Proposed Sagittal notation for orwell

The generator mapping of orwell is [<1, 0, 3, 1], <0, 7, -3, 8]>. Here is a sequence of orwell[31] arranged in 7 columns, so a fifth (-1, +7) is one step down, and a major third (+1, -3) is 3 steps to the left. You can see that the flag arithmetic for |) and /| works out nicely.

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

This can be extended in the obvious way, but with some gaps in the notation for orwell[53]. These gaps can be filled by using the apotome complement of /|) , which is (|\ . (Is there a convenient table of apotome complements somewhere?)

Summary of accidentals:

|) [6, -2, 0 -1> (+5, -22)
/| [-4, 4, -1> (-7, +31)
/|) [2, 2, -1, -1> (-2, +9)
(|\ [-13, 5, 1, 1> (-9, +40)
)||( [-3, -1, 2> (+3, -13)
||\ [-7, 3, 1> (-4, +18)

Follow this link for an illustration of one octave of an orwell[53] keyboard with the keys labeled in the suggested notation.

http://www.io.com/~hmiller/png/orwell-notation.png

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> ... (Is there a convenient table of
> apotome complements somewhere?)

Yep. The XH18 Sagittal paper
http://dkeenan.com/sagittal/Sagittal.pdf
has a complete list in Figure 13 (p. 22). Also, Table I (p. 8) has a
partial list of apotome complements.

The )/|\ symbol is self-complementing at sub-olympian JI levels of
precision. At the olympian (extreme precision) level the )/|\' is
the complement of )/|\; this is the only instance in which an
accented symbol is the complement of an unaccented one.

Herman, I believe that either you or Graham mentioned that you would
like unaccented symbols for ratios such as 21/20 and 28/27. The
promethean symbol set (which is the entire set of unaccented symbols)
notates these using the following symbols (in secondary roles):

Ratio Herculean Promethean
28/27 .(|\ |\\
21/20 .||) = #.!) )/||\ = #)!)
225/224 '|( ~|

When a symbol is used in a secondary role to notate a ratio, it's
notating a ratio different from the one that (strictly) defines the
symbol. In each of the above instances, the unaccented symbol is
being re-used for ratios that happen to be the *simplest ones* that
fall within the promethean-level symbol boundaries and are therefore
more likely to be the ratios intended (particularly if the
application is not above the 13 prime limit).

If I'm not mistaken, Scala gives the promethean-level JI notation
with the command "set notation saji2", so you can try these out.
(Note: the Sag_ji2.par file that this uses is not the latest; Dave &
I will be releasing new ji.par files shortly).

Whether or not you might want to use these symbols in this way
depends on several issues:

1) Are a larger number of unaccented symbols (promethean set) easier
to remember and use (i.e., simpler) than a smaller number (subset of
the herculean set) of symbol cores (symbols disregarding the accents)
plus up/down left accents?

2) Are you frequently going to refer to the full set of olympian
(right-accented) symbols? If so, then it's counterproductive to use
the promethean symbols for JI. Besides, failing to notice a left-
accent mark when reading JI symbols would result in an error of only
2 cents, which is not too significant. (The fruit of our latest
effort, soon to be released, includes a no-brainer method to convert
olympian-level to herculean-level symbols: simply drop the right
accents.)

3) However, if you're using the symbols to notate temperaments, then
you might not want accented symbols, since missing an accent could
result in an error somewhat greater than 2 cents. At least, you
would not want two different versions (e.g., both unaccented and
accented) of the same symbol core in your symbol set, if that could
be avoided, so it would be possible to have a few accented symbols
without the problem of misreading accents.

--George

George D. Secor wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>> ... (Is there a convenient table of >> apotome complements somewhere?)
> > Yep. The XH18 Sagittal paper
> http://dkeenan.com/sagittal/Sagittal.pdf
> has a complete list in Figure 13 (p. 22). Also, Table I (p. 8) has a > partial list of apotome complements.

I was hoping for a more recent one. Besides )/|\ , are there any new pairs outside of the list in Sagittal.pdf?

> Herman, I believe that either you or Graham mentioned that you would > like unaccented symbols for ratios such as 21/20 and 28/27. The > promethean symbol set (which is the entire set of unaccented symbols) > notates these using the following symbols (in secondary roles):
> > Ratio Herculean Promethean
> 28/27 .(|\ |\\
> 21/20 .||) = #.!) )/||\ = #)!)
> 225/224 '|( ~|

Perfect! I assume that )/||\ is meant to be )/|| ? As |\\ and )/|| are ordinarily 19-limit symbols, and the usual use for .(|\ and .||) will be for 7- and 11-limit temperaments, there shouldn't be much chance for confusion. I run across a need for G.||) frequently, and A.(|\ also comes up.

> Whether or not you might want to use these symbols in this way > depends on several issues:
> > 1) Are a larger number of unaccented symbols (promethean set) easier > to remember and use (i.e., simpler) than a smaller number (subset of > the herculean set) of symbol cores (symbols disregarding the accents) > plus up/down left accents?

I think the main use of unaccented (|\ and ||) is likely to be as apotome complements of /|) and |) , which will mainly be needed in larger temperaments. They can be replaced with mixed sagittal notation if the number of symbols becomes a problem. More frequently, I have use for .(|\ and especially .||) . Other accented symbols, such as .//| , do come up in some temperaments, but the unaccented //| is needed in others.

> 2) Are you frequently going to refer to the full set of olympian > (right-accented) symbols? If so, then it's counterproductive to use > the promethean symbols for JI. Besides, failing to notice a left-
> accent mark when reading JI symbols would result in an error of only > 2 cents, which is not too significant. (The fruit of our latest > effort, soon to be released, includes a no-brainer method to convert > olympian-level to herculean-level symbols: simply drop the right > accents.)

I haven't had much need for right-accented symbols. But then, I haven't done anything with high-limit JI.

> 3) However, if you're using the symbols to notate temperaments, then > you might not want accented symbols, since missing an accent could > result in an error somewhat greater than 2 cents. At least, you > would not want two different versions (e.g., both unaccented and > accented) of the same symbol core in your symbol set, if that could > be avoided, so it would be possible to have a few accented symbols > without the problem of misreading accents.

Technically .X\ could represent 2 steps of 10-ET, and X\ could represent 3 steps, a difference of 120 cents! That's an unlikely case, of course, but it does illustrate the point. Another issue that might come up is that accents might point in the wrong direction in some temperaments.

--- 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 convenient table of
> >> apotome complements somewhere?)
> >
> > Yep. The XH18 Sagittal paper
> > http://dkeenan.com/sagittal/Sagittal.pdf
> > has a complete list in Figure 13 (p. 22). Also, Table I (p. 8)
has a
> > partial list of apotome complements.
>
> I was hoping for a more recent one.

Figure 13 is the final list of unaccented symbols and complements.

> Besides )/|\ , are there any new
> pairs outside of the list in Sagittal.pdf?

)/|\ is in Figure 13, at the far right, where its position indicates
it to be self-complementing. It's defined as 392:405, ~56.482c,
which is <0.4c from 1/2 apotome.

> > Herman, I believe that either you or Graham mentioned that you
would
> > like unaccented symbols for ratios such as 21/20 and 28/27. The
> > promethean symbol set (which is the entire set of unaccented
symbols)
> > notates these using the following symbols (in secondary roles):
> >
> > Ratio Herculean Promethean
> > 28/27 .(|\ |\\
> > 21/20 .||) = #.!) )/||\ = #)!)
> > 225/224 '|( ~|
>
> Perfect! I assume that )/||\ is meant to be )/|| ?

Yes.

> As |\\ and )/|| are
> ordinarily 19-limit symbols, and the usual use for .(|\ and .||)
will be
> for 7- and 11-limit temperaments, there shouldn't be much chance
for
> confusion.

Yes, all of these less common unaccented symbols have higher prime-
limit definitions, so if your tuning or temperament is at a lower
prime limit, then these symbols will default to lower-prime
(secondary) roles.

> I run across a need for G.||) frequently, and A.(|\ also
> comes up.
>
> > Whether or not you might want to use these symbols in this way
> > depends on several issues:
> >
> > 1) Are a larger number of unaccented symbols (promethean set)
easier
> > to remember and use (i.e., simpler) than a smaller number (subset
of
> > the herculean set) of symbol cores (symbols disregarding the
accents)
> > plus up/down left accents?
>
> I think the main use of unaccented (|\ and ||) is likely to be as
> apotome complements of /|) and |) , which will mainly be needed in
> larger temperaments. They can be replaced with mixed sagittal
notation
> if the number of symbols becomes a problem. More frequently, I have
use
> for .(|\ and especially .||) . Other accented symbols, such
as .//| , do
> come up in some temperaments, but the unaccented //| is needed in
others.

