of ratios in the Scala archive that would be notated exactly
by these commas. I just looked up the actual numbers and there were
52 ocurrences for 385/384 and 51 for 441/440, so that's neither here
nor there.

I think that your "atomic" observation, plus the fact that 441/440
(which we'd call the 49:55-schisma) is closer to the sum-of-flags,
means that it should indeed be the primary role of .|( . What do you
think Mr Secor?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> >
> > > There was only one for which the symbol of the right size
doesn't
> > > currently have one of the commas you listed, as its primary
role,
> > > and that's the two schismas symbol .|(, for which we have
385/384 as
> > > its primary role.
> >
> > This suggests there may be a pretty good fit to atomic. 385/384 is
> > mapped by atomic to one semitone, down 49 schismas, so it isn't
what
> > we'd want to use for 2 schismas; but a 441/440 symbol fits nicely.
>
> OK. Well you saw that 385/384 isn't set in stone for this symbol.
> 441/440 was second in line. That tentative decision was based on
the
> count of ratios in the Scala archive that would be notated exactly
> by these commas. I just looked up the actual numbers and there were
> 52 ocurrences for 385/384 and 51 for 441/440, so that's neither
here
> nor there.
>
> I think that your "atomic" observation, plus the fact that 441/440
> (which we'd call the 49:55-schisma) is closer to the sum-of-flags,
> means that it should indeed be the primary role of .|( . What do
you
> think Mr Secor?

I think you have a winner!

--George

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> ...
> If you really had to do 2460-ET you'd have to invent a new kind of
> accent mark that is defined as some comma (specifically a
schismina)
> somewhere in the 0.3 to 1 cent range, preferably around 0.6 or 0.7
> of a cent, that doesn't vanish in the 12-fold schismas (atomic)
> temperament.^

Since the 5-schisma is 4deg2460, you would need *two* new accent
marks.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > ...
> > If you really had to do 2460-ET you'd have to invent a new kind
of
> > accent mark that is defined as some comma (specifically a
> schismina)
> > somewhere in the 0.3 to 1 cent range, preferably around 0.6 or
0.7
> > of a cent, that doesn't vanish in the 12-fold schismas (atomic)
> > temperament.^
>
> Since the 5-schisma is 4deg2460, you would need *two* new accent
> marks.

Yikes! Well fortunately no one needs to do it.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > ...
> > If you really had to do 2460-ET you'd have to invent a new kind of
> > accent mark that is defined as some comma (specifically a
> schismina)
> > somewhere in the 0.3 to 1 cent range, preferably around 0.6 or 0.7
> > of a cent, that doesn't vanish in the 12-fold schismas (atomic)
> > temperament.^
>
> Since the 5-schisma is 4deg2460, you would need *two* new accent
> marks.

What you want, evidently, is a symbol for [1, -51]; that is, 100 cents
minus 51 schismas. If you had something lying around for 2401/2400,
(0.72 cents) that would be perfect; or 3025/3024 or 4375/4374 would
do. If you also need something for four times that, or [4, -204],
notating 24192/24167 would do it, but I'm not clear why this would be
needed.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan"
<d.keenan@b...>
> > wrote:
> > > ...
> > > If you really had to do 2460-ET you'd have to invent a new
kind
> of
> > > accent mark that is defined as some comma (specifically a
> > schismina)
> > > somewhere in the 0.3 to 1 cent range, preferably around 0.6 or
> 0.7
> > > of a cent, that doesn't vanish in the 12-fold schismas
(atomic)
> > > temperament.^
> >
> > Since the 5-schisma is 4deg2460, you would need *two* new accent
> > marks.
>
> Yikes! Well fortunately no one needs to do it.

Wait a minute. We already get 1 cent resolution even though our
smallest flag (the accent) is about 2 cents, because of the way
things just happen to overlap. So we only need one other accent (say
a wavy one) to do 2460-ET. But I was wrong about what its size would
need to be. Ideally it would be in the range of about 0.33 cents to
0.5 cents.

Ideally it would be defined as whatever (in that size range)
vanishes most often between primary and secondary roles in the
Olympian notation. But of course this is all completely insane.

It might however shed some light on what commas to prefer for
primary roles in Olympian.

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

> Wait a minute. We already get 1 cent resolution even though our
> smallest flag (the accent) is about 2 cents, because of the way
> things just happen to overlap. So we only need one other accent (say
> a wavy one) to do 2460-ET. But I was wrong about what its size would
> need to be. Ideally it would be in the range of about 0.33 cents to
> 0.5 cents.

I'm not clear on where all this is coming from; however commas
which map to [1, -51] in atomic and seem the most plusible are
2401/2400, 0.72 cents; 3025/3024, 0.57 cents, 4096/4095, 0.42 cents
and 4375/4374, 0.40 cents. Presumably you'd choose 4375/4374.

A sumbol for [1, -51] would be very useful in atomic, as would one for
[-2, 103], which would mean a symbol for 1716/1715 would be nice to have.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> Wait a minute. We already get 1 cent resolution even though our
> smallest flag (the accent) is about 2 cents, because of the way
> things just happen to overlap. So we only need one other accent
(say
> a wavy one) to do 2460-ET. But I was wrong about what its size
would
> need to be. Ideally it would be in the range of about 0.33 cents to
> 0.5 cents.

Ah so, I see! But have you checked all 116 steps up to 1/2-apotome
to see if every step would be covered by a combination of flags and
accents? The region between 90 and 99 steps looks like it could be
troublesome.

I had hoped that Gene would have come up with an octave division
somewhere between 1600 and 2000. (You know, this *really is* insane!)

--George

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

> I had hoped that Gene would have come up with an octave division
> somewhere between 1600 and 2000. (You know, this *really is* insane!)

Why in there? We could use 1848 in connection with atomic; it is a
strong 11-limit temperament which supports atomic. I was picking on
2460 since it is a 17-27 limit temperament which tells me what mapping
to use
in higher limits, but if you want something in the range 1800 to 2000
which supports atomic, 1848 is absolutely perfect.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > I had hoped that Gene would have come up with an octave division
> > somewhere between 1600 and 2000. (You know, this *really is*
insane!)
>
> Why in there?

Because it takes the 5-schisma as 3 steps, and an additional pair
accent-mark symbols would serve for intermediate single-steps.

> We could use 1848 in connection with atomic; it is a
> strong 11-limit temperament which supports atomic. I was picking on
> 2460 since it is a 17-27 limit temperament which tells me what
mapping to use
> in higher limits, but if you want something in the range 1800 to
2000
> which supports atomic, 1848 is absolutely perfect.

No, the higher prime limit offered by 2460 is preferable.

--George

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

> > We could use 1848 in connection with atomic; it is a
> > strong 11-limit temperament which supports atomic. I was picking on
> > 2460 since it is a 17-27 limit temperament which tells me what
> mapping to use
> > in higher limits, but if you want something in the range 1800 to
> 2000
> > which supports atomic, 1848 is absolutely perfect.
>
> No, the higher prime limit offered by 2460 is preferable.

The val I gave for 1848 also gets to up to 23; it is equally well an
atomic val. Others are:

612: 51 periods and 1 generator
<612 970 1421 1718 2117 2265 2502 2600 2769|
[1, -51]-commas: 2401/2400, 3025/3024, 4096/4095, 4375/4375

This is actually the standard val up through the 19 limit.

1236: 103 periods and 2 generators
<1236 1959 2870 3470 4276 4573 5051 5250 5590|
[-2, 103]-commas: 1716/1715, 2080/2079

Standard through the 11-limit

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

Because of its 5-limit importance, we should also add this:

4296: 358 periods and 7 generators
<4296 6809 9975 12060 14861 15898 17561 18250 19435|
[7, -358]-commas: 85750/85683, 114345/114244, pirate |-90 -15 45>,
raider |71 -99 37>

Standard through the 7-limit

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
>
> > > We could use 1848 in connection with atomic; it is a
> > > strong 11-limit temperament which supports atomic. I was
picking on
> > > 2460 since it is a 17-27 limit temperament which tells me what
mapping to use
> > > in higher limits, but if you want something in the range 1800
to 2000
> > > which supports atomic, 1848 is absolutely perfect.
> >
> > No, the higher prime limit offered by 2460 is preferable.
>
> The val I gave for 1848 also gets to up to 23; it is equally well an
> atomic val.

As I said, going this high is bordering on insanity. I'll leave it
to you and Dave to settle on something.

> Others are:
>
> 612: 51 periods and 1 generator
> <612 970 1421 1718 2117 2265 2502 2600 2769|
> [1, -51]-commas: 2401/2400, 3025/3024, 4096/4095, 4375/4375
>
> This is actually the standard val up through the 19 limit.

And this is amount of resolution that Dave and I agreed on for high-
precision Sagittal, so it is nice that you have val-idated this
choice.

How do 494 and 1600 fare?

--George

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

> How do 494 and 1600 fare?

Fare how? They'd lead to different systems; hercules aka 224&270 is a
494-temperament, of course.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> > wrote:
> >
> > > > We could use 1848 in connection with atomic; it is a
> > > > strong 11-limit temperament which supports atomic. I was
> picking on
> > > > 2460 since it is a 17-27 limit temperament which tells me
what
> mapping to use
> > > > in higher limits, but if you want something in the range
1800
> to 2000
> > > > which supports atomic, 1848 is absolutely perfect.
> > >
> > > No, the higher prime limit offered by 2460 is preferable.
> >
> > The val I gave for 1848 also gets to up to 23; it is equally
well an
> > atomic val.
>
> As I said, going this high is bordering on insanity. I'll leave
it
> to you and Dave to settle on something.
>
> > Others are:
> >
> > 612: 51 periods and 1 generator
> > <612 970 1421 1718 2117 2265 2502 2600 2769|
> > [1, -51]-commas: 2401/2400, 3025/3024, 4096/4095, 4375/4375
> >
> > This is actually the standard val up through the 19 limit.
>
> And this is amount of resolution that Dave and I agreed on for
high-
> precision Sagittal, so it is nice that you have val-idated this
> choice.

In tuning, Gene wrote:
> Were you going to or did you add a lynchpin symbol for the lynchpin
> comma, 4096/4095; or alternatively for the ennealimmal commas,
> 2401/2400 or 4375/4374?

Sorry Gene. I guess you're waiting for a decision on this before
going any further with Olympian.

I don't see any need to add such a symbol, but if that's what it
takes to keep your interest, lets assume we have a second up/down-
pair of accents. Let's call the existing accent the "left accent" and
this new one the "right accent". So we add a 10th column to the
right of the ones for the existing 9 flags, and it can have values -
1, 0 1, independent of the others.

Let's give the right accent the primary role 4095:4096. Secondary
roles will no doubt include 4374:4375 and 3024:3025, but probably
not 2400:2401 since it is nearly twice as big.

Given that it's difficult to make different shapes of accent
distinct, and considering that we've already used up the full
character map available for a symbol-font, we can just reuse the
same accent symbols (represented in ASCII as the ' and . characters)
but put them to the right of any arrow symbol instead of the left.

The right accent is strictly a "semantic radicals" (the left accent
pretty much is too). That is, it doesn't chage the sound, only the
meaning.

The only thing is, we would need to change the existing single
charater '| and .! symbols in the TrueType font (for accents applied
directly to natural notes) to bare up and down shafts | and !, so
the accents could be put on either side of them. Then '| .! |' !.
would all be two-character symbols. It's not too late to do that.

One way to approach it would be to generate all 73 possible max-two-
at-a-time sets of ordinary flags on a single shaft, combined with
all 9 possible sets of accents, for a total of 657 compound single-
shaft symbols, and find the largest (atomic?) ET that you can notate
with them (keeping consistent with the existing comma definitions in
Table 1). I expect it will be at least 2460. You can ignore any
symbols larger than a half-apotome.

There will of course be many compuond symbols that notate the same
degree of this ET and we can use this to whittle down the set of
single shaft symbols to a minimum. We can only accomodate 26 of them
per half-apotome in the symbol-font map. And we should ensure that
they are such that we can still notate 1171-ET without using right-
accents (5.7.13-schismina accents).

Then having chosen a set of symbols to notate this ET (as way of
getting a fairly evenly spaced set, we should define comma ratios
for them all.

Another constraint on the symbol set is that we must be able to
notate 494-ET without using any accents.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> Another constraint on the symbol set is that we must be able to
> notate 494-ET without using any accents.

Ouch! How are you going to do that? There are 3 problems:

1) What are you going to use for 19 degrees? I was planning on '|~)
(as 5:13S+5s, 53248:54675), since I thought that we could allow an
accented symbol in the set as long as its unaccented version was not
also present (as I also expected would be done for the 47-step high-
precision JI that I was proposing).

2) Does )|~ get a new primary value (other than 19456:19683) so it
can be used for 8 degrees? You realize that )|~ can't be 2025:2048
(diaschisma), since that's already taken by ./| . Or should we
allow )|~ to represent a diaschisma at a non-accented-symbol level
for a less-than-19-limit application?

3) What do we use for 3 degrees? )|( as 7:11k (891:896) would be 4,
but the symbol arithmetic dictates 3. For 4deg we can have ~|
(2176:2187), but the best thing I can find for 3deg is '|(
(224:225). So we end up with 2 symbols in the 494 set differing by
an accent mark.

Also, we need a name for the high-precision symbol level that doesn't
use accents (or at least has no two symbols differing by an accent
mark). Before I suggested herculean for 47-step JI and promethean
for 58-step JI, but you've been using herculean for 58-steps. If
that's what you want to keep, then what name applies to 47 steps?

I thought I wouldn't be involved much in this discussion, but here I
am! 8-O

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > Another constraint on the symbol set is that we must be able to
> > notate 494-ET without using any accents.
>
> Ouch! How are you going to do that? There are 3 problems:

Oops. Sorry. I misremembered. Forget that, Gene.
...

> Also, we need a name for the high-precision symbol level that
doesn't
> use accents (or at least has no two symbols differing by an accent
> mark).

Is that what I'm remembering? No two symbols differing by an accent
mark?

> Before I suggested herculean for 47-step JI and promethean
> for 58-step JI,

That's fine.

> but you've been using herculean for 58-steps.

Have I? I thought I was just using Herculean for an unspecified high-
precision version of the notation, either 47 or 58 steps to the
apotome. Herculean for 47-step JI and Promethean for 58-step JI is
fine by me.

