Commas for MOS

The following commas are

(1) 5-limit
(2) tempered out by <7 11 16|
(3) Together give each shape of 7-note MOS

1L6s porcupine 250/243
2L5s mavila 135/128
3L4s dicot 25/24
4L3s sixix 3125/2916
5L2s meantone 81/80
6L1s enipucpop, tetracot 1125/1024, 20000/19683

Useful information, but I haven't decided where to put it.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> The following commas are
>
> (1) 5-limit
> (2) tempered out by <7 11 16|
> (3) Together give each shape of 7-note MOS
>
> 1L6s porcupine 250/243
> 2L5s mavila 135/128
> 3L4s dicot 25/24
> 4L3s sixix 3125/2916
> 5L2s meantone 81/80
> 6L1s enipucpop, tetracot 1125/1024, 20000/19683
>
> Useful information, but I haven't decided where to put it.

It's related to the edo-distinct stuff. The three edo-distinct 5-limit temperaments for 7 are

<<1 4 4||, associated to 81/80
<<2 1 -3||, associated to 25/24
<<3 5 1||, associated to 250/243

Without tuning information, we don't know if 81/80, for example, goes with 5L2s or 2L5s. Tuning tells us 5L2s, so for a 2L5s we have to find another member of the Fokker group with +- the right number, modulo 7, for complexity of 3 and the tuning range we want.

<<2 1 -3|| - <<1 4 4|| = <<1 -3 -7||; 135/128
<<2 1 -3|| + <<3 5 1|| = <<5 6 -2||; 3125/2916
2<2 1 -3|| - <<1 4 4|| = <<3 -2 -10||; 1125/1024
or
2<<1 4 4|| + <2 1 -3}} = <<4 9 5||; 20000/19683

On Mon, Nov 26, 2012 at 2:03 PM, genewardsmith <genewardsmith@...>
wrote:
>
> The following commas are
>
> (1) 5-limit
> (2) tempered out by <7 11 16|
> (3) Together give each shape of 7-note MOS
>
> 1L6s porcupine 250/243
> 2L5s mavila 135/128
> 3L4s dicot 25/24
> 4L3s sixix 3125/2916
> 5L2s meantone 81/80
> 6L1s enipucpop, tetracot 1125/1024, 20000/19683
>
> Useful information, but I haven't decided where to put it.

Here's 5-EDO for 2.3.7 <5 12 14|:
1L4s: slendric 1029/1024
2L3s: superpyth 64/63
3L2s: mother 28/27
4L1s: semaphore 49/48

I was going to do some for 2.5.7 6-EDO, but then I realized I didn't
want to open the can of worms that is subgroup temperament names just
yet. Back to Cangwu badness...

-Mike

These MOS are also 7&8, 7&9, 7&10, 7&11, 7&12, and 7&13 (respectively), regardless of commas, provided 2/1 is mapped to 1 period.

Worth noting, however, is that some of these commas are not distinct in generating the associated MOS; both 7&10 and 7&11 can temper out 25/24, for instance, and both 7&9 and 7&11 can temper out 135/128 (though it's worth noting that in 7&11, it's contorted, actually being "mohavila" temperament or whatever if you call the generator 11/9).

-Igs

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> The following commas are
>
> (1) 5-limit
> (2) tempered out by <7 11 16|
> (3) Together give each shape of 7-note MOS
>
> 1L6s porcupine 250/243
> 2L5s mavila 135/128
> 3L4s dicot 25/24
> 4L3s sixix 3125/2916
> 5L2s meantone 81/80
> 6L1s enipucpop, tetracot 1125/1024, 20000/19683
>
> Useful information, but I haven't decided where to put it.
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> These MOS are also 7&8, 7&9, 7&10, 7&11, 7&12, and 7&13 (respectively), regardless of commas, provided 2/1 is mapped to 1 period.

I don't know what you mean by "7&11 regardless of commas" since as I use the notation it's impossible for it to be regardless of commas. In my notation scheme, used on the optimal patent val page, 7&10 tempers out 25/24 and is dicot, and 7&11 tempers out 135/128 and is mavila.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > These MOS are also 7&8, 7&9, 7&10, 7&11, 7&12, and 7&13 (respectively), regardless of commas, provided 2/1 is mapped to 1 period.
>
> I don't know what you mean by "7&11 regardless of commas" since as I use the notation it's > impossible for it to be regardless of commas.

