Yahoo Tuning Groups Ultimate Backup tuning-math Extreme precison (Olympian) Sagittal

Hi Gene,

Sorry for gaving you a hard time about jargon on "tuning". But I
think you'd rather get it from me, now, than have it build up until
you get attacked by a serious mob of maurauding math-phobes. :-)

If you're not too pissed off with me to still be interested, here's
one way of approaching the extreme-precision Sagittal question.

The raw data you need is I think contained in Table 1 and Figure 3
of the XH paper.

http://dkeenan.com/sagittal/Sagittal.pdf

In table 1, you can ignore everything but the first comma listed for
each symbol (not necessarily the bold one, but the topmost one, the
one on the same line as the symbol). All the other roles for a
symbol are allowed to break, and be taken over by another (possibly
accented) symbol in the extreme-precision set.

And ignore the second row of symbols in Figure 3 since their
definitions are purely as apotome complements of those on the first
line no matter what set we are using.

The idea then is to treat each of the 9 flags (left and right of
barb, arc, scroll, boathook; plus accent) as a generator and find
the optimum value in cents for each so as to minimise the maximum
error over all the symbol/comma relationships in table 1.

There may be some other important symbol-comma relationships not
shown in table 1, that we don't want to break, but I'd like to see
what happens when we just base it on those first.

Due to the apotome complementarity we have one constraint on the
flag values initially, so we are down to 8 degrees of freedom. That
is that /| + |\ + (| + |) = apotome. Although you could ignore that
and see what happens.

Then once you have the value of each flag you can calculate the
value of each symbol in the top row of figure 3 and the same with up
accents and the same with down accents. And maybe figure out what 23-
limit comma might be assigned as the primary role of each.
Commas which notate more popular (simpler or lower-prime-limit)
ratios are to be preferred, as are commas which notate ratios
without having to go too far up or down the chain of fifths.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> The idea then is to treat each of the 9 flags (left and right of
> barb, arc, scroll, boathook; plus accent) as a generator and find
> the optimum value in cents for each so as to minimise the maximum
> error over all the symbol/comma relationships in table 1.

I can't decypher this well enough to tackle the problem, and looking
at the Sagittal symbols and trying to figure out flags is just going
to confuse me. Could you post, very explicitly, what you are seeking?
What are the specific symbol/comma relationships you want
approximated, in terms of the nine generators?

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> > The idea then is to treat each of the 9 flags (left and right of
> > barb, arc, scroll, boathook; plus accent) as a generator and find
> > the optimum value in cents for each so as to minimise the maximum
> > error over all the symbol/comma relationships in table 1.
>
> I can't decypher this well enough to tackle the problem, and looking
> at the Sagittal symbols and trying to figure out flags is just going
> to confuse me. Could you post, very explicitly, what you are
seeking?
> What are the specific symbol/comma relationships you want
> approximated, in terms of the nine generators?

Gene, I've also had trouble wrestling with this, but since I've had a
few months to ponder it, let me give you my take on it.

I believe that Dave is trying to arrive at a Sagittal JI notation
with the highest precision possible *without* introducing any new
symbol-elements beyond the left and right barb, arc, scroll, and
boathook in combination with no/up/down (5-schisma) accent marks.
This precision would hopefully be something on the order of 1-cent
increments. Dave has determined that 1171-ET is the highest one that
can be notated with these constraints, and the extreme-precision
(olympian) JI that he's proposing is modeled after that.

Personally, I feel that going very much beyond the precision offered
by 612-ET is overkill in the extreme and that the economy sacrificed
by using separate symbols for 13-limit consonances is a high price to
pay for this. Still, if someone really thinks they might want it,
then I have no objection to having it available.

BTW, I should mention that we *actually use* one of these olympian-
level symbols in the 12-relative (trojan) symbol set, for 5 degrees
of 144-edo. Since there will probably be other instances in which
one of these rarely used symbols will need to be pressed into
service, I'll concede that olympian symbol development can have
benefits.

Gene, please see what you can make of this. I don't think that
you'll be able to find a good alternative to 1171; 1224 was my choice
as best division 1-cent resolution, but it can't be done completely
with our existing symbol elements (although we come very close! -- we
lack only 35, 55, and 57 steps up to 1/2-apotome).

The best division I could find that makes a distinction between
351:351 and 5103:5120 (11:13s & 5:7s), between 45:46 and 1664:1701
(115S & 7:13S), and also between 6400:6561 and 39:40 (25S & 5:13S) is
742-ET (612+130). Instead of distinguishing between 1024:1053 and
35:36 (13M and 35M) and their complements (8192:8505 and 26:27, 13L
and 35L), it distinguishes between 1892:8505 and 512000:531441 (35L
and 125L, the latter being (81/80)^3). This retains the vanishing
status of the "linchpin" schismina 4095:4096. But it doesn't even
come close to the 1-cent resolution that Dave desired, so it is
doubtful that this one really buys us anything.

