Sagittal question

How does one express 49/36 in Sagittal? Is it the theoretical standard
that one should use double/triple symbols?

-A.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
> How does one express 49/36 in Sagittal?

Hi, Aaron. So glad you asked!

There's more than one answer, depending on what level of precision
you desire. If you use medium-precision (a/k/a athenian-level)
Sagittal JI in Scala (set nota SAJI1, taking C as 1/1), you'll get 3
spellings: F(|(, Gb\!!!/ or Gb\!/, and E)|||( or E#)|(, all of which
are approximations.

If you use high-precision (a/k/a herculean-level) Sagittal JI, there
are symbols that notate the 2 preferred spellings *exactly*: the ~|)
symbol is defined as 48:49 and the (/| symbol as 3969:4096.

So 49/36 of C would be notated exactly as F~|), and Gb(\!!! or Gb(\!,
while the 3rd spelling is approximated by E')|||( or E#')|(; however,
we may eventually have a way to notate this last spelling exactly in
an extreme-precision notation -- if anyone really cares.

> Is it the theoretical standard
> that one should use double/triple symbols?

The choice between single- or multiple-shaft symbols is simply a
matter of personal preference involving a trade-off between number of
symbols vs. conciseness. Since they can be be converted back and
forth unambiguously from one form to the other, either form is
suitable for theoretical applications.

--George

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
>
> How does one express 49/36 in Sagittal? Is it the theoretical standard
> that one should use double/triple symbols?

Oops, I suddenly realized what you meant by the second part. More than
one Sagittal accidental at a time is a no-no. They may be used only in
combination with conventional sharp and flat (including double) symbols.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
> wrote:
> >
> >
> > How does one express 49/36 in Sagittal? Is it the theoretical standard
> > that one should use double/triple symbols?
>
> Oops, I suddenly realized what you meant by the second part. More than
> one Sagittal accidental at a time is a no-no. They may be used only in
> combination with conventional sharp and flat (including double) symbols.

Now I see that you know what I mean---so my question is, why does
sagittal seem to limit the accurate perfect representation of free,
distant modulation? ;)

Best,
Aaron.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
wrote:
> > >
> > > How does one express 49/36 in Sagittal? Is it the theoretical
standard
> > > that one should use double/triple symbols?
> >
> > Oops, I suddenly realized what you meant by the second part.
More than
> > one Sagittal accidental at a time is a no-no. They may be used
only in
> > combination with conventional sharp and flat (including double)
symbols.
>
> Now I see that you know what I mean---so my question is, why does
> sagittal seem to limit the accurate perfect representation of free,
> distant modulation? ;)

I'm glad you used the word "seem", because you may be surprised to
find out how distantly Sagittal is able to represent some of these
exactly. For example, there are exact symbols for 7-limit
alterations such as 16384:16875 and 625:648 (each containing 5^4),
3072:3125 (containing 5^5), 864:875 (containing 5^3*7), 1024:1029 and
1715:1728 (each containing 7^3), 64827:65536 and 19208:19683 (each
containing 7^4), 243:245 (containing 5*7^2), and 32768:33075
(containing 5^2*7^2).

As I'm sure you're well aware, how one ultimately decides to go about
notating microtonal accidentals involves certain trade-offs: more
symbols (or combinations thereof) allow more commas to be notated
exactly (or greater precision, when they must be approximated) for
theoretical or electronic-music applications, but this can become so
complicated that it's virtually impossible to read on a manuscript in
a real-time performance.

Dave Keenan & I intended Sagittal to be a notation that would be
equally well suited to both theory and performance, and as we began
to ponder the problem of notating comma-accidentals containing
multiple primes >3, we rejected the idea of using multiple Sagittal
accidentals, because it becomes very difficult to judge the aggregate
size of an alteration, particularly when the symbols alter in
opposite directions. For example, notating something as simple as a
5:6:7 chord on C would result in Gbt/ for the top note (in Sagittal
shorthand), where b = apotome down (flat), / = /| 80:81 up, and t
= !) 63:64 down. Instead, we decided to create a separate symbol |(
for the difference between 63:64 and 81:80, i.e., the 5:7 kleisma
(5103:5120), so that 7/5 could be notated more simply as Gbc or Gb!( .

In the process, we found that the 17-kleisma (2176:2187) and 17-comma
(4096:4131) differed by almost exactly a 5:7 kleisma (<0.25 cents
difference), so we took advantage of that fact by using the
combination of flags for the 17-kleisma ~| and 5:7 kleisma |( in the
17-comma ~|( symbol (which is defined exactly as 17C, and not as
the "sum" of 5:7k and 17k). So you can easily judge the resultant
size of the symbol alteration by its component flags (which never
alter in opposite directions).

Likewise, we also created symbols -- /|\ (|) (|( (| )|( --
that notate all of the 11-limit consonances exactly, as well as other
symbols -- //| /|) (|\ -- that notate 5^2 and 5*7 exactly. These
are all members of the athenian-level (medium-precision) JI set.

