Hahn norm formula

For a note class represented by 3^a 5^b 7^c, I get that it should be

||(a,b,c)||_Hahn = max(|a|,|b|,|c|,|b+c|,|a+c|,|a+b|,|a+b+c|)

This is a pretty simple formula, and applicable to non-lattice points.
I'll take a look at scales which arise from it.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> For a note class represented by 3^a 5^b 7^c, I get that it should be
>
> ||(a,b,c)||_Hahn = max(|a|,|b|,|c|,|b+c|,|a+c|,|a+b|,|a+b+c|)
>
> This is a pretty simple formula, and applicable to non-lattice points.
> I'll take a look at scales which arise from it.

The number of notes in Hahn shell n turns out to be 10n^2+2; about
this the handbook of integer seqences says "Points on surface of
cuboctahedron (or icosahedron): a(0) = 1, for n> 0, a(n) = 10n^2 + 2
(coordination sequence for f.c.c. lattice)." So we seem to be doing
things right here. If we take the union of the Hahn shells, we get
lattice-point-centered Hahn scales. About the integer sequence for
this the handbook says "Centered icosahedral (or cuboctahedral)
numbers, also crystal ball sequence for f.c.c. lattice.
Comments: Called 'magic numbers' in some chemical contexts."

The latter stikes me as nifty. Crystal ball scales? Magic number
scales? I like the first especially.

Here are the first few Hahn shells:

Shell 0
[1]

Shell 1
[8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]

Shell 2
[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25,
9/8, 25/21, 60/49, 49/40, 32/25, 9/7, 64/49, 21/16, 49/36, 48/35,
25/18, 36/25,35/24, 72/49, 32/21, 49/32, 14/9, 25/16, 80/49, 49/30,
42/25, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18,
96/49, 49/25]

Shell 3
[126/125, 64/63, 128/125, 28/27, 256/245, 360/343, 343/320, 27/25,
160/147, 49/45, 192/175, 54/49, 125/112, 384/343, 245/216, 343/300,
147/128, 144/125, 125/108, 400/343, 75/64, 288/245, 147/125, 32/27,
343/288, 175/144, 128/105, 216/175, 56/45, 432/343, 63/50, 80/63,
125/98, 245/192, 35/27, 125/96, 98/75, 75/56, 343/256, 168/125, 27/20,
200/147, 175/128, 343/250, 480/343, 45/32, 64/45, 343/240, 500/343,
256/175, 147/100, 40/27, 125/84, 512/343, 112/75, 75/49, 192/125,
54/35, 384/245, 196/125, 63/40, 100/63, 343/216, 45/28, 175/108,
105/64, 288/175, 576/343, 27/16, 250/147, 245/144, 128/75, 343/200,
216/125, 125/72, 256/147, 600/343, 432/245, 343/192, 224/125, 49/27,
175/96, 90/49, 147/80, 50/27, 640/343, 343/180, 245/128, 27/14,
125/64, 63/32, 125/63]

Here are some crystal ball scales:

Ball 1 13 notes
[1, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]

Ball 2 55 notes
[1, 50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9,
28/25,9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49,
21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2,
32/21, 49/32, 14/9,25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4,
16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25]

Ball 3 147 notes
[1, 126/125, 64/63, 50/49, 49/48, 128/125, 36/35, 28/27, 25/24,
256/245,360/343, 21/20, 16/15, 15/14, 343/320, 27/25, 160/147, 49/45,
35/32, 192/175,54/49, 10/9, 125/112, 384/343, 28/25, 9/8, 245/216,
8/7, 343/300, 147/128, 144/125, 125/108, 400/343, 7/6, 75/64, 288/245,
147/125, 32/27, 25/21, 343/288, 6/5, 175/144, 128/105, 60/49, 49/40,
216/175, 56/45, 5/4, 432/343, 63/50, 80/63, 125/98, 245/192, 32/25,
9/7, 35/27, 125/96, 64/49, 98/75, 21/16, 4/3, 75/56,343/256, 168/125,
27/20, 200/147, 49/36, 175/128, 48/35, 343/250, 25/18, 480/343, 7/5,
45/32, 64/45, 10/7, 343/240, 36/25, 500/343, 35/24, 256/175, 72/49,
147/100, 40/27, 125/84, 512/343, 112/75, 3/2, 32/21, 75/49, 49/32,
192/125, 54/35, 14/9, 25/16, 384/245, 196/125, 63/40, 100/63, 343/216,
8/5, 45/28, 175/108, 80/49, 49/30, 105/64, 288/175, 5/3, 576/343,
42/25, 27/16, 250/147, 245/144, 128/75, 12/7, 343/200, 216/125,
125/72, 256/147, 600/343, 7/4, 432/245, 16/9, 25/14, 343/192, 224/125,
9/5, 49/27, 175/96, 64/35, 90/49, 147/80, 50/27, 640/343, 28/15, 15/8,
40/21, 343/180, 245/128, 48/25, 27/14, 35/18, 125/64,96/49, 49/25,
63/32, 125/63]

