Saggital chord notation

Hi there,

I've just added a Sagittal type chord notation to FTS for the next
release.

Here is how it works - I thought I'd use a simplified Saggital
here where bc means to go flat by a comma and #c to go sharp
by a comma. The ascii saggital symbols might not fit so well
with other symbols used to make chords particularly the / for note addition
in a chord.

Also letters are easy to read quickly in a chord symbol, while
a chord symbol with several saggital symbols in a single chord
mightn't be so easy to read. Of course if it could be
done in a proper font or written by hand then it would
be easier to read using the proper symbols. The bc and #c
notation could be useful even then as an intro notation
as a stepping stone towards using the proper symbols
for use by newbies to Sagittal chord symbols.

Then the list of comma symbols as I have it so far is:
c 81/80
s 64/63
u 33/32
t 27/26
e 128/125 diesis
d 2048/2025 diaschisma

which seemed to be the most urgent ones to support
to be able to notate the chords that one seems to
come across most often.

Then, to see most of them in play in a single chord,
1/1 5/4 3/2 7/4 9/4 11/4 13/4
could be notated as
C5p/bc3/bs7/9/#u11/bt13
just to show how the notation works.

The p at the start there says to start from the pythagorean scale as ones basis
for the tuning of the chord, as is usual in Saggital. If you left it out then
it would be understood as a twelve equal chord which we don't want.
But that's just for FTS users, some of whom will be new to all this
and may not be well up on microtonal - it can be used to make 12-et
music for those who want to use it that way for instance. So
it needs 12-et to be the standard interpretation
of the chords. For microtonal work it makes more sense
to take pythagorean as the preset and just have it all as pure
Saggital. But the bc etc is just a comma shift so could be used
with any tuning as a basis if one needs it.

I actually have a harmonic chord notation in the chord progressin player
which can be used to notate the same chord as Ch13, so most users
will use taht I expect, but this shows how the notation works.

Then for example,

1/1 6/5 4/3 3/2 12/7 2/1
is
C5p/#cb3/#s6/4

and
1/1 7/6 4/3 3/2 5/3 2/1
is
C5p/bsb3/bc6/4
and so on.

The diaschisma can be used to get a 25/16 from the nearest pythagorean pitch, in e.g.:
1/1 5/4 25/16
which can be notated as:Cp0/bc3/bbd6

Then to show the diesis in use:
1/1 6/5 36/25 216/125
would be
Cp0/#cb3/#e6

The Cp0 there says to just start from C without any extra notes.
The C5p starts with the fifth only as is the normal interpretation
of C5.

This is all programmed and working in FTS ready for the next release.
Easily changed though, for instance very easy to add new
comma symbols to it. I'm not sure though which others in the long list
at http://dkeenan.com/sagittal/Sagittal.pdf
are particularly useful for notating chords (rather than scales).

I suppose one could also have e.g. Cpbc with the bc working backwards
to make a 5/4 from the pythagorean 81/64, (similarly to use of m in
Cm) - similarly Cm#c to get the 6/5, and so on.
I may do that after thinking it over, if it seems
it won't be too confusing for users.

Interested in any comments on all this.

Robert

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
> Hi there,
>
> I've just added a Sagittal type chord notation to FTS for the next
> release.
>
> Here is how it works - I thought I'd use a simplified Saggital
> here where bc means to go flat by a comma and #c to go sharp
> by a comma. The ascii saggital symbols might not fit so well
> with other symbols used to make chords particularly the / for note
addition
> in a chord.

Hi Robert. Since you asked for comments, here goes.

I don't think it's a good idea to have two systems of character
shorthand, but I have looked at what you have and offer the following
observations.

The ASCII characters for sharp and flat symbols, "#" and "b", are
currently understood to indicate both *direction* and a specific of
*amount of alteration* (i.e., an apotome). You are proposing to use
them to indicate only *direction* for amounts of alteration
considerably less than an apotome. I think that this unusual
employment of the "#" and "b" characters could be rather confusing.

To specify the amount of alteration you are using characters with
meanings that are completely different from what Dave Keenan and I
have agreed upon for Sagittal shorthand (in which single characters
indicate *both* direction and amount of alteration). Your chord
notation should therefore not be referred to as "Sagittal" (please
note the correct spelling).