Okay, so I'll expand and sort (and correct) that table to read:

Ratio Herculean Promethean
------- ----------- -------------
225/224 '|( ~|
2835/2816 ')|( )~|
2048/2025 ./| )|~
3645/3584 '|) )|)
128/125 .//| ~|\\
59049/57344 '/|) (|~
28/27 .(|\ |\\
21/20 .||) = #.!) )/|| = #)!)

There are also some others that I should mention:
)| for 32805/32768 (5-schisma)
)~| for 2835/2816
~|( for 126/125 or 245/243
|~ for 413343/409600 or 105/104
~~| for 100/99 (it's defined as 99/98)
)/| for 531441/524288 (pythagorean comma)
(/| is defined as 405/392
|\) for 512/495 (defined as 8680203/8388608)

These 3-flag symbols (apotome complements) could also come in handy
for notating temperaments:
)//| is defined as 416/405
)|\\ is defined as 885735/851968

This will all be in our soon-to-be-released documentation.

> > 2) Are you frequently going to refer to the full set of olympian
> > (right-accented) symbols? If so, then it's counterproductive to
use
> > the promethean symbols for JI. Besides, failing to notice a left-
> > accent mark when reading JI symbols would result in an error of
only
> > 2 cents, which is not too significant. (The fruit of our latest
> > effort, soon to be released, includes a no-brainer method to
convert
> > olympian-level to herculean-level symbols: simply drop the right
> > accents.)
>
> I haven't had much need for right-accented symbols. But then, I
haven't
> done anything with high-limit JI.

They're handy for electronic music applications, particularly if you
want to transfer something notated in JI to a temperament. Small
differences in JI may translate to different tones in a temperament
(as I noted in the following).

> > 3) However, if you're using the symbols to notate temperaments,
then
> > you might not want accented symbols, since missing an accent
could
> > result in an error somewhat greater than 2 cents. At least, you
> > would not want two different versions (e.g., both unaccented and
> > accented) of the same symbol core in your symbol set, if that
could
> > be avoided, so it would be possible to have a few accented
symbols
> > without the problem of misreading accents.
>
> Technically .X\ could represent 2 steps of 10-ET, and X\ could
represent
> 3 steps, a difference of 120 cents! That's an unlikely case, of
course,
> but it does illustrate the point. Another issue that might come up
is
> that accents might point in the wrong direction in some
temperaments.

Yes, I've seen numerous examples of that, e.g., '| is -2deg26.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
> > Besides )/|\ , are there any new
> > pairs outside of the list in Sagittal.pdf?
>
> )/|\ is in Figure 13, at the far right, where its position indicates
> it to be self-complementing. It's defined as 392:405, ~56.482c,
> which is <0.4c from 1/2 apotome.

Herman or Graham, Sagittal.pdf was updated at the same time as the
font and character map spreadsheet were updated to version 2. You may
be reading an old version offline. Just google "sagittal notation" for
the latest.

> Okay, so I'll expand and sort (and correct) that table to read:
>
> Ratio Herculean Promethean
> ------- ----------- -------------
> 225/224 '|( ~|
> 2835/2816 ')|( )~|
> 2048/2025 ./| )|~
> 3645/3584 '|) )|)
> 128/125 .//| ~|\\
> 59049/57344 '/|) (|~
> 28/27 .(|\ |\\
> 21/20 .||) = #.!) )/|| = #)!)

George, there is no ~|\\ symbol.

> There are also some others that I should mention:
> )| for 32805/32768 (5-schisma)
> )~| for 2835/2816
> ~|( for 126/125 or 245/243
> |~ for 413343/409600 or 105/104
> ~~| for 100/99 (it's defined as 99/98)
> )/| for 531441/524288 (pythagorean comma)
> (/| is defined as 405/392
> |\) for 512/495 (defined as 8680203/8388608)

I don't think you made it clear in all cases above, when you were
giving a secondary role and if so, what the primary role was.

>
> These 3-flag symbols (apotome complements) could also come in handy
> for notating temperaments:
> )//| is defined as 416/405
> )|\\ is defined as 885735/851968

These are both primary roles. I think 3-flaggers should be avoided if
possible.

> This will all be in our soon-to-be-released documentation.

The primary roles of all the the symbols in the two lists above are
already given in the character map spreadsheet on the sagittal website.

-- Dave Keenan

George D. Secor wrote:
> 128/125 .//| ~|\\

Should that be ~|\ ?

> (/| is defined as 405/392

Wasn't that )/|\ ? I had 4096/3969 (or alternatively, 3720087/3604480) for (/| . Or did you settle on 1323/1280 for )/|\ ?

>> I haven't had much need for right-accented symbols. But then, I > haven't >> done anything with high-limit JI.
> > They're handy for electronic music applications, particularly if you > want to transfer something notated in JI to a temperament. Small > differences in JI may translate to different tones in a temperament > (as I noted in the following).

Yes, that's why I said "much" need (rather than "any" need). The issue comes up when working with Csound, but I rarely do much with that.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@>
wrote:
> ...
> > Okay, so I'll expand and sort (and correct) that table to read:
> >
> > Ratio Herculean Promethean
> > ------- ----------- -------------
> > 225/224 '|( ~|
> > 2835/2816 ')|( )~|
> > 2048/2025 ./| )|~
> > 3645/3584 '|) )|)
> > 128/125 .//| ~|\\
> > 59049/57344 '/|) (|~
> > 28/27 .(|\ |\\
> > 21/20 .||) = #.!) )/|| = #)!)
>
> George, there is no ~|\\ symbol.

Sorry, I was in too much of a hurry. That should have been:

Ratio Herculean Promethean
------- ----------- -------------
128/125 .//| ~|\

Anyway, I'm glad to see that Herman figured it out.

> > There are also some others that I should mention:
> > )| for 32805/32768 (5-schisma)
> > )~| for 2835/2816
> > ~|( for 126/125 or 245/243
> > |~ for 413343/409600 or 105/104
> > ~~| for 100/99 (it's defined as 99/98)
> > )/| for 531441/524288 (pythagorean comma)
> > (/| is defined as 405/392
> > |\) for 512/495 (defined as 8680203/8388608)
>
> I don't think you made it clear in all cases above, when you were
> giving a secondary role and if so, what the primary role was.

They're all secondary, except where the word "defined" is used.

> > These 3-flag symbols (apotome complements) could also come in
handy
> > for notating temperaments:
> > )//| is defined as 416/405
> > )|\\ is defined as 885735/851968
>
> These are both primary roles. I think 3-flaggers should be avoided
if
> possible.

Of course.

> > This will all be in our soon-to-be-released documentation.
>
> The primary roles of all the the symbols in the two lists above are
> already given in the character map spreadsheet on the sagittal
website.

Okay, I wasn't sure exactly what all was there, since I rarely visit
that website. :-)

--George

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
> > 128/125 .//| ~|\\
>
> Should that be ~|\ ?

Yep, sorry again.

> > (/| is defined as 405/392
>
> Wasn't that )/|\ ?

Oh, golly, you're right again. I've been working on a highly
detailed diagram of Sagittal symbols over the past couple of weeks,
and I think my eyes are starting to tell me that I've had enough of
this for a while.

I had 4096/3969 (or alternatively, 3720087/3604480)
> for (/| .

4096/3969 is correct.

Or did you settle on 1323/1280 for )/|\ ?

No, that ratio gets )/|\' -- the apotome-complement.

--George

George D. Secor wrote:

> ~~| for 100/99 (it's defined as 99/98)

Hmm, I'm looking at possible notation for 11-limit miracle and noticed a problem with this. 100/99 appears to be the most accurate and simplest Sagittal notation for (+4, -41) (TOP 16.98 cents). However, 99/98 is the most accurate notation for (-3, +31) (TOP 16.45 cents). Of course, these are so close to the same size that no one will notice, and they're the same in 72-ET. Both of them turn out to be needed for notating miracle[72] from a chain of 13 fifths G/||\ to A\!!/ -- (-3, +31) for four pitches, which can also be notated F/| C/| G/| D/| , and (+4, -41) for one pitch, A)|( . It's that A)|( that's the biggest problem here: the JI value of )|( is only 9.69 cents (much smaller than 16.98), and the other alternatives are the even smaller |( and the ambiguous ~|( .

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
>
> > ~~| for 100/99 (it's defined as 99/98)
>
> Hmm, I'm looking at possible notation for 11-limit miracle and
noticed a
> problem with this. 100/99 appears to be the most accurate and
simplest
> Sagittal notation for (+4, -41) (TOP 16.98 cents). However, 99/98
is the
> most accurate notation for (-3, +31) (TOP 16.45 cents). Of course,
these
> are so close to the same size that no one will notice, and they're
the
> same in 72-ET. Both of them turn out to be needed for notating
> miracle[72] from a chain of 13 fifths G/||\ to A\!!/ -- (-3, +31)
for
> four pitches, which can also be notated F/| C/| G/| D/| , and (+4, -
41)
> for one pitch, A)|( . It's that A)|( that's the biggest problem
here:
> the JI value of )|( is only 9.69 cents (much smaller than 16.98),
and
> the other alternatives are the even smaller |( and the ambiguous ~|
( .

