Yahoo Tuning Groups Ultimate Backup tuning Notating certain ETs in Sagittal

Concerning, namely, the lower-range ETs with "bad fifths" or "no fifths at all" etc.
Before the Committee fully delves into the issue of linear temperaments and which
nominals for which ones (which certainly will help a few of the ETs in question), I'd
like to voice my desire for

Notating 5-tET "preferably" as a subset of 50-tET (meantone, I'm told) is awfully
backwards if you treat the ETs as a "naturally occurring" sequence of divisions. (At
this point, throw your head back and laugh at the irony of using a "naturalness"
argument for ET's that are among the worst at approximating low-limit JI!)

Nevertheless, 5tet is not every 10th note out of some bigger scale with better fifths.
It is a closed circle of 720-cent fifths (or 240-cent 2nds or whatever). It should be
notated this way. Now let's get some better nominals than CGDAE (which was going
to come up anyway with 1/5-octave-period LT's, I know, I know).

Similar arguments for 7, 11, and 13 (suggested as subsets of 56, 22, and 26). It does
not make sense for simple systems to be born out of more complex ones. From a
practical standpoint, it made the job of assigning allll these different ETs notation
easier, but it makes comprehending each single system in a vacuum (which is what
musicians in any one system would do, right?) more difficult.

Since the Trojan (multiples of 12) set of symbols seems to work out so nicely, I was
wondering, if we had the right sets of nominals for 5 and 7, could all of their
multiples be worked out with the same harmonic consistency? (I'm thinking of that
beautiful Figure 10 in the Xenharmonikon paper.) Or are the sagittal accidentals so
tied to a series of near-enough-fifths that only a mess would result? What does
everybody think?

Jacob

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> Concerning, namely, the lower-range ETs with "bad fifths" or "no
fifths at all" etc.
> Before the Committee fully delves into the issue of linear
temperaments and which
> nominals for which ones (which certainly will help a few of the
ETs in question), I'd
> like to voice my desire for

Hi Jacob,

for what? I guess it's explained below.

> Notating 5-tET "preferably" as a subset of 50-tET (meantone, I'm
told) is awfully
> backwards if you treat the ETs as a "naturally occurring" sequence
of divisions. (At
> this point, throw your head back and laugh at the irony of using
a "naturalness"
> argument for ET's that are among the worst at approximating low-
limit JI!)
>
> Nevertheless, 5tet is not every 10th note out of some bigger scale
with better fifths.
> It is a closed circle of 720-cent fifths (or 240-cent 2nds or
whatever). It should be
> notated this way.

That's fine. Sometimes you want to look at it one way, and sometimes
the other. That's why we gave both notations. We only put
the "preferably"s in there for folks who had no idea of their own
about which notation to use.

> Now let's get some better nominals than CGDAE (which was going
> to come up anyway with 1/5-octave-period LT's, I know, I know).

Ok. But please explain what you find is wrong with using CGDAE (and
the corresponding staff positions) in the case of 5-tET.

> Similar arguments for 7, 11, and 13 (suggested as subsets of 56,
22, and 26). It does
> not make sense for simple systems to be born out of more complex
ones.

Yes it does. From a JI-approximation point of view it does. You just
don't happen to be adopting that point of view towards them, which
is fine.

So what's wrong with FCGDAE as a chain of fifths for 7-tET. How will
changing to different letters help?

> From a
> practical standpoint, it made the job of assigning all these
different ETs notation
> easier, but it makes comprehending each single system in a vacuum
(which is what
> musicians in any one system would do, right?) more difficult.

What is "it" here. What made it more difficult to comprehend? (a)
using the "preferably" notations (which you certainly don't have to
do) or (b) using conventional nominals and staff positions.

Of course, with 11 and 13 there is no "native-fifth" notation
because there is no native fifth. The choice is pretty much between
(a) new nominals (and new staves to match) or (b) notation as every
second note of 22 and 26. I expect many people will find (b)
preferable. And it's the only one we agree on how to do, at this
stage. But, as you say, we're working on (a).

Are you saying you don't even want to notate 6-ET as a subset of 12-
ET?

All the methods given in the Sagittal paper at least have the
advantage of working with conventional staves.

Please explain more about how you would like to use new nominals
and/or staves for 5, 7, 11, 13-ETs.

> Since the Trojan (multiples of 12) set of symbols seems to work
out so nicely, I was
> wondering, if we had the right sets of nominals for 5 and 7, could
all of their
> multiples be worked out with the same harmonic consistency? (I'm
thinking of that
> beautiful Figure 10 in the Xenharmonikon paper.)

