Notating Hemififths

This is an important temperament, so it should go onto the notation "to do" list.

As a 13-limit temperament, it has the following wedgie:

[2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22]

We have generators (half of a fifth, or a neutral third) in the following range:

12/41<65/222<53/181<41/140<70/239<29/99<46/157<63/215<17/58

The vals corresponding to 58, 99, 140 and 239 which cover it are as follows:

l58 := [58, 92, 135, 163, 201, 215]
l99 := [99, 157, 230, 278, 343, 367]
l140 := [140, 222, 325, 393, 485, 519]
l239 := [239, 379, 555, 671, 828, 886]

The errors for the first five odd primes using these vals are:

err58 := [1.493275, 6.789735, 3.587887, 7.302747, 7.748202]
err99 := [1.075302, 1.565075, .871064, 6.257815, 7.957188]
err140 := [.902142, -.599427, -.254477, 5.824914, 8.043769]
err239 := [.973869, .297166, .211751, 6.004233, 8.007905]

70/239 is poptimal in the 7 and 9 limits, 29/99 in the 11-limit, and
17/58 in the 13-limit. How to sort all of this out? How much would the notations differ?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> This is an important temperament, so it should go onto the
notation "to do" list.
>
> As a 13-limit temperament, it has the following wedgie:
>
> [2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22]
>
> We have generators (half of a fifth, or a neutral third) in the
following range:
>
> 12/41<65/222<53/181<41/140<70/239<29/99<46/157<63/215<17/58
>
> The vals corresponding to 58, 99, 140 and 239 which cover it are as
follows:
>
> l58 := [58, 92, 135, 163, 201, 215]
> l99 := [99, 157, 230, 278, 343, 367]
> l140 := [140, 222, 325, 393, 485, 519]
> l239 := [239, 379, 555, 671, 828, 886]
>
> The errors for the first five odd primes using these vals are:
>
> err58 := [1.493275, 6.789735, 3.587887, 7.302747, 7.748202]
> err99 := [1.075302, 1.565075, .871064, 6.257815, 7.957188]
> err140 := [.902142, -.599427, -.254477, 5.824914, 8.043769]
> err239 := [.973869, .297166, .211751, 6.004233, 8.007905]
>
> 70/239 is poptimal in the 7 and 9 limits, 29/99 in the 11-limit, and
> 17/58 in the 13-limit. How to sort all of this out? How much would
the notations differ?^

For 41, and 58: 3 = +2G, 11 = +5G, 13 = -1G

99, 140, 157 are not 1,3,11,13-consistent; they take 11 as one degree
lower than +5G and 13 as one degree higher than -1G.

181 is 1,3,11,13-consistent, but takes 11 as one degree lower than
+5G and 13 as one degree lower than -1G.

To put it another way, none of those that are larger than 58-ET take
either 9:11 or 13:16 as half of 2:3. It looks like we need something
below the 11 limit.

To digress for a moment, if you consider only 41 and 58, you might as
well include all of these:

9deg31 < 7deg24 < 12deg41 < 17deg58 < 5deg17 < 3deg10

which lets you notate using the 11-diesis:

1/1 27/22 3/2 18/11 9/8 11/8 27/16
C E\!/ G B\!/ D F/|\ A

But you're considering a narrower range that includes higher
divisions, so you want a 7-limit just ratio that's approximately half
of 2:3. The simplest one is 40:49, which is good for all the ET's
you gave except for 215 (which is not 1,7,49-consistent).
Fortunately, Dave has just proposed some symbols involving 7^2. I
was also looking (previously) for a decent symbol that would notate
half an apotome as nearly as possible (for such divisions as 99 and
311-ET), so it looks as if Dave's proposal of the symbol '|)) for the
5:49' diesis, 392:405, is going to be the answer for both. The
hemififth notation would then be (if we agree to use a period to
symbolize the -5' comma in ascii):

C E.!)) G B.!)) D F'|)) A C'|)) E etc.

--George

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

> > The vals corresponding to 58, 99, 140 and 239 which cover it are as
> follows:
> >
> > l58 := [58, 92, 135, 163, 201, 215]
> > l99 := [99, 157, 230, 278, 343, 367]
> > l140 := [140, 222, 325, 393, 485, 519]
> > l239 := [239, 379, 555, 671, 828, 886]
> For 41, and 58: 3 = +2G, 11 = +5G, 13 = -1G

> 99, 140, 157 are not 1,3,11,13-consistent; they take 11 as one degree
> lower than +5G and 13 as one degree higher than -1G.

That's why I gave the vals; you are supposed to use them, not the "standard" val.

> 181 is 1,3,11,13-consistent, but takes 11 as one degree lower than
> +5G and 13 as one degree lower than -1G.
>
> To put it another way, none of those that are larger than 58-ET take
> either 9:11 or 13:16 as half of 2:3. It looks like we need something
> below the 11 limit.

