Yahoo Tuning Groups Ultimate Backup tuning How to make any ET <26 sound as much like JI as possible

No, I'm not talking about TOP or other tuning optimizations, I'm talking about *use* optimization--how to use what each ET gives you, assuming pure octaves (since that's how most people use their ETs), in the optimal way, given a goal of "sounding like JI". This is basically what I've been researching over the last six or seven years, and I think I've reached the point where I have nothing more to learn and could conceivably "publish" my findings. Anyone think it would be a worthwhile effort on my part to write up a guide to using the smaller ETs as subgroup temperaments, conclusively demonstrating that all ETs have some plausible relationship to JI, and the majority have actually a very strong relationship to it? Mind, my findings are all empirical, I haven't developed any mathematical algorithms or even quantified my definition of "optimal", but I feel pretty confident in my results. The cutoff at 25-ET is only slightly arbitrary; it is primarily ETs below this cutoff that benefit the most from a subgroup approach, as anything above 25 can function plausibly as a full 13-limit temperament.

So, should I do it?

-Igs

🔗Chris Vaisvil <chrisvaisvil@...>

Hi Igs,

I would appreciate it.

Chris

On Mon, Oct 29, 2012 at 4:59 AM, cityoftheasleep <igliashon@...>wrote:

> **
>
>
> No, I'm not talking about TOP or other tuning optimizations, I'm talking
> about *use* optimization--how to use what each ET gives you, assuming pure
> octaves (since that's how most people use their ETs), in the optimal way,
> given a goal of "sounding like JI". This is basically what I've been
> researching over the last six or seven years, and I think I've reached the
> point where I have nothing more to learn and could conceivably "publish" my
> findings. Anyone think it would be a worthwhile effort on my part to write
> up a guide to using the smaller ETs as subgroup temperaments, conclusively
> demonstrating that all ETs have some plausible relationship to JI, and the
> majority have actually a very strong relationship to it? Mind, my findings
> are all empirical, I haven't developed any mathematical algorithms or even
> quantified my definition of "optimal", but I feel pretty confident in my
> results. The cutoff at 25-ET is only slightly arbitrary; it is primarily
> ETs below this cutoff that benefit the most from a subgroup approach, as
> anything above 25 can function plausibly as a full 13-limit temperament.
>
> So, should I do it?
>
> -Igs
>
>
>

🔗Wolf Peuker <wolfpeuker@...>

🔗Mike Battaglia <battaglia01@...>

On Mon, Oct 29, 2012 at 4:59 AM, cityoftheasleep <igliashon@...>
wrote:
>
> So, should I do it?

Your goal is basically to identify the most accurate subgroup
supported by some (supposedly bad) equal temperament, right?

I think you should do it. I'll be the first to admit that I was very
influenced by this idea when you started pushing it years ago as a way
to make use of "bad" EDOs, and at this point I think it's here to
stay. But unfortunately, nothing's really here to stay unless it's
written down in a formal paper. So if you're offering to do that, I
think it'd be a good idea.

-Mike

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >The cutoff at 25-ET is only slightly arbitrary; it is primarily ETs below this cutoff that benefit the most from a subgroup approach, as anything above 25 can function plausibly as a full 13-limit temperament.
>
> I think 29edo is the subgroup equal division par excellance, beating out even 17edo.
>

I don't disagree; as a 2.3.13/5.13/7.13/11.15/13 temperament, it's downright close to being a microtemperament. But I think it's also a 13-limit temperament par excellance as well, since it's the best 13-limit temperament below 31.

It goes without saying that the accuracy of ETs above 25 can indeed be improved by treating them as subgroup ETs, but in general that just means they go from "good" to "great", rather than from "useless" to "good" the way the lower ETs do. My goal is to turn people on to the idea that "simple" doesn't have to mean "intolerable error", and to remove the notion that the only ETs of interest to those who like the JI sound are those that are strong in the 3-limit.

-Igs

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Oct 29, 2012 at 4:59 AM, cityoftheasleep <igliashon@...>
> wrote:
> >
> > So, should I do it?
>
> Your goal is basically to identify the most accurate subgroup
> supported by some (supposedly bad) equal temperament, right?

Not necessarily "most"; under some criteria for "most accurate", 12-TET could come out looking like a 2.17 subgroup temperament (or something). Have you read Carl's writing on subgroups? He did a subgroup search up to the 17-limit (though he left out non-integer bases, an oversight he's expressed regret over), and I had many issues with the results. My search has been messy and empirical, and what I've identified are what I consider the "most sensible" subgroups supported by the small ETs. By "most sensible" I mean "best balance between accuracy and versatility". For some I'll doubtless include a couple different subgroups, starting with the one closest to "most accurate" and then introducing additional basis intervals that lower the accuracy but expand the chord vocabulary.

> I think you should do it. I'll be the first to admit that I was very
> influenced by this idea when you started pushing it years ago as a way
> to make use of "bad" EDOs, and at this point I think it's here to
> stay. But unfortunately, nothing's really here to stay unless it's
> written down in a formal paper. So if you're offering to do that, I
> think it'd be a good idea.

Alright; with your endorsement, as well as Chris's and Wolf's (Wolf being the unsung hero of the xenwiki), I'll go ahead with it. One of the main reasons I left the XA on facebook this time was because I wanted to divert my "music theory" energy toward something more productive. I've had so many papers half-finished for so long, and I think I'll be of more service to the community if I hunker down and finish them.

-Igs

🔗Jason Conklin <jason.conklin@...>

It looks like you've gotten all the encouragement you need, but I'll look
forward to reading these, as well. The practical (or perhaps 'applied' is a
better word) aspect of your older writings has been a welcome extension to
the more abstract approaches offered elsewhere.

It's probably good to have more of that available to beginners, too. The
"make temperament sound like JI" angle seems to offer a promising foothold,
even if people decide to explore different "values" down the road.
If I'm understanding your exchange with Gene, it sounds like you'll be
outlining an approach that will work (or at least give a starting point) if
I wanted to explore, say, 26 or 29edo on my own, in a similar way; is that
a valid statement?