The thing is, the fifth of 12-ET is so good that no better fifth
appears in its multiples until we get past 300-ET.

That is certainly not the case for 5 or 7-ET.

But one way to make the multiples of say 5-ET correspond, is to
derive them all as subsets of a common superset. Which is exactly
what happens with 5, 10 and 25 being derived as subsets of 50
(meantone). Unfortunately 15, 20 and 30 are not subsets of 50, but
subsets of 60 (which is a multiple of 12).

Similarly 7, 14 and 28 are subsets of 56, while 21 is a subset of 63
and 35 is a subset of 70.

But I understand that what you want to do is define the nominals not
as approximations to any particular ratio, or linear temperament
generator, but to the steps of the ET itself (this is equivalent to
the idea we discussed in the "Native nominals ..." thread for
notating linear temps with more than 5 periods per octave, which we
could drop back to "5 or more").

However, in order to do this we need to have a particular prime
mapping in mind for the "generator" of the nominals. Perhaps Paul or
Gene can suggest some for 5-ET and 7-ET.

> Or are the sagittal accidentals so
> tied to a series of near-enough-fifths that only a mess would
result? What does
> everybody think?

No, I don't believe they are so tied to a series of near-enough-
fifths that a mess would result, but this is as yet largely
untested. See footnote 2 on page 2 of the XH article.
http://dkeenan.com/sagittal/Sagittal.pdf

It's just that we haven't figured out a good general system for
notating chains of non-fifth/fourth generators yet. The aspect of
finding new nominals looks easier than figuring out how to represent
them on a staff.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Herman Miller <hmiller@IO.COM>

Jacob wrote:
> Since the Trojan (multiples of 12) set of symbols seems to work out so nicely, I was > wondering, if we had the right sets of nominals for 5 and 7, could all of their > multiples be worked out with the same harmonic consistency? (I'm thinking of that > beautiful Figure 10 in the Xenharmonikon paper.) Or are the sagittal accidentals so > tied to a series of near-enough-fifths that only a mess would result? What does > everybody think?
> > Jacob

7-ET can use the regular 7 nominals, but some of them are far from their usual values. If you set D of 7-ET equal to D of 12-ET, the ones that are farthest off are F (42.9 cents sharp) and B (42.9 cents flat). So you might want new nominals for F and B. 5-ET can use D E G A C, where E is 40 cents sharp and C is 40 cents flat.

The odd thing about 7-ET multiples is that the syntonic comma is negative. So if you use the 81;80 sagittal accidental as an actual 81;80, it will appear to point in the wrong direction. For instance, if you notate 28-ET as B=20, E=4, A=16, D=0, G=12, C=24, F=8, then the major third between 24 and 5 (28-ET has very good 9-step major thirds) would be written C E\! (where E\! represents a comma "below" E), even though E\! is actually a higher pitch than E.

So you could choose to notate the actual step size as an accidental: in the case of 28-ET this is close to 41/40 (42.9 cents). You could arbitrarily pick /|~ or /|) based on the size of the interval, without regard for its actual meaning, or you could just use the 50-cent arrow /|\ as a rough approximation.

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:
> But I understand that what you want to do is define the nominals not > as approximations to any particular ratio, or linear temperament > generator, but to the steps of the ET itself (this is equivalent to > the idea we discussed in the "Native nominals ..." thread for > notating linear temps with more than 5 periods per octave, which we > could drop back to "5 or more"). > > However, in order to do this we need to have a particular prime > mapping in mind for the "generator" of the nominals. Perhaps Paul or > Gene can suggest some for 5-ET and 7-ET.

5-ET doesn't have good thirds, but it does have a reasonable 7/4, which suggests a 7-limit temperament. There's a bunch of those that include 5-ET; here are a few possibilities:

[<1, 2, 3, 3|, <0, -2, -3, -1|] "bug" (aka "beep") (1/5 gen.)
[<1, 2, 2, 3|, <0, -2, 2, -1|] "number 57" (1/5 gen.)
[<1, 2, 4, 2|, <0, -1, -4, 2|] "dominant" (2/5 gen.)
[<1, 1, 1, 3|, <0, 3, 7, -1|] "gorgo" (1/5 gen.)
[<1, 2, 4, 3|, <0, -2, -8, -1|] "hemifourths" (1/5 gen.)
[<1, 2, 6, 2|, <0, -1, -9, 2|] "superpyth" (2/5 gen.)
[<1, 1, 0, 3|, <0, 3, 12, -1|] "mothra" (1/5 gen.)
[<1, 1, -1, 3|, <0, 3, 17, -1|] "rodan" (1/5 gen.)

