An explanation of hobbits

I have tried a few times to understand hobbit scales now, and can't
figure them out. I understand dwarf scales really well and how they're
constructed and what the algorithm is to create them, but the
following paragraph from tuning-math really stumped me:

/tuning-math/message/18164

> The standard Euclidean norm on weighted monzos provides a canonical Euclidean
> metric on interval space. Projecting othogonally from a set of intervals
> consisting of the commas + 2 in this metric defines a canonical projection map
> from the p-limit to a lattice of interval classes.

So your approach to tempering, then, is to start with a rank-n tuning,
and then come up with a projective version of that space by "looking
directly at" the comma, repeating for each comma?

> Given a val v tempering out
> the commas c, we choose the smallest lattice element q (ie, the one nearest the
> origin) for each i, 0 <= i < v[1] such that v(q) mod v[1] = i.

I don't understand this, what do you mean by "nearest the origin?" You
mean in the new projective space?

> This assumes q is
> expressed in terms of rational numbers, but since v tempers out the commas and
> we plan to temper the result that result is independent of the particular choice
> of q. The result is well-defined except when v[1] is an even number, where both
> q and 2/q have the same length. We may break the tie by choosing the smallest of
> the two tempered values; that is, the one less than 600 cents. Since the result
> is now unique up to choice of tuning, I propose calling it Temperament[n] where
> "Temperament" is the name of the temperament, and n is v[1].

I guess I don't really understand the concept of a val. A val seems to
be analogous to a monzo, except that it applies to tuning space rather
than interval space, right? I found the explanation here a bit
esoteric:

http://xenharmonic.wikispaces.com/Vals+and+Tuning+Space

but it's the only one I have so far. I guess I've never seen a plot of
actual tuning space before, just the "projective" tuning space plots
that Paul likes to post on facebook. Are vals basically just
coordinates for tuning space?

> When the temperament is rank two, this gives the n-note MOS. For ranks larger
than two, we get a scale in the temperament of the commas c which is more likely
to be constant structure and etc than most constructions. When the comma list is
empty we get a JI scale which is an alternative as a canonical choice to the
dwarf scales and which sometimes seems preferable (comparing the 34-note 5-limit
dwarf with the hobbit is illuminating in that regard.)

So if we're in meantone temperament, the MOS ends up being the
diatonic scale, but if we decide to not temper that comma, we end up
with the 5-limit JI scale?

-Mike

Also, if I understand correctly, couldn't you use this to also
generalize Paul's concept of there being "chromatic" and "commatic"
unison vectors? e.g. if you temper out all of the commas but one, that
one ends up being the "chromatic" vector. So if you take the diatonic
periodicity block and come up with the hobbit that leaves 25/24
untempered, you get the meantone diatonic scale, and if you also leave
81/80 untempered, then you get the JI 5-limit diatonic scale.

At least that's what I think you're saying, but I really need to
understand vals better first to get it I think.

