TM-reduced bases for ETs

hi Gene,

have you already posted the TM-reduced bases for a number
of common ETs in various prime-limits? if so, can you
point me to them or repost them?

right now, i'm particularly interested in 7-limit 41-ET.
i've tried putting various unison-vectors into my
software, but while i always get 41-ET for the temperament,
i keep getting 82-tone periodicity-blocks for the JI.
i don't know if these are cases of torsion (which is
what i suspect), or if there's a bug somewhere.

but aside from that specific example, i'd like to
have a nice list of TM-reduced bases for, say,
12, 13, 15, 16, 17, 19, 22, 24, 31, 34, 36, 41, 43, 46,
48, 50, 53, and 72-ETs, in the 5-, 7-, and 11-prime-limits.

(and please, label as much stuff as you can.)

thanks.

-monz

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

> have you already posted the TM-reduced bases for a number
> of common ETs in various prime-limits? if so, can you
> point me to them or repost them?

You can do exactly what I would need to do, namely search for them,
but I'll try to chase this down.

hi Gene,

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > have you already posted the TM-reduced bases for a number
> > of common ETs in various prime-limits? if so, can you
> > point me to them or repost them?
>
> You can do exactly what I would need to do, namely search for them,
> but I'll try to chase this down.

thanks ... mainly because i'm not even sure *how* to search
for them.

i'm thinking that a nice list would be good to add to the
"TM-reduced lattice basis" Dictionary page.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene,
>
>
> have you already posted the TM-reduced bases for a number
> of common ETs in various prime-limits? if so, can you
> point me to them or repost them?
>
> right now, i'm particularly interested in 7-limit 41-ET.
> i've tried putting various unison-vectors into my
> software, but while i always get 41-ET for the temperament,
> i keep getting 82-tone periodicity-blocks for the JI.
> i don't know if these are cases of torsion (which is
> what i suspect), or if there's a bug somewhere.
>
>
> but aside from that specific example, i'd like to
> have a nice list of TM-reduced bases for, say,
> 12, 13, 15, 16, 17, 19, 22, 24, 31, 34, 36, 41, 43, 46,
> 48, 50, 53, and 72-ETs, in the 5-, 7-, and 11-prime-limits.
>
> (and please, label as much stuff as you can.)
>
>
> thanks.
>
>
>
> -monz

hi monz, i found this:

/tuning-math/message/3151

but for some of the ETs you're asking about (namely the ones
inconsistent in the relevant limit), gene is going to assume the
standard val, even when better vals are available . . . to get
an "untainted" answer, you should provide a list of vals, not merely
a list of ETs.

thanks, paul! awesome!

i do want these for my webpages ... but i have to be honest,
the main reason i wanted them was so that i could have some
fun popping the unison-vectors into our software and seeing
the 3-D periodicity-blocks which result. so 7-limit is perfect!

:)

-monz

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Gene,
> >
> >
> > have you already posted the TM-reduced bases for a number
> > of common ETs in various prime-limits? if so, can you
> > point me to them or repost them?
> >
> > right now, i'm particularly interested in 7-limit 41-ET.
> > i've tried putting various unison-vectors into my
> > software, but while i always get 41-ET for the temperament,
> > i keep getting 82-tone periodicity-blocks for the JI.
> > i don't know if these are cases of torsion (which is
> > what i suspect), or if there's a bug somewhere.
> >
> >
> > but aside from that specific example, i'd like to
> > have a nice list of TM-reduced bases for, say,
> > 12, 13, 15, 16, 17, 19, 22, 24, 31, 34, 36, 41, 43, 46,
> > 48, 50, 53, and 72-ETs, in the 5-, 7-, and 11-prime-limits.
> >
> > (and please, label as much stuff as you can.)
> >
> >
> > thanks.
> >
> >
> >
> > -monz
>
> hi monz, i found this:
>
> /tuning-math/message/3151
>
> but for some of the ETs you're asking about (namely the
> ones inconsistent in the relevant limit), gene is going
> to assume the standard val, even when better vals are
> available . . . to get an "untainted" answer, you should
> provide a list of vals, not merely a list of ETs.

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

> hi monz, i found this:
>
> /tuning-math/message/3151

Thanks, Paul; this is exactly what's needed.

