Fun with the Leech Lattice

(Has this newsgroup moved or something?)

1^2 + 2^2 + 3^2 ... 24^2 = 70^2

We know that 35 is about 50 cents flat. So, 4900, is about 100 cents
flat.

4900, considered as E, is a very flat E.

Major becomes minor.

4900/4096 I would like to dub the Leech third.

Splitting 4900 into 100 and 49: 100/49 is G#/Ab, which
is the center of symmetry of the black white keyboard, along with D.

Just as G-G# is chromatic, G-Ab is diatonic, but in this case
we go to the 7-limit and consider also 49/48 instead of 16/15.
So now it is about the distinction between two kinds of chromatic
half-steps...

But back to the LL.

It's very elementary that the differences between perfect squares
are the odds. So the above sum is now

1
1 3
1 3 5
1 3 5 7
...
1 3 5 ... 47

This is just 24 * 1 + 23 * 5 + 22 * 7 ... 1 * 47 = 4900

Now taking the sums, and the differences of the sums:

24 24
69 45
110 41
147 37
180 33
209 29
234 25
255 21
272 17
285 13
294 9
299 5
300 1
297 -3
290 -7
279 -11
264 -15
245 -19
222 -23
195 -27
164 -31
129 -35
90 -39
47 -43
4900 -47

Here I have summed the first column, and have considered 0 - 47
to get -47 in the second column.

24 stands alone, the sum of the remaining positive differences is 276,
which is 23 * 12, the sum of the negative differences is -300,
which is -25 * 12, of course this is (23-25)* 12 which is -24
which with 24 of course is zero. Or just consider 300-300=0

M12->M24->Co1->Monster (from "Symmetry and the Monster")

If I could get through SPLAG I would have something better to offer,
and could offer better discourse regarding Lattices discussed
on this newsgroup.

As soon as I can get some time,

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> (Has this newsgroup moved or something?)
>
> 1^2 + 2^2 + 3^2 ... 24^2 = 70^2
>
> We know that 35 is about 50 cents flat. So, 4900, is about 100
cents
> flat.
>
> 4900, considered as E, is a very flat E.
>
> Major becomes minor.
>
> 4900/4096 I would like to dub the Leech third.
>
> Splitting 4900 into 100 and 49: 100/49 is G#/Ab, which
> is the center of symmetry of the black white keyboard, along with D.
>
> Just as G-G# is chromatic, G-Ab is diatonic, but in this case
> we go to the 7-limit and consider also 49/48 instead of 16/15.
> So now it is about the distinction between two kinds of chromatic
> half-steps...
>
> But back to the LL.
>
> It's very elementary that the differences between perfect squares
> are the odds. So the above sum is now
>
> 1
> 1 3
> 1 3 5
> 1 3 5 7
> ...
> 1 3 5 ... 47
>
> This is just 24 * 1 + 23 * 5 + 22 * 7 ... 1 * 47 = 4900
>
> Now taking the sums, and the differences of the sums:
>
> 24 24
> 69 45
> 110 41
> 147 37
> 180 33
> 209 29
> 234 25
> 255 21
> 272 17
> 285 13
> 294 9
> 299 5
> 300 1
> 297 -3
> 290 -7
> 279 -11
> 264 -15
> 245 -19
> 222 -23
> 195 -27
> 164 -31
> 129 -35
> 90 -39
> 47 -43
> 4900 -47
>
> Here I have summed the first column, and have considered 0 - 47
> to get -47 in the second column.
>
> 24 stands alone, the sum of the remaining positive differences is
276,
> which is 23 * 12, the sum of the negative differences is -300,
> which is -25 * 12, of course this is (23-25)* 12 which is -24
> which with 24 of course is zero. Or just consider 300-300=0
>
> M12->M24->Co1->Monster (from "Symmetry and the Monster")
>
> If I could get through SPLAG I would have something better to offer,
> and could offer better discourse regarding Lattices discussed
> on this newsgroup.
>
> As soon as I can get some time,
>
> PGH

Not much to add, just that 6144/6125 is a comma here, which
tempers out hemikleismic, and also that 300 is the 24th triangular
number

PGH

I'm afraid I haven't the foggiest what you're on about.
I know what the leach lattice is...

At 03:18 PM 8/7/2007, you wrote:
>(Has this newsgroup moved or something?)
>
>1^2 + 2^2 + 3^2 ... 24^2 = 70^2
>
>We know that 35 is about 50 cents flat. So, 4900, is about 100 cents
>flat.

