Standard definitions of Sagittal flags?

Is there a standard definition of each of the Sagittal flags, or does each Sagittal symbol have to be defined as a unit? I want to make sure I have the correct definitions of the symbols, without any of the schisminas (which cause problems when notating regular temperaments). For the more common Sagittal symbols I'm going by the definitions in the symbol map:

(http://dkeenan.com/sagittal/map/index.htm)

For commas not in that list, I've been using Scala, but the problem is that two different intervals (200/189 and 1323/1250, for example) can give the same Sagittal representation (||( even though they may be mapped differently in a given temperament. I'd like to know which one is "correct", if there is such a thing.

George D. Secor wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> >>Is there a standard definition of each of the Sagittal flags, > > > Yes.

Is this available in a file somewhere? (Or do you happen to know the specific thread where this was discussed?)

>>or does >>each Sagittal symbol have to be defined as a unit? > > > And yes. Each flag, by itself, has a standard (exact) definition for > JI (its "primary role"), and various combinations of flags (and > accents, if applicable) also have primary roles. For example, the > primary role of |( is 5103:5120 and that of ~| is 2176:2187, but the > symbol that consists of that combination of flags ~|( has a primary > role of 4096:4131, which is not exactly the sum of the two flags, but > a close approximation.

Well, this could complicate things a bit.

> The two intervals you've given are very close indeed to one another --
> so close that we don't even think that it would be necessary to > differentiate them, even in extreme-precision JI, in which I expect > we will be notating them with the /||~.. symbol (those two periods > are part of the symbol, a double right-accent, which amounts to ~0.73 > to 1.21 cents, typically ~0.83 cents), which has a primary role of > (believe it or not) 189:200. The same symbol would also represent > any other ratio ranging from 97.937 to 98.485 cents (the boundaries > arrived at by dividing an apotome into 233 parts, called "minas") in > a "secondary role". So both ratios are correct for this symbol, even > if one is represented in a secondary role.

....

> You mentioned the possibility that these ratios might be mapped > differently in a temperament. Did you have a particular temperament > in mind? Without any specific details, it would be difficult to > comment further, other than to say that there would probably be a way > to notate them differently without resorting to right-accents (and > possibly without left-accents), but that would depend on how many > tones are involved.

This particular example is from dimisept <4, 6, 9, 11], <0, 1, 1, 1]. In this mapping, 200/189 [3, -3, 2, -1> is mapped (+1, -2) and 1323/1250 [-1, 3, -4, 2> is mapped (0, +1). These are identical in 12-ET, but potentially different in other tunings of dimisept (95.62c vs. 101.46c in TOP dimisept).

The problem can be avoided by using /||) for 625/588 [-2, -1, 4, -2>, which also maps to (0, +1), but the standard value of /||) is [-21, 13, -1, 1>, which maps to (-4, +13). It's not likely that anyone would want to notate (-4, +13) in dimisept (!), so this may not be an issue. The best compromise may be to use |||( for 15/14, but that is ordinarily a 119.44 cent interval. I don't know how far it makes sense to stretch the sizes of these things.

Here's a link to the page I'm working on, so you can see where I'm going with this. I'll eventually lead up to the 24-nominal system and the issues with staff notation of 24 nominals, but I want to make sure that I get the Sagittal part of the notation right.

http://www.io.com/~hmiller/music/notation.html

Corrected message #14566 (now deleted):

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Is there a standard definition of each of the Sagittal flags,

Yes.

> or does
> each Sagittal symbol have to be defined as a unit?

And yes. Each flag, by itself, has a standard (exact) definition for
JI (its "primary role"), and various combinations of flags (and
accents, if applicable) also have primary roles. For example, the
primary role of |( is 5103:5120 and that of ~| is 2176:2187, but the
symbol that consists of that combination of flags ~|( has a primary
role of 4096:4131, which is not exactly the sum of the two flags, but
a close approximation.

> I want to make sure I
> have the correct definitions of the symbols, without any of the
> schisminas (which cause problems when notating regular
temperaments).
> For the more common Sagittal symbols I'm going by the definitions
in the
> symbol map:
>
> (http://dkeenan.com/sagittal/map/index.htm)
>
> For commas not in that list, I've been using Scala, but the problem
is
> that two different intervals (200/189 and 1323/1250, for example)
can
> give the same Sagittal representation (||( even though they may be
> mapped differently in a given temperament. I'd like to know which
one is
> "correct", if there is such a thing.

The two intervals you've given are very close indeed to one another --
so close that we don't even think that it would be necessary to
differentiate them, even in extreme-precision JI, in which I expect
we will be notating them with the /||~'' symbol (those two
apostrophes are part of the symbol, a double right-accent, which
amounts to ~0.73 to 1.21 cents, typically ~0.83 cents), which has a
primary role of (believe it or not) 189:200. The same symbol would
also represent any other ratio ranging from 97.937 to 98.485 cents
(the boundaries arrived at by dividing an apotome into 233 parts,
called "minas") in a "secondary role". So both ratios are correct
for this symbol, even if one is represented in a secondary role.

Should you really want to distinguish 189:200 from 1232:1250 in JI,
you might use (||(.. (a symbol with another double right-accent) for
the latter, which corresponds to the same number of minas.

Since Dave and I are still in the process of working out all the
intricacies of extreme-precision JI, none of this has yet been
implemented in Scala. All that is presently available there is
medium-precision (or athenian-level) JI, in which everything from
94.900 to 100.267 cents is represented by a single (unaccented)
symbol (||(, which has a primary role of 17:18, and which notates all
other ratios in that range in secondary roles.

You mentioned the possibility that these ratios might be mapped
differently in a temperament. Did you have a particular temperament
in mind? Without any specific details, it would be difficult to
comment further, other than to say that there would probably be a way
to notate them differently without resorting to right-accents (and
possibly without left-accents), but that would depend on how many
tones are involved.

--George

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@>
wrote:
> >
> >>Is there a standard definition of each of the Sagittal flags,
> >
> > Yes.
>
> Is this available in a file somewhere? (Or do you happen to know
the
> specific thread where this was discussed?)

The flag definitions are given in the Xenharmonikon 18 article,
reprinted here:
http://dkeenan.com/sagittal/Sagittal.pdf
in Table 1 (page 8). As the accompanying text (page 9, 2nd
paragraph) explains, "When more than one ratio (or role) is given for
a symbol in the table, the first one listed is the _primary role_ for
that symbol, i.e., the ratio that _exactly defines_ that symbol."
The bold-type listings in the table serve to emphasize how prime
harmonics are notated, but they are not necessarily the primary roles
for those symbols. As the text goes on to explain, "Primary comma
roles were chosen on the basis of which commas would be used to
notate the most _popular_ ratios, as determined from ratio occurrence
statistics obtained by Manuel Op de Coul from the Scala archive of
over 2000 historical and experimental tunings." We therefore
define /|) as notating 35:36 _exactly_ and 1024:1053
_approximately_. It was only when the right-accent of extreme-
precision (olympian-level) JI was introduced that we became able to
distinguish 1024:1053 with a separate symbol /|). _exactly defined_
by that ratio.

> >>or does
> >>each Sagittal symbol have to be defined as a unit?
> >
> > And yes. Each flag, by itself, has a standard (exact) definition
for
> > JI (its "primary role"), and various combinations of flags (and
> > accents, if applicable) also have primary roles. For example,
the
> > primary role of |( is 5103:5120 and that of ~| is 2176:2187, but
the
> > symbol that consists of that combination of flags ~|( has a
primary
> > role of 4096:4131, which is not exactly the sum of the two flags,
but
> > a close approximation.
>
> Well, this could complicate things a bit.

Perhaps, but fortunately it has been my experience that JI is
generally more complicated to notate than a temperament.