> but for some of the ETs you're asking about (namely the ones
> inconsistent in the relevant limit), gene is going to assume the
> standard val, even when better vals are available . . . to get
> an "untainted" answer, you should provide a list of vals, not
merely
> a list of ETs.

There isn't a problem with the ones I did in that posting--namely,
7-limit 9, 10, 12, 15, 19, 22, 27, 31, 41, 68, 72, 99, 130, 140. Monz
wants 13, 16, 17, 24, 34, 36, 43, 46, 48, 50, 53 as well, and also
the 5- and 11-limits. Some of these *will* lead to problems, so a val
is going to be required in some cases.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > hi monz, i found this:
> >
> > /tuning-math/message/3151
>
> Thanks, Paul; this is exactly what's needed.

you also had this followup:

"I accidentally left off

171: [2401/2400, 4375/4374, 32805/32768]

Wouldn't want to do that--look at those three high-powered commas!"

hello all,

i've just updated the Encyclopaedia of Tuning
entry for "TM-reduced lattice", to include examples
of TM-reduced bases for ETs that were posted here
by Gene at the beginning of November 2003.

http://tonalsoft.com/enc/tm-reduced-lattice.htm

feedback appreciated.

(paul, i know that i also need to quote your post
about how certain ETs can have multiple equally-good/bad
val mappings ... i will when i get more time.)

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hello all,
>
>
> i've just updated the Encyclopaedia of Tuning
> entry for "TM-reduced lattice", to include examples
> of TM-reduced bases for ETs that were posted here
> by Gene at the beginning of November 2003.

The title should be TM-reduced lattice basis, since the lattice is not
being reduced.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hello all,
>
>
> i've just updated the Encyclopaedia of Tuning
> entry for "TM-reduced lattice", to include examples
> of TM-reduced bases for ETs that were posted here
> by Gene at the beginning of November 2003.
>
> http://tonalsoft.com/enc/tm-reduced-lattice.htm
>
> feedback appreciated.

Right off the bat:

"A method for reducing the bases of a lattice."

should read

"A method for reducing the basis of a lattice."

TM-reduction results in a *single* basis for the lattice, not
multiple _bases_.

hi paul,

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

> > http://tonalsoft.com/enc/tm-reduced-lattice.htm

> Right off the bat:
>
> "A method for reducing the bases of a lattice."
>
> should read
>
> "A method for reducing the basis of a lattice."
>
> TM-reduction results in a *single* basis for the lattice, not
> multiple _bases_.

thanks. i'd really like to add a little more "regular English"
to the opening part of that definition, describing exactly
what TM-reduction does ... before heading into Gene's
mathematical definition. can you or anyone else help?

i'm thinking something like this:

"A method for reducing the basis of a lattice to its
most compact representation, with all unison-vectors
as small as possible in prime-space."

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > http://tonalsoft.com/enc/tm-reduced-lattice.htm
>
> > Right off the bat:
> >
> > "A method for reducing the bases of a lattice."
> >
> > should read
> >
> > "A method for reducing the basis of a lattice."
> >
> > TM-reduction results in a *single* basis for the lattice, not
> > multiple _bases_.
>
>
>
> thanks. i'd really like to add a little more "regular English"
> to the opening part of that definition, describing exactly
> what TM-reduction does ... before heading into Gene's
> mathematical definition. can you or anyone else help?
>
> i'm thinking something like this:
>
> "A method for reducing the basis of a lattice to its
> most compact representation, with all unison-vectors
> as small as possible in prime-space."

Well, you must mean the "a kernel lattice" rather than simply "a
lattice", since it's a kernel lattice in general whose basis consists
of commatic unison vectors. The TM-reduced basis of the ordinary 5-
limit lattice, which is not normally a kernel lattice, is {2:1, 3:1,
5:1}, and these are normally not unison vectors.

Otherwise, the above is roughly correct, as long as "prime space"
means "the Tenney lattice with taxicab metric". If your usually-
presented conception of "prime space" doesn't include an axis for 2
then the above will be misleading unless you clarify it.

