441-meantone-Sagittal

While obviously just using sharps and flats makes no sense for
notating 441-et, and hence 7-limit effective JI, one can use more
symbols and get a Saggital system based on the 1/4 comma meantone
fifth rather than the near-JI fifth of 441. To shart with, we have
sharps and flats, and also the Tartini-Sagittal version (I'm thinking
George's new version) of the diesis symbol.

Here are some 7-limit intervals which have Sagittal symbols to go
with them; I list the number of steps of 441, and then in the form +-
n, the number of meantone fifths to get to it. 256/245 has a symbol
since we use #, and "einsteinma" is a nonce name based on the fsct
that 99 is the temperament which tempers out 3136/3125, 5120/5103 and
6144/6125, and Einsteinium is element 99. It's probably a good idea
to have such a two-schisma symbol, as this is the amount the pure
fifth is sharp of the 1/4 comma fifth.

256/245 sharp/minor semitone 28 +7g
128/125 diesis 15 -12g
64/63 septimal comma 10 -155g
81/80 comma 8 -124g
15625/15552~225/224~1029/1024 kleisma 3 +174g
5120/5103~3136/3125~6144/6125 2 einsteinma -31g
32805/32768~65625/65536 schisma 1 +205g

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

> "einsteinma" is a nonce name based on the fsct
> that 99 is the temperament which tempers out 3136/3125, 5120/5103 and
> 6144/6125, and Einsteinium is element 99. It's probably a good idea
> to have such a two-schisma symbol, as this is the amount the pure
> fifth is sharp of the 1/4 comma fifth.

I think I'll propose "quark" as the name for 1/4-comma. In 1/4-comma
meantone, the fifth is flat by a quarter comma, a quar c, or quark. In
441-et, the quark is two schismas, and is the same as 5120/5103 (for
which Sagittal has a symbol.)

441-et is not the only interesting et where we can play this game, by
the way. 1783 is a super-accurate 5-limit tuning, and it has a 5^(1/4)
in it also. 789 is a denominator for a convergent to 5^(1/4), (1783
being a semiconvergent) and also gives a super-accurate version of 1/4-
comma meantone. Twice 789 is 1578, which is a very strong high limit
(29 and stuff) system, so we could do something like the 217 bicycle
chains notation idea with two chains. Other ideas along similar lines
involve 224, 684, 2019, 2460, 3125, 3566, 5144, 5585 and 6691, for
instance. 2460 already is a sort of universal system for Sagittal, and
this would be another bicycle-chain system, as 2460=12*205. Hence it is
12 cycles of the 205 approximation to 1/4 comma, separated by 100
cents, similar to but more elaborate than the 217 business George was
talking about.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> While obviously just using sharps and flats makes no sense for
> notating 441-et, and hence 7-limit effective JI, one can use more
> symbols and get a Saggital system based on the 1/4 comma meantone
> fifth rather than the near-JI fifth of 441. To shart with, we have
> sharps and flats, and also the Tartini-Sagittal version (I'm
thinking
> George's new version) of the diesis symbol.

That "new" symbol pair is well over a half-century old -- or did you
mean new to Sagittal?

Gene, I'm afraid that I've inadvertently opened a can of worms by
suggesting a 217-ET notation with nominals in a chain of 126-deg
(meantone-style) fifths. I never intended it to be used for anything
other than 217, and that only because:
1) it's a multiple of 31, and
2) it would be only for use with instruments specially built for 31-
ET,
and that only because:
3) 31 (as subset of 217) is so well suited to 7-limit adaptive JI,
having primes 5 and 7 in the chain of fifth-generators in the
meantone positions +4 and +10, respectively (which is what AKJ
wanted). This is not true of a chain of 256deg441 fifths, or even of
130deg224.

This type of notation would be conceptually similar to Johnny
Reinhard's 1200-ET notation in that the notes would be modified by
conventional sharps and flats (including doubles) and semi- and
sesqui- Tartini-style symbols. But it would be different in that:

1) It would be intended for specially designed 31-ET instruments
rather than conventional (12-ET) instruments (unfretted strings,
trombone, and voice excepted);
2) The notated steps of 1deg217 (~5.5 cents) are much coarser than
those of 1deg1200 (1 cent);
3) The number of steps of alteration would range from -3 to +3 rather
than -50 to +50;
4) The size of the alterations would range from about -17 to +17
cents rather than -50 to +50 cents, thus decreasing the amount of
pitch-bending required of the player by a factor of 3 (which would be
much less demanding);
5) Instead of numbers placed above the staff,.there would be 3 pairs
of symbols (on the staff, to the left of the notehead and
conventional symbol, if any) to specify the amount of alteration; and
finally,
6) It would not be capable of notating all tunings, but only those
which could be reasonably approximated by 217 (e.g., 21-limit JI).

