What should we call these "sagittal bridges"?

In the sagittal paper, S&K give us a table of symbols and tell
us "The symbols which are used to notate prime harmonic factors
relative to the natural Pythagorean notes are shown in bold type."
Since I don't have bold type, I've put the non-bold lines into
parenthesis. Everything in bold type, which are the most important
symbols of Sagittal, are what I wanted to call "sagittal bridges", so
the same is certainly logical and appropriate. Since George objects,
I'd really like to hear an alternative.

Nearly all of the rest of the symbols are for xenharmonic bridges
which aren't "sagittal". The three exceptions I've put an "nb",
for "not a bridge", next to. Those are for 2048/2025, 250/243, and
531441/512000. So I'd say the importance of xenharmonic bridges,
and "sagittal" bridges in particular, to Sagittal notation is clear
and strong.

George calls microtempering by 4096/4095 the lynchpin of Sagittal,
which for someone who is not concerned with the 13 limit it clearly
won't be. The idea of using microtempering is of course attractive,
and was the basis for my proposal to notate the 7 and 11 limits using
ennealimmal/hemiennealimmal. Other "schisminas" George mentions are,
if I am interpreting him correctly 2080/2079 and 256000/255879.
Putting these together gives a 13-limit planar microtemperament with
TM basis {2080/2079, 4096/4095, 4375/4374}. The most logical means of
extending this to a linear microtemperament seems to be to add
1716/1715 to the list; in this way we get a temperament supported by
224, 270 and 494 which I think has been mentioned but for which I
didn't have a name.

arrow {1716/1715, 2080/2079, 4096/4095, 4375/4374}
<<22 48 -38 -34 -54 25 -122 -130 -167 -223 -245 -303 36 -11 -61||
[<2 7 13 -1 1 -2|, <<0 -11 -24 19 17 27|]
[599.971, 208.896]

Table 1 Selected Sagittal Symbols

5 schisma 32805/32768 sb
19 schisma 513/512 sb
(5:7 kleisma 5102/5103 xb)
(11:13 kleisma 352/351 xb)
17 kleisma 2187/2176 sb
(7:11 kleisma 896/891 xb)
17 comma sb
23 comma 736/729 sb
(25 comma 2048/2025 nb)
19 comma 19683/19456 sb
5 comma 81/80 sb
7 comma 64/63 sb
(55 comma 55/54 xb)
(7:11 comma 45927/45056 xb)
(13:17 comma 52/51 xb)
29 comma 261/256 sb
(5:11 S-diesis 45/44 xb)
(7:13 S-diesis 1701/1664 xb)
(11:17 S-diesis 1408/1377 xb)
(35 M-diesis 36/35 xb)
13 M-diesis 1053/1024 sb
(125 M-diesis 250/243 nb)
11 M-diesis 33/32 sb
11 L-diesis 729/704 sb
(35 L-diesis 8505/8192 xb)
13 L-diesis 27/26 sb
(125 L-diesis 531441/512000 nb)

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> arrow {1716/1715, 2080/2079, 4096/4095, 4375/4374}
> <<22 48 -38 -34 -54 25 -122 -130 -167 -223 -245 -303 36 -11 -61||
> [<2 7 13 -1 1 -2|, <<0 -11 -24 19 17 27|]
> [599.971, 208.896]

Poptimal for the 13 limit is 305/1752; I'd give the 15-limit but I
havn't written the code. Obviously 494 would do fine.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> In the sagittal paper, S&K give us a table of symbols and tell
> us "The symbols which are used to notate prime harmonic factors
> relative to the natural Pythagorean notes are shown in bold type."
> Since I don't have bold type, I've put the non-bold lines into
> parenthesis. Everything in bold type, which are the most important
> symbols of Sagittal, are what I wanted to call "sagittal bridges",

They aren't necessarily the most important symbols (particularly for
the higher primes). They only happen to be the first ones that we
worked out, inasmuch as we began with the one-symbol-per-prime
philosophy. We moved away from that approach once we began trying to
notate JI tonality diamonds and found that we also needed symbols for
commas having *two* primes above 3. We then found that just about
everything beyond that can be approximated reasonably well by
existing symbols.

Since only about half the Sagittal symbols in the table (below) are
marked with "sb", I see no point in calling these "sagittal
bridges". And if we were to add the rest of the Sagittal symbols
(the least common ones) to the table, I think you might find
that "sb" symbols are in the minority.