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Gene, I've also had trouble wrestling with this, but since I've had a
> few months to ponder it, let me give you my take on it.

I was just hoping for a statement of the problem which did not require
me to read things I have to squint at and still can't make out. What
are the barbs, arcs, scrolls, boathooks and accent marks supposed to
do, in numerical terms?

> The best division I could find that makes a distinction between
> 351:351 and 5103:5120 (11:13s & 5:7s), between 45:46 and 1664:1701
> (115S & 7:13S), and also between 6400:6561 and 39:40 (25S & 5:13S) is
> 742-ET (612+130).

(5120/5103)/(352/351) = 2080/2079
(1701/1644)/(46/45) = 76545/76544
(6561/6400)/(40/39) = 256000/255879

The first and last are 13-limit schisminas, with a TM reduction
{2080/2079, 59319/59290}. The other is tiny, and drags us all of the
way up to the 23-limit; it TM reduces to {2025/2024, 2080/2079,
35000/34983}. This defines a temperament with six generators, but we
can cut that down to four if we ignore 17 and 19, which aren't in the
picture. However, we are looking ets in which *none* of these commas
vanish, which are quite common, though 742 with its best tuning isn't
one of them, so I'm a little confused. Possibilies include 270, 311,
684, 1178, 1506, 2190, 2684 etc. For the "linchpin" comma to vanish
while the rest of these do not is also not uncommon; 270, 311, 581,
764, 1012, 1106, 1236, 1506, 1600, 2742 etc. etc. It looks to me that
1506, a strong 13-17 limit system, would be a good choice; also 1600
and 2742 are examples of decent 23-limit systems which fit the bill
and which clearly would give good accuracy.

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I was just hoping for a statement of the problem which did not
require
> me to read things I have to squint at and still can't make out.
What
> are the barbs, arcs, scrolls, boathooks and accent marks supposed
to
> do, in numerical terms?

I understand it may be difficult to figure out the flag-
decomposition of some of the symbols in Figure 3 of the paper, but
those in Table 1 should be obvious enough.

I realise now that you should ignore Figure 3, and see if you come
up with the same new unaccented symbols that we did.

If you're having to squint to figure out the flag-composition of the
symbols in Table 1, then maybe you need to set the zoom on your PDF
reader to exactly 100%, or maybe 200%. But just in case, here they
are in the same order as in Table 1, in ASCII longhand:
'| )| |( ~| )|( ~|( |~ ./| )|~ /| |) |\ (| (|
( //| /|) /|\ (|) (|\

So you could set up 9 columns, one for each flag, with a row for
each symbol. The accent column would have either -1, 0 or +1, and
all the others would have 0, 1 or 2. For any symbol, the sum of the
numbers in the non-accent columns must never be greater than 2.

Remember it's only the first comma definition for each that has to
be preserved. And you only need to consider single-shaft upward
symbols smaller than a semiapotome (56.84 c).

I think you wrote elsewhere that you thought with this "Olympian"
Sagittal we were on a quest for the ultimate notation. That's not
really the case. It's just that we came up with symbols that did
everything we thought anyone actually needed to do, and then we
said, "Well we have these 9 flag types, there must be plenty of
spare capacity here. How far can we go with what we've got (for JI
and ETs)?"

The rules of the game are:

0. No new flag types.
1. Every flag must be assigned an approximate value in cents, and
every symbols must represent a comma close to its sum-of-flags value
(no more than about 0.5 c away).
2. Start with the symbols in Table 1 of
http://dkeenan.com/sagittal/Sagittal.pdf
and ensure they keep their primary comma roles (on same line as
symbol, not necc. in bold).
3. Allow new symbols to be formed by adding either an up or down
accent to any of the unaccented symbols in Table 1 or in the
Athenian or Trojan sets. See below.
4. No new unaccented symbols are to be created (from new pairings of
existing flags) if an accented version of an existing (Table 1 or
Athenian or Trojan) symbol will do.
5. If new symbols are created, they must have no more than two
ordinary flags (in the same direction) and one accent (up or down).

I'm pretty sure the only symbols in Athenian or Trojan but not in
Table 1 are )/| and /|~

Have fun.

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

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> If you're having to squint to figure out the flag-composition of the
> symbols in Table 1, then maybe you need to set the zoom on your PDF
> reader to exactly 100%, or maybe 200%. But just in case, here they
> are in the same order as in Table 1, in ASCII longhand:
> '| )| |( ~| )|( ~|( |~ ./| )|~ /| |) |\ (| (|
> ( //| /|) /|\ (|) (|\

If there are 9 flags, can't you just give a vector of nine integers or
something of that sort? Or if that is too much, what are the intervals
which go to each set of flags in your ascii notation? This stuff
should already be in the possession of you and George.

> So you could set up 9 columns, one for each flag, with a row for
> each symbol.

It would be easy enough if I had a table, in ascii, posted to this
list, which gave a rational number followed by a 9-vector on each line.