OK. I'm certifiable. I couldn't wait for Gene to look at it. I
checked whether we could notate 2460-ET, or rather 233-EDA, the 233
Equal Division of the Apotome, using the existing symbols (Figure 3)
plus the existing 5-schisma left accent, plus the new 5.7.13-
schismina (tridecimal schismina) right accent. The answer is yes
(sort of). The following is the assignment of steps to flags (just
over twice the number of cents in each comma, relative to
Pythagorean)

'| )| (| /| ~| |~ |\ |) |( |'
4 7 68 44 18 34 65 56 12 1

The "sort of" comes because the 7:11-kleisma is
no longer )|( but )|('

and the 29-comma is no longer (| but (|' or '|\
but that was only a secondary role for (| anyway.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> OK. I'm certifiable. I couldn't wait for Gene to look at it. I
> checked whether we could notate 2460-ET, or rather 233-EDA, the 233
> Equal Division of the Apotome, using the existing symbols (Figure 3)
> plus the existing 5-schisma left accent, plus the new 5.7.13-
> schismina (tridecimal schismina) right accent. The answer is yes
> (sort of). The following is the assignment of steps to flags (just
> over twice the number of cents in each comma, relative to
> Pythagorean)
>
> '| )| (| /| ~| |~ |\ |) |( |'
> 4 7 68 44 18 34 65 56 12 1

Of course, I was interested in atomic more than 2460 per se. From the
above we could deduce

4 ~ [0,1]: schisma, 5632/5625 or 32805/32768
7 ~ [-1, 53]: 513/512 (540/539, 4000/3993)
68 ~ [0,17]: 45927/45056 (4459/4374)
44 ~ [0,11]: 81/80 (245/242, 3125/3087)
18 ~ [2,-98]: 2187/2176 (196/195)
34 ~ [-2, 111] 736/729 ([2,-94]: 105/104 is also 34 steps of 2460)
65 ~ [1,-35]: 55/54
12 ~ [0, 3]: 5120/5103 (3136/3125)
1 ~ [1, -51]: 4096/4095 (4375/4374, 3025/3024, 2401/2400

Not exactly elegant, but certainly more than sufficient. A [0,7],
[0,8], [0,9], [0,16] or even [0,25] symbol would be nice, or as many
of them as are handy. For example we have a [0,17] symbol, and hence
can say 7/6 is [0,17] flat from eb/d# of 12-equal. Of course by
combining 1, 3, 11, and 17 we can get to these. Mention was made of a
441/440, which would give us a [0,2], 225/224 would be [0,4], we
already have 896/891 which is a [0,5], 126/125 would be [0,7], and
handy to have, 99/98 or 100/99 is [0,9], 50/49 or 56/55 would give me
my [0,16], and 36/35 would be a [0,25].

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
50/49 or 56/55 would give me
> my [0,16], and 36/35 would be a [0,25].

No, 50/49 is clearly much too much larger than to equate to 56/55; it
is in fact [0,18].

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > Another constraint on the symbol set is that we must be able to
> > > notate 494-ET without using any accents.
> >
> > Ouch! How are you going to do that? There are 3 problems:
>
> Oops. Sorry. I misremembered. Forget that, Gene.
> ...
> > Also, we need a name for the high-precision symbol level that
doesn't
> > use accents (or at least has no two symbols differing by an
accent
> > mark).
>
> Is that what I'm remembering? No two symbols differing by an
accent
> mark?

Since 47-step JI can't be done completely without accents, this is
the next best thing.

See my recent discussion re 47 and 58-step high-precision JI options:

/tuning/topicId_54244.html#54617

> > Before I suggested herculean for 47-step JI and promethean
> > for 58-step JI,
>
> That's fine.
>
> > but you've been using herculean for 58-steps.
>
> Have I? I thought I was just using Herculean for an unspecified
high-
> precision version of the notation, either 47 or 58 steps to the
> apotome.

I had the impression that it was a foregone conclusion that we would
be having a 58-step high-precision JI and that the 47-step JI was in
question.

> Herculean for 47-step JI and Promethean for 58-step JI is
> fine by me.

Hooray!

This, then, would almost certainly have to be the 47-step JI symbol
sequence:

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

This gives a JI notation that's 29-limit unique. Exact symbol
boundaries would still need to be established.

The 494-ET symbol sequence would be almost the same, but would
require one or two more single-shaft accented symbols:

Definitely: '|( in place of ~|
And possibly: ./| in place of )|~

In any event, these are good ones to have, since they represent
224:225 and 2025:2048, respectively.

--George

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > OK. I'm certifiable. I couldn't wait for Gene to look at it. I
> > checked whether we could notate 2460-ET, or rather 233-EDA, the
233
> > Equal Division of the Apotome, using the existing symbols
(Figure 3)
> > plus the existing 5-schisma left accent, plus the new 5.7.13-
> > schismina (tridecimal schismina) right accent. The answer is yes
> > (sort of). The following is the assignment of steps to flags
(just
> > over twice the number of cents in each comma, relative to
> > Pythagorean)
> >
> > '| )| (| /| ~| |~ |\ |) |( |'
> > 4 7 68 44 18 34 65 56 12 1
>
> Of course, I was interested in atomic more than 2460 per se. From
the
> above we could deduce
>
> 4 ~ [0,1]: schisma, 5632/5625 or 32805/32768
> 7 ~ [-1, 53]: 513/512 (540/539, 4000/3993)
> 68 ~ [0,17]: 45927/45056 (4459/4374)
> 44 ~ [0,11]: 81/80 (245/242, 3125/3087)
> 18 ~ [2,-98]: 2187/2176 (196/195)
> 34 ~ [-2, 111] 736/729 ([2,-94]: 105/104 is also 34 steps of 2460)
> 65 ~ [1,-35]: 55/54
> 12 ~ [0, 3]: 5120/5103 (3136/3125)
> 1 ~ [1, -51]: 4096/4095 (4375/4374, 3025/3024, 2401/2400
>
> Not exactly elegant, but certainly more than sufficient. A [0,7],
> [0,8], [0,9], [0,16] or even [0,25] symbol would be nice, or as
many
> of them as are handy.

The above do not represent symbols, but symbol-components called
flags. You put them together to make symbols. A symbol can have -1,
0 or 1 of each of the first or last flags (called accents) but of
the other eight combined, it can have a total of at most two, in the
same direction (up or down).

And in fact there's a further constraint due to the limitations of
Symbol fonts, that we can have at most 26 unaccented symbols between
the natural and the half-apotome. So we need to choose the
combinations carefully, and of course those 17 (from Table 1 plus
the two extra Trojan symbols) that I gave as 9-vectors, must be
present.

> For example we have a [0,17] symbol, and hence
> can say 7/6 is [0,17] flat from eb/d# of 12-equal.

I was wondering how 12-equal got in here, so I referred back to the
start of this thread and realised I've got my threads crossed.
/tuning-math/message/10853
I see now that this thread has to do with finding out if sagittal
accidentals can be used to notate the 12-fold schismas temperament
(atomic) relative to a set of nominals that correspond to 12-equal.

Of course I'm more concerned with looking at atomic for help in
defining the maximum precision notation for JI that's possible with
Sagittal as it currently stands, based on 7 nominals in a chain of
just fifths.

If the nominals are in 12-equal, and hence the notational fifth (the
approximation to 2:3 that the comma sizes depend on) is 700 cents
then this changes the size of the commas completely, and none of
what we have worked out above is valid. Instead, this would be an
extension of the Trojan or 12-relative notation. It would be filling
in the gaps in Figure 10 of the XH paper.

For 12-equal nominals, each flag must be assigned a number of steps
of 2460-ET based on the size of its defining comma (Table 1)
relative to the fifths (prime 3's) of 12-equal. In Paul Erlich's
notation, we need the size of d;n rather than d:n. So whatever the
exponent of 3 in the comma, the comma size will change by that many
times lg2(3/2)*1200 - 700 cents.

This raises the question of why anyone would want a notation for
something that is indistinguishable (by ear) from just intonation
(2460-ET) and yet does not use C:G to notate a just fifth, but
rather something differing from it by 4 steps.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > OK. I'm certifiable. I couldn't wait for Gene to look at it. I
> > > checked whether we could notate 2460-ET, or rather 233-EDA, the
233
> > > Equal Division of the Apotome, using the existing symbols
(Figure 3)
> > > plus the existing 5-schisma left accent, plus the new 5.7.13-
> > > schismina (tridecimal schismina) right accent.

Since the fifth of 2460-ET is accurate to within 0.004c (and thus the
apotome to within 0.026c), I don't imagine that there will be any
significant difference between basing the notation on 2460-ET and 233-
EDA.

> > > The answer is yes
> > > (sort of). The following is the assignment of steps to flags
(just
> > > over twice the number of cents in each comma, relative to
> > > Pythagorean)
> > >
> > > '| )| (| /| ~| |~ |\ |) |( |'
> > > 4 7 68 44 18 34 65 56 12 1
> >
> > Of course, I was interested in atomic more than 2460 per se. From
the
> > above we could deduce
> >
> > 4 ~ [0,1]: schisma, 5632/5625 or 32805/32768
> > 7 ~ [-1, 53]: 513/512 (540/539, 4000/3993)
> > 68 ~ [0,17]: 45927/45056 (4459/4374)
> > 44 ~ [0,11]: 81/80 (245/242, 3125/3087)
> > 18 ~ [2,-98]: 2187/2176 (196/195)
> > 34 ~ [-2, 111] 736/729 ([2,-94]: 105/104 is also 34 steps of 2460)
> > 65 ~ [1,-35]: 55/54
> > 12 ~ [0, 3]: 5120/5103 (3136/3125)
> > 1 ~ [1, -51]: 4096/4095 (4375/4374, 3025/3024, 2401/2400
> >
> > Not exactly elegant, but certainly more than sufficient. A [0,7],
> > [0,8], [0,9], [0,16] or even [0,25] symbol would be nice, or as
many
> > of them as are handy.
>
> The above do not represent symbols, but symbol-components called
> flags. You put them together to make symbols. A symbol can have -1,
> 0 or 1 of each of the first or last flags (called accents) but of
> the other eight combined, it can have a total of at most two, in
the
> same direction (up or down).
>
> And in fact there's a further constraint due to the limitations of
> Symbol fonts, that we can have at most 26 unaccented symbols
between
> the natural and the half-apotome. So we need to choose the
> combinations carefully, and of course those 17 (from Table 1 plus
> the two extra Trojan symbols) that I gave as 9-vectors, must be
> present.
>
> > For example we have a [0,17] symbol, and hence
> > can say 7/6 is [0,17] flat from eb/d# of 12-equal.
>
> I was wondering how 12-equal got in here, so I referred back to the
> start of this thread and realised I've got my threads crossed.
> /tuning-math/message/10853
> I see now that this thread has to do with finding out if sagittal
> accidentals can be used to notate the 12-fold schismas temperament
> (atomic) relative to a set of nominals that correspond to 12-equal.
>
> Of course I'm more concerned with looking at atomic for help in
> defining the maximum precision notation for JI that's possible with
> Sagittal as it currently stands, based on 7 nominals in a chain of
> just fifths.

Yes, and you were supposed to be calling the extreme-precision
notation *olympian*. (The gods are not pleased at this latest trend
in extreme-precision nomenclature, so I've taken the liberty of
modifying the subject line before Zeus gets wind of this.)

> If the nominals are in 12-equal, and hence the notational fifth
(the
> approximation to 2:3 that the comma sizes depend on) is 700 cents
> then this changes the size of the commas completely, and none of
> what we have worked out above is valid. Instead, this would be an
> extension of the Trojan or 12-relative notation. It would be
filling
> in the gaps in Figure 10 of the XH paper.

A notation of that sort would amount to the 12-ET subdivisionists of
Troy exacting revenge. This could get ugly. :-(

--George

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

> Yes, and you were supposed to be calling the extreme-precision
> notation *olympian*. (The gods are not pleased at this latest trend
> in extreme-precision nomenclature, so I've taken the liberty of
> modifying the subject line before Zeus gets wind of this.)

I was calling it valhalla because, not being based on a chain of
fifths for nominals, I wasn't sure it was olympian. I can see how Zeus
would get annoyed; if he's seen the opera I imagine he regards Wotan
as a one-eyed imposter and long-winded besides.

^--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > Yes, and you were supposed to be calling the extreme-precision
> > notation *olympian*. (The gods are not pleased at this latest
trend
> > in extreme-precision nomenclature, so I've taken the liberty of
> > modifying the subject line before Zeus gets wind of this.)
>
> I was calling it valhalla because, not being based on a chain of
> fifths for nominals, I wasn't sure it was olympian.

Ah, you're right, it wasn't olympian (and I see you managed to get
Dave confused about that, too).

> I can see how Zeus
> would get annoyed; if he's seen the opera I imagine he regards Wotan
> as a one-eyed imposter and long-winded besides.

Indeed.

--George

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> OK. I'm certifiable. I couldn't wait for Gene to look at it. I
> checked whether we could notate 2460-ET, or rather 233-EDA, the 233
> Equal Division of the Apotome, using the existing symbols (Figure
3)
> plus the existing 5-schisma left accent, plus the new 5.7.13-
> schismina (tridecimal schismina) right accent. The answer is yes
> (sort of). The following is the assignment of steps to flags (just
> over twice the number of cents in each comma, relative to
> Pythagorean)
>
> '| )| (| /| ~| |~ |\ |) |( |'
> 4 7 68 44 18 34 65 56 12 1
>
> The "sort of" comes because the 7:11-kleisma is
> no longer )|( but )|('

Not a bit deal, I would say, as long as they differ only by a right-
accent mark

> and the 29-comma is no longer (| but (|' or '|\
> but that was only a secondary role for (| anyway.

I would go for (|', since it differs from the lower-level symbol by a
right-accent mark.

I've check this over, and there is only one small hitch in this whole
thing, which causes a problem both in a continuous range of steps and
in a symbol complementation. Flag arithmetic makes |\\ 130deg, but
it wants to be 131deg, because:

1) It is supposed to be the complement of (|~ and
2) There would otherwise be no symbol for 135deg (to give a
continuous range of single-shaft symbols to 138deg).

Do we allow a lapse in symbol arithmetic for this one complement, or
what? In a pinch would could use )|| for 131deg, but it would be
nicer to have all single-shaft symbols in this range.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > '| )| (| /| ~| |~ |\ |) |( |'
> > 4 7 68 44 18 34 65 56 12 1
> >
> > The "sort of" comes because the 7:11-kleisma is
> > no longer )|( but )|('
>
> Not a bit deal, I would say, as long as they differ only by a
> right-accent mark

Agreed.

> > and the 29-comma is no longer (| but (|' or '|\
> > but that was only a secondary role for (| anyway.
>
> I would go for (|', since it differs from the lower-level symbol
> by a right-accent mark.

Agreed.

And now we can have the Olympian 13-diesis as /|). (down right
accent) instead of |~\

I think the Muses have been working on Gene and I, to double the
resolution of Olympian like this with no additional characters
required in the font, only replacing '| with | and .! with ! (bare
up and down shafts).

George, do you agree I should make this change to the font?

Thanks Gene, for telling us about 2460-ET and suggesting we have a
flag for 4095:4096, the 5.7.13-shismina, and friends.

> I've check this over, and there is only one small hitch in this
whole
> thing, which causes a problem both in a continuous range of steps
and
> in a symbol complementation. Flag arithmetic makes |\\ 130deg,
but
> it wants to be 131deg, because:
>
> 1) It is supposed to be the complement of (|~ and
> 2) There would otherwise be no symbol for 135deg (to give a
> continuous range of single-shaft symbols to 138deg).
>
> Do we allow a lapse in symbol arithmetic for this one complement,
> or what?

Yes. Absolutely. We already agreed to do this. It was already a
problem with the earlier 1171-ET/111-EDA Olympian proposal.

Any single shaft symbol larger than a half-apotome is defined as the
complement of a symbol smaller than a half-apotome, not in terms of
what its flags add up to, although we want these to be reasonably
close. We agreed to do this so that an unaccented symbol always has
an unaccented symbol as complement.

