Yahoo Tuning Groups Ultimate Backup tuning Re: Question on chords

> [Johnny Reinhard, TD 741.5]
>
> If the stasis of the overtone series does does not need to
> resolve, why should a straight ordering with the root in the
> bass need to resolve?

> [John deLaubenfels, TD 741.17]
>
> A question of degree, perhaps.

IMO, that's the most important point to make about this
discussion.

An idea that has developed slowly in the evolution of
my theories, mainly thru debate in this forum concerning
non-JI tunings, is this:

No matter *what* type of mathematical system is used for
a tuning, be it JI, ET, meantone, or any other method, there
seems to be a continuum of sonance, from perfect consonance
at one end (unison, 'octave', '5th', -->...) to an infinitude
of dissonance at the other end, and the *degree* of sonance
within this continuum seems to be indicated by the degree of
simplicity/complexity of numerical representation.

This idea has been well-developed in JI theory, particularly
in Partch's work, but doesn't seem to have been explored
very well in other tuning systems.

In addition, it seems to be a workable hypothesis when
describing individual intervals (i.e., dyads), but it has
become apparent that the situation concerning triads and
'higher-ads' is much more complicated, with many psycho-
acoustical factors coming into play which contradict
some of the results of numerical analysis.

-monz

Joseph L. Monzo San Diego monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
| 'I had broken thru the lattice barrier...' |
| -Erv Wilson |
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🔗Pierre Lamothe <plamothe@aei.ca>

Joseph L. Monzo wrote

<< No matter *what* type of mathematical system is used for
a tuning, be it JI, ET, meantone, or any other method, there
seems to be a continuum of sonance, from perfect consonance
at one end (unison, 'octave', '5th', -->...) to an infinitude
of dissonance at the other end, and the *degree* of sonance
within this continuum seems to be indicated by the degree of
simplicity/complexity of numerical representation. >>

I agree totally.

Besides, I have proposed a measure of that *degree* of sonance. For a
reduced ratio a/b, value of sonance would be

S = log2(ab) = log2(a) + log2(b)

Obviously, this mathematical definition exceeds largely what it can be
perceived (discrimination capacity). It's why I introduce a distinction
between partial ordered structure that can be perceived on sonance field
and total ordered structure that can be the most fruitfull.

It seems rather evident, in the first levels of Stern-Brocot tree, that we
can easily discrimine sonance of adjacent ratios and that this partial
ordered structure is accorded to perception. There exist similar trees
(with permutations on same level only) in which partial order is also
accorded to perception.

Among all possible total orders which can include partial order
perceivable, it seems that the abstract sonance field which corresponds to
mentionned definition is really the most congruent and fruitfull.

I have also weakly developed maths for tracking "invariant math index"
about chords under a set of forms like 3-5-9-15, 5-6-9-15, 6-9-10-15,
9-10-12-15, 10-12-15-18, 12-15-18-20 for min7 chord.

I wait to observe consensus on perception before to introduce actually
irrelevant abstract standpoint.

Pierre

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

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

> Besides, I have proposed a measure of that *degree* of sonance. For
a
> reduced ratio a/b, value of sonance would be
>
> S = log2(ab) = log2(a) + log2(b)
>
That is the same as Tenney's Harmonic Distance.

The problem is that it breaks down for high enough a and b (let alone
what happens for irrational ratios). For example, 30001:20000 is
clearly more consonant than 17:13, though S is much larger for the
first than for the second.

> Obviously, this mathematical definition exceeds largely what it can
be
> perceived (discrimination capacity).

Ah yes. That is why I use Tenney's Harmonic Distance (or some other
rational complexity measure) as a first step in defining harmonic
entropy, a sonance measure which takes perceptual discrimination
capacity into account, allowing a comparison of all types of
intervals,
rational or irrational.

> It's why I introduce a distinction
> between partial ordered structure that can be perceived on sonance
field
> and total ordered structure that can be the most fruitfull.

Could you please explain these concepts, with examples if possible?

🔗Pierre Lamothe <plamothe@aei.ca>

Paul

You wrote

<< The problem is that it breaks down for high enough a and b (let alone
what happens for irrational ratios). For example, 30001:20000 is
clearly more consonant than 17:13, though S is much larger for the
first than for the second. >>

It will be hard to find better example to illustrate that we have to pay
serious attention at theoric context. I don't propose sonance definition as
a measure in practical context. If 30001:20000 "is" very consonant, it's
clearly because 30001:20000 "is perceived" as 3:2 ratio. Rationality of
sonance, or rationality of width, are a matter of brain. 30001:20000 exists
in (abstract) space of acoustical measured values, but don't exist in brain
as category for sonance apprehension.

I've wanted to distinct partial order accessible by perception of total
possible orders to dissipate bunch of "malentendus" on historical
discussions about order of consonant tones. There is no total order of
sonance in music like quasi-total perceived order of width. What I expect
from a global sonance definition (compatible with perception) has been well
expicited: only perfect congruence and fruitfullness. With that, I gathered
many fruits like harmonical relations and matrix and found Stern-Brocot tree.