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> It would be easy enough if I had a table, in ascii, posted to this
> list, which gave a rational number followed by a 9-vector on each
line.

The more of our work you duplicate, the more chance of finding our
mistakes. But here it is, tab-delimited, with column headings.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

hi Dave,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> It would be easy enough if I had a table, in ascii,
> posted to this list, which gave a rational number
> followed by a 9-vector on each line.

if you would furnish me with a table of ASCII-sagittal
symbols with their ratios, i'll put it into the "sagittal"
definition webpage.

you might want to look at this and use the same format;
you'll find many of your ratios in it:

http://tonalsoft.com/enc/interval-list.htm

-monz

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > It would be easy enough if I had a table, in ascii, posted to this
> > list, which gave a rational number followed by a 9-vector on each
> line.
>
> The more of our work you duplicate, the more chance of finding our
> mistakes. But here it is, tab-delimited, with column headings.
>
> num den '| )| (| /| ~| |~ |\
> |) |( symbol comma name
> 32768 32805 1 0 0 0 0 0 0
> 0 0 '| 5-schisma
>
> etc. -- snip>

i've put a neat copy of it, with monzos added, into the
Encyclopaedia "sagittal" page.

-monz

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > It would be easy enough if I had a table, in ascii, posted to
this
> > > list, which gave a rational number followed by a 9-vector on
each line.
> >
> > The more of our work you duplicate, the more chance of finding
our
> > mistakes. But here it is, tab-delimited, with column headings.
> >
> > num den '| )| (| /| ~| |~ |\
> > |) |( symbol comma name
> > 32768 32805 1 0 0 0 0 0
0
> > 0 0 '| 5-schisma
> >
> > etc. -- snip>
>
> i've put a neat copy of it, with monzos added, into the
> Encyclopaedia "sagittal" page.

Monz, please delete the line for symbol /|~ (the one with the
question marks), since it isn't used in its primary definition in any
symbol set below the olympian-level (extreme-precision). In the
trojan (12-relative) set (for 144-edo) it represents 45:46 (a
secondary symbol definition), whereas in olympian it represents
1664:1701 (its primary symbol definition; but from the question marks
might I infer that Dave has second thoughts about this?).

Whatever the case, listing it here could be confusing to those not
interested in extreme-precision notation, since 1664:1701 is
approximated by (|( in all versions of Sagittal with less-than-
extreme precision.

--George

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

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> Monz, please delete the line for symbol /|~ (the one with the
> question marks),

Agreed.

> since it isn't used in its primary definition in any
> symbol set below the olympian-level (extreme-precision). In the
> trojan (12-relative) set (for 144-edo) it represents 45:46 (a
> secondary symbol definition), whereas in olympian it represents
> 1664:1701 (its primary symbol definition; but from the question
marks
> might I infer that Dave has second thoughts about this?).

Yes. The Scala-archive ratio counts are as follows

Comma Name Number of exactly-notatable ratio ocurrences
-------------------------------------------------------------------
524288:535815 245S 165
1664:1701 7:13S 145
45:46 5:23S 45

So it's clear that 5:23S must remain a secondary role, but it's not
so clear between 245S and 7:13S. The difference between these two
commas is only 0.36 cents, so the decision isn't all that important.

245 = 5 * 7 * 7

One consideration might be if there was another comma that could
notate one of these, that might be given the primary role for some
other symbol, so they could each have a primary role of some symbol
that could be used to notate them exactly. But no. The only other
commas worth considering for these are 245C and 7:13C and these are
both only secondary roles for the symbol ~|( whose primary role was
won hands-down by 17C, since it has 318 notatable ocurrences in the
Scala archive.

Another consideration is the absolute value of the "slope" of the
comma. This is a measure of its usefulness for notating
temperaments, as it gives the rate of change in the size of the
comma relative to the apotome, with changes in the size of the
notational fifth. Low slope is better. 245S has a slope of 4.7 while
7:13S has 2.7.

Let's wait and see what Gene comes up with.

> Whatever the case, listing it here could be confusing to those not
> interested in extreme-precision notation, since 1664:1701 is
> approximated by (|( in all versions of Sagittal with less-than-
> extreme precision.

Yes. And so is 245S (524288:535815)

I only included /|~ in that list for Gene because it is a flag-pair
that _must_ exist in Olympian, since it already exists in Trojan
(medium precision 12-ET-relative notation), but since we were forced
to use only a secondary role for it in Trojan (an extremely rare
event), its primary role remains open. All the other symbols I
listed have their primary roles set in stone by now.

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

Monz,

I just realised you should delete the line for the symbol )/| from
that table on your sagittal page, as well as the /|~ symbol.

Now I know why neither of these appeared in Table 1 of the XH paper.

)/| also does not necessarily have its primary role in the 12-
relative (trojan) notation. The role of 13-comma (6561:6656) might
well be more important.

So Gene, as for /|~, all you need to know about )/| for the purposes
of designing an Olympian notation, is that it must exist. You can
choose a different comma for it if you wish.