The pair |~\ and ((| are slightly worse than (|~ and |\\ according
to sum of flags. ((| is out by 2 steps of 2360-ET/233-EDA.

> In a pinch would could use )|| for 131deg,

No.

> but it would be
> nicer to have all single-shaft symbols in this range.

Agreed.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > '| )| (| /| ~| |~ |\ |) |( |'
> > > 4 7 68 44 18 34 65 56 12 1
> > >
> > > The "sort of" comes because the 7:11-kleisma is
> > > no longer )|( but )|('
> >
> > Not a bit deal, I would say, as long as they differ only by a
> > right-accent mark
>
> Agreed.
>
> > > and the 29-comma is no longer (| but (|' or '|\
> > > but that was only a secondary role for (| anyway.
> >
> > I would go for (|', since it differs from the lower-level symbol
> > by a right-accent mark.
>
> Agreed.
>
> And now we can have the Olympian 13-diesis as /|). (down right
> accent) instead of |~\

And likewise the 13L-diesis as (|\' . It's also nice to see that
51200:531441, or (80:81)^3, is (|\. , while the unaccented symbol (|\
is 8192:8505. Being able to notate schisminas will be great for
theoretical discussions (and also for the electronic composer who
wants a notation with 1/2-cent resolution that might actually be
*comprehensible* enough to real-live musicians that it could be sight-
read (although they would probably just ignore the accent marks).

> I think the Muses have been working on Gene and I, to double the
> resolution of Olympian like this with no additional characters
> required in the font, only replacing '| with | and .! with ! (bare
> up and down shafts).
>
> George, do you agree I should make this change to the font?

Absolutely!

> Thanks Gene, for telling us about 2460-ET and suggesting we have a
> flag for 4095:4096, the 5.7.13-shismina, and friends.

Yes, I would say that this is more than enough precision to satisfy
anyone's needs. Monz will have to add this one to his ET table,
citing Gene as discoverer and all three of us as advocates.

Here are some nice properties of 2460-ET:

1) The best fifth has < 0.004c error -- better than 612-ET. In fact,
every harmonic through 21 has less error than 612.

2) Of course, we already know that it's a multiple of 12, but it's
nice to note that if we were to notate 12-ET using olympian notation
that Sagittal symbols that are multiples of 4deg2460 would be used.
It's interesting to observe how many symbols are multiples of 4deg
and especially that the majority of those are ones that are
frequently used:

deg symbol ratio
--- ------ -----
4 '| 32768:32805
12 |( 5103:5120
36 ~~| 99:100
44 /| 80:81
56 |) 63:64
68 (| 45056:45927
80 (|( 44:45
88 //| 6400:6561
112 (/| 3969:4096
124 (|) 704:729

3) To my great delight, it has a fifth of 1426deg that comes within
0.02 cents of the fifth of 5/17-comma (equal-beating) meantone
temperament.

Do you think that anyone could possibly take exception to the
statement that 2460-ET, for all practical purposes, *is* JI?

> > I've check this over, and there is only one small hitch in this
whole
> > thing, which causes a problem both in a continuous range of steps
and
> > in a symbol complementation. Flag arithmetic makes |\\ 130deg,
but
> > it wants to be 131deg, because:
> >
> > 1) It is supposed to be the complement of (|~ and
> > 2) There would otherwise be no symbol for 135deg (to give a
> > continuous range of single-shaft symbols to 138deg).
> >
> > Do we allow a lapse in symbol arithmetic for this one complement,
> > or what?
>
> Yes. Absolutely. We already agreed to do this. It was already a
> problem with the earlier 1171-ET/111-EDA Olympian proposal.

Okay, agreed!

--George

hi George, Dave, Gene,

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

> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> >
> > Thanks Gene, for telling us about 2460-ET and
> > suggesting we have a flag for 4095:4096, the
> > 5.7.13-shismina, and friends.
>
> Yes, I would say that this is more than enough precision
> to satisfy anyone's needs. Monz will have to add this one
> to his ET table, citing Gene as discoverer and all three
> of us as advocates.

done.

http://tonalsoft.com/enc/eqtemp.htm

but i haven't really followed this thread closely ...
is my description of 2460 as "a good approximation to
high-limit JI" adequate? please advise.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> but i haven't really followed this thread closely ...
> is my description of 2460 as "a good approximation to
> high-limit JI" adequate? please advise.

It's (for instance) an excellent 27-limit temperament, but not better
than 1578. I was particularly interested in it because it supports the
atomic temperament, which can be described as 12&2460. This is a
temperament compatible with 12-equal nominals, and which can be seen
as a refinement of 612-equal. It also has the property of being
absurdly accurate in the 5-limit.

From George and Dave's point of view probably notating 1578 should be
considered also, in conjunction with 2460. Dave, a big fan of
microtemperaments, might want to ponder 1578&2460:

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

It has a period of 200 cents and a generator a comma between 64/63 and
81/80 in size, and I suppose it could be useful to consider, though
atomic looks more interesting to me.

Speaking of the 27-limit, can Sagittal notate 159, 282, 311 and 422?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> Speaking of the 27-limit, can Sagittal notate 159, 282, 311 and
> 422?

Sure. It eats them for breakfast. Probably doesn't even need to use
any accented symbols. 422 is the only one of those I don't recall
having looked at.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi George, Dave, Gene,
>
>
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > --- In tuning-math@yahoogroups.com, "Dave Keenan"
<d.keenan@b...>
> > wrote:
> > >
> > > Thanks Gene, for telling us about 2460-ET and
> > > suggesting we have a flag for 4095:4096, the
> > > 5.7.13-shismina, and friends.
> >
> > Yes, I would say that this is more than enough precision
> > to satisfy anyone's needs. Monz will have to add this one
> > to his ET table, citing Gene as discoverer and all three
> > of us as advocates.
>
>
> done.
>
> http://tonalsoft.com/enc/eqtemp.htm
>
>
> but i haven't really followed this thread closely ...
> is my description of 2460 as "a good approximation to
> high-limit JI" adequate? please advise.

No. Get rid of that. As Gene points out, it's not particularly
special in that regard. Just say
"The largest ET that can be notated in the Sagittal notation system."

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> > I think the Muses have been working on Gene and I, to double the
> > resolution of Olympian like this with no additional characters
> > required in the font, only replacing '| with | and .! with !
(bare
> > up and down shafts).
> >
> > George, do you agree I should make this change to the font?
>
> Absolutely!

Darn! I did it. Then when I tested it, I saw that it's not that
simple after all. Remember that the up and down left accents do not
merely slope in opposite directions, but are also offset vertically
in opposite direction from the staff position. The up left accent is
_above_ the horizontal center-line of the notehead while the down
left accent is _below_ it. So we cannot simply use an up left accent
as a down right accent, and a down left accent as an up right accent
without some possibly-undesirable consequences.

While the schisma up symbol looks like this

/| __ center line
| of notehead
|

the schismina up symbol looks like this

| __ center line
|\ of notehead
|

thereby giving mixed cues about its direction of alteration.

The other possible combinations of the two accents look like this:

schisma plus schismina

/| __ center line
|\ of notehead
|

schisma minus schismina

/|/ __ center line
| of notehead
|

The problem is particularly acute (or should that be grave) when you
see the ASCII longhand versions of these four _up_ symbols
'| schisma up
|. schismina _up_
'|. schisma _plus_ schismina up
'|' schisma _minus_ schismina up

If we could add new right accent characters to the font, having
the "correct" vertical positions (these would be well symbolised in
ASCII as back-quote ` and comma , ) the above symbols would then
look like this:

/| __ center line
| of notehead
|

|\ __ center line
| of notehead
|

/|\ __ center line
| of notehead
|

/| __ center line
|/ of notehead
|

'| schisma up
|` schismina up
'|` schisma plus schismina up
'|, schisma minus schismina up

The ambiguity of direction for the right accent is probably even
worse when it is away from the shaft. Compare

(no new accent characters)

|\
| \ \
|

|\ /
| \
|

|\. 55-comma _plus_ schismina
|\' 55-comma _minus_ schismina

with (new accent characters)

|\ \
| \
|

|\
| \ /
|

|\` 55-comma plus schismina
|\, 55-comma minus schismina

To add more characters to the font would require abandoning
the "Symbol" mapping. I'm not sure what the consequences of that
are, but I assume that all the non-text fonts used by the makers of
Finale and Sibelius are "Symbol" fonts for a good reason. Anyone
know?

So I suggest we simply reuse the left accents for now, and put up
with the strange vertical positioning, but I suggest we use the
backquote and comma in ASCII (lucky we decided not to use these
previously). There's also the question of what to do for the bitmap
version (as used in Scala).

There is one advantage of reusing the left accents on the right and
thereby having the "wrong" vertical positioning. It avoids lateral
confusability.

And lets face it, it's only theorists, or those with god-like sight
and hearing, who will use right accents (as "semantic radicals" like
in Chinese text).

Sibelius will actually allow you to individually alter the relative
positions of the characters making up a multi-character symbol, so
they could be corrected that way. So the question is, what is the
official vertical positioning of the right accents? The one you get
when you simply reuse the left accents? Or are they the lateral
mirror images of the left accents?

The updated font is available now at
http://dkeenan.com/sagittal/SagittalSAT.ttf

> 2) Of course, we already know that it's a multiple of 12, but it's
> nice to note that if we were to notate 12-ET using olympian
notation
> that Sagittal symbols that are multiples of 4deg2460 would be
used.

All this means is that we could already notate 12-ET relative to
Pythagorean using 612-ET/58-EDA notation (no right accents). Now it
should even be possible to notate the subtle distinctions between
fifths in a well temperament. But whether it makes sense to do so,
other than for theoretical purposes, is another question.

> Do you think that anyone could possibly take exception to the
> statement that 2460-ET, for all practical purposes, *is* JI?

Yes. For any given thing, you can generally find someone who will
take exception to it. But even if one accepts the statement, very
little is gained by doing so.

Fortunately we don't have to go there. The purpose of looking at
2460-ET/233-EDA is merely to ensure we have a fairly even
distribution of symbols. We have yet to define all these symbols.
But when we do, it will be in rational terms, not as degrees of an
ET or EDA.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > > I think the Muses have been working on Gene and I, to double
the
> > > resolution of Olympian like this with no additional characters
> > > required in the font, only replacing '| with | and .! with !
(bare
> > > up and down shafts).
> > >
> > > George, do you agree I should make this change to the font?
> >
> > Absolutely!
>
> Darn! I did it. Then when I tested it, I saw that it's not that
> simple after all. Remember that the up and down left accents do not
> merely slope in opposite directions, but are also offset vertically
> in opposite direction from the staff position. The up left accent
is
> _above_ the horizontal center-line of the notehead while the down
> left accent is _below_ it. So we cannot simply use an up left
accent
> as a down right accent, and a down left accent as an up right
accent
> without some possibly-undesirable consequences.
>
> While the schisma up symbol looks like this
>
> /| __ center line
> | of notehead
> |
>
> the schismina up symbol looks like this
>
> | __ center line
> |\ of notehead
> |

No, no, no!!!

I thought you would be doing it this way:

'| schisma up
'|' schisma _plus_ schismina up
|. schismina down
'|. schisma _plus_ schismina down

since it is only the *slope* of the accent and the *vertical
placement* that are indicating the direction of change. Remember
that neither the schisma nor schismina accent mark is intended to
form an arrow with the arrow-shaft line, since neither one actually
touches it.

So you can still use this:

> The updated font is available now at
> http://dkeenan.com/sagittal/SagittalSAT.ttf

> > Do you think that anyone could possibly take exception to the
> > statement that 2460-ET, for all practical purposes, *is* JI?
>
> Yes. For any given thing, you can generally find someone who will
> take exception to it. But even if one accepts the statement, very
> little is gained by doing so.

I don't think that anyone ever objected to the use of a subharmonic-
style division of the octave into 1024 parts on the Scalatron for JI
tunings when the pitches were *not* specially arranged so as to be
phase-locked (which was most of the time), so that pitch errors of up
to 0.8c (or interval errors up to 1.6c) were routinely accepted. But
if the Scalatron had used the pitches of 2460-ET for JI (max. 27-odd-
limit error < 0.32c), then one might take exception?

But I'm making this point to the wrong person (on the wrong list).
Better to take it to the main list or MMM, where we could start a
flame war. ;-)

> Fortunately we don't have to go there. The purpose of looking at
> 2460-ET/233-EDA is merely to ensure we have a fairly even
> distribution of symbols. We have yet to define all these symbols.
> But when we do, it will be in rational terms, not as degrees of an
> ET or EDA.

Yes, although, for purposes of convenience, they would be capable of
being expressed as degrees.

--George

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > but i haven't really followed this thread closely ...
> > is my description of 2460 as "a good approximation to
> > high-limit JI" adequate? please advise.
>
> It's (for instance) an excellent 27-limit temperament, but not
better
> than 1578.

2460 is much better than 1578 if you're trying to approximate the
*actual pitches* of JI (e.g., when trying to notate them), where
cumulative error enters the picture. In a chain of 12 fifths of 1578-
ET (with octave reduction) you overshoot the starting tone by
30deg1578, whereas the pythagorean comma in 1578 is 31deg.

Prime 5 in a JI tuning is not likely to be taken higher than the 5th
power, so its margin of error can be greater (up to 10% of a degree,
but hopefully less; in 2460-ET it's 5.7%). Higher primes will allow
more error, and since 2460 keeps the error well below 25% through the
23-limit, any prime within that limit can safely be taken to the 2nd
power.

> Speaking of the 27-limit, can Sagittal notate 159, 282, 311 and 422?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> Sure. It eats them for breakfast. Probably doesn't even need to use
> any accented symbols. 422 is the only one of those I don't recall
> having looked at.

For 159 we have already agreed on this symbol set (which you can also
see in Scala):

159: |( ~|( /| |) (|( //| /|\ (|) )||( ~||( ||) ||\ (||
( /||) /||\

We haven't yet taken the time to finalize symbols for the others.
The choices for most of the degrees are obvious (to Dave and me, that
is), and I can offer the following comments about those that aren't:

282 works quite well, requiring only one accented symbol: '|~)
although the pair '/|) and .(|\ might be preferable to (|~ and |\\

311 works extremely well, although it does require 2 accented
symbols: ./| and '(/|

Like Dave, I hadn't looked at 422. It's rather strange in that the 5-
schisma is 2 degrees, which causes some quirky things to occur with
accented symbols (which are required for a few of the degrees) -- but
we *can* notate it. While the division is 27-limit consistent, three
harmonics are in error by over 40% of a degree, something that would
give me second thoughts about using it.

--George

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

> > It's (for instance) an excellent 27-limit temperament, but not
> better
> > than 1578.
>
> 2460 is much better than 1578 if you're trying to approximate the
> *actual pitches* of JI (e.g., when trying to notate them), where
> cumulative error enters the picture. In a chain of 12 fifths of 1578-
> ET (with octave reduction) you overshoot the starting tone by
> 30deg1578, whereas the pythagorean comma in 1578 is 31deg.