/jc
On Oct 29, 2012 12:02 PM, "cityoftheasleep" <igliashon@...> wrote:

> **
>
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Mon, Oct 29, 2012 at 4:59 AM, cityoftheasleep <igliashon@...>
> > wrote:
> > >
> > > So, should I do it?
> >
> > Your goal is basically to identify the most accurate subgroup
> > supported by some (supposedly bad) equal temperament, right?
>
> Not necessarily "most"; under some criteria for "most accurate", 12-TET
> could come out looking like a 2.17 subgroup temperament (or something).
> Have you read Carl's writing on subgroups? He did a subgroup search up to
> the 17-limit (though he left out non-integer bases, an oversight he's
> expressed regret over), and I had many issues with the results. My search
> has been messy and empirical, and what I've identified are what I consider
> the "most sensible" subgroups supported by the small ETs. By "most
> sensible" I mean "best balance between accuracy and versatility". For some
> I'll doubtless include a couple different subgroups, starting with the one
> closest to "most accurate" and then introducing additional basis intervals
> that lower the accuracy but expand the chord vocabulary.
>
> > I think you should do it. I'll be the first to admit that I was very
> > influenced by this idea when you started pushing it years ago as a way
> > to make use of "bad" EDOs, and at this point I think it's here to
> > stay. But unfortunately, nothing's really here to stay unless it's
> > written down in a formal paper. So if you're offering to do that, I
> > think it'd be a good idea.
>
> Alright; with your endorsement, as well as Chris's and Wolf's (Wolf being
> the unsung hero of the xenwiki), I'll go ahead with it. One of the main
> reasons I left the XA on facebook this time was because I wanted to divert
> my "music theory" energy toward something more productive. I've had so many
> papers half-finished for so long, and I think I'll be of more service to
> the community if I hunker down and finish them.
>
> -Igs
>
>
>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Jason Conklin <jason.conklin@...> wrote:

> If I'm understanding your exchange with Gene, it sounds like you'll be
> outlining an approach that will work (or at least give a starting point) if
> I wanted to explore, say, 26 or 29edo on my own, in a similar way; is that
> a valid statement?

What I'm outlining isn't an approach or a method, it's the result of one. My approach isn't formulaic, though it could possibly be formalized. It's simply the result of many years of me poking and prodding these ETs to see what can be done with them.

If you wanted to explore 26 or 29-EDO, they're both excellent 13-limit temperaments; neither needs a subgroup approach to justify using them. 26 is fully 13-limit consistent, meaning if you just map the full 13-odd-limit diamond to their best approximations in 26, that mapping will be self-consistent (this doesn't work in all ETs--you might end up with the distance from the best 11/8 to the best 7/4 not being the best 14/11, or something). In fact, 26 is the smallest one in which you can do this for the 13-odd-limit. 29 is fully 15-limit consistent, and is the smallest 15-limit consistent ET as well.

HTH,

-Igs

🔗cityoftheasleep <igliashon@...>

Here's a preliminary list of the subgroups I've selected for each ET. Questions, comments, and suggestions are welcomed, before I get down to the actual writing.

Subgroups:

1-ET: 2-limit JI
2-ET: 2.7/5
3-ET: 2.5, 2.11/7.11/9
4-ET: 2.5/3.7/3.17/3
5-ET: 2.3.7
6-ET: 2.5.7.9
7-ET: 2.3.13, or 2.11/5.11/9
8-ET: 2.5/3.11/3.13/3
9-ET: 2.5.7/3.11/3.13/3
10-ET: 2.5.7, 2.7.13.15
11-ET: 2.7.9.11.15.17
12-ET: 2.3.5.7.9.15.17.19
13-ET: 2.5.9.13.17.21
14-ET: 2.11/5.11/7.11/9, or 2.3.7 (11)
15-ET: 2.3.5.7.11, or simply 2.5.7.11
16-ET: 2.5.7.13.19
17-ET: 2.3.7.9.11.13
18-ET: 2.5.9.7/3.11/3.13/3.17/3
19-ET: 2.3.5.7.9.13.15
20-ET: 2.7.11.13.15.19.27
21-ET: 2.5.7.9.11.13 (15)
22-ET: 2.3.5.7.9.11.15.17
23-ET: 2.9.13.15.17.21.33
24-ET: 2.3.5.9.11.13.15.17.19
25-ET: 2.5.7.9.17.19

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> No, I'm not talking about TOP or other tuning optimizations, I'm talking about *use* optimization--how to use what each ET gives you, assuming pure octaves (since that's how most people use their ETs), in the optimal way, given a goal of "sounding like JI". This is basically what I've been researching over the last six or seven years, and I think I've reached the point where I have nothing more to learn and could conceivably "publish" my findings. Anyone think it would be a worthwhile effort on my part to write up a guide to using the smaller ETs as subgroup temperaments, conclusively demonstrating that all ETs have some plausible relationship to JI, and the majority have actually a very strong relationship to it? Mind, my findings are all empirical, I haven't developed any mathematical algorithms or even quantified my definition of "optimal", but I feel pretty confident in my results. The cutoff at 25-ET is only slightly arbitrary; it is primarily ETs below this cutoff that benefit the most from a subgroup approach, as anything above 25 can function plausibly as a full 13-limit temperament.
>
> So, should I do it?
>
> -Igs
>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Here's a preliminary list of the subgroups I've selected for each ET. Questions, comments, and suggestions are welcomed, before I get down to the actual writing.
>
> Subgroups:
>
> 1-ET: 2-limit JI
> 2-ET: 2.7/5
> 3-ET: 2.5, 2.11/7.11/9 = 2.9/7.11/9
> 4-ET: 2.5/3.7/3.17/3
> 5-ET: 2.3.7
> 6-ET: 2.5.7.9
> 7-ET: 2.3.13, or 2.11/5.11/9 = 2.9/5.11/9
> 8-ET: 2.5/3.11/3.13/3
> 9-ET: 2.5.7/3.11/3.13/3
> 10-ET: 2.5.7, 2.7.13.15
> 11-ET: 2.7.9.11.15.17
> 12-ET: 2.3.5.7.9.15.17.19 = 2.3.5.7.17.19
> 13-ET: 2.5.9.13.17.21
> 14-ET: 2.11/5.11/7.11/9, or 2.3.7 (11)
> 15-ET: 2.3.5.7.11, or simply 2.5.7.11
> 16-ET: 2.5.7.13.19
> 17-ET: 2.3.7.9.11.13 = 2.3.7.11.13
> 18-ET: 2.5.9.7/3.11/3.13/3.17/3 = 2.9.5.21.33.39.51
> 19-ET: 2.3.5.7.9.13.15 = 2.3.5.7.13
> 20-ET: 2.7.11.13.15.19.27
> 21-ET: 2.5.7.9.11.13 (15)
> 22-ET: 2.3.5.7.9.11.15.17 = 2.3.5.7.11.17
> 23-ET: 2.9.13.15.17.21.33
> 24-ET: 2.3.5.9.11.13.15.17.19 = 2.3.5.11.13.17.19
> 25-ET: 2.5.7.9.17.19

🔗cityoftheasleep <igliashon@...>

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

> > 3-ET: 2.5, 2.11/7.11/9 = 2.9/7.11/9
> > 7-ET: 2.3.13, or 2.11/5.11/9 = 2.9/5.11/9

What are the advantages to writing them this way? If you don't like the inital way, why not 2.9/7.11/7, or 2.9/5.11/5?

