Re: Harmonic Entropy Concordance

Joseph,

In 13293 you wrote :

<< Why would this pair be so down the scale at 22-23, when the
4:5:6:7 is so very concordant it hardly beats at all! At
least in "hi fi play" mode?? The rankings *IN GENERAL* do
not sound like increasing discordance to me (with a few
exceptions) ... and I was just wondering why... (??) >>

I would like to open a new possible question about that. Since new Paul's
H.E. curve seems to establish a close correspondance with "perceptible
part" of Stern-Brocot tree, perhaps summing of dyadic harmonic entropy may
now be questionned.

I'll use here a numerical parallel. Attention, I'm not promoting with that
an alternative to Paul's theory. I hope it's clear that using of pure
numerical values for this parallel don't mean there is no need for a
smoothed upper-limited perceptive concept on which Paul is working. I'll
show how question arise in numbers before asking if relevant in Paul's
theory for ranking tetrads.

For each numerical tetrad a:b:c:d, we can associate the vector of 6 dyads
sonance (S(a:b), S(b:c), S(c:d), S(a:c), S(b:d), S(a:d)). Sonance is
logarithm in base 2 of complexity defined as n*d for irreductible fraction
n/d. (I use my proper terminology for I'm not sure if Tenny's distance is
the complexity or its logarithm).

I'll use for the following examples only complexities vector. (Sonances
vector can be obtained taking log values : lg(x) = ln(x)/ln(2))

4:5:6:7 == (4*5,5*6,6*7,2*3,5*7,4*7) = (20,30,42,20,6,28)
8:10:12:15 == (4*5,5*6,4*5,2*3,2*3,8*15) = (20,30,20,6,6,120)

We can see how factorization properties act here. Complexity of ratio Kn:Kd
is n*d and then reduced by square of common factor K. If all the values
a:b:c:d were relatively primes there could not exist reduction. In attempt
to conceive global "sonance" of a chord (and it's surely an aspect of the
perception) as a totalness applied to the 6 dyads sonance, question arise
on manner to reduce a six elements vector to give the best representative
scalar. Among possibilities we can note these sums :

arithmetic sum u + v + w ...

vectorial sum sqrt( u*u + v*v + w*w ... )

geometrical sum log u + log v + log w ...

harmonic sum 1/( 1/u + 1/v + 1/w ... )

Maybe there could exist in Paul's theory a choice inherent to the entropy
concept about which I have only a vague idea. This discussion has sense
only if it's not the case and if question would be possibly open.

Two properties of Stern-Brocot tree tend to point out distinct choices. The
first one is the fact that Stern-Brocot tree can be built, on the
logarithmic representation with distance concept acting on (w,s) dyads,
where w = (log n - log d) is the width and s = (log n + log d) is the
sonance of interval. Distance used is sqrt( w*w + s*s ). Starting with
(0,0) representing 1/1 and going up tracing for each dyad an arc towards
nearest left dyad and nearest right dyad, we consitute the Stern-Brocot
tree. This property don't exist on (n,d) dyads with same distance. On the
following representation of first octave subtree the thing is obvious
excepted for few cases resulting of compressed representation on y-axis.

http://www.aei.ca/~plamothe/pix/stern-brocot.gif

The other property is existence of characterized harmonic sums on
complexities as described, for example, in my post "Distribution variant"
at number 12928. Harmonic sum of complexities on each layer of Stern-Brocot
tree is equal to 1.

As phi joins additive and multiplicative properties (phi * phi = phi + 1),
Stern-Brocot tree appears a golden structure as a whole, blending additive
et multiplicative properties in such manner that we have often to consider
a complementary approach for each problem.

I leave theoritical considerations here waiting before to deepen to know if
relevant in ranking tetrads.

Returning to numerical example, a more empirical approach would ask how can
be balanced vectorial components ?

4:5:6:7 == (20,30,42,20,6,28) == (,,42,,28)
8:10:12:15 == (20,30,20,6,6,120) == (,,,6,120)

How is perceived a high dissonance isolated in remains consonant chord as
compared to less consonant but more balanced chord ?

Since I'm not musician and I don't have means to experiment myself I ask
simply the question.

Pierre Lamothe.