On the other hand, why couldn't you use the existing Sagittal
shorthand characters and thereby avoid the addition confusion
incurred by having different meanings for the comma-characters? I
will use your examples to show how this could be done.

> Then the list of comma symbols as I have it so far is:
> c 81/80
> s 64/63
> u 33/32
> t 27/26
> e 128/125 diesis
> d 2048/2025 diaschisma

We already have Sagittal up and down shorthand characters for all of
these -- and also for many more commas to the 23-limit. You would
only need to replace the slash character "/" that you're using as a
note delimiter by a dash "--", since a hyphen character is not used
in Sagittal shorthand (except in a user-defined application).

For your reference, here is a message that contains a table showing
the complete Sagittal ASCII shorthand:
/tuning/topicId_54793.html#54967

In the first example you gave:

> Then, to see most of them in play in a single chord,
> 1/1 5/4 3/2 7/4 9/4 11/4 13/4
> could be notated as
> C5p/bc3/bs7/9/#u11/bt13
> just to show how the notation works.
>
> The p at the start there says to start from the pythagorean scale
as ones basis
> for the tuning of the chord, as is usual in Saggital.

By "pythagorean scale" do you mean a pythagorean major scale
beginning on the particular tone designated as 1/1, or a pythagorean
scale on the natural notes (i.e., major scale beginning on C)?

> If you left it out then
> it would be understood as a twelve equal chord which we don't want.

It appears that you left out a note. I believe you meant this:
C5p/bc3/5/bs7/9/#u11/bt13

Sagittal shorthand characters would render this as:
C5p--\3--5--t7--9--^11--w13
which I find much easier to read than what you have, since there are
fewer characters to read for each note and the dashes separate the
notes more clearly.

In addition, I have a question. If that same chord were built on F,
would your notation indicate it as this (as alterations to a
pythagorean major scale beginning on F)?

F5p/bc3/5/bs7/9/#u11/bt13

Or would the unaltered 11th be a "B" (thereby requiring the 11th to
be flatted thus)?

F5p/bc3/5/bs7/9/#ub11/bt13

> The diaschisma can be used to get a 25/16 from the nearest
pythagorean pitch, in e.g.:
> 1/1 5/4 25/16
> which can be notated as:Cp0/bc3/bbd6

Should that be Cp0/bc3/bdb6? Using "b" to mean two different things
is especially confusing in this example. The existing Sagittal ASCII
shorthand characters could express this chord in two different ways,
with either an altered 5th or altered 6th degree of the scale:

C5p--\3--_#5
C5p--\3--'\b6

where _ represents a downward alteration of 6400:6561 (twice 80:81)
to a pythagorean G#, and
'\ represents a downward alteration of a diaschisma to a pythagorean
Ab. Thus you would have the ability to spell chords in more than one
way.

--George

Hi Robert,

Although I am not reading or replying to the list at the moment,
George Secor pointed me to your message
/tuning-math/message/11476
and his reply
/tuning-math/message/11491

While I encourage anything to do with standardisation, so we can all
understand each other better, and of course I support the use of
Sagittal, I actually go further than George in one direction.

My feeling is that sagittal, and accidentals in general, are for
notating pitches, not intervals, so the only place an accidental
should occur in a chord name/description is in giving the pitch of
the root. The rest is essentially a list of intervals, in which case
I would name them according to the scheme that uses terms like ,
minor, neutral, major, diminished, perfect, augmented, sub, super,
wide, narrow. As per their approximations of the 11-limit ratios
given here
http://dkeenan.com/Music/Miracle/MiracleIntervalNam
ing.txt

I don't think it is necessary to extend this system beyond the 11-
limit, as the fact of whether an interval actually comes out as 11-
limit or higher-limit JI or some ET or linear approximation, depends
on the specifics of the scale being used. The name of the chord can
still be based on the nearest 11-limit consonances.

How do you feel about that?