Herman, you've pointed up a problem we had hoped wouldn't come up
this soon. In an effort to keep the number of Sagittal symbols
manageable, Dave & I decided to draw the line at a place where
dropping the right accents would make a simple conversion of any
olympian-level (233-EDA, meaning equal division of the apotome into
233 equal parts or "minas") symbol to herculean level (roughly 58-
EDA, with modifications). This meant that only in a few instances
would we have two different symbols within the same mina (i.e.,
within the same 233-EDA boundaries). The upshot of all this is that
99:100 (which falls within the same mina as 98:99) doesn't have its
own symbol, so it will be notated by ~~|, which is defined as 98:99.

We do happen to have, however, in our back pocket a
separate "undocumented" symbol |~'' for 99:100 if you absolutely
cannot live without it. We didn't want to make this one official for
the simple reason that dropping the double-right accent mark then
makes |~ (defined as 729:736) the herculean-level symbol for 99:100,
which is much less accurate -- and less desirable, since we expect
that olympian-level JI notation is not likely to be used as often as
the simpler herculean or promethean levels of JI.

The reason for defining ~~| as 98:99 rather than 99:100 becomes
apparent once you look at how those ratios map to temperaments. In
the "best" octave divisions where the ~~| symbol is useful (e.g.,
198, 342, 388, 494, 518, 612, 742, 764) those two ratios map to the
same number of degrees, but in most of the "best" divisions where the
~~| symbol is useful but the number of degrees is different (e.g.,
301, 311, 441, 525, 653, 665) 99:100 generally maps to the same
number of degrees as 729:736; so it's appropriate for 99:100 to get
the symbol defined by 726:736. A second reason is that |~'' should
be smaller than ~~|, since the sum of the core |~ and the double-
right accent |'' (17.377c) is less than the sum of ~| and ~|
(17.459c).

Now I'm hoping that you're not going to tell me next that you need a
separate Sagittal symbol to distinguish 28431:28672 from 4096:4131
(although one is possible) or 6075:7272 from 1048576:1082575 (the
latter not so scary once you realize it's the apotome-complement of
495:512; however no separate symbol is possible). :-)

--George

George D. Secor wrote:

> We do happen to have, however, in our back pocket a > separate "undocumented" symbol |~'' for 99:100 if you absolutely > cannot live without it. We didn't want to make this one official for > the simple reason that dropping the double-right accent mark then > makes |~ (defined as 729:736) the herculean-level symbol for 99:100, > which is much less accurate -- and less desirable, since we expect > that olympian-level JI notation is not likely to be used as often as > the simpler herculean or promethean levels of JI.

Since 100/99 is only needed for one note near each end of a long chain of miracle, it might be best to avoid this problem by using the cross-shaft symbol X\ (295245/262144) and writing BY/ instead of A)|( or A|~'' . So a 73-note chain of miracle (from which either A\!!/ or G/||\ can be excluded to represent miracle[72]) might look like this (note the repeated notes between the end of one line and the start of the next):

........................ 36 A\!!/ 43 BY/ 50 B!!) 57 C\!/
57 C\!/ 64 C)||( 71 D\! 06 E\!!/ 13 F!!/ 20 F|) 27 G\!/
27 G\!/ 34 G)||( 41 A\! 48 B\!!/ 55 C!!/ 62 C|) 69 D\!/
69 D\!/ 04 D)||( 11 E\! 18 F 25 G!!/ 32 G|) 39 A\!/
39 A\!/ 46 A)||( 53 B\! 60 C 67 D!!/ 02 D|) 09 E\!/
09 E\!/ 16 F!) 23 F||\ 30 G 37 A!!/ 44 A|) 51 B\!/
51 B\!/ 58 C!) 65 C||\ 00 D 07 E!!/ 14 E|) 21 F/|\
21 F/|\ 28 G!) 35 G||\ 42 A 49 B!!/ 56 B|) 63 C/|\
63 C/|\ 70 D!) 05 D||\ 12 E 19 F/| 26 G)!!( 33 G/|\
33 G/|\ 40 A!) 47 A||\ 54 B 61 C/| 68 D)!!( 03 D/|\
03 D/|\ 10 E!) 17 E||\ 24 F/||\ 31 G/| 38 A)!!( 45 A/|\
45 A/|\ 52 B!) 59 B||\ 66 C/||\ 01 D/| 08 E)!!( 15 E/|\
15 E/|\ 22 F||) 29 FX\ 36 G/||\

Here, B!!) and F||) are using the apotome complements of |) although ~||( might be a better match ... the fact that B!!) : F|) represents a fifth makes this easier to remember. Also, B!!) matches the 72-ET notation. The rest of the notation, with the exception of X\ and )||( , also happens to match the 72-ET notation.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
>
> > We do happen to have, however, in our back pocket a
> > separate "undocumented" symbol |~'' for 99:100 if you absolutely
> > cannot live without it. We didn't want to make this one official
for
> > the simple reason that dropping the double-right accent mark then
> > makes |~ (defined as 729:736) the herculean-level symbol for
99:100,
> > which is much less accurate -- and less desirable, since we
expect
> > that olympian-level JI notation is not likely to be used as often
as
> > the simpler herculean or promethean levels of JI.
>
> Since 100/99 is only needed for one note near each end of a long
chain
> of miracle, it might be best to avoid this problem by using the
> cross-shaft symbol X\ (295245/262144) and writing BY/ instead of A)|
( or
> A|~'' . So a 73-note chain of miracle (from which either A\!!/ or
G/||\
> can be excluded to represent miracle[72]) might look like this
(note the
> repeated notes between the end of one line and the start of the
next):
>
> ........................ 36 A\!!/ 43 BY/ 50 B!!) 57 C\!/
> 57 C\!/ 64 C)||( 71 D\! 06 E\!!/ 13 F!!/ 20 F|) 27 G\!/
> 27 G\!/ 34 G)||( 41 A\! 48 B\!!/ 55 C!!/ 62 C|) 69 D\!/
> 69 D\!/ 04 D)||( 11 E\! 18 F 25 G!!/ 32 G|) 39 A\!/
> 39 A\!/ 46 A)||( 53 B\! 60 C 67 D!!/ 02 D|) 09 E\!/
> 09 E\!/ 16 F!) 23 F||\ 30 G 37 A!!/ 44 A|) 51 B\!/
> 51 B\!/ 58 C!) 65 C||\ 00 D 07 E!!/ 14 E|) 21 F/|\
> 21 F/|\ 28 G!) 35 G||\ 42 A 49 B!!/ 56 B|) 63 C/|\
> 63 C/|\ 70 D!) 05 D||\ 12 E 19 F/| 26 G)!!( 33 G/|\
> 33 G/|\ 40 A!) 47 A||\ 54 B 61 C/| 68 D)!!( 03 D/|\
> 03 D/|\ 10 E!) 17 E||\ 24 F/||\ 31 G/| 38 A)!!( 45 A/|\
> 45 A/|\ 52 B!) 59 B||\ 66 C/||\ 01 D/| 08 E)!!( 15 E/|\
> 15 E/|\ 22 F||) 29 FX\ 36 G/||\
>
> Here, B!!) and F||) are using the apotome complements of |)
although
> ~||( might be a better match ... the fact that B!!) : F|)
represents a
> fifth makes this easier to remember. Also, B!!) matches the 72-ET
> notation. The rest of the notation, with the exception of X\ and )||
( ,
> also happens to match the 72-ET notation.

Hi Herman,

The simplest ratio that tone 26, shown as G)!!(, makes with tone 0,
D, is 9/7, so why don't you notate that as G!!)? Likewise, tone 46,
shown as A)||( is 14/9, so it could be notated as either B!!!) or A.
(|\; however as 7/4 of C, B!!!) would seem to be the preferable
spelling (besides being an unaccented symbol).

How do you figure that X\ doesn't match the 72-ET notation?