> > 12-ET: 2.3.5.7.9.15.17.19 = 2.3.5.7.17.19

I deliberately included (or tried to include; there may have been some oversights) all odd identies in these, so that they can function as bases for a tonality diamond as well as for a subgroup temperament.

> > 18-ET: 2.5.9.7/3.11/3.13/3.17/3 = 2.9.5.21.33.39.51

Again, what's the advantage to writing it this way? Why not 2.7/3.11/3.13/3.15/3.17/3.27/3, if you don't like the original?

-Igs

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> > > 3-ET: 2.5, 2.11/7.11/9 = 2.9/7.11/9
> > > 7-ET: 2.3.13, or 2.11/5.11/9 = 2.9/5.11/9
>
> What are the advantages to writing them this way? If you
> don't like the inital way, why not 2.9/7.11/7, or 2.9/5.11/5?

Numerous bases can define the same subgroup. It's just
nice to use the same one every time so subgroups are easier
to recognize. Kinda like writing fractions in lowest terms.

> > > 12-ET: 2.3.5.7.9.15.17.19 = 2.3.5.7.17.19
>
> I deliberately included (or tried to include; there may have
> been some oversights) all odd identies in these, so that they
> can function as bases for a tonality diamond as well as for
> a subgroup temperament.

A basis for a group in which ordinary multiplication is
an operator, should have coprime elements (or it isn't a
basis in the usual sense).

-Carl

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > What are the advantages to writing them this way? If you
> > don't like the inital way, why not 2.9/7.11/7, or 2.9/5.11/5?
>
> Numerous bases can define the same subgroup. It's just
> nice to use the same one every time so subgroups are easier
> to recognize. Kinda like writing fractions in lowest terms.

So, this gets back to what Mike was asking: how do we pick the basis? Should we establish a norm of, say, using the intervals with the smallest Tenney Harmonic Distance that can define the subgroup?

> > > > 12-ET: 2.3.5.7.9.15.17.19 = 2.3.5.7.17.19
> >
> > I deliberately included (or tried to include; there may have
> > been some oversights) all odd identies in these, so that they
> > can function as bases for a tonality diamond as well as for
> > a subgroup temperament.
>
> A basis for a group in which ordinary multiplication is
> an operator, should have coprime elements (or it isn't a
> basis in the usual sense).

What is it, then? And what kind of problems will I encounter by including non-coprime elements? I notice I encounter no troubles inputting such bases into Graham's app....

-Igs

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > > What are the advantages to writing them this way? If you
> > > don't like the inital way, why not 2.9/7.11/7, or 2.9/5.11/5?
> >
> > Numerous bases can define the same subgroup. It's just
> > nice to use the same one every time so subgroups are easier
> > to recognize. Kinda like writing fractions in lowest terms.
>
> So, this gets back to what Mike was asking: how do we pick the basis?
> Should we establish a norm of, say, using the intervals with the smallest
> Tenney Harmonic Distance that can define the subgroup?

Gene has a procedure
http://xenharmonic.wikispaces.com/Normal+lists#x-Normal interval lists
that always returns the same basis for a subgroup, given any other
basis for that subgroup.

You're talking about a Tenney-reduced basis, which is often different
than Gene's normal interval list...

> > A basis for a group in which ordinary multiplication is
> > an operator, should have coprime elements (or it isn't a
> > basis in the usual sense).
>
> What is it, then? And what kind of problems will I encounter by including
> non-coprime elements? I notice I encounter no troubles inputting such
> bases into Graham's app....

The true rank of the subgroup won't always be obvious, for one
thing. What's the benefit of including them? -C.

🔗Carl Lumma <carl@...>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Sorry Igs, "comprime" isn't right... more like factor-free (no element
> can divide another element). -C.

"Coprime" isn't right (too strong), but "factor-free" isn't right either (too weak). Counterexample: 4.6.9 (not linearly independent, and therefore not a basis, even though no element divides another).

I don't know any way to say the desired property simpler than the awkward "linearly independent in the vector space of monzos".

Keenan

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
>
> > I don't know any way to say the desired property simpler than the awkward "linearly independent in the vector space of monzos".
>
> A set of elements in a free abelian group which generate the group and is of minimal cardinality is a basis.

I should add I was thinking of finite ranks here, which is the only kind we use. In general instead of cardinality you'd need to say you can't remove an element from the set and still generate the group.

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 11:40 AM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> > I don't know any way to say the desired property simpler than the
> > awkward "linearly independent in the vector space of monzos".
>
> A set of elements in a free abelian group which generate the group and is
> of minimal cardinality is a basis.

I think that what Keenan is saying is that he doesn't know of any nice
simple number-theoretic properties that correspond to a set of
rationals being linearly independent in the multiplicative group of
Q+.

Igs proposed that a set of rationals is linearly independent iff
they're coprime, but 2.6 is a counterexample to that; this condition
is sufficient but not necessary. Carl counterproposed that a set of
rationals is linearly independent iff none of them cleanly divides
into another, but 2.6.9 is a counterexample to that; this condition is
necessary but not sufficient.

The question is, if a set of rationals is linearly (multiplicatively)
independent, what does that mean from a number-theoretic standpoint in
terms of properties like rationals being coprime and so on?

-Mike

🔗cityoftheasleep <igliashon@...>

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

> The true rank of the subgroup won't always be obvious, for one
> thing. What's the benefit of including them? -C.

The rank of the subgroup is irrelevant for my purposes, because it's all tempered down to rank-1 anyway. The point of including them is they represent consonances of the system, and if I fail to include them it won't be obvious that they're intended to be included as consonances.

What I intend to do is define a tonality diamond from the basis intervals and map that diamond to intervals of each ET. Whenever two consonances get mapped to the same interval, that demonstrates one of the vanishing commas. All of the temperaments I will be using are consistent on their given subgroup, so this shouldn't be a problem.

-Igs

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > A set of elements in a free abelian group which generate the
> > group and is of minimal cardinality is a basis.
>
> I think that what Keenan is saying is that he doesn't know of
> any nice simple number-theoretic properties that correspond to a
> set of rationals being linearly independent in the multiplicative
> group of Q+.

Isn't Gene saying being of minimal cardinality is such
a property?

> Igs proposed that a set of rationals is linearly independent
> iff they're coprime,

In this thread, I think that was me.

-Carl

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 3:08 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > > A set of elements in a free abelian group which generate the
> > > group and is of minimal cardinality is a basis.
> >
> > I think that what Keenan is saying is that he doesn't know of
> > any nice simple number-theoretic properties that correspond to a
> > set of rationals being linearly independent in the multiplicative
> > group of Q+.
>
> Isn't Gene saying being of minimal cardinality is such
> a property?