Here are shells around a deep hole, the octadron around the hole is
the first one listed. I then take the union to get deep hole centered
scales. The number of inhabitants of shell n is 6n^2, and the total
number in a ball is n(n+1)(2n+1); this is six times the number of
stacked canon balls in a square pyramid--ie, 1^2+2^2+3^2+ ... + n^2 =
n(n+1)(2n+1)/6.

Shell 1 (hexany)
[1, 21/20, 6/5, 7/5, 3/2, 7/4]

Shell 2
[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 3
[50/49, 1029/1000, 49/45, 192/175, 54/49, 441/400, 10/9, 343/300,
144/125, 75/64, 288/245, 189/160, 25/21, 343/288, 175/144, 216/175,
56/45, 245/192, 1029/800, 64/49, 98/75, 75/56, 343/256, 175/128,
343/250, 441/320, 25/18, 45/32, 112/75, 189/125, 32/21, 75/49,
192/125, 384/245, 196/125, 45/28, 1029/640, 80/49, 288/175, 27/16,
245/144, 216/125, 432/245, 441/250, 16/9, 343/192, 224/125, 175/96,
90/49, 189/100, 40/21, 343/180, 245/128, 27/14]

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

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

Ball 3 84 notes
[1, 50/49, 49/48, 36/35, 1029/1000, 21/20, 15/14, 49/45, 35/32,
192/175,54/49, 441/400, 10/9, 28/25, 9/8, 8/7, 343/300, 144/125,
75/64, 288/245, 189/160, 25/21, 343/288, 6/5, 175/144, 216/175, 56/45,
63/50, 245/192, 9/7, 1029/800, 64/49, 98/75, 4/3, 75/56, 343/256,
175/128, 48/35, 343/250, 441/320, 25/18, 7/5, 45/32, 10/7, 36/25,
35/24, 147/100, 112/75, 3/2, 189/125, 32/21, 75/49, 49/32, 192/125,
384/245, 196/125, 63/40, 45/28, 1029/640, 80/49, 49/30, 288/175, 5/3,
27/16, 245/144, 216/125, 7/4, 432/245, 441/250, 16/9, 343/192,
224/125, 175/96, 90/49, 147/80, 28/15, 15/8, 189/100, 40/21, 343/180,
245/128,48/25, 27/14, 49/25]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

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

If we take the aproximate 7-limit consonances of this with commas less
than 10 cents, we get 2401/2400, 6144/6125, 225/224 and 1029/1024;
once again hemiwuerschmidt or miracle suggests itself.

> Ball 3 84 notes
> [1, 50/49, 49/48, 36/35, 1029/1000, 21/20, 15/14, 49/45, 35/32,
> 192/175,54/49, 441/400, 10/9, 28/25, 9/8, 8/7, 343/300, 144/125,
> 75/64, 288/245, 189/160, 25/21, 343/288, 6/5, 175/144, 216/175, 56/45,
> 63/50, 245/192, 9/7, 1029/800, 64/49, 98/75, 4/3, 75/56, 343/256,
> 175/128, 48/35, 343/250, 441/320, 25/18, 7/5, 45/32, 10/7, 36/25,
> 35/24, 147/100, 112/75, 3/2, 189/125, 32/21, 75/49, 49/32, 192/125,
> 384/245, 196/125, 63/40, 45/28, 1029/640, 80/49, 49/30, 288/175, 5/3,
> 27/16, 245/144, 216/125, 7/4, 432/245, 441/250, 16/9, 343/192,
> 224/125, 175/96, 90/49, 147/80, 28/15, 15/8, 189/100, 40/21, 343/180,
> 245/128,48/25, 27/14, 49/25]