After that, we addressed the problem of notating the more complex
combinations of primes that would occur in more distant modulations
by means of: 1) additional symbol cores (i.e., flag combinations) and
2) the addition of accent marks added to the left and/or right side
of a symbol core. Thus, the symbol core (|\, with various accent-
marks added, represents the following dieses exactly:

.(|\ 7L, 27:28
(|\. 125L, 51200:531441
(|\ 35L, 8192:8505
(|\' 13L, 26:27
'(|\. 5:77L, 77:80
'(|\ 175L, 2^28:3^13*5^2*7

There are also 2 or 3 other (more complicated) possibilities that I
haven't listed, simply because they haven't been finalized. (As I
write this, we're still working on some of the more esoteric symbol
definitions.) Observe that 125L and 13L differ from 35L by ~0.4
cents.

In a real-time performance at a fast tempo, a player will probably
choose to disregard the accent marks (and, assuming perfect
technique, will still come within 2 cents of the intended pitch). At
a slower tempo the left-accent marks (amounting to ~2 cents) may be
observed and fine-tuned by ear, while most right-accent marks will
probably be disregarded. In a theory text or an electronic music
application the accent marks will allow these fine distinctions to be
made.

I realize that the prime-number content of complex commas won't be
readily apparent in most of these accented symbols, but OTOH the size
alteration definitely will be (which may lead to some ideas for
harmonic bridging). If you work with these symbols over an extended
period of time, you'll memorize the ones you use most -- in the above
table, I already knew the prime content of the first four dieses off
the top of my head.

If your modulations are so extensive that you run out of symbols,
then would there be any problem with approximations that are
typically 0.2 cents or a maximum of 0.4 cents? Put another way, if
your computer gave you nothing finer than the pitches of 2460-ET,
would that be good enough for you?

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> If your modulations are so extensive that you run out of symbols,
> then would there be any problem with approximations that are
> typically 0.2 cents or a maximum of 0.4 cents? Put another way, if
> your computer gave you nothing finer than the pitches of 2460-ET,
> would that be good enough for you?

Given that I recently used 441-et for JI, it would be. ;)

More to the point--in JI, I tend to be a 7-limit kind of guy; I think
that for me the 'messyness' of using Sagittal starts to have
diminishing returns. I'm more inclined, were I to use acoustic
players, to notate in extended meantone with standard accidentals
(31-eq), and just instruct them to do adaptive JI on the fifths etc.
Of course, one would still have difficulties with notation if one were
using commas as melodic intervals, though. And still, one could argue,
as you have, that acoustic situations are going to be typically at or
greater than 2 errors anyway.....

That said, I appreciate the work you and Dave have done to give the
world Sagittal--it sure looks cool, and it's very complete. My one
critique would be it's inelegance (meaning the literal sense--lacking
simplicity--it's a lot of symbols to memorize, and it gets bloated
fast---a lot to ask of players who would be fish out of water to begin
with), but that reflects the real-world situation of any JI endeavor
that would 'leave Kansas' ;) --- but we should remember that there are
very few reasons to modulate far in JI, anyway....

Best,
Aaron.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:

> Given that I recently used 441-et for JI, it would be. ;)

Of course part of the fun with 441 is that it tempers out
2401/2400 and 4375/4374, and you can make use of that if you want.
Aside from that, it tempers out the 5-limit semithirds comma,
|38 -2 -15> and so does 5-limit semithirds more or less to perfection.
Right now I'm working with 118, and so with semithirds, by the way,
though I'm not treating it as a pure 5-limit system by any means. But
as a way of getting extreme 5-limit accuracy in a MOS, it's worth
mentioning that half of a major third of 441 as a generator, strung
out to, say, 25 or 31 notes, will give you that.

> but we should remember that there are
> very few reasons to modulate far in JI, anyway....

Why do you say that? One of the great things about nanotemperaments
like 441 is that they allow you to modulate all over the map without
having the linear growth in numerator and denominator (which will
choke Scala, among other problems) you would get in strict JI. If
anyone else is freely modulating in JI, raise your hand. How do you
manage it?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
> wrote:

> > but we should remember that there are
> > very few reasons to modulate far in JI, anyway....
>
> Why do you say that?

'Cause I wasn't thinking at the time. Although many claim that JI is
calm and liberates us from Western culture neurotic "gotta go
somewhere else" tendencies. Pooh on them.

> One of the great things about nanotemperaments
> like 441 is that they allow you to modulate all over the map
without
> having the linear growth in numerator and denominator (which will
> choke Scala, among other problems) you would get in strict JI.

And, apparantly, Sagittal gets choked too.

> If
> anyone else is freely modulating in JI, raise your hand. How do
you
> manage it?
>

Yeah, I want to know too. THe only far modullating composers I know
of off the top of my head are Ben Johnston and Toby Twining.

-A

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:

> And, apparantly, Sagittal gets choked too.

Sagittal would quickly die the Death of Impossible Notation if you
tried notating such a thing, yes. Nanontempering is really the way to
go here, and if you have sagittal notation for 441, 612, 2460 or
whatever else you are fond of and which would work for you, you'd need
to use that instead.

> Yeah, I want to know too. THe only far modullating composers I know
> of off the top of my head are Ben Johnston and Toby Twining.

