Yahoo Tuning Groups Ultimate Backup tuning Heuristic approach of coherence

HEURISTIC APPROACH OF COHERENCE

[ As preamble, it would be CLEAR that all what is written here has nothing
to do with practical tuning problems. Theme is purely COHERENCE.
Transposition, comma problems or enharmony haven't to emerge at this level.
In Descartes tradition, splitting contributes to understanding. I'll be
glad to debate at coherence level, but it would be too heavy for me to
discuss about what someone may presume as consequences for tuning. That
implies much more concepts. I use also results of Paul H. Erlich's theory
as starting point. I hope having catched the general ideas in my synthetic
references. If not, I assure that it wouldn't a disguised criticism and
that it would be due only to insufficient reading. ]

At first glance (I mean in apparence), harmonic entropy theory seems to
have few to say about scales structure since it concerns mainly smoothing
and upper limitation of dissonance curve on intervals continuum. It appears
like a "realistic" justification of Euler's principle about trend to
perceive intervals near simple ratio as this simple ratio. But curves
exhibit severe limitation at only few exceptional attractors among ratios.
On what, (Width, Sonance) logarithmic representation of Stern-Brocot tree
was given a congruent anticipation. So entropy theory seems to half-open
the door to introduce heuristic approach of COHERENCE. It's a basic concept
of macrotonal theory about scales structure.

In gammier theory, Width and Sonance, concerning intervals, are distinct
dimensions. The way mathematicians link the two (partially ordered)
perceptions, using irrationals, is a pitfall for representation. It's only
in mathematical heaven that Width and Sonance belong to same dimension.
This reduction, that implies infinity, is absolutely out of perception
possibilities.

Keeping in mind the irreductibility (by mean of perception) of the two
dimensions, we can observe, in chosen set of tones, that tones have well
distinct bidimensional comparitivity profiles. Since tones 9/8 and 10/9 are
in slow slope dissonance curve, they are more easily ordered as Width than
as Sonance. (However, choice between 9/8 and 10/9 in harmonic context
implies comparison with other tones, as we'll see, and then difference
intervals may be easily ordered as Sonance.) On the other hand, the most
consonant ratios have, inversely, better comparativity as Sonance than as
Width, mainly for exceptional reinforcements like it appears on dissonance
curve and also on representation in bidimensional tonal spaces.

Adding now a bit of macrotonal ingredients, we can build a gammier
structure in heuristic way. (I would like to emphasize that gammier theory
don't stand out a similar approach but have a much deeper mathematical
foundation. It's just much easier to look at gammier, in first approach,
without maths.)

Before starting to build a global COHERENT structure, we could look at an
intermediate level, how coherence question may arise with a known simple
example implying chordal concordance. I note that concordance measuring
approach of tetrads in entropy theory sensibly address similar problem.
I'll just add, in this first example, a tonal context (and not a global
context). I won't use here measured values but it will be easy to make
parallel with tetrads ranking problem and see why talking about concordance
rather than dissonance is required. Even if the term "concordance" I use in
French, as mathematical concept, is different of the first one, word
similarity is not fortuitous. Collective properties of tones are concerned
in both cases (even if level changes). Such properties can't be seen as
simple scalar sum of individual properties. Interaction is the dominant
factor.

Let C=(1) be the tonic in a set of tones where we find also F=(4/3),
G=(3/2), A=(5/3), B=(15/8). Which of D=(10/9) or D=(9/8) would be more
concordant ?
We have seen that 9/8 and 10/9 can hardly be ordered relatively to Sonance.
Is this fact implies difficulties to choose them in harmonic context ? Let
us compare the two chords (D F A) and (G B D). We seek for the best
concordant X in the two following cases, (X 4/3 5/3) and (3/2 15/8 X). We
have here, successively,

(tones) == (ratios) == (complexities vector)

(10/9 4/3 5/3) == (10:12:15) == (30,20,6)
( 9/8 4/3 5/3) == (27:32:40) == (864,20,1080)
( 3/2 4/3 20/9) == (108:135:160) == (20,864,1080)
( 3/2 4/3 9/4) == (4:5:6) == (20,30,6)

Obviously choice of D depends of chord choice.

