Subgroup temperaments

In light of the recent 24-tet-as-subgroup discussion, and after the
little spinoff that we had on Gene's pluto temperament thread, I've
started believing that exploring subgroups more would be extremely
productive for making use of some otherwise "useless" low-numbered
ET's. Well, at least, it would be useful for me.

For starters, I think that part of the 12-tet harmonic revolution
starting from Debussy onward stems from the realization that 12-tet
can rudimentarily imply 7, 17, and 19 (and maybe 11, if you believe in
that sort of thing). So you can say that they've been treating 12-tet
as a 3.5.7.(11).17.19 subgroup temperament. And while it is
questionable that dyads like 17/12 are really heard that way, the
17-ness definitely starts to pop out in larger structures such as
2:4:8:10:14:17:19.

(Or, for a trivial example, 17/12 is definitely heard as 17/12 in
8:9:10:11:12:13:14:15:16:17:18:19:etc.)

So it might be useful then to also think of 9, 11, 13, 18, etc edo's
as being really accurate for different subgroups. Or to find some
different rank-2 subgroup MOS's that exist in those scales that might
be useful candidates for a tonal system.

Have different subgroup temperaments like this been explored in depth?
I couldn't find much literature on it, although I saw the entry on
"clans" in the xenharmonic wiki and that was really cool.

-Mike

I just spent the last hour and a half playing around with 24-tet and I
think there's a lot of interesting regular mapping stuff to be found
in there.

For example, although 8:13 is pretty sharp, 10:13 is represented
within 4 cents of just, so you get pretty good 10:13:15's and
10:12:13:15's. Furthermore, a chain of 150 cents will give you an
approximation to 10:11:12:13:14, so there's a pretty good
10:11:12:13:14:15 in 24-tet.

I was trying to turn this into an MOS, but I'm not sure if it's
possible. 3-3-3-3-2 gives you 10:11:12:13:14:15, but is this contained
in some simpler superstructure? The best I can do is to treat it as
rank-3 right now where the generators are 150 cents, 700 cents, and
2/1.

There's also near-just 8:9:11:12's, which might be great as alternate
sonorities to build a tonic off of. Who needs 7/4 anyway?

-Mike

On Tue, Dec 28, 2010 at 6:34 PM, Mike Battaglia <battaglia01@...> wrote:
> In light of the recent 24-tet-as-subgroup discussion, and after the
> little spinoff that we had on Gene's pluto temperament thread, I've
> started believing that exploring subgroups more would be extremely
> productive for making use of some otherwise "useless" low-numbered
> ET's. Well, at least, it would be useful for me.
>
> For starters, I think that part of the 12-tet harmonic revolution
> starting from Debussy onward stems from the realization that 12-tet
> can rudimentarily imply 7, 17, and 19 (and maybe 11, if you believe in
> that sort of thing). So you can say that they've been treating 12-tet
> as a 3.5.7.(11).17.19 subgroup temperament. And while it is
> questionable that dyads like 17/12 are really heard that way, the
> 17-ness definitely starts to pop out in larger structures such as
> 2:4:8:10:14:17:19.
>
> (Or, for a trivial example, 17/12 is definitely heard as 17/12 in
> 8:9:10:11:12:13:14:15:16:17:18:19:etc.)
>
> So it might be useful then to also think of 9, 11, 13, 18, etc edo's
> as being really accurate for different subgroups. Or to find some
> different rank-2 subgroup MOS's that exist in those scales that might
> be useful candidates for a tonal system.
>
> Have different subgroup temperaments like this been explored in depth?
> I couldn't find much literature on it, although I saw the entry on
> "clans" in the xenharmonic wiki and that was really cool.
>
> -Mike
>

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

> Have different subgroup temperaments like this been explored in depth?

One problem with subgroup temperaments is that there are so many of them. You can cut that down some by insisting the subgroups be generated by primes, but as you go up in prime limit, that still comes to a lot. If we knock out just one prime it becomes easier yet. In the 7 limit, we can ignore no-twos, and no-sevens is just the 5-limit. That leaves no-threes, for which 31 is a great system, and no-fives, dominated by 36.

