Tuning-Math Digests messages 5501 - 5525

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Message: 5501

Date: Wed, 6 Nov 2002 00:14:59

Subject: Re: Tree zoom duals

From: monz

----- Original Message ----- 
From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Tuesday, November 05, 2002 11:55 PM
Subject: [tuning-math] Re: Tree zoom duals


> --- In tuning-math@y..., "monz" <monz@a...> wrote:
> 
> > >
> > > What would be really cool is a 3D applet, which let you look at
> > > a 7-limit picture from various directions. We could have either
> > > ets as points, linear temperaments as lines, and commas as planes,
> > > or a dual picture with commas as points, linear temperaments
> > > again as lines, and ets as planes.
> > 
> > 
> > 
> > this is all stuff that i've always intended from the beginning
> > to have available in my JustMusic software.
> > 
> > JustMusic software,  (c) 1999 by Joseph L. Monzo *
> 
> is that really true? do you have any of the ideas above referenced 
> anywhere?



what's on the webpage and in the archives of the Yahoo JustMusic group
is all i've ever made public about it.  the rest of my ideas are in
folders and folders of notes and old BASIC code ... stuff that i
haven't even looked at in over 5 years.




-monz
(*really* wishing that i could find that Visual C++ programmer!!)


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Message: 5503

Date: Wed, 06 Nov 2002 09:35:51

Subject: Re: Tree zoom duals

From: Carl Lumma

>The high-octane ets crowd in towards the origin, so you zoom in
>to find them. The high-octane commas expand out to infinity, so
>you would need to zoom out to include them.

Rad!

In both cases, the unzoomed versions show me what I want to see!

-Carl


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Message: 5505

Date: Thu, 7 Nov 2002 14:49:51

Subject: About metrics

From: Pierre Lamothe

It seems there is confusion about metrics.

A metric measure something. One have to reflect clearly what is
supposed to be measured and for that, to distinguish first the
pertinent entities.

At microtonal level, i.e. independently of any interactional system,
one have to reflect, the most appropriately possible, what corresponds
to the perception.
  It is well known there exist two irreducible manner to order a set
of pitch height intervals. One could call that
    the ordinary or melodic order and the harmonic order.
  There exist in this way two distinct distances to be measured, one
on each ordered structure. One could call that
    the melodic distance and the harmonic distance.
At macrotonal level, i.e. when rules are added on how interact the
elements, one have to reflect the geometry of the created global
entity. In pure JI, one can simply use the Euclidean distance, once is
determined the quadric or hyperquadric defining the scalar product and
the orthogonality on that space, created by choice.


Microtonal level


There is a unique representation of a rational interval X in an
appropriate primal basis <pi>. 
  X = <pi> (xi) = (p0^x0) (p1^x1) ... (pN^xN)
I argue that the simplest appropriate distances between the intervals
X and Y are
  Ordinary or Melodic Distance 
    | sum ( ( yi - xi ) log pi ) |
  Tenney or Harmonic Distance
    sum ( | yi - xi | log pi )
I remark also that the harmonic distance concept has no sense in the
real segment of pitch heights.


Macrotonal level


Suppose one choose to use heptatonic modes in prime 5-limit. One knows
that the simplest val is [7 11 16], which is simply
  Round ( 7 [log 2 log 3 log 5] / log 2 )
Using X = (w,x,y) rather than (x0,x1,x2), that means each interval has
now a class or degree D(X) = 7w + 11x + 16y, or simply 4x + 2y (mod 7)
in the octave.That means also the set of unison vectors, i.e. all the
intervals having the degree 0 in the octave, is a sublattice of the <3
5> ZxZ lattice.

Is it sufficient to define a system ? No.

Defining a system implies equally the choice of the minimal unison
vectors around the unison which determines the fundamental domain of
that system. In a plane, like here, that implies the choice of the six
" nearest " unison vectors giving an hexagonal shape in which the
unison, at the center, is the unique interval having the degree 0.

Nearest here implies an undefined macrotonal metric. One have to
define a macrotonal distance permitting to say which among such
vectors are the nearest of the unison. There are different
possibilities. Each choice determine a distinct system.

Suppose one choose to get the " simplest ".

Can one only observe the skewness of possible hexagons to determine
that? No.