"Number 57" looks like a nice simple map.

7-ET could be dicot, meantone, porcupine, or mavila.

[<1, 1, 2|, <0, 2, 1|] "dicot" (2/7 gen.)
[<1, 2, 4|, <0, -1, -4|] "meantone" (3/7 gen.)
[<1, 2, 3|, <0, -3, -5|] "porcupine" (1/7 gen.)
[<1, 2, 1|, <0, -1, 3|] "mavila" (3/7 gen.)

>>Or are the sagittal accidentals so >>tied to a series of near-enough-fifths that only a mess would > > result? What does > >>everybody think?
> > > No, I don't believe they are so tied to a series of near-enough-
> fifths that a mess would result, but this is as yet largely > untested. See footnote 2 on page 2 of the XH article.
> http://dkeenan.com/sagittal/Sagittal.pdf
> > It's just that we haven't figured out a good general system for > notating chains of non-fifth/fourth generators yet. The aspect of > finding new nominals looks easier than figuring out how to represent > them on a staff.

I've been giving that some thought. A 7-line staff could represent the basic chain-of-fifth nominals on the lines; the additional ones would be drawn attached above or below the lines but not touching the adjacent lines, or floating between the lines if more than 21 nominals are required. Alternatively, for the set of 22 nominals I've described, the lines could represent the evenly spaced A through G, with H-N attached below the line and P-V above the line; the half-octave O would be floating above or below the 7-line staff without a ledger line.

🔗monz <monz@attglobal.net>

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

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> >
> > > The thing is, the fifth of 12-ET is so good that no
> > > better fifth appears in its multiples until we get
> > > past 300-ET.
> >
> > The first better fifth which appears is 29-et. The next
> > convergent for log2(3/2) is 24/41, ie 41-et; which is
> > therefore also the better in terms of relative cents.
> > The next et to beat 12 in terms of logflat badness is
> > 53. I'm not getting 306 no matter how hard I try. :)

i have a graphic which shows this on my "perfect-5th" page:

http://tonalsoft.com/enc/p5.htm

> But 29 is not a multiple of 12. By "its multiples" I
> meant "multiples of 12-ET", i.e. 24-ET, 36-ET, 48-ET,
> 60-ET etc.

i think Paul has been working too hard on his paper.

-monz

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> 7-ET can use the regular 7 nominals, but some of them are far from
their
> usual values. If you set D of 7-ET equal to D of 12-ET, the ones
that
> are farthest off are F (42.9 cents sharp) and B (42.9 cents flat).
So
> you might want new nominals for F and B. 5-ET can use D E G A C,
where E
> is 40 cents sharp and C is 40 cents flat.
>
> The odd thing about 7-ET multiples is that the syntonic comma is
> negative. So if you use the 81;80 sagittal accidental as an actual
> 81;80, it will appear to point in the wrong direction. For
instance, if
> you notate 28-ET as B=20, E=4, A=16, D=0, G=12, C=24, F=8, then
the
> major third between 24 and 5 (28-ET has very good 9-step major
thirds)
> would be written C E\! (where E\! represents a comma "below" E),
even
> though E\! is actually a higher pitch than E.

Of course that goes completely against any Sagittal philosophy, and
dare I say common-sense. We would never suggest the use of a symbol
for a comma which is negative in some temperament. An arrow pointing
down to indicate a rise in pitch is just too horrible to
contemplate. I assume you were just making an academic point. :-)

At this point we would start to work our way from most common to
least common symbols until we find one for a comma whose size,
relative to the fifth size that we have chosen for notation
purposes, is best approximated by +1 degree of the ET.

The other requirement is that if there is more than one single-shaft
symbol pair required for the ET then the symbol-flag arithmetic must
be consistent between them. That is, it must be possible to assign a
fixed number of degress of the ET to each flag making up the symbols
used (left barb, right barb, left arc, right arc, left scroll, right
scroll, left boathook, right boathook, and accent). Negative flag-
values are allowed if you're desperate, but not negative values for
whole symbols.

> So you could choose to notate the actual step size as an
accidental: in
> the case of 28-ET this is close to 41/40 (42.9 cents). You could
> arbitrarily pick /|~ or /|) based on the size of the interval,
without
> regard for its actual meaning,

We hope that people will never choose a Sagittal symbol for some
purpose without regard for its actual meaning!

