no 1/1?

So what does it mean for a tuning to have no 1/1?

What's the difference between 1/1 and anything else? This has no effect on
the intervals, only on the pitches, so I don't understand why it is of any
significance at all, except possibly to cause software that assumes 1/1 to
define a reference pitch to function in a way that the scale designer
probably did not intend.

The following page:

http://tonalsoft.com/td/lumma/carl-cps.htm As far as this statement

contains this statement about CPS's:

> Notice there's no 1/1! All those pitches are measured from 1/1, but it's not
> in the tuning itself. That's one cool thing about CPSs -- they have no
> "center" tonality (unlike tonality diamonds).

and so therefore I have to ask what does 1/1 have to do with a center
tonality? You still have a set of pitches which could be sorted and each
divided by the lowest one, in order to obtain a tuning *with* a 1/1 having
the same intervals. How does that change the issue of center tonality? And
how should a piece of tuning software (or a synthesizer) that accepts scala
files as input behave when given a tuning that does not start with 1/1, and
also, how should such tunings be speicified in scala?

-Kurt

>and so therefore I have to ask what does 1/1 have to do with a center
>tonality? You still have a set of pitches which could be sorted and
>each divided by the lowest one, in order to obtain a tuning *with* a
>1/1 having the same intervals. How does that change the issue of
>center tonality? And how should a piece of tuning software (or a
>synthesizer) that accepts scala files as input behave when given a
>tuning that does not start with 1/1, and also, how should such tunings
>be speicified in scala?

This is indeed unclear from my primer, unless you happen to grok
Wilson's diagram (the link to which is currently broken, but which
can be seen at...

http://lumma.org/tuning/13,14.gif

...have a look as you read the following.)

Ratios written with a slash indicate pitches. Since we usually only
consider intervals important, any pitch in a scale may be called
"1/1" as long as its relationships to the other pitches in the scale
are preserved and the other pitches are respelled accordingly. If
the tuning contains a wolf, usually "1/1" is placed far away from it.
If the tuning contains a chain of fifths, often "1/1" is placed at
its center. This is merely a notational convenience.

The point with CPS scales is, there's no obvious place to put the
"1/1" just by looking at them on the lattice, because they are
symmetrical about a point which isn't on the lattice (for example,
there is no pitch at the center of an octahedron on the 7-limit
lattice).

There may be other criteria that help place the "1/1". But the
reason the 'centerless' claim is often made of CPS scales is because
they are often seen as the alternative to tonality diamonds for
extended just intonation. And tonality diamonds are highly
symmetrical about one of their own pitches. This pitch occurs in
every chord in the diamond, and there is no doubt it is the "1/1".
Someone once remarked he could always recognize Partch's music,
because it was "always in G". There is no such pitch in a CPS or
stellated CPS. Yet, CPS scales are generally competitive with
diamonds as far as chords/note goes in pure JI.

-Carl

on 1/11/04 3:13 AM, Carl Lumma <ekin@lumma.org> wrote:

>> and so therefore I have to ask what does 1/1 have to do with a center
>> tonality? You still have a set of pitches which could be sorted and
>> each divided by the lowest one, in order to obtain a tuning *with* a
>> 1/1 having the same intervals. How does that change the issue of
>> center tonality? And how should a piece of tuning software (or a
>> synthesizer) that accepts scala files as input behave when given a
>> tuning that does not start with 1/1, and also, how should such tunings
>> be speicified in scala?
>
> This is indeed unclear from my primer, unless you happen to grok
> Wilson's diagram (the link to which is currently broken, but which
> can be seen at...
>
> http://lumma.org/tuning/13,14.gif
>
> ...have a look as you read the following.)
>
> Ratios written with a slash indicate pitches. Since we usually only
> consider intervals important, any pitch in a scale may be called
> "1/1" as long as its relationships to the other pitches in the scale
> are preserved and the other pitches are respelled accordingly. If
> the tuning contains a wolf, usually "1/1" is placed far away from it.
> If the tuning contains a chain of fifths, often "1/1" is placed at
> its center. This is merely a notational convenience.
>
> The point with CPS scales is, there's no obvious place to put the
> "1/1" just by looking at them on the lattice, because they are
> symmetrical about a point which isn't on the lattice (for example,
> there is no pitch at the center of an octahedron on the 7-limit
> lattice).
>
> There may be other criteria that help place the "1/1". But the
> reason the 'centerless' claim is often made of CPS scales is because
> they are often seen as the alternative to tonality diamonds for
> extended just intonation. And tonality diamonds are highly
> symmetrical about one of their own pitches. This pitch occurs in
> every chord in the diamond, and there is no doubt it is the "1/1".
> Someone once remarked he could always recognize Partch's music,
> because it was "always in G". There is no such pitch in a CPS or
> stellated CPS. Yet, CPS scales are generally competitive with
> diamonds as far as chords/note goes in pure JI.
>
> -Carl

Ah, that's much better. Thanks.

-Kurt

hi Carl,

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> This is indeed unclear from my primer,

http://tonalsoft.com/td/lumma/carl-cps.htm

> unless you happen to grok Wilson's diagram (the link
> to which is currently broken, but which can be seen at...
>
> http://lumma.org/tuning/13,14.gif

thanks for pointing out the broken link. i've actually
simply added the diagram to the page now.