-Mike

On Tue, Jan 11, 2011 at 1:39 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
> I have tried a few times to understand hobbit scales now, and can't
> figure them out. I understand dwarf scales really well and how they're
> constructed and what the algorithm is to create them, but the
> following paragraph from tuning-math really stumped me:
>
> /tuning-math/message/18164
>
>> The standard Euclidean norm on weighted monzos provides a canonical Euclidean
>> metric on interval space. Projecting othogonally from a set of intervals
>> consisting of the commas + 2 in this metric defines a canonical projection map
>> from the p-limit to a lattice of interval classes.
>
> So your approach to tempering, then, is to start with a rank-n tuning,
> and then come up with a projective version of that space by "looking
> directly at" the comma, repeating for each comma?
>
>> Given a val v tempering out
>> the commas c, we choose the smallest lattice element q (ie, the one nearest the
>> origin) for each i, 0 <= i < v[1] such that v(q) mod v[1] = i.
>
> I don't understand this, what do you mean by "nearest the origin?" You
> mean in the new projective space?
>
>> This assumes q is
>> expressed in terms of rational numbers, but since v tempers out the commas and
>> we plan to temper the result that result is independent of the particular choice
>> of q. The result is well-defined except when v[1] is an even number, where both
>> q and 2/q have the same length. We may break the tie by choosing the smallest of
>> the two tempered values; that is, the one less than 600 cents. Since the result
>> is now unique up to choice of tuning, I propose calling it Temperament[n] where
>> "Temperament" is the name of the temperament, and n is v[1].
>
> I guess I don't really understand the concept of a val. A val seems to
> be analogous to a monzo, except that it applies to tuning space rather
> than interval space, right? I found the explanation here a bit
> esoteric:
>
> http://xenharmonic.wikispaces.com/Vals+and+Tuning+Space
>
> but it's the only one I have so far. I guess I've never seen a plot of
> actual tuning space before, just the "projective" tuning space plots
> that Paul likes to post on facebook. Are vals basically just
> coordinates for tuning space?
>
>> When the temperament is rank two, this gives the n-note MOS. For ranks larger
> than two, we get a scale in the temperament of the commas c which is more likely
> to be constant structure and etc than most constructions. When the comma list is
> empty we get a JI scale which is an alternative as a canonical choice to the
> dwarf scales and which sometimes seems preferable (comparing the 34-note 5-limit
> dwarf with the hobbit is illuminating in that regard.)
>
> So if we're in meantone temperament, the MOS ends up being the
> diatonic scale, but if we decide to not temper that comma, we end up
> with the 5-limit JI scale?
>
> -Mike
>

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

> So your approach to tempering, then, is to start with a rank-n tuning,
> and then come up with a projective version of that space by "looking
> directly at" the comma, repeating for each comma?

It's hardly tempering, as 2 is included with the commas. And the projection is not onto a projective space, byt an ordinary Euclidean subspace. The projection is chosen so that 2 and the commas project down to the zero vector.

> > Given a val v tempering out
> > the commas c, we choose the smallest lattice element q (ie, the one nearest the
> > origin) for each i, 0 <= i < v[1] such that v(q) mod v[1] = i.
>
> I don't understand this, what do you mean by "nearest the origin?" You
> mean in the new projective space?

On the new subspace.

> I guess I don't really understand the concept of a val. A val seems to
> be analogous to a monzo, except that it applies to tuning space rather
> than interval space, right?

Right, except that you don't need to embed things in ambient spaces. A val is simply a linear map from monzos to integers.

> So if we're in meantone temperament, the MOS ends up being the
> diatonic scale, but if we decide to not temper that comma, we end up
> with the 5-limit JI scale?

Sounds plausible, but it turns out my routines don't accept an empty list as input, so give me a moment to write something for JI hobbits.

On Wed, Jan 12, 2011 at 1:46 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > So your approach to tempering, then, is to start with a rank-n tuning,
> > and then come up with a projective version of that space by "looking
> > directly at" the comma, repeating for each comma?
>
> It's hardly tempering, as 2 is included with the commas. And the projection is not onto a projective space, byt an ordinary Euclidean subspace. The projection is chosen so that 2 and the commas project down to the zero vector.

What's the difference between a projective space and an ordinary
Euclidean subspace? Are normal projective spaces incompatible with the
L2 norm or something?

> > I guess I don't really understand the concept of a val. A val seems to
> > be analogous to a monzo, except that it applies to tuning space rather
> > than interval space, right?
>
> Right, except that you don't need to embed things in ambient spaces. A val is simply a linear map from monzos to integers.

OK, so Paul Erlich dropped this mysterious page on my Facebook wall
earlier this morning:

http://mathworld.wolfram.com/IntegerRelation.html

So if a val tempers out some comma, there will be an integer relation
between the val coefficients and the monzo coefficients, right?