--George

George D. Secor wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>>
>> ........................ 36 A\!!/ 43 BY/ 50 B!!) 57 C\!/
>> 57 C\!/ 64 C)||( 71 D\! 06 E\!!/ 13 F!!/ 20 F|) 27 G\!/
>> 27 G\!/ 34 G)||( 41 A\! 48 B\!!/ 55 C!!/ 62 C|) 69 D\!/
>> 69 D\!/ 04 D)||( 11 E\! 18 F 25 G!!/ 32 G|) 39 A\!/
>> 39 A\!/ 46 A)||( 53 B\! 60 C 67 D!!/ 02 D|) 09 E\!/
>> 09 E\!/ 16 F!) 23 F||\ 30 G 37 A!!/ 44 A|) 51 B\!/
>> 51 B\!/ 58 C!) 65 C||\ 00 D 07 E!!/ 14 E|) 21 F/|\
>> 21 F/|\ 28 G!) 35 G||\ 42 A 49 B!!/ 56 B|) 63 C/|\
>> 63 C/|\ 70 D!) 05 D||\ 12 E 19 F/| 26 G)!!( 33 G/|\
>> 33 G/|\ 40 A!) 47 A||\ 54 B 61 C/| 68 D)!!( 03 D/|\
>> 03 D/|\ 10 E!) 17 E||\ 24 F/||\ 31 G/| 38 A)!!( 45 A/|\
>> 45 A/|\ 52 B!) 59 B||\ 66 C/||\ 01 D/| 08 E)!!( 15 E/|\
>> 15 E/|\ 22 F||) 29 FX\ 36 G/||\
>>
>> Here, B!!) and F||) are using the apotome complements of |) > although >> ~||( might be a better match ... the fact that B!!) : F|) > represents a >> fifth makes this easier to remember. Also, B!!) matches the 72-ET >> notation. The rest of the notation, with the exception of X\ and )||
> ( , >> also happens to match the 72-ET notation.
> > Hi Herman,
> > The simplest ratio that tone 26, shown as G)!!(, makes with tone 0, > D, is 9/7, so why don't you notate that as G!!)?

Do you mean G!// ?

Let's see ... the simplest ratios of all four pitches notated with )!!( are:

(-1, +14) 9/7 G!// or G'(!/
(-1, +20) 27/14 D!// or D'(!/
(-2, +26) 36/25 A)!!(
(-3, +32) 27/25 E)!!(

Both )||( and |\\ are of similar accuracy and complexity. But G!// is much simpler than G)!!( (32/25).

There's something very appealing about just having a general rule "Always notate the simplest ratio" (relative to whatever your base pitch is: typically C, A, G, or D). Many of these can be notated without accents, but not all. The rest can be filled in by extending the pattern. For 7-limit miracle:

........................ 36 43 50 57
57 64 C)||( 71 06 13 20 F~|) 27 G\!)
27 G\!) 34 G)||( 41 A\! 48 55 62 C~|) 69 D\!)
69 D\!) 04 D)||( 11 E\! 18 F~| 25 G)\!! 32 G~|) 39 A\!)
39 A\!) 46 A|\\ 53 B\! 60 C 67 D)\!! 02 D~|) 09 E\!)
09 E\!) 16 F!) 23 F||\ 30 G 37 A)\!! 44 A|) 51 B)\!/
51 B)\!/ 58 C!) 65 C||\ 00 D 07 E)\!! 14 E|) 21
21 28 G!) 35 G)/|| 42 A 49 B!!/ 56 B|) 63 C/|)
63 C/|) 70 05 D)/|| 12 E 19 F/| 26 G!// 33 G/|)
33 G/|) 40 A~!) 47 A)/|| 54 B 61 C/| 68 D!// 03 D/|)
03 D/|) 10 E~!) 17 24 31 G/| 38 A)!!( 45 A/|)
45 A/|) 52 B~!) 59 66 01 D/| 08 E)!!( 15 E/|)
15 E/|) 22 29 36

and 11-limit, which has the same TOP tuning:

........................ 36 43 A~~| 50 57 C~!/
57 C~!/ 64 C)||( 71 06 13 E~~| 20 F(|( 27 G\!)
27 G\!) 34 G)||( 41 A\! 48 55 B)|( 62 C(|( 69 D\!)
69 D\!) 04 D)||( 11 E\! 18 F~| 25 G)!!~ 32 G(|( 39 A\!/
39 A\!/ 46 A|\\ 53 B\! 60 C 67 D)!!~ 02 D(|( 09 E\!/
09 E\!/ 16 F!) 23 F||\ 30 G 37 A)\!! 44 A|) 51 B\!/
51 B\!/ 58 C!) 65 C||\ 00 D 07 E)\!! 14 E|) 21 F/|\
21 F/|\ 28 G!) 35 G)/|| 42 A 49 B!!/ 56 B|) 63 C/|\
63 C/|\ 70 05 D)/|| 12 E 19 F/| 26 G!// 33 G/|\
33 G/|\ 40 A(!( 47 A)||~ 54 B 61 C/| 68 D!// 03 D/|\
03 D/|\ 10 E(!( 17 F)!( 24 31 G/| 38 A)!!( 45 A/|)
45 A/|) 52 B(!( 59 C~~! 66 01 D/| 08 E)!!( 15 E/|)
15 E/|) 22 F~||( 29 G~~! 36

> How do you figure that X\ doesn't match the 72-ET notation?

It wouldn't normally be used for 72-ET because you'd use the next note up: G\! instead of FX\ (which is a different note in miracle).

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@>
wrote:
> >>
> >> ........................ 36 A\!!/ 43 BY/ 50 B!!) 57 C\!/
> >> 57 C\!/ 64 C)||( 71 D\! 06 E\!!/ 13 F!!/ 20 F|) 27 G\!/
> >> 27 G\!/ 34 G)||( 41 A\! 48 B\!!/ 55 C!!/ 62 C|) 69 D\!/
> >> 69 D\!/ 04 D)||( 11 E\! 18 F 25 G!!/ 32 G|) 39 A\!/
> >> 39 A\!/ 46 A)||( 53 B\! 60 C 67 D!!/ 02 D|) 09 E\!/
> >> 09 E\!/ 16 F!) 23 F||\ 30 G 37 A!!/ 44 A|) 51 B\!/
> >> 51 B\!/ 58 C!) 65 C||\ 00 D 07 E!!/ 14 E|) 21 F/|\
> >> 21 F/|\ 28 G!) 35 G||\ 42 A 49 B!!/ 56 B|) 63 C/|\
> >> 63 C/|\ 70 D!) 05 D||\ 12 E 19 F/| 26 G)!!( 33 G/|\
> >> 33 G/|\ 40 A!) 47 A||\ 54 B 61 C/| 68 D)!!( 03 D/|\
> >> 03 D/|\ 10 E!) 17 E||\ 24 F/||\ 31 G/| 38 A)!!( 45 A/|\
> >> 45 A/|\ 52 B!) 59 B||\ 66 C/||\ 01 D/| 08 E)!!( 15 E/|\
> >> 15 E/|\ 22 F||) 29 FX\ 36 G/||\
> >>
> >> Here, B!!) and F||) are using the apotome complements of |)
although
> >> ~||( might be a better match ... the fact that B!!) : F|)
represents a
> >> fifth makes this easier to remember. Also, B!!) matches the 72-
ET
> >> notation. The rest of the notation, with the exception of X\
and )||( ,
> >> also happens to match the 72-ET notation.
> >
> > Hi Herman,
> >
> > The simplest ratio that tone 26, shown as G)!!(, makes with tone
0,
> > D, is 9/7, so why don't you notate that as G!!)?
>
> Do you mean G!// ?

No, I meant F|||), which I then hastily converted to G!!), but now I
see that's okay for 72-ET, but not Miracle in general.

> Let's see ... the simplest ratios of all four pitches notated
with )!!( are:
>
> (-1, +14) 9/7 G!// or G'(!/
> (-1, +20) 27/14 D!// or D'(!/
> (-2, +26) 36/25 A)!!(
> (-3, +32) 27/25 E)!!(

I would like the herculean-level (accented) G'(!/ instead of the
(promethean-level) symbol G!//, but that's strictly a personal
preference. Is it simpler to use no accents and more symbol cores or
left accents and fewer symbol cores? The case for each side hasn't
been completely stated, nor does the jury yet have enough members to
issue a verdict.

> Both )||( and |\\ are of similar accuracy and complexity. But G!//
is
> much simpler than G)!!( (32/25).

Another issue is: which is the *simpler symbol* and which results in
a *simpler notation*? I would rather see G)!!( than G!//, because:

1) The former is a more common symbol (athenian-level), hence easier
to recognize, and
2) The former results in fewer overall symbols in the notation (and
thus a simpler notation); if you're already using )!!( for some of
the tones, then why not use it, instead of !//, for the others?

