Yahoo Tuning Groups Ultimate Backup tuning-math Domes

Gene wrote this on XA

"In particular, it is very helpful not only in defining what "dome" or
class of blocks a given scale belongs to, but in defining
unambiguously precisely which one of that dome."

What did you mean by this? Are we using the word "dome" in the same way?

I originally used the word "dome" to refer to the different pitch
sets, up to translation, that you can get from the same set of unison
vectors by just shifting them around on the lattice. If you apply this
operation to a Fokker block you can obtain a new set which is not a
cyclic permutation of the first set (e.g. not a "mode" of the first
set). So these two are "domes" of the same Fokker block, just like
Ionian and Dorian are "modes" of the same scale.

If that's what you meant above, then how do you specify a dome from a
set of wedgies? For instance, the meantone-dicot block has 7 domes, I
believe, each of which has 7 modes. How do I unambiguously specify a
dome using wedgies?

"It is very useful for the problem of determining, given a scale, if
it is a Fokker block."

Also, I've been trying to figure out how to do this for a while - do
you have a method for it using wedgies?

Thanks,
Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

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

> If that's what you meant above, then how do you specify a dome from a
> set of wedgies? For instance, the meantone-dicot block has 7 domes, I
> believe, each of which has 7 modes. How do I unambiguously specify a
> dome using wedgies?

You take meantone = <<1 4 4|| and dicot = <<2 1 -3||, and take interior products with the notes of the scale, which give two 7-note ranges of contiguous generators from -6 to 0 to 0 to 6. See below!

> "It is very useful for the problem of determining, given a scale, if
> it is a Fokker block."
>
> Also, I've been trying to figure out how to do this for a while - do
> you have a method for it using wedgies?

Let's see if the zarlino scale is a Fokker block. Putting it into Scala tells us that it is epimorphic with val <7 11 16|, and the group it generates is 2.3.5. Looking <7 11 16| up on "Minkowski reduced bases for Fokker blocks" tells us that the meantone and dicot wedgies generate the Fokker group. Now we check Z-linear combinations of meantone and dicot to see if they are wedgies, and then if they give a Graham complexity to the Zarlino scale less than 7. For instance, we find

If we take the first coefficient of each of these seven vals, which is where they map 2, we get 2, 4, -1, 1, 3, 5, 0. Sorting that gives -1, 0, 1, 2, 3, 4, 5. 5-(-1)=6, so meantone gives the Zarlino scale a Graham complexity of 6. Moreover, dicot gives a Graham complexity of 6 also, with generators ranging from -2 to 4. And we also find that dicot-meantone = <<1 -3 -7|| = mavila gives a Graham complexity of 6 also, with a generator range from -4 to 2. So the Zarlino scale is not only a Fokker block, it is a wakalix. Moreover, since each pair of these three wedgies are a basis for the Fokker group, it's a denizen of three different domes: the dicot-meantone dome, the dicot-mavila dome, and the meantone-mavila dome.

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > If that's what you meant above, then how do you specify a dome from a
> > set of wedgies? For instance, the meantone-dicot block has 7 domes, I
> > believe, each of which has 7 modes. How do I unambiguously specify a
> > dome using wedgies?
>
> You take meantone = <<1 4 4|| and dicot = <<2 1 -3||, and take interior products with the notes of the scale, which give two 7-note ranges of contiguous generators from -6 to 0 to 0 to 6. See below!
>
> > "It is very useful for the problem of determining, given a scale, if
> > it is a Fokker block."
> >
> > Also, I've been trying to figure out how to do this for a while - do
> > you have a method for it using wedgies?
>
> Let's see if the zarlino scale is a Fokker block. Putting it into Scala tells us that it is epimorphic with val <7 11 16|, and the group it generates is 2.3.5. Looking <7 11 16| up on "Minkowski reduced bases for Fokker blocks" tells us that the meantone and dicot wedgies generate the Fokker group. Now we check Z-linear combinations of meantone and dicot to see if they are wedgies, and then if they give a Graham complexity to the Zarlino scale less than 7. For instance, we find
>
> meantoneV9/8 = <2 3 4|
> meantoneV5/4 = <4 6 8|
> meantoneV4/3 = <-1 -2 -4|
> meantoneV3/2 = <1 1 0|
> meantoneV5/3 = <3 4 4|
> meantoneV15/8 = <5 7 8|
> meantoneV2 = <0 -1 -4|
>
> If we take the first coefficient of each of these seven vals, which is where they map 2, we get 2, 4, -1, 1, 3, 5, 0. Sorting that gives -1, 0, 1, 2, 3, 4, 5. 5-(-1)=6, so meantone gives the Zarlino scale a Graham complexity of 6. Moreover, dicot gives a Graham complexity of 6 also, with generators ranging from -2 to 4. And we also find that dicot-meantone = <<1 -3 -7|| = mavila gives a Graham complexity of 6 also, with a generator range from -4 to 2. So the Zarlino scale is not only a Fokker block, it is a wakalix. Moreover, since each pair of these three wedgies are a basis for the Fokker group, it's a denizen of three different domes: the dicot-meantone dome, the dicot-mavila dome, and the meantone-mavila dome.

