Looking for scales with "trivalence property"

Hi tuners.

Once I figure this out, maybe I'll start finding completely new tetradic chord progressions -- but I don't want to start saying something in particular before I crack down the algorithm. That's why I'm asking this.

Suppose we have some three variables called "x|y|z" and three logarithmic scale steps, in cents for example, called "l|m|s". The period of the scale in cents is equal to "x*l + y*m + z*s" cents. Let's pick up an example where the variables x|y|z are equal to 12, 7, and 3, respectively. Now I'm trying to find such a permutation of the three step sizes where the resulting scale has the "trivalence property". And I'm not sure which permutations satisfy this and which ones don't. Anyone here who could help?

Thanks in advance.

Petr

On Fri, May 25, 2012 at 4:00 PM, Petr Pařízek <petrparizek2000@yahoo.com> wrote:
> Now I'm trying
> to find such a permutation of the three step sizes where the resulting scale
> has the "trivalence property". And I'm not sure which permutations satisfy
> this and which ones don't. Anyone here who could help?
>
> Thanks in advance.
>
> Petr

What's the trivalence property?

-Mike

Mike wrote:

> What's the trivalence property?

Three different interval sizes in each interval class. An example of such a thing is the 5-limit 7-tone diatonic.

An analogy in the 2D field are scales with the "Mihill's property" (I'm not sure about the spelling now) where there are exactly two sizes in each interval class.

Petr

On Fri, May 25, 2012 at 4:10 PM, Petr Pařízek <petrparizek2000@yahoo.com> wrote:
>
> Mike wrote:
>
> > What's the trivalence property?
>
> Three different interval sizes in each interval class. An example of such a
> thing is the 5-limit 7-tone diatonic.
>
> An analogy in the 2D field are scales with the "Mihill's property" (I'm not
> sure about the spelling now) where there are exactly two sizes in each
> interval class.

Most rank-3 Fokker blocks don't have your trivalence properly: rank-3
Fokker blocks tend to come in 4 sizes. You're looking for a special
type of Fokker block called a wakalix. These blocks have the property
that they can temper down to MOS in 3 different ways: for instance,
the JI major scale, which is a wakalix, becomes an MOS if you
eliminate either 81/80, 25/24, or 138/125.

Gene's compiled quite a database of wakalixes. I'm not sure anyone's
undertaken the question of which xL ym zs combinations can or can't be
wakalixes. This diagram might be helpful

/tuning-math/files/KeenanPepper/wakalix.svg

Note that the same pitch set appears in three Fokker blocks. Keenan
had a nice graphic showing that if you put them all together you get a
trihexagonal tiling of the lattice, but it seems to have disappeared.
I'll have to make another one...

-Mike

Mike wrote:

> the JI major scale, which is a wakalix, becomes an MOS if you
> eliminate either 81/80, 25/24, or 138/125.

I think you meant 135/128.

... Corresponding to l-m, m-s, and l-s, in this particular case. You could equally well temper out "s" itself, for example.

> Gene's compiled quite a database of wakalixes. I'm not sure anyone's
> undertaken the question of which xL ym zs combinations can or can't be
> wakalixes.

Well, if there's a way to nail this down somehow, then we're onto something ... But I'll keep waiting. -- In what form does the database exist?

> This diagram might be helpful
>
> /tuning-math/files/KeenanPepper/wakalix.svg

Some piece of text would be possible? I suspect I won't go into trying to OCR this stuff.

Petr

You might look at the databases at:
http://www.lucytune.com/scales/

and search for matching patterns in various fields.

On 25 May 2012, at 21:00, Petr Pařízek wrote:

> Hi tuners.
>
> Once I figure this out, maybe I'll start finding completely new tetradic
> chord progressions -- but I don't want to start saying something in
> particular before I crack down the algorithm. That's why I'm asking this.
>
> Suppose we have some three variables called "x|y|z" and three logarithmic
> scale steps, in cents for example, called "l|m|s". The period of the scale
> in cents is equal to "x*l + y*m + z*s" cents. Let's pick up an example where
> the variables x|y|z are equal to 12, 7, and 3, respectively. Now I'm trying
> to find such a permutation of the three step sizes where the resulting scale
> has the "trivalence property". And I'm not sure which permutations satisfy
> this and which ones don't. Anyone here who could help?
>
> Thanks in advance.
>
> Petr
>
>

Charles Lucy
lucy@...