Regards,
-- Dave Keenan

Hi Dave,

Ah the thing is that my chord progression player
can play exact pitches. So I must have
an exact pitch notation for the chords
- it must indicate what exact pitch to play
for each note if the user swithes on exact pitches.

When exact pitches are switched off, then it does
indeed play the nearest available pitch in the current scale.

For instance, Ch13 plays the nearest to a harmonic thirteenth
in the current scale, but yCh13 actually plays a
pure just intonation harmonic thirteenth chord rooted
on C.

So I need all the dieses for that.

So for instance, the user will be able to
play a comma pump sequence in Sagittal using:

y-C y-E\-\ y-A\-\ y-G\-\-t7 y-C\ y-C

The y there says to play the exact pitches
for the entire chord, whatever the current
scale is - just ignore the current scale.

Then in y-E\ the E\ there is a note name
in Sagittal, and is an E flattened by a comma
as you would expect.

The y-E\-\ is short for y-E\-\3
- if the number is omitted from the first
note addition in the symbol, it is taken
to be a 3, that's just to allow for more concise notation
of major / minor type chords.

The y-G\-\-t7 then plays a harmonic seventh.

Then, there's a radio button in the player
which can be used to set it to play pure j.i.
or to play an n-et version of Sagittal and you
can choose which n-et to use, so you will be able then
to play the same chord progression, without changing
anything, to hear what the same comma pump sequence
sounds like in any n-et.

Sorry, I can't go along with your idea that
13 limit and higher symbols aren't needed for chords
as I frequently use 13 limit ratios in chords
and would want to be able to indicate that an exact
13/8 is intended, and not some approximation to it.

Seventeen limit too sometimes, as in this nice
harmonic series based diatonic scale:

1/1 9/8 5/4 3/2 7/4 17/16 2/1

I see that you have a list of names for intervals
- okay - it isn't so concise but could use those.

So for instance

C/M3/sm7

for
C-/-t7

However, I much prefer the Sagittal symbol there for
the chord progression player, because it
shows you how far you have to adjust the pitch relative
to the 6th, so gives a performance direction,
and you don't need to remember that
/SM6 is a 6 sharp by a 64/63, and that
sm7 is a seventh flat by a 64/63 etc.

Also your naming system is limited to certain
pitches, and is fine for a tuning system
such as miracle presumably which doesn't
make fine pitch distinctions but how about

1/1 5/4 25/16

or the harmonic 13th etc,

It doesn't give us any way to notate those
at all.

While Sagittal gives total flexibility
- can notate any pure ratio and you can do
it in a way that gives performance directions
too.

I'm not arguing against your naming system
for miracle at all, and doubtless it is
useful for many situations but it seems
it can't be used in as general way
as Sagittal can be.

I need soemthing more general for the player.

I was so glad to find that Sagittal can
be used to notate the chords for the player
so concisely and clearly and with such flexibility
and do hope we can find a way to
resolve this.

Robert

Hi Dave,

Sorry, of course should be:

y-C-\ y-E\-\ y-A\-\ y-G\-\-t7

y-C\-\

y-C-\

for the comma pump chord progression.

Or to go round twice:

y-C-\ y-E\-\ y-A\-\ y-G\-\-t7

y-C\-\ y-E\\-\ y-A\\-\ y-G\\-\-t7

y-C\\-\

y-C-\

Robert

Hi Robert,

I hadn't meant to post to the list. I meant to email it to you
direct. But I figure I'd better continue to respond to the list on
this topic so other readers aren't left wondering what happened.

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
> Hi Dave,
>
> Ah the thing is that my chord progression player
> can play exact pitches. So I must have
> an exact pitch notation for the chords
> - it must indicate what exact pitch to play
> for each note if the user swithes on exact pitches.

Ah! I see. I had assumed the chords would always be in a given scale
(which would be unlikely to have two pitches close to the same
degree of 72-ET or the same 11-limit consonance).

> When exact pitches are switched off, then it does
> indeed play the nearest available pitch in the current scale.
...
> Then, there's a radio button in the player
> which can be used to set it to play pure j.i.
> or to play an n-et version of Sagittal and you
> can choose which n-et to use, so you will be able then
> to play the same chord progression, without changing
> anything, to hear what the same comma pump sequence
> sounds like in any n-et.