If the basis has more than two components, though, there is more than
one possible interpretation of "as small as possible", so for a truly
precise description one would have to peek into Gene's mathematical
definition.

hi paul,

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

> > > > http://tonalsoft.com/enc/tm-reduced-lattice.htm

> > thanks. i'd really like to add a little more "regular English"
> > to the opening part of that definition, describing exactly
> > what TM-reduction does ... before heading into Gene's
> > mathematical definition. can you or anyone else help?
> >
> > i'm thinking something like this:
> >
> > "A method for reducing the basis of a lattice to its
> > most compact representation, with all unison-vectors
> > as small as possible in prime-space."
>
> Well, you must mean the "a kernel lattice" rather than simply
> "a lattice", since it's a kernel lattice in general whose
> basis consists of commatic unison vectors. The TM-reduced basis
> of the ordinary 5-limit lattice, which is not normally a
> kernel lattice, is {2:1, 3:1, 5:1}, and these are normally
> not unison vectors.
>
> Otherwise, the above is roughly correct, as long as
> "prime space" means "the Tenney lattice with taxicab metric".
> If your usually-presented conception of "prime space" doesn't
> include an axis for 2 then the above will be misleading
> unless you clarify it.
>
> If the basis has more than two components, though, there
> is more than one possible interpretation of "as small as
> possible", so for a truly precise description one would
> have to peek into Gene's mathematical definition.

ok, thanks. i need a defintiion of "kernel lattice"
for the Encyclopaedia too, along with a good description
of what makes a "kernel lattice" different from one
that's not a kernel.

here is my current definition of "prime-space":
http://tonalsoft.com/enc/primespace.htm

please feel free to offer corrections, comments, etc.
on that, and to provide me with a "kernel lattice" definition,
and a good opening paragraph for the "TM-reduced basis"
definition. i thank you in advance.

my use of prime-space generally does not include 2 as
a prime-factor, but it certainly can and sometimes must
be included, depending on the precise nature of the tuning.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > > > http://tonalsoft.com/enc/tm-reduced-lattice.htm
>
> > > thanks. i'd really like to add a little more "regular English"
> > > to the opening part of that definition, describing exactly
> > > what TM-reduction does ... before heading into Gene's
> > > mathematical definition. can you or anyone else help?
> > >
> > > i'm thinking something like this:
> > >
> > > "A method for reducing the basis of a lattice to its
> > > most compact representation, with all unison-vectors
> > > as small as possible in prime-space."
> >
> > Well, you must mean the "a kernel lattice" rather than simply
> > "a lattice", since it's a kernel lattice in general whose
> > basis consists of commatic unison vectors. The TM-reduced basis
> > of the ordinary 5-limit lattice, which is not normally a
> > kernel lattice, is {2:1, 3:1, 5:1}, and these are normally
> > not unison vectors.
> >
> > Otherwise, the above is roughly correct, as long as
> > "prime space" means "the Tenney lattice with taxicab metric".
> > If your usually-presented conception of "prime space" doesn't
> > include an axis for 2 then the above will be misleading
> > unless you clarify it.
> >
> > If the basis has more than two components, though, there
> > is more than one possible interpretation of "as small as
> > possible", so for a truly precise description one would
> > have to peek into Gene's mathematical definition.
>
>
> ok, thanks. i need a defintiion of "kernel lattice"
> for the Encyclopaedia too, along with a good description
> of what makes a "kernel lattice" different from one
> that's not a kernel.

umm . . . it's not a lattice that's a kernel, it's the lattice
*formed* by a kernel. The kernel consists of all the commatic unison
vectors of a tuning -- in the case of an ET, it's all the occurences
of "0" in the bingo-card. As you can see by examining any of my bingo-
cards, the "0"s themselves form a lattice. This is true for the
kernels of higher-dimensional temperaments as well.

> here is my current definition of "prime-space":
> http://tonalsoft.com/enc/primespace.htm

objectification? as a series of axes?

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

> > here is my current definition of "prime-space":
> > http://tonalsoft.com/enc/primespace.htm
>
> objectification? as a series of axes?

My own preference would be for a more mathematically precise
definition, but I suppose normed real vector spaces is nothing Monz
wants to get into.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > here is my current definition of "prime-space":
> > > http://tonalsoft.com/enc/primespace.htm
> >
> > objectification? as a series of axes?
>
> My own preference would be for a more mathematically precise
> definition, but I suppose normed real vector spaces is
> nothing Monz wants to get into.

you guys all know that i'm way out of my league here.
i just want to get the best definitions i can get of
all the terms i'm using. prime-space is perhaps the
most important of all, so if any of you want to add to
my definition of it, please do.

-monz