Dave and I deliberately did not include all of the symbol definitions
in Table 1, because we wanted to leave the door open to changing the
exact definitions of a couple of the more obscure symbols (should
that seem desirable in the course of our ongoing discussions). As a
matter of fact, I'm happy to report that we've just resolved some
issues about which we've had a difference of opinion for quite some
time, so you'll be the very first one to receive our newly updated
table of Sagittal symbols (including ASCII shorthand) and their exact
definitions. Please handle this carefully; it's so new that the
paint's still wet! ;-)

ASCII shorthand, with extended-character decimal codes
long down up Comma and extended-character description
----- ---- -- ----------------------------------------
'| . ' 5-schisma, 32768:32805
)| ; " 19-schisma, 512:513
|( c r 5:7-kleisma, 5103:5120
~| s $ 17-kleisma, 2176:2187
)|( i * 7:11 kleisma, 891:896
)~| ¦ ¡ 143 comma, 143:144
166 161 broken bar, inverted exclamation mark
~|( a g 17-comma, 4096:4131
|~ z ~ 23-comma, 729:736
~~| § 11:49 comma, 98:99
167 182 section sign, paragraph sign
)|~ ÷ ¯ 19 comma, 19456:19683
247 175 division sign, macron
/| \ / 5-comma, 80:81
)/| ¢ 5:19 comma, 40960:41553
162 172 cents, not sign
|) t f 7-comma, 63:64
)|) ß þ 7:19 comma, 56:57
223 254 lowercase sz ligature, lowercase thorn (iceland)
|\ k y 55-comma, 54:55
(| j ? 7:11-comma, 45056:45927
~|) h p 49S-diesis, 48:49
/|~ ð ø 5:23S-diesis, 45:46
240 248 lowercase eth (iceland), lowercase o slash
(|( d q 5:11S-diesis, 44:45
~|\ £ µ 23S-diesis, 16384:16767
163 181 pound sterling, lowercase mu
//| _ = 25S-diesis, 6400:6561
)//| ± ¥ 5:13M-diesis, 6480:6656
177 165 plus or minus, yen
/|) u n 35M-diesis, 35:36
(|~ æ ª 11:19M-diesis, 171:176
230 170 lowercase ae ligature, female ordinal
/|\ v ^ 11M-diesis, 32:33
(/| ¿ ç 49M-diesis, 3969:4096
191 231 inverted question mark, lowercase c cedilla
)/|\ & % 5:49M-diesis, 392:405
|\) ¤ ° 49L-diesis, 8388608:8680203, or 7:55L, 1701:1760
164 176 general currency sign, degree sign
(|) o @ 11L-diesis, 704:729
|\\ « » 11:19L-diesis, 360448:373977
171 187 left angle-quote, right angle-quote
(|\ w m 35L-diesis), 8192:8505
)|\\ © ® 5:13L-diesis, 851968:885735
169 174 copyright, registered
, ` user-definable
- + user-definable
< > user-definable
[ ] user-definable
{ } user-definable

There are some minor (but significant) changes to the symbol set.
We've eliminated three symbols |~) |~\ |)) along with their
multiple-shaft versions, because we found that they were not all that
useful. We've replaced them with the 3-flag symbols )//| and )|\\,
which are an apotome-complement pair. Since ~| formerly had ||~) as
its apotome-complement, )//|| is now its new complement.

We've also introduced another 3-flag symbol )/|\ that replaces '(/|,
the reason being that we want to avoid left-accented symbols that
aren't in the athenian symbol set. We expect that this new (self-
complementing) symbol will be very useful for notating the semi-
apotome in temperaments (such as 99-ET).

If you've printed out the XH18 Sagittal article from the website,
you'll need to mark these changes to Figure 13 until we're able to
update it with these latest changes. The Sagittal font will also be
undergoing some changes in the days ahead -- please stay tuned!

> ...
> > You mentioned the possibility that these ratios might be mapped
> > differently in a temperament. Did you have a particular
temperament
> > in mind? Without any specific details, it would be difficult to
> > comment further, other than to say that there would probably be a
way
> > to notate them differently without resorting to right-accents
(and
> > possibly without left-accents), but that would depend on how many
> > tones are involved.
>
> This particular example is from dimisept <4, 6, 9, 11], <0, 1, 1,
1]. In
> this mapping, 200/189 [3, -3, 2, -1> is mapped (+1, -2) and
1323/1250
> [-1, 3, -4, 2> is mapped (0, +1). These are identical in 12-ET, but
> potentially different in other tunings of dimisept (95.62c vs.
101.46c
> in TOP dimisept).
>
> The problem can be avoided by using /||) for 625/588 [-2, -1, 4, -
2>,
> which also maps to (0, +1), but the standard value of /||) is [-21,
13,
> -1, 1>, which maps to (-4, +13). It's not likely that anyone would
want
> to notate (-4, +13) in dimisept (!), so this may not be an issue.
The
> best compromise may be to use |||( for 15/14, but that is
ordinarily a
> 119.44 cent interval. I don't know how far it makes sense to
stretch the
> sizes of these things.
>
> Here's a link to the page I'm working on, so you can see where I'm
going
> with this. I'll eventually lead up to the 24-nominal system and the
> issues with staff notation of 24 nominals, but I want to make sure
that
> I get the Sagittal part of the notation right.
>
> http://www.io.com/~hmiller/music/notation.html

Since I haven't been closely following the threads on the intricacies
of vector notation, this is not very familiar territory for me, so
I'm going to have to study this a bit. I think Dave Keenan will also
need to be involved.

It would save me some time if you would list some octave divisions
that correspond to dimisept, particularly one or two that have
200/189 and 1323/1250 mapped differently. I could take a look at how
we went (or would go) about notating those and offer some feedback
based on what I find.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
I've been using Scala, but the problem
> is
> > that two different intervals (200/189 and 1323/1250, for example)
> can
> > give the same Sagittal representation (||( even though they may be
> > mapped differently in a given temperament. I'd like to know which
> one is
> > "correct", if there is such a thing.
>
> The two intervals you've given are very close indeed to one another --
> so close that we don't even think that it would be necessary to
> differentiate them, even in extreme-precision JI...

They differ by the landscape comma, 250027/250000. This is tempered
out in, among other things, ennealimmal and 7-limit atomic. However,
the landscape comma is not tempered out by many other systems, for
instance meantone, miracle, orwell, magic, 19, 22, 31, 41, 46, 53, 58
or 68. If all we want is something good enough for 7-limit JI, I think
ennealimmal is already good enough, by the way. A nice notation for
ennealimmal could do about all that needs to be done so far as
notating the 7-limit goes. Great for those barbershop fellas if they
wanted to learn it.

George D. Secor wrote:

> The flag definitions are given in the Xenharmonikon 18 article, > reprinted here:
> http://dkeenan.com/sagittal/Sagittal.pdf
> in Table 1 (page 8).

In other words, the standard size of a flag is the same as the first listed size of the symbol with only a single flag?

Is there a standard for the difference between a single shaft arrow and the same arrow with a double or triple shaft?

> Dave and I deliberately did not include all of the symbol definitions > in Table 1, because we wanted to leave the door open to changing the > exact definitions of a couple of the more obscure symbols (should > that seem desirable in the course of our ongoing discussions). As a > matter of fact, I'm happy to report that we've just resolved some > issues about which we've had a difference of opinion for quite some > time, so you'll be the very first one to receive our newly updated > table of Sagittal symbols (including ASCII shorthand) and their exact > definitions. Please handle this carefully; it's so new that the > paint's still wet! ;-)

Thanks for the list; I'll look over these and see if I have any questions or comments after I check some of the more common temperaments (which could take a while.....)

> Since I haven't been closely following the threads on the intricacies > of vector notation, this is not very familiar territory for me, so > I'm going to have to study this a bit. I think Dave Keenan will also > need to be involved.
> > It would save me some time if you would list some octave divisions > that correspond to dimisept, particularly one or two that have > 200/189 and 1323/1250 mapped differently. I could take a look at how > we went (or would go) about notating those and offer some feedback > based on what I find.

Dimisept tempers out 36/35, 50/49, and so on (126/125, 360/343, 648/625 ... ). This includes 12-ET, 16-ET, and 28-ET.

From the mapping <4, 6, 9, 11], <0, 1, 1, 1]:

the octave (2/1) is 4 periods (and 0 generators),
3/1 is 6 periods and 1 generator up,
5/1 is 9 periods and 1 generator up,
7/1 is 11 periods and 1 generator up.

The period is 3 steps of 12-ET, 4 steps of 16-ET, or 7 steps of 28-ET.
The generator is 1 step of 12-ET and 16-ET, or 2 steps of 28-ET. This is also the size of interval that 1323/1250 maps to.

200/189 would be 1 step of 12-ET, 2 steps of 16-ET, or 3 steps of 28-ET.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
>
> > The flag definitions are given in the Xenharmonikon 18 article,
> > reprinted here:
> > http://dkeenan.com/sagittal/Sagittal.pdf
> > in Table 1 (page 8).
>
> In other words, the standard size of a flag is the same as the
first
> listed size of the symbol with only a single flag?

Yes.

> Is there a standard for the difference between a single shaft arrow
and
> the same arrow with a double or triple shaft?

Every triple-shaft symbol is exactly defined as the ratio for its
single-shaft counterpart "plus" an apotome (by "plus" I mean, of
course, that you should multiply the ratios).

Double-shaft symbols operate differently. They're defined as
the "difference" between an apotome and the single-shaft complement.
For example:

Since the complement of /| is ||\, and /| is 80:81, then ||\ is
2048:2187 "less" 80:81, or 128:135.
Since the complement of |) is ||), and |) is 63:64, then ||) is
2048:2187 "less" 63:64, or 131072:137781.
Since the complement of |\ is /||, and |\ is 54:55, then /|| is
2048:2187 "less" 54:55, or 56320:59049.

> > Dave and I deliberately did not include all of the symbol
definitions
> > in Table 1, because we wanted to leave the door open to changing
the
> > exact definitions of a couple of the more obscure symbols (should
> > that seem desirable in the course of our ongoing discussions).
As a
> > matter of fact, I'm happy to report that we've just resolved some
> > issues about which we've had a difference of opinion for quite
some
> > time, so you'll be the very first one to receive our newly
updated
> > table of Sagittal symbols (including ASCII shorthand) and their
exact
> > definitions. Please handle this carefully; it's so new that the
> > paint's still wet! ;-)
>
> Thanks for the list; I'll look over these and see if I have any
> questions or comments after I check some of the more common
temperaments
> (which could take a while.....)

