Another use for the pseudoinverse

Suppose we have a JI subgroup, such as 2.3.13/5. We have a corresponding matrix M whose rows are the monzos for 2, 3 and 13/5. We can then take the pseudoinverse of M, M`. If we have an interval which belongs to the subgroup, for example 338/225, we can find the monzo for that also, in this case that would be u = |1 -2 -2 0 0 2>. Then taking uM`, we get |1 -2 2>, which is the subgroup version of the monzo, by which I mean that 2^1 3^(-2) (13/5)^2 = 338/225.

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

>Then taking uM`, we get |1 -2 2>, which is the subgroup version of the monzo, by which I mean that 2^1 3^(-2) (13/5)^2 = 338/225.

What do people say to "svals" and "smonzos" for vals and monzos relativized to subgroups?

On Fri, Feb 11, 2011 at 2:45 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> >Then taking uM`, we get |1 -2 2>, which is the subgroup version of the monzo, by which I mean that 2^1 3^(-2) (13/5)^2 = 338/225.
>
> What do people say to "svals" and "smonzos" for vals and monzos relativized to subgroups?

I'm down with that, as long as "smonzo" is actually pronounced
"smonzo" and not "ess-monzo."

But a question - I've been using monzos like this for subgroups the
whole time. Is this just a formal derivation for it, or a deeper
purpose that I'm missing?

-Mike

>What do people say to "svals" and "smonzos" for vals and monzos
>relativized to subgroups?

'swonderful -Carl

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

> But a question - I've been using monzos like this for subgroups the
> whole time. Is this just a formal derivation for it, or a deeper
> purpose that I'm missing?

The "deep" fact you are missing is that I'm coding up subgroup stuff and wanted names to call the functions, etc. Last time I did that I ended up converting my private names into public ones, much to Paul' distress.

On Fri, Feb 11, 2011 at 3:36 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > But a question - I've been using monzos like this for subgroups the
> > whole time. Is this just a formal derivation for it, or a deeper
> > purpose that I'm missing?
>
> The "deep" fact you are missing is that I'm coding up subgroup stuff and wanted names to call the functions, etc. Last time I did that I ended up converting my private names into public ones, much to Paul' distress.

Why, it all makes sense now! Great scott.

Are you doing everything in Maple? Would it be in my best interest to
obtain a copy of that? I don't see why I'm working in MATLAB if
everyone else is working in Maple.

-Mike

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

> Are you doing everything in Maple? Would it be in my best interest to
> obtain a copy of that? I don't see why I'm working in MATLAB if
> everyone else is working in Maple.

I'm coding everything in Maple; I'll email a copy of my programs if you like. Graham is doing stuff in Python and Pari; so far as I know I'm the only one doing anything in Maple.

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

> I'm coding everything in Maple

And now that I've got subgroups all coded up in Maple, I still have the same problem: there are too many effing subgroups. Still, I'll toddle over to the Xenwiki and add something to the Archipelago page.

IT BEGINS -C.

Gene wrote:
> Suppose we have a JI subgroup, such as 2.3.13/5. We have a
> corresponding matrix M whose rows are the monzos for 2, 3 and 13/5. We
> can then take the pseudoinverse of M, M`. If we have an interval which
> belongs to the subgroup, for example 338/225, we can find the monzo
> for that also, in this case that would be u = |1 -2 -2 0 0 2>. Then
> taking uM`, we get |1 -2 2>, which is the subgroup version of the
> monzo, by which I mean that 2^1 3^(-2) (13/5)^2 = 338/225.

>--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>And now that I've got subgroups all coded up in Maple, I still have
>the same problem: there are too many effing subgroups. Still, I'll
>toddle over to the Xenwiki and add something to the Archipelago page.

Two possible-partial remedies have been floating around in my head.
One, tighter weighting to knock down the contribution of the higher
integers (and the multitude of fractions one can make with them) and
two, some notion of 'universal' temperaments, where the errors on the
primes of the complete p-limit are reasonably balanced (so they would
tend to place in top-10 lists for various subgroups)... One thing
regarding the latter that comes to mind for some reason is the
"kernel trick", which I forget the specifics of, but am looking
forward to going over my notes on. It looks like I've got tomorrow
off so it should be a good day

http://www.youtube.com/watch?v=QWfbGGZE07M

-C.

On Fri, Feb 11, 2011 at 7:59 PM, Carl Lumma <carl@lumma.org> wrote:
>
> IT BEGINS -C.