Besides, there is nothing more in (W,S) dyad than in (a,b) dyad. With
W=log2(a/b) and S=log2(ab), there is a bijection between them. None
information is lost or added. Context is purely mathematical.

You wrote (on partial and total orders)

<< Could you please explain these concepts, with examples if possible? >>

There is a total order on a set of elements if, for all pairs (a,b), we can
write a <= b or (non-exclusive) a >= b. Order is partial if the relation is
defined for only a subset of all pairs.

For example, I can translate and develop what I have soon written privately
in french. We can construct three total distinct orders on all pairs of
rationals in using criteria (ab) or (a+b) or max(a,b) for reduced ratio a/b
in comparaison relation. With criteria

(ab) 7/2 < 3/5 for (7x2) < (3X5)
(a+b) 7/2 > 3/5 for (7+2) > (3+5)
max(a,b) 7/2 > 3/5 for (7) > (5)

All rationals are completly ordered by one or other of these criteria. Two
ratios can be equal but all ratios are comparable. These total orders are
very different. Comparing 7/2, 3/5 et 7/1, we have (14)(15)(7) for (ab),
(9)(8)(8) for (a+b) and (7)(5)(7) for max(a,b).

But partial order of adjacent ratios in Stern-Brocot tree is compatible
with the three total orders defined. (As you know, adjacent ratios a/b and
c/d, on Stern-Brocot tree, are binded by relation ad - bc = 1).

What is important for music is what is perceived. But for mathematical
standpoint, congruence is important. The three criteria have interesting
properties. But the following examples showing octave translation pointed
to (ab) as best candidate for logarithmic transformation.

7/8 7/4 7/2 7/1 14/1 1/4 1/2 1 2/1 4/1

(ab) 56 28 14 7 14 4 2 1 2 4

(a+b) 15 11 9 8 15 5 3 1 3 5

max(a,b) 8 7 7 7 14 4 2 1 2 4

---------

Paul

I like very much exchange on the List, but for few days I will restraint my
participation to end text on Indian system.

What is written is yet on line. Text is in French, but figures, arrays and
equations are universal. So, you can look at section "Définition
universelle du shruti" for examples of S matrix which allows to express
clearly isomorphism of tones' class, and srutal basis in wich tones are
represented by unique coordinate vector. I hope to complete soon.

Pierre

🔗Carl Lumma <CLUMMA@NNI.COM>

🔗Joe Monzo <MONZ@JUNO.COM>

> [Pierre Lamothe, TD 743.13]
>
> What is important for music is what is perceived. But for
> mathematical standpoint, congruence is important. The three
> criteria have interesting properties. But the following examples
> showing octave translation pointed to (ab) as best candidate
> for logarithmic transformation.
>
>
> 7/8 7/4 7/2 7/1 14/1 1/4 1/2 1 2/1 4/1
>
> (ab) 56 28 14 7 14 4 2 1 2 4
>
> (a+b) 15 11 9 8 15 5 3 1 3 5
>
> max(a,b) 8 7 7 7 14 4 2 1 2 4
>

> [Paul Erlich, TD 744.3]
>
> Please explain why (ab) is the best candidate. It's certainly a
> very attractive one, in that it allows Tenney's geometrical
> interpretation of harmonic distance using a rectangular lattice
> with prime axes (including 2) and units along each axis
> proportional to the log of the corresponding prime.
>

Paul, isn't that explanatory last clause in your statement
precisely a description of my lattice formula? If it is (and
it seems to be to me), then doesn't my formula also 'allow
Tenney's geometrical interpretation of harmonic distance'?

I haven't read Tenney's work, and I haven't been following
this thread anywhere near as closely as I want to, as I feel
that it merits more attention than I have time for right now;
I hope to read what Pierre's written in French when I have
time to study it.

But that leapt out at me as being quite familiar.

-monz

🔗Joe Monzo <MONZ@JUNO.COM>

> [me, monz, TD 745.7]
>
> Paul, isn't that explanatory last clause in your statement
> precisely a description of my lattice formula?

Oops! No, it's not at all. My lattice formula uses 'a
rectangular lattice with prime axes (including 2) and units
along each axis proportional to the' *actual value* (*not*
the log) 'of the corresponding prime'.

Never mind... (that's what happens when you post at 2:31 am!)

-----

On the subject of communicating with Pierre:

I agree with Joe Pehrson that Pierre's less-than-perfect
attempts at English are not that bad, and that they certainly
have stimulated a lot of discussion among the mostly-English-
speaking (I'd guess probably over 95%) posters on this List.

But, that said, I encourage Pierre to provide elaborations
of his thoughts in French, so that he can expound unrestricted.
A few of us will be able to follow. Perhaps it would be best
if he provided bilingual posts.

Then again, considering the bandwidth limitations some users
are up against, maybe that's not a good idea...
votes for? votes against? ...

-monz