Yahoo Tuning Groups Ultimate Backup tuning-math Dual L1 norm deep hole scales

Here are the dual-L1 deep hole results Paul asked for.

Shell 1 radius 3 6 notes
[1, 21/20, 6/5, 7/5, 3/2, 7/4]

Shell 2 radius 6 8 notes
[7/6, 49/40, 5/4, 21/16, 8/5, 42/25, 12/7, 9/5]

Shell 3 radius 7 24 notes
[49/48, 36/35, 15/14, 35/32, 28/25, 9/8, 8/7, 63/50, 9/7, 4/3, 48/35,
10/7, 36/25, 35/24, 147/100, 49/32, 63/40, 49/30, 5/3, 147/80, 28/15,
15/8,
48/25, 49/25]

Shell 4 radius 9 6 notes
[441/400, 245/192, 98/75, 45/28, 288/175, 40/21]

Shell 5 radius 10 24 notes
[126/125, 25/24, 16/15, 343/320, 27/25, 147/128, 147/125, 60/49, 32/25,
168/125, 27/20, 49/36, 343/240, 72/49, 54/35, 14/9, 25/16, 105/64,
343/200,
25/14, 64/35, 35/18, 96/49, 63/32]

Ball 1 radius 3 6 notes
[1, 21/20, 6/5, 7/5, 3/2, 7/4]

Ball 2 radius 6 14 notes
[1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25,
12/7, 7/4,9/5]

Ball 3 radius 7 38 notes
[1, 49/48, 36/35, 21/20, 15/14, 35/32, 28/25, 9/8, 8/7, 7/6, 6/5, 49/40,
5/4, 63/50, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 36/25, 35/24, 147/100,
3/2,
49/32, 63/40, 8/5, 49/30, 5/3, 42/25, 12/7, 7/4, 9/5, 147/80, 28/15,
15/8,
48/25, 49/25]

Ball 4 radius 9 44 notes
[1, 49/48, 36/35, 21/20, 15/14, 35/32, 441/400, 28/25, 9/8, 8/7, 7/6,
6/5, 49/40, 5/4, 63/50, 245/192, 9/7, 98/75, 21/16, 4/3, 48/35, 7/5,
10/7,
36/25, 35/24, 147/100, 3/2, 49/32, 63/40, 8/5, 45/28, 49/30, 288/175,
5/3,
42/25, 12/7, 7/4, 9/5, 147/80, 28/15, 15/8, 40/21, 48/25, 49/25]

Ball 5 radius 10 68 notes
[1, 126/125, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 343/320, 27/25,
35/32, 441/400, 28/25, 9/8, 8/7, 147/128, 7/6, 147/125, 6/5, 60/49,
49/40,
5/4, 63/50, 245/192, 32/25, 9/7, 98/75, 21/16, 4/3, 168/125, 27/20,
49/36,
48/35, 7/5, 10/7, 343/240, 36/25, 35/24, 72/49, 147/100, 3/2, 49/32,
54/35,
14/9, 25/16, 63/40, 8/5, 45/28, 49/30, 105/64, 288/175, 5/3, 42/25, 12/7,
343/200, 7/4, 25/14, 9/5, 64/35, 147/80, 28/15, 15/8, 40/21, 48/25,
35/18,
96/49, 49/25, 63/32]

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> P.S. Why aren't they simply the "L1 deep hole" results? Why dual? I
> tend to think of the monzo space or lattice of notes itself as
> the 'standard' space, while the space of linear functionals on it
> (breeds) as its dual. Or am I misunderstanding?

Presumably L1 would mean

||3^a 5^b 7^c|| = |a|+|b|+|c|

This has a different geometry from that, certainly. "Dual" comes in
because if you use an L1 norm when measuring errors of tunings, dual
to that is the "Dual L1 norm" on note-classes.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > Ball 2 radius 6 14 notes
> > > [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25,
> > > 12/7, 7/4,9/5]
> >
> > Just as I suspected. We've finally constructed the Stellated
Hexany,
> > aka Mandala!
>
> Eh, I think that's turned up already. :)

Did it?

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > P.S. Why aren't they simply the "L1 deep hole" results? Why dual?
I
> > tend to think of the monzo space or lattice of notes itself as
> > the 'standard' space, while the space of linear functionals on it
> > (breeds) as its dual. Or am I misunderstanding?
>
> Presumably L1 would mean
>
> ||3^a 5^b 7^c|| = |a|+|b|+|c|
>
> This has a different geometry from that, certainly.

If that's your concern, I would say "Triangular L1 norm" or something
similar, for this.

> "Dual" comes in
> because if you use an L1 norm when measuring errors of tunings,

What do errors have to do with any of this? These are simply
symmetrical JI scales . . .

> dual
> to that is the "Dual L1 norm" on note-classes.

