Notating Sub-Group Vals?

When you give mappings of generators and periods for temperaments, like:
<0 1 4]
<1 1 0]

(for meantone), that's a val, right?

Whatever they're called, I like using them quite a bit, because they are a really compact way of presenting the structure and functionality of a temperament. I have decided that they will be included in my "Field Guide to Alternative EDOs". However, since the book is focused pretty heavily on triadic (and occasionally tetradic) harmony, I use sub-groups *heavily*, and I also want to give mappings for intervals within a single octave (since I'm focusing on triads that span less than an octave). So what I want to do is precede this notation with a list of the intervals being mapped, and also (since I'm writing for absolute novices) I want to make it obvious which part is the generator and which part is the period. Since I expect my book to be the first exposure to these ideas for a lot of people, I want to be very careful with the conventions I'm setting, so I'd like to know what the most correct way to notate this would be. What I've been doing in the first draft is this:

{interval, interval, interval...} g:<mapping mapping mapping...] p:<mapping mapping mapping...]

Is this okay or is there a better way?

-Igs

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> When you give mappings of generators and periods for temperaments, like:
> <0 1 4]
> <1 1 0]
>
> (for meantone), that's a val, right?

No, it's either a list of vals or a matrix, depending on how you want to look at it. You can call it a mapping, which is what Graham's site does.

>So what I want to do is precede this notation with a list of the intervals being mapped, and also (since I'm writing for absolute novices) I want to make it obvious which part is the generator and which part is the period.

You might find this useful:

http://xenharmonic.wikispaces.com/Smonzos+and+Svals

If you want to drop the part about using a canonical form for the subgroup you could still call the result a "mapping". Graham's program puts the subgroup generators above the corresponding columns of the mapping matrix, but takes input in the form of intervals separated by periods.

>Since I expect my book to be the first exposure to these ideas for a lot of people, I want to be very careful with the conventions I'm setting, so I'd like to know what the most correct way to notate this would be. What I've been doing in the first draft is this:
>
> {interval, interval, interval...} g:<mapping mapping mapping...] p:<mapping mapping mapping...]
>
> Is this okay or is there a better way?

I have no idea, since you don't give a specific example.
> -Igs
>

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

> No, it's either a list of vals or a matrix, depending on how you want to look at it. You can call it a mapping, which is what Graham's site does.
>

Cool. Mapping it is.

> I have no idea, since you don't give a specific example.

Okay, here's an example using meantone:

{2/1, 3/2, 5/4} g:<0 1 4] p:<1 0 -2]

Mostly I'm concerned about using the "right" kinds of brackets and such. What do you think? I'd prefer to do it as a matrix or a table, and will when I can, but I need to use in-line formatting sometimes.

-Igs

gene wrote:

>> When you give mappings of generators and periods for temperaments, like:
>> <0 1 4]
>> <1 1 0]
>>
>> (for meantone), that's a val, right?
>
>No, it's either a list of vals

In particular, it is two vals, one on top of the other.

igs wrote:

> So what I want to do is precede this notation with a list of the
> intervals being mapped, and also (since I'm writing for absolute
> novices) I want to make it obvious which part is the generator
> and which part is the period.

2 7 9
<0 1 4| generator
<1 1 0| period

Problems solved.

-Carl

igs wrote:

>Cool. Mapping it is.

It's a good name.

>Okay, here's an example using meantone:
>
>{2/1, 3/2, 5/4} g:<0 1 4] p:<1 0 -2]

That's gawdawful!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
> >{2/1, 3/2, 5/4} g:<0 1 4] p:<1 0 -2]
>
> That's gawdawful!

Okay, yes. Yes it is. Can you come up with a better in-line format, though? Maybe I should go for a totally different, non-val format?

-Igs

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> 2 7 9
> <0 1 4| generator
> <1 1 0| period

Except that everyone's been putting the period on top and the generator on the bottom. Why not stick to what Graham's site gives you?

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> > >{2/1, 3/2, 5/4} g:<0 1 4] p:<1 0 -2]
> >
> > That's gawdawful!
>
> Okay, yes. Yes it is. Can you come up with a better in-line format, though? Maybe I should go for a totally different, non-val format?

I'd do something like 2.3.5/4 [<1 0 -2|, <0 1 4|]. Except that since it's the five-limit, I can assume we are using actual vals and then I'd just write [<1 0 -4|, <0 1 4|] where the 2.3.5 is implicit.

igs wrote:
>> >{2/1, 3/2, 5/4} g:<0 1 4] p:<1 0 -2]
>>
>> That's gawdawful!
>
>Okay, yes. Yes it is. Can you come up with a better in-line format,

Why does it have to be inline? What's your vector, Victor? -Carl

gene wrote:
>> 2 7 9
>> <0 1 4| generator
>> <1 1 0| period
>
>Except that everyone's been putting the period on top and the
>generator on the bottom. Why not stick to what Graham's site gives you?

