Hanson and keemun notation

The minor third generator of hanson temperament divides the 3/1 into 6 equal parts, which means we need accidentals for 1, 2, and 3 commas. The symbols for 1 and 2 commas are obvious, but there are two choices for 3: .//| 128/125 and /|) 250/243. Ordinarily you might expect /|) to be the better choice, since if you carry out the temperament to 53 notes, you'll need //| (so leaving out the accent isn't an option). And in catakleismic temperament, /|) works out to be the same as in hanson. But it's a different interval in keemun. One way to deal with this might be to ignore the difference and simply avoid /|) in notating keemun. Alternatively, the accentless substitute ~|\\ could be used for .//| -- which would make keemun[34] look something like this:

... 08 F 17 A!!/ 26 C)!!( 01
01 D~|\\ 10 F)||( 19 A\! 28 C 03 E!!/ 12 G)!!( 21
21 A~|\\ 30 C)||( 05 E\! 14 G 23 B!!/ 32 D)!!( 07 F~!//
07 E~|\\ 16 G)||( 25 B\! 00 D 09 F/| 18 A)!!( 27 C~!//
27 B~|\\ 02 D)||( 11 F||\ 20 A 29 C/| 04 E)!!( 13 G~!//
13 22 A)||( 31 C||\ 06 E 15 G/| 24 B)!!( 33 D~!//
33 08 E)||( 17 G||\ 26 B

This could work, but another advantage of /|) for hanson is that the arithmetic works out.

/| [-4, 4, -1> (-5, +19)
/|) [1, -5, 3> (+4, -15)
)||( [-3, -1, 2> (-1, +4)
||\ [-7, 3, 1> (-6, +23)

Keemun has a different symbol arithmetic (in this case, the flags are simply added).

/| [-4, 4, -1> (-5, +19)
|) [6, -2, 0, -1> (+4, -15)
/|) [2, 2, -1, -1> (-1, +4)

So here's a case where a symbol might mean one thing in a 5-limit temperament, but something different in a related 7-limit temperament. But a case could be made that catakleismic is really the more closely related temperament. There are possible notations for keemun that avoid /|) and still have the arithmetic work out, but they use accented symbols, and (-6, +23) doesn't fit into the arithmetic. So far the only thing I've found that gets the arithmetic right for keemun is this:

'|( [-5, 2, 2, -1> (-5, +19)
.//| [7, 0, -3> (+4, -15)
/|) [2, 2, -1, -1> (-1, +4)
)/|\ [-3, 4, 1, -2> (-6, +23)

That's not very appealing, and it has the confusing /|) -- and even though the symbols add correctly, you can't tell that by looking at them. So it might just be better to forget about the symbol arithmetic and use a modified hanson notation for keemun, with |) taking the place of /|) .

/| [-4, 4, -1> (-5, +19)
|) [6, -2, 0, -1> (+4, -15)
)||( [-3, -1, 2> (-1, +4)
||\ [-7, 3, 1> (-6, +23)

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> The minor third generator of hanson temperament divides the 3/1
into 6
> equal parts, which means we need accidentals for 1, 2, and 3
commas. The
> symbols for 1 and 2 commas are obvious, but there are two choices
for 3:
> .//| 128/125 and /|) 250/243. Ordinarily you might expect /|) to be
the
> better choice, since if you carry out the temperament to 53 notes,
> you'll need //| (so leaving out the accent isn't an option). And in
> catakleismic temperament, /|) works out to be the same as in
hanson. But
> it's a different interval in keemun. One way to deal with this
might be
> to ignore the difference and simply avoid /|) in notating keemun.
> Alternatively, the accentless substitute ~|\\ could be used
for .//| --
> which would make keemun[34] look something like this:

Hmmm, I was wondering where you got ~|\\, because it should be ~|\
(which is defined as 23S, 16384:16767) that could be used for .//|,
since there is no ~|\\ symbol. All 3-flag symbols consist of the ):
flag combined with two straight flags.

Looking through my past messages, I see that I made a mistake in a
table I gave in message #16863, which should have been:

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

Sorry 'bout that!

Herman, I've temporarily uploaded a graphic file that shows symbol
definitions for all single-shaft non-right-accented (and also some of
the simpler right-accented) symbols in the bottom half of the
diagram. For best clarity, zoom to 100% or "actual" size.

/tuning-
math/files/secor/notation/SaBounds.gif

The top half of the diagram shows which symbols cover which portions
of the single-shaft spectrum at various JI levels, which will enable
you to make symbol conversions from one level to another. (Observe
that olympian-to-herculean symbol conversion consists of merely
dropping any right accents.) The symbol shaft for each ratio aligns
with the number of cents in the scale at the very top. The
background color hues (which match those in Microsoft Excel) are also
correlated with the highest prime factor of the non-right-accented
symbol (orange=2, 3, and 5; yellow=7; aqua=11; blue=13; violet=17;
green=19; pink=23).

This diagram will be going on the Sagittal website soon, along with
more documentation, which will list the symbol boundaries (in cents),
so you're getting a sneak preview.

--George

Hi Herman,

There is no ~|\\ symbol. It was a typo.