Yes, but all minimal cardinality means is that the basis is as small
as possible; so if you have a rank-2 subgroup, you know your basis
needs to have only two elements in it. Having a set of elements that
generates the group and is as small as possible is just the same as
saying you need a set of elements that generates the group and which
is linearly independent. It's just a restatement of the definition of
"basis" using different terms in a way that applies to free abelian
groups in general.

It seemed more like you guys were asking about a number-theoretic
connection that applies specifically to the multiplicative group of
positive rationals. If a set of rational numbers serves as a basis for
a subgroup of Q+, i.e. if it's linearly independent, how does this
property manifest itself as some relationship between the prime
factorizations of the elements of this set?

> > Igs proposed that a set of rationals is linearly independent
> > iff they're coprime,
>
> In this thread, I think that was me.

OK, my bad.

-Mike

🔗cityoftheasleep <igliashon@...>

So should I take this digression into group and/or number theory as endorsement for the bases I've chosen for each ET?

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Nov 1, 2012 at 11:40 AM, genewardsmith <genewardsmith@...>
> wrote:
> >
> > --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> >
> > > I don't know any way to say the desired property simpler than the
> > > awkward "linearly independent in the vector space of monzos".
> >
> > A set of elements in a free abelian group which generate the group and is
> > of minimal cardinality is a basis.
>
> I think that what Keenan is saying is that he doesn't know of any nice
> simple number-theoretic properties that correspond to a set of
> rationals being linearly independent in the multiplicative group of
> Q+.
>
> Igs proposed that a set of rationals is linearly independent iff
> they're coprime, but 2.6 is a counterexample to that; this condition
> is sufficient but not necessary. Carl counterproposed that a set of
> rationals is linearly independent iff none of them cleanly divides
> into another, but 2.6.9 is a counterexample to that; this condition is
> necessary but not sufficient.
>
> The question is, if a set of rationals is linearly (multiplicatively)
> independent, what does that mean from a number-theoretic standpoint in
> terms of properties like rationals being coprime and so on?
>
> -Mike
>

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 3:29 PM, cityoftheasleep <igliashon@...> wrote:
>
> So should I take this digression into group and/or number theory as endorsement for the bases I've chosen for each ET?
>
> -Igs

No, because something like 2.3.7.9.11.13 isn't a basis, since the "9"
is extraneous. Some of the things you listed aren't bases because of
that reason. You are of course free to list odd integers supported by
your subgroup, but I do wish you wouldn't use the a.b.c.d. ...
notation, since it looks like you're outlining an actual basis.

-Mike

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
>
> > The true rank of the subgroup won't always be obvious, for one
> > thing. What's the benefit of including them? -C.
>
> The rank of the subgroup is irrelevant for my purposes, because it's all tempered down to rank-1 anyway. The point of including them is they represent consonances of the system, and if I fail to include them it won't be obvious that they're intended to be included as consonances.
>
> What I intend to do is define a tonality diamond from the basis intervals and map that diamond to intervals of each ET. Whenever two consonances get mapped to the same interval, that demonstrates one of the vanishing commas. All of the temperaments I will be using are consistent on their given subgroup, so this shouldn't be a problem.

Ah, I think I finally understand what you mean by these dot-separated lists of numbers that's different from a mere subgroup basis. So, for

9-ET: 2.5.7/3.11/3.13/3

The "consonances" of the tonality diamond are

15/14 14/13 13/12 12/11 15/13 7/6 13/11 5/4 14/11 15/11

and their inversions and octave equivalents?

Some of the numbers are pretty high... for 23-ET do you really mean 33/32 to be a "consonance"?

Or perhaps you don't mean to imply octave equivalence... but that would be absurd because then for 9-ET 15/13 would be a "consonance" but 5/4 would not be. If octave equivalence is not implied, you better have a lot more even numbers in those dot-separated lists.

Keenan

🔗cityoftheasleep <igliashon@...>

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 4:14 PM, cityoftheasleep <igliashon@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > You are of course free to list odd integers supported by
> > your subgroup, but I do wish you wouldn't use the a.b.c.d. ...
> > notation, since it looks like you're outlining an actual basis.
>
> What should I use instead? Colons?
>
> -Igs

I don't know; I'm still trying to figure out what it is you're after.
The questions Keenan just asked are pertinent, for instance. What,
exactly, do you want to list? Every single (octave-equivalent) thing
in a certain odd-limit supported by the subgroup, or something else?

-Mike

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> Ah, I think I finally understand what you mean by these dot-separated lists of numbers that's different from a mere subgroup basis. So, for
>
> 9-ET: 2.5.7/3.11/3.13/3
>
> The "consonances" of the tonality diamond are
>
> 15/14 14/13 13/12 12/11 15/13 7/6 13/11 5/4 14/11 15/11
>
> and their inversions and octave equivalents?

Yes, I think so.

> Some of the numbers are pretty high... for 23-ET do you really mean > 33/32 to be a "consonance"?

Dyadically? No. But one of the main points of this approach is that we are defining consonance more broadly than mere dyads. That 33/32 can form consonant harmonies with 9, 15, and 21; a chord such as 8:9:15:21:33 is indeed very consonant, as dropping the 8 makes it 3:5:7:11. I've long loathed the obsessive focus on dyads and triads when defining consonances, considering there's not much point in using extended JI at all if you're primarily interested in dyads and triads.

> Or perhaps you don't mean to imply octave equivalence... but that
> would be absurd because then for 9-ET 15/13 would be a "consonance"
> but 5/4 would not be. If octave equivalence is not implied, you
> better have a lot more even numbers in those dot-separated lists.

I do mean to imply octave equivalence.

-Igs

🔗cityoftheasleep <igliashon@...>

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

> The questions Keenan just asked are pertinent, for instance. What,
> exactly, do you want to list? Every single (octave-equivalent) thing
> in a certain odd-limit supported by the subgroup, or something else?

I want to list all the identities necessary to generate a tonality diamond which contains all octave-equivalent identities that might appear in all the most significant (by my rough estimation) consonant harmonies possible in each ET.

-Igs

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 4:27 PM, cityoftheasleep <igliashon@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The questions Keenan just asked are pertinent, for instance. What,
> > exactly, do you want to list? Every single (octave-equivalent) thing
> > in a certain odd-limit supported by the subgroup, or something else?
>
> I want to list all the identities necessary to generate a tonality diamond
> which contains all octave-equivalent identities that might appear in all the
> most significant (by my rough estimation) consonant harmonies possible in
> each ET.

This can't be what you want because it sounds like you want a linearly
independent basis. But, you're saying you want both 3 and 9 involved?
If you just want to list the ratios (identities?) necessary to
generate a tonality diamond, then 3 also generates 9, so there's no
reason to put in 9. Is there some additional criterion that you have
here?

