Necklaces

This concept might be of use:

http://en.wikipedia.org/wiki/Necklace_(combinatorics)

Kalle Aho

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> This concept might be of use:
>
> http://en.wikipedia.org/wiki/Necklace_(combinatorics)
>
> Kalle Aho

I recently implemented the FKM algorithm for generating
necklaces in Scheme, to search for rational well temperaments.
"Necklace" is, I suppose a decent alternative to abscale.

I can feel a poll coming on...

-Carl

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>> This concept might be of use:
>>
>> http://en.wikipedia.org/wiki/Necklace_(combinatorics)
>>
>> Kalle Aho
> > I recently implemented the FKM algorithm for generating
> necklaces in Scheme, to search for rational well temperaments.
> "Necklace" is, I suppose a decent alternative to abscale.
> > I can feel a poll coming on...

They don't look like scales to me. The main property is that they're circular (hence the metaphor) and you can start on any element. That's like cyclic permutations of scales. So "diatonic scales" would be a necklace but not "the (C) major scale". I think that works if you make step sizes the alphabet.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >> This concept might be of use:
> >>
> >> http://en.wikipedia.org/wiki/Necklace_(combinatorics)
> >>
> >> Kalle Aho
> >
> > I recently implemented the FKM algorithm for generating
> > necklaces in Scheme, to search for rational well temperaments.
> > "Necklace" is, I suppose a decent alternative to abscale.
> >
> > I can feel a poll coming on...
>
> They don't look like scales to me. The main property is
> that they're circular (hence the metaphor) and you can start
> on any element. That's like cyclic permutations of scales.
> So "diatonic scales" would be a necklace but not "the (C)
> major scale". I think that works if you make step sizes the
> alphabet.
>
> Graham

Speaking of alphabets, Jon Wild wrote me offlist to let me
know that there's already a name for something very similar
to what we're on about -- "words"...

http://en.wikipedia.org/wiki/Word_(group_theory)

Jon provided these references:

http://web.auth.gr/cim08/cim08_abstracts/035_CIM08_abstracts.pdf
http://www.mcm2007.info/pdf/sun1-clampitt.pdf

I haven't looked at it yet, but knowing the fetishism that
goes on in the scale theory crowd, the first thing to check
is that there aren't extra symmetries implied that we
don't want. . .

-Carl

Carl Lumma wrote:

> Speaking of alphabets, Jon Wild wrote me offlist to let me
> know that there's already a name for something very similar
> to what we're on about -- "words"...
> > http://en.wikipedia.org/wiki/Word_(group_theory)
>
> Jon provided these references:
> > http://web.auth.gr/cim08/cim08_abstracts/035_CIM08_abstracts.pdf
> http://www.mcm2007.info/pdf/sun1-clampitt.pdf
> > I haven't looked at it yet, but knowing the fetishism that
> goes on in the scale theory crowd, the first thing to check
> is that there aren't extra symmetries implied that we
> don't want. . .

They're being mathematically precise, so of course they have to be fetishistic about their definitions. The interesting thing is that they're bringing more music-related theory into known mathematics. This paper is interesting:

http://www.mcm2007.info/pdf/sun0a-domingues.pdf

They say that well formed scales (either a period of an MOS or equally tempered) written as step sizes are the same as "Christoffel words". So, effectively, the properties of MOS scales are already being studied by mathematicians.

Figure 2 at the end of that paper shows a "Christoffel tree". I'm not sure what it means, but could it be the scale tree?

Graham

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > Carl Lumma wrote:
> > > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> > >> This concept might be of use:
> > >>
> > >> http://en.wikipedia.org/wiki/Necklace_(combinatorics)
> > >>
> > >> Kalle Aho
> > >
> > > I recently implemented the FKM algorithm for generating
> > > necklaces in Scheme, to search for rational well temperaments.
> > > "Necklace" is, I suppose a decent alternative to abscale.
> > >
> > > I can feel a poll coming on...
> >
> > They don't look like scales to me. The main property is
> > that they're circular (hence the metaphor) and you can start
> > on any element. That's like cyclic permutations of scales.
> > So "diatonic scales" would be a necklace but not "the (C)
> > major scale". I think that works if you make step sizes the
> > alphabet.
> >
> > Graham
>
> Speaking of alphabets, Jon Wild wrote me offlist to let me
> know that there's already a name for something very similar
> to what we're on about -- "words"...
>
> http://en.wikipedia.org/wiki/Word_(group_theory)

I think Clampitt is actually using a different thing
even if he calls them "words":

http://en.wikipedia.org/wiki/String_(computer_science)

BTW, necklaces are equivalence classes of strings where rotations
are taken as equivalent.

Kalle Aho

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> > Jon provided these references:
> >
> > http://web.auth.gr/cim08/cim08_abstracts/035_CIM08_abstracts.pdf
> > http://www.mcm2007.info/pdf/sun1-clampitt.pdf
> >
> > I haven't looked at it yet, but knowing the fetishism that
> > goes on in the scale theory crowd, the first thing to check
> > is that there aren't extra symmetries implied that we
> > don't want. . .
>
> They're being mathematically precise, so of course they have
> to be fetishistic about their definitions.

They also have a history of using musically meaningless
group transformations as the basis of paper after paper,
in an incestuous circle of citations.

> The interesting thing is that they're bringing more
> music-related theory into known mathematics. This paper
> is interesting:
>
> http://www.mcm2007.info/pdf/sun0a-domingues.pdf
>
> They say that well formed scales (either a period of an MOS
> or equally tempered) written as step sizes are the same as
> "Christoffel words". So, effectively, the properties of MOS
> scales are already being studied by mathematicians.

Good find. I'll read it presently.

One thing I'd like to return to is John Chalmers' point,
which I fear may have been lost in the madness. While the
list of 2nds with the tuning does let you generate the
entire rank-order matrix, it's not immediately obvious
looking at two different tunings of the same list of 2nds
whether they expand to the same matrix. It would be nice
to have a compact way to identify the entire matrix...

-Carl