The exhaustive subgroup search?

-Mike

On Fri, Feb 11, 2011 at 8:10 PM, Carl Lumma <carl@lumma.org> wrote:
>
> Two possible-partial remedies have been floating around in my head.
> One, tighter weighting to knock down the contribution of the higher
> integers (and the multitude of fractions one can make with them) and
> two, some notion of 'universal' temperaments, where the errors on the
> primes of the complete p-limit are reasonably balanced (so they would
> tend to place in top-10 lists for various subgroups)...

There's also the concept of building the subgroup around some kind of
target chord. This is what regular full-limit JI is about anyway - we
determine the error of chords relative to some chord like 4:5:6 or
4:5:6:7. It's relatively easy to determine what chords are important
and useful and which ones aren't.

For example, in the case of the 2.3.13/5 subgroup, we're building
things around the 10:13:15 chord, which is an obviously xenharmonic,
low-complexity chord with a fifth on the outer dyad to start tinkering
around. If you aren't willing to work with subgroups, then your only
option to investigate this chord is the 13-limit, which means lots of
good temperaments are missed.

There are tons of subgroups but probably not that many that generate
reasonable target chords of a certain complexity. Another good one
that I've been exploring is 4:7:9:11, which leads to the 2.3.7.11 and
2.7.9.11 subgroups. And within the latter, the 11-note MOS I posted a
while ago works about as well as Pajara does for the 7-limit: medium
error, lots of good tetrads everywhere, etc.

It may also be worthwhile to start optimizing tunings for "minor
chords" that aren't just the utonal inverse of whatever target chord
you're shooting for.

-Mike

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

> There's also the concept of building the subgroup around some kind of
> target chord. This is what regular full-limit JI is about anyway - we
> determine the error of chords relative to some chord like 4:5:6 or
> 4:5:6:7. It's relatively easy to determine what chords are important
> and useful and which ones aren't.

I like this plan. Why not make a list of useful chords?

> There are tons of subgroups but probably not that many that generate
> reasonable target chords of a certain complexity. Another good one
> that I've been exploring is 4:7:9:11, which leads to the 2.3.7.11 and
> 2.7.9.11 subgroups. And within the latter, the 11-note MOS I posted a
> while ago works about as well as Pajara does for the 7-limit: medium
> error, lots of good tetrads everywhere, etc.

You tend to get 1/2 of a full 11-limit et: 6, 11, 135, etc. I like the microtemperament supported by 135, 353, 488, 623 but no one else will, I presume.

On 12 February 2011 04:45, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> I'm coding everything in Maple; I'll email a copy of my programs if you like. Graham is doing stuff in Python and Pari; so far as I know I'm the only one doing anything in Maple.

Right, I have old code in Python and I've done some more advanced
algebra in Pari/GP. As both are free, it would be nice if everybody
else were using them. They're both involved in Sage, which is also
free, but bloated, so I don't currently have it installed.

The GP code, along with comments and examples to remind me what I was
doing, is here:

http://x31eq.com/parametric.gp

Graham

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> I like this plan. Why not make a list of useful chords?

Funny you should mention that. I'm slowly making my way through various harmonic series triads that span less than a 2/1 but not less than a 5/4, using harmonics 16:18:20:21:22:24:25:26:27:28:30:32 (i.e. 13-prime limit), making little notes about how they feel to me--kinda like a wine-tasting or something. I initially thought I'd go up to the 19-limit, but I found most of the chords I tried with 17's or 19's in them to sound "unjust". I'd love to do tetrads too, but that would be a staggering amount of work. I've already tested around 60 chords and I've barely scratched the surface.

I doubt these notes will be of much interest to anyone but myself, but my goal is to pick out some favorites, then compare with their utonal counterparts to see if there is a significant difference in character between them such that they might function as analogs to 4:5:6 and 1/(4:5:6). THEN, using Carl's spreadsheet (which hopefully can be modified to handle fractional-octave periods in the near future) and a list of EDO approximations to this chunk of the harmonic series, I'll be able to find not just the EDOs that give good approximations of these chords, but also the MOS scales in each EDO that give these chords the lowest complexity.

Hopefully I can finish this project in the next year or so, as I want to integrate it with the rest of the material in "A Field Guide to Alternative EDOs".

-Igs

On Fri, Feb 11, 2011 at 11:01 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > There's also the concept of building the subgroup around some kind of
> > target chord. This is what regular full-limit JI is about anyway - we
> > determine the error of chords relative to some chord like 4:5:6 or
> > 4:5:6:7. It's relatively easy to determine what chords are important
> > and useful and which ones aren't.
>
> I like this plan. Why not make a list of useful chords?

