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

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--- 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

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--- 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.