Your notation system assumes you are using the mapping which rounds n*log2(p) to the nearest integer for primes p and division n?

> To digress for a moment, if you consider only 41 and 58, you might as
> well include all of these:
>
> 9deg31 < 7deg24 < 12deg41 < 17deg58 < 5deg17 < 3deg10

You are wandering out of the good range for the temperament we are trying to notate.

> which lets you notate using the 11-diesis:
>
> 1/1 27/22 3/2 18/11 9/8 11/8 27/16
> C E\!/ G B\!/ D F/|\ A

Can you give me what the 11-diesis is?

> But you're considering a narrower range that includes higher
> divisions, so you want a 7-limit just ratio that's approximately half
> of 2:3. The simplest one is 40:49, which is good for all the ET's
> you gave except for 215 (which is not 1,7,49-consistent).

(49/40)^2/(3/2) = 2401/2400 is a comma of Hemififths, so it works for any of the mappings I gave including the 239 val.

> Fortunately, Dave has just proposed some symbols involving 7^2. I
> was also looking (previously) for a decent symbol that would notate
> half an apotome as nearly as possible (for such divisions as 99 and
> 311-ET), so it looks as if Dave's proposal of the symbol '|)) for the
> 5:49' diesis, 392:405, is going to be the answer for both.

405/392 is seven generator steps of Hemififths, down two octaves. This kind of description is independent of the exact tuning, so perhaps it could be used to figure notations out.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
>
> > > The vals corresponding to 58, 99, 140 and 239 which cover it
are as
> > follows:
> > >
> > > l58 := [58, 92, 135, 163, 201, 215]
> > > l99 := [99, 157, 230, 278, 343, 367]
> > > l140 := [140, 222, 325, 393, 485, 519]
> > > l239 := [239, 379, 555, 671, 828, 886]
> > For 41, and 58: 3 = +2G, 11 = +5G, 13 = -1G
>
> > 99, 140, 157 are not 1,3,11,13-consistent; they take 11 as one
degree
> > lower than +5G and 13 as one degree higher than -1G.
>
> That's why I gave the vals; you are supposed to use them, not
the "standard" val.

I don't know what those numbers are supposed to be indictating, so we
have a language barrier. Until now there was no pressing need for me
to learn your mathematical notation for some of these things, but I
did jot down a reference that I saw in a recent message that some of
these things are explained beginning in message #682, so I have some
reading to do. Are there any other things that might be helpful?

Anyway, in looking at the problem, I'm just trying to arrive at
symbols that will notate whatever is in your range of generators,
including as many of the ET's as possible.

> > 181 is 1,3,11,13-consistent, but takes 11 as one degree lower
than
> > +5G and 13 as one degree lower than -1G.
> >
> > To put it another way, none of those that are larger than 58-ET
take
> > either 9:11 or 13:16 as half of 2:3. It looks like we need
something
> > below the 11 limit.
>
> Your notation system assumes you are using the mapping which rounds
n*log2(p) to the nearest integer for primes p and division n?

That's correct.

> > To digress for a moment, if you consider only 41 and 58, you
might as
> > well include all of these:
> >
> > 9deg31 < 7deg24 < 12deg41 < 17deg58 < 5deg17 < 3deg10
>
> You are wandering out of the good range for the temperament we are
trying to notate.
>
> > which lets you notate using the 11-diesis:
> >
> > 1/1 27/22 3/2 18/11 9/8 11/8 27/16
> > C E\!/ G B\!/ D F/|\ A
>
> Can you give me what the 11-diesis is?

It's 32:33. For 27/22 and 18/11 rational sagittal notation would use
the symbol (!), which represents the 11' diesis, 704:729. But since
726:729 vanishes in these divisions, the 11 diesis symbol suffices.

Since we're still lacking a formal presentation of the sagittal
notation, we have another language barrier. I made a file that
provides a quick reference to ascii simulations of single-shaft
sagittal symbols for you (and anyone else interested):

/tuning-
math/files/secor/notation/quickref.txt

This includes our latest notation for the down and up 5' comma
(or "schisma", 32768:32805), which will *not* be appearing in my XH18
paper. So maybe now these symbols won't seem so cryptic.

> > But you're considering a narrower range that includes higher
> > divisions, so you want a 7-limit just ratio that's approximately
half
> > of 2:3. The simplest one is 40:49, which is good for all the
ET's
> > you gave except for 215 (which is not 1,7,49-consistent).
>
> (49/40)^2/(3/2) = 2401/2400 is a comma of Hemififths, so it works
for any of the mappings I gave including the 239 val.

Yes.

