Proposed definitions--convex, block

A scale is *convex* if every octave equivalence class contained in the convex hull
http://mathworld.wolfram.com/ConvexHull.html
of the classes of the scale is itself a class of the scale.

A scale is a *block* iff it is epimorphic and convex.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 15, 2002 1:38 PM
> Subject: [tuning-math] Proposed definitions--convex, block
>
>
> A scale is *convex* if every octave equivalence class contained in the
convex hull
> http://mathworld.wolfram.com/ConvexHull.html
> of the classes of the scale is itself a class of the scale.
>
> A scale is a *block* iff it is epimorphic and convex.

when you say "block", do you mean "periodicity-block"?

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> when you say "block", do you mean "periodicity-block"?

Right--you can prove convexity entails a regular covering, by the way.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > when you say "block", do you mean "periodicity-block"?
>
> Right

in other words, gene is proposing a strict limitation on how far
he'll allow you to transpose notes of a fokker periodicity block by
its unison vectors, before the periodicity block ceases to
look 'blocky' anymore. i believe gene also has a related definition
of 'semiblock', which would include things like my pentachordal
decatonics. right, gene?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, February 15, 2002 3:06 PM
> Subject: [tuning-math] Re: Proposed definitions--convex, block
>
>
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > when you say "block", do you mean "periodicity-block"?
> >
> > Right
>
> in other words, gene is proposing a strict limitation on how far
> he'll allow you to transpose notes of a fokker periodicity block by
> its unison vectors, before the periodicity block ceases to
> look 'blocky' anymore. i believe gene also has a related definition
> of 'semiblock', which would include things like my pentachordal
> decatonics. right, gene?

cool -- thanks for the explanation, paul.

before we get into semiblocks, can i have a few examples of
"convex" (and nonconvex?) for the Dictionary entries? thanks.

http://www.ixpres.com/interval/dict/convex.htm

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> in other words, gene is proposing a strict limitation on how far
> he'll allow you to transpose notes of a fokker periodicity block by
> its unison vectors, before the periodicity block ceases to
> look 'blocky' anymore.

Actually, I'm not. I'm simply saying the blocky shape can't have concavities.

i believe gene also has a related definition
> of 'semiblock', which would include things like my pentachordal
> decatonics. right, gene?

Except that under this definition, which allows a block to be something much more general than a Fokker block, the pentachordal decatonics could easily be blocks--Genesis Minus turned out to be, so why not pentachordal decatonics? I could calculate the convex hull and check.