Excellent. Yes this is certainly in the spirit of Sagittal. In
effect what you are doing for your exact pitch mode is letting the
user change the "notational fifth" (between 4/7 to 3/5 octave) which
is used to interpret the A to G nominals and sagittal accidentals.

Some ETs have more than one reasonable choice for the notational
fifth, one being their nearest (but poor) approximation to a just
fifth and the other being a more reasonable 1,3,9-consistent
approximation in a multiple of that ET. In Scala these are called
SAxx and SAxxN where xx is replaced by the ET number and the one
with the "N" is the "nearest fifth" or "native-fifth" or "non-
preferred" one.

> Sorry, I can't go along with your idea that
> 13 limit and higher symbols aren't needed for chords
> as I frequently use 13 limit ratios in chords
> and would want to be able to indicate that an exact
> 13/8 is intended, and not some approximation to it.

Right. What I said only applies if the notes of the chords are
always taken from a scale with reasonable melodic properties. In
that case the scale would determine whether say a neutral sixth (N6)
came out as an 11:18 or an 8:13 (or something else nearby).

But I'm more wrong (wronger?) than that, because I realise now that
my interval naming scheme could be improved by going out to the 17-
limit (since 72-ET is 17-limit consistent) and giving the simplest
ratio for _every_ degree of 72-ET. In that case the narrow neutral
sixth (nN6) would be 8:13.

But for precise specification of the notes of a chord without
reference to a scale (except maybe for specifying the size of the
notational fifth and octave) then yes, the sagittal accidentals do
look like a reasonable solution.

But of course the terms minor, major, diminished, perfect, augmented
in referring to intervals has been in use for centuries, and many
microtonalists already use Fokker's extension of these which adds
sub, super, and neutral. It's only the wide and narrow that are new
(and even here Scala has been using some of them for quite some
time). I didn't invent them. I just collected them and smoothed out
some bumps.

So you might want to consider them as an optional alternative to the
use of sagittal accidentals for modifying intervals.

How do you think people should pronounce these sagittally modified
intervals?

> So for instance
>
> C/M3/sm7
>
> for
> C-/-t7

Yes. Or just Csm7. The slashes seem unnecessary and there are the
usual abbreviations whereby thirds and fifths don't need to be
explictly mentioned if they are major and perfect respectively, and
one can write "no 3" or "no 5" if necessary.

> However, I much prefer the Sagittal symbol there for
> the chord progression player, because it
> shows you how far you have to adjust the pitch relative
> to the 6th, so gives a performance direction,
> and you don't need to remember that
> /SM6 is a 6 sharp by a 64/63, and that
> sm7 is a seventh flat by a 64/63 etc.

I'm not sure I understand the distinction here. Instead of
remembering what "sm" and "SM" mean, don't you just have to remember
what "t" and "f" mean? Why is one better than the other?

> Also your naming system is limited to certain
> pitches, and is fine for a tuning system
> such as miracle presumably which doesn't
> make fine pitch distinctions but how about
>
> 1/1 5/4 25/16
>
> or the harmonic 13th etc,
>
> It doesn't give us any way to notate those
> at all.

As I said above, the harmonic 13th is the obvious interpretation
of "narrow neutral third" (nN3). But you're right about 25/16. In
that context it is clearly an augmented fifth (A5) but that's also a
subminor sixth and my list would have that interpreted as 9:14.

There is no doubt that Sagittal in its full generality can make
finer pitch distinctions than anything else. Thanks to Gene, we're
now able to use it to distinguish pitches 0.5 cents apart! But this
is not the case for the limited set you are using, which is the set
that can be represented by single ASCII characters - the Athenian
(medium-precision) set plus a few others.

But admittedly this does have a resolution of around 5 cents, while
the simple miracle/72-ET based interval-naming system only has a
resolution of 17 cents. It can however be extended by the use of the
qualifiers "Pythagorean", "septimal", "undecimal", "tridecimal",
etc. As described in my earlier attempt

http://dkeenan.com/Music/IntervalNaming.htm

> I was so glad to find that Sagittal can
> be used to notate the chords for the player
> so concisely and clearly and with such flexibility
> and do hope we can find a way to
> resolve this.