In designing ET notations we have been very strict about flag
arithmetic, at least in the case of the single-shaft symbols.

If a notation includes a multiflag symbol, _and_ it includes the
single-flag symbol for each flag making up that multiflag symbol, then
the sum of the values of the single-flag symbols must equal the value
of the multiflag symbol.

e.g. if you use /|) and /| and |) then
/| + |) must equal /|)

If that cannot be the case in some temperament, then one of the
symbols should be replaced with another that does not cause flag
arithmetic violation.

I don't believe we've ever violated that principle for single shaft
symbols in a temperament notation. And in JI notations such violations
are always less than 0.5 cents.

We even tried to maintain it with the double-shaft apotome
complements, where the second shaft is considered to have the same
value as (|), but in a few very cases we violated it for double shaft
symbols because it was just too difficult to do otherwise.

I think the same principle should be applied to LT notations.

I note that a stronger condition is possible, and desirable. That is,
we could require that even though some flag never appears on its own,
if it appears in more than one symbol, it must be possible to assign
it a single value that is consistent with all its appearances.

-- Dave Keenan

Dave Keenan wrote:
> Hi Herman,
> > There is no ~|\\ symbol. It was a typo.
> > In designing ET notations we have been very strict about flag
> arithmetic, at least in the case of the single-shaft symbols.
> > If a notation includes a multiflag symbol, _and_ it includes the
> single-flag symbol for each flag making up that multiflag symbol, then
> the sum of the values of the single-flag symbols must equal the value
> of the multiflag symbol.
> > e.g. if you use /|) and /| and |) then > /| + |) must equal /|)
> > If that cannot be the case in some temperament, then one of the
> symbols should be replaced with another that does not cause flag
> arithmetic violation.

That seems reasonable. That pretty much rules out some symbols, though. E.g., )|( would either represent [-5, 2, 2, -1> or [1, -3, 1, -1, 0, 0, 0, 1>, and it would be a matter of luck whether [7, -4, 0, 1, -1> is the same interval in the temperament or not. Unless the temperament never uses )| or |(, it wouldn't be able to use )|(. The same goes for )~| , ~~| , )|~ , ~|\ , etc. Unlike ET's, there isn't a fixed limit for how far it might be necessary to carry out a notation for open-ended temperaments, although there are clear practical limits in most cases. So, although it should be possible to identify all the symbols that will be needed for some temperament, it's always possible that someone might need one or two more for playing in remote keys, and the arithmetic might not always work out. Clearly, though, it's an advantage for a notation if the flag arithmetic does work out.

Typically the problem with symbol arithmetic isn't with the flags, but with the temperament. The sensisept notation I suggested uses /| and |) which add properly to /|), and /| + /| also adds to //| . But adding |) to //| results in )||( . This isn't a problem since there isn't a //|) symbol, but if I happen to run across a set of symbols that adds correctly, I'd consider using that (unless it violates some other constraint, such as intervals not going negative).

> I don't believe we've ever violated that principle for single shaft
> symbols in a temperament notation. And in JI notations such violations
> are always less than 0.5 cents.
> > We even tried to maintain it with the double-shaft apotome
> complements, where the second shaft is considered to have the same
> value as (|), but in a few very cases we violated it for double shaft
> symbols because it was just too difficult to do otherwise.
> > I think the same principle should be applied to LT notations.
> > I note that a stronger condition is possible, and desirable. That is,
> we could require that even though some flag never appears on its own,
> if it appears in more than one symbol, it must be possible to assign
> it a single value that is consistent with all its appearances.
> > -- Dave Keenan

That's putting a lot of conditions on that might not be possible to satisfy (and will certainly complicate the process of finding workable notations for temperaments). If the flags always added consistently, the tempered flags would also be consistent, but that isn't always the case. As it is, there's a lot of trial and error that still needs to be done by hand to check these things; it would be easy to miss potential problems.

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

> That seems reasonable. That pretty much rules out some symbols, though.
> E.g., )|( would either represent [-5, 2, 2, -1> or [1, -3, 1, -1, 0, 0,
> 0, 1>, and it would be a matter of luck whether [7, -4, 0, 1, -1> is
the
> same interval in the temperament or not. Unless the temperament never
> uses )| or |(, it wouldn't be able to use )|(.

Just to be sure we're saying the same thing. If )| + |( =/= )|( then
one can still use any two of the three symbols.

-- Dave Keenan

Dave Keenan wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> >> That seems reasonable. That pretty much rules out some symbols, though. >> E.g., )|( would either represent [-5, 2, 2, -1> or [1, -3, 1, -1, 0, 0, >> 0, 1>, and it would be a matter of luck whether [7, -4, 0, 1, -1> is
> the >> same interval in the temperament or not. Unless the temperament never >> uses )| or |(, it wouldn't be able to use )|(. > > Just to be sure we're saying the same thing. If )| + |( =/= )|( then
> one can still use any two of the three symbols.
> > -- Dave Keenan

True; there could be a case where one of the three symbols is unusable for other reasons (tempered out, negative, or beyond the range needed for notation). In a case like that, the other two symbols can be used.