-Mike

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> If you just want to list the ratios (identities?) necessary to
> generate a tonality diamond, then 3 also generates 9, so there's no
> reason to put in 9. Is there some additional criterion that you have
> here?

3 does not generate 9 when forming a tonality diamond. Look at Partch's tonality diamonds, and you'll see 9 is separate from 3. Tonality diamonds are formed by division, not multiplication (unless you count multiplication by 2 to bring the members into a single octave).

-Igs

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 4:41 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > If you just want to list the ratios (identities?) necessary to
> > generate a tonality diamond, then 3 also generates 9, so there's no
> > reason to put in 9. Is there some additional criterion that you have
> > here?
>
> 3 does not generate 9 when forming a tonality diamond. Look at Partch's tonality diamonds, and you'll see 9 is separate from 3. Tonality diamonds are formed by division, not multiplication (unless you count multiplication by 2 to bring the members into a single octave).

I guess that what you want is to find a set S of vectors whereby the
set of differences S-S fits into a Weil/Kees-bounded subset of the
octave-equivalent lattice. Maybe Gene knows some strategy for working
that out.

All I can say is, use whatever notation you want, but IMO it's
confusing if you use the a.b.c.etc notation and equate this with a
basis for a subgroup, because it's a totally different thing.

-Mike

🔗Graham Breed <gbreed@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I guess that what you want is to find a set S of vectors whereby the
> set of differences S-S fits into a Weil/Kees-bounded subset of the
> octave-equivalent lattice. Maybe Gene knows some strategy for working
> that out.

I believe I have already found what I want to find. I do not see how the groups I posted are problematic in meeting my stated goals, nor do I see any suggestions for improving them in regards to their ability to meet my stated goals.

> All I can say is, use whatever notation you want, but IMO it's
> confusing if you use the a.b.c.etc notation and equate this with a
> basis for a subgroup, because it's a totally different thing.

Perhaps it's different on some deep theoretical level, but in terms of commas, complexity rankings, and abilities to describe consonant musical intervals within a tuning system, I fail to see how they are different in any way I should care about.

-Igs

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> 3 does not generate 9 when forming a tonality diamond. Look at
> Partch's tonality diamonds, and you'll see 9 is separate from 3.
> Tonality diamonds are formed by division, not multiplication
> (unless you count multiplication by 2 to bring the members into a
> single octave).

The point of a tonality diamond isn't to produce consonant
dyads, it's to show a relationship between consonant chords
(that requires very few notes in JI).

It seems to me that you do want subgroups. The point of a
subgroup is that we know how to look at its basis and deduce
things about the chords it contains. Otherwise, you're either
talking about a single chord (numbers separated by colons)
or some list of single chords. If you look at something like
the weighted dyadic error of a chord, it won't be terribly far
off the T2 error of subgroups containing it. The more chords
you list (or the bigger), the closer the two will become.

Partch's tonality diamond contains chords, and 4:5:6:9:11
does sound different than 4:5:6:11. If you really are
interested in one or the other, you can't beat directly
measuring its error. If you want both, you should use the
T2 error of 2.3.5.11.

-Carl

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Nov 1, 2012 at 5:08 PM, Carl Lumma <carl@...> wrote:
> >
> > Isn't that what you were asking for? I mean, it's true you
> > still need a reduction algorithm at the end of the day, but
> > it doesn't look like there's any way out of that.
>
> Not me, I was just trying to clear up a miscommunication... I'm busy
> trying to figure out subgroup temperament badness here.
>
> -Mike

I'm curious why the hairy ball trick doesn't work for
complexity. I feel like I figured this out once, but I've
apparently forgotten it. -Carl

🔗cityoftheasleep <igliashon@...>

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

> The point of a tonality diamond isn't to produce consonant
> dyads, it's to show a relationship between consonant chords
> (that requires very few notes in JI).

I thought the point of a tonality diamond is to illustrate the intervallic gamut entailed by a single consonant chord under octave equivalence.

> It seems to me that you do want subgroups. The point of a
> subgroup is that we know how to look at its basis and deduce
> things about the chords it contains. Otherwise, you're either
> talking about a single chord (numbers separated by colons)
> or some list of single chords. If you look at something like
> the weighted dyadic error of a chord, it won't be terribly far
> off the T2 error of subgroups containing it. The more chords
> you list (or the bigger), the closer the two will become.

I think what I do want is a list of chords, usually with a single member. The idea being, take the largest consonant chord you can make in the ET, and then look at the intervals entailed by the chord, and map them to the ET.

> Partch's tonality diamond contains chords, and 4:5:6:9:11
> does sound different than 4:5:6:11. If you really are
> interested in one or the other, you can't beat directly
> measuring its error.

Indeed. I think that is the best approach. However, I'm still confused as to how this approach is different than regular tempering, since we still end up with a list of commas tempered out, and a generator mapping....

-Igs

🔗Mike Battaglia <battaglia01@...>

🔗Keenan Pepper <keenanpepper@...>

Okay, Igs, given that what you're really interested in is not abstract infinite groups, but tonality diamonds and chords, here's how I'd present the information:

> 1-ET: 1
> 2-ET: 5:7
> 3-ET: 1:5, 7:9:11
> 4-ET: 3:5:7:17
> 5-ET: 1:3:7
> 6-ET: 1:5:7:9
> 7-ET: 1:3:13, or 5:9:11
> 8-ET: 3:5:11:13
> 9-ET: 3:7:11:13:15
> 10-ET: 1:5:7, 1:7:13:15
> 11-ET: 1:7:9:11:15:17
> 12-ET: 1:3:5:7:9:15:17:19
> 13-ET: 1:5:9:13:17:21
> 14-ET: 5:7:9:11, or 1:3:7(:11)
> 15-ET: 1:3:5:7:11, or simply 1:5:7:11
> 16-ET: 1:5:7:13:19
> 17-ET: 1:3:7:9:11:13
> 18-ET: 3:7:11:13:15:17:27
> 19-ET: 1:3:5:7:9:13:15
> 20-ET: 1:7:11:13:15:19:27
> 21-ET: 1:5:7:9:11:13(:15)
> 22-ET: 1:3:5:7:9:11:15:17
> 23-ET: 1:9:13:15:17:21:33
> 24-ET: 1:3:5:9:11:13:15:17:19
> 25-ET: 1:5:7:9:17:19

What this means is that those chords are "consonant", so if you make a tonality diamond from all their intervals that's the appropriate tonality diamond. (Octave equivalence is always assumed.)

Each of these chords implies a specific subgroup, but as Gene and others pointed out, if the only thing you want to specify is the subgroup then you should give each basis in a standard form (which means, first of all, making sure that it actually *is* a basis). These chords contain more information than the mere subgroups.

Does that sound like what you really want to be saying?