Since the dual of L1 is L-inf, is this actually a "Triangular L-inf
norm" on note-classes?

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > > wrote:
> > >
> > > > Ball 2 radius 6 14 notes
> > > > [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25,
> > > > 12/7, 7/4,9/5]
> > >
> > > Just as I suspected. We've finally constructed the Stellated
> Hexany,
> > > aka Mandala!
> >
> > Eh, I think that's turned up already. :)
>
> Did it?

Sure. It came up as the scale of the unit cube in connection with
chord-lattice scales, and again as the union of the first two
Euclidean shells around a deep hole.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > "Dual" comes in
> > because if you use an L1 norm when measuring errors of tunings,
>
> What do errors have to do with any of this? These are simply
> symmetrical JI scales . . .

Quite a lot, as these correspond to the ways we measure error. The
Hahn norm corresponds to the minimax error, the Euclidean norm to the
rms error, and this "dual L1" gadget to the L1 error measurement which
you were advocating.

> > dual
> > to that is the "Dual L1 norm" on note-classes.
>
> Since the dual of L1 is L-inf, is this actually a "Triangular L-inf
> norm" on note-classes?

No, because "L-inf" is an L-inf on errors of consonances, and hence
has the symmetry properties deriving from that.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> > > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > <gwsmith@s...>
> > > > wrote:
> > > >
> > > > > Ball 2 radius 6 14 notes
> > > > > [1, 21/20, 7/6, 6/5, 49/40, 5/4, 21/16, 7/5, 3/2, 8/5,
42/25,
> > > > > 12/7, 7/4,9/5]
> > > >
> > > > Just as I suspected. We've finally constructed the Stellated
> > Hexany,
> > > > aka Mandala!
> > >
> > > Eh, I think that's turned up already. :)
> >
> > Did it?
>
> Sure. It came up as the scale of the unit cube in connection with
> chord-lattice scales, and again as the union of the first two
> Euclidean shells around a deep hole.

Oops -- missed that.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > "Dual" comes in
> > > because if you use an L1 norm when measuring errors of tunings,
> >
> > What do errors have to do with any of this? These are simply
> > symmetrical JI scales . . .
>
> Quite a lot, as these correspond to the ways we measure error.

The raison d'etre for these scales would seem to be if someone's
looking for a good JI scale to use, without regard to its melodic
structure. What does he or she care how we measure error?

> The
> Hahn norm corresponds to the minimax error,

I thought the "Hahn scales" we were coming up with were minimax in
terms of note-classes. So wouldn't the dual of that be L1 in terms of
error?

> the Euclidean norm to the
> rms error, and this "dual L1" gadget to the L1 error measurement
which
> you were advocating.

Don't get it.

> > > dual
> > > to that is the "Dual L1 norm" on note-classes.
> >
> > Since the dual of L1 is L-inf, is this actually a "Triangular L-
inf
> > norm" on note-classes?
>
> No, because "L-inf" is an L-inf on errors of consonances, and hence
> has the symmetry properties deriving from that.

Don't get it. But either way, dual or not, wouldn't "L1" indicate a
rectangular, rather than triangular, lattice or metric? You were the
one to bring this up but I don't see how duality ameliorates it.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> The raison d'etre for these scales would seem to be if someone's
> looking for a good JI scale to use, without regard to its melodic
> structure. What does he or she care how we measure error?

It seems to me the duality provides another and convincing raison d'etre.

> > The
> > Hahn norm corresponds to the minimax error,
>
> I thought the "Hahn scales" we were coming up with were minimax in
> terms of note-classes. So wouldn't the dual of that be L1 in terms of
> error?

The situation is confusing, because we've got three different (at
minimum) norms to contend with. We have a norm which is Linf (minimax)
or Euclidean (rms.) This leads to *another* norm by applying it to the
consonances.

Starting with the Euclidean norm (norm #1) (x^2+y^2+z^2)^(1/2), we
apply it to 3,5,7,5/3,7/3,7/5 and get the following:

x^2+y^2+z^2+(y-x)^2+(z-x)^2+(z-y)^2 = 3(x^2+y^2+z^2)-2(xy+xz+yz)

Taking the square root of this is norm #2. Now form the symmetric
matrix for the above, and take the inverse:

[[3,-1,-1],[-1,3,-1],[-1,-1,3]]^(-1) =
[[1/2,1/4,1/4],[1/4,1/2,1/4],[1/4,1/4,1/2]]

The quadratic form for this last is

(a^2+b^2+c^2+ab+ac+bc)/2, which gives us norm #2; of course we can
rescale by multiplying by 2.

The same situation, three *different* norms, we find if we start with
norm #1 being the L_inf norm. Then norm #2 has a unit ball which is
the convex hull of the twelve consonances--ie, a cuboctahedron. From
the 14 faces of this we get norm #3.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > The raison d'etre for these scales would seem to be if someone's
> > looking for a good JI scale to use, without regard to its melodic
> > structure. What does he or she care how we measure error?
>
> It seems to me the duality provides another and convincing raison
d'etre.