If I had to just pick one to start messing around with at first, it
would probably be 4:7:9:11. Something that has a 3/2 involved might
probably be better.

I have a few ideas about what I think are good qualities to shoot for
in a target chord, which might hopefully lead to a way to enumerate
all possible chords that have those features.

One thing that I think is a good quality for our initial search to
start things off, although I won't say this is the only way to go, is
to mess with target chords that are "rooted." 4:5:6 is rooted, for
example, because its fundamental is octave-equivalent to the lowest
note in the chord. This isn't just a nice little triviality, but I
think is important for octave equivalent scales: it makes it so that
if you double the root down an octave or two, that you also happen to
be fleshing out an overall low-complexity chord. There will no doubt
be lots of awesome subgroups that don't involve rooted triads (I think
they'll be a lot more intense, actually), but I predict that the
rooted ones will be particularly magical.

So if you're working with 4:5:6, for instance, and you double the root
down an octave or two, you get 1:2:4:5:6. If you're working with
5:6:7, on the other hand, you get 5:10:12:24:28, which is more
complex, and it sounds more dissonant to me too. To me, 5:6:7 sounds
like it might be the upper partials of 4:5:6:7, but 5:10:12:24:28
sounds like a sinister and ominous "diminished"-style chord. In my
ears, making the root octave-equivalent in this case changes the
nature and sonority of the triad. Anyway, that's just my personal
impression of this.

What about the barbados triad? I would count a sonority like 10:13:15
(or 10:12:15) as being quasi-rooted, even though technically it isn't,
because psychoacoustically speaking, that 3/2 is so strong that you
tend to hear the VF of this chord as 5. Might as well include those
too. I find that fully rooted triads tend to sound more abstractly
"major" and quasi-rooted ones tend to sound less so, and I don't think
anyone would disagree with this.

A few interesting rank-3 options to start things off would be 4:7:9,
4:7:11, 8:9:11, 8:10:11, 10:11:15, 10:13:15, and 16:19:24. Maybe
6:7:11 as well, although that might be a bit darker. There's probably
tons of really creative options I'm missing.

I think the rank-4 options are far more interesting: 4:7:9:11,
4:7:11:13, 8:9:10:11, 8:9:11:12, 8:9:11:13 (!), 10:11:13:15 (!),
10:11:14:15 seem pretty magical. 10:11:12:15 and 10:12:13:15 make for
some interesting ways to tie this into the 5-limit too. 10:13:15:18 is
pretty interesting.

The list of tetrads just really goes on and on, but let's stop it there for now.

> > There are tons of subgroups but probably not that many that generate
> > reasonable target chords of a certain complexity. Another good one
> > that I've been exploring is 4:7:9:11, which leads to the 2.3.7.11 and
> > 2.7.9.11 subgroups. And within the latter, the 11-note MOS I posted a
> > while ago works about as well as Pajara does for the 7-limit: medium
> > error, lots of good tetrads everywhere, etc.
>
> You tend to get 1/2 of a full 11-limit et: 6, 11, 135, etc. I like the microtemperament supported by 135, 353, 488, 623 but no one else will, I presume.

Are there interesting puns to be had here?

-Mike

On Sat, Feb 12, 2011 at 4:56 AM, Graham Breed <gbreed@gmail.com> wrote:
>
> On 12 February 2011 04:45, genewardsmith <genewardsmith@sbcglobal.net> wrote:
>
> > I'm coding everything in Maple; I'll email a copy of my programs if you like. Graham is doing stuff in Python and Pari; so far as I know I'm the only one doing anything in Maple.
>
> Right, I have old code in Python and I've done some more advanced
> algebra in Pari/GP. As both are free, it would be nice if everybody
> else were using them. They're both involved in Sage, which is also
> free, but bloated, so I don't currently have it installed.
>
> The GP code, along with comments and examples to remind me what I was
> doing, is here:
>
> http://x31eq.com/parametric.gp

I've never worked in GP. I should probably start sometime soon. Maybe
reading this will help me figure out Cangwu badness better.

-Mike

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

> If I had to just pick one to start messing around with at first, it
> would probably be 4:7:9:11. Something that has a 3/2 involved might
> probably be better.

Was your name for the 11&28 temperament in 2.7.9.11 "machine"?