Keenan

🔗Chris Vaisvil <chrisvaisvil@...>

As someone pretty ignorant of tuning theory I wish there was a table that
said

If you play scale (tuning) steps 0:4:8:10 in 11 edo you will come close to
approximating a harmonic series chord of 0:3:6:7

I'm thinking the table below is telling me just what prime limits are
capable with in the ET.

If they really are scale steps I am unsure how to interpret 2-ET: 5:7

On Thu, Nov 1, 2012 at 7:50 PM, Keenan Pepper <keenanpepper@...>wrote:

> **
>
>
>
>
> Okay, Igs, given that what you're really interested in is not abstract
> infinite groups, but tonality diamonds and chords, here's how I'd present
> the information:
>
> > 1-ET: 1
> > 2-ET: 5:7
> > 3-ET: 1:5, 7:9:11
> > 4-ET: 3:5:7:17
> > 5-ET: 1:3:7
> > 6-ET: 1:5:7:9
> > 7-ET: 1:3:13, or 5:9:11
> > 8-ET: 3:5:11:13
> > 9-ET: 3:7:11:13:15
> > 10-ET: 1:5:7, 1:7:13:15
> > 11-ET: 1:7:9:11:15:17
> > 12-ET: 1:3:5:7:9:15:17:19
> > 13-ET: 1:5:9:13:17:21
> > 14-ET: 5:7:9:11, or 1:3:7(:11)
> > 15-ET: 1:3:5:7:11, or simply 1:5:7:11
> > 16-ET: 1:5:7:13:19
> > 17-ET: 1:3:7:9:11:13
> > 18-ET: 3:7:11:13:15:17:27
> > 19-ET: 1:3:5:7:9:13:15
> > 20-ET: 1:7:11:13:15:19:27
> > 21-ET: 1:5:7:9:11:13(:15)
> > 22-ET: 1:3:5:7:9:11:15:17
> > 23-ET: 1:9:13:15:17:21:33
> > 24-ET: 1:3:5:9:11:13:15:17:19
> > 25-ET: 1:5:7:9:17:19
>
> What this means is that those chords are "consonant", so if you make a
> tonality diamond from all their intervals that's the appropriate tonality
> diamond. (Octave equivalence is always assumed.)
>
> Each of these chords implies a specific subgroup, but as Gene and others
> pointed out, if the only thing you want to specify is the subgroup then you
> should give each basis in a standard form (which means, first of all,
> making sure that it actually *is* a basis). These chords contain more
> information than the mere subgroups.
>
> Does that sound like what you really want to be saying?
>
> Keenan
>
> _
>

🔗Mike Battaglia <battaglia01@...>

🔗Chris Vaisvil <chrisvaisvil@...>

For certain, no. I could only guess that subgroups are actually harmonic
series prime limits.

On Thu, Nov 1, 2012 at 8:46 PM, Mike Battaglia <battaglia01@...>wrote:

> **
>
>
> On Thu, Nov 1, 2012 at 8:45 PM, Chris Vaisvil <chrisvaisvil@...>
> wrote:
>
> >
> > As someone pretty ignorant of tuning theory I wish there was a table that
> > said
> >
> > If you play scale (tuning) steps 0:4:8:10 in 11 edo you will come close
> to
> > approximating a harmonic series chord of 0:3:6:7
>
> If I say "11-EDO supports the 2.7.9.11 subgroup pretty accurately," do
> you know what that means?
>
> -Mike
>
>
>

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 9:10 PM, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> For certain, no. I could only guess that subgroups are actually harmonic
> series prime limits.

Subgroups expand the concept of a prime limit. For instance, say you
want the 7-limit, but you don't care about prime 5; you just want
primes 2, 3, and 7. So that's the 2.3.7 subgroup. Or say that you want
the 7-limit, but you don't care about 3/1 but you do care about 9/1.
Then that's the 2.5.7.9 subgroup. Or, say that you want primes 2 and
3, and the composite interval 7/5; that's the 2.3.7/5 subgroup.

The rule for any subgroup is that if you multiply or divide intervals,
that's also in the subgroup. So for the 2.3.7 subgroup, 7*3 = 21/1 is
in the subgroup, as is 7/(2*3) = 7/6. And for the 2.3.7/5 subgroup,
2/(7/5 * 7/5) = 50/49 is in the subgroup, and so on. They're infinite
lattices of intervals.

11-EDO happens to be a decent temperament for the 2.7.9.11 subgroup.

-Mike

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 9:24 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Thu, Nov 1, 2012 at 5:37 PM, Carl Lumma <carl@...> wrote:
> > >
> > > I'm curious why the hairy ball trick doesn't work for
> > > complexity. I feel like I figured this out once, but I've
> > > apparently forgotten it. -Carl
> >
> > LMFAO, the what? I see you've found my Reddit account.
> >
> > -Mike
>
> Eh? Your reddit account was obvious from the start, but
> what does that have to do with the above? -Carl

I thought you were referencing something I posted on Reddit. What is
the hairy ball trick? Are you talking about the hairy ball theorem?

http://en.wikipedia.org/wiki/Hairy_ball_theorem

It says you can't comb a hairy ball flat.

-Mike

🔗Chris Vaisvil <chrisvaisvil@...>

Alright this is reasonable - is this on the wiki? If not it should go there
in the n00bs section.

On Thu, Nov 1, 2012 at 9:21 PM, Mike Battaglia <battaglia01@...>wrote:

> **
>
>
> On Thu, Nov 1, 2012 at 9:10 PM, Chris Vaisvil <chrisvaisvil@...>
> wrote:
> >
>
> > For certain, no. I could only guess that subgroups are actually harmonic
> > series prime limits.
>
> Subgroups expand the concept of a prime limit. For instance, say you
> want the 7-limit, but you don't care about prime 5; you just want
> primes 2, 3, and 7. So that's the 2.3.7 subgroup. Or say that you want
> the 7-limit, but you don't care about 3/1 but you do care about 9/1.
> Then that's the 2.5.7.9 subgroup. Or, say that you want primes 2 and
> 3, and the composite interval 7/5; that's the 2.3.7/5 subgroup.
>
> The rule for any subgroup is that if you multiply or divide intervals,
> that's also in the subgroup. So for the 2.3.7 subgroup, 7*3 = 21/1 is
> in the subgroup, as is 7/(2*3) = 7/6. And for the 2.3.7/5 subgroup,
> 2/(7/5 * 7/5) = 50/49 is in the subgroup, and so on. They're infinite
> lattices of intervals.
>
> 11-EDO happens to be a decent temperament for the 2.7.9.11 subgroup.
>
> -Mike
>
>
>

🔗Carl Lumma <carl@...>

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

> > Eh? Your reddit account was obvious from the start, but
> > what does that have to do with the above? -Carl
>
> I thought you were referencing something I posted on Reddit. What is
> the hairy ball trick? Are you talking about the hairy ball theorem?
>
> http://en.wikipedia.org/wiki/Hairy_ball_theorem
>
> It says you can't comb a hairy ball flat.