I would say the Pythagorean comma of 1578 is 30 degrees, but then I'd
say the Pythagorean comma of 19 was -1 degrees. The point remains that
2460 is a much better 3-limit temperament, though not a better 5-limit
temperament, than 1578. I agree that there is something attractive
about temperaments which are good in the 3-limit for notational
purposes, especially if they are divisible by 12. That means that for
notational purposes 72, 612, 1848 and 2460 are interesting, and 2460
takes us as far in the prime limit direction as I personally would
want to go (can we hear the 29 limit?)

If you leave 72 off the list and add 12, you find you are always
getting the atomic temperament, which makes it interesting also. 72
and 612 gives us hemiennealimmal, 72 with 2460 a sort of jumped-up
atomic, and with 1848 a quarter-tone nominal system. Of these only
hemiennealimmal is interesting as a notation alternative to atomic
since it is so relatively simple, yet suffices through the 11 limit
for JI (TOP error 0.05 cents, rms error .2 cents) but nine or 18
nominals take it off in its own strange direction. I've proposed it,
but I don't know if it will ever find a use.

> Like Dave, I hadn't looked at 422. It's rather strange in that the 5-
> schisma is 2 degrees, which causes some quirky things to occur with
> accented symbols (which are required for a few of the degrees) -- but
> we *can* notate it. While the division is 27-limit consistent, three
> harmonics are in error by over 40% of a degree, something that would
> give me second thoughts about using it.

I's a pretty decent 35-limit system, actually. If you put it together
with 311, you get a 31-limit linear temperament with 32/31 as a
generator, which the world may not be ready for, but who knows?

Why is 40% error a problem considering you are dividing the octave
into 422 parts, which is a lot of parts?

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> No, no, no!!!
>
> I thought you would be doing it this way:

I've inserted ASCII-graphic approximations of what yours will look
like near a bare shaft with the TrueType font.

> '| schisma up

/|
|
|

> '|' schisma _plus_ schismina up

/|/
|
|

This is what I thought of as schisma _minus_ schismina. You've
certainly proved my assertion that it's ambiguous.

> |. schismina down

Let's show all _up_ symbols for comparison. So your schismina up
would be |'

|/
|
|

This wasn't used in my scheme.

> '|. schisma _plus_ schismina down

This description has got to be wrong. Surely you mean this to be an
_up_ symbol representing a change of schisma _minus_ shismina. As
such, it looks obvious enough in ASCII, but with the TrueType font.
It looks like this.

/|
|\
|

> since it is only the *slope* of the accent and the *vertical
> placement* that are indicating the direction of change. Remember
> that neither the schisma nor schismina accent mark is intended to
> form an arrow with the arrow-shaft line, since neither one
> actually touches it.

I beg to differ. We agreed that Bosanquet's slashes were somewhat
ambiguous in direction until they were placed in relation to an
arrow shaft. I conceive of the left accent as a miniature version of
the left barb. This is also a handy mnemonic for the fact that they
are both 5-commas.

Whether it is touching or only almost-touching makes no difference
to my perception of it as an arrow. Only when it is spaced far away
from the shaft by the presence of a left flag, does it cease to look
like a part-arrowhead to me.

A sloping line has no inherent up or down direction. It can be given
one by imposing a sideways direction on it (either left-to-right or
right-to-left). I agree that in the absence of other cues this ought
to be the direction of reading, left-to-right.

The upward right barb is considered as sloping upward from right to
left (the opposite of Bosanquet's slashes), because it is in
relation to an arrow shaft. Putting a slight break between it and
the shaft would not suddenly make it seem to point the other way, at
least not for me. It merely makes it look like a stencilled version
of an arrow, like the "Government Property" mark sometimes seen on
packing cases.

When the right accent is not spaced well away from the shaft by an
intervening right flag, it seems to me to be part of an arrow, a
miniature right barb.

> So you can still use this:
>
> > The updated font is available now at
> > http://dkeenan.com/sagittal/SagittalSAT.ttf

Sure. It's just a question of how. We'd better get some other
opinions.

I've uploaded a MsWord file with the options to
http://dkeenan.com/sagittal/RightAccents.doc

You will need to first install the latest version of the Sagittal
font as above.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > > It's (for instance) an excellent 27-limit temperament, but not
better
> > > than 1578.
> >
> > 2460 is much better than 1578 if you're trying to approximate the
> > *actual pitches* of JI (e.g., when trying to notate them), where
> > cumulative error enters the picture. In a chain of 12 fifths of
1578-
> > ET (with octave reduction) you overshoot the starting tone by
> > 30deg1578, whereas the pythagorean comma in 1578 is 31deg.
>
> I would say the Pythagorean comma of 1578 is 30 degrees, but then
I'd
> say the Pythagorean comma of 19 was -1 degrees.

Yes, I agree -- and sorry, I misspoke. I meant to say that the
interval of 1578 nearest in size to the pythagorean comma is 31deg,
which disagrees with its functional size (as determined by a chain of
12 fifths) of 31deg.

> The point remains that
> 2460 is a much better 3-limit temperament, though not a better 5-
limit
> temperament, than 1578.

Huh? Judging by this table, I would say that the numbers for 5-limit
consonances look better for 2460:

Error as %
Interval 1578 2460
2:3 -7.1 -0.8
4:5 -0.3 +5.7
3:5 +6.8 +6.5

1578 might do better at the 7 limit, but 2460 would retake the lead
at the 9 limit. Likewise, 1578 would do better at 11, but 2460 would
surpass it at 13 and maintain its lead through 27.

> I agree that there is something attractive
> about temperaments which are good in the 3-limit for notational
> purposes, especially if they are divisible by 12.

Yes, it gives everyone a familiar frame of reference.

> That means that for
> notational purposes 72, 612, 1848 and 2460 are interesting, and 2460
> takes us as far in the prime limit direction as I personally would
> want to go (can we hear the 29 limit?)

We can hear the 29 limit under certain circumstances, as in a
24:29:34 triad. I've experimented with some heptatonic JI scales
using selected harmonics, in which most of the 3rds differ by 5.

But I agree that it's generally safe to stop at a prime-limit of 23,
taking a hint from the fact that this is the first time that there
are 5 integers between this and the next prime.

> If you leave 72 off the list and add 12, you find you are always
> getting the atomic temperament, which makes it interesting also. 72
> and 612 gives us hemiennealimmal, 72 with 2460 a sort of jumped-up
> atomic, and with 1848 a quarter-tone nominal system. Of these only
> hemiennealimmal is interesting as a notation alternative to atomic
> since it is so relatively simple, yet suffices through the 11 limit
> for JI (TOP error 0.05 cents, rms error .2 cents) but nine or 18
> nominals take it off in its own strange direction. I've proposed it,
> but I don't know if it will ever find a use.
>
> > Like Dave, I hadn't looked at 422. It's rather strange in that
the 5-
> > schisma is 2 degrees, which causes some quirky things to occur
with
> > accented symbols (which are required for a few of the degrees) --
but
> > we *can* notate it. While the division is 27-limit consistent,
three
> > harmonics are in error by over 40% of a degree, something that
would
> > give me second thoughts about using it.
>
> I's a pretty decent 35-limit system, actually. If you put it
together
> with 311, you get a 31-limit linear temperament with 32/31 as a
> generator, which the world may not be ready for, but who knows?
>
> Why is 40% error a problem considering you are dividing the octave
> into 422 parts, which is a lot of parts?

Error accumulates if one of those primes is combined with another
having only a small error, which results in inconsistencies, e.g.,
422 is not 1,3,13,39-consistent. If I'm going to use one of these
large-numbered divisions to compose electronic music or as a tuning-
measure for theoretical purposes, I would want my intervals
to "behave" better than this.

--George

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > No, no, no!!!
> >
> > I thought you would be doing it this way:
>
> I've inserted ASCII-graphic approximations of what yours will look
> like near a bare shaft with the TrueType font.
>
> > '| schisma up
>
> /|
> |
> |
>
> > '|' schisma _plus_ schismina up
>
> /|/
> |
> |
>
> This is what I thought of as schisma _minus_ schismina. You've
> certainly proved my assertion that it's ambiguous.

Particularly in ascii, where an apostrophe often shows no slope at
all (or if you use Microsoft Word, you get the wrong slope when these
get converted to single quotes).

> > |. schismina down
>
> Let's show all _up_ symbols for comparison. So your schismina up
> would be |'
>
> |/
> |
> |
>
> This wasn't used in my scheme.
>
> > '|. schisma _plus_ schismina down
>
> This description has got to be wrong. Surely you mean this to be an
> _up_ symbol representing a change of schisma _minus_ shismina.

"Schisma _plus_ schismina down" and "Schisma _minus_ schismina up"
mean the same thing, just as "4 plus -1" and 4 minus 1" mean the same
thing. I used the former description because that tells how the
graphics are to be combined.

> As
> such, it looks obvious enough in ASCII,

which is where I would assume that it would be used most often. Do
you think that live performers are going to read (or need) schisminas?

> but with the TrueType font.
> It looks like this.
>
> /|
> |\
> |
>
> > since it is only the *slope* of the accent and the *vertical
> > placement* that are indicating the direction of change. Remember
> > that neither the schisma nor schismina accent mark is intended to
> > form an arrow with the arrow-shaft line, since neither one
> > actually touches it.
>
> I beg to differ.

Hmmm, looks like we're back to the good old days, once again debating
the appearance of symbols here on tuning-math. :-)

> We agreed that Bosanquet's slashes were somewhat
> ambiguous in direction until they were placed in relation to an
> arrow shaft.

or until they are also given a vertical placement (which I think
should be the main distinction). You would have to imagine a very
thin man (the shaft) facing to the right with his arm (the schismina
accent) extended and pointing either upward or downward.

> I conceive of the left accent as a miniature version of
> the left barb. This is also a handy mnemonic for the fact that they
> are both 5-commas.

Yes, but in spite of both the *upward slope* and *arrow-up* cues, we
decided that we still needed *vertical placement* to distinguish up
from down sufficiently. Making the vertical placement for the right-
side accent marks to mean the *opposite* would result in confusion.

> Whether it is touching or only almost-touching makes no difference
> to my perception of it as an arrow. Only when it is spaced far away
> from the shaft by the presence of a left flag, does it cease to
look
> like a part-arrowhead to me.
>
> A sloping line has no inherent up or down direction. It can be
given
> one by imposing a sideways direction on it (either left-to-right or
> right-to-left). I agree that in the absence of other cues this
ought
> to be the direction of reading, left-to-right.
>
> The upward right barb is considered as sloping upward from right to
> left (the opposite of Bosanquet's slashes), because it is in
> relation to an arrow shaft. Putting a slight break between it and
> the shaft would not suddenly make it seem to point the other way,
at
> least not for me. It merely makes it look like a stencilled version
> of an arrow, like the "Government Property" mark sometimes seen on
> packing cases.
>
> When the right accent is not spaced well away from the shaft by an
> intervening right flag, it seems to me to be part of an arrow, a
> miniature right barb.

With your (b) option (below) this would be the only cue as to
direction, since |. would have both slope and vertical placement
indicating down, not up. For a right accent used in combination with
a symbol having a right flag, that cue would be lost, whereas slope
and vertical placement would still be effective if they were the
indicators of direction.

> > So you can still use this:
> >
> > > The updated font is available now at
> > > http://dkeenan.com/sagittal/SagittalSAT.ttf
>
> Sure. It's just a question of how. We'd better get some other
> opinions.
>
> I've uploaded a MsWord file with the options to
> http://dkeenan.com/sagittal/RightAccents.doc
>
> You will need to first install the latest version of the Sagittal
> font as above.

I don't like (b) at all (for reasons given above). Both (a) and (c)
are fine with me, but wouldn't (c) require two more characters in the
font, and (even if you find space for them in the font) would it be
worth the trouble to provide them if users are presented with the
additional burden to exercise care not to mix them up with the left-
accent characters? KISS!

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> >
> > > 2460 is much better than 1578 if you're trying to approximate
the
> > > *actual pitches* of JI (e.g., when trying to notate them),
where
> > > cumulative error enters the picture. In a chain of 12 fifths
of 1578-
> > > ET (with octave reduction) you overshoot the starting tone by
> > > 30deg1578, whereas the pythagorean comma in 1578 is 31deg.
> >
> > I would say the Pythagorean comma of 1578 is 30 degrees, but then
I'd
> > say the Pythagorean comma of 19 was -1 degrees.
>
> Yes, I agree -- and sorry, I misspoke. I meant to say that the
> interval of 1578 nearest in size to the pythagorean comma is 31deg,
> which disagrees with its functional size (as determined by a chain
of
> 12 fifths) of 31deg.

Oops!!! I was concentrating so much on the wording that I got the
number wrong; that last paragraph should have been:

<< Yes, I agree -- and sorry, I misspoke. I meant to say that the
interval of 1578 nearest in size to the pythagorean comma is 31deg,
which disagrees with its functional size (as determined by a chain of
12 fifths) of 30deg. >>

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > You've certainly proved my assertion that it's ambiguous.
>
> Particularly in ascii, where an apostrophe often shows no slope at
> all (or if you use Microsoft Word, you get the wrong slope when
> these get converted to single quotes).

Huh? You get the _correct_ slope. It's an upward left accent, it
slopes upward from left to right. So do curly apostrophes or single-
quotes.

> > > '|. schisma _plus_ schismina down
> >
> > This description has got to be wrong. Surely you mean this to be
> > an _up_ symbol representing a change of schisma _minus_
> > shismina.
>
> "Schisma _plus_ schismina down" and "Schisma _minus_ schismina up"
> mean the same thing, just as "4 plus -1" and 4 minus 1" mean the
> same thing. I used the former description because that tells how
> the graphics are to be combined.

This would have been clear to me, if you had written "schisma _up_
plus schismina down", but you gave no explicit direction for the
schisma. So what would you call it's inversion? "schisma down plus
schismina"?

My intention was of course that the "up" or "down", appearing always
at the end, indicates the overall direction of the symbol, not that
of an individual flag or accent, and that the rest of the name
indicates the magnitude. But I will use hyphens to make that clearer
in future.

|` schismina up

!, schismina down

'|, schisma-minus-schismina up, or schisma-schismina up

.!` schisma-minus-schismina down, or schisma-schismina down

'| schisma up

.! schisma down

'|` schisma-plus-schismina up, or schisma+schismina up

.!, schisma-plus-schismina down, or schisma+schismina down

OK?

> > As such, it looks obvious enough in ASCII,
>
> which is where I would assume that it would be used most often.
> Do you think that live performers are going to read (or need)
> schisminas?

Of course not. I already said as much. But that's not the only
reason to want them typeset, or as graphics, whether appearing on a
staff, or in text. People who write theory sometimes like to have
articles published, in journals and on the web; or hadn't you
noticed. :-)

So we cannot allow the tail to wag the dog. We must first decide on
the detailed graphical form, and only then should we look at how
best to approximate that in ASCII. We should not worry, for now,
about whether it can be added to the TrueType font or not. I've used
the above ASCII form, for now, since we both accept version (c), as
shown in
http://dkeenan.com/sagittal/RightAccents.doc

If you have just joined us, to view the above correctly you will
need to install the latest version of the font
http://dkeenan.com/sagittal/SagittalSAT.ttf

> > I beg to differ.
>
> Hmmm, looks like we're back to the good old days, once again
> debating the appearance of symbols here on tuning-math. :-)

Hee hee. And this time I think its obvious to everyone that we're
completely crazy. 8-P... Arguing over symbols to represent half a
cent. Ho ho ho.