> or you could just use the 50-cent arrow
> /|\ as a rough approximation.

Apollo's arrow isn't always 50 cents, although it is often a half-
apotome (half-sharp/flat). Its constant meaning across all tunings
is 32;33 (the 11-diesis). Remember Apollo-11. That is the difference
between the best approximation to the interval 8:11 and the best
approximation to 3:4.

Even though any particular user may not be concerned about ratio
approximations at all, we ask that these still be respected. It's
the glue that holds the whole system together, across all tunings.

As it turns out, for notating 28-ET while using the 7-ET fifth as
the notational fifth, the arrow (left and right barbs) is an
excellent choice for notating 1deg28. Since it _is_ the difference
between the best approximation to the interval 8:11 and the best
approximation to 3:4 in this temperament.

A symbol for 2 degrees is harder. We'd like a double-shaft symbol
for this, to take the place of the apotome symbol (double-shaft
arrow) since the apotome vanishes with this fifth size.

I don't have my spreadsheet set up to check these against ETs
because normally we can stop at the apotome.

-- Dave Keenan

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Jacob <jbarton@rice.edu>

> > Nevertheless, 5tet is not every 10th note out of some bigger scale
> with better fifths.
> > It is a closed circle of 720-cent fifths (or 240-cent 2nds or
> whatever). It should be
> > notated this way.
>
> That's fine. Sometimes you want to look at it one way, and sometimes
> the other. That's why we gave both notations. We only put
> the "preferably"s in there for folks who had no idea of their own
> about which notation to use.

Good point, two completely valid ways of approaching it. But why are subsets of larger
ETs "preferred"? I dunno, we ought to ask someone who has no idea what they'd prefer,
how they'd approach it. Anyone?

> Ok. But please explain what you find is wrong with using CGDAE (and
> the corresponding staff positions) in the case of 5-tET.

Well, you got two big holes, F and B. And 5tet degrees 2 and 4 are closer to 12tet F and
B flat. More sensible, I think would be (Easley Blackwood's???) treating E and F as
enharmonics, and B and C also. Even more sensible to me would be a nice four-line staff
with the middle space defined as C (maybe with different nominal names, I don't really
care I guess). However, the sense of this is yet to be tested with musicians, so... (At this
point I wish someone could respond with firsthand experience in 5tet or 7tet ensembles.
I know they exist! Just not here.)

I guess what I'm going for in a score that has 5 or 7 is something that's transparent
enough, simple enough (I don't want people reading it to say hey, this is a complicated
tuning, because it ain't), but just alien enough for you to pause and think, ooh man,
these ain't your grandma's CGDAE. On the other hand, why should five notes per octave
require five different sagittal accidentals, as in the "preferred" notation?

The 7 "white key" representation of 7tet is great except that it looks normal. Perhaps
some wacko modified treble clef would be enough in this case.

> So what's wrong with FCGDAE as a chain of fifths for 7-tET. How will
> changing to different letters help?

Only to help the disorientation, which maybe I'm aiming for too much of, we'll see.

> What is "it" here. What made it more difficult to comprehend? (a)
> using the "preferably" notations (which you certainly don't have to
> do) or (b) using conventional nominals and staff positions.

The use of subsets of larger ETs is mainly what I am questioning.

> Are you saying you don't even want to notate 6-ET as a subset of 12-
> ET?

Ouch! Tough call. Maybe I'm okay with it because it only needs one or two accidentals.

> But one way to make the multiples of say 5-ET correspond, is to
> derive them all as subsets of a common superset. Which is exactly
> what happens with 5, 10 and 25 being derived as subsets of 50
> (meantone). Unfortunately 15, 20 and 30 are not subsets of 50, but
> subsets of 60 (which is a multiple of 12).

I see. But would it not be better if everything but 25 were derived from 60? Is it not
better for at least 10, 15, and 20 to agree on how to represent 5? Or are there other
things to consider...

> But I understand that what you want to do is define the nominals not
> as approximations to any particular ratio, or linear temperament
> generator, but to the steps of the ET itself (this is equivalent to
> the idea we discussed in the "Native nominals ..." thread for
> notating linear temps with more than 5 periods per octave, which we
> could drop back to "5 or more").

I realize that mentioning all this right now is rather silly since the brains 'round here just
started working on it, albeit on LTs primarily. I don't know if this means that instead of
being held up to JI, ETs will be held up to "optimal generators" for assorted prime-limit
systems. It makes more sense for ETs to be noted for what they are.