Minor tone 10/9 is clearly more concordant in given minor triad.
Major tone 9/8 is clearly more concordant in given major triad.

If interaction is important at chord level let imagine how interaction of
tones and chords could be important at global level.

We can now facing global level. I recall it's an heuristic process and that
we have shown quickly but clearly, I hope, that tones may have very
different bidimensional comparativity profiles.

To build our structure, let us begin with choice, after {1}, of the four
best consonances in first octave {5/4, 4/3, 3/2, 5/3}. We understand easily
that harmonic interactions implying these tones will be the most
determining of all context (due to extreme reinforcement). Besides, we
understand easily that limiting free combinatory of these tones would be an
artifice (possibly relevant only at stylistic level). Now, before adding
new tones, let us see which tones are soon implied by total free use of the
four best consonances.

After {1} and {5/4, 4/3, 3/2, 5/3} we see minimaly that the three steps in
the series are forcely soon implied. And we understand immediately that the
comparativity profile of these intervals are very different. The steps
{16/15, 9/8, 10/9} appears mostly as Width than as Sonance. And here, not
only the tones can be ordered, but 16/15 appears easily in Width as half of
9/8 or 10/9. The approximative Width relation 4-5-7-9 (12-tET) for the best
consonances is less immediate than 1-2 for the steps. Using more than an
octave implies also all inverses : {8/5, 6/5} and {15/8, 16/9, 9/5}. Since
we have soon 13 tones, let us imagine that it could be complete and let us
see how musical modes could be related to this chosen context.

The tones collected are :

{1, 16/15, 10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, 9/5, 15/8}

Let us define a musical mode as contiguous unidirectional path through
octave. This simple definition has no sense without context because, and
only because, the term contiguous require a context. We have to determine
previously what is contiguous and what is not. In a previously chosen
context like our set of 13 ratios, contiguous is obviously defined. Some
tones can't be divided by lower tones in the set. Excepted {1}, these tones
are the prime tones of the set, and a contiguous path has only steps pick
out among the prime tones. Here we have only {16/15, 10/9, 9/8} as
contiguous steps.

Analysing all possibilities, we find 16 modes in this context. All these
modes have 7 degrees. Much of these modes are well-known like the dual
Doristi diatonic mode (main mode of Aristoxenus System presented here in
its original descending form) and Zarlino major mode.

2 16/9 8/5 3/2 4/3 6/5 16/15 1

1 9/8 5/4 4/3 3/2 5/3 15/8 2

Something seems magic with that and arbitrary. As heuristic process, it's
effectively arbitrary. It's easy to misunderstand, by pure impression, that
Zarlino context is promoted. Zarlino context has no privilege in the
theory. Also, it's not easy to detect what are the operating concepts and
how they could permit to build much more contexts.

I could soon pointed out these concepts but I'm not certain that heuristic
way is the good one on the List. I'll wait for feedback before to progress
in that way. If bells won't soon be ringing it will be better to proceed
more formally or yet with questions like "What is a musical scale ?".

I would like to terminate with an aspect of all musical contexts described
by the (gammier) theory and applied here to our 13 ratios. There exist a
matrix analog to Partch's diamonds for each COHERENT context. But
generators are not limited by the concept of N-limit. Not only this matrix
gives the ratios, but the algebric composition law on these ratios is
totally defined by this matrix. I give here two matrix. The first one
generates more easily the 13 ratios and the (factorization) lattice of
musical modes. This matrix generate the context as LARGE CHORDOID and as
such don't give the complete composition law. The second matrix generates
the same context as CHORDOID and then defines the complete composition law.
The first matrix is ordered to allow direct reading of the lattice (degrees
are on diagonals). The second matrix is ordered to show constitutive
harmonic properties.