Don't think I saw that post. I just meant the Hahn-Banach
theorem trick. I think of it as basically the same as the
hairy ball theorem.

-Carl

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 9:30 PM, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> Alright this is reasonable - is this on the wiki? If not it should go
> there in the n00bs section.

A subgroup section would be nice for beginners but I don't have time
to add it now.

Anyway, you asked about figuring out what steps in 11-EDO approximate
what intervals. So if 11-EDO supports the 2.7.9.11 subgroup, I can
just give you the mappings for 2/1, 7/1, 9/1, and 11/1, and you can
mix and match them to get what intervals you want, right. So for
instance, 2/1 is 11 steps, 7/1 is 31 steps, 9/1 is 35 steps, and 11/1
is 38 steps. So then 11/9 is 38-35 = 3 steps. That should be enough
information for you to get all the intervals.

-Mike

🔗Chris Vaisvil <chrisvaisvil@...>

Thank you, makes perfect sense, and I literally created a n00b page from
this.

On the weekend I'll make it presentable.

On Thu, Nov 1, 2012 at 9:34 PM, Mike Battaglia <battaglia01@...>wrote:

> **
>
>
> On Thu, Nov 1, 2012 at 9:30 PM, Chris Vaisvil <chrisvaisvil@...>
> wrote:
> >
>
> > Alright this is reasonable - is this on the wiki? If not it should go
> > there in the n00bs section.
>
> A subgroup section would be nice for beginners but I don't have time
> to add it now.
>
> Anyway, you asked about figuring out what steps in 11-EDO approximate
> what intervals. So if 11-EDO supports the 2.7.9.11 subgroup, I can
> just give you the mappings for 2/1, 7/1, 9/1, and 11/1, and you can
> mix and match them to get what intervals you want, right. So for
> instance, 2/1 is 11 steps, 7/1 is 31 steps, 9/1 is 35 steps, and 11/1
> is 38 steps. So then 11/9 is 38-35 = 3 steps. That should be enough
> information for you to get all the intervals.
>
> -Mike
>
>
>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
>
>
> Okay, Igs, given that what you're really interested in is not
> abstract infinite groups, but tonality diamonds and chords, here's > how I'd present the information:

Excellent.

> What this means is that those chords are "consonant", so if you
> make a tonality diamond from all their intervals that's the
> appropriate tonality diamond. (Octave equivalence is always
> assumed.)

Does that mean you're assuming there will always be a "2/2" square on the tonality diamond, thus allowing, say, the 2-ET diamond to contain both 7/5 and 10/7?

> Each of these chords implies a specific subgroup, but as Gene and
> others pointed out, if the only thing you want to specify is the
> subgroup then you should give each basis in a standard form (which
> means, first of all, making sure that it actually *is* a basis).
> These chords contain more information than the mere subgroups.
>
> Does that sound like what you really want to be saying?

Yes, I think so.

-Igs

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 9:38 PM, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> Thank you, makes perfect sense, and I literally created a n00b page from
> this.
>
> On the weekend I'll make it presentable.

OK, so rather than write all of that out in English, though, we can
just use a simple notation. So if 2/1 is 11 steps, 7/1 is 31 steps,
9/1 is 35 steps, and 11/1 is 38 steps, then we can just condense all
that as follows:

<11 31 35 38|

where it's understood in this particular case that the coefficients
represent how many steps map to 2/1, 7/1, 9/1 and 11/1, respectively.
This is called a val, and this is why we use them; so we can figure
out how many steps every interval maps to. So 9/7 in the above case is
35-31 = 4 steps.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 1, 2012 at 9:31 PM, Carl Lumma <carl@...> wrote:
>
> Don't think I saw that post. I just meant the Hahn-Banach
> theorem trick. I think of it as basically the same as the
> hairy ball theorem.

The Hahn-Banach trick can be used to show that the TE badness of 2.3.5
81/80 is the same as 2.3.5.7 81/80. But the 2.9.5 81/80 temperament
is, I believe, only half as bad as 2.3.5 81/80 - at least the one
where the generators are 2/1 and 9/8. But, the contorted and insane
2.9.5 81/80 temperament where the generators are 2/1 and an unmapped
sqrt(9/1) has the same TE badness as 2.9.5 81/80.

I dub 2.9.5 81/80 "wholetone" temperament. Here's what we get

Meantone: http://x31eq.com/cgi-bin/rt.cgi?ets=12_19&limit=2.3.5 -
complexity 0.710802, error 1.582221
Wholetone: http://x31eq.com/cgi-bin/rt.cgi?ets=12_19&limit=2.9.5 -
complexity 0.355401, error 1.582221
Insane contorted 2.9.5 meantone:
http://x31eq.com/cgi-bin/rt.cgi?ets=12_19&limit=2.9.5 - complexity
0.710802, error 1.582221

I'm looking to design something that takes the complexity of the
subgroup into account as well. So to start, I'm looking at TE
complexity * TE error * TE subgroup complexity, and for now I'm just
defining "TE subgroup complexity" as the L2 norm of the multimonzo
defined by the subgroup. Ironically, this would even it out so that
2.3.5 meantone and 2.9.5 wholetone now have the same complexity,
though I'm not sure that was what I was looking for.

-Mike

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
> > What this means is that those chords are "consonant", so if you
> > make a tonality diamond from all their intervals that's the
> > appropriate tonality diamond. (Octave equivalence is always
> > assumed.)
>
> Does that mean you're assuming there will always be a "2/2" square on the tonality diamond, thus allowing, say, the 2-ET diamond to contain both 7/5 and 10/7?

There will always be a 1/1 square in the tonality diamond, yes.

Keenan

🔗Carl Lumma <carl@...>

Mike Battaglia <battaglia01@...> wrote:

> > Don't think I saw that post. I just meant the Hahn-Banach
> > theorem trick. I think of it as basically the same as the
> > hairy ball theorem.
>
> The Hahn-Banach trick can be used to show that the TE badness
> of 2.3.5 81/80 is the same as 2.3.5.7 81/80. But the 2.9.5 81/80
> temperament is, I believe, only half as bad as 2.3.5 81/80
//
> I dub 2.9.5 81/80 "wholetone" temperament. Here's what we get
> Meantone: http://x31eq.com/cgi-bin/rt.cgi?ets=12_19&limit=2.3.5 -
> complexity 0.710802, error 1.582221
> Wholetone: http://x31eq.com/cgi-bin/rt.cgi?ets=12_19&limit=2.9.5 -
> complexity 0.355401, error 1.582221

It's telling me the complexity of the second one is 0.710802 also.
(?)