I do it.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > If your modulations are so extensive that you run out of symbols,
> > then would there be any problem with approximations that are
> > typically 0.2 cents or a maximum of 0.4 cents? Put another way,
if
> > your computer gave you nothing finer than the pitches of 2460-ET,
> > would that be good enough for you?
>
> Given that I recently used 441-et for JI, it would be. ;)
>
> More to the point--in JI, I tend to be a 7-limit kind of guy; I
think
> that for me the 'messyness' of using Sagittal starts to have
> diminishing returns. I'm more inclined, were I to use acoustic
> players, to notate in extended meantone with standard accidentals
> (31-eq), and just instruct them to do adaptive JI on the fifths etc.

I've thought about this sort of thing before, in connection with
hypothetical acoustic instruments built in 31-ET -- a fusion of two
ideas in the Sagittal paper, using 3 pairs of accidentals to do
the "instructing":
http://dkeenan.com/sagittal/Sagittal.pdf
See the section on Adaptive JI (page 18) and the mention of 217-ET
(7*31) in footnote 15 (page 20).

It would differ from the adaptive JI shown in Figure 11 in that the
chain of nominals would be actual 1/4-comma meantone fifths (or
126deg217) instead of just 5ths. My idea is to notate a 31-ET subset
of 217-ET with the Tartini-Couper symbols (top line of Fig. 3) and
indicate (or instruct regarding) the remaining tones of 217 with
additional accidentals (as multiples of 1/4-comma alterations, i.e.,
1/7's of 1deg31). These additional accidentals could be the first 3
symbols in the 217-ET set: |( ~| ~|( -- the actual symbols are on
the next-to-last line of Fig. 9 on p. 16. (There is a problem with
this, however: under our existing standards, this would not be a
legitimate use of the Sagittal accidentals, so I'll have to discuss
this with Dave, to see whether we should sanction this usage; its
simplicity would seem to make a strong argument in favor.) Since 217
is 21-limit consistent, you could go quite a bit beyond the 7 limit,
should the need arise.

> Of course, one would still have difficulties with notation if one
were
> using commas as melodic intervals, though.

The essence of the legitimacy problem is that the comma-symbols would
be used melodically rather than harmonically (and the harmonically
correct alternatives I looked for don't look very pretty, either).
But I don't know if this is what you meant by that last statement.

> And still, one could argue,
> as you have, that acoustic situations are going to be typically at
or
> greater than 2 errors anyway.....
>
> That said, I appreciate the work you and Dave have done to give the
> world Sagittal--it sure looks cool, and it's very complete.

Thanks. We always appreciate a few kind words every now & then.

> My one
> critique would be it's inelegance (meaning the literal sense--
lacking
> simplicity--it's a lot of symbols to memorize, and it gets bloated
> fast---a lot to ask of players who would be fish out of water to
begin
> with), but that reflects the real-world situation of any JI endeavor
> that would 'leave Kansas' ;) --- but we should remember that there
are
> very few reasons to modulate far in JI, anyway....

With Sagittal we've tried to offer options that will suit individual
purposes, such as the ability to trade off complexity against some
other parameter. If you're working at the 7 limit, for example, then
it would be possible to expand the use of "smart defaults" (i.e., the
ability to associate the same symbol with different commas, depending
on the context) to various prime limits, so that an 11M-diesis
(32:33) symbol, for example, would instead be interpreted as a 49M-
diesis (4969:4096).

Another (perhaps better) possibility would be to make substitutions
in the medium-precision (athenian) symbol set for 7-limit JI so that
7-limit symbols are maximized. Thus the actual 49M & 49L symbols
would replace the 11M & 11L symbols, and the 49S symbol would replace
5:11S. This is something I haven't discussed with Dave, so I'm just
thinking "out loud" here; I have a hunch he'll convince me that the
promethean symbol set (still not quite finalized) will do just as
well for your purposes, probably with 7-limit smart defaults applied.

(If you don't follow everything I've said here, don't worry, because
I figure I've rattled on too long. Dave will eventually read this,
and then we'll hash it out off-list. So many details ...)

With the passage of time new concerns will be raised, and we have to
be open to addressing those issues as they come up, so your
constructive comments are very helpful and highly appreciated.

Best,

--George

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

> I've thought about this sort of thing before, in connection with
> hypothetical acoustic instruments built in 31-ET -- a fusion of two
> ideas in the Sagittal paper, using 3 pairs of accidentals to do
> the "instructing":
> http://dkeenan.com/sagittal/Sagittal.pdf
> See the section on Adaptive JI (page 18) and the mention of 217-ET
> (7*31) in footnote 15 (page 20).

It seems to me that Tartini has a big advantage over your system, in
that it uses single symbols and familiar notation together. You get
single symbols from pure sagittal, and familiar notation from mixed,
but what you really want is a sharp/flat pair, a double sharp/double
flat pair, a diesis up and down pair, and a sharp+diesis, flat-diesis
pair, and Tartini does that pretty well.

> It would differ from the adaptive JI shown in Figure 11 in that the
> chain of nominals would be actual 1/4-comma meantone fifths (or
> 126deg217) instead of just 5ths.