--George

Siemen Terpstra has worked out notations for many of
the good high Equal Divisions of the Octave. If
anyone is interested, I can look for a 441-EDO
notation in my foot-tall box of Siem's unpublished
papers, or I can try to contact him (he's on a trip in
the far east).

Mark Rankin

--- "George D. Secor" <gdsecor@yahoo.com> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@...>
> wrote:
> >
> > While obviously just using sharps and flats makes
> no sense for
> > notating 441-et, and hence 7-limit effective JI,
> one can use more
> > symbols and get a Saggital system based on the 1/4
> comma meantone
> > fifth rather than the near-JI fifth of 441. To
> shart with, we have
> > sharps and flats, and also the Tartini-Sagittal
> version (I'm
> thinking
> > George's new version) of the diesis symbol.
>
> That "new" symbol pair is well over a half-century
> old -- or did you
> mean new to Sagittal?
>
> Gene, I'm afraid that I've inadvertently opened a
> can of worms by
> suggesting a 217-ET notation with nominals in a
> chain of 126-deg
> (meantone-style) fifths. I never intended it to be
> used for anything
> other than 217, and that only because:
> 1) it's a multiple of 31, and
> 2) it would be only for use with instruments
> specially built for 31-
> ET,
> and that only because:
> 3) 31 (as subset of 217) is so well suited to
> 7-limit adaptive JI,
> having primes 5 and 7 in the chain of
> fifth-generators in the
> meantone positions +4 and +10, respectively (which
> is what AKJ
> wanted). This is not true of a chain of 256deg441
> fifths, or even of
> 130deg224.
>
> This type of notation would be conceptually similar
> to Johnny
> Reinhard's 1200-ET notation in that the notes would
> be modified by
> conventional sharps and flats (including doubles)
> and semi- and
> sesqui- Tartini-style symbols. But it would be
> different in that:
>
> 1) It would be intended for specially designed 31-ET
> instruments
> rather than conventional (12-ET) instruments
> (unfretted strings,
> trombone, and voice excepted);
> 2) The notated steps of 1deg217 (~5.5 cents) are
> much coarser than
> those of 1deg1200 (1 cent);
> 3) The number of steps of alteration would range
> from -3 to +3 rather
> than -50 to +50;
> 4) The size of the alterations would range from
> about -17 to +17
> cents rather than -50 to +50 cents, thus decreasing
> the amount of
> pitch-bending required of the player by a factor of
> 3 (which would be
> much less demanding);
> 5) Instead of numbers placed above the staff,.there
> would be 3 pairs
> of symbols (on the staff, to the left of the
> notehead and
> conventional symbol, if any) to specify the amount
> of alteration; and
> finally,
> 6) It would not be capable of notating all tunings,
> but only those
> which could be reasonably approximated by 217 (e.g.,
> 21-limit JI).
>
> --George
>
>
>

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--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@...> wrote:
>
> Siemen Terpstra has worked out notations for many of
> the good high Equal Divisions of the Octave. If
> anyone is interested, I can look for a 441-EDO
> notation in my foot-tall box of Siem's unpublished
> papers, or I can try to contact him (he's on a trip in
> the far east).
>
> Mark Rankin

Hi Mark,

Yes, George and I would be interested in any such large ET notations.
We're unlikely to use the specific symbols but may well use their
semantics (with credit given), in whatever we finally recommend as the
off-the-shelf Sagittal notations for these. I say "off-the-shelf"
because various people will have various purposes in using such large
ETs and may have good reasons for variations from whatever we
recommend, but at least all would have a common point of departure.

Here's a possible set of saggital accidentals (in the long ASCII
representation) for successive degrees of 441-EDO. It assumes that
nominals, sharps and flats are in a chain of near-just fifths.

'| |( '|( )~| ')~| |~ .|\ |\ '|\ |) '|)
1 2 3 4 5 6 7 8 9 10 11

(| ~|) (|( .//| //| '//| /|) '/|) (/| '(/|
12 13 14 15 16 17 18 19 20 21

The 5-schisma accents are characters 38 (grave) and 250 (acute). The
cores (unaccented symbols) are characters 146, 149, 151, 154, 156,
159, 160, 162, 164, 166, 169 in the new Sagittal-2 font.

Note that the flag arithmetic is consistent, e.g. /| + |) = /|) in
degrees, where /| is the 5-comma and |) the 7-comma and /|) the
35-M-diesis.

-- Dave K