I just labeled what Igs gave. I agree period should go on top. -C.

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

>
Except that everyone's been putting the period on top and the generator on the bottom. Why not stick to what Graham's site gives you?
>

Really? Man, I swear I've seen it the other way around in other places...but man, this makes Graham's site make a lot more sense now.

-Igs

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> >
> Except that everyone's been putting the period on top and the generator on the bottom. Why not stick to what Graham's site gives you?
> >
>
> Really? Man, I swear I've seen it the other way around in other places...but man, this makes Graham's site make a lot more sense now.

On old postings. Graham and I both have Hermite normal form on the brain.

"genewardsmith" <genewardsmith@sbcglobal.net> wrote:

> > Except that everyone's been putting the period on top
> > and the generator on the bottom. Why not stick to what
> > Graham's site gives you?
>
> On old postings. Graham and I both have Hermite normal
> form on the brain.

I've always put the period first because I put the octave
first in JI matrices. The period is a generalization of
the octave. People have suggested I put the generator
mapping first, because it's what you use to recognize the
temperament class. They have a point. There are only two
options and you have to pick one.

I don't think there's any chance of confusion because the
generator mapping has a leading zero.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>People have suggested I put the generator
> mapping first, because it's what you use to recognize the
> temperament class. They have a point. There are only two
> options and you have to pick one.

There are more than two options in rank higher than two, and in those cases you obviously want to order the primes 2, 3, 5, etc.

I think I'm going to opt for the generator first, because the period can be ignored when 1 period is an octave due to octave equivalence anyway.

-Igs

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "genewardsmith" <genewardsmith@...> wrote:
>
> > > Except that everyone's been putting the period on top
> > > and the generator on the bottom. Why not stick to what
> > > Graham's site gives you?
> >
> > On old postings. Graham and I both have Hermite normal
> > form on the brain.
>
> I've always put the period first because I put the octave
> first in JI matrices. The period is a generalization of
> the octave. People have suggested I put the generator
> mapping first, because it's what you use to recognize the
> temperament class. They have a point. There are only two
> options and you have to pick one.
>
> I don't think there's any chance of confusion because the
> generator mapping has a leading zero.
>
>
> Graham
>

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> I think I'm going to opt for the generator first, because the period can be ignored when 1 period is an octave due to octave equivalence anyway.

So you plan on using a system different than what Graham or the Xenwiki uses, even though you yourself became confused by the issue when using Graham's site? It's just going to confuse things not to use a consistent standard.

Okay, okay. I'll list the period first. I guess that does have some validity because it tells someone right away whether they're dealing with an octave-repeating scale or not.

Just FYI also, I've opted to use tables/spreadsheets to display mappings rather than ascii-friendly text for the sake of aesthetics and avoiding the use of monospaced fonts. I think it looks better and is easier to read, and since I have the luxury of doing it in OpenOffice I figure I might as well.

-Igs

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > I think I'm going to opt for the generator first, because the period can be ignored when 1 period is an octave due to octave equivalence anyway.
>
> So you plan on using a system different than what Graham or the Xenwiki uses, even though you yourself became confused by the issue when using Graham's site? It's just going to confuse things not to use a consistent standard.
>

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Okay, okay. I'll list the period first. I guess that does have some validity because it tells someone right away whether they're dealing with an octave-repeating scale or not.

Wow, Igs, if you were Paul Erlich we'd just be getting started!

> Just FYI also, I've opted to use tables/spreadsheets to display mappings rather than ascii-friendly text for the sake of aesthetics and avoiding the use of monospaced fonts. I think it looks better and is easier to read, and since I have the luxury of doing it in OpenOffice I figure I might as well.

So far as I know, I'm the only one who hates spreadsheets, and that's not because I can't read them but because when people post Excel spreadsheets, they never seem to work right for me, and I can't be bothered to find out why.

>> Okay, okay. I'll list the period first. I guess that does have
>>some validity because it tells someone right away whether they're
>>dealing with an octave-repeating scale or not.
>
>Wow, Igs, if you were Paul Erlich we'd just be getting started!

http://en.wikipedia.org/wiki/Parkinson%27s_Law_of_Triviality

-Carl

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

> > Okay, okay. I'll list the period first. I guess that does have some validity because it tells someone right away whether they're dealing with an octave-repeating scale or not.
>
> Wow, Igs, if you were Paul Erlich we'd just be getting started!

It's really not that big of a deal which one comes first, if you ask me. Does Paul have a stronger opinion on the matter?

> So far as I know, I'm the only one who hates spreadsheets, and that's not because I can't
> read them but because when people post Excel spreadsheets, they never seem to work
> right for me, and I can't be bothered to find out why.

Lucky for you, my "spreadsheets" are just going to be part of the PDF with the rest of the text.

-Igs

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> It's really not that big of a deal which one comes first, if you ask me. Does Paul have a stronger opinion on the matter?

I'm not aware he has any opinion on the matter, but his opinions tend to be pretty strong.