HyperMOS

Finally, I'll take the time to say something on this
fascinating thread...

() Does "HyperMOS" refer to the generalized higher-D
MOS's of Paul's hypothesis, or are they another type?

() Paul, you said that 'as far as we can tell, CS
and PB are the same thing'. But aren't there many
irrational CS scales that don't have any sensible
PB interp?

() Paul's (hi, Paul!) hypothesis post may be the most
feverishly dense and impressive post I've seen since
his original harmonic entropy post in '97. I don't
think a notion of HyperMOS is nearly as important as
harmonic entropy, but who knows.... I'll quote the
pertinent parts here, and maybe say something more
later when (a) I understand more of it and (b) I don't
have an apartment full of boxes to unpack. For now,
please accept my once-over comments:

>Take an n-dimensional lattice, and pick n independent unison
>vectors. Use these to divide the lattice into parallelograms or
>parallelepipeds or hyperparallelepipeds, Fokker style. Each one
>contains an identical copy of a single scale (the PB) with N notes.

Paul, did you have to use the letter "n" twice? You seem to want
to use case to distinguish them... I'll hold you to that.

>Any vector in the lattice now corresponds to a single generic
>interval in this scale no matter where the vector is placed (if
>the PB is CS, which it normally should be).

Check.

>Now suppose all but one of the unison vectors are tempered out.
>The "wolves" now divide the lattice into parallel strips, or
>layers, or hyper-layers. The "width" of each of these, along the
>direction of the chromatic unison vector (the one that remains
>untempered), is equal to the length of exactly one of this
>chromatic unison vector.

Check.

>Now let's go back to "any vector in the lattice". This vector,
>added to itself over and over, will land one back at a pitch in the
>same equivalence class as the pitch one started with, after N
>iterations (and more often if the vector represents a generic
>interval whose cardinality is not relatively prime with N).

Okay.

>In general, the vector will have a length that is some fraction
>M/N of the width of one strip/layer/hyperlayer, measured in the
>direction of this vector (NOT in the direction of the chromatic
>unison vector). M must be an integer, since after N iterations,
>you're guaranteed to be in a point in the same equivalence class as
>where you started, hence you must be an exact integer M
>strips/layers/hyperlayers away. As a special example, the generator
>has length 1/N of the width of one strip/layer/hyperlayer, measured
>in the direction of the generator.

I don't see why M/N yet, but I can see what's happening.

>Anyhow, each occurence of the vector will cross either floor(M/N)
>or ceiling(M/N) boundaries between strips/layers/hyperlayers. Now,
>each time one crosses one of these boundaries in a given direction,
>one shifts by a chromatic unison vector. Hence each specific
>occurence of the generic interval in question will be shifted by
>either floor(M/N) or ceiling(M/N) chromatic unison vectors. Thus
>there will be only two specific sizes of the interval in question,
>and their difference will be exactly 1 of the chromatic unison
>vector. And since the vectors in the chain are equally spaced and
>the boundaries are equally spaced, the pattern of these two sizes
>will be an MOS pattern.

Spot on, for the 1-D case I can vidi!!!

-Carl

--- In tuning-math@y..., carl@l... wrote:
> Finally, I'll take the time to say something on this
> fascinating thread...
>
> () Does "HyperMOS" refer to the generalized higher-D
> MOS's of Paul's hypothesis, or are they another type?

I don't know . . . it's really a term you coined for something we may
not have unambiguously defined yet.

>
> () Paul, you said that 'as far as we can tell, CS
> and PB are the same thing'. But aren't there many
> irrational CS scales that don't have any sensible
> PB interp?

I mean the CSs that Kraig has drawn from Wilson's work, all of which
are JI scales.
>
> () Paul's (hi, Paul!) hypothesis post may be the most
> feverishly dense and impressive post I've seen since
> his original harmonic entropy post in '97. I don't
> think a notion of HyperMOS is nearly as important as
> harmonic entropy, but who knows.... I'll quote the
> pertinent parts here, and maybe say something more
> later when (a) I understand more of it and (b) I don't
> have an apartment full of boxes to unpack. For now,
> please accept my once-over comments:

OK, but note that none of this relates to "HyperMOS", merely to the
Hypothesis which relates PBs to MOS scales. I think it's important
because many modern music theorists are obsessed with MOS, which they
call well-formed or deep scales, but the Hypothesis _derives_ the
concept from basic JI considerations.
>
> >Take an n-dimensional lattice, and pick n independent unison
> >vectors. Use these to divide the lattice into parallelograms or
> >parallelepipeds or hyperparallelepipeds, Fokker style. Each one
> >contains an identical copy of a single scale (the PB) with N notes.
>
> Paul, did you have to use the letter "n" twice? You seem to want
> to use case to distinguish them... I'll hold you to that.

OK. Sorry.
>
> >Any vector in the lattice now corresponds to a single generic
> >interval in this scale no matter where the vector is placed (if
> >the PB is CS, which it normally should be).
>
> Check.
>
> >Now suppose all but one of the unison vectors are tempered out.
> >The "wolves" now divide the lattice into parallel strips, or
> >layers, or hyper-layers. The "width" of each of these, along the
> >direction of the chromatic unison vector (the one that remains
> >untempered), is equal to the length of exactly one of this
> >chromatic unison vector.
>
> Check.
>
> >Now let's go back to "any vector in the lattice". This vector,
> >added to itself over and over, will land one back at a pitch in the
> >same equivalence class as the pitch one started with, after N
> >iterations (and more often if the vector represents a generic
> >interval whose cardinality is not relatively prime with N).
>
> Okay.
>
> >In general, the vector will have a length that is some fraction
> >M/N of the width of one strip/layer/hyperlayer, measured in the
> >direction of this vector (NOT in the direction of the chromatic
> >unison vector). M must be an integer, since after N iterations,
> >you're guaranteed to be in a point in the same equivalence class as
> >where you started, hence you must be an exact integer M
> >strips/layers/hyperlayers away. As a special example, the generator
> >has length 1/N of the width of one strip/layer/hyperlayer, measured
> >in the direction of the generator.
>
> I don't see why M/N yet, but I can see what's happening.

M is just the number of times you cross a "wolf" or period boundary
before you get back to the note you started with. N is the number of
steps before you get back to the note you started with. Now do you
see why the length is M/N units in the relevant direction?
>
> >Anyhow, each occurence of the vector will cross either floor(M/N)
> >or ceiling(M/N) boundaries between strips/layers/hyperlayers. Now,
> >each time one crosses one of these boundaries in a given direction,
> >one shifts by a chromatic unison vector. Hence each specific
> >occurence of the generic interval in question will be shifted by
> >either floor(M/N) or ceiling(M/N) chromatic unison vectors. Thus
> >there will be only two specific sizes of the interval in question,
> >and their difference will be exactly 1 of the chromatic unison
> >vector. And since the vectors in the chain are equally spaced and
> >the boundaries are equally spaced, the pattern of these two sizes
> >will be an MOS pattern.
>
> Spot on, for the 1-D case I can vidi!!!

I'll have to make an illustration for the 2-D case, as I promised
Monz. But this will still be a (1-D) MOS. Whatever n is, if you
temper out n-1 unison vectors, you're left with a (1-D) MOS.

>> () Does "HyperMOS" refer to the generalized higher-D
>> MOS's of Paul's hypothesis, or are they another type?
>
>I don't know . . . it's really a term you coined for something we
>may not have unambiguously defined yet.

Wow! I don't even remember seeing it before Dave's post.
All those years of sniffing glue must be starting to
catch up with me...

>> () Paul, you said that 'as far as we can tell, CS
>> and PB are the same thing'. But aren't there many
>> irrational CS scales that don't have any sensible
>> PB interp?
>
> I mean the CSs that Kraig has drawn from Wilson's work, all of
> which are JI scales.

Thanks (just checking).

> OK, but note that none of this relates to "HyperMOS", merely to the
> Hypothesis which relates PBs to MOS scales. I think it's important
> because many modern music theorists are obsessed with MOS, which
> they call well-formed or deep scales, but the Hypothesis _derives_
> the concept from basic JI considerations.

Yeah- that's what makes it so exciting! I've never been a big
fan of MOS myself, until now...