For linear temperaments, 1029/1024 is a great no-fives comma, and 3136/3125 is excellent for no-threes. We can add the 11-limit by, for instance, using 176/175 as a no-threes comma or 896/891 or 243/242 for no-fives. Putting these together gives some no-threes or no-fives rank two temperaments, which we can also explore using edos. For example, putting 176/175 with 3136/3125, we either get a rank three temperament or a (very closely related) no-threes temperament. For some optimized tunings the tuning will be the same for the no-threes part. In this example, the generator is ~28/25 and 31 or 37 can be used as tunings; it could be called the no-threes 31&37 tuning, but another way to describe it is as the no-threes reduction of hemiwuerschmidt. 37edo is an excellent tuning both for the 11-limit and the extension of this temperament to the 13 limit, and MOS of size 7, 13, 19, 25, 31 and 37 are available. The 13 note MOS gives plenty of scope, but the 7 note MOS is already quite interesting. So there's an example.

To make it easier yet, decide in advance which subgroup you want to look at. Now it's very manageable!

On Tue, Dec 28, 2010 at 7:14 PM, Mike Battaglia <battaglia01@...> wrote:
> I just spent the last hour and a half playing around with 24-tet and I
> think there's a lot of interesting regular mapping stuff to be found
> in there.
>
> For example, although 8:13 is pretty sharp, 10:13 is represented
> within 4 cents of just, so you get pretty good 10:13:15's and
> 10:12:13:15's. Furthermore, a chain of 150 cents will give you an
> approximation to 10:11:12:13:14, so there's a pretty good
> 10:11:12:13:14:15 in 24-tet.
>
> I was trying to turn this into an MOS, but I'm not sure if it's
> possible. 3-3-3-3-2 gives you 10:11:12:13:14:15, but is this contained
> in some simpler superstructure? The best I can do is to treat it as
> rank-3 right now where the generators are 150 cents, 700 cents, and
> 2/1.

I figured something out. Guess I'm getting better with the math here.
So this is a rank-2, 13 limit temperament that eliminates the
following commas:

50/49
121/120
144/143
169/168

This yields an apparently unnamed "8 & 16" temperament in Graham's
temperament finder. This temperament has a period of 1/8 of octave,
and has an MOS at 16 notes. The generator is 50 cents (or 100 cents if
you want 3/2) and the period is 150 cents.

So you end up with the following symmetrical scale, which is something
that I had stumbled upon about a long time ago but never thought to
analyze via regular mapping:

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 out of 24

This is, in a sense, diminished[8] (superdiminished[16]?) smushed into
half the space. It is extremely rich in 13-limit harmony, and is an
intuitive higher-limit extension of the diminished[8] octatonic scale.
Just like diminished[8] has 4:5:6 and 10:12:15 chords over the same
root, this scale has 8:10:11:12:13 and 10:11:12:13:15 chords over the
same root. So this works really well as a 2.3.5.11.13 subgroup
temperament. It doesn't fare as well with 7, but if you want to map
7/5 to 600 cents and 7/4 to 1000 cents then you can. Keep in mind that
with 24-equal, the 17's and 19's are near-just and you have yourself a
2.3.5.11.13.17.19 subgroup temperament, although you'd probably have
to be really clever to get the 17 and 19 to "ring out" so to speak.

This 16-note MOS might work really well if treated like we treat
miracle[21], in the sense that it's a "chromatic" scale with 8-equal
function as the "base" MOS to build melodies around and/or alter. I
think there will be lots of really useful subscales that come out of
this scale as well.

-Mike

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

> One problem with subgroup temperaments is that there are
> so many of them. You can cut that down some by insisting
> the subgroups be generated by primes, but as you go up in
> prime limit, that still comes to a lot. If we knock out
> just one prime it becomes easier yet. In the 7 limit, we
> can ignore no-twos, and no-sevens is just the 5-limit.
> That leaves no-threes, for which 31 is a great system,
> and no-fives, dominated by 36.

I assisted Carl with a search like this some time ago.
It's complicated because you have to decide how to compare
temperaments of different sub-prime limits. But I remember
that no-threes 31 came out very well. Maybe the best
overall. There'll be details in the tuning-math archive.