You lost me. 126/217 = 18/31, so this is just contorted 31-et
meantone.

My idea is to notate a 31-ET subset
> of 217-ET with the Tartini-Couper symbols (top line of Fig. 3) and
> indicate (or instruct regarding) the remaining tones of 217 with
> additional accidentals (as multiples of 1/4-comma alterations,
i.e.,
> 1/7's of 1deg31). These additional accidentals could be the first
3
> symbols in the 217-ET set: |( ~| ~|( -- the actual symbols are
on
> the next-to-last line of Fig. 9 on p. 16.

OK, now you have a basis for a 217 notation, but what does this have
to do with meantone? 217, by the way, makes sense as a parakleismic
(99&118 temperament) system, with a generator being a minor third a
little less than half a cent flat. I continue to think that a
generator-based notation system might be better; that is, have
separate basic schemes for fifth, major third or minor sixth, minor
third or major sixth, etc as generators.

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>If you use medium-precision (a/k/a athenian-level)
> Sagittal JI in Scala (set nota SAJI1, taking C as 1/1), you'll get 3
> spellings: F(|(, Gb\!!!/ or Gb\!/, and E)|||( or E#)|(, all of
> which are approximations.

George, I think you meant
F(|(
G\!!!/ or Gb\!/
E)|||( or E#)|(

(you had a flat with the G triple-shaft)

> If you use high-precision (a/k/a herculean-level) Sagittal JI, there
> are symbols that notate the 2 preferred spellings *exactly*: the ~|)
> symbol is defined as 48:49 and the (/| symbol as 3969:4096.
>
> So 49/36 of C would be notated exactly as F~|), and Gb(\!!! or
> Gb(\!, while the 3rd spelling is approximated by E')|||( or E#')|(;

Again
F~|)
G(\!!! or Gb(\!
E')|||( or E#')|(

Clearly the F spelling is preferred in both cases when C is 1/1.

-- Dave Keenan

Aaron wrote:
>How does one express 49/36 in Sagittal? Is it the theoretical standard
>that one should use double/triple symbols?

Hi Aaron,

As George mentioned, the simplest notation for 49/36 uses the boathook-and-arc symbol ~|) for the 49-small-diesis (48:49) which unfortunately does not appear in the version 1 TrueType font or character map currently on the Sagittal website. It appears tenth from the right in the top row of Figure 13 of the XH18 paper, and in the development version 2 postscript font that is distributed with Hudson Lacerda's MicroABC, but without documentation in either case.

I have been very remiss in not finishing the documentation i.e. character map for Sagittal 2, and will make an effort today. -- Sorry Robin Perry, your guitar will have to be delayed a little longer.

>Now I see that you know what I mean---so my question is, why does
>sagittal seem to limit the accurate perfect representation of free,
>distant modulation? ;)

>More to the point--in JI, I tend to be a 7-limit kind of guy; I think
>that for me the 'messyness' of using Sagittal starts to have
>diminishing returns.

Sagittal can be used in so many different ways. I suspect the problems you are finding are only with one particular way of using Sagittal (as well as not yet having the complete system at your disposal).

Can you tell us, the "messyness of using Sagittal", as compared with what other notation system?

We should probably distinguish semantics from symbolisation here. Sagittal basically gives you a set of symbols for commas (whether tempered or not). It is the most extensive, logical and beautiful such set ever devised -- this is the symbolisation. But there are many different ways to use commas (tempered or not) to notate scales -- this is the semantics.

If you insist on absolute accuracy with unlimited modulation in JI then no notation can do better than let you stack up multiple accidentals. I assume this is what Gene means by "the Death of Impossible Notations".

Consider that conventional notation gives a single symbol for double-sharp but not triple sharp. This is perfectly reasonable, even in Pythagorean tuning people rarely modulate so far as to need more. But if they do, they are free to choose whether to simply stack up more sharps or flats, or whether to notice that after 29 or 41 or 53 or 306 (or whatever) modulations they are back very close to their starting point and might as well start reusing shorter notations. The same goes for sagittal, but this time with comma symbols and higher prime limits.

You are free to stack up multiple sagittals if that's what you really want to do. There are no Sagittal police. :-) But we strongly discourage it as it will quickly become unreadable and we suspect that people are often not aware of all the alternatives (although clearly you are aware of some, with mention of 31-edo and 441-edo). Which is why George said it is a "no no". And I suppose we worry that the sight of long strings of Sagittals against a single note might give people the wrong idea about Sagittal and hinder its acceptance.

There are situations where it does not need to be readable in real time, e.g. for purposes of theoretical explanation or analysis or indeed composition. And it certainly can make sense to use two sagittals in cases where the first can be considered to be tightly bound to the nominal, conceptually a compound nominal, due to modulation. But we would very strongly discourage more than two sagittals per note.

>I'm more inclined, were I to use acoustic
>players, to notate in extended meantone with standard accidentals
>(31-eq), and just instruct them to do adaptive JI on the fifths etc.