Can you outline this raison d'etre, please? I have no idea what it
could be.

> > > The
> > > Hahn norm corresponds to the minimax error,
> >
> > I thought the "Hahn scales" we were coming up with were minimax
in
> > terms of note-classes. So wouldn't the dual of that be L1 in
terms of
> > error?
>
> The situation is confusing, because we've got three different (at
> minimum) norms to contend with. We have a norm which is Linf
(minimax)
> or Euclidean (rms.) This leads to *another* norm by applying it to
the
> consonances.
>
> Starting with the Euclidean norm (norm #1) (x^2+y^2+z^2)^(1/2),

How is this used below?

> we
> apply it to 3,5,7,5/3,7/3,7/5 and get the following:
>
> x^2+y^2+z^2+(y-x)^2+(z-x)^2+(z-y)^2 = 3(x^2+y^2+z^2)-2(xy+xz+yz)
>
> Taking the square root of this is norm #2.

This is the rms error criterion (or rms 'loss function'), right?

> Now form the symmetric
> matrix for the above, and take the inverse:
>
> [[3,-1,-1],[-1,3,-1],[-1,-1,3]]^(-1) =
> [[1/2,1/4,1/4],[1/4,1/2,1/4],[1/4,1/4,1/2]]
>
> The quadratic form for this last is
>
> (a^2+b^2+c^2+ab+ac+bc)/2, which gives us norm #2; of course we can
> rescale by multiplying by 2.

So you're saying the dual of rms error is euclidean norm in the
symmetric oct-tet lattice, yes?

> The same situation, three *different* norms, we find if we start
with
> norm #1 being the L_inf norm. Then norm #2 has a unit ball which is
> the convex hull of the twelve consonances--ie, a cuboctahedron. From
> the 14 faces of this we get norm #3.

I guess I must have been wrong above. What's the difference between
this latter #2 and #3? And what about duality and the fact that on
your Tenney page, you say that the dual of L1 is L_inf, but today you
seem to be saying something different (is it the triangularity of the
lattice that alters the situation)?

Gene, it takes a huge amount of guesswork to navigate your posts.
Being concise is all well and good, but taking things step-by-step
will reap many rewards in human understanding, I predict.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > The raison d'etre for these scales would seem to be if
someone's
> > > looking for a good JI scale to use, without regard to its
melodic
> > > structure. What does he or she care how we measure error?
> >
> > It seems to me the duality provides another and convincing raison
> d'etre.
>
> Can you outline this raison d'etre, please? I have no idea what it
> could be.
>
> > > > The
> > > > Hahn norm corresponds to the minimax error,
> > >
> > > I thought the "Hahn scales" we were coming up with were minimax
> in
> > > terms of note-classes. So wouldn't the dual of that be L1 in
> terms of
> > > error?
> >
> > The situation is confusing, because we've got three different (at
> > minimum) norms to contend with. We have a norm which is Linf
> (minimax)
> > or Euclidean (rms.) This leads to *another* norm by applying it
to
> the
> > consonances.
> >
> > Starting with the Euclidean norm (norm #1) (x^2+y^2+z^2)^(1/2),
>
> How is this used below?
>
> > we
> > apply it to 3,5,7,5/3,7/3,7/5 and get the following:
> >
> > x^2+y^2+z^2+(y-x)^2+(z-x)^2+(z-y)^2 = 3(x^2+y^2+z^2)-2(xy+xz+yz)
> >
> > Taking the square root of this is norm #2.
>
> This is the rms error criterion (or rms 'loss function'), right?
>
> > Now form the symmetric
> > matrix for the above, and take the inverse:
> >
> > [[3,-1,-1],[-1,3,-1],[-1,-1,3]]^(-1) =
> > [[1/2,1/4,1/4],[1/4,1/2,1/4],[1/4,1/4,1/2]]
> >
> > The quadratic form for this last is
> >
> > (a^2+b^2+c^2+ab+ac+bc)/2, which gives us norm #2; of course we can
> > rescale by multiplying by 2.
>
> So you're saying the dual of rms error is euclidean norm in the
> symmetric oct-tet lattice, yes?
>
> > The same situation, three *different* norms, we find if we start
> with
> > norm #1 being the L_inf norm. Then norm #2 has a unit ball which
is
> > the convex hull of the twelve consonances--ie, a cuboctahedron.
From
> > the 14 faces of this we get norm #3.
>
> I guess I must have been wrong above. What's the difference between
> this latter #2 and #3? And what about duality and the fact that on
> your Tenney page, you say that the dual of L1 is L_inf, but today
you
> seem to be saying something different (is it the triangularity of
the
> lattice that alters the situation)?

Still wondering about all of this,
Paul