> > We agreed that Bosanquet's slashes were somewhat
> > ambiguous in direction until they were placed in relation to an
> > arrow shaft.
>
> or until they are also given a vertical placement

Yes. But a vertical placement relative to what? The notehead or the
shaft?

If the notehead, then I think this is insufficient on its own, as it
is a bit too subtle.

If the shaft, then we cannot use this to indicate the case where the
schismina must be subtracted from the schisma (to obtain the
magnitude), since I assume we are not willing to have accents at
both ends of a shaft

/|
|
|/

because the notehead to which it applied would then become
ambiguous. Indeed this is just what we already don't do (does that
make sense?) for say the diaschisma symbol ./| . So it has to
be "relative to the notehead".

> (which I think should be the main distinction).

I'm afraid that doesn't work for me. For me it's important, but
not "main".

> You would have to imagine a very
> thin man (the shaft) facing to the right with his arm (the
schismina
> accent) extended and pointing either upward or downward.

I don't understand. Why would I have to imagine this? I don't
imagine a man at all. I imagine a stencilled arrow pointing upward,
where the right half of the arrowhead can either be in its normal
orientation or be flipped vertically to indicate that it is intended
to partly cancel the effect of the left half.

> Yes, but in spite of both the *upward slope* and *arrow-up* cues,
> we decided that we still needed *vertical placement* to
> distinguish up from down sufficiently.

Agreed. Although I do not believe there is such a thing as
an "upward slope" except in relation to a direction of travel. For
me, it is travel toward the shaft that gives it an upward direction,
and so I don't see this as being independent of the "arrowness" cue.

> Making the vertical placement for the right-
> side accent marks to mean the *opposite* would result in confusion.

Agreed. It's just that this was my favoured fallback position if we
couldn't have option (c). I guess I see the arrow cue (or the
direction as given by travel toward the shaft) as stronger than
vertical position relative to the notehead. This is certainly the
case for me when the right accent is next to the shaft (with no
intervening right flag), but I admit that the vertical position cue
would tend to take over when the right accent is closer to a
notehead than it is to a shaft. However, if you think it will never
be used on a staff then this would be irrelevant since there will be
no notehead, and the letter name will be to the _left_ of the
accidental.

> > A sloping line has no inherent up or down direction. It can be
> > given one by imposing a sideways direction on it (either left-to-
> > right or right-to-left). I agree that in the absence of other
> cues this ought to be the direction of reading, left-to-right.

Note that I said, "In the absence of other cues". For me, the
presence of a shaft, anywhere, is such a cue, and results in the
direction of travel being towards the shaft, from either side.

> With your (b) option (below) this would be the only cue as to
> direction, since |. would have both slope and vertical placement
> indicating down, not up. For a right accent used in combination
> with a symbol having a right flag, that cue would be lost, whereas
> slope and vertical placement would still be effective if they were
> the indicators of direction.

Of course I agree that vertical placement is working for us in (a)
and (c), but against us in (b).

However I disagree about what you call "slope". I believe "slope" is
working for us in (b) and (c) but not in (a), as is "arrowness".

The idea that a slope has a definite direction up or down, in the
way that you seem to assume, seems exceedingly odd to me. If I were
to point at a distant hill or mountain /\ and solemnly tell my
companions that the slope on the left goes up, while the one on the
right goes down, or to a distant canyon or crater \/ and tell them
that the slope on the left goes down, while the slope on the right
goes up, they would assume I was either joking or mad. The usual
convention is that all sides of a mountain go up /\ and all sides of
a crater go down \/, unless of course you're standing on the top of
the mountain, or the bottom of the crater. But these are somewhat
special positions that don't seem to relate to how we are viewing
our symbols, and in any case these still result in both sides being
considered to go in the _same_ vertical direction relative to the
viewer, either both up or both down.

Admittedly it's unclear what this /|/ should be viewed as, in terms
of mountains and craters. But when we go from geography to archery,
it's clear that /\ and \/ are transferred intact, since both sides
of

/|\
|

are considered to point up, while both sides of

|
\|/

point down. And so surely the left side of this points up, while its
right side points down

/|/
|

which is why I can't accept (a).

> I don't like (b) at all (for reasons given above). Both (a) and
> (c) are fine with me,

Well it looks like (c) is the only one that everyone will find
acceptable.

> but wouldn't (c) require two more characters in the font,

Maybe so. But we knew the TrueType font was full when we wrote that
the system was "extensible", so clearly we weren't intending that to
stop us. The TrueType font is not the system, as evidenced by the
fact that Scala doesn't use it, but uses bitmaps instead. Would you
prohibit Scala from using a bitmapped version of (c) just because we
can't fit the extra accents in the TrueType font?

Strictly speaking we don't need the 5 conventional accidentals in
the font at all, since any notation software will have other fonts
containing these. Jacob Barton didn't use them with his Sibelius
scripts, although I've just asked him to use at least the Sagittal
font's version of the conventional double-sharp x since I think the
one in Sibelius' standard font (Opus) looks terrible when combined
with Sagittals (having strokes that flare gradually as they depart
from the center, most unlike crossed arrow-shafts). But if we ever
got desperate we could replace say the double-flat and the sharp.

But didn't you wonder how I made option (c) in that Word document? I
didn't add any new accents to the font. I just used Word's
formatting abilities to raise and lower the existing accents. The
correct amount is one eighth of the Sagittal point size. This can be
done even more easily in Sibelius than it can in Word since it only
has to be done once. I'm downloading the Finale demo to see if it
can do it too.

So I propose we use the down left-accent symbol for the up right-
accent and vice versa, and offset them vertically to the correct
positions, giving option (c).

When using Sagittal symbols in text, with 12 point Arial (or
Helvetica) for the letter names, we already need to set the Sagittal
in 18 point and move it up 3 points. A right accent will merely need
an adjustment on the amount by which it is raised, to either 5
points, or 1 point depending on whether it is an up or down right
accent. (18 pt/8 = 2.25 pt but Word seems to round to half points).

> and (even if you find space for them in the font) would it be
> worth the trouble to provide them if users are presented with the
> additional burden to exercise care not to mix them up with the
> left-accent characters? KISS!

Same to you, you cheeky boy! ;-)

Seriously, that's a good point. I agree the font should stay as it
is now. But we'll need new bitmaps for Scala.

This is also a good reason _not_ to reuse the characters ' and . for
the right accents in ASCII, but rather to use ` and ,

Do you agree?

-- Dave Keenan

I checked out Finale. Although it can do multi-character
accidentals, there's no way to change vertical offsets.

So if we ever encounter anyone who needs to use right accents in
Finale we can make them a special version of the font, with sharp
and double-flat (or something) changed to the corectly offset right
accents.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> >
> > > You've certainly proved my assertion that it's ambiguous.
> >
> > Particularly in ascii, where an apostrophe often shows no slope
at
> > all (or if you use Microsoft Word, you get the wrong slope when
> > these get converted to single quotes).
>
> Huh? You get the _correct_ slope. It's an upward left accent, it
> slopes upward from left to right. So do curly apostrophes or single-
> quotes.

No, try typing this in Word: '|' and you'll get this: u|c ,
where "c" looks like a superscripted comma and "u" like a
superscripted upside-down-comma. For a schisma-plus-schismina you
would want c|u (but then again, do you really want those curly
characters?).

> > > > '|. schisma _plus_ schismina down
> > >
> > > This description has got to be wrong. ...
>
> My intention was of course that the "up" or "down", appearing
always
> at the end, indicates the overall direction of the symbol, not that
> of an individual flag or accent, and that the rest of the name
> indicates the magnitude. But I will use hyphens to make that
clearer
> in future.

Aha! Now I understand.

> |` schismina up
>
> !, schismina down
>
> '|, schisma-minus-schismina up, or schisma-schismina up
>
> .!` schisma-minus-schismina down, or schisma-schismina down
>
> '| schisma up
>
> .! schisma down
>
> '|` schisma-plus-schismina up, or schisma+schismina up
>
> .!, schisma-plus-schismina down, or schisma+schismina down
>
> OK?

I would say use + and - (minus sign, not a hyphen). I'm beginning to
think that we should change the term "schismina" to "atom", so we
could simply write s+a up, s-a up, s+a down, and s-a down. Then
4095:4096 could be called a 13-atom. (I'm sure Gene would like this.)

Wait a minute! Isn't there already an atom of Kirnberger? Yes, look
here:
http://tonalsoft.com/enc/eqtemp.htm
This gives "atomic" temperament as the last entry in a table in
connection with divisions 12, 600, 612, 3072, 3684, and 4296.

Hey, Gene, what gives? Do we look for another name for ratios
corresponding to a single degree of 2460-ET? If so, how about
something related to "olympian"?

> > > As such, it looks obvious enough in ASCII,
> >
> > which is where I would assume that it would be used most often.
> > Do you think that live performers are going to read (or need)
> > schisminas?
>
> Of course not. I already said as much. But that's not the only
> reason to want them typeset, or as graphics, whether appearing on a
> staff, or in text. People who write theory sometimes like to have
> articles published, in journals and on the web; or hadn't you
> noticed. :-)

My point is that it's not critical that schismina accents be read
accurately in *real time*, inasmuch as the reader should have time to
take a good look to be sure of the meaning.

> So we cannot allow the tail to wag the dog. We must first decide on
> the detailed graphical form, and only then should we look at how
> best to approximate that in ASCII. We should not worry, for now,
> about whether it can be added to the TrueType font or not. I've
used
> the above ASCII form, for now, since we both accept version (c), as
> shown in
> http://dkeenan.com/sagittal/RightAccents.doc

Although I still prefer version (a), for the reasons given below.

> If you have just joined us, to view the above correctly you will
> need to install the latest version of the font
> http://dkeenan.com/sagittal/SagittalSAT.ttf
>
> > > I beg to differ.
> >
> > Hmmm, looks like we're back to the good old days, once again
> > debating the appearance of symbols here on tuning-math. :-)
>
> Hee hee. And this time I think its obvious to everyone that we're
> completely crazy. 8-P... Arguing over symbols to represent half a
> cent. Ho ho ho.

Do you think there's anyone else out there crazy enough to want to
bother to add their 4 schisminas?

Hey, it's all a matter of opinion whether attention to minute detail
is important, but we happen to think that it pays off. To quote
Emerson: "Difference from me is the measure of absurdity." (From the
essay, "Uses of Great Men")

> > > We agreed that Bosanquet's slashes were somewhat
> > > ambiguous in direction until they were placed in relation to an
> > > arrow shaft.

Not ambiguous actually, just not distinguishable easily or reliably
enough to be read in real time.

> > or until they are also given a vertical placement
>
> Yes. But a vertical placement relative to what? The notehead or the
> shaft?

Yes, and yes.

> If the notehead, then I think this is insufficient on its own, as
it
> is a bit too subtle.

But it would be also relative to the shaft, positioned either near
the end or near the middle.

> If the shaft, then we cannot use this to indicate the case where
the
> schismina must be subtracted from the schisma (to obtain the
> magnitude), since I assume we are not willing to have accents at
> both ends of a shaft
>
> /|
> |
> |/

The combined amount of alteration serves to determine the position of
the shaft, so the schisma accent would go near the end, and the
schismina accent would go near the *middle* (not the other end).
Option (c) would have schisma-minus-schismina thus:

/|
|/
|

and option (a) would have it thus:

/|
|\
|

In either case the differing vertical placement of the schismina
accent (from the position of the schisma accent) would make it clear
that it's altering in the opposite direction from the schisma. The
vertical placement would also provide a cue whenever any flags are
used in the symbol.

> because the notehead to which it applied would then become
> ambiguous. Indeed this is just what we already don't do (does that
> make sense?) for say the diaschisma symbol ./| . So it has to
> be "relative to the notehead".
>
> > (which I think should be the main distinction).
>
> I'm afraid that doesn't work for me. For me it's important, but
> not "main".
>
> > You would have to imagine a very
> > thin man (the shaft) facing to the right with his arm (the
schismina
> > accent) extended and pointing either upward or downward.
>
> I don't understand. Why would I have to imagine this?

Because that's what you would need to imagine for option (a) when a
right accent occurs next to a shaft. The stick-man is facing in the
direction that one reads the music.

> I don't
> imagine a man at all.

Then I would say that you're having a failure of imagination, which
has been known to lead to serious consequences. ;-(

> I imagine a stencilled arrow pointing upward,
> where the right half of the arrowhead can either be in its normal
> orientation or be flipped vertically to indicate that it is
intended
> to partly cancel the effect of the left half.

This works only half of the time, when there's no right flag
between. Remember that these are accent marks and not necessarily
arrowheads (see next comment).

> > Yes, but in spite of both the *upward slope* and *arrow-up* cues,
> > we decided that we still needed *vertical placement* to
> > distinguish up from down sufficiently.
>
> Agreed. Although I do not believe there is such a thing as
> an "upward slope" except in relation to a direction of travel. For
> me, it is travel toward the shaft that gives it an upward
direction,
> and so I don't see this as being independent of the "arrowness"
cue.

When we read (either music or text) we travel left to right.

Did you not make the point some time ago that these accents are
*acute* and *grave*, which have upward and downward meaning in spoken
languages? Only option (a) abides by those meanings.

> > Making the vertical placement for the right-
> > side accent marks to mean the *opposite* would result in
confusion.
>
> Agreed. It's just that this was my favoured fallback position if we
> couldn't have option (c). I guess I see the arrow cue (or the
> direction as given by travel toward the shaft) as stronger than
> vertical position relative to the notehead. This is certainly the
> case for me when the right accent is next to the shaft (with no
> intervening right flag), but I admit that the vertical position cue
> would tend to take over when the right accent is closer to a
> notehead than it is to a shaft. However, if you think it will never
> be used on a staff then this would be irrelevant since there will
be
> no notehead, and the letter name will be to the _left_ of the
> accidental.

I didn't say that it would never be used, only that the vertical
position would be more valuable than the arrow cue when you consider
that this would be used mostly in ascii.

> > > A sloping line has no inherent up or down direction. It can be
> > > given one by imposing a sideways direction on it (either left-
to-
> > > right or right-to-left). I agree that in the absence of other
> > cues this ought to be the direction of reading, left-to-right.
>
> Note that I said, "In the absence of other cues". For me, the
> presence of a shaft, anywhere, is such a cue, and results in the
> direction of travel being towards the shaft, from either side.

So we have the following as cues:
1) Perception of accent mark as arrowhead
2) Vertical placement (relative to notehead & shaft)
3) Perception of accent mark as acute or grave

In the graphic cues 2) and 3) are *always* present, whereas cue 1) is
weakened if there is a right flag between the shaft and right accent.

In ascii cue 2) by itself is sufficient to clarify the direction,
while cues 1) and 3) are of little value, regardless of whether the
same or differing characters are used for left and right accents.
The non-slanting apostrophe of Courier font or the curly comma and
apostrophe of Times Roman do not readily indicate a line of constant
slope.

> > With your (b) option (below) this would be the only cue as to
> > direction, since |. would have both slope and vertical placement
> > indicating down, not up. For a right accent used in combination
> > with a symbol having a right flag, that cue would be lost, whereas
> > slope and vertical placement would still be effective if they
were
> > the indicators of direction.
>
> Of course I agree that vertical placement is working for us in (a)
> and (c), but against us in (b).

