Yahoo Tuning Groups Ultimate Backup tuning-math Re: Sagittal Chord Notation

Hi Dave,

> 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.

Rightio. Can discuss other things offlist.

> 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).

Yes.

> > 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.

Rightio, thanks. I need to make the same distinction in the
Gui.

> > 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).

Yes if it was just for notating nearest pitches indeed,
then you can assume that most normal scales won't have
consecutive pitches at such tiny intervals apart!

> > 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.

Yes I understand that.

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

Yes it could be a useful optional alternative indeed, thanks.

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

I don't know, have you thought about that? It's a bit of a puzzle
I suppose. So far I've not needed to talk about them to someone
else, but you need some word to use when you do. Have you had
any thoughts about pronunciation of the Sagittal accidentals?

> > 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.

> Yes indeed, if the chord is purely in your notation then
> you won't need the slashes.

> 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?

Don't know, might just be slightly more familiarity.
But sub minor or super major do apply to specific intervals
while t and f are accidentals, so it is like the distinction
between minor, and flat, where the flat sign is more
general because it can be applied to any note while the
major and minor is somewhat less general because it
refers to a particular system of pitches rather than
being a note modifier. both have their place of course.

...

> 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.

Great thanks. I'll c.c. this to you so you don't need to reply to
this list. A lot of this sort of detailed stuff is best done
off-list anyway.

Thanks,

Robert

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Dave,

> The primary consideration in choosing single-ASCII characters was
> that they should _look_ as much as possible like the actual
> graphical or true-type symbol. The graphical symbols are primary.
> The assumption was that the reader was already familiar with the
> graphical symbols or soon would be, and that they would eventually
> just read the ASCII _as_ the corresponding graphical. i.e. they
> would look at the ASCII, but see an image of the graphical symbol in
> their "mind's eye". I guess we should put together a chart to
> facilitate that.

Yes I can help in FTS by having an optino to switch between the
graphical symbols and the abbreviations so user can show
it either way.

> But for now, here are a few rules to help remember at least the
> direction up or down.

> First try to see the ascii character as consiting of a single
> vertical shaft with a flag or flags on it, chosen from straight,
> convex, concave and wavy (although sometimes there is no part of the
> character corresponding to the shaft). That often gives you the
> direction up or down. For example the following are all up

> ?frnmpyqg

> and these are all down

> juvwthkdb

> Another way of looking at these is if you were to cut the lowercase
> letters out of cardboard (in a san serif font such as Arial or
> Helvetica) and try to balance them about their midline, would they
> fall toward their top side? i.e. are they top-heavy? If so, then
> they are upward pointing, otherwise (whether they would fall to the
> bottom or balance perfectly) they are downward. This is already the
> case with "b" as the flat symbol.

Rightio. I'll explain this in the help. Yes that's clear.

> If a pair consists of a lowercase letter and a special character.
> The lowercase letter is always down and the special character is
> always up. This is already the case with b for flat and # for sharp.
> These pairs are

> b#
> s$
> i*
> z~
> j?
> v^
> o@

Rightio.

> In a pair consisting of two special characters, the vertical
> placement of the character gives the direction. The high placed ones
> are up while the low placed are down. As in these pairs

> ,`
> .'
> ;"
> _=

Yes those are easy to remember.

> > The notation window will have a list of all the symbols
> > so that it is easy to look them up from within the program.
> > Also the user can show them on the picture of the p.c.keyboard
> > and swap between the various possibilities.

> Great. It would be good if you can display the actual true-type
> symbols as well as the ASCII.

Hope so, don't see why not. Need to have both because
the user will need to be able to type in teh symbols on the
keyboard so will need to press ascii character keys to do that
and probably will just get them displayed the same wya though
later could do it so optionally you type the ascii and see teh Sagittal
perhaps.

> > I do like the Sagittal system of accidental signs
> > and this wasn't meant as any kind of attempt to
> > try and change or improve on it. I just found
> > they weren't working with chord symbols, but your
> > short symbols do indeed solve that problem.

> Great.

> > 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 since this is a graphic, there's really no excuse for
> not using the actual sagittal symbols rather than the ASCII
> characters, is there?