> > You tend to get 1/2 of a full 11-limit et: 6, 11, 135, etc. I like the microtemperament supported by 135, 353, 488, 623 but no one else will, I presume.
>
> Are there interesting puns to be had here?

Just a microtemperament with a pretty low complexity.

On Sat, Feb 12, 2011 at 9:42 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > If I had to just pick one to start messing around with at first, it
> > would probably be 4:7:9:11. Something that has a 3/2 involved might
> > probably be better.
>
> Was your name for the 11&28 temperament in 2.7.9.11 "machine"?

Yeah, although I was also calling the 2.3.7.11.13 one machine too.
It's another archipelago-type scenario. Maybe this one could be the
industrial complex.

> > > You tend to get 1/2 of a full 11-limit et: 6, 11, 135, etc. I like the microtemperament supported by 135, 353, 488, 623 but no one else will, I presume.
> >
> > Are there interesting puns to be had here?
>
> Just a microtemperament with a pretty low complexity.

My subgroup mojo is not strong enough to delve into this. What MOS's
does it support? I hope you're not saying it's low complexity if the
first MOS is 135 notes.

-Mike

On Sat, Feb 12, 2011 at 9:42 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > If I had to just pick one to start messing around with at first, it
> > would probably be 4:7:9:11. Something that has a 3/2 involved might
> > probably be better.
>
> Was your name for the 11&28 temperament in 2.7.9.11 "machine"?

Also, to clarify what I was talking about with "rooted" triads, here's
the simplest and most effective example yet.

Take 9:11:13. Now, if you double that 9 down an octave or two, you get
9:18:36:44:52. It also sounds like crap. Try it.

Now take 9:11:13 and instead, an octave or two down, put the actual VF
there. So you get 2:4:9:11:13. Yay. Sounds much better.

For all of the triads we're used to, these are the same, because for
2:3, 4:5:6, 4:5:6:7, 4:5:6:7:9:11, if you double the root, you're also
doubling the VF. Otherwise, for all of these subgroups, that might not
be the case. That's all for me.

-Mike

The "rootedness" thing doesn't seem to me to have much of an effect on mood to my ears. It has an effect on timbre, though; chords where the intervals all point to the same VF sound fuller and deeper, because you can actually hear the VF.

But isn't 16:19:24 supposed to be a "rooted" triad? And what about 12:15:18--yeah, you can reduce that to 4:5:6 and say the 12 is the 1, but you can also make it 4:12:15:18 and whoa, it's a whole different ball-game! It seems to me like rootedness is something that you can really "play around" with in music, something that varies greatly depending on voicing. When you start inverting chords, you end up with chords like 3:4:5, 5:6:8, etc. that don't fit your definition of rootedness. And there's also the question of how much you can temper before even the most "rooted" chord loses that property--has that question ever been satisfactorily answered?

-Igs

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Feb 12, 2011 at 9:42 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > If I had to just pick one to start messing around with at first, it
> > > would probably be 4:7:9:11. Something that has a 3/2 involved might
> > > probably be better.
> >
> > Was your name for the 11&28 temperament in 2.7.9.11 "machine"?
>
> Also, to clarify what I was talking about with "rooted" triads, here's
> the simplest and most effective example yet.
>
> Take 9:11:13. Now, if you double that 9 down an octave or two, you get
> 9:18:36:44:52. It also sounds like crap. Try it.
>
> Now take 9:11:13 and instead, an octave or two down, put the actual VF
> there. So you get 2:4:9:11:13. Yay. Sounds much better.
>
> For all of the triads we're used to, these are the same, because for
> 2:3, 4:5:6, 4:5:6:7, 4:5:6:7:9:11, if you double the root, you're also
> doubling the VF. Otherwise, for all of these subgroups, that might not
> be the case. That's all for me.
>
> -Mike
>

On Sat, Feb 12, 2011 at 11:19 PM, cityoftheasleep
<igliashon@sbcglobal.net> wrote:
>
> The "rootedness" thing doesn't seem to me to have much of an effect on mood to my ears. It has an effect on timbre, though; chords where the intervals all point to the same VF sound fuller and deeper, because you can actually hear the VF.

You're telling me that you don't hear a difference in mood between
2:4:9:11:13 and 9:18:36:44:52?

> But isn't 16:19:24 supposed to be a "rooted" triad? And what about 12:15:18--yeah, you can reduce that to 4:5:6 and say the 12 is the 1, but you can also make it 4:12:15:18 and whoa, it's a whole different ball-game!