So I guess we agree that (b) is eliminated.

> However I disagree about what you call "slope". I believe "slope"
is
> working for us in (b) and (c) but not in (a), as is "arrowness".

And the acute-grave slope distinction works for us in (a), but not in
(b) or (c), if we look at these as "accent marks" rather
than "arrowheads".

> ... And so surely the left side of this points up, while its
> right side points down
>
> /|/
> |
>
> which is why I can't accept (a).

But on the other hand, if you see them as accent marks, they look
alike.

> > I don't like (b) at all (for reasons given above). Both (a) and
> > (c) are fine with me,
>
> Well it looks like (c) is the only one that everyone will find
> acceptable.
>
> > but wouldn't (c) require two more characters in the font,
>
> Maybe so. But we knew the TrueType font was full when we wrote that
> the system was "extensible", so clearly we weren't intending that
to
> stop us. The TrueType font is not the system, as evidenced by the
> fact that Scala doesn't use it, but uses bitmaps instead. Would you
> prohibit Scala from using a bitmapped version of (c) just because
we
> can't fit the extra accents in the TrueType font?

I think Scala will have to conform to whatever we have in the font.

> Strictly speaking we don't need the 5 conventional accidentals in
> the font at all, since any notation software will have other fonts
> containing these. Jacob Barton didn't use them with his Sibelius
> scripts, although I've just asked him to use at least the Sagittal
> font's version of the conventional double-sharp x since I think the
> one in Sibelius' standard font (Opus) looks terrible when combined
> with Sagittals (having strokes that flare gradually as they depart
> from the center, most unlike crossed arrow-shafts). But if we ever
> got desperate we could replace say the double-flat and the sharp.
>
> But didn't you wonder how I made option (c) in that Word document?
I
> didn't add any new accents to the font. I just used Word's
> formatting abilities to raise and lower the existing accents.

Yeah, I figured you did something like that.

> The
> correct amount is one eighth of the Sagittal point size. This can
be
> done even more easily in Sibelius than it can in Word since it only
> has to be done once. I'm downloading the Finale demo to see if it
> can do it too.
>
> So I propose we use the down left-accent symbol for the up right-
> accent and vice versa, and offset them vertically to the correct
> positions, giving option (c).
>
> When using Sagittal symbols in text, with 12 point Arial (or
> Helvetica) for the letter names, we already need to set the
Sagittal
> in 18 point and move it up 3 points. A right accent will merely
need
> an adjustment on the amount by which it is raised, to either 5
> points, or 1 point depending on whether it is an up or down right
> accent. (18 pt/8 = 2.25 pt but Word seems to round to half points).
>
> > and (even if you find space for them in the font) would it be
> > worth the trouble to provide them if users are presented with the
> > additional burden to exercise care not to mix them up with the
> > left-accent characters? KISS!
>
> Same to you, you cheeky boy! ;-)
>
> Seriously, that's a good point. I agree the font should stay as it
> is now. But we'll need new bitmaps for Scala.

Yes, for sai.png and sal.png .

> This is also a good reason _not_ to reuse the characters ' and .
for
> the right accents in ASCII, but rather to use ` and ,
>
> Do you agree?

I came to the opposite conclusion: You would be presenting the user
with the burden to exercise care not to mix up accent characters,
e.g., by mistakenly typing `| or ,| or |' or |. or `|' or `|. or ...,
such that a computer might not process the character stream properly.

I don't think you completely got my point. Keep It Simple, Stu_d!:
1) By using only 2 characters instead of 4 in ascii, thereby:
a) Having *matching* characters on the left and right sides for the
*same* direction of alteration, and
b) Making it unnecessary to remember which characters go on the
left side and which go on the right; and
2) By eliminating problems with the font:
a) To make space for two more characters, or
b) To require special character alignment to create the graphic.

And if we agree to use only 2 characters for the accents, I would
propose replacing . with , for both the schisma down and schismina
down. The period character tends to be difficult to see next to an
arrow-shaft in some fonts.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> I checked out Finale. Although it can do multi-character
> accidentals, there's no way to change vertical offsets.
>
> So if we ever encounter anyone who needs to use right accents in
> Finale we can make them a special version of the font, with sharp
> and double-flat (or something) changed to the corectly offset right
> accents.

I would say that it looks like one more reason not to use option (c).

--George

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

> Wait a minute! Isn't there already an atom of Kirnberger? Yes, look
> here:
> http://tonalsoft.com/enc/eqtemp.htm
> This gives "atomic" temperament as the last entry in a table in
> connection with divisions 12, 600, 612, 3072, 3684, and 4296.
>
> Hey, Gene, what gives? Do we look for another name for ratios
> corresponding to a single degree of 2460-ET? If so, how about
> something related to "olympian"?

We could try "olympum" for the noun form, I suppose. "Olympic" should
be the adjective, so we can help take back the language from the
Olympic Committee, who think they own the word. I suppose you might
call 9801/9800 and the like, which vanish in atomic, atoms.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> Here are some nice properties of 2460-ET:
>
> 1) The best fifth has < 0.004c error -- better than 612-ET. In
fact,
> every harmonic through 21 has less error than 612.
>
> 2) Of course, we already know that it's a multiple of 12 ...
>
> 3) To my great delight, it has a fifth of 1426deg that comes within
> 0.02 cents of the fifth of 5/17-comma (equal-beating) meantone
> temperament.

4) It's also a multiple of 41, so instruments of flexible pitch built
for 41-ET could be pitch-bent by the player in 60ths of a degree (a
nice number, divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30) to
achieve olympian JI.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:

> No, try typing this in Word: '|' and you'll get this: u|c ,
> where "c" looks like a superscripted comma and "u" like a
> superscripted upside-down-comma. For a schisma-plus-schismina you
> would want c|u (but then again, do you really want those curly
> characters?).

If I was using Word, I would use the real thing - the Sagittal font.

If I was typing the ASCII approximation of schisma+schismina
(anywhere) I wouldn't type '|' but rather '|` which comes out with
the right slopes even if Word does turn ' into a curly quote. Both
of what you call "c" and "u" above give me the general impression of
sloping upward to the right, so of course what you call "c" is wrong
for the schismina up, but what you call "u" is fine for the schisma
up. But you're right, I would turn off curly quotes in Word if I
wanted to show the ASCII approximation.

> I would say use + and - (minus sign, not a hyphen). I'm beginning
to
> think that we should change the term "schismina" to "atom", so we
> could simply write s+a up, s-a up, s+a down, and s-a down. Then
> 4095:4096 could be called a 13-atom. (I'm sure Gene would like
this.)

No. The only already-named atom is far far smaller than a schismina.
It's only 0.015 cents. Calling something as large as 1.8 cents
an "atom" doesn't seem right. I'd like to keep that name for a
category even smaller than a schismina, with maybe even another
category intervening.

In any case, we will eventually be naming all these symbol as simply
X-schisminas, where X is some combination of primes greater than 3.

> Wait a minute! Isn't there already an atom of Kirnberger? Yes,
look
> here:
> http://tonalsoft.com/enc/eqtemp.htm
> This gives "atomic" temperament as the last entry in a table in
> connection with divisions 12, 600, 612, 3072, 3684, and 4296.
>
> Hey, Gene, what gives? Do we look for another name for ratios
> corresponding to a single degree of 2460-ET? If so, how about
> something related to "olympian"?

Yes it's strange that 2460-ET was not included on that list, but I
thought you knew that the "atomic temperament" of this thread was
the one in which, in the 5-limit version, only the atom of
Kirnberger vanishes.

> My point is that it's not critical that schismina accents be read
> accurately in *real time*, inasmuch as the reader should have time
> to take a good look to be sure of the meaning.

Agreed. But I still think we should make it as easy as possible, and
not let Symbol font limitations influence us too much.

> Do you think there's anyone else out there crazy enough to want to
> bother to add their 4 schisminas?

Hee hee.

> But it would be also relative to the shaft, positioned either near
> the end or near the middle.

True.

> Because that's what you would need to imagine for option (a) when
a
> right accent occurs next to a shaft. The stick-man is facing in
the
> direction that one reads the music.

I still don't get it. When I'm reading the music, I'm not _in_ the
page.

> Then I would say that you're having a failure of imagination,
which
> has been known to lead to serious consequences. ;-(

OK. Well I _can_ imagine myself in the page, but it isn't something
I would have thought of doing myself (nor do I suspect many others),
so I don't see what bearing it has on this discussion.

> > I imagine a stencilled arrow pointing upward,
> > where the right half of the arrowhead can either be in its
normal
> > orientation or be flipped vertically to indicate that it is
> intended
> > to partly cancel the effect of the left half.
>
> This works only half of the time, when there's no right flag
> between. Remember that these are accent marks and not necessarily
> arrowheads (see next comment).

It still works for me when there are right flags between.

> When we read (either music or text) we travel left to right.

That's an interesting point. When reading music the overall
direction is certainly left tro right. But what is read first, the
accidental or the note? It would make sense from an information
processing point of view to get the big distinctions first and then
take in the finer ones. But the accidental is to the _left_ of the
note.

But I really expect the sagittal symbol, with accents, to be treated
as a single symbol. No one asks in which direction we read a
letter "i". Do you read the dot first, or the vertical stroke? Or
the "Q". Do you read the tail first or last? The questions don't
make sense. So I'll drop "direction of travel" and rely only
on "arrowness", which has exactly the same effect as a "direction of
travel" toward the shaft, but a different explanation.

> Did you not make the point some time ago that these accents are
> *acute* and *grave*, which have upward and downward meaning in
spoken
> languages? Only option (a) abides by those meanings.

That's correct. Darn! :-)

> I didn't say that it would never be used, only that the vertical
> position would be more valuable than the arrow cue when you
consider
> that this would be used mostly in ascii.

OK. Well, I'm agreeing to use vertical position like this, by
agreeing to option (c).

> > Note that I said, "In the absence of other cues". For me, the
> > presence of a shaft, anywhere, is such a cue, and results in the
> > direction of travel being towards the shaft, from either side.
>
> So we have the following as cues:
> 1) Perception of accent mark as arrowhead
> 2) Vertical placement (relative to notehead & shaft)
> 3) Perception of accent mark as acute or grave
>
> In the graphic cues 2) and 3) are *always* present, whereas cue 1)
is
> weakened if there is a right flag between the shaft and right
accent.
>

Yes. But 3) is only very weak to start with, since acutes and graves
usually appear _above_ the thing they modify. And that thing is
usually a _letter_, not an arrow.

We both agree we want cue 2). But I'm afraid I still find cue 1)
stronger than cue 3).

Even with a right barb in place, say for (|\` doesn't the right
accent seem to be _helping_ the right barb when it is nearly
parallel to it (particularly if it is kerned closer)?

> In ascii cue 2) by itself is sufficient to clarify the direction,
> while cues 1) and 3) are of little value, regardless of whether
the
> same or differing characters are used for left and right accents.

Agreed. Except that this can remind you what the standard is for the
graphical case.

> The non-slanting apostrophe of Courier font or the curly comma and
> apostrophe of Times Roman do not readily indicate a line of
constant
> slope.

They do to me. The non-slanting apostrophe is certainly constant, it
just is neither upward-to-the-left nor upward-to-the-right. Although
the comma (and curly apostrophe although I don't know why we're
including it since it isn't ASCII) is not constant, at least in
Courier it is downward-to-the-left (or upward-to-the-right) while
the ASCII back-quote (far upper left of US keyboard) is clearly
upward to the left.

> So I guess we agree that (b) is eliminated.

Yes. Unless more voters turn out and vote it number 1. But I suspect
it could only ever be number 2 for anyone.

> > However I disagree about what you call "slope". I
believe "slope"
> is
> > working for us in (b) and (c) but not in (a), as is "arrowness".
>
> And the acute-grave slope distinction works for us in (a), but not
in
> (b) or (c), if we look at these as "accent marks" rather
> than "arrowheads".

Yes, but only if.

> > ... And so surely the left side of this points up, while its
> > right side points down
> >
> > /|/
> > |
> >
> > which is why I can't accept (a).
>
> But on the other hand, if you see them as accent marks, they look
> alike.

Agreed. But I see them as arrowheads. Accent marks go _above_
things, and you never have two of them applied to the same thing. Or
am I just being stubborn?

> > The TrueType font is not the system, as evidenced by the
> > fact that Scala doesn't use it, but uses bitmaps instead. Would
you
> > prohibit Scala from using a bitmapped version of (c) just
because
> we
> > can't fit the extra accents in the TrueType font?
>
> I think Scala will have to conform to whatever we have in the font.

I disagree, given that Word and Sibelius can easily work around this.

> > This is also a good reason _not_ to reuse the characters ' and .
> for
> > the right accents in ASCII, but rather to use ` and ,
> >
> > Do you agree?
>
> I came to the opposite conclusion: You would be presenting the
user
> with the burden to exercise care not to mix up accent characters,
> e.g., by mistakenly typing `| or ,| or |' or |. or `|' or `|.
or ...,
> such that a computer might not process the character stream
properly.
>

Hey, there's no problem with making the software forgiving of such
mistakes on input, but we should still use the single most
representative version for output, and promote it for human to human
communication.

As a case in point I asked Jacob Barton two days ago if he would
modify his Sibelius script so when the user enters mixed notation it
doesn't matter if they put the sagittal to the right or the left of
the conventional.

It can also be arranged for software to ignore whether the shaft is
a bar or exclamation mark when the symbol contains directed flags
(barbs or left accents) and they all agree on a direction.

But of course this doesn't alter what the standard form actually is.

> I don't think you completely got my point. Keep It Simple, Stu_d!:

ROTFL :-) Stud? I didn't know you thought of me in _that_ way. ;-)

> 1) By using only 2 characters instead of 4 in ascii, thereby:
> a) Having *matching* characters on the left and right sides for
the
> *same* direction of alteration, and
> b) Making it unnecessary to remember which characters go on the
> left side and which go on the right; and
> 2) By eliminating problems with the font:
> a) To make space for two more characters, or
> b) To require special character alignment to create the graphic.
>
> And if we agree to use only 2 characters for the accents, I would
> propose replacing . with , for both the schisma down and schismina
> down. The period character tends to be difficult to see next to
an
> arrow-shaft in some fonts.

It seems we are at an impasse. Let it rest for a while.

> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > I checked out Finale. Although it can do multi-character
> > accidentals, there's no way to change vertical offsets.
> >
> > So if we ever encounter anyone who needs to use right accents in
> > Finale we can make them a special version of the font, with
sharp
> > and double-flat (or something) changed to the corectly offset
right
> > accents.
>
> I would say that it looks like one more reason not to use option
(c).

An extremely tiny, almost insignificant reason.

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

> > Wait a minute! Isn't there already an atom of Kirnberger? Yes,
> look
> > here:
> > http://tonalsoft.com/enc/eqtemp.htm
> > This gives "atomic" temperament as the last entry in a table in
> > connection with divisions 12, 600, 612, 3072, 3684, and 4296.
> >
> > Hey, Gene, what gives? Do we look for another name for ratios
> > corresponding to a single degree of 2460-ET? If so, how about
> > something related to "olympian"?
>
> Yes it's strange that 2460-ET was not included on that list, but I
> thought you knew that the "atomic temperament" of this thread was
> the one in which, in the 5-limit version, only the atom of
> Kirnberger vanishes.