> so
> the same is certainly logical and appropriate. Since George
objects,
> I'd really like to hear an alternative.
>
> Nearly all of the rest of the symbols are for xenharmonic bridges
> which aren't "sagittal". The three exceptions I've put an "nb",
> for "not a bridge", next to. Those are for 2048/2025, 250/243, and
> 531441/512000. So I'd say the importance of xenharmonic bridges,
> and "sagittal" bridges in particular, to Sagittal notation is clear
> and strong.
>
> George calls microtempering by 4096/4095 the lynchpin of Sagittal,
> which for someone who is not concerned with the 13 limit it clearly
> won't be.

This is the point at which the Sagittal symbol "economy" begins, in
that no more symbols are required to notate 15-limit consonances than
are required for 11-limit consonances. Thus, ratios of 13 (except in
extreme-precision Sagittal) are not exact, but rather very close (sub-
cent) approximations of more complex 11-limit ratios.

The flip side of this is that *all* of the 11-limit consonances in
Sagittal are notated *exactly*, beginning with medium-precision (or
athenian-level) Sagittal JI.

> The idea of using microtempering is of course attractive,
> and was the basis for my proposal to notate the 7 and 11 limits
using
> ennealimmal/hemiennealimmal. Other "schisminas" George mentions
are,
> if I am interpreting him correctly 2080/2079 and 256000/255879.
> Putting these together gives a 13-limit planar microtemperament with
> TM basis {2080/2079, 4096/4095, 4375/4374}. The most logical means
of
> extending this to a linear microtemperament seems to be to add
> 1716/1715 to the list; in this way we get a temperament supported
by
> 224, 270 and 494 which I think has been mentioned but for which I
> didn't have a name.

FYI, our names for Sagittal symbol sets (which correlate to levels of
JI precision) have names that you may be interested in:

"spartan" - the lowest level at which we would recommend notating JI,
essentially a mapping to 130-ET

"athenian" - the symbol set for medium-precision JI and 224-ET (note:
medium-precision JI is *not* a 224-ET mapping!!!)

"herculean" - the symbol set for high-precision JI, with no symbols
differentiated by accent marks (with a resolution on the order of 494-
ET, it uses symbols which better distinguish primes 19 and 23; but
this is *not* a 494-ET mapping!!!)

"promethean" - the symbol set for high-precision JI, with some pairs
of symbols differentiated by accent marks (with a resolution on the
order of 612-ET, tends to favor complex 13-limit JI; again, *not* a
612-ET mapping!!!)

"olympian" the symbol set for extreme-precision JI, with pairs of
symbols differentiated by accent marks (with a resolution on the
order of 1171-ET)

Please note: Dave and I have put off the question of whether there
should be two different high-precision versions of Sagittal JI (but
if not, then the "herculean" designation would probably be used for
JI with ~612-ET resolution).

However, if Dave decides to agree with me about the above
classifications, then I would like to suggest the name "herculean"
temperament for the linear temperament with generator ~39:44 and
period of 1/2-octave that defines the 224-270-494 family.

--George

> arrow {1716/1715, 2080/2079, 4096/4095, 4375/4374}
> <<22 48 -38 -34 -54 25 -122 -130 -167 -223 -245 -303 36 -11 -61||
> [<2 7 13 -1 1 -2|, <<0 -11 -24 19 17 27|]
> [599.971, 208.896]
>
> Table 1 Selected Sagittal Symbols
>
> 5 schisma 32805/32768 sb
> 19 schisma 513/512 sb
> (5:7 kleisma 5102/5103 xb)
> (11:13 kleisma 352/351 xb)
> 17 kleisma 2187/2176 sb
> (7:11 kleisma 896/891 xb)
> 17 comma sb
> 23 comma 736/729 sb
> (25 comma 2048/2025 nb)
> 19 comma 19683/19456 sb
> 5 comma 81/80 sb
> 7 comma 64/63 sb
> (55 comma 55/54 xb)
> (7:11 comma 45927/45056 xb)
> (13:17 comma 52/51 xb)
> 29 comma 261/256 sb
> (5:11 S-diesis 45/44 xb)
> (7:13 S-diesis 1701/1664 xb)
> (11:17 S-diesis 1408/1377 xb)
> (35 M-diesis 36/35 xb)
> 13 M-diesis 1053/1024 sb
> (125 M-diesis 250/243 nb)
> 11 M-diesis 33/32 sb
> 11 L-diesis 729/704 sb
> (35 L-diesis 8505/8192 xb)
> 13 L-diesis 27/26 sb
> (125 L-diesis 531441/512000 nb)