One caveat about that list I gave you: there are a few symbol
definitions in that list that have not been finalized, specifically:

)~| could be 71680:72171 instead of 143:144
~~| could be 99:100 instead of 98:99
)/| could be 6561:6656 instead of 40960:41553
/|~ could be 1664:1701 instead of 45:46

It's probably best that you avoid these for now, since they're among
the least-used symbols.

Another thing that you should take note of is that it's possible to
use a symbol in a secondary role to notate a particular interval in a
temperament, *if* that same symbol would also be valid in its primary
role for the same interval in that temperament. For an example, see
the 99-ET notation that I just posted for Gene:

/tuning-math/message/14572

You'll see that we use the 23-comma symbol |~ for 1deg99 (and also
2deg189), but we're using it to notate the 7-limit ratio 243:245 (a
secondary role, which is valid in that it results in the same mapping
as the primary role).

> > Since I haven't been closely following the threads on the
intricacies
> > of vector notation, this is not very familiar territory for me,
so
> > I'm going to have to study this a bit. I think Dave Keenan will
also
> > need to be involved.
> >
> > It would save me some time if you would list some octave
divisions
> > that correspond to dimisept, particularly one or two that have
> > 200/189 and 1323/1250 mapped differently. I could take a look at
how
> > we went (or would go) about notating those and offer some
feedback
> > based on what I find.
>
> Dimisept tempers out 36/35, 50/49, and so on (126/125, 360/343,
648/625
> ... ). This includes 12-ET, 16-ET, and 28-ET.
>
> From the mapping <4, 6, 9, 11], <0, 1, 1, 1]:
>
> the octave (2/1) is 4 periods (and 0 generators),
> 3/1 is 6 periods and 1 generator up,
> 5/1 is 9 periods and 1 generator up,
> 7/1 is 11 periods and 1 generator up.

Now I get it -- thanks!

> The period is 3 steps of 12-ET, 4 steps of 16-ET, or 7 steps of 28-
ET.
> The generator is 1 step of 12-ET and 16-ET, or 2 steps of 28-ET.
This is
> also the size of interval that 1323/1250 maps to.
>
> 200/189 would be 1 step of 12-ET, 2 steps of 16-ET, or 3 steps of
28-ET.

So you don't seem to be using very many tones, but you want to use
accidentals that will be valid across all of the temperaments in the
dimisept family, right?

--George

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

> Since the complement of /| is ||\, and /| is 80:81, then ||\ is
> 2048:2187 "less" 80:81, or 128:135.
> Since the complement of |) is ||), and |) is 63:64, then ||) is
> 2048:2187 "less" 63:64, or 131072:137781.
> Since the complement of |\ is /||, and |\ is 54:55, then /|| is
> 2048:2187 "less" 54:55, or 56320:59049.

Why do you use the colon? It seems to me this notation is incorrect,
since you are notating a rational number, which can be either greater
than or less than 1.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
>
> > Since the complement of /| is ||\, and /| is 80:81, then ||\ is
> > 2048:2187 "less" 80:81, or 128:135.
> > Since the complement of |) is ||), and |) is 63:64, then ||) is
> > 2048:2187 "less" 63:64, or 131072:137781.
> > Since the complement of |\ is /||, and |\ is 54:55, then /|| is
> > 2048:2187 "less" 54:55, or 56320:59049.
>
> Why do you use the colon? It seems to me this notation is incorrect,
> since you are notating a rational number, which can be either
greater
> than or less than 1.

I'm using the convention of notating intervals with the numbers
separated by a colon (as opposed to pitches, in which the numbers are
usually separated by a slash). I've put the lower number on the left
in order to be consistent with the manner in which interval ratios
containing more than two integers (i.e., chords) are almost always
spelled: from the bottom up, e.g., 3:4:5 or 5:6:7:9.

In any interval the smallest number is always understood to be the
lowest pitch, so there should not be any confusion in putting the
smaller number to the left of the colon for an interval.

But you think this is not proper mathematical etiquette?

--George

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

> > Why do you use the colon? It seems to me this notation is incorrect,
> > since you are notating a rational number, which can be either
> greater
> > than or less than 1.
>
> I'm using the convention of notating intervals with the numbers
> separated by a colon (as opposed to pitches, in which the numbers are
> usually separated by a slash).

But these aren't intervals in the sense of unsigned intervalic
distance, but they are adjustments up or down from a particular note.
They are, in fact, rational numbers, making the ":" notation incorrect
it seems to me.

George D. Secor wrote:
> Every triple-shaft symbol is exactly defined as the ratio for its > single-shaft counterpart "plus" an apotome (by "plus" I mean, of > course, that you should multiply the ratios).

Makes sense.

> Double-shaft symbols operate differently. They're defined as > the "difference" between an apotome and the single-shaft complement. > For example:
> > Since the complement of /| is ||\, and /| is 80:81, then ||\ is > 2048:2187 "less" 80:81, or 128:135.
> Since the complement of |) is ||), and |) is 63:64, then ||) is > 2048:2187 "less" 63:64, or 131072:137781.
> Since the complement of |\ is /||, and |\ is 54:55, then /|| is > 2048:2187 "less" 54:55, or 56320:59049.

Okay, I think I have most of the ones that I might potentially need for now, so I'll leave it at that.

> One caveat about that list I gave you: there are a few symbol > definitions in that list that have not been finalized, specifically:
> > )~| could be 71680:72171 instead of 143:144
> ~~| could be 99:100 instead of 98:99
> )/| could be 6561:6656 instead of 40960:41553
> /|~ could be 1664:1701 instead of 45:46
> > It's probably best that you avoid these for now, since they're among > the least-used symbols.

With the exception of ~~| (which won't come up until I start looking at the 11-limit temperaments), I probably won't be using these any time soon. I'm mainly looking at 7-limit commas for the moment.

> So you don't seem to be using very many tones, but you want to use > accidentals that will be valid across all of the temperaments in the > dimisept family, right?

What I'm trying to do is determine how to use Sagittal notation to notate a particular class of "regular" temperaments, temperaments built from two sizes of generators with a defined generator mapping. This is one temperament that illustrates the potential problem with schisminas, but there are ways to avoid the conflict in this case, and it's possible that it might not happen to be a problem. But in order to know this, I'd have to test all temperaments that may theoretically be of interest.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> What I'm trying to do is determine how to use Sagittal notation to
> notate a particular class of "regular" temperaments, temperaments built
> from two sizes of generators with a defined generator mapping.

I'm with Herman; it seems to me that notating 7-limit rank two
temperaments is basic, and I wish more attention was paid to by George
and Dave.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
>
> > > Why do you use the colon? It seems to me this notation is
incorrect,
> > > since you are notating a rational number, which can be either
greater
> > > than or less than 1.
> >
> > I'm using the convention of notating intervals with the numbers
> > separated by a colon (as opposed to pitches, in which the numbers
are
> > usually separated by a slash).
>
> But these aren't intervals in the sense of unsigned intervalic
> distance, but they are adjustments up or down from a particular
note.
> They are, in fact, rational numbers, making the ":" notation
incorrect
> it seems to me.

Are they? Here's an example from my last message to Herman:

> Since the complement of /| is ||\, and /| is 80:81, then ||\ is
> 2048:2187 "less" 80:81, or 128:135.

All I see there is interval arithmetic. There's nothing there about
making adjustments to notes, and I didn't intend to be indicating any
particular direction of alteration when I used the ||\ symbol. I
could just as easily have used !!/ , or I could have given both
symbols. Anyway, I thought that it would be clear from the operator
("less") that the numbers would be converted to fractions, division
performed, and the result converted back to a ratio.

--George

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@> wrote:
>
> > What I'm trying to do is determine how to use Sagittal notation
to
> > notate a particular class of "regular" temperaments, temperaments
built
> > from two sizes of generators with a defined generator mapping.
>
> I'm with Herman; it seems to me that notating 7-limit rank two
> temperaments is basic, and I wish more attention was paid to by
George
> and Dave.

I spent quite a bit of time last night looking at dimisept, trying to
figure out which Sagittal accidentals I would use to notate it. (See
my next reply to Herman.)

--George

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
> ...
> > One caveat about that list I gave you: there are a few symbol
> > definitions in that list that have not been finalized,
specifically:
> >
> > )~| could be 71680:72171 instead of 143:144
> > ~~| could be 99:100 instead of 98:99
> > )/| could be 6561:6656 instead of 40960:41553
> > /|~ could be 1664:1701 instead of 45:46
> >
> > It's probably best that you avoid these for now, since they're
among
> > the least-used symbols.
>
> With the exception of ~~| (which won't come up until I start
looking at
> the 11-limit temperaments), I probably won't be using these any
time
> soon. I'm mainly looking at 7-limit commas for the moment.