We're splitting hairs here. If you're talking about 4:12:15:18, then
you're talking about building a subgroup around the octave-equivalent
10:12:15:18 tetrad, and that subgroup happens to be the 5-limit.

> It seems to me like rootedness is something that you can really "play around" with in music, something that varies greatly depending on voicing. When you start inverting chords, you end up with chords like 3:4:5, 5:6:8, etc. that don't fit your definition of rootedness.

We are again splitting hairs. 3:4:5 is a rooted triad that's in
inversion. So is 5:6:8. 5:6:7 is not, and neither is 6:7:10 or
7:10:12. If you're building a temperament around 4:5:6 then you're
building it around 3:4:5 and 5:6:8 by definition. 5:6:9 might be an
interesting case, but this was a simple rule of thumb to get us
started anyway.

> And there's also the question of how much you can temper before even the most "rooted" chord loses that property--has that question ever been satisfactorily answered?

I fail to see how this is relevant. There's also the question of how
much you can temper 4:5:6 before it stops sounding like 4:5:6. And the
question is resolved by just not using temperaments that are so
inaccurate that the target chords no longer resemble what they're
supposed to be.

If you have a 4:5:6 triad and it stops sounding "rooted," that means
that doubling the bottom note down a few octaves will suddenly start
making it sound a lot worse. I don't know when this will happen, and
it probably has to do with the triadic field of attraction for 4:5:6.
But why must we change the subject? There will be good temperaments
and bad ones. If a temperament is so inaccurate that 4:7:9:11 doesn't
sound like it points to 1 or 2 anymore, then it doesn't make sense to
map the resulting tetrad as 4:7:9:11.

-Mike

On Sat, Feb 12, 2011 at 11:40 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
> On Sat, Feb 12, 2011 at 11:19 PM, cityoftheasleep
> <igliashon@sbcglobal.net> wrote:
>>
>> The "rootedness" thing doesn't seem to me to have much of an effect on mood to my ears. It has an effect on timbre, though; chords where the intervals all point to the same VF sound fuller and deeper, because you can actually hear the VF.
>
> You're telling me that you don't hear a difference in mood between
> 2:4:9:11:13 and 9:18:36:44:52?

To elaborate on this even further, I think that even subgroups that
produce nonrooted triads will be useful. I just don't think those
triads will sound "major" like the rooted ones will if we're talking
about octave-equivalent music.

As Gene said, there are just too many subgroups right now. We still
sometimes discover interesting 5-limit temperaments and it's been
what, almost a decade since this was all invented? So while I can
think of some interesting uses for a subgroup like 8.11/9.13/11, or
whatever the reduced form of it might be, I think that it might be
more useful to look at something like 8.9.11.13 first, or else I don't
think you're going to have much of a "major" chord.

One interesting thing might be to come up with a 8.11/9.13/11 scale,
and then deliberately use a different transposition of that scale in
the bottom register to avoid this. I think that would be really neat
and I've thought about doing that for quite some time.

-Mike

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

> My subgroup mojo is not strong enough to delve into this. What MOS's
> does it support? I hope you're not saying it's low complexity if the
> first MOS is 135 notes.

Rather than post all the info here I started a Xenwiki page:
http://xenharmonic.wikispaces.com/Subgroup+temperaments

The 1-9/8-11/8-7/4 tetrad has Graham complexity 19, and the 1-11/8-7/4 triad complexity 16, hence there are lots of chords within the 31 note MOS and some within the 21 note MOS. Tuning error is well under a cent so I'm counting it as a microtemperament.

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

> The 1-9/8-11/8-7/4 tetrad has Graham complexity 19, and the 1-11/8-7/4 triad
> complexity 16, hence there are lots of chords within the 31 note MOS and some within the > 21 note MOS. Tuning error is well under a cent so I'm counting it as a microtemperament.

Of course, if you're in 11-EDO, that same tetrad has a complexity of 4, though the error is considerably higher.

-Igs

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > The 1-9/8-11/8-7/4 tetrad has Graham complexity 19, and the 1-11/8-7/4 triad
> > complexity 16, hence there are lots of chords within the 31 note MOS and some within the > 21 note MOS. Tuning error is well under a cent so I'm counting it as a microtemperament.
>
> Of course, if you're in 11-EDO, that same tetrad has a complexity of 4, though the error is considerably higher.

If you are in machine temperament, the 11edo can be converted to a MOS and the overall error somewhat reduced by playing the 11-note scale on your handy dandy 28edo guitar.