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

> They aren't necessarily the most important symbols (particularly
for
> the higher primes). They only happen to be the first ones that we
> worked out, inasmuch as we began with the one-symbol-per-prime
> philosophy. We moved away from that approach once we began trying
to
> notate JI tonality diamonds and found that we also needed symbols
for
> commas having *two* primes above 3.

OK. I've just now concluded that 11 symbol pairs should pretty well
do you for arrow/hercules; I think I'll produce a list of them and
ask if and how Sagittal notates them.

> > George calls microtempering by 4096/4095 the lynchpin of
Sagittal,
> > which for someone who is not concerned with the 13 limit it
clearly
> > won't be.
>
> This is the point at which the Sagittal symbol "economy" begins, in
> that no more symbols are required to notate 15-limit consonances
than
> are required for 11-limit consonances.

I think it begins with 4375/4374, a comma of arrow/hercules, which
expresses the 7-limit in terms of the 5-limit; this schismina is a
consequence of the equivalences you already use in 13-limit Sagittal.

Thus, ratios of 13 (except in
> extreme-precision Sagittal) are not exact, but rather very close
(sub-
> cent) approximations of more complex 11-limit ratios.

This means that you are using a system in the 11 and 7 limits which
differs from the 13-limit, since the 11-limit or 7-limit systems you
can derive from the 13-limit planar temperament will be 11 and 7-
limit planar temperaments, ending up at 4375/4374-planar in the 7-
limit. In any case I think an arrow/hercules level or version of
Sagittal would be clearly worth working out; evidently it would be a
herculean level but probably not a herculean task.

> However, if Dave decides to agree with me about the above
> classifications, then I would like to suggest the name "herculean"
> temperament for the linear temperament with generator ~39:44 and
> period of 1/2-octave that defines the 224-270-494 family.

This is, of course, exactly what I've been calling "arrow". I
suggest "hercules" rather than "herculean"; anyone else want to chime
in?

> > arrow/hercules {1716/1715, 2080/2079, 4096/4095, 4375/4374}
> > <<22 48 -38 -34 -54 25 -122 -130 -167 -223 -245 -303 36 -11 -61||
> > [<2 7 13 -1 1 -2|, <<0 -11 -24 19 17 27|]
> > [599.971, 208.896]

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > In the sagittal paper, S&K give us a table of symbols and tell
> > us "The symbols which are used to notate prime harmonic factors
> > relative to the natural Pythagorean notes are shown in bold
type."
> > Since I don't have bold type, I've put the non-bold lines into
> > parenthesis. Everything in bold type, which are the most
important
> > symbols of Sagittal, are what I wanted to call "sagittal
bridges",

It's natural to assume that bold entries are more important, (sorry
about that) but we explain in the text that the definition of each
symbol (i.e. the ratio that it is considered to notate exactly) is
the one listed first in Table 1. These are the one's most likely to
need notating. They are generally the lowest prime limit, or lowest
product complexity.

George Secor wrote:
> FYI, our names for Sagittal symbol sets (which correlate to levels
of
> JI precision) have names that you may be interested in:
>
> "spartan" - the lowest level at which we would recommend notating
JI,
> essentially a mapping to 130-ET
>
> "athenian" - the symbol set for medium-precision JI and 224-ET
(note:
> medium-precision JI is *not* a 224-ET mapping!!!)
>
> "herculean" - the symbol set for high-precision JI, with no
symbols
> differentiated by accent marks (with a resolution on the order of
494-
> ET, it uses symbols which better distinguish primes 19 and 23; but
> this is *not* a 494-ET mapping!!!)
>
> "promethean" - the symbol set for high-precision JI, with some
pairs
> of symbols differentiated by accent marks (with a resolution on
the
> order of 612-ET, tends to favor complex 13-limit JI; again, *not*
a
> 612-ET mapping!!!)
>
> "olympian" the symbol set for extreme-precision JI, with pairs of
> symbols differentiated by accent marks (with a resolution on the
> order of 1171-ET)
>
> Please note: Dave and I have put off the question of whether
there
> should be two different high-precision versions of Sagittal JI
(but
> if not, then the "herculean" designation would probably be used
for
> JI with ~612-ET resolution).
>
> However, if Dave decides to agree with me about the above
> classifications, then I would like to suggest the name "herculean"
> temperament for the linear temperament with generator ~39:44 and
> period of 1/2-octave that defines the 224-270-494 family.