Yes should be easy to do them. Best tohave option to show them
either way so that user can see also the accidentals corresponding
to what they need to type on the keyboard for the accidentals.
Could be preset to show the proper accidentals depending on what
seems would be best.

> Also, I expect George has pointed this out, but in the mixed
> notation the flats and sharps are always closer to the letter-name
> or notehead than are the sagittals, so we always go from large pitch
> difference (the letter nominal or staff position) to medium pitch
> difference (the sharp or flat or doubles thereof) to small pitch
> difference (sagittal accidental). On the staff this sequence runs
> right to left but in text with letter names it runs left to right.
> So Ebt rather than Etb. And so it is naturally read as "E flat 7-
> comma down" rather than "E 7-comma down flat".

Oh right I'd been doing it the other way around.

> Combining sagittals with diatonic interval numbers isn't something
> George and I have considered before. But since it is traditional for
> the flat or sharp to be on the right in that case, it makes sense to
> me that the sagittal should be leftmost in this case. So for example
> tb3 pronounced naturally as "7-comma down minor third" or
> indeed "sub minor third".

Rightio. Yes I think that makes more sense.

> You could decide between 125:128 versus (80:81)^2 on the basis of
> which gives the simplest ratio relative to the root. So when applied
> to certain intervals it would be interpreted one way, and when
> applied to others it would be interpreted the other way. Or you
> could get fancy and look at relationships to notes other than the
> root as well, but that sounds too hard.

I can't do that because the user may on occasion actually intend
the more complex ratio for some reason. They may not be using the
most likely or expected note relationships - as a composer
the user can do anything.

> Seriously. I think you goofed in figuring that the E is already a
> 7:9 from C? It is 4/14-oct which is 343 cents. A 7:9 is 435 cents
> and corresponds to Ef as you would expect.

Yes I did. I was confusing it with an 11/9 which is something
I do sometimes.

> But I wondered, thinking forwards
> to possiblee later developments, is there
> any work done on a non octave version of
> Sagittal?
>
> For instance I suppose one might do a
> 3/1 based Saggital with the 2/1 in place
> of 3/1 as the generating interval for the
> note names or something.

No need.

> > How would you notate Bohlen Pierce in
> > Sagittal? Or Wendy Carlos's
> > Gamma?

> Easy.

> The BP generator is 3^(1/13) or about 146.3 cents. There are about
> 8.2 of these in an octave. We can get rid of that .2 by multiplying
> by 5. 8.2 * 5 = 41. So we would notate BP in the same way as we
> would notate every fifth step of 41-ET. The notational octave in
> this case would be 3^(41/65) (~1199.7 cents) and the "notational
> fifth" would be 3^(24/65) (~702.3 cents).

> The gamma generator is 35.1 cents. There are about 34.188 of these
> to a just octave. That .188 is close enough to 0.2 or 1/5. If we
> multiply 34.2 by 5 we get 171. So Wendy Carlos' Gamma scale would be
> notated the same as every fifth step of 171-ET. The notational
> octave is exactly 34.2 gamma generators and the notational fifth is
> 21 gamma generators.

> Notice that we are not approximating BP with 41-ET or Gamma with 171-
> ET but we are notating these scales _exactly_ by correctly
> specifying our notational octave and fifth.

Ah the thing there is that at least one way to use these
scales is to treat the 3/1 or 3/2 as the interval of equivalence
in place of the octave. When I've done work composing
in non octave scales I've followed that approach of
treating the non octave as an interval of equivalence.

The idea is that since there are no 2s in the scale
the ear can come to recognise 3/1 instead as an equivalence
as the nearest it can find to an equivalence. So therefore
pitches shifted up or down by a 3/1 (or 3/2 in case of
Gamma) should have the same name. In the case of the e.t
scales then the 3/1 (or 3/2) is the only pure ratio around.

Paul tells me that in Bohlen Pierce the generator is 7/3
or just 7. So maybe something can be done by using 7 there
analogously to the use of 3 to generate the pythagorean scale

> But since it is traditional for the flat or sharp to be to the left
> of the number in that case, it makes sense to me that the sagittal
should be leftmost in this case.