> > Fortunately, Dave has just proposed some symbols involving 7^2.
I
> > was also looking (previously) for a decent symbol that would
notate
> > half an apotome as nearly as possible (for such divisions as 99
and
> > 311-ET), so it looks as if Dave's proposal of the symbol '|)) for
the
> > 5:49' diesis, 392:405, is going to be the answer for both.
>
> 405/392 is seven generator steps of Hemififths, down two octaves.
This kind of description is independent of the exact tuning, so
perhaps it could be used to figure notations out.

Yes, I would like to see a notation that will work for the whole
range of generators. After taking a closer look at this I see that 5
is +25G (generator steps) and 7 is +13G. I said that 215 isn't
1,7,49-consistent, but now I see that the real reason that this
notation won't apply to it is that ~4:5 (taken as +25G) is so large
that it isn't the best 4:5 of 215.

--George

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

> > Your notation system assumes you are using the mapping which
rounds
> n*log2(p) to the nearest integer for primes p and division n?
>
> That's correct.

that's terrible -- for example, for 64-equal in the 5-limit, this is
a notably poor choice.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"
> <gdsecor@y...> wrote:
>
> > > Your notation system assumes you are using the mapping which
> rounds
> > n*log2(p) to the nearest integer for primes p and division n?
> >
> > That's correct.
>
> that's terrible -- for example, for 64-equal in the 5-limit, this
is
> a notably poor choice.

But this isn't the whole story. We have two ways of notating 64-ET.

The first uses its native fifth in conjunction with the 13 diesis
(1024:1053) and 13' diesis (26:27) symbols:

64: /|) (|\ /||\

But our preferred method notates it as a subset of 128-ET:

128: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||(
(||~ /||\

This notation uses the 5 and 7 commas, the 11 and 11' dieses, and the
(|( symbol (valid in all three of its comma roles), in addition to
the 19 and 17' commas and the 11:19 diesis. This was not a
particularly easy one for us, and it was only after spending a lot of
time on a lot of different ETs and constantly fine-tuning our symbol
definitions and symbol selection procedure that Dave and I came to an
agreement on this as the 128-ET standard symbol set.

--George

--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
> ... We have two ways of notating 64-ET.
>
> The first uses its native fifth in conjunction with the 13 diesis
> (1024:1053) and 13' diesis (26:27) symbols:
>
> 64: /|) (|\ /||\
>
> But our preferred method notates it as a subset of 128-ET:
>
> 128: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||(
> (||~ /||\
>
> This notation uses the 5 and 7 commas,

Oops, sorry! There's no 7 comma there.

> the 11 and 11' dieses, and the
> (|( symbol (valid in all three of its comma roles),

as the 5:11, 7:13, and 11:17 commas; it's highly desirable for
symbols that have multiple roles to be valid in all of them, and
that, in conjunction with a low product complexity for its primary
comma role, nailed this symbol for 4 degrees.

> in addition to
> the 19 and 17' commas and the 11:19 diesis.

These higher primes were unavoidable. We pick the ones that are
least unusual, and sometimes there's only a single possiblity.

--George

--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"
> > <gdsecor@y...> wrote:
> >
> > > > Your notation system assumes you are using the mapping which
> > rounds
> > > n*log2(p) to the nearest integer for primes p and division n?
> > >
> > > That's correct.
> >
> > that's terrible -- for example, for 64-equal in the 5-limit, this
> is
> > a notably poor choice.
>
> But this isn't the whole story. We have two ways of notating 64-ET.
>
> The first uses its native fifth in conjunction with the 13 diesis
> (1024:1053) and 13' diesis (26:27) symbols:
>
> 64: /|) (|\ /||\
>
> But our preferred method notates it as a subset of 128-ET:
>
> 128: )| ~|( /| (|( (|~ /|\ (|) )|| ~||( ||\ (||(
> (||~ /||\
>
> This notation uses the 5 and 7 commas, the 11 and 11' dieses, and
the
> (|( symbol (valid in all three of its comma roles), in addition to
> the 19 and 17' commas and the 11:19 diesis. This was not a
> particularly easy one for us, and it was only after spending a lot
of
> time on a lot of different ETs and constantly fine-tuning our
symbol
> definitions and symbol selection procedure that Dave and I came to
an
> agreement on this as the 128-ET standard symbol set.
>
> --George

well, 64 is an interesting case. depending on how you map the 5-limit
consonances, 64 can either be:

(a) meantone (80:81 vanishing) -- narrow perfect fifth, narrow major
third, ;

(b) both diminished (648:625 vanishing) and escapade
(4294967296:4271484375 vanishing) -- narrow perfect fifth, wide major
third;

(c) kleismic (15625:15552 vanishing) -- wide perfect fifth, wide
major third.

i imagine the consequences for notation could be frightening . . .
and this is without even proceeding to the 7-limit . . .