31-edo with adaptive JI is a perfectly fine way to do it. That's the semantic level. Now why not use the Sagittal symbols for 31-edo. The only difference from conventional is that we can choose to use semi-sharp/flat symbols /|\ \!/ instead of double sharps or flats whenever it would make more sense. That's the symbolisation level.
C C^ C# Db Dv D
versus
C Dbb C# Db Cx D

>Of course, one would still have difficulties with notation if one were
>using commas as melodic intervals, though. And still, one could argue,
>as you have, that acoustic situations are going to be typically at or
>greater than 2 errors anyway.....

As George mentioned, 0.2 cent errors will be the ultimate limit of Sagittal, not 2 cent errors. But the 2 cent level -- Athenian -- should be adequate for most purposes, being effectively an unequal division of the apotome into 21 parts.

The higher precision levels have not yet been published as they have not been finalised and any input such as yours is very welcome. George and I discuss the details almost daily (with occasional gaps of a month or two). We've been going at it for years now. Maybe it's time to lay out the remaining problems for wider discussion on tuning-math again -- after we've published Sagittal 2 and its character map.

>That said, I appreciate the work you and Dave have done to give the
>world Sagittal--it sure looks cool, and it's very complete. My one
>critique would be it's inelegance (meaning the literal sense--lacking
>simplicity--it's a lot of symbols to memorize, and it gets bloated
>fast---a lot to ask of players who would be fish out of water to begin
>with), but that reflects the real-world situation of any JI endeavor
>that would 'leave Kansas' ;) --- but we should remember that there are
>very few reasons to modulate far in JI, anyway....

Yes there are a lot of symbols for completeness. But some of those symbols are there to fill a need that occurs maybe once in the entire Scala scale archive of over 2000 scales (20,000 pitches). You can forget about them. Unfortunately they still take up one slot in the font, and one slot in any table of symbols, which makes them look like they are just as important. We have tried to address this in the new character map by making the less important symbols smaller. If they were really made proportional to their popularity you wouldn't see them at all, but we'll use some kind of log scale.

It's entirely your choice as to how many symbols to actually use, e.g. at what point to rely on approximations. But there is yet another alternative that does _not_ rely on approximations.

Smart defaults
--------------
You may not be aware that in the sagittal notation of JI it is perfectly acceptable to define the same symbol as having a different comma meaning depending on which nominal and sharps or flats it occurs with, provided all such commas are within the "capture zone" of the symbol. This enormously increases the number of JI pitches notatable _exactly_ with only the 12 symbol pairs of Athenian. See
http://dkeenan.com/sagittal/whatpitch.txt

And since you're only doing 7-limit, you could do the same thing using only the 3 symbol pairs of the 72-edo symbol set. You can then provide a table or staff showing what each combination of nominal, sharps or flats and sagittal means as a ratio, to accompany your score. The player will probably still just use adaptive JI, but with this method she doesn't have to. You would not be notating 72-edo, it would be strict JI, but using the same symbol set as 72-edo for a (possibly unequal) division of the apotome into 6 parts.

I would be very keen to know whether this works for you. How far can one go in notating strict 7-limit JI using only 3 symbol pairs?

Let us know if you need help with this. Do you prefer mixed or pure sagittal?

I think you will find that Sagittal is very elegant indeed.

-- Dave

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
> It seems to me that Tartini has a big advantage over your system, in
> that it uses single symbols and familiar notation together. You get
> single symbols from pure sagittal, and familiar notation from mixed,
> but what you really want is a sharp/flat pair, a double sharp/double
> flat pair, a diesis up and down pair, and a sharp+diesis, flat-diesis
> pair, and Tartini does that pretty well.

Gidday Gene,

Sure. That's the case for 31-edo (or 17, 24, 38) where you only need
semi and sesqui apotome symbols (in addition to conventional).

George and I both love the Tartini-Couper symbols, but as we say in
the XH18 article, they don't generalise well symbolically to finer
divisions or JI. And if you're going to have to learn different
symbols later (that fit the logic and aesthetics of the wider sagittal
system) then why not learn them right from the start.

However, you should know that the forthcoming Sagittal-2 font includes
the Tartini-Couper symbols in a style that is compatible with both
conventional and sagittal symbols. It has the added advantage that the
Couper backwards flat is narrower than the forwards flat so it
actually _looks_ like it represents a smaller alteration and minimises
left-right confusability.

-- Dave Keenan

--- In tuning@yahoogroups.com, Dave Keenan <d.keenan@...> wrote:
>
> Aaron wrote:
> >How does one express 49/36 in Sagittal? Is it the theoretical standard
> >that one should use double/triple symbols?
>
> Hi Aaron,
>
> As George mentioned, the simplest notation for 49/36 uses the
boathook-and-arc symbol ~|) for the 49-small-diesis (48:49) which
unfortunately does not appear in the version 1 TrueType font or
character map currently on the Sagittal website. It appears tenth from
the right in the top row of Figure 13 of the XH18 paper, and in the
development version 2 postscript font that is distributed with Hudson
Lacerda's MicroABC, but without documentation in either case.

Good!....too bad about no documentation, though. Alas for microabc,
too. It's a good package, but a little thin in docs, too. Although I
have learned a great deal looking at the *.abh, *.abc files, etc, to
see how the macros work, and end up being backslash refs to numerical
table codes of symbols in the font.