Graham

On Tue, Dec 28, 2010 at 7:36 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Have different subgroup temperaments like this been explored in depth?
>
> One problem with subgroup temperaments is that there are so many of them. You can cut that down some by insisting the subgroups be generated by primes, but as you go up in prime limit, that still comes to a lot. If we knock out just one prime it becomes easier yet. In the 7 limit, we can ignore no-twos, and no-sevens is just the 5-limit. That leaves no-threes, for which 31 is a great system, and no-fives, dominated by 36.

No-5's seems particularly interesting to me because of 17-tet. Did
36-tet end up lower in badness than 17?

> For linear temperaments, 1029/1024 is a great no-fives comma, and 3136/3125 is excellent for no-threes. We can add the 11-limit by, for instance, using 176/175 as a no-threes comma or 896/891 or 243/242 for no-fives. Putting these together gives some no-threes or no-fives rank two temperaments, which we can also explore using edos. For example, putting 176/175 with 3136/3125, we either get a rank three temperament or a (very closely related) no-threes temperament. For some optimized tunings the tuning will be the same for the no-threes part. In this example, the generator is ~28/25 and 31 or 37 can be used as tunings; it could be called the no-threes 31&37 tuning, but another way to describe it is as the no-threes reduction of hemiwuerschmidt. 37edo is an excellent tuning both for the 11-limit and the extension of this temperament to the 13 limit, and MOS of size 7, 13, 19, 25, 31 and 37 are available. The 13 note MOS gives plenty of scope, but the 7 note MOS is already quite interesting. So there's an example.

These are great options, thanks for this! I'll spend some time going
through these tonight.

Have you messed with no-7's in the 11-limit? I think that might be
another really good option to consider. No-5's and no-7's I think
would also be really interesting, as it leads to the aforementioned
8:9:11:12 tetrads as base sonorities.

Perhaps this field will really start to take off once we have tetradic
entropy to work with, which I think will yield some really unexpected
concordant sonorities.

> To make it easier yet, decide in advance which subgroup you want to look at. Now it's very manageable!

I think 2.3.7, 2.3.5.11, and 2.3.11 are on the list for me. Things
will probably get a lot more xenharmonic when 3 is removed, as it will
force me to really figure out a few things about how modulation works,
but this is a good start. A list of 2.3.7 commas sorted by badness is
probably what I'm after, but I'm not sure how to generate that kind of
a list (is this a "normal comma list?"). I would assume that 64/63 is
lowest in badness.

-Mike

Mike Battaglia <battaglia01@...> wrote:

> I think 2.3.7, 2.3.5.11, and 2.3.11 are on the list for
> me. Things will probably get a lot more xenharmonic when
> 3 is removed, as it will force me to really figure out a
> few things about how modulation works, but this is a good
> start. A list of 2.3.7 commas sorted by badness is
> probably what I'm after, but I'm not sure how to generate
> that kind of a list (is this a "normal comma list?"). I
> would assume that 64/63 is lowest in badness.

There's a list of 2.3.7 temperaments here:

http://x31eq.com/cgi-bin/pregular.cgi?limit=2.3.7&error=5.0

You'll have to work out how to get the commas from them.
Dominant tempers out 64:63.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> There's a list of 2.3.7 temperaments here:
>
> http://x31eq.com/cgi-bin/pregular.cgi?limit=2.3.7&error=5.0
>
> You'll have to work out how to get the commas from them.
> Dominant tempers out 64:63.

Getting the commas from them is straightforward with the right tools, because when you click on the name of the temperament, the mapping is shown. You can also click to make the search parameters more or less fussy about tuning accuracy. When you come to this page directly and not through the temperament finder front end it's not so clear the parametric badness being used is set by a choice of the user, but the results more or less tell you that.

Setting the error to 1, 2, 3, or 4 cents, the best 2.5.7 system is the 6&31 system, the same as what I called 31&37, for which the comma is 3136/3125; the corresponding rank three temperament is hemimean. For 1, 2, or 3 cents the best 2.5.7 system is slendric, the 5&36 system, for which the comma is 1029/1024, with corresponding rank three temperament gamelismic (should it be slendric?) You can do the same in the 11-limit, etc, of course. For 2.3.7.11, for instance, you get no-fives miracle with commas 243/242 and 1029/1024 from 1 cent, and no-fives hemififths with commas 243/242 and 896/891 from 2 or 3 cents.