Dave and I should have that one figured out by the time you need it.

> > So you don't seem to be using very many tones, but you want to
use
> > accidentals that will be valid across all of the temperaments in
the
> > dimisept family, right?
>
> What I'm trying to do is determine how to use Sagittal notation to
> notate a particular class of "regular" temperaments, temperaments
built
> from two sizes of generators with a defined generator mapping. This
is
> one temperament that illustrates the potential problem with
schisminas,
> but there are ways to avoid the conflict in this case, and it's
possible
> that it might not happen to be a problem. But in order to know
this, I'd
> have to test all temperaments that may theoretically be of interest.

As I mentioned to Gene in my last message, I spent a bit of time
looking at dimisept last night and couldn't help but wonder why
you're looking at such complex ratios for your notational semantics.
Correct me if I'm wrong, but it seems to me that the objective should
be to specify a reasonably simple ratio (and corresponding symbol) to
correspond to each of the smallest intervals in the temperament,
which will be the accidentals in the notation. I thought it should
be easier to come up with some simpler ratios than what you have.

For period and generator (0, +1) you have 1323/1250; I found 21/20.
For (1, -2) you have 200/189; I found 28/27. You also need a ratio
for (1, -3), which is the difference between the other two, or 81/80,
but, oops! -- the joke's on me, because 81/80 turned out to be (-1,
3). There's a problem in that the larger interval in the temperament
is represented by the smaller ratio, so try again. :-(

Using a spreadsheet that calculates the number of degrees each
Sagittal symbol would be in any division of the octave, I found three
simple 7-limit ratios that maintain the expected order of size:

(per, gen) deg12 deg16 deg28 ratio exact symbol
(1, -3) 0 1 1 64/63 |)
(0, 2) 1 1 2 49/48 ~|)
(1, -2) 1 2 3 28/27 .(|\

I see that the next tone would be 9/8, which you're probably going to
cover with a separate nominal. If you prefer a notation without any
accent marks, there's at least one way to do it (with promethean-
level symbols, where |\\ would represent 27:28 and the other two
symbols would remain the same).

So will this do it?

--George

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

> Are they? Here's an example from my last message to Herman:
>
> > Since the complement of /| is ||\, and /| is 80:81, then ||\ is
> > 2048:2187 "less" 80:81, or 128:135.
>
> All I see there is interval arithmetic. There's nothing there about
> making adjustments to notes, and I didn't intend to be indicating any
> particular direction of alteration when I used the ||\ symbol.

The sentence becomes nonsensical if you interpret it as undirected
intervals, and I don't see how you can claim that ||\, which notates a
n adjustment *up*, does not indicate any particular direction. A
notation which did not show direction, such as using the same symbol
for sharp as for flat, would be next to useless.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
>
> > Are they? Here's an example from my last message to Herman:
> >
> > > Since the complement of /| is ||\, and /| is 80:81, then ||\ is
> > > 2048:2187 "less" 80:81, or 128:135.
> >
> > All I see there is interval arithmetic. There's nothing there
about
> > making adjustments to notes, and I didn't intend to be indicating
any
> > particular direction of alteration when I used the ||\ symbol.
>
> The sentence becomes nonsensical if you interpret it as undirected
> intervals,

In other words, you think it would be nonsense for me to say that a
minor 3rd is the difference between a perfect 5th and a major 3rd, if
I neglected to indicate whether each of those intervals was directed
up or down (which is exactly what I'm doing in this sentence!)?

> and I don't see how you can claim that ||\, which notates an
> adjustment *up*, does not indicate any particular direction.

I didn't say that the symbol doesn't indicate any particular
direction, but that (for the purpose of illustrating how symbol
complements relate to interval arithmetic) I was interested only in
the size (or absolute value) of the intervals the symbols represented
and not in which direction the symbols were pointing (so long as they
all pointed in the same direction, so as not to give the impression
that direction might be of any significance).

> A
> notation which did not show direction, such as using the same symbol
> for sharp as for flat, would be next to useless.

Yes, I agree, but as long as one is simply comparing the sizes of two
things, the particular direction in which they happen to be oriented
at the time we make the measurements is irrelevant.

Can we spend our time on something more productive, please?

--George

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

> In other words, you think it would be nonsense for me to say that a
> minor 3rd is the difference between a perfect 5th and a major 3rd, if
> I neglected to indicate whether each of those intervals was directed
> up or down (which is exactly what I'm doing in this sentence!)?

No, I'm saying that "/| is 80:81" is nonsense, since 80:81 is an
undirected interval.

George D. Secor wrote:

> For period and generator (0, +1) you have 1323/1250; I found 21/20. > For (1, -2) you have 200/189; I found 28/27. You also need a ratio > for (1, -3), which is the difference between the other two, or 81/80, > but, oops! -- the joke's on me, because 81/80 turned out to be (-1, > 3). There's a problem in that the larger interval in the temperament > is represented by the smaller ratio, so try again. :-(

My preferred representation of (0, +1) (from the symbols represented in the Sagittal font) is actually 15/14, although 21/20 is about as good. I start by trying to find symbols in the standard font that are close to the size of the intervals in the TOP tuning of the temperament (as a somewhat arbitrary but generally useful convention). For (+1, -2) I have 16/15, and 225/224 for (-1, +3). The basic idea is that, where several different intervals are represented by the same number of generators, I'd like to use the one that is around the right size. But I like the idea of taking complexity into account and giving a higher priority to the simpler ratios.

I also considered 160/147 for (+1, -2), which was the best I could find in the existing font that didn't need a schisma accent. I'd almost like to say that the schisma accents could be dropped for these kind of temperaments, but I seem to remember a couple of cases where they were needed.

The 200/189 is actually about the right size in TOP dimisept, but I mentioned it mainly to illustrate the issue with (what I thought at the time was due to) schisminas. Still, it might have been better just to point out that 4375/4374, for instance, is mapped to (+1, -2) and 2401/2400 to (0, +1). I've had cases where 250/243 and 36/35 (which differ by 4375/4374) are mapped differently. I got around this by assuming 250/243 for 5-limit temperaments and 36/35 for 7-limit temperaments.

> Using a spreadsheet that calculates the number of degrees each > Sagittal symbol would be in any division of the octave, I found three > simple 7-limit ratios that maintain the expected order of size:
> > (per, gen) deg12 deg16 deg28 ratio exact symbol
> (1, -3) 0 1 1 64/63 |)
> (0, 2) 1 1 2 49/48 ~|)
> (1, -2) 1 2 3 28/27 .(|\
> > I see that the next tone would be 9/8, which you're probably going to > cover with a separate nominal. If you prefer a notation without any > accent marks, there's at least one way to do it (with promethean-
> level symbols, where |\\ would represent 27:28 and the other two > symbols would remain the same).

I've considered using 28/27 for pajara, semaphore, and possibly negrisept, so it'd be great to have an unaccented symbol for it! Other accented symbols that come up frequently are 21/20 and 16/15. 27/25 and 128/125 could also have some uses.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
>
> > In other words, you think it would be nonsense for me to say
that a
> > minor 3rd is the difference between a perfect 5th and a major
3rd, if
> > I neglected to indicate whether each of those intervals was
directed
> > up or down (which is exactly what I'm doing in this sentence!)?
>
> No, I'm saying that "/| is 80:81" is nonsense, since 80:81 is an
> undirected interval.

Gidday Gene,

I understand your point here. But please cut us a little slack. We
long ago adopted the shorthand convention of only giving the up-
symbol when we mean the up-down pair, since in sagittal the
transformation is so obvious. You will notice that we always give
the pair in the case of the single-ASCII characters.

I don't think most musician's are comfortable with the idea of
negative intervals. They would rather give an unsigned magnitude and
then give the direction separately as "up" or "down" when it matters.

And just as soon as we figure out how to go _sideways_ in pitch
instead of just up and down, we'll use a full polar representation
(unsigned magnitude and angle) for intervals. ;-)

I guess I weaseled out of your objection last time by claiming that
n:m could be read as the rational number max(m,n)/min(m,n) in the
context of tuning.

I still think that's mostly true. But the main driver behind the
convention is just that we think it's useful to musicians to use
colon versus slash to distinguish intervals from pitches.

I remember an acrimonious argument in the early days of the tuning
list, before you joined, Gene, that in the end turned out to be
based purely on a misunderstanding where someone was making claims
about a certain interval while the other was reading it as a pitch,
or vice versa.

-- Dave Keenan

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...>
wrote:
> My preferred representation of (0, +1) (from the symbols
represented in
> the Sagittal font) is actually 15/14, although 21/20 is about as
good. I
> start by trying to find symbols in the standard font that are
close to
> the size of the intervals in the TOP tuning of the temperament (as
a
> somewhat arbitrary but generally useful convention). For (+1, -2)
I have
> 16/15, and 225/224 for (-1, +3). The basic idea is that, where
several
> different intervals are represented by the same number of
generators,
> I'd like to use the one that is around the right size. But I like
the
> idea of taking complexity into account and giving a higher
priority to
> the simpler ratios.