Gene, everything you're saying is crystal clear to me, but you are using the word "dome" in a different way than Mike meant when he came up with it. The dicot-meantone system is not itself a dome; it's a set of 7 different domes.

"Dome" is supposed to be analogous to "mode". A 7-note scale has 7 different modes. The set of all the modes doesn't really have a name - it's just called "the scale" as a whole, or "the set of modes".

Similarly, a pair of temperaments that generate the Fokker group, like meantone+dicot, *has* 7 different domes (each of those with 7 modes each, for a total of 49). The set of all 7 domes (which you're calling "a dome" contrary to our usage) doesn't really have a name yet. You can come up with a name - perhaps something like "domeset"?

With that terminology you would say: "this wakalix is a dome of three different domesets: the dicot-meantone domeset, the dicot-mavila domeset, and the meantone-mavila domeset."

Keenan

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Gene, everything you're saying is crystal clear to me, but you are using the word "dome" in a different way than Mike meant when he came up with it. The dicot-meantone system is not itself a dome; it's a set of 7 different domes.

In which case, you've named the less important thing and left the more important thing without a name. I don't like "domeset", and am sorry I am going to have to go change all the references to "domes" on the Xenwiki. But to what?

> Similarly, a pair of temperaments that generate the Fokker group, like meantone+dicot, *has* 7 different domes (each of those with 7 modes each, for a total of 49). The set of all 7 domes (which you're calling "a dome" contrary to our usage) doesn't really have a name yet. You can come up with a name - perhaps something like "domeset"?

So a "dome" is just a scale up to modal equivalence? I guess that does need a name also.

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
>
> > With that terminology you would say: "this wakalix is a dome of three different domesets: the dicot-meantone domeset, the dicot-mavila domeset, and the meantone-mavila domeset."
>
> How does "arena" grab you? "This wakalix is a dome of three different arenas: the dicot-meantone arena, the dicot-mavila arena, and the meantone-mavila arena."

Sounds fine to me!

As long as we get to say "dome" for "chromatically altered version of a scale that's part of the same arena, but not simply a modal transposition".

Keenan

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 3:44 AM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > Similarly, a pair of temperaments that generate the Fokker group, like
> > meantone+dicot, *has* 7 different domes (each of those with 7 modes each,
> > for a total of 49). The set of all 7 domes (which you're calling "a dome"
> > contrary to our usage) doesn't really have a name yet. You can come up with
> > a name - perhaps something like "domeset"?
>
> So a "dome" is just a scale up to modal equivalence? I guess that does
> need a name also.

A dome is something specifically related to a Fokker block.

If you look at all of the scales that you can get with the 25/24 and
81/80 unison vectors which contain 1/1, you'll find that you get 49
different scales. If we consider scales which are modally equivalent
to be the same "dome," then the playing field reduces to 7 fundamental
"domes" which you can get out of the 25/24 and 81/80 Fokker block.
Each dome of this block is a collection of 7 scales which are modally
equivalent. However, every dome is modally independent from every
other dome of the block.

I had simply wanted to say before that all of the 49 different scales
in the 25/24 and 81/80 Fokker block are just the different -modes- of
that block, using a much deeper and generalized notion of the concept
of "mode." People complained that "mode" already meant cyclic rotation
though, so I came up with the word "dome" to denote this other, new
thing instead, so that the whole set of 49 scales can be classified
into 7 different domes with 7 modes each.

Whatever you want to call it, the idea is that these are the building
blocks of rank-3 modal harmony, which because of there being multiple
domes for one Fokker block is simply more complex and likely more
musically interesting than rank-2 modal harmony.

The term that I've been throwing around in my head for the collection
of domes and modes that makes up a Fokker block is the "modal system"
for that block, which is a name that I don't mind if we change. You
suggested "arena," but I can't tell if your word "arena" is synonymous
with the "modal system" for a Fokker block or with the phrase "Fokker
block" itself. Which is it?