A much more plausible list would be 12, 612, 1848, 2460, 4296, 4320,
11664, 16572. The encyclopedia lists things we don't need, and misses
stuff which seems downright obligatory.

> > My point is that it's not critical that schismina accents be read
> > accurately in *real time*, inasmuch as the reader should have time
> > to take a good look to be sure of the meaning.
>
> Agreed. But I still think we should make it as easy as possible, and
> not let Symbol font limitations influence us too much.
>
> > Do you think there's anyone else out there crazy enough to want to
> > bother to add their 4 schisminas?
>
> Hee hee.
>
> > But it would be also relative to the shaft, positioned either near
> > the end or near the middle.
>
> True.
>
> > Because that's what you would need to imagine for option (a) when
> a
> > right accent occurs next to a shaft. The stick-man is facing in
> the
> > direction that one reads the music.
>
> I still don't get it. When I'm reading the music, I'm not _in_ the
> page.
>
> > Then I would say that you're having a failure of imagination,
> which
> > has been known to lead to serious consequences. ;-(
>
> OK. Well I _can_ imagine myself in the page, but it isn't something
> I would have thought of doing myself (nor do I suspect many others),
> so I don't see what bearing it has on this discussion.
>
> > > I imagine a stencilled arrow pointing upward,
> > > where the right half of the arrowhead can either be in its
> normal
> > > orientation or be flipped vertically to indicate that it is
> > intended
> > > to partly cancel the effect of the left half.
> >
> > This works only half of the time, when there's no right flag
> > between. Remember that these are accent marks and not necessarily
> > arrowheads (see next comment).
>
> It still works for me when there are right flags between.
>
> > When we read (either music or text) we travel left to right.
>
> That's an interesting point. When reading music the overall
> direction is certainly left tro right. But what is read first, the
> accidental or the note? It would make sense from an information
> processing point of view to get the big distinctions first and then
> take in the finer ones. But the accidental is to the _left_ of the
> note.
>
> But I really expect the sagittal symbol, with accents, to be treated
> as a single symbol. No one asks in which direction we read a
> letter "i". Do you read the dot first, or the vertical stroke? Or
> the "Q". Do you read the tail first or last? The questions don't
> make sense. So I'll drop "direction of travel" and rely only
> on "arrowness", which has exactly the same effect as a "direction of
> travel" toward the shaft, but a different explanation.
>
> > Did you not make the point some time ago that these accents are
> > *acute* and *grave*, which have upward and downward meaning in
> spoken
> > languages? Only option (a) abides by those meanings.
>
> That's correct. Darn! :-)
>
> > I didn't say that it would never be used, only that the vertical
> > position would be more valuable than the arrow cue when you
> consider
> > that this would be used mostly in ascii.
>
> OK. Well, I'm agreeing to use vertical position like this, by
> agreeing to option (c).
>
> > > Note that I said, "In the absence of other cues". For me, the
> > > presence of a shaft, anywhere, is such a cue, and results in the
> > > direction of travel being towards the shaft, from either side.
> >
> > So we have the following as cues:
> > 1) Perception of accent mark as arrowhead
> > 2) Vertical placement (relative to notehead & shaft)
> > 3) Perception of accent mark as acute or grave
> >
> > In the graphic cues 2) and 3) are *always* present, whereas cue 1)
> is
> > weakened if there is a right flag between the shaft and right
> accent.
> >
>
> Yes. But 3) is only very weak to start with, since acutes and graves
> usually appear _above_ the thing they modify. And that thing is
> usually a _letter_, not an arrow.
>
> We both agree we want cue 2). But I'm afraid I still find cue 1)
> stronger than cue 3).
>
> Even with a right barb in place, say for (|\` doesn't the right
> accent seem to be _helping_ the right barb when it is nearly
> parallel to it (particularly if it is kerned closer)?
>
> > In ascii cue 2) by itself is sufficient to clarify the direction,
> > while cues 1) and 3) are of little value, regardless of whether
> the
> > same or differing characters are used for left and right accents.
>
> Agreed. Except that this can remind you what the standard is for the
> graphical case.
>
> > The non-slanting apostrophe of Courier font or the curly comma and
> > apostrophe of Times Roman do not readily indicate a line of
> constant
> > slope.
>
> They do to me. The non-slanting apostrophe is certainly constant, it
> just is neither upward-to-the-left nor upward-to-the-right. Although
> the comma (and curly apostrophe although I don't know why we're
> including it since it isn't ASCII) is not constant, at least in
> Courier it is downward-to-the-left (or upward-to-the-right) while
> the ASCII back-quote (far upper left of US keyboard) is clearly
> upward to the left.
>
> > So I guess we agree that (b) is eliminated.
>
> Yes. Unless more voters turn out and vote it number 1. But I suspect
> it could only ever be number 2 for anyone.
>
> > > However I disagree about what you call "slope". I
> believe "slope"
> > is
> > > working for us in (b) and (c) but not in (a), as is "arrowness".
> >
> > And the acute-grave slope distinction works for us in (a), but not
> in
> > (b) or (c), if we look at these as "accent marks" rather
> > than "arrowheads".
>
> Yes, but only if.
>
> > > ... And so surely the left side of this points up, while its
> > > right side points down
> > >
> > > /|/
> > > |
> > >
> > > which is why I can't accept (a).
> >
> > But on the other hand, if you see them as accent marks, they look
> > alike.
>
> Agreed. But I see them as arrowheads. Accent marks go _above_
> things, and you never have two of them applied to the same thing. Or
> am I just being stubborn?
>
> > > The TrueType font is not the system, as evidenced by the
> > > fact that Scala doesn't use it, but uses bitmaps instead. Would
> you
> > > prohibit Scala from using a bitmapped version of (c) just
> because
> > we
> > > can't fit the extra accents in the TrueType font?
> >
> > I think Scala will have to conform to whatever we have in the font.
>
> I disagree, given that Word and Sibelius can easily work around this.
>
> > > This is also a good reason _not_ to reuse the characters ' and .
> > for
> > > the right accents in ASCII, but rather to use ` and ,
> > >
> > > Do you agree?
> >
> > I came to the opposite conclusion: You would be presenting the
> user
> > with the burden to exercise care not to mix up accent characters,
> > e.g., by mistakenly typing `| or ,| or |' or |. or `|' or `|.
> or ...,
> > such that a computer might not process the character stream
> properly.
> >
>
> Hey, there's no problem with making the software forgiving of such
> mistakes on input, but we should still use the single most
> representative version for output, and promote it for human to human
> communication.
>
> As a case in point I asked Jacob Barton two days ago if he would
> modify his Sibelius script so when the user enters mixed notation it
> doesn't matter if they put the sagittal to the right or the left of
> the conventional.
>
> It can also be arranged for software to ignore whether the shaft is
> a bar or exclamation mark when the symbol contains directed flags
> (barbs or left accents) and they all agree on a direction.
>
> But of course this doesn't alter what the standard form actually is.
>
> > I don't think you completely got my point. Keep It Simple, Stu_d!:
>
> ROTFL :-) Stud? I didn't know you thought of me in _that_ way. ;-)
>
> > 1) By using only 2 characters instead of 4 in ascii, thereby:
> > a) Having *matching* characters on the left and right sides for
> the
> > *same* direction of alteration, and
> > b) Making it unnecessary to remember which characters go on the
> > left side and which go on the right; and
> > 2) By eliminating problems with the font:
> > a) To make space for two more characters, or
> > b) To require special character alignment to create the graphic.
> >
> > And if we agree to use only 2 characters for the accents, I would
> > propose replacing . with , for both the schisma down and schismina
> > down. The period character tends to be difficult to see next to
> an
> > arrow-shaft in some fonts.
>
> It seems we are at an impasse. Let it rest for a while.
>
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > I checked out Finale. Although it can do multi-character
> > > accidentals, there's no way to change vertical offsets.
> > >
> > > So if we ever encounter anyone who needs to use right accents in
> > > Finale we can make them a special version of the font, with
> sharp
> > > and double-flat (or something) changed to the corectly offset
> right
> > > accents.
> >
> > I would say that it looks like one more reason not to use option
> (c).
>
> An extremely tiny, almost insignificant reason.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> [GS:]
> > I would say use + and - (minus sign, not a hyphen). I'm
beginning to
> > think that we should change the term "schismina" to "atom", so we
> > could simply write s+a up, s-a up, s+a down, and s-a down. Then
> > 4095:4096 could be called a 13-atom. (I'm sure Gene would like
this.)
>
> No. The only already-named atom is far far smaller than a
schismina.
> It's only 0.015 cents. Calling something as large as 1.8 cents
> an "atom" doesn't seem right. I'd like to keep that name for a
> category even smaller than a schismina, with maybe even another
> category intervening.

Yes, I saw that once I found "atom of Kirnberger" in Monz's
encyclopaedia. The point is that the term "atom" can't be used for
1deg2460, because an atom is supposed to *vanish* in an atomic
temperament.

What I would really like is a new term that could be used for a
single degree of 2460-ET for a convenient unit of measure at the
olympian level, in the same manner that cents and heptamerides have
been used in the past. It might be "schismina" (in the same manner
as "comma" has been used for degrees of 53-ET), but I would prefer
something more concise that does not begin with the letter "s" (or k,
c, d, m, or l).

If anyone has a suggestion for a name that would relate to an ancient-
Greek theme, I would appreciate it.

> > My point is that it's not critical that schismina accents be read
> > accurately in *real time*, inasmuch as the reader should have time
> > to take a good look to be sure of the meaning.
>
> Agreed. But I still think we should make it as easy as possible,
and
> not let Symbol font limitations influence us too much.
>
> > Do you think there's anyone else out there crazy enough to want
to
> > bother to add their 4 schisminas?
>
> Hee hee.

So I don't think there will be anyone else to cast a vote (not even
Apollo, who left after making his announcement).

But what about the symbol font limitations? If we find ourselves at
an impasse, should they then be allowed to cast the deciding vote?

> ...
> > I don't think you completely got my point. Keep It Simple,
Stu_d!:
>
> ROTFL :-) Stud? I didn't know you thought of me in _that_ way. ;-)

I purposely made that last word ambiguous so you could replace the
underscore with whatever *you thought* was fitting. ;-)

> ...
> It seems we are at an impasse. Let it rest for a while.

Okay, once I'm done replying.

We've been mostly discussing the ease of *reading* the accents, but
some of my arguments had to do with *writing* the accents, i.e., ease
of choosing which character (in either ascii or the font) to use.

> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > I checked out Finale. Although it can do multi-character
> > > accidentals, there's no way to change vertical offsets.
> > >
> > > So if we ever encounter anyone who needs to use right accents
in
> > > Finale we can make them a special version of the font, with
sharp
> > > and double-flat (or something) changed to the corectly offset
right
> > > accents.

Besides making the procedure for choosing right-side accents more
*difficult* (see following paragraph), this also makes it *different*
depending on what software you're using.

The greater difficulty involves an additional challenge for the
original user who is creating the output. With option (c) there
would be two pairs of characters in the font that look exactly the
same *except* for their vertical placement (extremely small -- only 3
pixels different in vertical placement in the Scala graphic files),
which might confuse the user into choosing an inappropriate one for a
given side of the shaft. While software could spot an inappropriate
character, it could not be made to replace the correct one
automatically, because there would be two possibilities, e.g., if the
user places ' to the right of the shaft, was the confusion due to the
wrong *slope* or the wrong *vertical placement*. The user would need
to be prompted in this instance, but would Finale (or Sibelius, for
that matter) do that?

With option (a) the user only needs to observe which *direction* the
accent mark is slanting, regardless of which side of the shaft it is
on: / for an upward alteration or \ for downward, according to the
convention already established when reading a mathematical graph.
Requiring the user to choose from among 4 characters is more
complicated than if there are only 2. Let's Keep It Simpler for the
Stupefied (and possibly freaked-out after wading through all of the
symbols in the font) original *user* who is *creating* the output.

Are there any other crazies out there who would care to comment/vote?

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> What I would really like is a new term that could be used for a
> single degree of 2460-ET for a convenient unit of measure at the
> olympian level, in the same manner that cents and heptamerides
have
> been used in the past. It might be "schismina" (in the same
manner
> as "comma" has been used for degrees of 53-ET), but I would prefer
> something more concise that does not begin with the letter "s" (or
k,
> c, d, m, or l).

I don't think we need anything more concise that schismina or any
abbreviation shorter than "sa". No one is going to use this anyway,
at least no mere mortals. But we can certainly consider the
term "schismina", when unqualified by any ratio, to refer to
4095:4096, in the same way that an unqualified schisma refers to the
5-schisma.

> But what about the symbol font limitations? If we find ourselves
at
> an impasse, should they then be allowed to cast the deciding vote?

No way! In ten years time it will seem ridiculous if we let a
temporary technical limitation influence us to go against what is
best for the _reader_.

I thought we could get right accents for free with what we already
had. It turns out we can't. Do we really still want to even bother
with them at this stage. Why not instead write a manifesto for the
future, when the font limitations have been removed. What would be
the ideal way of doing them? Maybe they should be curved or wavy?
Maybe they should go on the left?

> Okay, once I'm done replying.

Yeah. And once I'm done replying to your reply. :-)

> We've been mostly discussing the ease of *reading* the accents,
but
> some of my arguments had to do with *writing* the accents, i.e.,
ease
> of choosing which character (in either ascii or the font) to use.

I think such arguments are spurious. One should always do what is
best for the reader if there is such a conflict. This is because
there is only ever one author, but the number of readers is
potentially unlimited. This is the argument I am contantly having
with authors on this list who want to make up, and exlusively use,
short meaningless names for things the moment they have thought of
them (and then change the name a few months later).

> Besides making the procedure for choosing right-side accents more
> *difficult* (see following paragraph), this also makes it
*different*
> depending on what software you're using.

OK. Let's redo the font mapping from scratch now that we have the
wilson symbols and the right accent, and leave room for another
pair. We can leave out all four standard accidentals bb b # x, and
lay out a section with the bare shafts and accents all together to
make it as easy as possible for the writer.

There aren't so many users yet, that they couldn't cope. For
Sibelius users we just have to give them another library to import
when they install the new font version and everything will be
unchanged. We haven't made a library for Finale yet, so it doesn't
matter.

> The greater difficulty involves an additional challenge for the
> original user who is creating the output. With option (c) there
> would be two pairs of characters in the font that look exactly the
> same *except* for their vertical placement (extremely small --
only 3
> pixels different in vertical placement in the Scala graphic
files),
> which might confuse the user into choosing an inappropriate one
for a
> given side of the shaft. While software could spot an
inappropriate
> character, it could not be made to replace the correct one
> automatically, because there would be two possibilities, e.g., if
the
> user places ' to the right of the shaft, was the confusion due to
the
> wrong *slope* or the wrong *vertical placement*. The user would
need
> to be prompted in this instance, but would Finale (or Sibelius,
for
> that matter) do that?