> No, I don't believe they are so tied to a series of near-enough-
> fifths that a mess would result, but this is as yet largely
> untested. See footnote 2 on page 2 of the XH article.
> http://dkeenan.com/sagittal/Sagittal.pdf

Righto. Who knows what the future holds.

Godspeed,
Jacob

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_54398.html#54455

> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > 7-ET can use the regular 7 nominals, but some of them are far
from
> their
> > usual values. If you set D of 7-ET equal to D of 12-ET, the ones
> that
> > are farthest off are F (42.9 cents sharp) and B (42.9 cents
flat).
> So
> > you might want new nominals for F and B. 5-ET can use D E G A C,
> where E
> > is 40 cents sharp and C is 40 cents flat.
> >
> > The odd thing about 7-ET multiples is that the syntonic comma is
> > negative. So if you use the 81;80 sagittal accidental as an
actual
> > 81;80, it will appear to point in the wrong direction. For
> instance, if
> > you notate 28-ET as B=20, E=4, A=16, D=0, G=12, C=24, F=8, then
> the
> > major third between 24 and 5 (28-ET has very good 9-step major
> thirds)
> > would be written C E\! (where E\! represents a comma "below" E),
> even
> > though E\! is actually a higher pitch than E.
>
> Of course that goes completely against any Sagittal philosophy, and
> dare I say common-sense. We would never suggest the use of a symbol
> for a comma which is negative in some temperament. An arrow
pointing
> down to indicate a rise in pitch is just too horrible to
> contemplate. I assume you were just making an academic point. :-)
>
>
***One of the greatest things about the Sagittal notation is the
*clarity* with which it indicates *direction.* That is one thing
that is *far and away* better than Ezra Sims' notation for 72-tET,
and one of the reasons that I am adopting Sagittal-Wilson...

J. Pehrson

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:

/tuning/topicId_54398.html#54460

> > > Nevertheless, 5tet is not every 10th note out of some bigger
scale
> > with better fifths.
> > > It is a closed circle of 720-cent fifths (or 240-cent 2nds or
> > whatever). It should be
> > > notated this way.
> >
> > That's fine. Sometimes you want to look at it one way, and
sometimes
> > the other. That's why we gave both notations. We only put
> > the "preferably"s in there for folks who had no idea of their own
> > about which notation to use.
>
> Good point, two completely valid ways of approaching it. But why
are subsets of larger
> ETs "preferred"? I dunno, we ought to ask someone who has no idea
what they'd prefer,
> how they'd approach it. Anyone?
>
> > Ok. But please explain what you find is wrong with using CGDAE
(and
> > the corresponding staff positions) in the case of 5-tET.
>
> Well, you got two big holes, F and B. And 5tet degrees 2 and 4 are
closer to 12tet F and
> B flat. More sensible, I think would be (Easley Blackwood's???)
treating E and F as
> enharmonics, and B and C also. Even more sensible to me would be a
nice four-line staff
> with the middle space defined as C (maybe with different nominal
names, I don't really
> care I guess). However, the sense of this is yet to be tested with
musicians, so... (At this
> point I wish someone could respond with firsthand experience in
5tet or 7tet ensembles.
> I know they exist! Just not here.)
>

***My guess is they don't use "staff notation" at all...

J. Pehrson

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> > But one way to make the multiples of say 5-ET correspond, is to
> > derive them all as subsets of a common superset. Which is
exactly
> > what happens with 5, 10 and 25 being derived as subsets of 50
> > (meantone). Unfortunately 15, 20 and 30 are not subsets of 50,
but
> > subsets of 60 (which is a multiple of 12).
>
> I see. But would it not be better if everything but 25 were
derived from 60? Is it not
> better for at least 10, 15, and 20 to agree on how to represent 5?
Or are there other
> things to consider...

I don't think so. I think this is just a case where someone with
actual personal interest in a tuning can make a valuable
contribution. I think you're right. It makes good sense to notate 5,
10, and 15 (as well as 20 and 30) as subsets of 60 rather than 50,
since 60 is a multiple of 12.

Having several hundred ETs to notate, we fairly blindly applied a
rule that the preferred notation should use the smallest multiple
that was 1,3,9-consistent, which is to say the smallest multiple
where C:D, G:A etc. were the best available approximation to 8:9.