Rightio I understood anyway, thanks.

> > 24:25 is )||( or \\!# or =#

> That should have been _#

> 24:25 is )||( or \\!# or _#

Rightio, makes sense. I've put it in like that.

Thanks,

Robert

🔗Robert Walker <robertwalker@ntlworld.com>

Hi George,

> Longhand shorthand
> down up down up comma
> ---- ---- ---- ---- ----------
> '\! ./| '\ ./ diaschisma
> .\! '/| .\ '/ pythagorean comma
> '\\! .//| '_ .= meantone diesis 125:128
> .!( '|( .c 'r 224:225
> '(!/ .(|\ 'w .m 27:28
> '!!) .||) b'f #.t 20:21

Thanks, I've added these.

> Using accent characters in combination with the other shorthand
> characters will give you JI pitch increments of around 2 cents, and I
> expect that you would have many more intervals possible than you will
> have interval names for. But the use of the accent characters would
> enable the FTS player to produce many more pitches with high
> precision.

Thanks. Yes it is particularly useful to have the
Pythagorean comma for intance, for the 17 tone Arabic
pythagorean scale. With these new accidentals it
can do many scales using single accidentals now.
I imagine it will give cover many of the applications.

I think in some situations it could be useful
to use multiple accidentals for just intonation
work, for instance in a pythagorean MOS it seems natural
to use multiple pythagorean commas.
In comma pumps it seems natural to use
multiple //.

Perhasp the situation where multiple accidentals might
be most needed is for cases where you have tonic
drift, so the second / could be thought
of as the accidental applied to the entire
pythagorean scale shifted by a / by tonic
drift.

That might even be useful in n-ets as a way of
showing composer intentions in some situations.

I find that most of the time it does find
single accidentals but there are a few
more complex j.i. intervals that require
multiple accidentals still. I'm not
sure quite what needs to be done.

As far as user input is concerned
I'll want to allow any number of accidentals
because some users might enter them like
that, and I like to program on the assumption
that the user is right in this situation if one
can because you never know what they might need
the program for, and one doesn't want to place
limitations on the input they can use if
one can avoid it - on the principle that the
user is always right sort of thing (where
possible).

For display one idea is to make it so that it is
preset to show only one accidental for approximate
pitches. For pure ratios it can go up to (say)
three accidentals (user set). Possibly one might be
better there to require three identical
accidentals but that isn't quite right
since maybe the user will be using the
j.i. retuning feature or the option to retune
the tonic via a midi in control channel or
controller, and have managed
to accumulate e.g. a comma shift +
a septimal diesis shift by way of
tonic drift, or indeed maybe more than
one of either, in which case they
would need to see both types of accidental.

Or could make it so the preset is just
one accidental and it needs to be changed
explicitly for those who want
to explore comma pumps and such like.
maybe have an "Exploring Comma drift" preset button
or some such.

Anyway the principle anyway is clear and
I'll make it so that it is a single
accidental where it is feasible to do that.
Hope that's okay. I can understand
the reasoning behind it because
asking a performer to make an adjustment
of a single accidental is one thing, and
they can learn the sizes of the various accidentals
they need - also asking for a schisma adjustment
on top of that isn't such a big thing
- if they are up to that type of precision
then presumably they will have no trouble
doing that extra adjustment.

But asking them to make two such adjustments
combined (on top of the flat or sharp
which isn't asking much on its own)
is perhaps asking a bit much especially
as the notation is primarily intended
to be easy to sight read isn't it.
I want to go along with that and
it will be great if perhaps some FTS users by using
the program and becoming familiar with
the notation as a result find
it somewhat easier to read scores and perform
scores accurately written in Sagittal notation.

Which is a good reason for trying to avoid
multiple accidentals in the program, apart
from some special situations such as tonic drift
where I think multiple accidentals may add to
clarity. But interested to hear what your
take on it is.

Thanks,

Robert

Re: Sagittal Chord Notation

🔗Robert Walker <robertwalker@ntlworld.com>

🔗Robert Walker <robertwalker@ntlworld.com>

🔗Robert Walker <robertwalker@ntlworld.com>

🔗Gene Ward Smith <gwsmith@svpal.org>