Consider it resolved. Go ahead with the sagittal stuff. I am
confident that you understand the sagittal system well and will make
a good job of it. I'm happy to help in any way, but you will need to
email me since I'm not reading the lists.

-- Dave Keenan

Hi George and David,

Glad that the matter of using Sagittal in chord symbols is resolved.
I had a first read through of your replies and they have given a lot
of material to think about. Something urgent came up so I haven't
had time to replay today. More later,

Thanks,

Robert

Hi George,

>
> In fact I've added in the longer sagittal symbols already.

> That's good, because there are some that don't have a shorthand.
> Have you added them in both the pure and mixed versions?

I'm not sure of this distinction.

> > One thing I might do is to retain the abbreviations for a learner
> > mode, now that they are already programmed in. Probably
> > preset to show the short ascii symbols, but user can switch
> > to abbreviations if they find it helpful when learning the
> > notation.

> Just so you make it very clear which is which, so as not to confuse
> the user.

Certainly will do! I also don't want to confuse the user!

Actually I have found that with only a little use one soon
gets to learn the accidentals you use frequently.

> > Here is David Canright's thirteen limit scale using them:
> >
> > http://www.robertinventor.com/sagittal.png
> >
> > 1 13/12 9/8 7/6 5/4 4/3 11/8 3/2 13/8 5/3 7/4 11/6 2/1

> Very nice. But 5/4 and 5/3 should be E\ and A\ (backslashes, with
> downward slope), respectively. Also, 11/6 should be Bo rather than
> B@, since the alteration is *downward* from a pythagorean B. It does
> take a little practice to read these quickly, but only a little.
> Even though I haven't used the shorthand very much, I was able to
> spot these right away.

Yes that's right. Got something back to front. It's fixed now.

> > I didn't find these in your list, maybe they can be got
> > by combining others in the list:
> >
> > 128/125 diesis

> Yes, but a shorthand character may be combined only with a 5-schisma
> accent mark. The 125S-diesis is equal to two 5-commas 80:81 less a 5-
> schisma (32768:32805). The Sagittal symbols for this are .//| up
> and '\\! down, which is .= and '_ , respectively, in the shorthand
> notation. We used the equal sign for the up-symbol because it has
> two strokes, and the best thing we could find for the down-symbol was
> the underscore.

Okay thanks. I've added those to the list.

More later.

Robert

Hi Dave and George,

Yes I got confused about 14-et. I was thnking about
11/9 actually. For some reason I tend to confuse
that with 9/7.

Since the undecimal comma vanishes I assume that
it will manage fine with an 11/9 and notate that as
just an E. While the 9/7 will be fine too.

Robert

Hi George,

> In the shorthand version of your "Sagittal Accidentals" table you
> should replace the "#cs" and "bcs" entries for 25/24 with #_ and b=,
> but the long entries you have for 25/24 in the upper table are
> correct. In the two tables I suggest that you replace the column
> headings "sharp" and "flat" with "up" and "down". The Sagittal
> entries for 40/39 should be file://| and \\! (long), = and ^ (short); we
> call this one the 5:13S-diesis. Should you be curious about the
> exact boundaries between small, medium, and large dieses, etc., see
> the following:

Made all the corrections ,thanks.

> /tuning/topicId_53869.html#54584

> > For the 1053/1024 or 35/36, the chord progression
> > player has to play one or the other in its just intonation
> > realisation of the chords, so I'm not sure yet what to
> do about that.

> There is also 250/243 (an apotome less three 5-commas) to be taken
> into account. We have given the symbol /|) a primary (or exact)
> definition of 35:36 and have allowed it to approximate the other two
> ratios (with errors ~0.4c). Similarly, we define (|\ as exactly
> 8192:8505, and that symbol will also approximate 26:27 and
> 51200:531441. I rather doubt that there are any JI fanatics out
> there that will be able to hear a 0.4-cent error. (But in case there
> are, please be advised that we, with a lot of help from Gene,
> recently laid the groundwork for the extreme-precision version of the
> Sagittal notation that has resolution fine enough to notate 2460-ET,
> or steps of less than 1/2 cent! Scary, eh?)