Simplest can only have a microtonal sense since the macrotonal metric
is not yet chosen. It's where the harmonic distance has to be used.
Comparing the harmonic norm (i.e. harmonic distance from unison) for
all intervals of each hexagons, it appears clearly here that the
simplest case corresponds to unison vector U = 81/80, V = 25/24,  U+V
and the three inverses. What is well-known to correspond at the
Zarlino system with modes using 16/15, 10/9 and 9/8 as steps.

An Euclidean space is defined as a vector space with a scalar product,
which in turn is defined by the choice of a bilinear form
corresponding to a quadric or hyperquadric. The appropriate choice to
reflect the geometric properties of such tonal systems is to fit the
quadric on the chosen minimal unison vectors.

The ellipse 
  3 x^2 + 13 y^2 + 3 xy = 49 
passes on the six chosen unison vectors. Using then the matrix M =
[(3, 3/2) (3/2, 13)], the scalar product
  (X|Y) = X M Y
define an appropriate structure of Euclidean space with the
correlative Euclidean distance, so there exist an orthonormal basis in
which the ellipse is a circle of radius 7, and the distance of (x,y)
from unison, for instance, is defined as sqrt (x^2 + y^2).


Pierre


[This message contained attachments]


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Message: 5506

Date: Thu, 07 Nov 2002 20:46:49

Subject: Re: from the realms of private correspondence

From: Carl Lumma

>>Also, still don't get how anything sensible could satisfy
>>property 1, or how my suggestion is a cop-out.
> 
>Ordinary Euclidean distance does.

I must not understand property 1.  If I have colinear
points a b c, how is the sum of AB and BC >= AC?

-Carl


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Message: 5507

Date: Thu, 07 Nov 2002 09:21:12

Subject: from the realms of private correspondence

From: Carl Lumma

[Paul wrote...]
>>>>You're wondering how to define the taxicab distance between
>>>>three points? The shortest path that connects the subgraph...
>>>>wouldn't punish all the dyads, and a fully-connecting path
>>>>would be equiv. to the sum of the dyadic Tenney HD... maybe
>>>>the area of the enclosed polygon... for a chord like 4:5:25,
>>>>this leads to the problem that you can only compare it to other
>>>>5-limit-only chords. Well, I give up.
>>> 
>>>actually, i figured this out once. it's proportional to the
>>>total edge length of the hyper-rectangle defined by the points.
>>>but that doesn't make it a metric!
>>
>>1. Can you give an example for 4:5:25?
>
>ok, in that case the hyper-rectangle collapses down to two
>dimensions, but the total edge length is unaffected. so no real
>problem there. the vertices of the hyper-rectangle are the
>pitches comprising the (non-octave reduced) euler genus whose
>factors are the notes in your chord.

Come again?  What points in the lattice does the rectangle
intersect?

>>>do you know what "metric" means?
>>
>>According to mathworld, it's a function that satisfies the
>>following:
>>
>>1. g(x,y) + g(y,z) >= g(x,z)
>>triangle inequality
>>
>>2. g(x,y) = g(y,x)
>>symmetric
>>
>>3. g(x,x) = 0
>>
>>4. g(x,y) = 0 implies x = y.
>>
>>1. Makes no sense to me. does Tenney HD satisfy it for dyads?
>
>once again, carl -- tenney HD is a metric for *pitches*, not
>dyads (notwithstanding your "cop-out").

When did you ever say that?  What does it mean?  What meaning
do pitches have in terms of concordance?

>>4. This would depend on the lack of chords sharing the same
>>set of factors. Without it, mathworld says we have a
>>"pseudometric".
>
>Tenney's HD is a metric, not a pseudometric. property 4 implies
>that any pitch has a zero harmonic distance from itself --
>that's all.

No, that's property 3.  Property 4 says any time you see zero
distance you measuring the distance from a pitch to itself.

Also, still don't get how anything sensible could satisfy
property 1, or how my suggestion is a cop-out.

>>>...with 5:3 v. 6:5, doesn't it seem wrong that adding span
>>>should be able to change the relative concordance (10:3 v.
>>>6:5)?
>>
>>not to me. we're not just adding span, we're also adding
>>complexity. ask george secor what he thinks.

Heya George... out there?  Do you find 5:3 or 6:5 more concordant?
Howabout 10:3 and 6:5?

-Carl


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Message: 5508

Date: Thu, 07 Nov 2002 20:51:08

Subject: Re: from the realms of private correspondence

From: Carl Lumma

>I say 5:3 is more consonant; my experience has indicated that 
>consonance of intervals between 1:1 and 2:1 can be ordered by
>the size of the product of the integers in their ratio

How long have you been using this product limit, may I ask?

>Howabout 10:3 and 6:5?
> 
>This one's a tougher call.  The product is the same, but the span
>is so different that we're beginning to compare apples with
>oranges.

That's my interpretation of span -- it dilutes concordance without
adding discordance.  But according to the product limit, we've
just added discordance to 5:3 by stretching it by an octave.

>Are we talking about comparing these with the lower tones being
>the same pitch, or the same upper tones, or an average of the two?
>You could even compare a 6:5 (with lower tone on middle C) with a
>6:5 two octaves lower, and I think you would agree that the lower
>one is more muddy, i.e., less consonant, so range of pitch also
>enters into the picture.

You haven't been talking to Dave Keenan, have you?

My feeling is that we want a measure that works when comparing
dyads in the same pitch range, and that product limit always
was such a measure.

>Now to answer your question, I think I would judge them to be
>equally consonant if the average pitch for each of the two
>intervals were the same.

Noted.  Thanks!

-Carl


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Message: 5511

Date: Thu, 07 Nov 2002 21:44:21

Subject: Fwd: Re: from the realms of private correspondence

From: Carl Lumma

> It is my hypothesis that this is one of the complications of
> dissonance that led to pitch grammars in which some intervals
> are understood to be dissonant, even if they are not
> especially rough, or lacking a strongly defined tone of
> reference/precedence.

Your hypothesis is that because psychoacoustic dissonance
varies from listener to listener, music had to develop a
grammar to make the perception of dissonance more... uniform?

-Carl


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Message: 5513

Date: Thu, 07 Nov 2002 22:17:59

Subject: Re: from the realms of private correspondence

From: Carl Lumma

>I don't understand how the term "limit" got into your question,
>or is this what others have called it?

It comes from that it can be used as an alternative to odd limit.
Interestingly, IIRC Paul showed odd-limit is as close to the
product thing as we can get in an octave-equivalent measure.  Is
that right, Paul?

>I've used the product of the integers for almost as long as I've
>been studying alternative tunings -- since the mid-1960s.

I got it from Denny Genovese, who was using it at least since
the mid 80's, and maybe since the mid 70's.  When did Tenney
come up with his HD?

>>That's my interpretation of span -- it dilutes concordance
>>without adding discordance.  But according to the product limit,
>>we've just added discordance to 5:3 by stretching it by an
>>octave.
> 
>Depends on which direction you stretch it.  Transpose the bottom
>tone downward and you'll add discordance, but transpose it upward
>and you won't add nearly as much

Transposing the bottom tone upward flips the ratio.  Perhaps you
meant the top tone upward?

Listening just now (in two registers), I find 10:3 less concordant
than 5:3, and 12:5 more concordant than 6:5.  6:5 suffers unfairly
perhaps from too little span.

>>You haven't been talking to Dave Keenan, have you?
>
>Not about this.  I picked this up from Herm Helmholtz, but have not 
>had the opportunity to discuss it with him either.

:)

-Carl


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Message: 5521

Date: Fri, 08 Nov 2002 20:05:54

Subject: Re: from the realms of private correspondence

From: Carl Lumma

> I haven't had any desire to delve too deeply into the all of the 
> variables that would be considered in a scientific treatment of
> the subject (too many other things to do) and am happy to leave
> that to others to pursue.  I gave my opinion only because you
> asked for it.  :)

For sure.  Thanks again.

-Carl


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Message: 5523

Date: Fri, 08 Nov 2002 20:15:13

Subject: Re: from the realms of private correspondence

From: Carl Lumma

>>In which case, I'd think property 4 is the only one my cop-out
>>doesn't meet, and Tenney HD also doesn't meet it.  If this
>>really isn't true, I'm hoping someone will refute it.
> 
>What's Tenney HD?

Tenney Harmonic Distance.  Note that it is only defined for
dyads.  I attempted to extend it to triads.  Paul claims that
in so doing, I removed its metric status.

>If you mean
> 
>|| 3^a 5^b 7^c || = |a/log(3)| + |b/log(5)| + |c/log(7)|
> 
>or something like that,

That's right, except that its |a * log(3)|... etc, and it
includes 2's.  So it is equivalent to just log(a*b) for a
dyad a:b in lowest terms.  I generalized it to log(a*b*c)
for a triad a:b:c.

>it is a vector space norm and automatically induces a metric
>by d(X, Y) = ||X - Y||

Not sure I follow that... 

-Carl


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