--- In tuning@egroups.com, Pierre Lamothe <plamothe@a...> wrote:

http://www.egroups.com/message/tuning/13315
>
> Joseph,
>
> In 13293 you wrote :
>
> << Why would this pair be so down the scale at 22-23, when the
> 4:5:6:7 is so very concordant it hardly beats at all! At
> least in "hi fi play" mode?? The rankings *IN GENERAL* do
> not sound like increasing discordance to me (with a few
> exceptions) ... and I was just wondering why... (??) >>
>
> I would like to open a new possible question about that. Since new
Paul's H.E. curve seems to establish a close correspondance with
"perceptible part" of Stern-Brocot tree, perhaps summing of dyadic
harmonic entropy may now be questionned.
>

I believe, at least insofar as "ordering of consonance" is concerned,
there is some question of Paul's method, as evidenced by the
experiment at the Tuning Lab:

http://artists.mp3s.com/artists/140/tuning_lab.html

All the tetrads are now "ordered" by, apparently, the exponential
Tenney method... and yet, there is no "perceptible auditory"
ordering... I don't hear it. Something is still wrong.

>
> arithmetic sum u + v + w ...
>
> vectorial sum sqrt( u*u + v*v + w*w ... )
>
> geometrical sum log u + log v + log w ...
>
> harmonic sum 1/( 1/u + 1/v + 1/w ... )
>

I see by this, Pierre, that you are saying there would be possibly
many different ways of "summing" the diadic concordances. Possibly
we just don't have the right one yet.

*OR* maybe we just don't have the correct mathematical model, yet.
Paul has been claiming that there should be a more complex model that
investigates the sonance of the tetrads as a whole... but it is
beyond the mathematical or programming expertise of the participants.

Well... I just went to music school, so I can't help! I'm hoping
that between *YOU* Pierre, Paul, David Keenan and John
deLaubenfels... people who really know some math, that we can get to
the "bottom" of this!....

>
> How is perceived a high dissonance isolated in remains consonant
chord as compared to less consonant but more balanced chord ?
>
> Since I'm not musician and I don't have means to experiment myself
I ask simply the question.
>

I'm not certain if you can hear the "experiment" at the Lab, Pierre
with your computer:

http://artists.mp3s.com/artists/140/tuning_lab.html

But, it seems to me that the tetrads with even *ONE* isolated
discordance seem more dissonant *OVERALL* than ones with more
"general discordance."

Anybody concur, disagree??
___________ ___ __ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <
josephpehrson@c...> wrote:
> >
> > arithmetic sum u + v + w ...
> >
> > vectorial sum sqrt( u*u + v*v + w*w ... )
> >
> > geometrical sum log u + log v + log w ...
> >
> > harmonic sum 1/( 1/u + 1/v + 1/w ... )
> >
>
> I see by this, Pierre, that you are saying there would be possibly
> many different ways of "summing" the diadic concordances. Possibly
> we just don't have the right one yet.

Taking the maximum is still another possibility.
However, remember that all of these possibilities will
leave 4:5:6:7 and 1/7:1/6:1/5:1/4 in a tie.
>
> *OR* maybe we just don't have the correct mathematical model, yet.
> Paul has been claiming that there should be a more complex model
that
> investigates the sonance of the tetrads as a whole... but it is
> beyond the mathematical or programming expertise of the
participants.

I'm guessing that for simple enough tetrads, the
product of the numbers in the lowest-terms
representation (e.g., 4:5:6, 10:12:15) will be the relevant
measure, with some smoothing coming into play
beyond the simplest tetrads.

> But, it seems to me that the tetrads with even *ONE* isolated
> discordance seem more dissonant *OVERALL* than ones with more
> "general discordance."