I'd like to show just intonations using an exact repressentation
where possible. It isn't really about hearing it though 0.4
cents could perhaps be heard in some situations couting beats in long sustained
chords using pure tones - and I wouldn't like to rule out the
possibility that some may have ears sensitive enough to make
such distinctions, I know it is something that is very variable
and the norm even amonst microtonal specialist performers
is surely far larger than that but who knows how far the
tail of the distribution extends.

But it is just a matter of keeping the ideas
clear - that if the composer wrote a 13/8 or whatever, that's
what they intended and not an approximation, just as
one would like to write 2/1 if they intend an octave and
not 1202 cents or whatever.

That's particularly relevent because I'd like
to have an option to let FTS show the scales in
Sagittal and let teh user edit them and enter them
in Sagittal. Thought it might be nice to have a
"Show current scale in Sagittal" option.
Then the user will be nice to be able to enter the likes of
C D E\ F G A\ B\ ]C (where ] is my octaving symbol)
to specify a scale in FTS and then to be
able to switch to / from Sagittal.

It would have to be a mixed notation so that it
will show ratios as ratios if it can't see how to
represent them exactly in Sagittal.

Anyway there are bound to be ratios that can't
be represented eventually. It's just a matter
of maximising that. I think I need to have
an exact Sagittal that can be used to represent
a large subset of the available pure ratios.
If need by I can just take whatever is the most
useful example of those equivalent intervals
or let the user select that, and say that is
the interpretation to be used for the exact
Sagittal. But it would be nice if these
equivalent ratios could be distinguished
in some way perhaps.

Perhaps if there is no existing way to
distinguish them I can make these the
preset "user defined variables" which
I plan to include as options since you have
included that feature in the Sagittal single
character accidentals. I wonder if that
would be acceptable.

anyway details of this are perhaps best discussed
offlist...

> > ...
> > I've also made it so user can choose to interpret in an
> > n-et with the interpretation of all the commas in that
> > et so that the commas and dieses can vanish or change sign
> > etc. There's a radio button the user can use to switch
> > to an n-et interpretation of Sagittal for any n-et they
> > like. But I haven't done anything about implementing the
> > n-et recommendations as that would involve a lot of programming
> > so can do that later if it is needed.

> Have you looked at the file sag_et.par that comes with Scala? This
> is one of the files that Dave and I maintain and update
> periodically. It contains the Sagittal symbol sequences for various
> ET's. You're welcome to use it for FTS, if you wish. (This reminds
> me that the 14-ET symbol sequence in it needs to be revised, as I
> note below.)

Thanks, I didn't know about those. Will look into it when I
get to that stage and I'm sure it will make the task much easier..

> > It will avoid using negative and vanishing commas when making
> > accidentals up itself at least. But allow anything in user
> > input - so for instance you can enter a chord sequence
> > and then change teh n-et and see how it gets re-interpreted
> > with e.g. zero or negative comma etc.

> Sounds pretty nifty!

Hope so. The n-et part isn't programmed completely yet
but I'm looking forward to find out what happens when
it is done :-).

> We do use the @ character for the 11L-diesis (704:729) up, but I
> don't think that one would be used very often. But when you parse
> this, you could write the code to interpret the second of two @
> characters without an intervening numeric character as a Sagittal
> character.

That's okay it's resolved now with the idea to
make - a Sagittal notation indicator.

> My immediate impression of "bb" is as a double-flat, another reason
> why I would discourage using it in combination with a comma-character
> to indicate direction.

Yes I'm not so keen on my abbreviations notation now. Considering possibly
leaving it out altogether, though I still think that possibly it might
help a newbie, but then again maybe it might confuse more than it helps
because of the need to unlearn some things such as association of
u with unidecimal which we don't want to have.

> No. I explained above how we arrived at it. Could the presence of a
> dash be used as an indication *not* to ignore an underscore character?

Yes, I've done it that way now.