> There's something very appealing about just having a general rule
> "Always notate the simplest ratio" (relative to whatever your base
pitch
> is: typically C, A, G, or D). Many of these can be notated without
> accents, but not all. The rest can be filled in by extending the
> pattern. For 7-limit miracle:
>
> ........................ 36 43 50 57
> 57 64 C)||( 71 06 13 20 F~|) 27 G\!)
> 27 G\!) 34 G)||( 41 A\! 48 55 62 C~|) 69 D\!)
> 69 D\!) 04 D)||( 11 E\! 18 F~| 25 G)\!! 32 G~|) 39 A\!)
> 39 A\!) 46 A|\\ 53 B\! 60 C 67 D)\!! 02 D~|) 09 E\!)
> 09 E\!) 16 F!) 23 F||\ 30 G 37 A)\!! 44 A|) 51 B)\!/
> 51 B)\!/ 58 C!) 65 C||\ 00 D 07 E)\!! 14 E|) 21
> 21 28 G!) 35 G)/|| 42 A 49 B!!/ 56 B|) 63 C/|)
> 63 C/|) 70 05 D)/|| 12 E 19 F/| 26 G!// 33 G/|)
> 33 G/|) 40 A~!) 47 A)/|| 54 B 61 C/| 68 D!// 03 D/|)
> 03 D/|) 10 E~!) 17 24 31 G/| 38 A)!!( 45 A/|)
> 45 A/|) 52 B~!) 59 66 01 D/| 08 E)!!( 15 E/|)
> 15 E/|) 22 29 36
>
> and 11-limit, which has the same TOP tuning:
>
> ........................ 36 43 A~~| 50 57 C~!/
> 57 C~!/ 64 C)||( 71 06 13 E~~| 20 F(|( 27 G\!)
> 27 G\!) 34 G)||( 41 A\! 48 55 B)|( 62 C(|( 69 D\!)
> 69 D\!) 04 D)||( 11 E\! 18 F~| 25 G)!!~ 32 G(|( 39 A\!/
> 39 A\!/ 46 A|\\ 53 B\! 60 C 67 D)!!~ 02 D(|( 09 E\!/
> 09 E\!/ 16 F!) 23 F||\ 30 G 37 A)\!! 44 A|) 51 B\!/
> 51 B\!/ 58 C!) 65 C||\ 00 D 07 E)\!! 14 E|) 21 F/|\
> 21 F/|\ 28 G!) 35 G)/|| 42 A 49 B!!/ 56 B|) 63 C/|\
> 63 C/|\ 70 05 D)/|| 12 E 19 F/| 26 G!// 33 G/|\
> 33 G/|\ 40 A(!( 47 A)||~ 54 B 61 C/| 68 D!// 03 D/|\
> 03 D/|\ 10 E(!( 17 F)!( 24 31 G/| 38 A)!!( 45 A/|)
> 45 A/|) 52 B(!( 59 C~~! 66 01 D/| 08 E)!!( 15 E/|)
> 15 E/|) 22 F~||( 29 G~~! 36

Looks like you're having a lot of fun figuring all this out, but
yikes, we've created a monster! ;-) We now have all these symbols
and so many different ways to notate virtually the same things. What
is the poor performer going to think of all this?

I expect that we'd end up with 72-ET notation, once it came down to
reading notes for a live performance. The reason I prefer using
accented symbols is that they're generally going to have athenian-
level symbol cores, with which I'm hoping more sophisticated
performers will eventually become familiar. Throwing in a smattering
of left-accents isn't likely to generate much confusion; at worst a
performer will simply ignore them, at the price of ~2 cents of error.

> > How do you figure that X\ doesn't match the 72-ET notation?
>
> It wouldn't normally be used for 72-ET because you'd use the next
note
> up: G\! instead of FX\ (which is a different note in miracle).

Okay, I see.

--George

George D. Secor wrote:
> I would like the herculean-level (accented) G'(!/ instead of the > (promethean-level) symbol G!//, but that's strictly a personal > preference. Is it simpler to use no accents and more symbol cores or > left accents and fewer symbol cores? The case for each side hasn't > been completely stated, nor does the jury yet have enough members to > issue a verdict.

Right, that's the big question. I prefer unaccented symbols for the simpler ratios, but someone working with the 19-limit may have different preferences. As it is, '(!/ seems to come up far more frequently than the unaccented (!/ (although (!/. with a right accent does come in handy for 13/8), and the 7- and 11-limit temperaments I've been looking at don't have any other need for 19-limit symbols like !// .

I've also been thinking about keeping things simple by using an ET notation -- 72-ET is an obvious choice (one that already has a bit of a following), but 171-ET looks very attractive. For those who don't want to be overwhelmed in the full depths of Sagittal details, those subsets of the notation should be perfectly adequate for many different tuning systems.

>> Both )||( and |\\ are of similar accuracy and complexity. But G!// > is >> much simpler than G)!!( (32/25).
> > Another issue is: which is the *simpler symbol* and which results in > a *simpler notation*? I would rather see G)!!( than G!//, because:
> > 1) The former is a more common symbol (athenian-level), hence easier > to recognize, and
> 2) The former results in fewer overall symbols in the notation (and > thus a simpler notation); if you're already using )!!( for some of > the tones, then why not use it, instead of !//, for the others?

Yes, that's probably the main reason I settled on )||( .

> Looks like you're having a lot of fun figuring all this out, but > yikes, we've created a monster! ;-) We now have all these symbols > and so many different ways to notate virtually the same things. What > is the poor performer going to think of all this?

That's the problem with notating temperaments with any JI-based notation. So it would be nice to establish some recommendations for some of the temperaments we expect are going to be more frequently used. And ultimately, although it would be nice to have a general set of rules that applies to all temperaments, a custom-built notation like this one will be far easier to remember.

.................. 36 A\!!/ 43 BY/ 50 B!!) 57 C\!/
57 C\!/ 64 C)||( 71 D\! 06 E\!!/ 13 F!!/ 20 F|) 27 G\!/
27 G\!/ 34 G)||( 41 A\! 48 B\!!/ 55 C!!/ 62 C|) 69 D\!/
69 D\!/ 04 D)||( 11 E\! 18 F 25 G!!/ 32 G|) 39 A\!/
39 A\!/ 46 A)||( 53 B\! 60 C 67 D!!/ 02 D|) 09 E\!/
09 E\!/ 16 F!) 23 F||\ 30 G 37 A!!/ 44 A|) 51 B\!/
51 B\!/ 58 C!) 65 C||\ 00 D 07 E!!/ 14 E|) 21 F/|\
21 F/|\ 28 G!) 35 G||\ 42 A 49 B!!/ 56 B|) 63 C/|\
63 C/|\ 70 D!) 05 D||\ 12 E 19 F/| 26 G)!!( 33 G/|\
33 G/|\ 40 A!) 47 A||\ 54 B 61 C/| 68 D)!!( 03 D/|\
03 D/|\ 10 E!) 17 E||\ 24 F/||\ 31 G/| 38 A)!!( 45 A/|\
45 A/|\ 52 B!) 59 B||\ 66 C/||\ 01 D/| 08 E)!!( 15 E/|\
15 E/|\ 22 F||) 29 FX\ 36 G/||\

> I expect that we'd end up with 72-ET notation, once it came down to > reading notes for a live performance. The reason I prefer using > accented symbols is that they're generally going to have athenian-
> level symbol cores, with which I'm hoping more sophisticated > performers will eventually become familiar. Throwing in a smattering > of left-accents isn't likely to generate much confusion; at worst a > performer will simply ignore them, at the price of ~2 cents of error.

As it turns out, this is almost 72-ET notation as it is, with a few small differences. I'd prefer not to have to use the double sharp symbols, but the alternative would be a special unrelated symbol for just those two notes. That's the one change I might want to make -- A)|( in place of BY/ , and similarly G)!( in place of FX\ . The only additional pair not in 72-ET notation is )||( )!!( , which will be familiar from many other usages in other temperaments.

Double sharp and flat symbols aren't much of a problem in scales the size of 12-ET, but they can really get confusing with scales this size. It's bad enough that F!!/ is in there among the E's, between E and E|) , and even after you get to F!) there's still another E, E||\ . Hmm, that discontinuity can be fixed by notating E)|( and B)|( in place of F!!/ and C!!/ , and similarly with F)!( C)!( in place of E||\ B||\ . Then G)!( and A)|( wouldn't be all by themselves. But that gets further from 72-ET notation.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
> > I would like the herculean-level (accented) G'(!/ instead of the
> > (promethean-level) symbol G!//, but that's strictly a personal
> > preference. Is it simpler to use no accents and more symbol
cores or
> > left accents and fewer symbol cores? The case for each side
hasn't
> > been completely stated, nor does the jury yet have enough members
to
> > issue a verdict.
>
> Right, that's the big question. I prefer unaccented symbols for the
> simpler ratios, but someone working with the 19-limit may have
different
> preferences. As it is, '(!/ seems to come up far more frequently
than
> the unaccented (!/ (although (!/. with a right accent does come in
handy
> for 13/8), and the 7- and 11-limit temperaments I've been looking
at
> don't have any other need for 19-limit symbols like !// .
>
> I've also been thinking about keeping things simple by using an ET
> notation -- 72-ET is an obvious choice (one that already has a bit
of a
> following), but 171-ET looks very attractive. For those who don't
want
> to be overwhelmed in the full depths of Sagittal details, those
subsets
> of the notation should be perfectly adequate for many different
tuning
> systems.