> I realize that mentioning all this right now is rather silly since
the brains 'round here just
> started working on it, albeit on LTs primarily. I don't know if
this means that instead of
> being held up to JI, ETs will be held up to "optimal generators"
for assorted prime-limit
> systems. It makes more sense for ETs to be noted for what they
are.

Not silly at all. Exactly the right time to mention it.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > ...
> > The odd thing about 7-ET multiples is that the syntonic comma is
> > negative. So if you use the 81;80 sagittal accidental as an
actual
> > 81;80, it will appear to point in the wrong direction. For
instance, if
> > you notate 28-ET as B=20, E=4, A=16, D=0, G=12, C=24, F=8, then
the
> > major third between 24 and 5 (28-ET has very good 9-step major
thirds)
> > would be written C E\! (where E\! represents a comma "below" E),
even
> > though E\! is actually a higher pitch than E.
>
> Of course that goes completely against any Sagittal philosophy, and
> dare I say common-sense. We would never suggest the use of a symbol
> for a comma which is negative in some temperament. An arrow
pointing
> down to indicate a rise in pitch is just too horrible to
> contemplate. I assume you were just making an academic point. :-)
>
> At this point we would start to work our way from most common to
> least common symbols until we find one for a comma whose size,
> relative to the fifth size that we have chosen for notation
> purposes, is best approximated by +1 degree of the ET.
>
> The other requirement is that if there is more than one single-
shaft
> symbol pair required for the ET then the symbol-flag arithmetic
must
> be consistent between them. That is, it must be possible to assign
a
> fixed number of degress of the ET to each flag making up the
symbols
> used (left barb, right barb, left arc, right arc, left scroll,
right
> scroll, left boathook, right boathook, and accent). Negative flag-
> values are allowed if you're desperate, but not negative values for
> whole symbols.
>
> > So you could choose to notate the actual step size as an
accidental: in
> > the case of 28-ET this is close to 41/40 (42.9 cents). You could
> > arbitrarily pick /|~ or /|) based on the size of the interval,
without
> > regard for its actual meaning,
>
> We hope that people will never choose a Sagittal symbol for some
> purpose without regard for its actual meaning!

Yes!

> > or you could just use the 50-cent arrow
> > /|\ as a rough approximation.
>
> Apollo's arrow isn't always 50 cents, although it is often a half-
> apotome (half-sharp/flat). Its constant meaning across all tunings
> is 32;33 (the 11-diesis). Remember Apollo-11. That is the
difference
> between the best approximation to the interval 8:11 and the best
> approximation to 3:4.
>
> Even though any particular user may not be concerned about ratio
> approximations at all, we ask that these still be respected. It's
> the glue that holds the whole system together, across all tunings.

Yes!

> As it turns out, for notating 28-ET while using the 7-ET fifth as
> the notational fifth, the arrow (left and right barbs) is an
> excellent choice for notating 1deg28. Since it _is_ the difference
> between the best approximation to the interval 8:11 and the best
> approximation to 3:4 in this temperament.

Until you proceed to 2 degrees:

> A symbol for 2 degrees is harder. We'd like a double-shaft symbol
> for this, to take the place of the apotome symbol (double-shaft
> arrow) since the apotome vanishes with this fifth size.
>
> I don't have my spreadsheet set up to check these against ETs
> because normally we can stop at the apotome.

Because the apotome vanishes in 28-ET and double-shaft symbols are
defined as apotome-complements), it would be a bit of a problem to
figure out how to use double-shaft symbols for 28-ET. I suppose you
could use the apotome-complement of a single-shaft symbol that's -2
degrees, e.g., )||~. But then there's the problem of what to use for
3 degrees (yikes!).

If we look at 21-ET, which has larger steps than 28, we find that we
already came up with this symbol sequence (with all single-shaft
symbols):

21: |) |\ with naturals F C G D A E B

These same symbols are also valid for 28-ET, and they can be
logically extended to 3 degrees thus:

28: |) |\ |\) with naturals F C G D A E B

This keeps it simple and thus easy to remember.

Now we already have this for 14-ET:
14: |) with naturals F C G D A E B
but we might want to change it to this (which is also valid) to make
it compatible with the 28-ET notation:

14: |\ with naturals F C G D A E B

Jacob and Herman, it's good that you're bringing up these questions
now, because you're more closely involved with the ins and outs of
these divisions than either Dave or myself. Having so many symbols
in Sagittal means that there will frequently be several ways to
notate any given degree of any ET, and we want to arrive at symbol
sequences that are going to make the most sense.

--George