For these we get 2401/2400, 589824/588245, 6144/6125, 5120/5103,
3136/3125, 16875/16807, 225/224, 15625/15552 and 1029/1024. The first,
second third and fifth commas are commas of hemiwuerschimidt;
5120/5103 isn't, but putting it together with the other four among the
smallest five commas leads to 99-et. Putting it all into 99-et shrinks
the scale down to 62 notes with steps of sizes 1, 2, and 3 99-equal steps.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > For a note class represented by 3^a 5^b 7^c, I get that it should
be
> >
> > ||(a,b,c)||_Hahn = max(|a|,|b|,|c|,|b+c|,|a+c|,|a+b|,|a+b+c|)
> >
> > This is a pretty simple formula, and applicable to non-lattice
points.
> > I'll take a look at scales which arise from it.
>
> The number of notes in Hahn shell n turns out to be 10n^2+2; about
> this the handbook of integer seqences says "Points on surface of
> cuboctahedron (or icosahedron): a(0) = 1, for n> 0, a(n) = 10n^2 + 2
> (coordination sequence for f.c.c. lattice)." So we seem to be doing
> things right here. If we take the union of the Hahn shells, we get
> lattice-point-centered Hahn scales. About the integer sequence for
> this the handbook says "Centered icosahedral (or cuboctahedral)
> numbers, also crystal ball sequence for f.c.c. lattice.
> Comments: Called 'magic numbers' in some chemical contexts."
>
> The latter stikes me as nifty. Crystal ball scales? Magic number
> scales? I like the first especially.
>
>
> Here are the first few Hahn shells:
>
> Shell 0
> [1]
>
> Shell 1
> [8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]

Union of shell 0 and shell 1 is known as the 7-limit Tonality Diamond.

Thanks for your great work here, Gene.

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

> Union of shell 0 and shell 1 is known as the 7-limit Tonality Diamond.
>
> Thanks for your great work here, Gene.

Thanks! Not only has Crystal Ball One been seen often before, Crystal
Ball Two has already appeared--it is exactly the 55 note scale which I
posted in message 9922, in connection with stepwise harmonizing scales.
Despite its apparent theoretical interest, Crystal Ball Two should
suffice as a name, as I at least don't believe in scales with steps of
size 2401/2400. Tempering it out leads to 51 notes, which is how many
it has in hemiwuerschmidt and ennealimmal also.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Union of shell 0 and shell 1 is known as the 7-limit Tonality
Diamond.
> >
> > Thanks for your great work here, Gene.
>
> Thanks! Not only has Crystal Ball One been seen often before,
Crystal
> Ball Two has already appeared--it is exactly the 55 note scale
which I
> posted in message 9922, in connection with stepwise harmonizing
scales.
> Despite its apparent theoretical interest, Crystal Ball Two should
> suffice as a name,

Paul Hahn might have called it the 7-limit radius 2 scale or perhaps
more likely, the Level 2 7-limit Diamond.

Now what do you get if you center around a major tetrad (shallow
hole)?

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

> > Despite its apparent theoretical interest, Crystal Ball Two should
> > suffice as a name,
>
> Paul Hahn might have called it the 7-limit radius 2 scale or perhaps
> more likely, the Level 2 7-limit Diamond.

Since apparently it is called a cystal ball in some nonmusical
connection already, it seems to me better to stick with that, on your
own principle of established names.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > Despite its apparent theoretical interest, Crystal Ball Two
should
> > > suffice as a name,
> >
> > Paul Hahn might have called it the 7-limit radius 2 scale or
perhaps
> > more likely, the Level 2 7-limit Diamond.
>
> Since apparently it is called a cystal ball in some nonmusical
> connection already, it seems to me better to stick with that, on
your
> own principle of established names.

In nonmusical connections?

Can someone briefly explain shell and hull, and the difference?
The mathworld definitions are, as usual, obtuse, and it isn't
clear which shell definition is in use here. And has anyone
noticed how mathworld is slowly becoming a Mathematica help file?

My guess (based on the "generalization of an annulus" definition
of shell) is that in 3-D, a hull is a surface while a shell is a
volume. Is this correct?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Can someone briefly explain shell and hull, and the difference?

Gene uses "shell" to mean the set of notes at a given "distance" from
the origin. I know what a "convex hull" is but don't know what "hull"
in general means, if anything.

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

> Now what do you get if you center around a major tetrad (shallow
> hole)?

At first you get exactly the same result as using the Euclidean norm.
It is only when we get to the fourth populated shell that we see a
difference--this is a funny one, with only four notes in it. The
Euclidean norm would put these notes together with some from shell 5.
It would seem that the Hahn norm is not always drawing less refined
distinctions.

