kernels for JI?

hi Gene,

in your definition of "kernel" which is what i am
currently using for the Encyclopaedia webpage

http://tonalsoft.com/enc/index2.htm?kernel.htm

you apply the concept only to temperaments.

isn't it equally valid for finite JI pitch-sets?
(should i be calling them pitch-groups?)

this is posing a bit of a problem for me right now.
i've just added some info about Tanaka's work in
these two pages:

http://tonalsoft.com/enc/index2.htm?pblock.htm
http://tonalsoft.com/enc/index2.htm?kleisma.htm

and on both of them i give a graphic illustrating
his diagram of pitches, with the "periodic parallelogram"
(periodicity-block) clearly shown in the center.

Tanaka does indeed go on after describing his
53-tone JI parallelogram, to construct 53-et as an
excellent closed (circular) approximation to it.
so there, the 53-tone JI block *is* a kernel,
according to your definition.

but the first part of the presentation is given
entirely in JI, and it seems to me that the 53-tone
parallelogram is still a kernel, especially given
the way he shows the mathematical operations which
define how the *other* parallelograms relate to the
central one.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> hi Gene,
>
>
> in your definition of "kernel" which is what i am
> currently using for the Encyclopaedia webpage
>
> http://tonalsoft.com/enc/index2.htm?kernel.htm
>
> you apply the concept only to temperaments.
>
>
> isn't it equally valid for finite JI pitch-sets?
> (should i be calling them pitch-groups?)

If these "JI pitch-sets" are epimorphic scales (which they seem to be
judging by the examples) then they have a val, and the val defines a
temperament, namely the equal temperament with the val as mapping.
Hence, they also have a kernel in the sense that the val has a kernel.

> Tanaka does indeed go on after describing his
> 53-tone JI parallelogram, to construct 53-et as an
> excellent closed (circular) approximation to it.
> so there, the 53-tone JI block *is* a kernel,
> according to your definition.

No, because they are not being mapped to the unison. What they are are
class representatives. We have a mapping from the 5-limit to the
cyclic group of order 53, and two 5-limit intervals q and r are
equivalent under the mapping (call it "Tanaka") if Tanaka(q) =
Tanaka(r). If we pick a particular set of 53 elements to represent
each equivalence class, we have class representatives. Fokker blocks
do this in an intelligent way.