For notating temperaments, wouldn't 171-ET be appropriate only if
it's a member of the particular temperament class under
consideration? (I saw that your next message addresses 171
specifically, so I'll respond to that later.)

If you're talking about notating JI, then there are some nice equal
divisions of the apotome (EDA's) into which alternative "spartan"
symbol sets could be mapped to serve the purpose:

Spartan (11-limit, 6-EDA): /| |) /|\ ||) ||\ /||\
Spartan (7-limit, 11-EDA): |( /| |) //| /|) (|\ )||( ||)
||\ /||) /||\
Spartan (13-limit, 12-EDA): |( /| |) |\ /|) /|\ (|\ /|| ||)
||\ /||) /||\
Spartan (13-limit, 13-EDA): |( /| |) |\ /|) /|\ (|) (|\ /||
||) ||\ /||) /||\

These correspond to 72-ET, 118-ET, 130-ET, and 142-ET, respectively.
If you're using exponents of prime 5 with an absolute value >2, then
you'll want (|\ to be 3 times as many steps as /|, which would prefer
11-EDA. But if the absolute value of prime 7 exponent >1, then
you're better off going to 21-EDA:

Athenian (21-EDA): |( )|( ~|( /| |) (| (|( //| /|) /|\
(|) (|\ )||( ~||( )||~ ||) ||\ (||( //|| /||) /||\

If you're staying within the 7 prime limit, you can may want to make
some symbol substitutions:

21-EDA, 7-limit: |( ~| ~|( /| |) (| ~|) //| /|) (/|
|\) (|\ )||( ~~|| )||~ ||) ||\ (||( )||\\ /||) /||\
where:
~| represents 1024:1029 in a secondary role, since it's not valid
here as 2176:2187; the symbol ~|. is defined as 1024:1029, so we've
simply dropped the right accent;
~|( represents 125:126 or 243:245; (this symbol didn't change, but
it's used here in a secondary role); ~|(.. is defined as 125:126 and
~|(. is defined as 243:245, so again we're simply dropping right
accents;
~|) represents 48:49, as defined; and
(/| represents 3969:4096, as defined.

The 5-schisma, 32768:32805, vanishes in each of the above, so ratios
that define left-accented symbols may be notated by dropping the left
accents.

> >> Both )||( and |\\ are of similar accuracy and complexity. But
G!// is
> >> much simpler than G)!!( (32/25).
> >
> > Another issue is: which is the *simpler symbol* and which results
in
> > a *simpler notation*? I would rather see G)!!( than G!//,
because:
> >
> > 1) The former is a more common symbol (athenian-level), hence
easier
> > to recognize, and
> > 2) The former results in fewer overall symbols in the notation
(and
> > thus a simpler notation); if you're already using )!!( for some
of
> > the tones, then why not use it, instead of !//, for the others?
>
> Yes, that's probably the main reason I settled on )||( .
>
> > Looks like you're having a lot of fun figuring all this out, but
> > yikes, we've created a monster! ;-) We now have all these
symbols
> > and so many different ways to notate virtually the same things.
What
> > is the poor performer going to think of all this?
>
> That's the problem with notating temperaments with any JI-based
> notation. So it would be nice to establish some recommendations for
some
> of the temperaments we expect are going to be more frequently used.
And
> ultimately, although it would be nice to have a general set of
rules
> that applies to all temperaments, a custom-built notation like this
one
> will be far easier to remember.
>
> .................. 36 A\!!/ 43 BY/ 50 B!!) 57 C\!/
> 57 C\!/ 64 C)||( 71 D\! 06 E\!!/ 13 F!!/ 20 F|) 27 G\!/
> 27 G\!/ 34 G)||( 41 A\! 48 B\!!/ 55 C!!/ 62 C|) 69 D\!/
> 69 D\!/ 04 D)||( 11 E\! 18 F 25 G!!/ 32 G|) 39 A\!/
> 39 A\!/ 46 A)||( 53 B\! 60 C 67 D!!/ 02 D|) 09 E\!/
> 09 E\!/ 16 F!) 23 F||\ 30 G 37 A!!/ 44 A|) 51 B\!/
> 51 B\!/ 58 C!) 65 C||\ 00 D 07 E!!/ 14 E|) 21 F/|\
> 21 F/|\ 28 G!) 35 G||\ 42 A 49 B!!/ 56 B|) 63 C/|\
> 63 C/|\ 70 D!) 05 D||\ 12 E 19 F/| 26 G)!!( 33 G/|\
> 33 G/|\ 40 A!) 47 A||\ 54 B 61 C/| 68 D)!!( 03 D/|\
> 03 D/|\ 10 E!) 17 E||\ 24 F/||\ 31 G/| 38 A)!!( 45 A/|\
> 45 A/|\ 52 B!) 59 B||\ 66 C/||\ 01 D/| 08 E)!!( 15 E/|\
> 15 E/|\ 22 F||) 29 FX\ 36 G/||\

Yep, that's pretty simple. With any temperament you have to deal
with the fact that, because various tones may have multiple ratio-
identities, multiple symbols are therefore possible. In limiting the
number of symbols (in order to simplify the notation) you have to
favor some of those identities over others.

> > I expect that we'd end up with 72-ET notation, once it came down
to
> > reading notes for a live performance. The reason I prefer using
> > accented symbols is that they're generally going to have athenian-
> > level symbol cores, with which I'm hoping more sophisticated
> > performers will eventually become familiar. Throwing in a
smattering
> > of left-accents isn't likely to generate much confusion; at worst
a
> > performer will simply ignore them, at the price of ~2 cents of
error.
>
> As it turns out, this is almost 72-ET notation as it is, with a few
> small differences. I'd prefer not to have to use the double sharp
> symbols, but the alternative would be a special unrelated symbol
for
> just those two notes. That's the one change I might want to make --
A)|(
> in place of BY/ , and similarly G)!( in place of FX\ . The only
> additional pair not in 72-ET notation is )||( )!!( , which will be
> familiar from many other usages in other temperaments.

Yes. Anything in the athenian-level symbol set is a safe choice.

> Double sharp and flat symbols aren't much of a problem in scales
the
> size of 12-ET, but they can really get confusing with scales this
size.
> It's bad enough that F!!/ is in there among the E's, between E and
E|) ,
> and even after you get to F!) there's still another E, E||\ . Hmm,
that
> discontinuity can be fixed by notating E)|( and B)|( in place of
F!!/
> and C!!/ , and similarly with F)!( C)!( in place of E||\ B||\ .
Then
> G)!( and A)|( wouldn't be all by themselves. But that gets further
from
> 72-ET notation.

I think you're justified in doing that if it serves to simplify
things.

--George

George D. Secor wrote:

> For notating temperaments, wouldn't 171-ET be appropriate only if > it's a member of the particular temperament class under > consideration? (I saw that your next message addresses 171 > specifically, so I'll respond to that later.)

What I've noticed is that I run across many of the same basic ratios frequently when notating temperaments. When I look at the sizes of the accidentals for 7-limit temperaments, they end up being close to 171-ET intervals. Typical examples: 225/224, 126/125, 245/243, 81/80, 64/63, 686/675, 50/49, 49/48, 36/35, 250/243, 28/27, 25/24, 21/20, 135/128. I don't know whether there's any point to that, but I have some sympathy for the problem of newcomers seeing the vast numbers of Sagittal symbols and scaring them away. So I guess I was trying to see if I could find some way to narrow down a set of symbols to use in 7-limit temperaments. It sounded like an interesting approach, but may turn out to be more confusing than just learning the full notation system. The problem is that the *tempered* accidentals aren't anywhere near 171-ET.

> If you're talking about notating JI, then there are some nice equal > divisions of the apotome (EDA's) into which alternative "spartan" > symbol sets could be mapped to serve the purpose:
> > Spartan (11-limit, 6-EDA): /| |) /|\ ||) ||\ /||\
> Spartan (7-limit, 11-EDA): |( /| |) //| /|) (|\ )||( ||) > ||\ /||) /||\
> Spartan (13-limit, 12-EDA): |( /| |) |\ /|) /|\ (|\ /|| ||) > ||\ /||) /||\
> Spartan (13-limit, 13-EDA): |( /| |) |\ /|) /|\ (|) (|\ /|| > ||) ||\ /||) /||\

I guess the 171-ET set could be considered a 16-EDA notation. I only used up to +13, though.