?? English please.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> I'm afraid I haven't the foggiest what you're on about.
> I know what the leach lattice is...
>
> At 03:18 PM 8/7/2007, you wrote:
> >(Has this newsgroup moved or something?)
> >
> >1^2 + 2^2 + 3^2 ... 24^2 = 70^2
> >
> >We know that 35 is about 50 cents flat. So, 4900, is about 100 cents
> >flat.
>
> ?? English please.
>
> -Carl

It is in fact, "Leech". Wikipedia's where to start, even though
I read "Symmetry and the Monster" by Mark Ronan too. What's the
"Leach" lattice?

The importance of the Leech Lattice relates to the Monster group,
and "Moonshine". (Well, not Moonshine directly). I'd better shut
up before I say anything really dumb..

The sum of the squares of 1 through 24 equal 70 squared. It's
the only integer that does this. Leech Lattice is based on
M24, a group which itself is based on S(5,8,24). The Mathieu groups:
I think they are important for tuning, you might want to check out
Noam Elkie's website, he has a good illustration of 11 hexads
used in M12 (S(5,6,12), to be exact)

As you might expect, M12 and M24 are related, although not
easily. As you also might expect, they can be used for
12t-ET and 24t-ET respectively.

Music theory -> Physics/Math -> Cosmology!

But back to my post, which may be "vain" and "in vain",

I look for beautiful patterns, I don't claim to have found
anything. I would recommend surfing on the Leech Lattice, many
cosmologists think the Universe could be based on it!
(Via the Monster etc)

But in terms of music theory, some of the lattice stuff on
this newsgroup is covered in SPLAG, which also has a lot
of material on the LL. (Sphere Packings Lattices and Groups,
Conway et al.) It's definitely one of the hardest books I
have ever attempted to study. I also own the "ATLAS" of Finite
Groups, what a cool thing that is...

But although it may be trivial, look again:

1
1 3
1 3 5

each row adds to a square in 1^2 + 2^2 ... 24^2
each column produces something too, a product.

Here's more stuff and nonsense:

1
1 3
1 3 5
1 3 5 7
...
1 3 5 ... 47

gives

24 * 1 + 23 * 3 + 22 * 5... 1 *47

Now let's extrapolate

25 * -1 (off the edge, but you'll see why)
24 * 1
23 * 3
....
1 * 47
0 * 49 (off the other edge)

Let's set m in the middle: 12 * 25, recalcuate 25 as zero and
merely count position of the odds, so go from 12 to -12 (just
atrocious, I admit...)

Using the 1 in the second row as a reference point for n, we can find
all the differences between these sums with 1 + 4m. Now m = 12 - n
so this equals 49 - 4n

You see 25 at the top, and 49 at the bottom are off the two edges. The
first phantom difference is 49, then 45, 41, 37, ...1, -3, -7 ... -47
The products go up to 300 in the middle and then back down. Another
curiousity is that 300 is the 24th triangular number (it's used
in LL stuff, I googled it) which is 1 + 2 + 3 + 4 + 5 ... + 24

The whole point is merely to show the prevalence of 49 and 25.
It's also in 70^2=4900. Taking 100/49 is a bit of a leap, but
may be significant, consider that G#/Ab are in the center of
the black-white key system, and that there are 7 whites and 5 blacks

More nonsense:

50/49 is tempered out in 12t-ET 7-limit tuning. It lands on G#/Ab, the
distinction between 25/24 and 49/48 semitones. Not unlike
the diesis going from Ab -> G# which relates diatonicity to
chromaticism...(15/16 and 25/24). This is more 7-limity

If I wasn't so darn tired all the time I might be able to
do something a lot better than this.

This is mostly to stimulate discussion.

A little self-deprecation: ("It's a Wonderful Life")

Nick: Tuning-math is a forum where real mathematicians
can solve a hard tuning-math problem fast! We don't need
a bunch of "characters" in here for "atmosphere"

George Bailey: It's okay, Nick, he's with me

Well I'm no angel, but then I haven't been thrown out --- not
yet

PGH

Paul H. wrote...

>The sum of the squares of 1 through 24 equal 70 squared. It's
>the only integer that does this.

70 is the only integer that's the sum of squares of...
consecutive integers or....?

>Leech Lattice is based on M24, a group which itself is based
>on S(5,8,24). The Mathieu groups:
>I think they are important for tuning, you might want to check out
>Noam Elkie's website, he has a good illustration of 11 hexads
>used in M12 (S(5,6,12), to be exact)

It's not coming up.