Hi Herman,

A couple of things to note here:

1. Rather than comparing the size of the needed alteration with the
size of the _untempered_ symbol commas, you can first temper the
factors of 2 and 3 in the symbol commas, according to the
temperament you're notating, (provided they are not _too_ heavily
tempered), _then_ look at the sizes. For any given sagittal notation
we can speak of its notational octave size and its notational fifth
size.

However, we do of course like to maintain the _order_ of symbol
sizes for the temperament so that they agree with the untempered
order (which is also the order of the symbols in the font).

In the case of untempered octaves we don't allow the notational
fifth size to go outside of the range from the fifth of 7-EDO
(685.714286 c) to that of 5-EDO (720 c). I don't believe we have
ever thought about how far we would let the octaves go. I guess
we'll worry about that if it ever becomes a problem.

2. It's not the complexity of the comma that matters here in
determining priority, but rather the complexity of the (simplest)
intervals that will be notated by using a symbol for that comma.

-- Dave Keenan

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

> In the case of untempered octaves we don't allow the notational
> fifth size to go outside of the range from the fifth of 7-EDO
> (685.714286 c) to that of 5-EDO (720 c). I don't believe we have
> ever thought about how far we would let the octaves go. I guess
> we'll worry about that if it ever becomes a problem.

Why is Sagittal only used with fifths? Since JI can be noted using
Sagittal symbols without making use of nominals, octaves or staff
lines at all, it can be noted using any given choice of nominals and
staff lines also.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@>
wrote:
>
> > In the case of untempered octaves we don't allow the notational
> > fifth size to go outside of the range from the fifth of 7-EDO
> > (685.714286 c) to that of 5-EDO (720 c). I don't believe we have
> > ever thought about how far we would let the octaves go. I guess
> > we'll worry about that if it ever becomes a problem.
>
> Why is Sagittal only used with fifths?

Clearly it isn't only used with fifths. Herman is using it otherwise.

Sagittal absorbs the tempering of the first two primes -- up to a
point. I spoke of octaves and fifths because that is the usual basis
for the first two primes, from a musical point of view, as opposed
to a mathematical one.

Ask an average musician what the two most consonant intervals are.
She will say "The octave [1:2] and the fifth [2:3]". You may
ask "What about the twelfth [1:3]?". She will likely say, "But
that's just octave-equivalent to a fifth."

If the first two primes are tempered too much, then the sizes of the
Sagittal comma symbols change so much that their size-ordering
becomes too different from their untempered order.

> Since JI can be noted using
> Sagittal symbols without making use of nominals,

I wasn't aware of that. Please explain.

> octaves or staff
> lines at all, it can be noted using any given choice of nominals
and
> staff lines also.

I certainly hope so.

But as to how much we have biased it towards chains of
fourths/fifths I won't really know until I make a concerted effort
to apply it otherwise myself.

I figure a tough challenge would be a temperament where there is no
fourth or fifth (685.7 to 720 c) between any of the (multi-)MOS
nominals. Can you supply such a temperament?

-- Dave Keenan

On 3/6/06, Dave Keenan <d.keenan@bigpond.net.au> wrote:
[...]
> I figure a tough challenge would be a temperament where there is no
> fourth or fifth (685.7 to 720 c) between any of the (multi-)MOS
> nominals. Can you supply such a temperament?

There are good {2,5,7} sytems that temper out 50/49 and 3136/3125.

The 50/49 one has TOP generator 213.14 and period 598.45, and the
3136/3125 one has TOP generator 195.19 and period 1199.74. Their
intersection is 6-edo, so they both have 6-note MOSs which can be
considered "whole tone" scales.

Keenan Pepper

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

> > Since JI can be noted using
> > Sagittal symbols without making use of nominals,
>
> I wasn't aware of that. Please explain.

Anything above the three-limit, and up to the point you quit adding
symbols for primes, can be notated using Sagittal symbols only. It's
not a practical idea, but possible in theory. The reason for this is
that the 5-limit can be notated using Sagittal symbols only, and the
p-limit symbols are all p-bridges.

In the 5-limit, if a is the apotome, s is the shisma, and d is the
comma of Didymus, we have:

2 = a^12 s^(-7) d^(-7)
3 = a^19 s^(-11) d^(-11)
5 = a^28 s^(-16) d^(-17)

Now by adding symbols for higher primes, we can notate those primes
also. For instnace, if c = 64/63, then

7 = a^34 s^(-20) d^(-20) c ^(-1)

> I figure a tough challenge would be a temperament where there is no
> fourth or fifth (685.7 to 720 c) between any of the (multi-)MOS
> nominals. Can you supply such a temperament?

If I understood what your question meant, I might.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
>
> > For period and generator (0, +1) you have 1323/1250; I found
21/20.
> > For (1, -2) you have 200/189; I found 28/27. You also need a
ratio
> > for (1, -3), which is the difference between the other two, or
81/80,
> > but, oops! -- the joke's on me, because 81/80 turned out to be (-
1,
> > 3). There's a problem in that the larger interval in the
temperament
> > is represented by the smaller ratio, so try again. :-(
>
> My preferred representation of (0, +1) (from the symbols
represented in
> the Sagittal font) is actually 15/14, although 21/20 is about as
good. I
> start by trying to find symbols in the standard font that are close
to
> the size of the intervals in the TOP tuning of the temperament (as
a
> somewhat arbitrary but generally useful convention). For (+1, -2) I
have
> 16/15, and 225/224 for (-1, +3). The basic idea is that, where
several
> different intervals are represented by the same number of
generators,
> I'd like to use the one that is around the right size. But I like
the
> idea of taking complexity into account and giving a higher priority
to
> the simpler ratios.

Yes, I agree that the simplicity of the ratios should have priority
over actual size.

I now agree that 16/15 is the best choice for (+1, -2), but I think
that 15/14 for (0, +1) and 225/224 for (-1, +3) make sense only if
the tone at (-1, +3) is not higher in pitch than the origin (0, 0),
which is not the case with 16-ET and 28-ET. For those I would now
suggest 21/20 for (0, +1) and 64/63 for (1, -3). I have a table
(below) showing what the symbols for each of these ratios would be.

Another observation I made is that for 20-ET, which is shown in the
dimisept horogram in Paul Erlich's middle-path paper, the tone
closest in pitch to the 3rd harmonic doesn't map to (+6, +1). Would
that, in your opinion, exclude 20-ET from the dimisept family of
temperaments, or should I think that this might be evidence of a
Dimisept family feud? ;-)

> I also considered 160/147 for (+1, -2), which was the best I could
find
> in the existing font that didn't need a schisma accent. I'd almost
like
> to say that the schisma accents could be dropped for these kind of
> temperaments, but I seem to remember a couple of cases where they
were
> needed.

I would also recommend that the simplicity of the symbol carry least
priority in determining the notational semantics for temperaments.
If you make it a point to choose simple ratios, then the meanings of
the symbols will be easier to interpret.

I need to add that I wrote the foregoing before I saw Dave Keenan's
following comment:

> 2. It's not the complexity of the comma that matters here in
> determining priority, but rather the complexity of the (simplest)
> intervals that will be notated by using a symbol for that comma.

I think that makes good sense. You'll have to look at how that
factors into all of this.

> > ... If you prefer a notation without any
> > accent marks, there's at least one way to do it (with promethean-
> > level symbols, where |\\ would represent 27:28 and the other two
> > symbols would remain the same).
>
> I've considered using 28/27 for pajara, semaphore, and possibly
> negrisept, so it'd be great to have an unaccented symbol for it!
Other
> accented symbols that come up frequently are 21/20 and 16/15. 27/25
and
> 128/125 could also have some uses.

Your desire for unaccented symbols brings up an interesting
question. Consider the following options:
a) a symbol set consisting entirely of unaccented symbols, vs.
b) a symbol set consisting of a fewer number of symbols, most of
which may be used in combination with left-accents?
Is a notation that uses symbols entirely from set a) necessarily
simpler and/or more desirable than one that uses symbols entirely
from set b)?

To answer this, I submit the following points in favor of set a)
and/or against set b):
1) There are no accents to be bothered with in a).
2) Symbols in b) could be misinterpreted and/or become invalid due to
the inadvertent or careless omission of accents.
3) Two symbols representing ratios differing by a 5-schisma would be
easier to tell apart (particularly when read in real time) in a),
since they would have different symbol cores.

(A "symbol core" is defined as a symbol shaft (or shafts) along with
its attached flags, considered apart from any accent marks.)

And these are the points in favor of set b) and/or against set a):
1) There are fewer symbol cores to memorize for option b) than for a).
2) Symbol cores used in a) frequently have primary roles that are
more obscure (and therefore less memorable) than the corresponding
ones used in b).
3) It's possible to notate more ratios exactly (i.e., in primary
roles) with b) than with a), which is another way of saying that b)
offers a higher resolution of pitch than a).
4) More of the most popular ratios may be notated exactly (in primary
roles) with b) than with a), making b) the "cleaner" (more rigorous)
option.
5) Shorthand notation for a) would require the use of extended ASCII
characters much more often than for b).
6) In those frequent cases where the set of b)-symbols selected to
notate a tuning does not contain two symbols with the same core
(i.e., differing only by an accent), it would not be necessary to
watch for accents when reading in real time.