-Mike

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

> > So if we're in meantone temperament, the MOS ends up being the
> > diatonic scale, but if we decide to not temper that comma, we end up
> > with the 5-limit JI scale?
>
> Sounds plausible, but it turns out my routines don't accept an empty list as input, so give me a moment to write something for JI hobbits.
>

It turns out my routine does work for JI hobbits, as I recalled, but I fed it bad input. Here's some JI hobbits ("jobbits") for your listening pleasure:

! jobbit7_5.scl
! = mavchrome1, helmholz, trab7, diff7b
5-limit 7-note JI hobbit
7
!
16/15
5/4
4/3
3/2
8/5
15/8
2/1

! jobbit12_5.scl
! = albion.scl
12-note 5-limit JI hobbit
12
!
16/15
9/8
6/5
5/4
4/3
64/45
3/2
8/5
5/3
16/9
15/8
2/1

! jobbit12_5i.scl
! = malcolm.scl, verdi2.scl
Inverse jobbit12_5
12
!
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
5/3
16/9
15/8
2/1

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

> What's the difference between a projective space and an ordinary
> Euclidean subspace? Are normal projective spaces incompatible with the
> L2 norm or something?

It's a different kind of space. In the projective plane, two lines always intersect, in 3-space, two planes always intersect, etc. It's topologically compact. Various other differences.

> OK, so Paul Erlich dropped this mysterious page on my Facebook wall
> earlier this morning:
>
> http://mathworld.wolfram.com/IntegerRelation.html

Yeah, I'm afraid I couldn't help responding to that since you said not to.

> So if a val tempers out some comma, there will be an integer relation
> between the val coefficients and the monzo coefficients, right?

Integer relations means where one of the things is a list of real numbers, so you wouldn't ordinary call it an integer relation, as it's such a special case.

On Wed, Jan 12, 2011 at 3:08 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What's the difference between a projective space and an ordinary
> > Euclidean subspace? Are normal projective spaces incompatible with the
> > L2 norm or something?
>
> It's a different kind of space. In the projective plane, two lines always intersect, in 3-space, two planes always intersect, etc. It's topologically compact. Various other differences.

OK, I get it. So to create a projective subspace, you draw lines going
outward "conically" (or I guess "spherically" maybe) from some central
point, right? Kind of like a perspective view. But to create a
Euclidean subspace, you'd take more of an isometric view and draw
vectors going out from some central line, right?

> > OK, so Paul Erlich dropped this mysterious page on my Facebook wall
> > earlier this morning:
> >
> > http://mathworld.wolfram.com/IntegerRelation.html
>
> Yeah, I'm afraid I couldn't help responding to that since you said not to.

Haha, that was Paul's directive. I guess I've been barraging the poor
guy with too many random ideas lately, and he just couldn't take it
anymore.

> > So if a val tempers out some comma, there will be an integer relation
> > between the val coefficients and the monzo coefficients, right?
>
> Integer relations means where one of the things is a list of real numbers, so you wouldn't ordinary call it an integer relation, as it's such a special case.

You mean it's a special case because we're talking about two sets of
integers, not one set of reals and one set of integers?

I thought the whole point of the article, as it relates to tuning
theory, is that these algorithms are ways of finding out what commas
are tempered out by what vals. You took it in a different direction
though with your post on my facebook wall. How has this concept been
used in the past?

-Mike

On Wed, Jan 12, 2011 at 3:01 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> ! jobbit7_5.scl
> ! = mavchrome1, helmholz, trab7, diff7b
> 5-limit 7-note JI hobbit
> 7
> !
> 16/15
> 5/4
> 4/3
> 3/2
> 8/5
> 15/8
> 2/1

It spit out 16/15 and 15/8 instead of 9/8 and 16/9? Why do you think
this is...? What commas did you use as projection zero vectors here?

-Mike

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

> OK, I get it. So to create a projective subspace, you draw lines going
> outward "conically" (or I guess "spherically" maybe) from some central
> point, right? Kind of like a perspective view.

Yes, except it's not a subspace, it's a different kind of space.

> But to create a
> Euclidean subspace, you'd take more of an isometric view and draw
> vectors going out from some central line, right?

You'd take a flat through the origin:

http://en.wikipedia.org/wiki/Euclidean_subspace
http://en.wikipedia.org/wiki/Flat_(geometry)

> > > OK, so Paul Erlich dropped this mysterious page on my Facebook wall
> > > earlier this morning:
> > >

> > Integer relations means where one of the things is a list of real numbers, so you wouldn't ordinary call it an integer relation, as it's such a special case.
>
> You mean it's a special case because we're talking about two sets of
> integers, not one set of reals and one set of integers?

Right.