> I have been very remiss in not finishing the documentation i.e.
character map for Sagittal 2, and will make an effort today. -- Sorry
Robin Perry, your guitar will have to be delayed a little longer.

My advice would be to take paying customers first (assuming Robin is),
but that's your business.... ;)

<snipped>

> Can you tell us, the "messyness of using Sagittal", as compared with
what other notation system?

Well, a meantone system has the advantage of just having Pythagorean
notation cover the 5-limit and 7-limit, assuming you can do adaptive
JI if you want it. i.e., I can write c-e-g-a#, (or as I write in
micro_composer c-e-g-a^ or c-e-g-^a) and people know I mean 'harmonic
seventh chord'. Looks much better than C-E\-G-B!) for instance (I hope
I got the ascii-sagittal correct)

> We should probably distinguish semantics from symbolisation here.
Sagittal basically gives you a set of symbols for commas (whether
tempered or not). It is the most extensive, logical and beautiful such
set ever devised -- this is the symbolisation. But there are many
different ways to use commas (tempered or not) to notate scales --
this is the semantics.

Yes I agree, it's beautiful, but big. That's my main consideration. Of
course, you address the smart default option, which is promising, below.

> If you insist on absolute accuracy with unlimited modulation in JI
then no notation can do better than let you stack up multiple
accidentals. I assume this is what Gene means by "the Death of
Impossible Notations".

Yes, and I think microabc/abc, unfortunately doesn't allow a modular
approach to stacking accidentals. Unless they've got a new development
version which does. Hudson would know...Hudson?

> Consider that conventional notation gives a single symbol for
double-sharp but not triple sharp. This is perfectly reasonable, even
in Pythagorean tuning people rarely modulate so far as to need more.
But if they do, they are free to choose whether to simply stack up
more sharps or flats, or whether to notice that after 29 or 41 or 53
or 306 (or whatever) modulations they are back very close to their
starting point and might as well start reusing shorter notations. The
same goes for sagittal, but this time with comma symbols and higher
prime limits.

Inaccuracy and JI don't mix well as concepts, even though in the real
world of performance, inaccuracy is reality. It just seems a typical
JI fundamentalist (which I am *not*) would want unlimited precision of
expression. I guess I'm only arguing that hypothetical case, and
perhaps arguing it for the electronic/MIDI point-of-view as well.

> You are free to stack up multiple sagittals if that's what you
really want to do. There are no Sagittal police. :-) But we strongly
discourage it as it will quickly become unreadable and we suspect that
people are often not aware of all the alternatives (although clearly
you are aware of some, with mention of 31-edo and 441-edo). Which is
why George said it is a "no no". And I suppose we worry that the sight
of long strings of Sagittals against a single note might give people
the wrong idea about Sagittal and hinder its acceptance.

Very true...and it seems already that it's an uphill battle to
convince people that this is the best notational standard, *as is*.

> There are situations where it does not need to be readable in real
time, e.g. for purposes of theoretical explanation or analysis or
indeed composition. And it certainly can make sense to use two
sagittals in cases where the first can be considered to be tightly
bound to the nominal, conceptually a compound nominal, due to
modulation. But we would very strongly discourage more than two
sagittals per note.

Good call.

<snipped adaptive 31-eq part>

> Smart defaults
> --------------
> You may not be aware that in the sagittal notation of JI it is
perfectly acceptable to define the same symbol as having a different
comma meaning depending on which nominal and sharps or flats it occurs
with, provided all such commas are within the "capture zone" of the
symbol. This enormously increases the number of JI pitches notatable
_exactly_ with only the 12 symbol pairs of Athenian. See
> http://dkeenan.com/sagittal/whatpitch.txt
>
> And since you're only doing 7-limit, you could do the same thing
using only the 3 symbol pairs of the 72-edo symbol set. You can then
provide a table or staff showing what each combination of nominal,
sharps or flats and sagittal means as a ratio, to accompany your
score. The player will probably still just use adaptive JI, but with
this method she doesn't have to. You would not be notating 72-edo, it
would be strict JI, but using the same symbol set as 72-edo for a
(possibly unequal) division of the apotome into 6 parts.

Great...this makes sense, and I would guess, holds the future
sink-or-swim status of the adoption of Sagittal.

> I would be very keen to know whether this works for you. How far can
one go in notating strict 7-limit JI using only 3 symbol pairs?
>
> Let us know if you need help with this. Do you prefer mixed or pure
sagittal?

Mixed, by far. I see the intellectual *appeal* of pure, but in the
real world, players are going to continue to be trained to play/think
in the traditional way. Building on that, instead of knocking it down
and starting over, seems to me to be a gentler, more practical way.

> I think you will find that Sagittal is very elegant indeed.

I would say it's promising, provided certain user-flexibilities are
implemented, and this as much depends on software implementations of
the standard as anything else.

Best,
Aaron.

In a message dated 3/1/2007 3:57:23 A.M. Eastern Standard Time,
tuning@yahoogroups.com writes:

Yeah, I want to know too. THe only far modullating composers I know
of off the top of my head are Ben Johnston and Toby Twining.