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 2:05 AM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> Let's see if the zarlino scale is a Fokker block. Putting it into Scala
> tells us that it is epimorphic with val <7 11 16|, and the group it
> generates is 2.3.5. Looking <7 11 16| up on "Minkowski reduced bases for
> Fokker blocks" tells us that the meantone and dicot wedgies generate the
> Fokker group. Now we check Z-linear combinations of meantone and dicot to
> see if they are wedgies, and then if they give a Graham complexity to the
> Zarlino scale less than 7. For instance, we find
>
> meantoneV9/8 = <2 3 4|
> meantoneV5/4 = <4 6 8|
> meantoneV4/3 = <-1 -2 -4|
> meantoneV3/2 = <1 1 0|
> meantoneV5/3 = <3 4 4|
> meantoneV15/8 = <5 7 8|
> meantoneV2 = <0 -1 -4|
>
> If we take the first coefficient of each of these seven vals, which is
> where they map 2, we get 2, 4, -1, 1, 3, 5, 0. Sorting that gives -1, 0, 1,
> 2, 3, 4, 5. 5-(-1)=6, so meantone gives the Zarlino scale a Graham
> complexity of 6.

This is a really neat idea, but how did you come up with the notion of
taking the interior product of meantone and each note in the scale,
and then looking at the 2/1-coefficient in the resulting vals? Why the
2/1 coefficient, and how does that relate to the Graham complexity?

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

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

> A dome is something specifically related to a Fokker block.

That's not the definition Keenan has on the Xenwiki, which I just read. I think your precise definition deserves a word.

> The term that I've been throwing around in my head for the collection
> of domes and modes that makes up a Fokker block is the "modal system"
> for that block, which is a name that I don't mind if we change. You
> suggested "arena," but I can't tell if your word "arena" is synonymous
> with the "modal system" for a Fokker block or with the phrase "Fokker
> block" itself. Which is it?

It's the same. We also might want to name quasi-Fokker blocks, where the scale types are allowed to be something other than MOS. I might find I want to look at 19-note scales where via godzilla and meantone they temper to a MOS, but via keemun to a 19+1. By the way, you aksed me about the direction of generators for this sort of notation, and it occurs to me that since I am using wedgies provide canonical descriptions, the generator g we want is the one where (WVg)[1] is positive.

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 1:40 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > A dome is something specifically related to a Fokker block.
>
> That's not the definition Keenan has on the Xenwiki, which I just read. I
> think your precise definition deserves a word.

The word for my precise definition is "dome." The wiki page is
incorrect, and we had a discussion about it on XH chat a while ago but
forgot to change the page. I've changed it to what we agreed on, which
is what Keenan and I have both said above.

If we want to come up with a word for an "alternate form of a scale
which is not a mode," whatever that might mean, that word needs to not
be "dome," or else the term loses its meaning.

> > The term that I've been throwing around in my head for the collection
> > of domes and modes that makes up a Fokker block is the "modal system"
> > for that block, which is a name that I don't mind if we change. You
> > suggested "arena," but I can't tell if your word "arena" is synonymous
> > with the "modal system" for a Fokker block or with the phrase "Fokker
> > block" itself. Which is it?
>
> It's the same. We also might want to name quasi-Fokker blocks, where the
> scale types are allowed to be something other than MOS. I might find I want
> to look at 19-note scales where via godzilla and meantone they temper to a
> MOS, but via keemun to a 19+1. By the way, you aksed me about the direction
> of generators for this sort of notation, and it occurs to me that since I am
> using wedgies provide canonical descriptions, the generator g we want is the
> one where (WVg)[1] is positive.

Unless I'm doing it wrong, it looks like this will always give the
chroma-negative generator, right? Is there some advantage to doing it
this way? If it doesn't matter either way, why not always pick the
chroma-positive one, e.g. the one where (W V g)[1] is negative, to
bring it into alignment with UDP and so on?

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

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

> Unless I'm doing it wrong, it looks like this will always give the
> chroma-negative generator, right?

No, it picks the generator based on the temperament only, without reference to the size of the MOS. This has the advantage that the generator is always the same for different sizes of MOS.

Some standard generators using this system:

meantone: ~3/2
mavila: ~3/2
helmholtz/garibaldi: ~3/2

Etc etc, always 3/2, never 4/3

hanson: ~6/5
porcupine: ~9/5
magic: ~5/4
pajara: ~15/14, ~3/2
orwell: ~7/6
miracle: ~15/14

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> The word for my precise definition is "dome." The wiki page is
> incorrect, and we had a discussion about it on XH chat a while ago but
> forgot to change the page. I've changed it to what we agreed on, which
> is what Keenan and I have both said above.
>
> If we want to come up with a word for an "alternate form of a scale
> which is not a mode," whatever that might mean, that word needs to not
> be "dome," or else the term loses its meaning.