> I thought the whole point of the article, as it relates to tuning
> theory, is that these algorithms are ways of finding out what commas
> are tempered out by what vals.

If you toss a val into the PSLQ machine, you get a comma out, but you want all of them.

You took it in a different direction
> though with your post on my facebook wall. How has this concept been
> used in the past?

Finding polynomials from a single root, or finding polynomials with a root close to a given value are useful when you are trying to do stuff like Jacques Dudon does.

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

> It spit out 16/15 and 15/8 instead of 9/8 and 16/9? Why do you think
> this is...? What commas did you use as projection zero vectors here?

The only "comma" is 2. The reason for 16/15 in place of 9/8 goes back to why no one claims Tenney-Euclidean height is a more accurate measure than Tenney height despite its clear mathematical advantages. If you take a rectangular grid with the short blocks log2(3) long and the long blocks log2(5) long, then the distance to 9 is log2(9). The taxicab distance to 15 is log2(15), which is larger. However, as the crow flies the distance is sqrt(log2(3)^2 + log2(5)^2), which is actually less than log2(9). So the jobbit goes for 16/15 rather than 9/8.

On Wed, Jan 12, 2011 at 6:06 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > It spit out 16/15 and 15/8 instead of 9/8 and 16/9? Why do you think
> > this is...? What commas did you use as projection zero vectors here?
>
> The only "comma" is 2. The reason for 16/15 in place of 9/8 goes back to why no one claims Tenney-Euclidean height is a more accurate measure than Tenney height despite its clear mathematical advantages. If you take a rectangular grid with the short blocks log2(3) long and the long blocks log2(5) long, then the distance to 9 is log2(9). The taxicab distance to 15 is log2(15), which is larger. However, as the crow flies the distance is sqrt(log2(3)^2 + log2(5)^2), which is actually less than log2(9). So the jobbit goes for 16/15 rather than 9/8.

That's silly. But I see what's going on. So "hobbit" scales are, in a
sense, partially tempered periodicity blocks, yes?

So an alternative way to phrase the algorithm is, for example, if you
want a 7-note 7-limit hobbit, first go to 7-tet, which tempers out
25/24 and 81/80. Then to get a hobbit, detemper 7-tet by one of these
commas, so you end up with either meantone[7] or dicot[7]. If you want
a jobbit, detemper 7-tet by both of these commas.

And then, the reason that we're getting 16/15 and 15/8 here, is that
originally 7-tet equates 16/15 and 9/8 - as you detemper 25/24, the
algorithm picks the "simplest" interval for each equated pair, and
because TE-height seems to be all screwed up it picks 16/15 over 9/8.

Am I right?

-Mike

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

> That's silly. But I see what's going on. So "hobbit" scales are, in a
> sense, partially tempered periodicity blocks, yes?

Right.

> So an alternative way to phrase the algorithm is, for example, if you
> want a 7-note 7-limit hobbit, first go to 7-tet, which tempers out
> 25/24 and 81/80. Then to get a hobbit, detemper 7-tet by one of these
> commas, so you end up with either meantone[7] or dicot[7]. If you want
> a jobbit, detemper 7-tet by both of these commas.

Except the real interest of hobbits is not jobbits; you need to go to at least rank 3 to make the thing worthwhile. You could start with an 11-limit val.

> And then, the reason that we're getting 16/15 and 15/8 here, is that
> originally 7-tet equates 16/15 and 9/8 - as you detemper 25/24, the
> algorithm picks the "simplest" interval for each equated pair, and
> because TE-height seems to be all screwed up it picks 16/15 over 9/8.

Right.

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

> The only "comma" is 2. The reason for 16/15 in place of
> 9/8 goes back to why no one claims Tenney-Euclidean
> height is a more accurate measure than Tenney height
> despite its clear mathematical advantages. If you take a
> rectangular grid with the short blocks log2(3) long and
> the long blocks log2(5) long, then the distance to 9 is
> log2(9). The taxicab distance to 15 is log2(15), which is
> larger. However, as the crow flies the distance is
> sqrt(log2(3)^2 + log2(5)^2), which is actually less than
> log2(9). So the jobbit goes for 16/15 rather than 9/8.