Well, that would put the major seventh chord very low
in the rankings, against my perceptions. But looking at
the most discordant interval in a chord seems to have
made some sense to Partch, who used odd-limit, or
something very similar to octave-invariant harmonic
entropy, as his measure of intervallic discordance. In
fact, I used this definition when making this post: http://
www.cix.co.uk/~gbreed/erlichs.htm, which led Graham
Breed to make the following page: http://
www.cix.co.uk/~gbreed/ass.htm, which you should
read now. Basically, this would suggest that the most
concordant tetrads would be ranked as follows (with all
four inversions of each chord implicitly equal):

1.
4:5:6:7
1/7:1/6:1/5:1/4

2.
all 4-note subsets of 5:6:7:8:9 that include 9
all 4-note subsets of 1/9:1/8:1/7:1/6:1/5 that include 1/9
10:12:15:18
12:14:18:21

3.
all 4-note subsets of 6:7:8:9:10:11 that include 11
all 4-note subsets of 1/11:1/10:1/9:1/8:1/7:1/6 that
include 1/11
22:24:33:36

4.
all 4-note subsets of 7:8:9:10:11:12:13 that include 13
all 4-note subsets of 1/13:1/12:1/11:1/10:1/9:1/8:1/7 that
include 1/13
24:26:36:39

and the major seventh chord would be below all of
these . . .

Joseph, why don't you tell us what you think the most
glaring errors are in the ranking . . .

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13344

In
> fact, I used this definition when making this post: http://
> www.cix.co.uk/~gbreed/erlichs.htm, which led Graham
> Breed to make the following page: http://
> www.cix.co.uk/~gbreed/ass.htm, which you should
> read now.

Actually, I do remember this website where Graham Breed covers his
"ass" [Anomolous Saturated Suspensions]

Basically, this would suggest that the most
> concordant tetrads would be ranked as follows (with all
> four inversions of each chord implicitly equal):
>
> 1.
> 4:5:6:7
> 1/7:1/6:1/5:1/4
>
> 2.
> all 4-note subsets of 5:6:7:8:9 that include 9
> all 4-note subsets of 1/9:1/8:1/7:1/6:1/5 that include 1/9
> 10:12:15:18
> 12:14:18:21
>
> 3.
> all 4-note subsets of 6:7:8:9:10:11 that include 11
> all 4-note subsets of 1/11:1/10:1/9:1/8:1/7:1/6 that
> include 1/11
> 22:24:33:36
>
> 4.
> all 4-note subsets of 7:8:9:10:11:12:13 that include 13
> all 4-note subsets of 1/13:1/12:1/11:1/10:1/9:1/8:1/7 that
> include 1/13
> 24:26:36:39
>
> and the major seventh chord would be below all of
> these . . .
>

Well, this almost seems like an almost "obvious" way to go about
things... The question I have at this moment regards "odd" as
contrasted with "prime" limit.

If I am understanding this properly, using an "odd" limit
classification would allow INDIVIDUAL dissonant intervals to appear
in overall lower limit consonances, whereas, the "prime" limit would
pretty much "level out" the consonances to a greater degree of
similarity... and would not allow the odd (literally) dissonance to
be incorporated...

Am I getting that at all.. or is that off??

Oh... and yes, I am going to go over the tetrads... but I can't do it
at home with "low-fi." You know how THAT sounds!! (At least for THIS
exercise...)
______________ ____ __ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <
josephpehrson@c...> wrote:

[snip]

> Well, this almost seems like an almost "obvious" way to go about
> things...

But does it agree with your perceptions?

The question I have at this moment regards "odd" as
> contrasted with "prime" limit.
>
> If I am understanding this properly, using an "odd" limit
> classification would allow INDIVIDUAL dissonant intervals to appear
> in overall lower limit consonances, whereas, the "prime" limit
would
> pretty much "level out" the consonances to a greater degree of
> similarity... and would not allow the odd (literally) dissonance to
> be incorporated...

Quite the opposite -- maybe the above wasn't so
obvious after all. For example, the chord 400:512:625 is
5-prime limit, but contains _only_ dissonant intervals
(by any reasonable definition). Meanwhile, the
saturated chords and otonality and utonality subsets
listed above don't have _any_ dissonant intervals within
their respective odd limits.

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13385

> Quite the opposite -- maybe the above wasn't so
> obvious after all. For example, the chord 400:512:625 is
> 5-prime limit, but contains _only_ dissonant intervals
> (by any reasonable definition). Meanwhile, the
> saturated chords and otonality and utonality subsets
> listed above don't have _any_ dissonant intervals within
> their respective odd limits.

OK. So then could you please present some kind of summary, if it is
possible, that would help me understand the idea of "prime" as
contrasted with "odd" limit as it applies to generalized concordance
and discordance?? Or does that not pertain??

____________ ___ __ _
Joseph Pehrson