-A

1200 ET allows for free modulation acceptable for all tunings discussed.
The notation is simple cents notation. Polymicrotonal composition can include
freely modulating just intonation. The AFMM uses a standard cents notation
for its performances, when possible. The mere mixing of tunings on a single
program requires it. The numbers used are from 1-49, each placed above note
heads, often in combination with a 50-cent symbol for quarter sharp and
quarter flat.

Please take note: there is no quartertone bias to this notation in terms of
its actual usage.

Johnny

(back from the edge)

<BR><BR><BR>**************************************<BR> AOL now offers free
email to everyone. Find out more about what's free from AOL at
http://www.aol.com.

Johnny--

This makes sense. 1200-et is indeed accurate enough for any acoustic
situation (some might say overkill, since you could probably get away
with the 72-et sixth-tones), and even 100% accuracy I would imagine is
possible, should the tempo be slow enough (e.g. long drone pieces)
with the player locking into a harmonic.

When you create a score, do you subvert fingering functions to add the
numerals?

Best,
Aaron

--- In tuning@yahoogroups.com, Afmmjr@... wrote:
>
>
>
> In a message dated 3/1/2007 3:57:23 A.M. Eastern Standard Time,
> tuning@yahoogroups.com writes:
>
> Yeah, I want to know too. THe only far modullating composers I know
> of off the top of my head are Ben Johnston and Toby Twining.
>
> -A
>
>
>
>
>
> 1200 ET allows for free modulation acceptable for all tunings
discussed.
> The notation is simple cents notation. Polymicrotonal composition
can include
> freely modulating just intonation. The AFMM uses a standard cents
notation
> for its performances, when possible. The mere mixing of tunings on
a single
> program requires it. The numbers used are from 1-49, each placed
above note
> heads, often in combination with a 50-cent symbol for quarter sharp
and
> quarter flat.
>
> Please take note: there is no quartertone bias to this notation in
terms of
> its actual usage.
>
> Johnny
>
> (back from the edge)
>
>
> <BR><BR><BR>**************************************<BR> AOL now
offers free
> email to everyone. Find out more about what's free from AOL at
> http://www.aol.com.
>

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...>
wrote:

> Well, a meantone system has the advantage of just having Pythagorean
> notation cover the 5-limit and 7-limit, assuming you can do adaptive
> JI if you want it. i.e., I can write c-e-g-a#, (or as I write in
> micro_composer c-e-g-a^ or c-e-g-^a) and people know I mean 'harmonic
> seventh chord'.

Any 7-limit temperament with a fifth as generator ("brigeable") can do
that, in particular garibaldi. Unfortunately, C-Fb-G-Cbb just doesn't
fit our expectations, though it has the advantage that you can use
Pythagorean tuning for it if you want.

--- In tuning@yahoogroups.com, Afmmjr@... wrote:

> 1200 ET allows for free modulation acceptable for all tunings
discussed.
> The notation is simple cents notation. Polymicrotonal composition
can include
> freely modulating just intonation.

I dunno. I think I prefer 3600 et notation. That happens to be an
excellent ennealimmal tuning, and its 7-limit comma basis is in fact
2401/2400, 4375/4374 and something so absurdly complex there's no point
in writing it down.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >If you use medium-precision (a/k/a athenian-level)
> > Sagittal JI in Scala (set nota SAJI1, taking C as 1/1), you'll
get 3
> > spellings: F(|(, Gb\!!!/ or Gb\!/, and E)|||( or E#)|(, all of
> > which are approximations.
>
> George, I think you meant
> ...
> G\!!!/ or Gb\!/
> ...
> (you had a flat with the G triple-shaft)

Yeah, sorry!

> > If you use high-precision (a/k/a herculean-level) Sagittal JI,
there
> > are symbols that notate the 2 preferred spellings *exactly*: the
~|)
> > symbol is defined as 48:49 and the (/| symbol as 3969:4096.
> >
> > So 49/36 of C would be notated exactly as F~|), and Gb(\!!! or
> > Gb(\!, while the 3rd spelling is approximated by E')|||( or E#')|
(;
>
> Again
> ...
> G(\!!! or Gb(\!
> ...

Yeah, sorry again!

> Clearly the F spelling is preferred in both cases when C is 1/1.

At least I got that one right. :-)

--George

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@>
> wrote:
>
> > Well, a meantone system has the advantage of just having Pythagorean
> > notation cover the 5-limit and 7-limit, assuming you can do adaptive
> > JI if you want it. i.e., I can write c-e-g-a#, (or as I write in
> > micro_composer c-e-g-a^ or c-e-g-^a) and people know I mean 'harmonic
> > seventh chord'.
>
> Any 7-limit temperament with a fifth as generator ("brigeable") can do
> that, in particular garibaldi. Unfortunately, C-Fb-G-Cbb just doesn't
> fit our expectations, though it has the advantage that you can use
> Pythagorean tuning for it if you want.
>

Yes. Garibaldi (a.k.a. 7-limit schismic) would be more precise than
meantone and the notation problem is solved by adding the single pair
of sagittal accidentals for the 5-comma (which happens to be the same
size as the 7-comma in Garibaldi). C-E\-G-Bb\

-- Dave Keenan

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

> Yes. Garibaldi (a.k.a. 7-limit schismic) would be more precise than
> meantone and the notation problem is solved by adding the single pair
> of sagittal accidentals for the 5-comma (which happens to be the same
> size as the 7-comma in Garibaldi). C-E\-G-Bb\

That looks pretty practical to me. You can extend to the 11-limit by
adding 385/384 as a comma, and then F// is your 11/8. C-D-E\-F//-G-Bb\
is the complete 11-limit otonality. If someone wanted an alternative to
meantone which would be relatively easy to notate, and to read the
notation for, this looks recommendable.