Aw c'mon George. Anyone who is sophisticated enough to actually need
and want to use schismina accents can probably put their shoes on
the correct feet nine days out of ten. We just need to provide the
documentation for them on the website, showing them what it is
supposed to look like.

However, as I said, we could make right accents actually look
different (e.g. concave or wavy).

> With option (a) the user only needs to observe which *direction*
the
> accent mark is slanting, regardless of which side of the shaft it
is
> on: / for an upward alteration or \ for downward, according to the
> convention already established when reading a mathematical graph.

Sigh. But this convention is already violated for right flags, which
is how I perceive the right accent. Violated in favour
of "arrowness". Doesn't symmetry demand this?

> Requiring the user to choose from among 4 characters is more
> complicated than if there are only 2. Let's Keep It Simpler for
the
> Stupefied (and possibly freaked-out after wading through all of
the
> symbols in the font) original *user* who is *creating* the output.

I suggest we leave all the unaccented sagittals and natural where
they are in the character map, but where we currently have, at the
start of the font

bb b - # x

we replace it with

<spare> . ! , -

and where we currently have, at the end of the font

. ! + | '

we replace it with

+ ' | ` <spare>

(- and + are the wilson 5-comma symbols).

That should avoid confusion for the writer, even if the right
accents are just mirror images of the left ones, as the up accents
are on their correct sides of the up shaft in the character map, and
they are nowhere near the down accents which are on the correct
sides of the down shaft.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > What I would really like is a new term that could be used for a
> > single degree of 2460-ET for a convenient unit of measure at the
> > olympian level, in the same manner that cents and heptamerides
have
> > been used in the past. It might be "schismina" (in the same
manner
> > as "comma" has been used for degrees of 53-ET), but I would
prefer
> > something more concise that does not begin with the letter "s"
(or k,
> > c, d, m, or l).
>
> I don't think we need anything more concise that schismina or any
> abbreviation shorter than "sa".

I wanted a single letter, such as "a" (for atom), that could be used
for an interval abbreviation, so that 4095:4096 would be 13a, or so
that we could easily write equations such as 13M = 35M-a. All the
other Sagittal symbols have single-letter abbreviations, so what's so
difficult about coming up with a single letter for this one?

> No one is going to use this anyway,
> at least no mere mortals.

So why are we arguing about something that no one will use? ;-)

Still, theorists might use it.

> But we can certainly consider the
> term "schismina", when unqualified by any ratio, to refer to
> 4095:4096, in the same way that an unqualified schisma refers to
the
> 5-schisma.

Agreed.

> > But what about the symbol font limitations? If we find ourselves
at
> > an impasse, should they then be allowed to cast the deciding vote?
>
> No way! In ten years time it will seem ridiculous if we let a
> temporary technical limitation influence us to go against what is
> best for the _reader_.
>
> I thought we could get right accents for free with what we already
> had. It turns out we can't. Do we really still want to even bother
> with them at this stage. Why not instead write a manifesto for the
> future, when the font limitations have been removed. What would be
> the ideal way of doing them? Maybe they should be curved or wavy?
> Maybe they should go on the left?
>
> > Okay, once I'm done replying.
>
> Yeah. And once I'm done replying to your reply. :-)

And me to yours, etc., etc.. ;-(

But please bear with me.

> > We've been mostly discussing the ease of *reading* the accents,
but
> > some of my arguments had to do with *writing* the accents, i.e.,
ease
> > of choosing which character (in either ascii or the font) to use.
>
> I think such arguments are spurious. One should always do what is
> best for the reader if there is such a conflict.

But the problem is that we can't seem to agree on what's best for the
reader. You see accent marks making arrows in combination with the
shaft and I see them as being acute or grave.

On the other hand, I would say that choice (a) has a *huge* advantage
for the writer, and, taken together with the fact that it doesn't
create a font-space problem, is the simplest way to implement
schisminas in the notation. Why go out of our way to provide
additional characters for something that we think hardly anybody will
ever use?

> This is because
> there is only ever one author, but the number of readers is
> potentially unlimited. This is the argument I am contantly having
> with authors on this list who want to make up, and exlusively use,
> short meaningless names for things the moment they have thought of
> them (and then change the name a few months later).

So we make a decision and stick with it, as long as it makes *some*
kind of sense.

> > Besides making the procedure for choosing right-side accents more
> > *difficult* (see following paragraph), this also makes it
*different*
> > depending on what software you're using.
>
> OK. Let's redo the font mapping from scratch now that we have the
> wilson symbols and the right accent, and leave room for another
> pair. We can leave out all four standard accidentals bb b # x,

Do you think no one would ever want to use those symbols? ;-)

> and
> lay out a section with the bare shafts and accents all together to
> make it as easy as possible for the writer.
>
> There aren't so many users yet, that they couldn't cope. For
> Sibelius users we just have to give them another library to import
> when they install the new font version and everything will be
> unchanged. We haven't made a library for Finale yet, so it doesn't
> matter.

Why complicate things with separate fonts for separate applications?

> > The greater difficulty involves an additional challenge for the
> > original user who is creating the output. With option (c) there
> > would be two pairs of characters in the font that look exactly
the
> > same *except* for their vertical placement (extremely small --
only 3
> > pixels different in vertical placement in the Scala graphic
files),
> > which might confuse the user into choosing an inappropriate one
for a
> > given side of the shaft. While software could spot an
inappropriate
> > character, it could not be made to replace the correct one
> > automatically, because there would be two possibilities, e.g., if
the
> > user places ' to the right of the shaft, was the confusion due to
the
> > wrong *slope* or the wrong *vertical placement*. The user would
need
> > to be prompted in this instance, but would Finale (or Sibelius,
for
> > that matter) do that?
>
> Aw c'mon George. Anyone who is sophisticated enough to actually
need
> and want to use schismina accents can probably put their shoes on
> the correct feet nine days out of ten. We just need to provide the
> documentation for them on the website, showing them what it is
> supposed to look like.

Putting accents on multitudes of symbols is more tedious than putting
on shoes (two and you're done!). Once your eyes get tired,
everything starts looking too much alike. Given fewer opportunities
to make mistakes (through fewer choices), and you'll make fewer
mistakes.

> However, as I said, we could make right accents actually look
> different (e.g. concave or wavy).

I tried it, and it's not worth the trouble. Up and down versions of
other shapes are more difficult to tell apart than are sloped accent
marks, given the requirement that they must not be any larger than
our existing accent marks.

> > With option (a) the user only needs to observe which *direction*
the
> > accent mark is slanting, regardless of which side of the shaft it
is
> > on: / for an upward alteration or \ for downward, according to
the
> > convention already established when reading a mathematical graph.
>
> Sigh. But this convention is already violated for right flags,
which
> is how I perceive the right accent. Violated in favour
> of "arrowness". Doesn't symmetry demand this?

Why would we want symmetry? With option (c) it introduces lateral
confusability: of '| with |` and .| with |, (except in ascii). With
option (a) lateral confusability is eliminated (except in ascii).

Besides, you're proposing to violate the acute-vs.-grave convention.
If you recall, right flags already have the function of indicating
the direction of 5-comma alteration from pythagorean tones in the
pure notation, e.g., ||\ before a note is a sharp altered by \! . So
the reader may already know about observing arrow-ness for overall
direction of alteration and the slope of a single barb (either right
or left) for the direction of 5-comma alteration.

> > Requiring the user to choose from among 4 characters is more
> > complicated than if there are only 2. Let's Keep It Simpler for
the
> > Stupefied (and possibly freaked-out after wading through all of
the
> > symbols in the font) original *user* who is *creating* the output.
>
> I suggest we leave all the unaccented sagittals and natural where
> they are in the character map, but where we currently have, at the
> start of the font
>
> bb b - # x
>
> we replace it with
>
> <spare> . ! , -
>
> and where we currently have, at the end of the font
>
> . ! + | '
>
> we replace it with
>
> + ' | ` <spare>
>
> (- and + are the wilson 5-comma symbols).

Now you've sacrificed the convenience of having the accents near each
other in the font in order to address a potential problem with
confusion.

> That should avoid confusion for the writer, even if the right
> accents are just mirror images of the left ones,

i.e., lateral mirroring will *not* cause confusion. Too bad I
couldn't have quoted this back to you 2.5 years ago when you first
brought up the issue of lateral confusability. ;-)

> as the up accents
> are on their correct sides of the up shaft in the character map,
and
> they are nowhere near the down accents which are on the correct
> sides of the down shaft.

That's for the writer. It doesn't apply to the reader, who can't
tell where the characters are located in the font?

Sorry to be giving you such a hard time about this, but I wanted to
follow through with the established procedure of giving additional
reasons for making a decision until a consensus is reached (or until
one of us wears the other one out. :-)

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> I wanted a single letter, such as "a" (for atom), that could be
used
> for an interval abbreviation, so that 4095:4096 would be 13a, or
so
> that we could easily write equations such as 13M = 35M-a. All the
> other Sagittal symbols have single-letter abbreviations, so what's
so
> difficult about coming up with a single letter for this one?

So use "i" or "n" for schismINa.

> But the problem is that we can't seem to agree on what's best for
the
> reader. You see accent marks making arrows in combination with
the
> shaft and I see them as being acute or grave.

The only way to settle it would be by some kind of survey. But I'd
been embarrassed to ask anyone to spend their time on something of
so little use.

> On the other hand, I would say that choice (a) has a *huge*
advantage
> for the writer,

I don't see that.

> and, taken together with the fact that it doesn't
> create a font-space problem, is the simplest way to implement
> schisminas in the notation. Why go out of our way to provide
> additional characters for something that we think hardly anybody
will
> ever use?

That's a good point.

> > This is because
> > there is only ever one author, but the number of readers is
> > potentially unlimited. This is the argument I am contantly
having
> > with authors on this list who want to make up, and exlusively
use,
> > short meaningless names for things the moment they have thought
of
> > them (and then change the name a few months later).
>
> So we make a decision and stick with it, as long as it makes
*some*
> kind of sense.

How about we just decide that schismina accents go on the left, and
if its effect is in the same direction as the arrow shaft it goes
nearer the tip otherwise nearer the middle and we don't care which
way they slope. Suit yourself.

> Do you think no one would ever want to use those symbols? ;-)

OK. I meant almost no one.

> Why complicate things with separate fonts for separate
applications?
>

No. You misunderstood. There would only be one font. But as well as
the font, you have to provide a library that maps from an indirect
internal description like "accidental 1", "accidental 2" etc. to the
actual font and characters (multicharacter symbols are allowed). The
library formats are different for Finale and Sibelius because the
manufacturers made them that way.

> > However, as I said, we could make right accents actually look
> > different (e.g. concave or wavy).
>
> I tried it, and it's not worth the trouble. Up and down versions
of
> other shapes are more difficult to tell apart than are sloped
accent
> marks, given the requirement that they must not be any larger than
> our existing accent marks.

OK. So I guess that elimiates the possibility of putting them on the
left where accentness and arrowness coincide.

> Sorry to be giving you such a hard time about this, but I wanted
to
> follow through with the established procedure of giving additional
> reasons for making a decision until a consensus is reached (or
until
> one of us wears the other one out. :-)

You win. I'm worn out. :-)

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > I wanted a single letter, such as "a" (for atom), that could be
used
> > for an interval abbreviation, so that 4095:4096 would be 13a, or
so
> > that we could easily write equations such as 13M = 35M-a. All
the
> > other Sagittal symbols have single-letter abbreviations, so
what's so
> > difficult about coming up with a single letter for this one?
>
> So use "i" or "n" for schismINa.

Okay. I suppose that "n" would be preferable, since that's the one
letter that doesn't occur in "schisma".

> > But the problem is that we can't seem to agree ...
>
> > > However, as I said, we could make right accents actually look
> > > different (e.g. concave or wavy).
> >
> > I tried it, and it's not worth the trouble. Up and down versions
of
> > other shapes are more difficult to tell apart than are sloped
accent
> > marks, given the requirement that they must not be any larger
than
> > our existing accent marks.
>
> OK. So I guess that elimiates the possibility of putting them on
the
> left where accentness and arrowness coincide.

Yes, that's exactly what I concluded.

> > Sorry to be giving you such a hard time about this, but I wanted
to
> > follow through with the established procedure of giving
additional
> > reasons for making a decision until a consensus is reached (or
until
> > one of us wears the other one out. :-)
>
> You win. I'm worn out. :-)

Uh -- just one more question. Would you consider consider changing
the ascii schisma/schismina-down character "." to ",", since "."
tends to be difficult to see in certain fonts? I don't remember
offhand if Manuel had any problems handling a comma in the ascii
notation character string, but I do remember that he didn't want
anything ending with a period, because he uses that as a delimiter
for octave-number.

Manuel?

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > You win. I'm worn out. :-)
>
> Uh -- just one more question. Would you consider consider
> changing ...

Oh. So while I'm lying on the ground exhausted, you just want to
give me one more kick. :-)

> ... the ascii schisma/schismina-down character "." to ",",
> since "." tends to be difficult to see in certain fonts?

So use a different font. You expend all this effort to convince me
that the right accent should slope the same way as the left accent
and then you want to use an ASCII character that slopes the other
way!?

I guess I wouldn't mind too much if "," and "`" were used for the
right accents, but I wouldn't want them used on the left.

> I don't remember
> offhand if Manuel had any problems handling a comma in the ascii
> notation character string, but I do remember that he didn't want
> anything ending with a period, because he uses that as a delimiter
> for octave-number.

Manuel already had to deal with a trailing period in the mixed short
form of sagittal, for the 5-schisma down symbol itself. I forget
what was decided, but I certainly wouldn't mind if the comma was
required as a substitute in that case. I suspect it only required a
slightly smarter parser (or a special case) to handle the two
consecutive periods. I can't see why it would be a problem for
output, only input.

But here's a problem that's independent of Scala. If we use the same
characters for the left and right accents in pure and mixed-long
ASCII, then when they are used without shafts in the mixed-short
(single-character) ASCII there will be some ambiguity. I guess we
can simply say that there are _no_ single-character ASCII symbols
for schisminas, so they _always_ represent the 5-schisma in that
form.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > You win. I'm worn out. :-)
> >
> > Uh -- just one more question. Would you consider consider
> > changing ...
>
> Oh. So while I'm lying on the ground exhausted, you just want to
> give me one more kick. :-)

Sorry. I've been rather stressed lately myself and have been less
thoughtful than I should be.

> > ... the ascii schisma/schismina-down character "." to ",",
> > since "." tends to be difficult to see in certain fonts?
>
> So use a different font. You expend all this effort to convince me
> that the right accent should slope the same way as the left accent
> and then you want to use an ASCII character that slopes the other
> way!?
>
> I guess I wouldn't mind too much if "," and "`" were used for the
> right accents, but I wouldn't want them used on the left.

Okay, forget it. We'll stick with the period.

> ...
> But here's a problem that's independent of Scala. If we use the
same
> characters for the left and right accents in pure and mixed-long
> ASCII, then when they are used without shafts in the mixed-short
> (single-character) ASCII there will be some ambiguity. I guess we
> can simply say that there are _no_ single-character ASCII symbols
> for schisminas, so they _always_ represent the 5-schisma in that
> form.

Yes. It seems that we have to keep reminding ourselves that hardly
anyone will ever need to notate schisminas. :-)

--George