>As you might expect, M12 and M24 are related, although not
>easily. As you also might expect, they can be used for
>12t-ET and 24t-ET respectively.

How do 12-ET represent the symmetries of a group like M12?

>The whole point is merely to show the prevalence of 49 and 25.
>It's also in 70^2=4900. Taking 100/49 is a bit of a leap, but
>may be significant, consider that G#/Ab are in the center of
>the black-white key system,

In the center??

>and that there are 7 whites and 5 blacks

which are close to (70 49) / 10, I presume?

>If I wasn't so darn tired all the time I might be able to
>do something a lot better than this.

I know the feeling.

>This is mostly to stimulate discussion.

Gene's worked with moonshine theory...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Paul H. wrote...
>
> >The sum of the squares of 1 through 24 equal 70 squared. It's
> >the only integer that does this.
>
> 70 is the only integer that's the sum of squares of...
> consecutive integers or....?

24 is the only integer, where Sigma{1,n} x^2 equals a perfect square,
in this case 70 squared. n=24 of course. The trivial case is "1"
(1^2=1^2)

> >Leech Lattice is based on M24, a group which itself is based
> >on S(5,8,24). The Mathieu groups:
> >I think they are important for tuning, you might want to check out
> >Noam Elkie's website, he has a good illustration of 11 hexads
> >used in M12 (S(5,6,12), to be exact)
>
> It's not coming up.

http://www.math.harvard.edu/~elkies/m12.pdf
http://www.math.harvard.edu/~elkies/

> >As you might expect, M12 and M24 are related, although not
> >easily. As you also might expect, they can be used for
> >12t-ET and 24t-ET respectively.
>
> How do 12-ET represent the symmetries of a group like M12?

Just the opposite. M12 can be used to define a 12t-ET system.
In the illustration above, 11 hexads, and their inverses,
and 12 transpositions = 264 hexads which canvas all pentads
exactly twice. This is a Double Steiner System. A SSS is more
typical, but I haven't found one yet, that has the cool
tranpositional property of Dr. Elkies' DSS (where 22 hexad types
contain 66 pentads exactly twice, at each of the 12 transposes!)

> >The whole point is merely to show the prevalence of 49 and 25.
> >It's also in 70^2=4900. Taking 100/49 is a bit of a leap, but
> >may be significant, consider that G#/Ab are in the center of
> >the black-white key system,
>
> In the center??

Along with D, G# is the center of symmetry of the 7/5 MOS
>
> >and that there are 7 whites and 5 blacks
>
> which are close to (70 49) / 10, I presume?

Now I have no idea what you are saying? What is (70 49)?

> >If I wasn't so darn tired all the time I might be able to
> >do something a lot better than this.
>
> I know the feeling.
>
> >This is mostly to stimulate discussion.
>
> Gene's worked with moonshine theory...

Yes, I know. The primes of the Monster are very interesting

PGH

>

At 06:54 AM 8/9/2007, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>>
>> Paul H. wrote...
>>
>> >The sum of the squares of 1 through 24 equal 70 squared. It's
>> >the only integer that does this.
>>
>> 70 is the only integer that's the sum of squares of...
>> consecutive integers or....?
>
>24 is the only integer, where Sigma{1,n} x^2 equals a perfect square,
>in this case 70 squared. n=24 of course. The trivial case is "1"
>(1^2=1^2)

OK.

>> How do 12-ET represent the symmetries of a group like M12?
>
>Just the opposite. M12 can be used to define a 12t-ET system.
>In the illustration above, 11 hexads, and their inverses,
>and 12 transpositions = 264 hexads which canvas all pentads
>exactly twice. This is a Double Steiner System. A SSS is more
>typical, but I haven't found one yet, that has the cool
>tranpositional property of Dr. Elkies' DSS (where 22 hexad types
>contain 66 pentads exactly twice, at each of the 12 transposes!)

Oh, this stuff again.

>> >The whole point is merely to show the prevalence of 49 and 25.
>> >It's also in 70^2=4900. Taking 100/49 is a bit of a leap, but
>> >may be significant, consider that G#/Ab are in the center of
>> >the black-white key system,
>>
>> In the center??
>
>Along with D, G# is the center of symmetry of the 7/5 MOS

What's the 7/5 MOS?

>> >and that there are 7 whites and 5 blacks
>>
>> which are close to (70 49) / 10, I presume?
>
>Now I have no idea what you are saying? What is (70 49)?

A list of things I intend to divide by 10.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 06:54 AM 8/9/2007, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >>
> >> Paul H. wrote...
> >>
> >> >The sum of the squares of 1 through 24 equal 70 squared. It's
> >> >the only integer that does this.
> >>
> >> 70 is the only integer that's the sum of squares of...
> >> consecutive integers or....?
> >
> >24 is the only integer, where Sigma{1,n} x^2 equals a perfect
square,
> >in this case 70 squared. n=24 of course. The trivial case is "1"
> >(1^2=1^2)
>
> OK.
>
> >> How do 12-ET represent the symmetries of a group like M12?
> >
> >Just the opposite. M12 can be used to define a 12t-ET system.
> >In the illustration above, 11 hexads, and their inverses,
> >and 12 transpositions = 264 hexads which canvas all pentads
> >exactly twice. This is a Double Steiner System. A SSS is more
> >typical, but I haven't found one yet, that has the cool
> >tranpositional property of Dr. Elkies' DSS (where 22 hexad types
> >contain 66 pentads exactly twice, at each of the 12 transposes!)
>
> Oh, this stuff again.
>
> >> >The whole point is merely to show the prevalence of 49 and 25.
> >> >It's also in 70^2=4900. Taking 100/49 is a bit of a leap, but
> >> >may be significant, consider that G#/Ab are in the center of
> >> >the black-white key system,

A dreamt a solution to this, but I forgot when I woke up. I don't
know if I can justify throwing part of this in the denominator.

However, I was looking at partial sums of 1^2 + 2^2 ..24^4

1) They are all based on primes up to 47
2) There is a pattern to which primes are used
3) It resembles the same pattern of primes I found with my (4n, 4)
Z-relation grid, of all things! More on that later. If anyone cares

> >> In the center??
> >
> >Along with D, G# is the center of symmetry of the 7/5 MOS
>
> What's the 7/5 MOS?

My bad, I just meant the MOS with 7 whites and 5 blacks:

See Graham's "mapping by steps" and "mapping by period and generator"

http://x31eq.com/temper/twoet.html

> >> >and that there are 7 whites and 5 blacks
> >>
> >> which are close to (70 49) / 10, I presume?
> >
> >Now I have no idea what you are saying? What is (70 49)?
>
> A list of things I intend to divide by 10.

I don't follow. What are you dividing by 10, and why?

Thx

PGH

>1) They are all based on primes up to 47
>2) There is a pattern to which primes are used
>3) It resembles the same pattern of primes I found with my (4n, 4)
>Z-relation grid, of all things! More on that later. If anyone cares
>
>> >> In the center??
>> >
>> >Along with D, G# is the center of symmetry of the 7/5 MOS
>>
>> What's the 7/5 MOS?
>
>My bad, I just meant the MOS with 7 whites and 5 blacks:
>
>See Graham's "mapping by steps" and "mapping by period and generator"
>
>http://x31eq.com/temper/twoet.html

What of it?

>> >> >and that there are 7 whites and 5 blacks
>> >>
>> >> which are close to (70 49) / 10, I presume?
>> >
>> >Now I have no idea what you are saying? What is (70 49)?
>>
>> A list of things I intend to divide by 10.
>
>I don't follow. What are you dividing by 10, and why?

I'm asking you what 7/5 has to do with M12 or whatever.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >1) They are all based on primes up to 47
> >2) There is a pattern to which primes are used
> >3) It resembles the same pattern of primes I found with my (4n, 4)
> >Z-relation grid, of all things! More on that later. If anyone cares
> >
> >> >> In the center??
> >> >
> >> >Along with D, G# is the center of symmetry of the 7/5 MOS
> >>
> >> What's the 7/5 MOS?
> >
> >My bad, I just meant the MOS with 7 whites and 5 blacks:
> >
> >See Graham's "mapping by steps" and "mapping by period and
generator"
> >
> >http://x31eq.com/temper/twoet.html
>
> What of it?
>
> >> >> >and that there are 7 whites and 5 blacks
> >> >>
> >> >> which are close to (70 49) / 10, I presume?
> >> >
> >> >Now I have no idea what you are saying? What is (70 49)?
> >>
> >> A list of things I intend to divide by 10.
> >
> >I don't follow. What are you dividing by 10, and why?
>
> I'm asking you what 7/5 has to do with M12 or whatever.
>
> -Carl

Well, M12 (and Steiner (5,6,12) is all about pentachords
and hexachords (actually "pentads" and "hexads"). Well,
complements of pentachords are septachords, so there you
have 7 and 5. The piano keyboard layout is 7 and 5. My
hexachord system uses 35 hexachords. 70^2 is in the LL
and finally 100/49 lands right on G#.

the 5th and 7th overtone land on E and Bb, if you take
a linear division of the octave you get 12 13 14 15 .. 24
(The lower octaves are just 6 7 8 9 10 11 12, 3 4 5 6, and
then you get 2 and 1 on F). I believe all these things
tie together...