> > Also the dash too - I know it is a particularly clear notation
> > so that's a strong incentive to support it
> > - but users will be used to using for chords as in
> > C5/7 etc and the player supports that for all
> > the other chords, and I have to support that
> > as well certainly because one objective for the player
> > is that if anyone pastes any chord progression such as
> > they might find on a web page etc into the player,
> > then it should be able to play it right away with
> > no changes, to make it easy for complete newbies to use.

> Aha!

> > So, it is a bit awkward to have
> > to say that you can't use the / with the Sagittal
> > notation, and do a special case for it somehow.
> > But hopefully one can come to some suitable solution.

> Perhaps you could write a scan-and-replace routine that would convert
> those slashes to dashes (or whatever you settle on for note
> delimiters). Otherwise, I have some other ideas:

> 1) We kept "<" and ">" free for user-defined applications, so
> perhaps ">" could be used for /| and "<" for \! . However, the
> directionality of the symbols is not very intuitive, and this
> character pair is presently used for the 7-comma in HEWM notation, so
> this would not seem to be a very good solution.

> 2) Use "+" and "-" for the 5-comma up and down and keep the slashes
> for your note delimiters. The directionality is certainly easy to
> remember, and the characters would tie in with both HEWM notation and
> the Sagittal-Wilson option given near the end of our paper.

> 3) Use "--" as a sort of delimiter to indicate that whatever
characters are included in a chord containing "--" should, whenever
> possible, be interpreted as Sagittal shorthand characters. (You
> mentioned something like this to me off-list.)

Yes, in fact I found the pre-parser was quite easy to do.

Anything prefixed by a dash apart from a V or a I (upper or lower
case) switches on sagittal for that chord and sets it to
treat dashes as sagittal dashes rather than roman numeral
ones. Hopefuly there won't be any confusions resulting from
this. Maybe I should do a bit more intelligent checking,
e.g.-I or -V with the I or V capitalised woudl be a sure
indciation of roman numerals rather than Sagittal and could
override anything else which might be a typo. While
-- would be a sure indication of Sagittal and could
be used to switch on Sagittal if perchance the user
only wanted to use i and v as accidentals (admittedly
unlikely to be needed)

> We have two different notations for 14-ET. One approach notates 14-
> ET as a subset of 56-ET (for which the symbol sequence is given in
> Figure 8 of the Sagittal paper). This is evidently the approach you
> are referring to, since a chain of four 5ths of 56-ET brings us to
> 20deg56, which is 5deg14. In 56-ET |) is calculated to be 1deg (by
> going two best 4ths of 23deg up, then down 1deg to 45deg56, which is
> the closest approximation to 7/4), but the wide fifths of 56-ET do
> not result in E|) being the best approximation to 9/7. In essence,
> the problem is that 56-ET is not consistent at the 9-limit.

Well it wasn't actualy as it was just a slip up for 11/9, but
that's the sort of point I wanted to ask about.

> The other approach uses a chain of native fifths for the 7 naturals
> (i.e., a 7-ET circle), and a single pair of Sagittal accidentals is
> then used to alter these up or down to indicate the other 7-ET
> circle. With a chain of 4 fifths and octave reduction, the E is 4
> degrees (~343c) above C, and raising it by one degree would give us
> E|), ~429c. So the 14-ET native-fifth notation would give you what
> you want. However, since releasing our theoretical paper, we have
> concluded that to make the notation consistent with a 28-ET native-
> fifth notation we should instead use the symbol pair |\ and !/ for a
> single degree of 14-ET. The single degree of 28-ET then gets the
> symbol pair |) and !) , which was determined by going upward by two
> best 4ths (of 12deg28) to Bb (~1029c) and then lowering that by
> 1deg28 to arrive at the best approximation of 7/4 in 28-ET (23deg),
> which is then notated as Bb!) . Due to the fact that that 28-ET is
> not 9-limit consistent, this does not result in E|) being the best
> approximation to 9/7 in 28-ET.

Rightio. So you just have to live with it basically.

> > The Sagittal
> > entries for 40/39 should be file://| and \\! (long), = and ^ (short);

> should be:

> The Sagittal entries for 40/39 should be file://| and \\! (long), = and _
(short);

Thanks, I had the wrong values for these in fact, fixed.

Robert