Now who's going to do it? I suppose this could be my chance to write
something which I produce an actual score for; I've been working with
118 but it's not garibaldi-friendly. 94 is and it's on my list of
things to try.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
> That looks pretty practical to me. You can extend to the 11-limit by
> adding 385/384 as a comma, and then F// is your 11/8. C-D-E\-F//-G-Bb\
> is the complete 11-limit otonality. If someone wanted an alternative to
> meantone which would be relatively easy to notate, and to read the
> notation for, this looks recommendable.

Yes. Well done. And I assume you are aware of the single Sagittal
symbol for the double 5-comma or 25-small-diesis up. You can see it at
character code 164 in the new spreadsheet.
/tuning/topicId_70117.html#70117

In ASCII we normally write it as //| (long) or = (short) but // is
certainly unambiguous when we know we are only using one sagittal per
note.

> Now who's going to do it? I suppose this could be my chance to write
> something which I produce an actual score for; I've been working with
> 118 but it's not garibaldi-friendly. 94 is and it's on my list of
> things to try.

Sounds like a great idea. :-)

-- Dave Keenan

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
>
> --- In tuning@yahoogroups.com, Dave Keenan <d.keenan@> wrote:
...
> but without documentation
>
> Good!....too bad about no documentation, though.

It's now well and truly documented! Thanks for the nudge.
/tuning/topicId_70117.html#70117

> > Can you tell us, the "messyness of using Sagittal", as compared
> with what other notation system?
>
> Well, a meantone system has the advantage of just having Pythagorean
> notation cover the 5-limit and 7-limit, assuming you can do adaptive
> JI if you want it. i.e., I can write c-e-g-a#, (or as I write in
> micro_composer c-e-g-a^ or c-e-g-^a) and people know I mean
> 'harmonic seventh chord'. Looks much better than C-E\-G-B!) for
> instance

Sure. But both of those are valid Sagittal. So your issue isn't really
with Sagittal. Sagittal would also let you write the meantone version
as C-E-G-Bbv.

(I hope I got the ascii-sagittal correct).

Yes you did. If you want the single-ASCII-character form for the
7-comma symbol it is "t" for down and "f" for up. And of course the
long form for 5-comma down is simply "\!".

> Yes I agree, it's beautiful, but big. That's my main consideration.
> Of course, you address the smart default option, which is promising,
> below.

> Yes, and I think microabc/abc, unfortunately doesn't allow a modular
> approach to stacking accidentals. Unless they've got a new development
> version which does. Hudson would know...Hudson?

Hudson has been very responsive to requests. But you may need to email
him direct.

> Inaccuracy and JI don't mix well as concepts, even though in the
> real world of performance, inaccuracy is reality. It just seems a
> typical JI fundamentalist (which I am *not*) would want unlimited
> precision of expression. I guess I'm only arguing that hypothetical
> case, and perhaps arguing it for the electronic/MIDI point-of-view
> as well.

We are very well aware of that point of view and have thoroughly
catered for it (as well as every other point of view). That's why we
have so many symbols! And then there are the accent marks (which are
still not fully documented).

But then we have people, like you, who see all those symbols and feel
an urge to run screaming. And I don't blame you at all. It seems all
we can do is to keep repeating, "But you don't have to use them or
learn them. Most things can be done with only 3 symbol pairs /| |)
and /|\ ".

I've tried to make it not so scary in the new spreadsheet by giving
the rare and obscure symbols smaller row heights so it's easier to
ignore them, and hiliting the most important symbols.

> Very true...and it seems already that it's an uphill battle to
> convince people that this is the best notational standard, *as is*.

It seems rather that it is difficult to convince people to use a
universal notational standard. Once they are convinced of that there
are really only two contenders and they don't really compete with each
other, in fact they can be used simultaneously. Those are Sagittal and
Reinhard cents notation.

It is also hard to convey just how flexible Sagittal really is.

> > Smart defaults
> > --------------
...
> Great...this makes sense, and I would guess, holds the future
> sink-or-swim status of the adoption of Sagittal.

I suspect Sagittal is a lot more robust than that, but I encourage you
to develop this way of using Sagittal. Or consider the
schismic/Garibaldi Sagittal notation that Gene has proposed as a step
up from meantone.

> > Let us know if you need help with this. Do you prefer mixed or
> > pure sagittal?
>
> Mixed, by far.

I tend to think in the mixed notation too. George is the pure man. :-)

-- Dave Keenan