Kernel?

It would be nice if 'kernel' had a readily accessible definition.

I have looked through the Tonalsoft pages (not there) and the archives
and found only this:

/tuning/topicId_55471.html#55587

> > kernel - I haven't a clue.
>
> The kernel of a temperament is all of the intervals it sends to the
> unison; these form another group.

Correct?

~~~T~~~

On 7/14/06, Tom Dent <stringph@gmail.com> wrote:
>
> It would be nice if 'kernel' had a readily accessible definition.
>
> I have looked through the Tonalsoft pages (not there) and the archives
> and found only this:
>
> /tuning/topicId_55471.html#55587
>
> > > kernel - I haven't a clue.
> >
> > The kernel of a temperament is all of the intervals it sends to the
> > unison; these form another group.
>
> Correct?

Yes. "Kernel" is a mathematical term with many specific meanings, but
the common basic idea is "the set of elements that are mapped to the
identity". The kernel of a temperament is the set of all the intervals
that are mapped to the unison, i.e. "tempered out". The sum and
difference of any two intervals in the kernel is another interval in
the kernel.

Keenan

Wow, that's a glaring omission.
I could have sworn that i made a page in
the Encyclopedia for "kernel".

Gene, Graham ... can you guys help me with this?

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> It would be nice if 'kernel' had a readily accessible
> definition.
>
> I have looked through the Tonalsoft pages (not there)
> and the archives and found only this:
>
> /tuning/topicId_55471.html#55587
>
> > > kernel - I haven't a clue.
> >
> > The kernel of a temperament is all of the intervals it
> > sends to the unison; these form another group.
>
> Correct?
>
> ~~~T~~~
>

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Wow, that's a glaring omission.
> I could have sworn that i made a page in
> the Encyclopedia for "kernel".
>
> Gene, Graham ... can you guys help me with this?

I suggest you first define "regular temperament" and then define
"kernel". "Regular temperament" can mean either the concrete regular
tuning, which is where every interval in the tuning is expressible as
a product of a finte set of factors, and "logical" regular
temperament, which is a phrase I'm taking from your usage of "logical
meantone", and which is a more abstract object representable by a
wedgie. Both of these can be considered as defined by mappings from
just intonation.

Having reread that paragraph, maybe you'd better not start by defining
regular temperament. :( What about simply saying the kernel of a
temperament consists of all of the intervals which are equated to the
unison in that temperament?

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> >>Wow, that's a glaring omission.
>>I could have sworn that i made a page in
>>the Encyclopedia for "kernel".
>>
>>Gene, Graham ... can you guys help me with this?
> > > I suggest you first define "regular temperament" and then define
> "kernel". "Regular temperament" can mean either the concrete regular
> tuning, which is where every interval in the tuning is expressible as
> a product of a finte set of factors, and "logical" regular
> temperament, which is a phrase I'm taking from your usage of "logical
> meantone", and which is a more abstract object representable by a
> wedgie. Both of these can be considered as defined by mappings from
> just intonation.
> > Having reread that paragraph, maybe you'd better not start by defining
> regular temperament. :( What about simply saying the kernel of a
> temperament consists of all of the intervals which are equated to the
> unison in that temperament?

A kernel is all the intervals that map to a unison. That gets rid of temperaments altogether, so a notation for JI can still have a kernel. I still say the kernel is the set of all unison vectors so if you have a definition of unison vectors, or any synonym, you can refer to that.

For another omission, Monz, I think you have an entry for "vapro" but nothing for the projective space they're supposed to exist in. I noticed this because I want to talk about projective space in my errors and complexities PDF but I don't have anything good to link to. The mathematical definitions of projective space are too general.

Graham

Hi Graham,

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>
> A kernel is all the intervals that map to a unison. That
> gets rid of temperaments altogether, so a notation for JI
> can still have a kernel.
> I still say the kernel is the set of all unison vectors so
> if you have a definition of unison vectors, or any synonym,
> you can refer to that.

Thanks. Yes, i do have a definition for unison-vector,
and also one for the related concept of xenharmonic-bridge.

> For another omission, Monz, I think you have an entry for
> "vapro" but nothing for the projective space they're supposed
> to exist in. I noticed this because I want to talk about
> projective space in my errors and complexities PDF but I
> don't have anything good to link to. The mathematical
> definitions of projective space are too general.

I think i can agree with that, but i don't understand the
mathematical definitions of projective space well enough
to say for sure. ;-)

I was talking with Paul Erlich a couple of years ago when
i wrote the definition of "vapro", but we didn't continue
that discussion long enough for me to put together definitions
for all of the related concepts. I would like to return to
that ... maybe now is a good time.

(... by "now" i mean in a few days, when i'm finished with
my current composing project.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> For another omission, Monz, I think you have an entry for "vapro" but
> nothing for the projective space they're supposed to exist in. I
> noticed this because I want to talk about projective space in my errors
> and complexities PDF but I don't have anything good to link to. The
> mathematical definitions of projective space are too general.

What's really sad is that Grassmannians are relevant to all this, and
they are a kind of generalized projective space.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
>
> > For another omission, Monz, I think you have an entry
> > for "vapro" but nothing for the projective space they're
> > supposed to exist in. I noticed this because I want to
> > talk about projective space in my errors and complexities
> > PDF but I don't have anything good to link to. The
> > mathematical definitions of projective space are too
> > general.
>
> What's really sad is that Grassmannians are relevant to
> all this, and they are a kind of generalized projective space.

I know about Grassmann Algebra, but what exactly are
"Grassmannians"? And what's sad about them being relevant?

If you can add anything to my "vapro" page and the other
pages related to it, or give me the text to create ones
that should be in the Encyclopedia but aren't, i'll be
happy to do the updates ... on Tuesday.

PS -- Gene, i sent a few emails to you at gmail, asking
for a huge favor.

-monz
http://tonalsoft.com
Tonescape microtonal music

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> PS -- Gene, i sent a few emails to you at gmail, asking
> for a huge favor.

I just saw that you responded to me on tuning-math, so
disregard that.

-monz
http://tonalsoft.com
Tonescape microtonal music