My conclusion is that there are more advantages with option b) than
a).

Here's a table showing the symbols for each of the reasonably simple
(most popular) ratios we've mentioned, using the herculean (accented)
and promethean (unaccented) symbol set options:

ratio herculean promethean
224:225 '|( ~|
63:64 |) |)
48:49 ~|) ~|)
125:128 .//| ~|\
27:28 .(|\ |\\
20:21 .||) )/||
15:16 ./||\ (||~
14:15 |||( |||(
25:27 ./||| )|||~

Note that every single one of the herculean symbols has an athenian
symbol core (and regular ASCII shorthand), while 2/3 of the
promethean symbols have non-athean cores (and 1/3 would require
extended ASCII shorthand characters).

Herculean-level notation does not entirely avoid the more unusual
promethean symbols, but those come into play with increased ratio
complexity and/or prime limit.

--George

Dave Keenan wrote:

> I figure a tough challenge would be a temperament where there is no > fourth or fifth (685.7 to 720 c) between any of the (multi-)MOS > nominals. Can you supply such a temperament?

Father would be an obvious choice (TOP tuning <1185.869125, 1924.351908, 2819.124590], map <1, 2, 2], <0, -1, 1]), but it wouldn't be much of a loss if this couldn't be notated as a temperament, since it's only marginally one. (Otherwise it can be notated the same as sensipent up to 8 notes, which is about the limit of its usefulness.)

Mavila is on the edge, with TOP tuning 685.03c for the fifth, and 1206.55c for the octave. If you don't mind things like C-E being a minor third and D-F being a major third, you can use a chain of fourths to notate mavila.

The diminished temperaments have the issue that a chain of fourths notation by itself can only reach every other period, but that can be overcome by notating (-1, +3) as an accidental; 81/80 is one possibility consistent with diminished temperament, so one period above D would be notated F /| .

Dave Keenan wrote:
> A couple of things to note here:
> > 1. Rather than comparing the size of the needed alteration with the > size of the _untempered_ symbol commas, you can first temper the > factors of 2 and 3 in the symbol commas, according to the > temperament you're notating, (provided they are not _too_ heavily > tempered), _then_ look at the sizes. For any given sagittal notation > we can speak of its notational octave size and its notational fifth > size.

That's an interesting idea. Applying it to porcupine temperament gives 81/80 as the size of the (+1, -7) interval (60.68c in TOP tuning), and 135/128 for (+2, -14). So porcupine[15] with a chain of fourths notation would go like this:

D/| E F||\ G/| A B\! C/| D E\! F/| G A\! B!!/ C D\!

If you arrange these on a 5-limit grid you can see that this makes a lot of sense. I'm not sure if this would be the best idea if used with a specialized set of porcupine[7] or porcupine[8] nominals, though. Porcupine[8] can use 1280:1323 )/|\ as an accidental for (-1, +8), but this is really a 101.64 cent interval; 16/15 is what I find without accounting for the factors of 2 and 3.

George D. Secor wrote:
> Yes, I agree that the simplicity of the ratios should have priority > over actual size.
> > I now agree that 16/15 is the best choice for (+1, -2), but I think > that 15/14 for (0, +1) and 225/224 for (-1, +3) make sense only if > the tone at (-1, +3) is not higher in pitch than the origin (0, 0), > which is not the case with 16-ET and 28-ET. For those I would now > suggest 21/20 for (0, +1) and 64/63 for (1, -3). I have a table > (below) showing what the symbols for each of these ratios would be.

Hmm, it's unfortunate that an interval like (+1, -3) is positive in some tunings of the temperament and negative in others.

> Another observation I made is that for 20-ET, which is shown in the > dimisept horogram in Paul Erlich's middle-path paper, the tone > closest in pitch to the 3rd harmonic doesn't map to (+6, +1). Would > that, in your opinion, exclude 20-ET from the dimisept family of > temperaments, or should I think that this might be evidence of a > Dimisept family feud? ;-)

That's just the limit of consistency of that particular tuning of the temperament; (+5, +4) is a better 3/1 than (+6, +1), so any more than 16 notes is inconsistent. You could theoretically keep going 8, 12, 20, 32, 44, etc. The consistency is better around the size of 28-ET, which is one reason why TOP might not be the best tuning for this temperament. But short of a brute force computation of all the possible tunings up to a specified accuracy, I don't know how to find the tuning with the best consistency. If I could, possibly that would be the best reference tuning for a notation.

> I would also recommend that the simplicity of the symbol carry least > priority in determining the notational semantics for temperaments. > If you make it a point to choose simple ratios, then the meanings of > the symbols will be easier to interpret.
> > I need to add that I wrote the foregoing before I saw Dave Keenan's > following comment:
> > >>2. It's not the complexity of the comma that matters here in >>determining priority, but rather the complexity of the (simplest) >>intervals that will be notated by using a symbol for that comma.
> > > I think that makes good sense. You'll have to look at how that > factors into all of this.

Of course, that could result in different notations depending on the choice of nominals (since you'd have to check every possible nominal + accidental combination to find the simplest ratio).

> Your desire for unaccented symbols brings up an interesting > question. Consider the following options:
> a) a symbol set consisting entirely of unaccented symbols, vs.
> b) a symbol set consisting of a fewer number of symbols, most of > which may be used in combination with left-accents?
> Is a notation that uses symbols entirely from set a) necessarily > simpler and/or more desirable than one that uses symbols entirely > from set b)?

There probably aren't many cases where more than one pair of the ratios you'd want to use in notating a temperament are the same symbol with and without a schisma accent. There's a few cases with one pair that could use a schisma accent.

> To answer this, I submit the following points in favor of set a) > and/or against set b):
> 1) There are no accents to be bothered with in a).
> 2) Symbols in b) could be misinterpreted and/or become invalid due to > the inadvertent or careless omission of accents.
> 3) Two symbols representing ratios differing by a 5-schisma would be > easier to tell apart (particularly when read in real time) in a), > since they would have different symbol cores.

Note however that a 5-schisma is a complex interval and could end up being tempered quite severely (even negative in some cases).

> (A "symbol core" is defined as a symbol shaft (or shafts) along with > its attached flags, considered apart from any accent marks.)
> > And these are the points in favor of set b) and/or against set a):
> 1) There are fewer symbol cores to memorize for option b) than for a).

For any given temperament, there will be the same number of symbol combinations that have to be memorized; it's just that some of them will be accented symbols.

> 2) Symbol cores used in a) frequently have primary roles that are > more obscure (and therefore less memorable) than the corresponding > ones used in b).
> 3) It's possible to notate more ratios exactly (i.e., in primary > roles) with b) than with a), which is another way of saying that b) > offers a higher resolution of pitch than a).

But the extra intervals are likely to be well outside of the range that's useful to notate, since they differ by such a complex comma.

> 4) More of the most popular ratios may be notated exactly (in primary > roles) with b) than with a), making b) the "cleaner" (more rigorous) > option.

Most of the popular ratios (with a few exceptions like 16/15) don't need schisma accents to be notated exactly. It tends to be the more complex intervals, which are likely to be tempered to the point where they sound like some other interval anyway, that need schisma accents.

> 5) Shorthand notation for a) would require the use of extended ASCII > characters much more often than for b).
> 6) In those frequent cases where the set of b)-symbols selected to > notate a tuning does not contain two symbols with the same core > (i.e., differing only by an accent), it would not be necessary to > watch for accents when reading in real time.
> > My conclusion is that there are more advantages with option b) than > a).

That seems like a reasonable conclusion, although I'm not sure that the advantage is really all that much in the case of temperaments. I'd mainly want to leave the accents in for consistency, in the cases where I can't find unaccented symbols.

> Here's a table showing the symbols for each of the reasonably simple > (most popular) ratios we've mentioned, using the herculean (accented) > and promethean (unaccented) symbol set options:
> > ratio herculean promethean
> 224:225 '|( ~|
> 63:64 |) |)
> 48:49 ~|) ~|)
> 125:128 .//| ~|\
> 27:28 .(|\ |\\
> 20:21 .||) )/||
> 15:16 ./||\ (||~
> 14:15 |||( |||(
> 25:27 ./||| )|||~
> > Note that every single one of the herculean symbols has an athenian > symbol core (and regular ASCII shorthand), while 2/3 of the > promethean symbols have non-athean cores (and 1/3 would require > extended ASCII shorthand characters).

The unreserved ASCII symbols could be defined individually for each temperament if necessary.