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can be found at:

http://www.lullabies.co.uk

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Gene's compiled quite a database of wakalixes. I'm not sure anyone's
> undertaken the question of which xL ym zs combinations can or can't be
> wakalixes.

I figured this out already; I thought you knew, Mike.

xL ym zs can be a triple wakalix if and only if either:

x=y, x=z, or y=z

OR

{x,y,z} = {1,2,4} (any permutation)

In other words, there are normal triple wakalixes where at least two kinds of steps occur the same number of times, and then there are super special triple wakalixes with 7 notes per period: aabacab.

Keenan

Keenan wrote:

> x=y, x=z, or y=z
> > OR
> > {x,y,z} = {1,2,4} (any permutation)

Woohoo, I think I'm gonna save this somewhere! :-) Thanks.

Petr

I wrote:

> Woohoo, I think I'm gonna save this somewhere! :-) Thanks.

BTW: If the x,y,z were 12,7,3, do you think it is also possible to satisfy the requirement then?

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> I wrote:
>
> > Woohoo, I think I'm gonna save this somewhere! :-) Thanks.
>
> BTW: If the x,y,z were 12,7,3, do you think it is also possible to satisfy
> the requirement then?

No, impossible.

Keenan

I wrote:

> > BTW: If the x,y,z were 12,7,3, do you think it is also possible to > > satisfy
> > the requirement then?
>
> No, impossible.

Is that because of the number of steps = 22? Or is that because of the coefficient ratios of 12:7:3? Or why is that?

Petr

On May 27, 2012, at 3:25 AM, Keenan Pepper <keenanpepper@...> wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Gene's compiled quite a database of wakalixes. I'm not sure anyone's
> undertaken the question of which xL ym zs combinations can or can't be
> wakalixes.

I figured this out already; I thought you knew, Mike.

xL ym zs can be a triple wakalix if and only if either:

x=y, x=z, or y=z

OR

{x,y,z} = {1,2,4} (any permutation)

In other words, there are normal triple wakalixes where at least two kinds
of steps occur the same number of times, and then there are super special
triple wakalixes with 7 notes per period: aabacab.

Keenan

Oh yeah! Man, that was a while ago.

-Mike

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> I wrote:
>
> > > BTW: If the x,y,z were 12,7,3, do you think it is also possible to
> > > satisfy
> > > the requirement then?
> >
> > No, impossible.
>
> Is that because of the number of steps = 22? Or is that because of the
> coefficient ratios of 12:7:3? Or why is that?

It's because it doesn't satisfy either of the criteria (two numbers or equal, or the numbers are a multiple of {1,2,4}).

Here we have three numbers which are all distinct, and are not a multiple of {1,2,4}, so it's impossible to have a max-variety-3 scale with those numbers of steps. All such scales have at least one generic interval class with 4 or more specific intervals.

Keenan

"3, 7, 12"
looks like Peter is trying to use some mystical set of numbers, which i totally sympathize with, or a big coincidence~
--

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

Steinhaus showed that with a single generator, on cannot have more than 3 size intervals, most of theserepeat two sets of intervals too.
--

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

Kraig wrote:

> "3, 7, 12"
> looks like Peter is trying to use some mystical set of numbers, which i
> totally sympathize with, or a big coincidence~

It's just popped in my mind because I've been experimenting with multiples of 25/24, 128/125, 81/80. If I'm not mistaken, 12l + 7m + 3s then gives 2/1.

Petr

--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:
>
> Steinhaus showed that with a single generator, on cannot have more than
> 3 size intervals, most of theserepeat two sets of intervals too.

Could you elaborate on this? I don't understand what Steinhaus showed.

Keenan

He's saying that a scale formed by a generator which isn't an MOS will have
three step sizes at most and never more than that. It's called the Three
Gap Theorem.

-Mike

On Jun 2, 2012, at 1:58 PM, Keenan Pepper <keenanpepper@...> wrote:

--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:
>
> Steinhaus showed that with a single generator, on cannot have more than
> 3 size intervals, most of theserepeat two sets of intervals too.

Could you elaborate on this? I don't understand what Steinhaus showed.

Keenan

Kraig wrote:

> Steinhaus showed that with a single generator, on cannot have more than
> 3 size intervals, most of theserepeat two sets of intervals too.

Except that here I wasn't using a single generator, nor was I aiming to get one.