>>>In general, the vector will have a length that is some fraction
>>>M/N of the width of one strip/layer/hyperlayer, measured in the
>>>direction of this vector (NOT in the direction of the chromatic
>>>unison vector). M must be an integer, since after N iterations,
>>>you're guaranteed to be in a point in the same equivalence class
>>>as where you started, hence you must be an exact integer M
>>>strips/layers/hyperlayers away. As a special example, the
>>>generator has length 1/N of the width of one
>>>strip/layer/hyperlayer, measured in the direction of the
>>>generator.
>>
>> I don't see why M/N yet, but I can see what's happening.
>
> M is just the number of times you cross a "wolf" or period boundary
> before you get back to the note you started with. N is the number
> of steps before you get back to the note you started with. Now do
> you see why the length is M/N units in the relevant direction?

Yup. I've got some other questions though. I'll:

() Wait for the vis you promised Monz.
() Think about 'em on my own for a while.

> I'll have to make an illustration for the 2-D case, as I promised
> Monz. But this will still be a (1-D) MOS. Whatever n is, if you
> temper out n-1 unison vectors, you're left with a (1-D) MOS.

Right. What's killing me is what happens when you don't...

-Carl

> From: <carl@lumma.org>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, August 06, 2001 4:04 PM
> Subject: [tuning-math] Re: HyperMOS
>
>
> > OK, but note that none of this relates to "HyperMOS", merely to the
> > Hypothesis which relates PBs to MOS scales. I think it's important
> > because many modern music theorists are obsessed with MOS, which
> > they call well-formed or deep scales, but the Hypothesis _derives_
> > the concept from basic JI considerations.
>
> Yeah- that's what makes it so exciting! I've never been a big
> fan of MOS myself, until now...

Paul, I'm *deeply* interested in this too!

Can you lay the whole thing out (what you have so far... I realize
it's "in progress") simply and clearly, like a "Gentle Introduction"?
With *lots* and lots of visuals. I'll be happy to host it somewhere
at Sonic Arts... I think a Dictionary entry for "HyperMOS" would be
a good gateway to it. Guys?

> Yup. I've got some other questions though. I'll:
>
> () Wait for the vis you promised Monz.
> () Think about 'em on my own for a while.
>
> > I'll have to make an illustration for the 2-D case, as I promised
> > Monz. But this will still be a (1-D) MOS. Whatever n is, if you
> > temper out n-1 unison vectors, you're left with a (1-D) MOS.

Is a (1-D) MOS what Erv referred to as "extended linear mapping"?

And I don't remember you promising this to me!
(years of hanging out with Herbert catching up with me)

> Right. What's killing me is what happens when you don't...

Me too!

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: <carl@l...>
> > To: <tuning-math@y...>
> > Sent: Monday, August 06, 2001 4:04 PM
> > Subject: [tuning-math] Re: HyperMOS
> >
> >
> > > OK, but note that none of this relates to "HyperMOS", merely to
the
> > > Hypothesis which relates PBs to MOS scales. I think it's
important
> > > because many modern music theorists are obsessed with MOS, which
> > > they call well-formed or deep scales, but the Hypothesis
_derives_
> > > the concept from basic JI considerations.
> >
> > Yeah- that's what makes it so exciting! I've never been a big
> > fan of MOS myself, until now...
>
> Paul, I'm *deeply* interested in this too!
>
> Can you lay the whole thing out (what you have so far... I realize
> it's "in progress") simply and clearly, like a "Gentle
Introduction"?
> With *lots* and lots of visuals. I'll be happy to host it somewhere
> at Sonic Arts... I think a Dictionary entry for "HyperMOS" would be
> a good gateway to it. Guys?

The problem is, at present, we don't know exactly what HyperMOS will
end up meaning, or if we'll even end up agreeing on a meaning. And
again, the Hypothesis doesn't concern HyperMOS, just MOS.

As soon as I have time, I'll get cracking on those visuals.

> Is a (1-D) MOS what Erv referred to as "extended linear mapping"?

By (1-D) MOS, I just mean MOS (as opposed to HyperMOS). An MOS is a
linear tuning (since there is a single generator along with an
associated interval of repetition), carried out to some number of
notes such that there are only two step sizes.