The commas can be obtained from either what Graham calls "equal temperament mappings" or "reduced mapping".

On Tue, Dec 28, 2010 at 8:24 PM, Graham Breed <gbreed@...> wrote:
>
> Mike Battaglia <battaglia01@...> wrote:
>
> > I think 2.3.7, 2.3.5.11, and 2.3.11 are on the list for
> > me. Things will probably get a lot more xenharmonic when
> > 3 is removed, as it will force me to really figure out a
> > few things about how modulation works, but this is a good
> > start. A list of 2.3.7 commas sorted by badness is
> > probably what I'm after, but I'm not sure how to generate
> > that kind of a list (is this a "normal comma list?"). I
> > would assume that 64/63 is lowest in badness.
>
> There's a list of 2.3.7 temperaments here:
>
> http://x31eq.com/cgi-bin/pregular.cgi?limit=2.3.7&error=5.0
>
> You'll have to work out how to get the commas from them.
> Dominant tempers out 64:63.

!!!!!!!!! You can do that? Do you realize the torturous mental math
I've been subjecting myself to over the past month thinking about
this?

How did I miss this?

Also, doesn't superpyth temper out 64:63? Or is the idea that
superpyth and dominant both temper out 64:63?

-Mike

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

> Also, doesn't superpyth temper out 64:63? Or is the idea that
> superpyth and dominant both temper out 64:63?

Superpyth tempers out 64/63 and 243/243, and dominant, the linear temperament, tempers out 64/63 and 36/35.

http://xenharmonic.wikispaces.com/Archytas+clan

On Tue, Dec 28, 2010 at 11:01 PM, genewardsmith
<genewardsmith@...t> wrote:
>
> Getting the commas from them is straightforward with the right tools, because when you click on the name of the temperament, the mapping is shown. You can also click to make the search parameters more or less fussy about tuning accuracy. When you come to this page directly and not through the temperament finder front end it's not so clear the parametric badness being used is set by a choice of the user, but the results more or less tell you that.

How do you get the comma? Wedge product the two vals under "reduced mapping?"

-Mike

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

> How do you get the comma? Wedge product the two vals under "reduced mapping?"

That would work. The method in general I'm using at the moment is only easy to implement if you have the right software tools. I compute the Frobenius projection matrix P, get I-P, where I is the identity matrix, clear denominators, and desaturate. But I'm certainly not claiming that's the only way to go about it. What software have you got? Maybe Graham has this stuff worked out using Pari.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > How do you get the comma? Wedge product the two vals under "reduced mapping?"
>
> That would work.

I should expand on this--you then get the triprime commas, and reduce the whole mess. Trouble is, that's a recipe for rank two; the other works for rank anything.

On Wed, Dec 29, 2010 at 2:59 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > How do you get the comma? Wedge product the two vals under "reduced mapping?"
>
> That would work. The method in general I'm using at the moment is only easy to implement if you have the right software tools. I compute the Frobenius projection matrix P, get I-P, where I is the identity matrix, clear denominators, and desaturate. But I'm certainly not claiming that's the only way to go about it. What software have you got? Maybe Graham has this stuff worked out using Pari.

I have MATLAB, but I'm away from the computer that has it. I'm pretty
good with web programming though, so if it's simple enough to do in
Javascript then maybe I could put a quick web app up to do it. The
math you mentioned is far over my head, so I'd have to do some reading
up on it. I've never heard of a Frobenius projection matrix before.

-Mike

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

> I have MATLAB, but I'm away from the computer that has it.

You should check to see if it has pseudoinverses, Smith normal form, and Hermite normal form.

I'm pretty
> good with web programming though, so if it's simple enough to do in
> Javascript then maybe I could put a quick web app up to do it.

Stick with MATLAB.

> I've never heard of a Frobenius projection matrix before.

It's originally a tuning-math thing. Scala now finds the Frobenius tuning for you, but the tuning isn't really important, it's the matrix giving the tuning that works the magic.

http://xenharmonic.wikispaces.com/Tenney-Euclidean+Tuning

http://xenharmonic.wikispaces.com/Fractional+monzos

> -Mike
>

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

>I compute the Frobenius projection matrix P, get I-P, where I is the identity matrix, clear denominators, and desaturate.

I should have said "saturate". I've been intending to write a Xenwiki article on saturation, maybe this will prod me.

Graham wrote:
>
> I assisted Carl with a search like this some time ago.
> It's complicated because you have to decide how to compare
> temperaments of different sub-prime limits. But I remember
> that no-threes 31 came out very well. Maybe the best
> overall. There'll be details in the tuning-math archive.

It was discussed on this list more recently. And I've now
got a page up explaining it

http://lumma.org/music/theory/subgroup/

where results and code can be downloaded in plain text or
Excel format. Detailed are four ways of approaching the
issue, the 3rd probably being the one for answering which
system is best for some unspecified basis of integers.
That appears to be a 41-ET-like thing with basis 5.13.17.

-Carl

On Wed, Dec 29, 2010 at 4:18 AM, Carl Lumma <carl@...> wrote:
>
> Graham wrote:
> >
> > I assisted Carl with a search like this some time ago.
> > It's complicated because you have to decide how to compare
> > temperaments of different sub-prime limits. But I remember
> > that no-threes 31 came out very well. Maybe the best
> > overall. There'll be details in the tuning-math archive.
>
> It was discussed on this list more recently. And I've now
> got a page up explaining it
>
> http://lumma.org/music/theory/subgroup/
>
> where results and code can be downloaded in plain text or
> Excel format. Detailed are four ways of approaching the
> issue, the 3rd probably being the one for answering which
> system is best for some unspecified basis of integers.
> That appears to be a 41-ET-like thing with basis 5.13.17.

Thanks, this is perfect! I'm about to pass out while typing, so I'll
have to check tomorrow. But this is a comprehensive analysis of
exactly the type of thing I was looking for.

-Mike

"genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia
> <battaglia01@...> wrote:
>
> > How do you get the comma? Wedge product the two vals
> > under "reduced mapping?"
>
> That would work. The method in general I'm using at the
> moment is only easy to implement if you have the right
> software tools. I compute the Frobenius projection matrix
> P, get I-P, where I is the identity matrix, clear
> denominators, and desaturate. But I'm certainly not
> claiming that's the only way to go about it. What
> software have you got? Maybe Graham has this stuff worked
> out using Pari.

Right, the function "matkerint" returns some unison
vectors, that you can then simplify with lattice
reduction. But where there's only one comma, it's
much simpler than that. You get the unison vector by
fiddling with the second row of the reduced mapping.

Graham

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

> I figured something out. Guess I'm getting better with the math here.
> So this is a rank-2, 13 limit temperament that eliminates the
> following commas:
>
> 50/49
> 121/120
> 144/143
> 169/168
>
> This yields an apparently unnamed "8 & 16" temperament in Graham's
> temperament finder. This temperament has a period of 1/8 of octave,
> and has an MOS at 16 notes. The generator is 50 cents (or 100 cents if
> you want 3/2) and the period is 150 cents.

You can take the generator as an approximate 3/2 if you like, in which case the POTE generator is 698.358 cents; subtracting 600 gives 98.358. 100 would be the value for 24edo, which works better for tuning this temperament than 16. It looks as if you could arrive at something much more accurate by deep sixing the 50/49, but that won't work, as another basis for the commas is 36/35, 50/49, 66/65, 143/140.

Mike wrote:

> > http://lumma.org/music/theory/subgroup/
> >
> > where results and code can be downloaded in plain text or
> > Excel format. Detailed are four ways of approaching the
> > issue, the 3rd probably being the one for answering which
> > system is best for some unspecified basis of integers.
> > That appears to be a 41-ET-like thing with basis 5.13.17.
>
> Thanks, this is perfect! I'm about to pass out while typing, so I'll
> have to check tomorrow. But this is a comprehensive analysis of
> exactly the type of thing I was looking for.

For those without Excel, I've made a version available
via Google Docs

https://spreadsheets.google.com/ccc?key=0Ajr43tP50gqqdEdRVUtEdXJCa2JYb29RZ19HZU5RdkE

-Carl