Shells

Radius 3/4 Notes 4
[1, 5/4, 3/2, 7/4]

Radius 3/2 Notes 12
[21/20, 15/14, 35/32, 7/6, 6/5, 21/16, 7/5, 10/7, 35/24, 5/3, 12/7,15/8]

Radius 7/4 Notes 12
[49/48, 25/24, 9/8, 8/7, 49/40, 9/7, 4/3, 49/32, 25/16, 8/5,
25/14, 9/5]

Radius 9/4 Notes 4
[60/49, 105/64, 42/25, 35/18]

Radius 5/2 Notes 36
[50/49, 36/35, 10/9, 28/25, 147/128, 75/64, 25/21, 175/144, 63/50,
245/192, 75/56, 49/36, 175/128, 48/35, 25/18, 45/32, 36/25, 72/49,
147/100, 75/49,14/9, 63/40, 45/28, 80/49, 49/30, 245/144, 175/96,
90/49, 147/80, 28/15, 40/21, 245/128, 48/25, 96/49, 49/25, 63/32]

Radius 11/4 Notes 24
[16/15, 343/320, 27/25, 54/49, 125/112, 343/288, 125/98, 32/25,
125/96,64/49, 343/256, 27/20, 343/240, 125/84, 32/21, 54/35, 27/16,
343/200, 125/72,16/9, 343/192, 64/35, 27/14, 125/64]

Scales

Radius 3/4 Notes 4
[1, 5/4, 3/2, 7/4]

Radius 3/2 Notes 16
[1, 21/20, 15/14, 35/32, 7/6, 6/5, 5/4, 21/16, 7/5, 10/7, 35/24, 3/2,
5/3, 12/7, 7/4, 15/8]

Radius 7/4 Notes 28
[1, 49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 49/40, 5/4,
9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5, 5/3, 12/7,
7/4, 25/14, 9/5, 15/8]

Radius 9/4 Notes 32
[1, 49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 60/49,
49/40, 5/4, 9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5,
105/64, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 15/8, 35/18]

Radius 5/2 Notes 68
[1, 50/49, 49/48, 36/35, 25/24, 21/20, 15/14, 35/32, 10/9, 28/25, 9/8,
8/7, 147/128, 7/6, 75/64, 25/21, 6/5, 175/144, 60/49, 49/40, 5/4,
63/50, 245/192, 9/7, 21/16, 4/3, 75/56, 49/36, 175/128, 48/35, 25/18,
7/5, 45/32, 10/7, 36/25, 35/24, 72/49, 147/100, 3/2, 75/49, 49/32,
14/9, 25/16, 63/40, 8/5, 45/28, 80/49, 49/30, 105/64, 5/3, 42/25,
245/144, 12/7, 7/4, 25/14, 9/5, 175/96,90/49, 147/80, 28/15, 15/8,
40/21, 245/128, 48/25, 35/18, 96/49, 49/25, 63/32]

Radius 11/4 Notes 92
[1, 50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 343/320, 27/25,
35/32, 54/49, 10/9, 125/112, 28/25, 9/8, 8/7, 147/128, 7/6, 75/64,
25/21, 343/288, 6/5, 175/144, 60/49, 49/40, 5/4, 63/50, 125/98,
245/192, 32/25, 9/7, 125/96, 64/49, 21/16, 4/3, 75/56, 343/256, 27/20,
49/36, 175/128, 48/35, 25/18, 7/5, 45/32, 10/7, 343/240, 36/25, 35/24,
72/49, 147/100, 125/84, 3/2, 32/21, 75/49,49/32, 54/35, 14/9, 25/16,
63/40, 8/5, 45/28, 80/49, 49/30, 105/64, 5/3, 42/25, 27/16, 245/144,
12/7, 343/200, 125/72, 7/4, 16/9, 25/14, 343/192, 9/5, 175/96, 64/35,
90/49, 147/80, 28/15, 15/8, 40/21, 245/128, 48/25, 27/14, 35/18,
125/64, 96/49, 49/25, 63/32]

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Can someone briefly explain shell and hull, and the difference?
> The mathworld definitions are, as usual, obtuse, and it isn't
> clear which shell definition is in use here.

The relevant mathworld definition for hull is the one for convex hull;
at least, I haven't noticed anyone using another. "Shell" is used in
connection with lattices to mean sets of points all at the same
distance from a given point (which need not be a lattice point.) A
spherical shell could then be found that only the shell points lie in.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> At first you get exactly the same result as using the Euclidean norm.
> It is only when we get to the fourth populated shell that we see a
> difference--this is a funny one, with only four notes in it. The
> Euclidean norm would put these notes together with some from shell 5.
> It would seem that the Hahn norm is not always drawing less refined
> distinctions.

Message 9756 is relevant to this; adding those four extra notes gives
us the 3x3x3 chord cube.