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 . . .