-monz

>> This is indeed unclear from my primer,
>
>http://tonalsoft.com/td/lumma/carl-cps.htm
>
>> unless you happen to grok Wilson's diagram (the link
>> to which is currently broken, but which can be seen at...
>>
>> http://lumma.org/tuning/13,14.gif
>
>thanks for pointing out the broken link. i've actually
>simply added the diagram to the page now.

Thanks monz! Now if you could just change the 6 in
"6-dimensional object" to a 5 (that was my mistake)...

-Carl

[I wrote...]
>Thanks monz! Now if you could just change the 6 in
>"6-dimensional object" to a 5 (that was my mistake)...

Actually, though these are apparently projections of
5-D polytopes, he is effectively getting 6 dimensions
out of them. Nobody's ever made a positive ID on these
to my knowledge (Wilson did them intuitively).

-Carl

[I wrote...]
>>Thanks monz! Now if you could just change the 6 in
>>"6-dimensional object" to a 5 (that was my mistake)...
>
>Actually, though these are apparently projections of
>5-D polytopes, he is effectively getting 6 dimensions
>out of them. Nobody's ever made a positive ID on these
>to my knowledge (Wilson did them intuitively).

Actually, I searched the archives, and a positive ID was
made, by George Olshevsky. The Eikosany is a "uniform
dodecatetron", a 5-D polytope. So monz, the 6 should be
changed to a 5.

For those interested in History, here are some snippets
from my search....

[me]
>Yes- I believe those are true 6-D projections, but I find the 5-fold
>stuff of the "Treetoad" and "Pascal's Triangle of CPSs" easier to use.

[Paul]
>Carl, they're all 6-D projections.

[me]
>My own naive belief (based on what Wilson said to me,^1 and what little
>experience I have with these matters^2) is that the 5-fold projections
>are 2-D shadows of 6-D structures, with the 6-D structures rotated so
>they look the same as 5-D structures in 2-D. For example, you can rotate
>a 4-D cube so it looks like the shadow of 3-D cube in 2-D.

[Paul]
>Right.

[me]
>^1 Wilson called the 6-factor stuff with the 1 in the center of the
>pentagon, "cheating". If you look at the Pascal's triangle of CPSs,
>he uses the same projection for the x)5 CPSs as the x)6 ones.
>
>^2 The centered-pentagon stuff shares the 5-fold symmetry of the
>triakontahedron mapping and the Penrose tilings, which can all be
>projected, I believe I've read, from polytopes of no higher dimension
>than five.

[Paul]
>Unless you use the "shadow" trick you mentioned above!

[Dave]
>Good detective work re Dekany = Rectified Pentachoron. I take it that
>"rectified" means vertices in place of edge-midpoints and edges
>between what were midpoints of edges that shared a vertex.
>
>Notice that an octagon is a rectified tetrahedron.
>
>Here's a bit of a recap for readers who may have just tuned in.
>
>Take good ol' Pascal's triangle.
>
> 1
> 1 1
> 1 2 1
> 1 3 3 1
> 1 4 6 4 1
> 1 5 10 10 5 1
>1 6 15 20 15 6 1
>
>This gives us the number of ways of choosing N things from M. e.g.
>5choose2 and 5choose3 are both 10.
>
>As Erv Wilson realised, the musical application comes when the things
>we are choosing are small integers to be multiplied together to give the
>frequencies of the notes of a scale. Musically, the disadvantages
>probably outweigh the advantages of going beyond 6 factors, given the
>limitations of the human auditory system.
>
>Leaving off all the 1's around the outside and translating into musical
>terms (Wilson's) we get.
> dyad
> triad triad
> tetrad hexany tetrad
> pentad dekany dekany pentad
>hexad pentadekany eikosany pentadekany hexad
>
>To better understand the symmetries of each we want to translate them
>into regular geometrical objects. Leaving off the right half of the
>table (which is just a mirror image of the left) we get:
>
> dimensionality
> line-segment 1D
> triangle 2D
> tetrahedron octahedron 3D
> pentachoron rectified-pentachoron 4D
>hexa4ron ?pentadekany? ?eikosany? 5D
>
>Those on the left side of each row are also called "simplexes". e.g.
>the tetrahedron is the 3D simplex, the pentachoron is the 4D simplex.
>
>My guess is that the pentadekany is the rectified-hexa4ron. Where
>"rectified" means vertices in place of edge-midpoints, and edges
>between what were midpoints of edges that shared a vertex.
>
>And I guess the eikosany is the <something-else>-ified hexa4ron, where
><something-else>-ified means vertices in place of face-midpoints, and
>edges between what were midpoints of faces that shared an edge.
>
>Can anyone fill in the question marks below?
>
>Musical Number of Geometrical name(s)
>name vert edge face cell 4ron 5ron
>-----------------------------------------------------------------------
>Dyad 2 1 Line-segment, 1D-simplex
>Triad 3 3 1 Triangle, 2D-simplex
>Tetrad 4 6 4 1 Tetrahedron, 3D-simplex
>Hexany 6 12 8 1 Octahedron, rect-3D-simplex
>Pentad 5 10 10 5 1 Pentachoron, 4D-simplex
>Dekany 10 30 30 10 1 Rect-Pentachoron,
> rect-4D-simplex
>Hexad 6 15 20 15 6 1 Hexa4ron, 5D-simplex
>Pentadekany 15 ? ? ? ? 1 ?
>Eikosany 20 90 120 60? 10? 1 ?

[Paul]

>>>I am a musical theorist, and studying your page, I came to the
>>>realization that the Dispentachoron is a musical structure known as
>>>a "Dekany". There's an interesting musical structure known as an
>>>"Eikosany", which would be a 5-dimensional uniform polytope with
>>>
>>> 0 Pentachora
>>> 10 Dispentachora
>>> 30 Octahedra
>>> 30 Tetrahedra
>>> 120 Triangles
>>> 90 Edges
>>> 20 Vertices
>>>
>>>Do you know anything about this 5-dimension figure, particularly,
>>>any good ways of depicting its structure?
>
>George, I goofed. That should be 12 Dispentachora!
>
>>So in particular, your five-dimensional "eikosany" is the uniform
>>dodecatetron. It has twelve (not 10--please note correction; I
>>believe your other element counts are correct)
>
>You got me!!! And yes, the others are correct.
>
>>Its vertex figure is a triangular duoprism whose six cells are the
>>vertex figures of the six dispentachora that come together at each
>>vertex.
>
>uh-oh . . .
>
>>Whereas the ten vertices of a dispentachoron are located at the
>>midpoints of the ten edges of a regular pentachoron, the 20 vertices
>>of the uniform dodecatetron are located at the centroids of the
>>20 triangles of a regular hexatetron (the 5D simplex, or
>>tetrahedron-analog).
>
>I'm falling in a 5-dimesional chasm!
>
>>The Coxeter-Dynkin symbol
>>for the dodecatetron is
>>
>>o-----o----(o)----o-----o
>>
>>and its symmetry group is the hexatetric group [3,3,3,3], of order 720.
>
>720, huh? There are several musicians I know who will be interested in
>knowing that.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >>The Coxeter-Dynkin symbol
> >>for the dodecatetron is
> >>
> >>o-----o----(o)----o-----o
> >>
> >>and its symmetry group is the hexatetric group [3,3,3,3], of
order 720.
> >
> >720, huh? There are several musicians I know who will be
interested in
> >knowing that.

I'd be interested to know where this thread came from. Why am I
seeing this for the first time?

>> >>The Coxeter-Dynkin symbol
>> >>for the dodecatetron is
>> >>
>> >>o-----o----(o)----o-----o
>> >>
>> >>and its symmetry group is the hexatetric group [3,3,3,3], of
>order 720.
>> >
>> >720, huh? There are several musicians I know who will be
>interested in
>> >knowing that.
>
>I'd be interested to know where this thread came from. Why am I
>seeing this for the first time?

As I said, it came from the archives.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> This is indeed unclear from my primer,
> >
> >http://tonalsoft.com/td/lumma/carl-cps.htm
> >
> >> unless you happen to grok Wilson's diagram (the link
> >> to which is currently broken, but which can be seen at...
> >>
> >> http://lumma.org/tuning/13,14.gif
> >
> >thanks for pointing out the broken link. i've actually
> >simply added the diagram to the page now.
>
> Thanks monz! Now if you could just change the 6 in
> "6-dimensional object" to a 5 (that was my mistake)...

done.

-monz

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> So what does it mean for a tuning to have no 1/1?
>
> What's the difference between 1/1 and anything else? This has no
effect on
> the intervals, only on the pitches, so I don't understand why it is
of any
> significance at all,

You're absolutely right.

>And
> how should a piece of tuning software (or a synthesizer) that
accepts scala
> files as input behave when given a tuning that does not start with
1/1, and
> also, how should such tunings be speicified in scala?

By rotating them so that they do start with 1/1.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Carl,
>
>
>
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > This is indeed unclear from my primer,
>
> http://tonalsoft.com/td/lumma/carl-cps.htm
>
> > unless you happen to grok Wilson's diagram (the link
> > to which is currently broken, but which can be seen at...
> >
> > http://lumma.org/tuning/13,14.gif
>
>
>
> thanks for pointing out the broken link. i've actually
> simply added the diagram to the page now.
>
>
>
> -monz

Hey guys, maybe the same thing could be done for *my* CPS article,
since it keeps referencing one and only one diagram over and over?

Thanks,
Paul

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> [I wrote...]
> >Thanks monz! Now if you could just change the 6 in
> >"6-dimensional object" to a 5 (that was my mistake)...
>
> Actually, though these are apparently projections of
> 5-D polytopes, he is effectively getting 6 dimensions
> out of them.

What do you mean?

> Nobody's ever made a positive ID on these
> to my knowledge

Huh? What kind of ID did you have in mind? Certainly the polytopes
themselves have been ID'd, we had a mathematician versed in polychora
here for a little while . . .