Octoid and the {5,7,9,11}-symmetrical lattice

If D(q) is the symmetrical lattice distance of interval class q from
the origin, then a logflat badness measure related to this lattice is
cents(q)*D(q). Using this metric, commas with values under 1000 are
540/539, 3025/3024, 9801/9800; and putting these together gives us the
octoid temperament. Hence, octoid can be expected to be a good
temperament for tempering scales of ball type for this lattice in a
suitable range. More complex commas are 3294225/3294172 and
|1 26 0 2 10>, for really large, nanotempered scales.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> If D(q) is the symmetrical lattice distance of interval class q from
> the origin, then a logflat badness measure related to this lattice is
> cents(q)*D(q).

Sorry, I meant cents(q)*D(q)^4.

>If D(q) is the symmetrical lattice distance of interval class q from
>the origin, then a logflat badness measure related to this lattice is
>cents(q)*D(q). Using this metric, commas with values under 1000 are
>540/539, 3025/3024, 9801/9800; and putting these together gives us the
>octoid temperament. Hence, octoid can be expected to be a good
>temperament for tempering scales of ball type for this lattice in a
>suitable range. More complex commas are 3294225/3294172 and
>|1 26 0 2 10>, for really large, nanotempered scales.

I searched for "symmetrical lattice" and found several messages of
me asking you to define it, but no answer. Is it merely A3 or Z3,
or does it depend on what metric is defined on top of it, or...?

-Carl

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

> I searched for "symmetrical lattice" and found several messages of
> me asking you to define it, but no answer. Is it merely A3 or Z3,
> or does it depend on what metric is defined on top of it, or...?

By a {5,7,9,11}-symmetrical lattice I mean that these form a symmetrical
simplex, and a corresponding A4 lattice. Since I've left 3 off, we get
the same arragement of points, displaced by the 3, for the complete
lattice. Hence we end up with a metric where the 3 is half the size of
5, 7, 9, or 11, and we can use

|| |* a b c d> || = sqrt(a^2 + 2a(b+c+d) + 4(b^2+c^2+d^2+bc+cd+db))

as the norm defining the lattice metric.

>> I searched for "symmetrical lattice" and found several messages of
>> me asking you to define it, but no answer. Is it merely A3 or Z3,
>> or does it depend on what metric is defined on top of it, or...?
>
>By a {5,7,9,11}-symmetrical lattice I mean that these form a symmetrical
>simplex, and a corresponding A4 lattice. Since I've left 3 off, we get
>the same arragement of points, displaced by the 3, for the complete
>lattice. Hence we end up with a metric where the 3 is half the size of
>5, 7, 9, or 11, and we can use
>
>|| |* a b c d> || = sqrt(a^2 + 2a(b+c+d) + 4(b^2+c^2+d^2+bc+cd+db))
>
>as the norm defining the lattice metric.

I ran across this...

http://cst-www.nrl.navy.mil/lattice/struk.jmol/a4.html

...is this a projection of A4 or completely unrelated or?

-Carl

hi Gene,

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

> If D(q) is the symmetrical lattice distance of
> interval class q from the origin, then a logflat
> badness measure related to this lattice is
> cents(q)*D(q). Using this metric, commas with
> values under 1000 are 540/539, 3025/3024, 9801/9800;
> and putting these together gives us the octoid temperament.
> Hence, octoid can be expected to be a good temperament
> for tempering scales of ball type for this lattice in a
> suitable range. More complex commas are 3294225/3294172
> and |1 26 0 2 10>, for really large, nanotempered scales.

Gene, the stuff you post here has always been nearly
incomprehensible to me, but now, in failing to keep up
with what you've been writing about ball scales, i'm
totally lost.

i've searched the archives for an explanation of
"symmetrical lattice distance", only to find numerous
posts from Carl asking you to define it. please help
me out here.

i've seen references to "Tenney lattice", "symmetrical
lattice", etc. ... what are the differences between these?
can you please give examples? preferably, a single scale
shown on many of the different lattices ... but i'm not
even sure if i'm formulating this correctly ... perhaps
different lattices can only produce different scales?

-monz

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

> http://cst-www.nrl.navy.mil/lattice/struk.jmol/a4.html
>
> ...is this a projection of A4 or completely unrelated or?

This isn't a lattice, and mathematicians would call it D4+. The
mathematicians names for the basic lattices are taken from the
classification of simple Lie algebras, and I don't know if anyone but
mathematicians and physicists dig that.

>> http://cst-www.nrl.navy.mil/lattice/struk.jmol/a4.html
>>
>> ...is this a projection of A4 or completely unrelated or?
>
>This isn't a lattice, and mathematicians would call it D4+.

Thanks for the ID.

-Carl

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

> i've searched the archives for an explanation of
> "symmetrical lattice distance", only to find numerous
> posts from Carl asking you to define it. please help
> me out here.

Let's suppose |* a b c d> represents an 11-limit pitch class. Then we
can measure the symmetrical size of it by

|| |* a b c d> || = sqrt(a^2 + 2a(b+c+d) + 4(b^2+c^2+d^2+bc+cd+db))

If u and v are two such pitch class monzos, then the distance between
them is

d(u, v) = ||u - v||

> i've seen references to "Tenney lattice", "symmetrical
> lattice", etc. ... what are the differences between these?

The Tenney lattice is based on a non-Euclidean distance measure, which
measures the lattice distance in terms of the minimal number of
consonant steps it take to get from one pitch class to another.

> can you please give examples? preferably, a single scale
> shown on many of the different lattices ... but i'm not
> even sure if i'm formulating this correctly ... perhaps
> different lattices can only produce different scales?

I don't know what you are asking exactly, which is why I took so long
to respond. I can't show things on a Tenney lattice, since it is
non-Euclidean and doesn't fit into ordinary space, and I can't show
the 11-limit symmetrical lattice, since it is four dimensional. For
the rest, you are the guy we all rely on to show lattices.