Well, personally I don't see why we couldn't leave it open-ended. But that's a really minor quibble - I'm totally fine with the current version.

Keenan

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 4:10 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > Unless I'm doing it wrong, it looks like this will always give the
> > chroma-negative generator, right?
>
> No, it picks the generator based on the temperament only, without
> reference to the size of the MOS. This has the advantage that the generator
> is always the same for different sizes of MOS.

Oh yeah, duh.

> Some standard generators using this system:
>
> meantone: ~3/2
> mavila: ~3/2
> helmholtz/garibaldi: ~3/2
>
> Etc etc, always 3/2, never 4/3

Man, wtf. There has to be something wrong with my "interior" routine then.

I'm getting

>> interior([1 4 4], [-1 1 0])
ans =
-1 -1 0

>> interior([1 4 4], [2 -1 0])
ans =
1 2 4

And I assume that you're getting the opposite above?

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 4:16 PM, Keenan Pepper <keenanpepper@gmail.com>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> > The word for my precise definition is "dome." The wiki page is
> > incorrect, and we had a discussion about it on XH chat a while ago but
> > forgot to change the page. I've changed it to what we agreed on, which
> > is what Keenan and I have both said above.
> >
> > If we want to come up with a word for an "alternate form of a scale
> > which is not a mode," whatever that might mean, that word needs to not
> > be "dome," or else the term loses its meaning.
>
> Well, personally I don't see why we couldn't leave it open-ended. But
> that's a really minor quibble - I'm totally fine with the current version.

What did the open-ended definition mean, exactly? Under that
definition, would melodic minor and harmonic minor be "domes" of the
3-limit Fokker block that tempers out 2187/2048?

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> What did the open-ended definition mean, exactly? Under that
> definition, would melodic minor and harmonic minor be "domes" of the
> 3-limit Fokker block that tempers out 2187/2048?

No, those are MODMOSes, not domes.

Here's something that I would definitely like to call a "dome" but doesn't fit your strict Fokker-block-based definition: Suppose that instead of a parallelogram I choose some other shape on the lattice, for example a hexagon. Now I move that shape around and look at the scales formed from the points inside it. Some of them are modes of each other; others are not modes but... what? You're telling me I can't call them domes, but they seem just like domes to me.

Keenan

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 6:16 PM, Keenan Pepper <keenanpepper@gmail.com>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> > What did the open-ended definition mean, exactly? Under that
> > definition, would melodic minor and harmonic minor be "domes" of the
> > 3-limit Fokker block that tempers out 2187/2048?
>
> No, those are MODMOSes, not domes.
>
> Here's something that I would definitely like to call a "dome" but doesn't
> fit your strict Fokker-block-based definition: Suppose that instead of a
> parallelogram I choose some other shape on the lattice, for example a
> hexagon. Now I move that shape around and look at the scales formed from the
> points inside it. Some of them are modes of each other; others are not modes
> but... what? You're telling me I can't call them domes, but they seem just
> like domes to me.
>
> Keenan

OK, those can be domes too. I thought your initial objection was that
you wanted things like z-related scales to be domes, but this is very
different.

So how does one define this precisely? Just replace "Fokker block"
with "polytope with sides that are bound vectors with integral
coordinates" or something?

-Mike

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > OK, those can be domes too. I thought your initial objection was that
> > you wanted things like z-related scales to be domes, but this is very
> > different.
> >
> > So how does one define this precisely? Just replace "Fokker block"
> > with "polytope with sides that are bound vectors with integral
> > coordinates" or something?
>
> What about any fundamental domain of the translational symmetry group of the lattice? In other words, any tile?

Sorry, ambiguity! Of course this should say fundamental domain of the translational symmetry group of the *chroma* lattice. Not the basic JI or temperament lattice.

Keenan

🔗Mike Battaglia <battaglia01@gmail.com>

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> I'm still slightly confused - how does this apply to things like
> hexagons? Do the vertices of the hexagon, when tiled, still count as
> forming a "lattice" in the usual sense?

No, all the vertices do not form a lattice. But the hexagon is a fundamental domain, which means that it contains exactly one point from each equivalence class of points whose differences are lattice vectors (as long as you're careful with the boundary of the hexagon).