Petr

that is fine. What property are you trying to get, what musical property does it give you you find interesting.

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> Kraig wrote:
>
> > Steinhaus showed that with a single generator, on cannot have more than
> > 3 size intervals, most of theserepeat two sets of intervals too.
>
> Except that here I wasn't using a single generator, nor was I aiming to get
> one.
>
> Petr
>

http://anaphoria.com/MOSSteinhaus.pdf

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, kraiggrady <kraiggrady@> wrote:
> >
> > Steinhaus showed that with a single generator, on cannot have more than
> > 3 size intervals, most of theserepeat two sets of intervals too.
>
> Could you elaborate on this? I don't understand what Steinhaus showed.
>
> Keenan
>

It seems there would be areas of Chalmers' Tritriadic scales that would fit in this area too.

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> that is fine. What property are you trying to get, what musical property does it give you you find interesting.
>
> --- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@> wrote:
> >
> > Kraig wrote:
> >
> > > Steinhaus showed that with a single generator, on cannot have more than
> > > 3 size intervals, most of theserepeat two sets of intervals too.
> >
> > Except that here I wasn't using a single generator, nor was I aiming to get
> > one.
> >
> > Petr
> >
>

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, kraiggrady <kraiggrady@> wrote:
> >
> > Steinhaus showed that with a single generator, on cannot have more than
> > 3 size intervals, most of theserepeat two sets of intervals too.
>
> Could you elaborate on this? I don't understand what Steinhaus showed.

Steinhaus conjectured the Three Distances Theorem:

http://demonstrations.wolfram.com/ThreeDistanceTheorem/

Basically one only has to find the set of any three numbers that add up to the number of tones one wants in the scale. for instance if we want 9 tones we have the following sets and their permutations
1 1 7, 1 2 6, 1 3 5, 1 4 4, 2 2 5, 2 3 4
that is the simple part.

what is easily overlooked is that in the case of 7 with the 1 1 5, 1 2 4, 1 3 3, 2 2 3 is that if one looks at the different unit cycles one will find examples of the other sets. for instance the tritriadic ones will always be of the 5 1 1 variety when we look at the cycle of 4 or 5 units.

there is no reason why one could not have a 22 tone scale with any of the permutations of 3 7 12 for L m S

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@> wrote:
> >
> > I wrote:
> >
> > > Woohoo, I think I'm gonna save this somewhere! :-) Thanks.
> >
> > BTW: If the x,y,z were 12,7,3, do you think it is also possible to satisfy
> > the requirement then?
>
> No, impossible.
>
> Keenan
>

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> there is no reason why one could not have a 22 tone scale with any of the permutations of 3 7 12 for L m S

Right... but all such scales have at least one generic interval category (number of steps) for which there are at least *four* specific intervals.

For example, the steps (1-step intervals) could be of only three kinds: L, m, and s, but the 2-step intervals of four kinds. Or the 1-step and 2-step intervals could both be only of three kinds, but the 3-step intervals of four kinds.

Petr's question was about scales for which *all* the interval classes have only three kinds, and it's impossible for that two happen with 3, 7, 12. (If you don't believe me, try to find a counterexample!)

Keenan

I couldn't see the importance of whether the three were just or not so i played with this one for example
i did with proportions .618 ,1, 1.618?
in cents
160.6
99.3
160.6
259.9
160.6
99.3
259.9
there would be a tendency between break down the 259.9 interval into two parts in which we have a type of phi pelog as a 7 out of 9 cycle.

one also has the other sets available which would lead to different sizes.

In the long run work with a set of proportions between the L m S would i think be more productive in actual sound in sounding good. seems best to have s+m > L
in ETs-
Erv's faux pelog in 12 et also produces scales of three different sizes. the same idea could be apply to all i imagine by having a single generator that varies between two sizes.

end
--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> It seems there would be areas of Chalmers' Tritriadic scales that would fit in this area too.
>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
> >
> > that is fine. What property are you trying to get, what musical property does it give you you find interesting.
> >
> > --- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@> wrote:
> > >
> > > Kraig wrote:
> > >
> > > > Steinhaus showed that with a single generator, on cannot have more than
> > > > 3 size intervals, most of theserepeat two sets of intervals too.
> > >
> > > Except that here I wasn't using a single generator, nor was I aiming to get
> > > one.
> > >
> > > Petr
> > >
> >
>