It's a sign that you shouldn't use a rectangular grid for
octave-equivalent space. Triangular lattices are at least
a decade old now, and you never argue against them.

With the Kees-Euclidean metric, I get sqrt(log2(15)^2 +
log2(3)^2 + log2(5)^2) as 4.8 and sqrt(2)*log2(9) as 4.5.
So 9:8 is the simpler octave-equivalent interval. 10:9 is
simpler still (sqrt(log2(9/5)^2 + log2(9)^2 +
log2(5)^2)=4.0), which is where the factorization bias comes
in).

6:5 has a Kees-Euclidean score of 2.9 but is equivalent to
16:15 on an octave-equivalent rectangular lattice.

Graham

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

> It's a sign that you shouldn't use a rectangular grid for
> octave-equivalent space. Triangular lattices are at least
> a decade old now, and you never argue against them.

If I didn't originate the use of such metrics in music, who did? Of course its implicit in lattice diagrams. But you don't define "Kees-Euclidean" as used below; it seems it is the symmetrical metric x1^2+x2^2+...xn^2+(x1+x2+...+x^n)^2 applied to weighted monzos? Sounds like a good idea; I'll code it up.

On Thu, Jan 13, 2011 at 1:55 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > That's silly. But I see what's going on. So "hobbit" scales are, in a
> > sense, partially tempered periodicity blocks, yes?
>
> Right.

So it looks like they generalize MOS. Then here's two questions:
1) If you temper all of the commas in the PB, you end up with all
interval classes having one specific interval size. If you temper all
of the commas in the PB except for one of them, you end up with all
classes having two specific interval sizes. Does the pattern continue?
e.g. if you start with an 11-limit val and have, say, a 31-note
periodicity block, and then you decide not to temper out two of the
commas, will you end up with three step sizes all across the board?
2) Is the 12-tet diatonic scale a hobbit? It is rank one and tempers
out 128/125, but can still be viewed as a de-tempering of 7-tet. Is
there some val for which the 12-tet diatonic scale is a hobbit?

> > And then, the reason that we're getting 16/15 and 15/8 here, is that
> > originally 7-tet equates 16/15 and 9/8 - as you detemper 25/24, the
> > algorithm picks the "simplest" interval for each equated pair, and
> > because TE-height seems to be all screwed up it picks 16/15 over 9/8.
>
> Right.

I am displeased. Is there no way to scale the axes to get around this?
I was going to ask if it would be the same for an octave-inequivalent
lattice, but I guess that would still make 15/1 more consonant than
9/1. I am displeased!

What about the 1/n^2 + 1/d^2 metric we were talking about a month ago?
This is obviously far from being proven as "the" psychoacoustically
valid measure of consonance, but it was an interesting idea to throw
around.

That would translate to (n^2 + d^2)/(n^2d^2). So if we're going to
still scale the axes by log(p), then you'd have - whoops, gotta go,
conference is starting.

-Mike

"genewardsmith" <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed
> <gbreed@...> wrote:
>
> > It's a sign that you shouldn't use a rectangular grid
> > for octave-equivalent space. Triangular lattices are
> > at least a decade old now, and you never argue against
> > them.
>
> If I didn't originate the use of such metrics in music,
> who did? Of course its implicit in lattice diagrams. But
> you don't define "Kees-Euclidean" as used below; it seems
> it is the symmetrical metric
> x1^2+x2^2+...xn^2+(x1+x2+...+x^n)^2 applied to weighted
> monzos? Sounds like a good idea; I'll code it up.

What such metrics and why does it matter who thought of
them first? I got triangular lattices from Paul Erlich.
Probably Mathieu was drawing them before. Dr Crotch put
triangular connections on a rectangular diagram in 1830. I
don't know what he was drawing. The printed diagrams seem
to be inconsistent because what Bosanquet gives ("On Some
Points in the Harmony of Perfect Consonances") is different
to the Google Books edition.

I explained Kees-Euclidean complexity on the 3rd of
December. It corrects for the size of the odd ratio in
line with Kees Expressibility. I think that's the same as
what you say bar the ^ in x^n.

I still don't have a temperamental version so it isn't
ready for hobbits.

Graham