> These correspond to 72-ET, 118-ET, 130-ET, and 142-ET, respectively. > If you're using exponents of prime 5 with an absolute value >2, then > you'll want (|\ to be 3 times as many steps as /|, which would prefer > 11-EDA. But if the absolute value of prime 7 exponent >1, then > you're better off going to 21-EDA:
> > Athenian (21-EDA): |( )|( ~|( /| |) (| (|( //| /|) /|\
> (|) (|\ )||( ~||( )||~ ||) ||\ (||( //|| /||) /||\

I do have the occasional need for ratios like 49/30 (myna) and 49/36 (ennealimmal). So that's worth looking into.

> If you're staying within the 7 prime limit, you can may want to make > some symbol substitutions:
> > 21-EDA, 7-limit: |( ~| ~|( /| |) (| ~|) //| /|) (/|
> |\) (|\ )||( ~~|| )||~ ||) ||\ (||( )||\\ /||) /||\

It might be useful to have a 7-limit specific notation for 7-limit temperaments if you're not using the higher-limit approximations. E.g., my proposed miracle notation is 11-limit:

/| [-4, 4, -1> (-3, +31)
)|( [7, -4, 0, 1, -1> (+4, -41)
|) [6, -2, 0, -1> (+1, -10)
/|\ [-5, 1, 0, 0, 1> (-2, +21)
)||( [-3, -1, 2> (+2, -20)
||\ [-7, 3, 1> (-1, +11)
/||\ [-11, 7> (-4, +42)

but you could substitute 7-limit accidentals for a 7-limit miracle notation.

|( [10, -6, 1, -1> (+4, -41)
/|) [2, 2, -1, -1> (-2, +21)

>> Double sharp and flat symbols aren't much of a problem in scales > the >> size of 12-ET, but they can really get confusing with scales this > size. >> It's bad enough that F!!/ is in there among the E's, between E and > E|) , >> and even after you get to F!) there's still another E, E||\ . Hmm, > that >> discontinuity can be fixed by notating E)|( and B)|( in place of > F!!/ >> and C!!/ , and similarly with F)!( C)!( in place of E||\ B||\ . > Then >> G)!( and A)|( wouldn't be all by themselves. But that gets further > from >> 72-ET notation.
> > I think you're justified in doing that if it serves to simplify > things.

Yes, now that I think about it I like this better.

........................ 36 A\!!/ 43 A)|( 50 B!!) 57 C\!/
57 C\!/ 64 C)||( 71 D\! 06 E\!!/ 13 E)|( 20 F|) 27 G\!/
27 G\!/ 34 G)||( 41 A\! 48 B\!!/ 55 B)|( 62 C|) 69 D\!/
69 D\!/ 04 D)||( 11 E\! 18 F 25 G!!/ 32 G|) 39 A\!/
39 A\!/ 46 A)||( 53 B\! 60 C 67 D!!/ 02 D|) 09 E\!/
09 E\!/ 16 F!) 23 F||\ 30 G 37 A!!/ 44 A|) 51 B\!/
51 B\!/ 58 C!) 65 C||\ 00 D 07 E!!/ 14 E|) 21 F/|\
21 F/|\ 28 G!) 35 G||\ 42 A 49 B!!/ 56 B|) 63 C/|\
63 C/|\ 70 D!) 05 D||\ 12 E 19 F/| 26 G)!!( 33 G/|\
33 G/|\ 40 A!) 47 A||\ 54 B 61 C/| 68 D)!!( 03 D/|\
03 D/|\ 10 E!) 17 F)!( 24 F/||\ 31 G/| 38 A)!!( 45 A/|\
45 A/|\ 52 B!) 59 C)!( 66 C/||\ 01 D/| 08 E)!!( 15 E/|\
15 E/|\ 22 F||) 29 G)!( 36 G/||\

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> George D. Secor wrote:
>
> > For notating temperaments, wouldn't 171-ET be appropriate only if
> > it's a member of the particular temperament class under
> > consideration? (I saw that your next message addresses 171
> > specifically, so I'll respond to that later.)
>
> What I've noticed is that I run across many of the same basic
ratios
> frequently when notating temperaments. When I look at the sizes of
the
> accidentals for 7-limit temperaments, they end up being close to
171-ET
> intervals. Typical examples: 225/224, 126/125, 245/243, 81/80,
64/63,
> 686/675, 50/49, 49/48, 36/35, 250/243, 28/27, 25/24, 21/20,
135/128. I
> don't know whether there's any point to that, but I have some
sympathy
> for the problem of newcomers seeing the vast numbers of Sagittal
symbols
> and scaring them away. So I guess I was trying to see if I could
find
> some way to narrow down a set of symbols to use in 7-limit
temperaments.
> It sounded like an interesting approach, but may turn out to be
more
> confusing than just learning the full notation system. The problem
is
> that the *tempered* accidentals aren't anywhere near 171-ET.
>
> > If you're talking about notating JI, then there are some nice
equal
> > divisions of the apotome (EDA's) into which alternative "spartan"
> > symbol sets could be mapped to serve the purpose:
> >
> > Spartan (11-limit, 6-EDA): /| |) /|\ ||) ||\ /||\
> > Spartan (7-limit, 11-EDA): |( /| |) //| /|) (|\ )||( ||)
> > ||\ /||) /||\
> > Spartan (13-limit, 12-EDA): |( /| |) |\ /|) /|\ (|\ /||
||)
> > ||\ /||) /||\
> > Spartan (13-limit, 13-EDA): |( /| |) |\ /|) /|\ (|)
(|\ /||
> > ||) ||\ /||) /||\
>
> I guess the 171-ET set could be considered a 16-EDA notation. I
only
> used up to +13, though.

Yep. See my previous reply.

> > These correspond to 72-ET, 118-ET, 130-ET, and 142-ET,
respectively.
> > If you're using exponents of prime 5 with an absolute value >2,
then
> > you'll want (|\ to be 3 times as many steps as /|, which would
prefer
> > 11-EDA. But if the absolute value of prime 7 exponent >1, then
> > you're better off going to 21-EDA:
> >
> > Athenian (21-EDA): |( )|( ~|( /| |) (| (|( //| /|) /|\
> > (|) (|\ )||( ~||( )||~ ||) ||\ (||( //|| /||) /||\
>
> I do have the occasional need for ratios like 49/30 (myna) and
49/36
> (ennealimmal). So that's worth looking into.

In case you needed to distinguish between those two, and also have
both ~|( and (|( in your symbol set, then you could use:
26-EDA, 13-limit JI: |( )|( ~|( ~~| /| |) |\ ~|) (|
( //| /|) /|\ )/|\
(|) (|\ )||( ~||( ~~|| /|| ||) ||\ ~||) (||
( //|| /||) /||\

What's particularly nice about this one is that the entire sequence
of double-shaft symbols has flags that exactly match a portion of the
sequence of single-shaft symbols (a desirable feature that's
mentioned in the Sagittal paper). I think that this should also be
used as the standard symbol set for 270-ET.

If it seems that this gets more and more complicated as we go along,
you may find that many temperaments will each require only a
relatively small subset of the above accidentals.

> > If you're staying within the 7 prime limit, you can may want to
make
> > some symbol substitutions:
> >
> > 21-EDA, 7-limit: |( ~| ~|( /| |) (| ~|) //| /|) (/|
> > |\) (|\ )||( ~~|| )||~ ||) ||\ (||( )||\\ /||) /||\
>
> It might be useful to have a 7-limit specific notation for 7-limit
> temperaments if you're not using the higher-limit approximations.
E.g.,
> my proposed miracle notation is 11-limit:
>
> /| [-4, 4, -1> (-3, +31)
> )|( [7, -4, 0, 1, -1> (+4, -41)
> |) [6, -2, 0, -1> (+1, -10)
> /|\ [-5, 1, 0, 0, 1> (-2, +21)
> )||( [-3, -1, 2> (+2, -20)
> ||\ [-7, 3, 1> (-1, +11)
> /||\ [-11, 7> (-4, +42)
>
> but you could substitute 7-limit accidentals for a 7-limit miracle
notation.
>
> |( [10, -6, 1, -1> (+4, -41)
> /|) [2, 2, -1, -1> (-2, +21)

I worked out 11-limit miracle accidentals (with some left-accented
symbols) some time ago, with:
./||\ for 16/15 at +1 secor
|) for 64/63 at -10 secors
.||) for 21/20 at +11 secors
.(|\ for 28/27 at -20 secors
/|\ for 33/32 at +21 secors
.||\ for 256/243 at -30 secors
/| for 81/80 at +31 secors
~~| for 100/99 at -41 secors.

If you drop all of the left accents and change ~~| to ~|(, you'll get
the athenian-level symbols. (I noticed that you used 896/891 instead
of 100/99, which is not a bad idea.)

I just noticed that the first one could have been:
|||( for 15/14 at +1 secor
so as not to conflict with
/||\ for 2187/2048 at +42 secors
if the accents are dropped. However, that doesn't give 15/8 at -1
secor, so it's probably best to skip that one.

> >> Double sharp and flat symbols aren't much of a problem in scales
the
> >> size of 12-ET, but they can really get confusing with scales
this size.
> >> It's bad enough that F!!/ is in there among the E's, between E
and E|) ,
> >> and even after you get to F!) there's still another E, E||\ .
Hmm, that
> >> discontinuity can be fixed by notating E)|( and B)|( in place of
F!!/
> >> and C!!/ , and similarly with F)!( C)!( in place of E||\ B||\ .
Then
> >> G)!( and A)|( wouldn't be all by themselves. But that gets
further from
> >> 72-ET notation.
> >
> > I think you're justified in doing that if it serves to simplify
> > things.
>
> Yes, now that I think about it I like this better.
>
> ........................ 36 A\!!/ 43 A)|( 50 B!!) 57 C\!/
> 57 C\!/ 64 C)||( 71 D\! 06 E\!!/ 13 E)|( 20 F|) 27 G\!/
> 27 G\!/ 34 G)||( 41 A\! 48 B\!!/ 55 B)|( 62 C|) 69 D\!/
> 69 D\!/ 04 D)||( 11 E\! 18 F 25 G!!/ 32 G|) 39 A\!/
> 39 A\!/ 46 A)||( 53 B\! 60 C 67 D!!/ 02 D|) 09 E\!/
> 09 E\!/ 16 F!) 23 F||\ 30 G 37 A!!/ 44 A|) 51 B\!/
> 51 B\!/ 58 C!) 65 C||\ 00 D 07 E!!/ 14 E|) 21 F/|\
> 21 F/|\ 28 G!) 35 G||\ 42 A 49 B!!/ 56 B|) 63 C/|\
> 63 C/|\ 70 D!) 05 D||\ 12 E 19 F/| 26 G)!!( 33 G/|\
> 33 G/|\ 40 A!) 47 A||\ 54 B 61 C/| 68 D)!!( 03 D/|\
> 03 D/|\ 10 E!) 17 F)!( 24 F/||\ 31 G/| 38 A)!!( 45 A/|\
> 45 A/|\ 52 B!) 59 C)!( 66 C/||\ 01 D/| 08 E)!!( 15 E/|\
> 15 E/|\ 22 F||) 29 G)!( 36 G/||\

Okay!

--George

George D. Secor wrote:
> I worked out 11-limit miracle accidentals (with some left-accented > symbols) some time ago, with:
> ./||\ for 16/15 at +1 secor
> |) for 64/63 at -10 secors
> .||) for 21/20 at +11 secors
> .(|\ for 28/27 at -20 secors
> /|\ for 33/32 at +21 secors
> .||\ for 256/243 at -30 secors
> /| for 81/80 at +31 secors
> ~~| for 100/99 at -41 secors.
> > If you drop all of the left accents and change ~~| to ~|(, you'll get > the athenian-level symbols. (I noticed that you used 896/891 instead > of 100/99, which is not a bad idea.)

Yes, I think I mentioned the problem with ~~| in a previous post (the standard interpretation of 99/98 ends up as +31 secors). The alternative of ~|( as 245/243 might be slightly better, but could be confused with 126/125.

> I just noticed that the first one could have been:
> |||( for 15/14 at +1 secor
> so as not to conflict with
> /||\ for 2187/2048 at +42 secors
> if the accents are dropped. However, that doesn't give 15/8 at -1 > secor, so it's probably best to skip that one.

So you'd use the -30 secors instead of +42? I wonder if there's any way to get that without an accent.

-10 secors has a few alternatives, but I see that we both settled on 64/63, so that should be fine.

I considered .||) or )/|| 21/20 for +11, but I prefer ||\ -- for one thing because it doesn't need an accent, but also for the spelling of major and minor triads in the "white note" keys. These factors also had an influence on )||( instead of .(|\ for -20, but )||( also represents a simpler interval 25/24. Both )||( and .(|\ have almost the same accuracy and complexity, so it's really a matter of taste.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@> wrote:
> > ...
> > I guess the 171-ET set could be considered a 16-EDA notation. I
only
> > used up to +13, though.
>
> Yep. See my previous reply.
>
> > > These correspond to 72-ET, 118-ET, 130-ET, and 142-ET,
respectively.
> > > If you're using exponents of prime 5 with an absolute value >2,
then
> > > you'll want (|\ to be 3 times as many steps as /|, which would
prefer
> > > 11-EDA. But if the absolute value of prime 7 exponent >1, then
> > > you're better off going to 21-EDA:
> > >
> > > Athenian (21-EDA): |( )|( ~|( /| |) (| (|( //| /|) /|\
> > > (|) (|\ )||( ~||( )||~ ||) ||\ (||( //|| /||) /||\
> >
> > I do have the occasional need for ratios like 49/30 (myna) and
49/36
> > (ennealimmal). So that's worth looking into.
>
> In case you needed to distinguish between those two, and also have
> both ~|( and (|( in your symbol set, then you could use:
> 26-EDA, 13-limit JI: |( )|( ~|( ~~| /| |) |\ ~|) (|
( //| /|) /|\ )/|\
> (|) (|\ )||( ~||( ~~|| /|| ||) ||\ ~||) (||
( //|| /||) /||\
>
> What's particularly nice about this one is that the entire sequence
> of double-shaft symbols has flags that exactly match a portion of
the
> sequence of single-shaft symbols (a desirable feature that's
> mentioned in the Sagittal paper). I think that this should also be
> used as the standard symbol set for 270-ET.
>
> If it seems that this gets more and more complicated as we go
along,
> you may find that many temperaments will each require only a
> relatively small subset of the above accidentals.

Something I subsequently noticed about 26-EDA that significantly
complicates the situation is that the 5-schisma (32768:32805) doesn't
vanish, so I wouldn't want to go beyond 21-EDA for a simple
notation. I wrote:

> > > If you're staying within the 7 prime limit, you can may want to
make
> > > some symbol substitutions:
> > >
> > > 21-EDA, 7-limit: |( ~| ~|( /| |) (| ~|) //| /|) (/|
> > > |\) (|\ )||( ~~|| )||~ ||) ||\ (||( )||\\ /||) /||\

On second thought, in those substitutions I think it would be better
not to replace /|\ and (|) with (/| and |\), respectively, since:

1) /|\ and (|) are much common symbols than (/| and |\);
2) (/| would notate 7^2 only as an alternate spelling, and there is
already a symbol ~|) to notate 7^2 in its preferred spelling, so its
replacement by /|\ is not a great loss;
3) |\) is seldom needed, so its replacement by (|) is not much of an
issue;
4) (/| would also have been used instead of )/|\ for 392:405 (they
differ by a 5-schisma), whereas /|\, which is frequently used to
represent 1/2 apotome in temperaments, is a natural stand-in for )/|\
(which is actually even closer to 1/2 apotome).

It's also debatable whether ~| should be there instead of )|(,
especially since its apotome complement )||\\ is such a complicated
symbol. The only good reasons for having ~| are:

1) It notates 7^3 (in the role of 1024:1029, which is the definition
of ~|., so we're merely dropping a right accent mark). However, the
only other reasonably simple 7-limit comma that ~| could represent is
1715:1728, but that one isn't even important enough to have an
olympian-level symbol definition.

2) |( ~| ~|( is an easily memorized symbol sequence.

> > It might be useful to have a 7-limit specific notation for 7-
limit
> > temperaments if you're not using the higher-limit
approximations. ...

Herman, I think I need to limit my participation in this discussion
from this point on, because the notation of temperament classes is
turning out to be more complicated than I had anticipated, not only
in the sheer multitude of temperament classes to be notated, but also
in the problems involved in determining the most appropriate symbols
for their notation. These are issues that are quite distinct from
our (i.e., Dave's and my) original goal of devising and defining
symbols to notate both JI and temperaments, which task (after nearly
6 years) is now complete, save for the release of some remaining
(very helpful) documentation. I'd really like to get back to
composing some music (and perhaps some writing), now that I have
notations for all of the tunings I would ever want to use (and many
more, besides).

Since you have demonstrated a good grasp of Sagittal principles and a
much better knowledge of (and interest in) the great variety of
temperaments that have popped up over the past several years, I think
it would be wise to leave it to you and others to work out the
intricacies of symbol selection. Dave & I will still be around, of
course, to answer questions and offer suggestions, but it's been a
long haul with an ever-increasing load of responsibilities, and,
quite frankly, we need a break.

Therefore, we feel it's time that we started sharing the burden of
further development and support of the notation with others in the
tuning community (such as you and Graham) who are willing and able to
step in and lend a hand. Please give this some thought (and others
interested who are reading this) -- we would be very grateful.

--George