I'm not too interested in this kind of naming of such complex
temperaments.

I'm still not sure if we need both the 494-ET res and the 612-ET res
JI notation systems (they are so close together in resolution), but
it seems a good idea to have a set which is the largest that doesn't
use accents (the 494 Herculean).

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> > FYI, our names for Sagittal symbol sets (which correlate to
levels of
> > JI precision) have names that you may be interested in:
> > ...
> > "herculean" - the symbol set for high-precision JI, with no
symbols
> > differentiated by accent marks (with a resolution on the order of
494-
> > ET, it uses symbols which better distinguish primes 19 and 23;
but
> > this is *not* a 494-ET mapping!!!)
> >
> > "promethean" - the symbol set for high-precision JI, with some
pairs
> > of symbols differentiated by accent marks (with a resolution on
the
> > order of 612-ET, tends to favor complex 13-limit JI; again, *not*
a
> > 612-ET mapping!!!)
> > ...
> > Please note: Dave and I have put off the question of whether
there
> > should be two different high-precision versions of Sagittal JI
(but
> > if not, then the "herculean" designation would probably be used
for
> > JI with ~612-ET resolution).
> >
> > However, if Dave decides to agree with me about the above
> > classifications, then I would like to suggest the
name "herculean"
> > temperament for the linear temperament with generator ~39:44 and
> > period of 1/2-octave that defines the 224-270-494 family.
> ...
> I'm still not sure if we need both the 494-ET res and the 612-ET
res
> JI notation systems (they are so close together in resolution), but
> it seems a good idea to have a set which is the largest that
doesn't
> use accents (the 494 Herculean).

This division of the apotome into 47 steps would actually require one
accented symbol, '|~) -- two if you count its apotome-
complement .~|| -- but it would not contain the unaccented versions
of these). So it would need to be described as having no symbols
differing only by accent marks.

The other nice feature about this symbol set would be of interest to
those (like me) who want to use harmonics above 17. The *exact*
symbols for the 19-schisma (512:513), 19-comma (19456:19683), 23-
comma (729:736), and 23S-diesis (16384:16767) are all in this set
(for exact alternate spellings for harmonics and subharmonics 19 and
23).

In the 612-ET-res symbol set (of 58 steps per apotome) I believe that
we were taking the approach of using fewer flag combinations along
with accents in order to arrive at a JI symbol set with a lower prime
limit. Instead of the 19-comma )|~ and 23S-diesis symbols ~|\, there
would be symbols for the diaschisma ./| and meantone diesis .//|.

The 47-step high-precision JI notation happens to be *extremely*
important to me, since I wish to notate high-prime-limit JI rather
than extended 7, 11, or 13-limit JI. If we have already gone to the
trouble of defining exact symbols for 19C and 23S, then why can't I
have a high-precision JI symbol set that provides them?

My objective is to provide composers who use different approaches to
JI with symbol sets that will best accommodate their varying needs.
I suspect that most performers won't want to read parts in high-
precision JI and that these will end up being translated into medium-
resolution athenian (or even lower-resolution spartan) level JI for
performance (or with a little luck, perhaps 58-step JI can be done in
such a way that, in most instances, it becomes athenian-level JI if
the accent marks are ignored). But the composer or theorist is apt
to be more picky about using symbols that clearly convey the harmonic
function of each tone, and these higher-resolution symbol sets are
very useful for that purpose.

Part of the Sagittal philosophy is to provide plenty of versatility
in the form of options that may be important to only a limited number
of users, implementing these in such a way that they don't complicate
matters for the majority of users, who are going to have little or no
interest in those options.

Dave, we should let those who are reading this in-house debate of
ours know that we haven't yet finalized our 58-step high-precision JI
symbol set (whereas the 47-step one is fairly straightforward --
almost a no-brainer!) and that this was the primary reason that we
omitted any detailed discussion of high-precision JI in our XH18
article.

--George