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> On 3/6/06, Dave Keenan <d.keenan@...> wrote:
> [...]
> > I figure a tough challenge would be a temperament where there is no
> > fourth or fifth (685.7 to 720 c) between any of the (multi-)MOS
> > nominals. Can you supply such a temperament?
>
> There are good {2,5,7} sytems that temper out 50/49 and 3136/3125.
>
> The 50/49 one has TOP generator 213.14 and period 598.45, and the
> 3136/3125 one has TOP generator 195.19 and period 1199.74. Their
> intersection is 6-edo, so they both have 6-note MOSs which can be
> considered "whole tone" scales.

Thanks Keenan,

So I take it the intended mappings are:
2 3 5 7

<2 x 5 6] periods (598.45 c)
<0 x -1 -1] generators (213.14 c)

<1 x 2 2] periods (1199.74 c)
<0 x 2 5] generators (195.19 c)

(multi-)MOS cardinalities (improper):
twin 213c 6 (10 16 22) 28 (34 62 90) 118
195c 6 (7 13 19 25 31) 37 43

D-centered 24-EDO-based compound nominals (same for both temps):
O bB C D E #F where O = G#/Ab

Twin-chain temp has chains arranged:
C D E
#F O bB

We need to choose a mapping of prime 3 for notational purposes since symbols for 3-free commas are very rare in the sagittal system. One way to do this is to extend the set of nominals (i.e. extend the chain(s) of TOP generators and map them to their nearest 24-EDO degrees) until a fifth (or fourth) is notated by them (e.g. C:G, bE:bB, ^D:^A), even though we will not use that many nominals in the final system.

Note that when I talk of 24-EDO here, I'm actually using the 24-fold equal division of the TOP tempered octave.

By doing that, we find that the smallest absolute numbers of generators that give nominal-notatable fifths (or prime 3's) result in the following mappings:

<2 1 5 6] periods (598.45 c)
<0 6 -1 -1] generators (213.14 c)

<1 -1 2 2] periods (1199.74 c)
<0 16 2 5] generators (195.19 c)

Since this process allows fifths ranging from around 675c to around 725c you may choose to use a slightly more complex mapping of prime 3 to get better notational fifths. This generally corresponds to choosing between the standard mappings of various EDOs such as those with the cardinalities given as MOS above. For example, in the case of the 195 c temp 16 gens to prime-3 corresponds to 37-EDO, while -21 gens to the prime-3 corresponds to 43-EDO.

Existing notations for those ETs can provide a shortcut to finding valid symbols for accidentals. See the Xenharmonikon article on the Sagittal website. Unfortunately we haven't given any notation for 37-ET and we haven't given a native-fifth notation for 28-ET.

Now we can apply the above mappings to the 7-limit commas represented by the sagittal symbols, to find any in the right size range that correspond to the number of generators and periods required for the accidentals.

If you get desperate you might even choose to determine a mapping for the next highest prime and add 11-limit symbols to your list.

The smallest accidentals required when 6 nominals are used are:
twin 213c 40.97 c ( 3 gens -2 periods)
195c 28.60 c (-6 gens +1 period )

This one works for the 195c temp with the above 3-mapping.
~|) 49-small-diesis 48:49 [-4 -1 0 2> untempered size 35.7 c

I haven't found an unaccented 7-limit symbol that works for the twin 213c temp with the above 3-mapping.

Interestingly, this comma is valid for both temperaments and doesn't require you to specify a 3-mapping because it's 3-free.
.//| 125-small-diesis 125:128 [7 0 -3 0> untempered size 41.06 c
Unfortunately it has an accented symbol, which we'd generally like to avoid, except for very complex temperaments (or maybe very weird ones).

You will presumably want accidentals for twice and 3 times that amount as well.

The best I can find for the double accidental for twin 213c is
.||) 5:7-limma 20:21 [-2 1 -1 1>

If I had more time I'd looking at those other 3-mappings.

Yes. These are tough ones, but so are their representative ETs 28, 37, 43.

-- Dave Keenan

(Re {2,5,7} temperaments tempering out 50/49 and 3136/3125)

What about this?: The 50/49 one is every other note of pajara, and the
3136/3125 is every other note of meantone. Then they should be easy to
notate, right?

Keenan

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
> > Yes, I agree that the simplicity of the ratios should have
priority
> > over actual size.
> >
> > I now agree that 16/15 is the best choice for (+1, -2), but I
think
> > that 15/14 for (0, +1) and 225/224 for (-1, +3) make sense only
if
> > the tone at (-1, +3) is not higher in pitch than the origin (0,
0),
> > which is not the case with 16-ET and 28-ET. For those I would
now
> > suggest 21/20 for (0, +1) and 64/63 for (1, -3). I have a table
> > (below) showing what the symbols for each of these ratios would
be.
>
> Hmm, it's unfortunate that an interval like (+1, -3) is positive in
some
> tunings of the temperament and negative in others.

My feelings, exactly.

> > ...
> > I need to add that I wrote the foregoing before I saw Dave
Keenan's
> > following comment:
> >
> >>2. It's not the complexity of the comma that matters here in
> >>determining priority, but rather the complexity of the (simplest)
> >>intervals that will be notated by using a symbol for that comma.
> >
> > I think that makes good sense. You'll have to look at how that
> > factors into all of this.
>
> Of course, that could result in different notations depending on
the
> choice of nominals (since you'd have to check every possible
nominal +
> accidental combination to find the simplest ratio).

It makes me wonder whether it's advisable to specify "standard"
notations for some of these temperaments without actually attempting
to write some music for them.

> > Your desire for unaccented symbols brings up an interesting
> > question. Consider the following options:
> > a) a symbol set consisting entirely of unaccented symbols, vs.
> > b) a symbol set consisting of a fewer number of symbols, most of
> > which may be used in combination with left-accents?
> > Is a notation that uses symbols entirely from set a) necessarily
> > simpler and/or more desirable than one that uses symbols entirely
> > from set b)?
>
> There probably aren't many cases where more than one pair of the
ratios
> you'd want to use in notating a temperament are the same symbol
with and
> without a schisma accent. There's a few cases with one pair that
could
> use a schisma accent.

Yes, that's what I would expect.

> > ...
> > 3) Two symbols representing ratios differing by a 5-schisma would
be
> > easier to tell apart (particularly when read in real time) in a),
> > since they would have different symbol cores.
>
> Note however that a 5-schisma is a complex interval and could end
up
> being tempered quite severely (even negative in some cases).

Hmmm, in that case having two symbols differing only by a left-accent
could be downright confusing. That's an argument in favor of the
unaccented symbols, option a), one I hadn't thought of. Good call!

> > ...
> > My conclusion is that there are more advantages with option b)
than
> > a).
>
> That seems like a reasonable conclusion, although I'm not sure that
the
> advantage is really all that much in the case of temperaments. I'd
> mainly want to leave the accents in for consistency, in the cases
where
> I can't find unaccented symbols.

Okay, I just thought it would be good to present some of the pros and
cons.

> > Here's a table showing the symbols for each of the reasonably
simple
> > (most popular) ratios we've mentioned, using the herculean
(accented)
> > and promethean (unaccented) symbol set options:
> >
> > ratio herculean promethean
> > 224:225 '|( ~|
> > 63:64 |) |)
> > 48:49 ~|) ~|)
> > 125:128 .//| ~|\
> > 27:28 .(|\ |\\
> > 20:21 .||) )/||
> > 15:16 ./||\ (||~
> > 14:15 |||( |||(
> > 25:27 ./||| )|||~
> >
> > Note that every single one of the herculean symbols has an
athenian
> > symbol core (and regular ASCII shorthand), while 2/3 of the
> > promethean symbols have non-athean cores (and 1/3 would require
> > extended ASCII shorthand characters).
>
> The unreserved ASCII symbols could be defined individually for each
> temperament if necessary.

Yes, good point!

--George

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> But short of a brute force computation of all the possible tunings
up to
> a specified accuracy, I don't know how to find the tuning with the best
> consistency. If I could, possibly that would be the best reference
> tuning for a notation.

The first step is always definitional. What is your definition of "the
best consistency"?

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

> Yes. These are tough ones, but so are their representative ETs 28,
37, 43.

This seems like a good example of why paying attention to R2
temperaments is a good idea. For 28, the obvious R2 temperament is
diminished (dimisept.) For 37, the obvious temperament is
porcupine. For 43, the obvious temperament is of course meantone.

--- In tuning-math@yahoogroups.com, "Keenan Pepper"
<keenanpepper@...> wrote:
>
> (Re {2,5,7} temperaments tempering out 50/49 and 3136/3125)
>
> What about this?: The 50/49 one is every other note of pajara, and
the
> 3136/3125 is every other note of meantone. Then they should be
easy to
> notate, right?

If you're willing to do them that way, then yes!

This is exactly the question of choosing the best "notational fifth"
(or the notational mapping of the 3 prime).

What you're suggesting is what we call using a "non-native" fifth,
i.e. one that is not actually generated by the given generator, but
by some aliquot part of it, usually a half or a third. We often do
that with ETs when the native fifth(s) is(are) very far from just.
e.g. we suggest notating 28-ET as every other note of 56-ET (56-ET
being an ET that supports pajara).