At this point I admit it looks more like numerology than
number theory, but I still think there are just too many coincidences

PGH

I am going to move over to sci.math and discuss more on the LL
if you are interested, it's obviously not a real "tuning-math"
subject, yet.

>I am going to move over to sci.math and discuss more on the LL
>if you are interested, it's obviously not a real "tuning-math"
>subject, yet.

The LL may be useful in modeling tone space.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >I am going to move over to sci.math and discuss more on the LL
> >if you are interested, it's obviously not a real "tuning-math"
> >subject, yet.
>
> The LL may be useful in modeling tone space.
>
> -Carl

Right. It's a Lattice! SPLAG is the best resource, just rather
difficult. Especially with no MS or PhD :)

Lattices and packings go together of course

>> >I am going to move over to sci.math and discuss more on the LL
>> >if you are interested, it's obviously not a real "tuning-math"
>> >subject, yet.
>>
>> The LL may be useful in modeling tone space.
>
>Right. It's a Lattice! SPLAG is the best resource, just rather
>difficult. Especially with no MS or PhD :)
>
>Lattices and packings go together of course

Yes. -Carl

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

> Gene's worked with moonshine theory...

Gene also can't figure out the relevsnce of all of this. In terms of
simple groups, the Leech lattice is very closely associated to the
Conway groups, but I don't know where to go with that either.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
>
> > Gene's worked with moonshine theory...
>
> Gene also can't figure out the relevsnce of all of this. In terms
of
> simple groups, the Leech lattice is very closely associated to the
> Conway groups, but I don't know where to go with that either.

Darn. Nothing one can do with the Lorentzian lattice either? I was
hoping that 0^ + 1^ ..24^2=70^2 had some relevance to 24-tET,
or perhaps 12 linear divisions of the octave. (I can't remember
what they call that on tuning, ADO?).

Strangely, even though this subject matter is so advanced
mathematically, perhaps this is a better discussion for plain tuning?

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >
> > > Gene's worked with moonshine theory...
> >
> > Gene also can't figure out the relevsnce of all of this. In terms
> of
> > simple groups, the Leech lattice is very closely associated to
the
> > Conway groups, but I don't know where to go with that either.
>
> Darn. Nothing one can do with the Lorentzian lattice either? I was
> hoping that 0^ + 1^ ..24^2=70^2 had some relevance to 24-tET,
> or perhaps 12 linear divisions of the octave. (I can't remember
> what they call that on tuning, ADO?).
>
> Strangely, even though this subject matter is so advanced
> mathematically, perhaps this is a better discussion for plain
tuning?

Actually, I'd like to trim this way down. Forget the LL, Conway
groups, etc.

How about a just a discussion of pyramidal numbers?

We've discussed triangular numbers, squared numbers, squared
triangular numbers, Pell's equation, and so forth.

I would think pyramidal numbers must have some relevance.

And then maybe... the 24th Pyramidal number

Okay, going over to tuning would be a bad idea, unless I can
talk about ADO and so forth which is real popular "over there"

Humbly,

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:

> How about a just a discussion of pyramidal numbers?

The pyramidal number function is

pyr(n) = n(n+1)(2n+1)/6

and

pyr(n)-1 = (n-1)(2n^2+5n+6).

This isn't quite as well adapted as squares or triangles for producing
interesting results, but for instance 385 is neither square nor
triangular, but is pyramidal, and 385/384 is an important 11-limit
comma.

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

> pyr(n)-1 = (n-1)(2n^2+5n+6)/6

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
>
> > pyr(n)-1 = (n-1)(2n^2+5n+6)/6

Right, so "Without dividing by 3" in my "pattern"

Obtain

(2n+1)(n+1)(n/2) for even
(2n+1)(n)((n+1)/2) for odd

This equals

n^3 + (3/2)n^2 + (1/2)n

Therefore

(1/3)n^3 + (1/2)n^2 + (1/6)n

for pyr(n)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
>
> > How about a just a discussion of pyramidal numbers?
>
> The pyramidal number function is
>
> pyr(n) = n(n+1)(2n+1)/6
>
> and
>
> pyr(n)-1 = (n-1)(2n^2+5n+6).
>
> This isn't quite as well adapted as squares or triangles for
producing
> interesting results, but for instance 385 is neither square nor
> triangular, but is pyramidal, and 385/384 is an important 11-limit
> comma.
>
Missed this post. Thanks, are there perhaps triangular pyramidal
numbers, or squared pyramidal numbers, or squared triangular
pyramidal numbers?

You don't have to answer that!

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:

> Missed this post. Thanks, are there perhaps triangular pyramidal
> numbers, or squared pyramidal numbers, or squared triangular
> pyramidal numbers?

Square pyramidal numbers, of course, we know about. Both square and
triangular pyramidal numbers give rise to an elliptic curve, which by
Mordell's theorem has only a finite number of integer points on it. So
there can be only a finite number of solutions.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@>
> wrote:
>
> > Missed this post. Thanks, are there perhaps triangular pyramidal
> > numbers, or squared pyramidal numbers, or squared triangular
> > pyramidal numbers?
>
> Square pyramidal numbers, of course, we know about. Both square and
> triangular pyramidal numbers give rise to an elliptic curve, which by
> Mordell's theorem has only a finite number of integer points on it.
So
> there can be only a finite number of solutions.
>

We already knew pyr(1) and pyr(24) are the only square pyramidal
numbers. It looks like pyr(1), pyr(5), pyr(6) and pyr(85) are the only
triangular pyramidal numbers. It would be a straightforward exercise
for me to prove it, I presume, given what I know about elliptic curves.

The corresponding commas are 55/54, 91/90 and 208335/208334, of limits
11, 13 and 647.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> > wrote:
> >
> > > Missed this post. Thanks, are there perhaps triangular pyramidal
> > > numbers, or squared pyramidal numbers, or squared triangular
> > > pyramidal numbers?
> >
> > Square pyramidal numbers, of course, we know about. Both square
and
> > triangular pyramidal numbers give rise to an elliptic curve,
which by
> > Mordell's theorem has only a finite number of integer points on
it.
> So
> > there can be only a finite number of solutions.
> >
>
> We already knew pyr(1) and pyr(24) are the only square pyramidal
> numbers. It looks like pyr(1), pyr(5), pyr(6) and pyr(85) are the
only
> triangular pyramidal numbers. It would be a straightforward
exercise
> for me to prove it, I presume, given what I know about elliptic
curves.
>
> The corresponding commas are 55/54, 91/90 and 208335/208334, of
limits
> 11, 13 and 647.

Interesting. Backtracking a little, Pell's equation can find
squared triangular numbers, and you mentioned it vis-a-vis
superparticular ratios, so I will study that, obviously there
are no squared triangular pyramidal numbers, I'll have to look
at Mordell's theorem for SPN and TPN. Of course, you just listed them.

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:

> I'll have to look
> at Mordell's theorem for SPN and TPN. Of course, you just listed them.

I should have said Siegel's theorem. Neither theorem is about square or
triangular numbers, but elliptic curves with rational coefficients.
Mordell's theorem says that the (group theoretic) rank of such a curve
is finite, and Siegel's theorem says that there are only a finite
number of integer points on it. If we have, for example, the elliptic
curve

y^2 = x(x+1)(2x+1)/6

then by Siegel's theorem there are only a finite number of pairs of
integers (x, y) satisfying the equation.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
>
> > I'll have to look
> > at Mordell's theorem for SPN and TPN. Of course, you just listed
them.
>
> I should have said Siegel's theorem. Neither theorem is about
square or
> triangular numbers, but elliptic curves with rational coefficients.
> Mordell's theorem says that the (group theoretic) rank of such a
curve
> is finite, and Siegel's theorem says that there are only a finite
> number of integer points on it. If we have, for example, the
elliptic
> curve
>
> y^2 = x(x+1)(2x+1)/6
>
>
> then by Siegel's theorem there are only a finite number of pairs of
> integers (x, y) satisfying the equation.
>

I will study all these theorems, Mordell, Siegel, and the work
of St-slash-ormer and Lehmer, just been way to busy. I hesistate
to post now being so busy because my posts aren't very good! But
I do want to ask:

What does Pell's equation have to do with all the superparticular
things you have been discussing lately?

And one nonsense thing:

Anything to (2n+1)(n+1)(n)/6 (which I now see is the best formula for
my "pattern") which you define as an elliptic curve when set to y^2,

and it's relationship to ((2n+1)/(2n)) /((n+1)/(n)) (49/50, when n=24)

which are two superparticular fractions...

PGH