convex

Could someone explain to me what 'convex' means in relationship to scales or as a property.

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> Could someone explain to me what 'convex' means in relationship to
> scales or as a property.

Here's a non-rigorous explanation.

Plot your scale on a lattice with one dimension for each generator.

Think of a two-dimensional case. Draw a polygon that joins up the
outermost notes of the scale so there are no notes of the scale
_outside_ that polygon. Check that the polygon is convex, i.e. if it
has any indentations (concavities) then bridge across them until there
are no more. (e.g. a pentagram is not convex while a regular pentagon
is.) This is called the convex-hull of the set of points.

Now if there are any points _inside_ the convex-hull that do not
represent notes of the scale, then we say the scale is not convex.

In the 3-dimensional case you will be joining up planar faces between
triangles of points until you have a convex polyhedron. If there are
points inside that that are not in the scale then the scale is not
convex.

There are analogous conditions for more than 3 dimensions but of
course these are not directly visualisable.

-- Dave Keenan

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> Could someone explain to me what 'convex' means in relationship to
> scales or as a property.

Suppose you took a 5-limit JI scale with octave periodicity, so that the
notes of the scale are equivalent to a finite set of lattice points on
the 5-limit lattice of pitch classes. Given two lattice points, we can
draw a line segemnt between them, and convexity would require that
every lattice point on that line segment belongs to the scale.
Moreover, given three non-collinear points we can draw the triangle
with those verticies, and convexity requires that every lattice point
contained in the triagle is a part of the scale.

Here's World of Mathematics on convexity:

http://mathworld.wolfram.com/Convex.html

Here is Wikipedia:

http://en.wikipedia.org/wiki/Convex

If we have n p-limit pitch classes in the scale, t1, ... tn,
represented by ratios of odd integers, and if e1 ... en are
nonnegative real numbers such that e1+...+en=1, and if
t1^e1 * t2^e2 * ... * tn^en is a p-limit pitch class, then the scale
is convex iff all such pitch classes belong to it. We can if we like
deal with pitches rather than pitch classes with this definition.

The thread on convexity and Dave Keenan's thread on the definition of
just intonation overlap in an interesting way, because of the fact
that convexity is not preserved under tempering. If we take a septimal
hexany, for example 1-15/14-5/4-10/7-3/2-12/7, and project it via the
breed, or 2401/2400 temperament, down to a plane, the original
octahedron, a convex scale, is projected to a hexagon which is *not*
convex. However, if we adopt Keenan's ideas on what JI is, the two are
really the same JI scale. Convexity is not preserved by the fact that
in practical terms they are the same, because mathematically they are not.

We can take the convex closure of the scale thus projected, and get a
scale I've termed the octone:

1-15/14-49/40-5/4-10/7-3/2-12/7-7/4

Aside from convexity, the octone has the pleasant property of having
both an otonal and a utonal tetrad. In the tuning I gave, the utonal
tetrad is off by 2401/2400, but we can regard everything in sight as
being in 441 equal if we like, in which case there is no difference.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> The thread on convexity and Dave Keenan's thread on the definition of
> just intonation overlap in an interesting way, because of the fact
> that convexity is not preserved under tempering. If we take a septimal
> hexany, for example 1-15/14-5/4-10/7-3/2-12/7, and project it via the
> breed, or 2401/2400 temperament, down to a plane, the original
> octahedron, a convex scale, is projected to a hexagon which is *not*
> convex. However, if we adopt Keenan's ideas on what JI is, the two are
> really the same JI scale.

You've gone beyond my definition here. It only says they are both JI
scales. It doesn't say they are the _same_ scale.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> You've gone beyond my definition here. It only says they are both JI
> scales. It doesn't say they are the _same_ scale.

If you can't tell two scales apart, what would you call that?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > You've gone beyond my definition here. It only says they are both JI
> > scales. It doesn't say they are the _same_ scale.
>
> If you can't tell two scales apart, what would you call that?

I find it conceivable that you could not tell the difference between
the just intervals that the two scales have in common, but you might
be able to tell that some other interval is just in one scale (the
nanotempered one) but not the other.

But lets say they are indistinguishable. Surely then all the stuff
about non-convexity on projection, applies just as much to both of
them, or else one of them is _almost_ convex after projection. Is it
the projection or the nano-tempering, that causes the non-convexity?
In any case, no one should expect that convexity applies to the
goodness of musical scales in a black-and-white fashion.

Convexity (or near-convexity) is simply a consequence of wanting to
maximise connectivity, i.e. wanting to get as many consonant intervals
as possible with our few notes. And this is not because we value
consonance more highly than dissonance, but simply because consonance
is a "recessive gene". Dissonance is pretty much unavoidable in any
scale with more than about 6 notes, but consonance has to be selected for.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> But lets say they are indistinguishable. Surely then all the stuff
> about non-convexity on projection, applies just as much to both of
> them, or else one of them is _almost_ convex after projection.