So I guess the best advice for notating linear temperaments
remains, "First look at the standard notations for the ETs that
support it well".

-- Dave Keenan

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> > >>But short of a brute force computation of all the possible tunings
> > up to > >>a specified accuracy, I don't know how to find the tuning with the best >>consistency. If I could, possibly that would be the best reference >>tuning for a notation.
> > > The first step is always definitional. What is your definition of "the
> best consistency"?

In this context I'm defining consistency as not having a better mapping of one of the prime intervals in a contiguous chain of generators. Take for instance 1/4-comma meantone, with 503.42c fourths and 1200.00c octaves. After (+2, -1), you don't get a better mapping of 3 until you get to (-11, +30). So you can have up to 30 notes in a chain of fourths without getting a better mapping; the consistency of this tuning in the 5-limit is 30. TOP meantone has better consistency in the 5-limit; in this case it's the mapping of 5/1 at (+17, -35) that breaks the consistency. The tuning with the best consistency would be the one with the most notes in the chain of fourths that doesn't have a better mapping of any of the prime intervals (up to the specified prime limit).

George D. Secor wrote:

> It makes me wonder whether it's advisable to specify "standard" > notations for some of these temperaments without actually attempting > to write some music for them.

Yes; well, as long as we have a system without any ambiguous intervals, different composers could choose whatever notations make sense to them without possibility of confusion. That's why the schismina situation still bugs me a little, although situations where it could cause a problem are rare. It might happen that it only makes sense to use 243:250 as an accidental in 5-limit temperaments such as srutal, hanson, or superpyth, where it can't conflict with 35:36 or 1024:1053. But without testing a bunch of temperaments, it's hard to know how likely this situation is. So far it hasn't seemed to be a problem....

It would be nice to have lots of actual musical examples to test the notation with, but getting comfortable with composing in a temperament isn't easy without a convenient notation to work with. I've got my own ad hoc notations that need a chart in order to understand what's going on. Using a piano roll is one possibility for composing, but you can't easily look at that and transcribe from it, since all the rows look alike. Of course, you have the same problem if you're trying to make a MIDI file from a notated score....

And if you want to document the sorts of things that are interesting and effective about a temperament, to encourage the writing of more music, it helps to have some way to notate it in a way that makes sense to musicians....

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Dave Keenan <d.keenan@...> wrote:
> > >>Yes. These are tough ones, but so are their representative ETs 28,
> > 37, 43.
> > This seems like a good example of why paying attention to R2
> temperaments is a good idea. For 28, the obvious R2 temperament is
> diminished (dimisept.)

Good point, although diminished isn't a very nice one to notate. Another possibility is negri, with a generator of 3 steps of 28-ET, notating (+1, -9) and (-1, +10) as accidentals. (+1, -9) could be notated 128/125 or 135/128, and (-1, +10) as 25/24.

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

> So I guess the best advice for notating linear temperaments
> remains, "First look at the standard notations for the ETs that
> support it well".

I keep thinking that's backwards; linear temperaments should be
notated first, and ets afterwards.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
>
> > So I guess the best advice for notating linear temperaments
> > remains, "First look at the standard notations for the ETs that
> > support it well".
>
> I keep thinking that's backwards; linear temperaments should be
> notated first, and ets afterwards.

The problem with that view is that most ET's are members of more than
one linear-temperament family. For example, 19-ET may be interpreted
as being generated either by 5ths (meantone family), by major 3rds,
by minor 3rds, by hemi-fourths, etc.

I think Dave was making the statement in the context of notating
linear temperaments using nominals in a chain of 5ths. Granted
that's not always the "best" way to do it for a specific purpose, but
it's a relatively simple approach if you need a single "standard way"
of notating an ET for a performer.

--George

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

> The problem with that view is that most ET's are members of more than
> one linear-temperament family. For example, 19-ET may be interpreted
> as being generated either by 5ths (meantone family), by major 3rds,
> by minor 3rds, by hemi-fourths, etc.

I'd pick the one which most closely corresponds to the music I'm
writing, except for meantone systems like 19, where I'd just use
standard notation.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@>
wrote:
> >
> > > So I guess the best advice for notating linear temperaments
> > > remains, "First look at the standard notations for the ETs
that
> > > support it well".

> I think Dave was making the statement in the context of notating
> linear temperaments using nominals in a chain of 5ths.

No. I was definitely considering other nominal chains. If you need
an accidental corresponding to a specific number of degrees of the
representative ET, it doesn't matter what your nominals are. The
only thing to watch out for is the case where, for example you need
an accidental corresponding to say 1 degree of 31-ET and it has to
correspond to say 7 generators, but when you apply the mapping to
the comma used to notate one degree of 31-ET, it turns out to be not
6 generators but 37 generators, or -25 generators. i.e. some number
that is the same modulo 31, but not the right number when
considering the open-chain linear temperament.

For Miracle temperament you certainly wouldn't want to use anything
other than the symbol for 2 degrees of 72-ET, the 7-comma (63:64)
symbol |) , as the first accidental for notating Miracle with a
chain of 10 nominals.

-- Dave

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
> > > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@>
wrote:
> > >
> > > > So I guess the best advice for notating linear temperaments
> > > > remains, "First look at the standard notations for the ETs
that
> > > > support it well".
>
> > I think Dave was making the statement in the context of notating
> > linear temperaments using nominals in a chain of 5ths.
>
> No. I was definitely considering other nominal chains. If you need
> an accidental corresponding to a specific number of degrees of the
> representative ET, it doesn't matter what your nominals are. The
> only thing to watch out for is the case where, for example you need
> an accidental corresponding to say 1 degree of 31-ET and it has to
> correspond to say 7 generators, but when you apply the mapping to
> the comma used to notate one degree of 31-ET, it turns out to be
not
> 6 generators but 37 generators, or -25 generators. i.e. some number
> that is the same modulo 31, but not the right number when
> considering the open-chain linear temperament.
>
> For Miracle temperament you certainly wouldn't want to use anything
> other than the symbol for 2 degrees of 72-ET, the 7-comma (63:64)
> symbol |) , as the first accidental for notating Miracle with a
> chain of 10 nominals.

Yes, I agree. However:

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
>
> > The problem with that view is that most ET's are members of more
than
> > one linear-temperament family. For example, 19-ET may be
interpreted
> > as being generated either by 5ths (meantone family), by major
3rds,
> > by minor 3rds, by hemi-fourths, etc.
>
> I'd pick the one which most closely corresponds to the music I'm
> writing, except for meantone systems like 19, where I'd just use
> standard notation.

So if the performers in a hypothetical 31-ET ensemble were looking
for music to perform, and someone pointed out that it's possible to
play a blackjack piece in 31-ET, do you think the performers would
prefer that the parts be notated in a miracle notation or in a
meantone-plus-semi-apotome notation?

Or suppose that the 31-ET piece were written in such a way as to
contain passages that were distinctly "miracle" and others that were
distinctly heptatonic? Should we confuse the performers by changing
the basis for the notation every few measures?

--George

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

> > I'd pick the one which most closely corresponds to the music I'm
> > writing, except for meantone systems like 19, where I'd just use
> > standard notation.
>
> So if the performers in a hypothetical 31-ET ensemble were looking
> for music to perform, and someone pointed out that it's possible to
> play a blackjack piece in 31-ET, do you think the performers would
> prefer that the parts be notated in a miracle notation or in a
> meantone-plus-semi-apotome notation?

They would almost certainly be used to using meantone plus something
or other, and would prefer sticking to it, which is what I suggested.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:
> So if the performers in a hypothetical 31-ET ensemble were looking
> for music to perform, and someone pointed out that it's possible to
> play a blackjack piece in 31-ET, do you think the performers would
> prefer that the parts be notated in a miracle notation or in a
> meantone-plus-semi-apotome notation?
>
> Or suppose that the 31-ET piece were written in such a way as to
> contain passages that were distinctly "miracle" and others that were
> distinctly heptatonic? Should we confuse the performers by changing
> the basis for the notation every few measures?

In this case, the composer might well use a decimal miracle notation
for composing or analysing, but provide 31-ET notation for performers.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
> wrote:
> > So if the performers in a hypothetical 31-ET ensemble were
looking
> > for music to perform, and someone pointed out that it's possible
to
> > play a blackjack piece in 31-ET, do you think the performers
would
> > prefer that the parts be notated in a miracle notation or in a
> > meantone-plus-semi-apotome notation?
> >
> > Or suppose that the 31-ET piece were written in such a way as to
> > contain passages that were distinctly "miracle" and others that
were
> > distinctly heptatonic? Should we confuse the performers by
changing
> > the basis for the notation every few measures?
>
> In this case, the composer might well use a decimal miracle
notation
> for composing or analysing, but provide 31-ET notation for
performers.

My thoughts, exactly. A real boon to the composer would be notation
software that would allow non-standard staves, nominals, and
accidentals, which could translate and print parts for other
configurations and/or tunings. (Dream on!)

--George