> I'm looking to design something that takes the complexity of the
> subgroup into account as well. So to start, I'm looking at TE
> complexity * TE error * TE subgroup complexity, and for now I'm
> just defining "TE subgroup complexity" as the L2 norm of the
> multimonzo defined by the subgroup.

TE complexity is already weighted... doesn't that capture
the complexity of (each element of) the subgroup basis?

-Carl

🔗Mike Battaglia <battaglia01@...>

On Fri, Nov 2, 2012 at 4:47 AM, Carl Lumma <carl@...> wrote:
>
> > I dub 2.9.5 81/80 "wholetone" temperament. Here's what we get
> > Meantone: http://x31eq.com/cgi-bin/rt.cgi?ets=12_19&limit=2.3.5 -
> > complexity 0.710802, error 1.582221
> > Wholetone: http://x31eq.com/cgi-bin/rt.cgi?ets=12_19&limit=2.9.5 -
> > complexity 0.355401, error 1.582221
>
> It's telling me the complexity of the second one is 0.710802 also.
> (?)

Oops, wrong link; should have been
http://x31eq.com/cgi-bin/rt.cgi?ets=13_19&limit=2.9.5. I pasted the
third link twice.

> TE complexity is already weighted... doesn't that capture
> the complexity of (each element of) the subgroup basis?

The Hahn-Banach stuff from before shows that TE error will be the same
either way. So, since TE badness is TE error * TE complexity, we need
only look at complexity here.

The main issue in this case seems to be that moving from 2.3.5 to
2.9.5 actually changes the generators, which cuts the complexity of
5/1 in half.

I also note that it's hard to rectify this with the notion that the
multimonzo tempered out should be the same (in weighted coordinates
it's still |-4 6.340 -2.322>, which is just |-4 4 -1>). This looks
like a thing with the way that Gene's "dual" is defined on the wiki;
it isn't invariant under a change of basis. The L2 norms of the
multimonzo and multival associated with a temperament will be
identical if the coordinates aren't weighted, but once you weight
things, one will be a multiple of the other.

-Mike

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> The main issue in this case seems to be that moving from 2.3.5 to
> 2.9.5 actually changes the generators, which cuts the complexity of
> 5/1 in half.

Quick comment -- I can get a wedgie from the commas, and get
complexity from the wedgie. So shouldn't complexity stay
the same?

> I also note that it's hard to rectify this with the notion that the
> multimonzo tempered out should be the same (in weighted coordinates
> it's still |-4 6.340 -2.322>, which is just |-4 4 -1>).

Heh, yeah. Late. for. work.

> This looks
> like a thing with the way that Gene's "dual" is defined on the wiki;
> it isn't invariant under a change of basis. The L2 norms of the
> multimonzo and multival associated with a temperament will be
> identical if the coordinates aren't weighted, but once you weight
> things, one will be a multiple of the other.

Hmm.... -C.

🔗Mike Battaglia <battaglia01@...>

On Fri, Nov 2, 2012 at 4:18 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > The main issue in this case seems to be that moving from 2.3.5 to
> > 2.9.5 actually changes the generators, which cuts the complexity of
> > 5/1 in half.
>
> Quick comment -- I can get a wedgie from the commas, and get
> complexity from the wedgie. So shouldn't complexity stay
> the same?

I see you saw my explanation of this below, but there's another
comment as well: the thing you're getting from the commas is a
multimonzo, and the thing you're getting from the supporting ETs is a
multival, and even if you assume that we fix this scaling issue some
way or another, the two will only have the same norm if the norm is
L2. Otherwise, if you're using Tenney height, the natural norm induced
on multimonzos is also L1, and the norm induced on multivals is Linf.

So every temperament has -two- complexity measures - the Lp complexity
of the kernel (as a multimonzo), and the Lq complexity of the
supporting vals (as a multival), where Lq is dual to Lp, and these two
complexity measures tell you two subtly different things about the
temperament. Once upon a time I was going to work Tp complexity out to
follow the work we did on Tp error, but then I came down with a bad
case of Weil fever. I'll keep working on superbadness for now, but it
looks like we may have to flesh this out first.

-Mike

🔗Mike Battaglia <battaglia01@...>

I worked this out. The short story is

1) now that we're working with complex subgroups, weighted coordinates
are the root of all evil
2) if you use unweighted coordinates but a "weighted norm" (the Tp
norm), then it becomes much simpler. (note the norm will have to be
more than just "weighted" for more complex subgroups, but distorted in
more complex ways)
3) the real truth is that there are two types of Tp complexity -
"kernel complexity" and "mapping complexity", or something - and for
almost all values of p, these two do NOT agree. They're both useful
and have subtly different musical interpretations, which I'm still
tying down the nuances of.
4) for the T2 (TE) norm and only the T2 norm, these two types of
complexity DO agree - but one is a scaled version of the other.

This is what's really going on above. When it comes to defining TE
complexity for prime-limits, you get lucky: you can define it as the
norm of the multimonzo OR the multival associated with a temperament,
and you get the same rankings for everything; one just ends up being a
scaled version of the other. This is because the T2 (TE) norm is
related to the L2 norm, and the L2 norm is magic and special.

So, we looked at this situation and were like "hey, why don't these
two measures agree perfectly?" - but, we actually got lucky this time
that the only disagreement was a simple scaling factor. They usually
agree even less, and if we were using the T1 norm instead of the T2
norm, they wouldn't have agreed at all.

The real takehome point here is that there are two types of Tp
complexity. You're a fan of the thing I called kernel complexity,
whereas Graham's a fan of the other one. Paul's ruined everything by
using the T1 norm on monzos and then taking the T1 norm of the
multival, but he's only doing that because he had the idea that the
norm of the multival and multimonzo should be the same (because the
coefficients are the same up to sign). So it's a brave new world now.

A longer exposition about all that is here:

/tuning-math/message/21054

-Mike

🔗Carl Lumma <carl@...>

Sorry for the delay. Now that I've seen Cloud Atlas,
I can get on with my life...

> I see you saw my explanation of this below, but there's another
> comment as well: the thing you're getting from the commas is a
> multimonzo, and the thing you're getting from the supporting ETs
> is a multival, and even if you assume that we fix this scaling
> issue some way or another, the two will only have the same norm
> if the norm is L2. Otherwise, if you're using Tenney height, the
> natural norm induced on multimonzos is also L1, and the norm
> induced on multivals is Linf.

That only L2 is self-dual in this fashion is well known.

I don't know what you mean by weighted coordinates vs a
weighted norm... that is, I can only think of one thing
these two terms could mean... but I'll pop over to
tuning-math now and I'm sure all will be made clear.

-Carl