In other words, if you pick an arbitrary point inside the hexagon, then that point, together with the *corresponding* points in all the tiled copies of the hexagon, forms a lattice.

Keenan

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 7:11 PM, Keenan Pepper <keenanpepper@gmail.com>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
> > I'm still slightly confused - how does this apply to things like
> > hexagons? Do the vertices of the hexagon, when tiled, still count as
> > forming a "lattice" in the usual sense?
>
> No, all the vertices do not form a lattice. But the hexagon is a
> fundamental domain, which means that it contains exactly one point from each
> equivalence class of points whose differences are lattice vectors (as long
> as you're careful with the boundary of the hexagon).
>
> In other words, if you pick an arbitrary point inside the hexagon, then
> that point, together with the *corresponding* points in all the tiled copies
> of the hexagon, forms a lattice.

So if I understand correctly, this allow us to talk about domes of
non-convex sets, so long as they tile the lattice, right?

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 7:40 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > So how does one define this precisely? Just replace "Fokker block"
> > with "polytope with sides that are bound vectors with integral
> > coordinates" or something?
>
> I like my suggestion of extending MOS to other shapes better.

Are you talking about your quasi-Fokker block suggestion? It seems
like Keenan's group-theoretic definition would handle those sorts of
objects, right?

-Mike

🔗Keenan Pepper <keenanpepper@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

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

> This is a really neat idea, but how did you come up with the notion of
> taking the interior product of meantone and each note in the scale,
> and then looking at the 2/1-coefficient in the resulting vals? Why the
> 2/1 coefficient, and how does that relate to the Graham complexity?

Recall that the rank two wedgie defines a 2-map, ie a bilinear map, on pairs of wedgies/intervals. If W is the wedgie and g is the generator, then for linear temperaments W(2,g) is +-1, and I'm saying normalize so that W(2,g)=1. (WVg)[1] is just another way of denoting W(2,g), which I used mostly because WVs for scale intervals s defines the abstract MOS. If you are doing computations of Graham complexity, it's actually faster to use W(2,s) directly.

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 10:33 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > This is a really neat idea, but how did you come up with the notion of
> > taking the interior product of meantone and each note in the scale,
> > and then looking at the 2/1-coefficient in the resulting vals? Why the
> > 2/1 coefficient, and how does that relate to the Graham complexity?
>
> Recall that the rank two wedgie defines a 2-map, ie a bilinear map, on
> pairs of wedgies/intervals. If W is the wedgie and g is the generator, then
> for linear temperaments W(2,g) is +-1, and I'm saying normalize so that
> W(2,g)=1. (WVg)[1] is just another way of denoting W(2,g), which I used
> mostly because WVs for scale intervals s defines the abstract MOS. If you
> are doing computations of Graham complexity, it's actually faster to use
> W(2,s) directly.

Whoa, that's a very neat idea! OK, so that at least works as an
NP-complexity algorithm to see if a scale is a Fokker block. I wonder
if an algorithm exists for this in P.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 10:49 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > Are you talking about your quasi-Fokker block suggestion? It seems
> > like Keenan's group-theoretic definition would handle those sorts of
> > objects, right?
>
> It would? I thought he was talking about things like hexagonal tilings.

If I understood his definition correctly, then anything that Paul's
calling a "periodicity block" would suffice, as well as things which
are like periodicity blocks but which have torsion.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 4:10 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> No, it picks the generator based on the temperament only, without
> reference to the size of the MOS. This has the advantage that the generator
> is always the same for different sizes of MOS.
>
> Some standard generators using this system:
>
> meantone: ~3/2

OK, so this is cool. Is there any practical advantage to doing it this
way vs doing something simpler, like picking the generator within the
octave that has the lowest Tenney height?

Lowest Tenney height gives

meantone: ~3/2
mavila: ~3/2
helmholtz/garibaldi: ~3/2
hanson: ~5/3
porcupine: ~9/5
magic: ~5/4
pajara: ~15/14
orwell: ~7/6
miracle: ~15/14

Equally reasonable choices, but easier to calculate.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Apr 4, 2012 at 11:52 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > OK, so this is cool. Is there any practical advantage to doing it this
> > way vs doing something simpler, like picking the generator within the
> > octave that has the lowest Tenney height?
>
> It's certainly easier if you are giving exact definitions for Fokker
> blocks.

OK, we'll just go with that then.

> Speaking of which, why does "dome" only refer to Fokker blocks?

It doesn't; my initial definition was overly restrictive. The thing
that Keenan said is spot on and should allow for all that stuff.

-Mike