It's a function of what you take the generators to be. The hexany is
in a lattice with generators {3,5,7}; however, we can take a lattice
with generators of (close approximations to) 49/49 and 10/7, and then
some scale which is audibly very close to the hexany, and identical to
it if we decide to tune both to 441 equal, will no longer be convex.
Considered as a set of notes in the circle of major thirds for 441,
the notes clearly are not contiguous and thus in that picture not convex.

We can take the same 441 notes to the octave and arrange them so that
they are in a Fokker-block like structure in three dimensions, and
look at everything in terms of {3,5,7}. Or we can arrange the notes
in two dimensions, with neutral thirds and 10/7 generators. Or we can
use neutral thirds and 27/25 (1/9 octave) generators, and wrap it
around. We can, in other words, impose a multitude of different
structures on the *same* 441 notes of the octave, and what structure
we choose will determine what we think is convex or not convex.

> Convexity (or near-convexity) is simply a consequence of wanting to
> maximise connectivity, i.e. wanting to get as many consonant intervals
> as possible with our few notes.

Exactly, and this remains true even in the very simple case of a MOS.
But we should not fail to note that there may be many different ways
to organize the same notes, leading to different theories about what
is convex, and hence different choices as to what seems optimal.

Hanson, who is the one who took the chain of 6/5s thought of 19 as a temperment as far as i know. Possibly Haverstick might have more on this.
JI is made of ratios relations and it stands to reason some where there will be relations to various ETs and likewise Erv has a few recurrent sequences that are indistinguishable for some ETs. ( we are talking hundredth of a cent. one gets 12 ET BTW
The point is each of these three is a continuum and at the extremes they all over lap.
I have tended to view the recurrent sequences , if formed by a series of whole number as JI.
These scales cannot be latticed at all as the generators, changes with each successive step.
i included it in the same way i tend to include much performance art as theater, in that it does more to progress theater , than to cleave it hanging out by itself. In fact it has influenced theater quite a bit.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> I have tended to view the recurrent sequences , if formed by a series
> of whole number as JI.
> These scales cannot be latticed at all as the generators, changes with
> each successive step.

Kraig,

Where can we read more about these "recurrent sequences"? Or can you
explain what they are, and give an example. They do sound like a
useful borderline case for what might be considered a JI scale.

hmmm, a long time ago, back in 1999 or so, I posted about 19 as a
tuning based on the highly concordant 3:4:5 triad . As a temperament
I'm not so sure how useful this 11-tone 2D scales that tempers out
the relatively small 15625/15552 is though ?

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> Hanson, who is the one who took the chain of 6/5s thought of 19 as
a
> temperment as far as i know. Possibly Haverstick might have more on
this.
> JI is made of ratios relations and it stands to reason some where
there
> will be relations to various ETs and likewise Erv has a few
recurrent
> sequences that are indistinguishable for some ETs. ( we are talking
> hundredth of a cent. one gets 12 ET BTW
> The point is each of these three is a continuum and at the
extremes
> they all over lap.
> I have tended to view the recurrent sequences , if formed by a
series
> of whole number as JI.
> These scales cannot be latticed at all as the generators, changes
with
> each successive step.
> i included it in the same way i tend to include much performance
art as
> theater, in that it does more to progress theater , than to cleave
it
> hanging out by itself. In fact it has influenced theater quite a
bit.
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "daniel_anthony_stearns"
<daniel_anthony_stearns@y...> wrote:

> hmmm, a long time ago, back in 1999 or so, I posted about 19 as a
> tuning based on the highly concordant 3:4:5 triad . As a temperament
> I'm not so sure how useful this 11-tone 2D scales that tempers out
> the relatively small 15625/15552 is though ?

It's very useful in the 5-limit, not so useful in the 7-limit where
you must choose between high 7-limit error or high complexity of
7-limit inervals. Even so it makes the grade as a 7-limit temperament
worthy of mention.

I believe Haverstick has been using this scale for years for one.
Same with the 15.
You are not one to pick out scales that can't be used.
all in all i do not particularly like 19 very much, but both these scales are probably the most interesting to me. thr 4-5-6 doesn't happen for me at all. Message: 5 Date: Tue, 31 May 2005 01:26:58 -0000
From: "daniel_anthony_stearns" <daniel_anthony_stearns@yahoo.com>
Subject: Re: Convexity and just intonation

hmmm, a long time ago, back in 1999 or so, I posted about 19 as a tuning based on the highly concordant 3:4:5 triad . As a temperament I'm not so sure how useful this 11-tone 2D scales that tempers out the relatively small 15625/15552 is though ?

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles