Constant Structures

Is there any general method for finding all or some of the constant
structures inside a finite set of tones which is not a constant
structure itself?

I'm aware of the opposite idea of filling out the gaps between
adjacent tones in a structure. That is what Partch did to get his 43-
tone scale. What if we instead of that omitted some of the tones to
get a constant structure?

I'm particularly interested in finding constant structures other than
hexanies contained in the 1-3-5-7-9-11-13-15 Hebdomekontany.

Wishing happier times,
Kalle

--- In tuning@y..., kalleaho@m... wrote:
> Is there any general method for finding all or some of the constant
> structures inside a finite set of tones which is not a constant
> structure itself?

Should be possible using PB theory.

>
> I'm aware of the opposite idea of filling out the gaps between
> adjacent tones in a structure. That is what Partch did to get his
43-
> tone scale. What if we instead of that omitted some of the tones to
> get a constant structure?

The most famous instance of this: if you choose only one of the pair
(11/10, 10/9), and only one of the pair (20/11, 9/5), you have a big
41-tone Constant Structures scale!

> I'm particularly interested in finding constant structures other
than
> hexanies contained in the 1-3-5-7-9-11-13-15 Hebdomekontany.

Most hexanies are _not_ constant structures. Perhaps you mean a
different concept?

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., kalleaho@m... wrote:
> > Is there any general method for finding all or some of the
constant
> > structures inside a finite set of tones which is not a constant
> > structure itself?
>
> Should be possible using PB theory.
>

How?

> >
> > I'm aware of the opposite idea of filling out the gaps between
> > adjacent tones in a structure. That is what Partch did to get his
> 43-
> > tone scale. What if we instead of that omitted some of the tones
to
> > get a constant structure?
>
> The most famous instance of this: if you choose only one of the
pair
> (11/10, 10/9), and only one of the pair (20/11, 9/5), you have a
big
> 41-tone Constant Structures scale!
>

Yes, but I was thinking about not filling the gaps att all! I was
thinking about constant structure subsets of different harmonic
structures like Tonality Diamonds and Combination Product Sets.

> > I'm particularly interested in finding constant structures other
> than
> > hexanies contained in the 1-3-5-7-9-11-13-15 Hebdomekontany.
>
> Most hexanies are _not_ constant structures. Perhaps you mean a
> different concept?

Really? Which ones are not?

According to Joe Monzo's tuning terms index Constant Structure is a
tuning system where each interval occurs always subtended by the same
number of steps.

Trying with Scala reveals that most hexanies in the scale
archive are indeed Constant Structures. The 1.3.5.9 hexany
for example not. The command used was
ITERATE "SHOW DATA" hexa*

One way of getting a CS subset is picking tones at a regular
distance. With a bit of trial and error I found a 31-tone
CS subset this way:
EQUAL 31
LOAD hebdome1 1
REPLACE/MODEL 1
This one is barely CS with a margin of 0.177 cents.

Manuel

--- In tuning@y..., kalleaho@m... wrote:
> --- In tuning@y..., paul@s... wrote:
> > --- In tuning@y..., kalleaho@m... wrote:
> > > Is there any general method for finding all or some of the
> constant
> > > structures inside a finite set of tones which is not a constant
> > > structure itself?
> >
> > Should be possible using PB theory.
> >
>
> How?

All JI CS scales seem to be PBs, and all "reasonable" PBs seem to be
JI CS scales. So one would look for various sets of unison vectors
that could "fit" inside your finite set of tones.
> > >
> > > I'm aware of the opposite idea of filling out the gaps between
> > > adjacent tones in a structure. That is what Partch did to get
his
> > 43-
> > > tone scale. What if we instead of that omitted some of the
tones
> to
> > > get a constant structure?
> >
> > The most famous instance of this: if you choose only one of the
> pair
> > (11/10, 10/9), and only one of the pair (20/11, 9/5), you have a
> big
> > 41-tone Constant Structures scale!
> >
>
> Yes, but I was thinking about not filling the gaps att all! I was
> thinking about constant structure subsets of different harmonic
> structures like Tonality Diamonds and Combination Product Sets.

OK -- I haven't heard too much of that going on. Mostly Erv Wilson
found CS supersets of CPS scales . . .
>
> > > I'm particularly interested in finding constant structures
other
> > than
> > > hexanies contained in the 1-3-5-7-9-11-13-15 Hebdomekontany.
> >
> > Most hexanies are _not_ constant structures. Perhaps you mean a
> > different concept?
>
> Really? Which ones are not?

Sorry -- it seems that most or all of the hexanies in the 1-3-5-7-9-
11-13-15 Hebdomekontany are CS. Wow! I was of how improper most of
them are, and of non-CS examples such as Robert Walker's hexatonic
scale (which required factors as high as 27, IIRC).
>
> According to Joe Monzo's tuning terms index Constant Structure is a
> tuning system where each interval occurs always subtended by the
same
> number of steps.

Correct. If you choose a scale at random, with no two intervals
identical to one another, it is thus CS.

Maybe Gene can help?

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> Trying with Scala reveals that most hexanies in the scale
> archive are indeed Constant Structures. The 1.3.5.9 hexany
> for example not.

Thanks, Manuel. I thought I was going crazy there for a moment.

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> Trying with Scala reveals that most hexanies in the scale
> archive are indeed Constant Structures. The 1.3.5.9 hexany
> for example not. The command used was
> ITERATE "SHOW DATA" hexa*
>
> One way of getting a CS subset is picking tones at a regular
> distance. With a bit of trial and error I found a 31-tone
> CS subset this way:
> EQUAL 31
> LOAD hebdome1 1
> REPLACE/MODEL 1
> This one is barely CS with a margin of 0.177 cents.
>
> Manuel

Thanks Manuel!

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Correct. If you choose a scale at random, with no two intervals
> identical to one another, it is thus CS.

> Maybe Gene can help?

I'm not sure what help you are asking for--to come up with an
algorithm for finding CS subscales, or for a definition which doesn't
allow highly random scales.