No it isn't. The "X&Y" notation doesn't specify a basis. The commas you'd get for 7&9 on the 2.7.19.27 subgroup will be completely different than the 5-limit commas, obviously, yet the MOS pattern will come out the same. The basis is specified independent of the notation, but you can leave the basis out entirely (with the exception of 2) and the notation will still specify an MOS just as well as it does with the basis included.

-Igs

On Thu, Nov 29, 2012 at 12:56 PM, cityoftheasleep <igliashon@...>
wrote:
>
> No it isn't. The "X&Y" notation doesn't specify a basis. The commas you'd
> get for 7&9 on the 2.7.19.27 subgroup will be completely different than the
> 5-limit commas, obviously, yet the MOS pattern will come out the same. The
> basis is specified independent of the notation, but you can leave the basis
> out entirely (with the exception of 2) and the notation will still specify
> an MOS just as well as it does with the basis included.

Are you using 7&9 to stand for vals here, or something else? I know
Gene's talking about vals, but I think you're talking about something
more general.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Are you using 7&9 to stand for vals here, or something else? I know
> Gene's talking about vals, but I think you're talking about something
> more general.

Yes, <7| and <9|. What else do you need to get MOS scales out of these pairs of ETs?

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> > >
> > > These MOS are also 7&8, 7&9, 7&10, 7&11, 7&12, and 7&13 (respectively), regardless of commas, provided 2/1 is mapped to 1 period.
> >
> > I don't know what you mean by "7&11 regardless of commas" since as I use the notation it's > impossible for it to be regardless of commas.
>
> No it isn't. The "X&Y" notation doesn't specify a basis.

I said as I use it, and it does when I use it. I ought to know, right? And I think Graham does something similar.

The commas you'd get for 7&9 on the 2.7.19.27 subgroup will be completely different than the 5-limit commas, obviously, yet the MOS pattern will come out the same.

Depends on the tuning, doesn't it?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> I said as I use it, and it does when I use it. I ought to know, right? And I think Graham does > something similar.

So how do you use it? I'm confused how you could use that notation in a way such that the basis is always specified, without actually specifying the basis. And without a basis, how can it have commas specified?

> The commas you'd get for 7&9 on the 2.7.19.27 subgroup will be completely different than the 5-limit commas, obviously, yet the MOS pattern will come out the same.
>
> Depends on the tuning, doesn't it?

It always depends on the tuning, so I'm not sure how this is different than your original post, where you associated commas with MOS scales.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> > I said as I use it, and it does when I use it. I ought to know, right? And I think Graham does > something similar.
>
> So how do you use it? I'm confused how you could use that notation in a way such that the basis is always specified, without actually specifying the basis. And without a basis, how can it have commas specified?

I always assume patent vals, unless there is a wart stuck on the number. Context provides the prime limit.

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

> I always assume patent vals, unless there is a wart stuck on the number. Context provides > the prime limit.

What if context doesn't provide prime limit?

-Igs

On Thu, Nov 29, 2012 at 4:31 PM, cityoftheasleep <igliashon@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Are you using 7&9 to stand for vals here, or something else? I know
> > Gene's talking about vals, but I think you're talking about something
> > more general.
>
> Yes, <7| and <9|. What else do you need to get MOS scales out of these
> pairs of ETs?

Nothing really. Graham's done work relating maximally even scales to
the scale tree, so that 7&9 is also the 7 note maximally even scale in
9-EDO, which with the usual 5-limit mapping is what we'd call
mavila[7].

A generalized version of this approach lets you take a triple of
numbers and have them specify a rank > 2 scales, though there's more
than one way to interpret the thing that it's specifying, and it's not
trivial to figure out how to generalize the scale tree to the 3D case.

-Mike

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > I always assume patent vals, unless there is a wart stuck on the number. Context provides > the prime limit.
>
> What if context doesn't provide prime limit?

Then give it.