15 1 9 5

15 1 16/15 6/5 4/3

1 15/8 1 9/8 5/4

9 5/3 16/9 1 10/9

5 3/2 8/5 9/5 1

9/8 15/8
/ \ / \
/ 5/4 5/3 \
/ / \ / \ \
1 - 10/9 4/3 - 3/2 9/5 - 2
\ \ / \ / /
\ 6/5 8/5 /
\ / \ /
16/15 16/9

1 5 3 15 9

1 1 5/4 3/2 15/8 9/4

5 4/5 1 6/5 3/2 9/5

3 2/3 5/6 1 5/4 3/2

15 8/15 2/3 4/5 1 6/5

9 4/9 5/9 2/3 6/5 1

I imagine that most members can appreciate the coming closer of Partch and
constitutive classical harmony through this diamond a few magically built.

Pierre Lamothe

🔗Pierre Lamothe <plamothe@aei.ca>

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Pierre Lamothe <plamothe@aei.ca>

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

🔗Pierre Lamothe <plamothe@aei.ca>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

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

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

> (10/9 4/3 5/3) == (10:12:15) == (30,20,6)
> ( 9/8 4/3 5/3) == (27:32:40) == (864,20,1080)

Those look correct, but . . .

> ( 3/2 4/3 20/9) == (108:135:160) == (20,864,1080)
> ( 3/2 4/3 9/4) == (4:5:6) == (20,30,6)

I believe you mean 15/8 instead of 4/3 in the
two chords above.

> To build our structure, let us begin with choice, after {1}, of the
four
> best consonances in first octave {5/4, 4/3, 3/2, 5/3}.

Those would be the four best concordances
against the 1/1 in the octave above it.

Or if you also went into the octave below,
the four best concordances against the 1/1
would be an octave below 6/5, 4/3, 3/2,
and 8/5. If you then allowed octave-
equivalence, you have a total of seven tones
per octave (this seems to help lead to what
you do below -- let me know if it is a
misinterpretation).

>
> After {1} and {5/4, 4/3, 3/2, 5/3} we see minimaly that the three
steps in
> the series are forcely soon implied.

I don't understand this. Why do the first five
tones (or seven, if you go into the lower
octave as well) get chosen only based on
their concordance against the 1/1, and not
based on their concordance with one
another, while subsequent tones are not
chosen based on their concordance with 1/1
but only based on their concordance with
other tones? It seems you need to justify
your choice of the number five (or seven) as
a delimiter between the two stages of scale
formation. It certainly would seem hard to
justify based on considerations of the history
of musical practice. (In case anyone's
confused, the five note scale
1/1 5/4 4/3 3/2 5/3
and the seven note scale
1/1 6/5 5/4 4/3 3/2 8/5 5/3
have nothing to do with the pentatonic and
heptatonic scales of historical importance --
neither Pierre nor I are suggesting that they
do).

And we understand immediately that the
> comparativity profile of these intervals are very different. The
steps
> {16/15, 9/8, 10/9} appears mostly as Width than as Sonance. And
here, not
> only the tones can be ordered, but 16/15 appears easily in Width as
half of
> 9/8 or 10/9. The approximative Width relation 4-5-7-9 (12-tET) for
the best
> consonances is less immediate than 1-2 for the steps.

I'm having trouble understanding the
implications of Width-Sonance Comparativity
(?).

Using more than an
> octave implies also all inverses : {8/5, 6/5} and {15/8, 16/9,
9/5}. Since
> we have soon 13 tones, let us imagine that it could be complete and
let us
> see how musical modes could be related to this chosen context.
>
> The tones collected are :
>
> {1, 16/15, 10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 16/9, 9/5, 15/8}
>
> Let us define a musical mode as contiguous unidirectional path
through
> octave.

Now we're on to a third stage -- first there
was the stage where seven tones were
determined due to their concordance with
1/1; then there was a stage where six more
tones were added due to their concordance
with the existing tones (and I don't see a
rigorous rule for this choice, though Partch
used it illustratively and Pierre probably has
a Width-Sonance Comparitivity rule for it
(???)); now there is a stage where one
chooses a path through the thirteen tones
we have so far. I might be comfortable with
this degree of complexity of evolution if it
corresponded with a sequence of known
desiderata in the history of musical practice,
but as it stands, I can as yet see no
justification for such contortions. :) :) :)

> This simple definition has no sense without context because, and
> only because, the term contiguous require a context. We have to
determine
> previously what is contiguous and what is not. In a previously
chosen
> context like our set of 13 ratios, contiguous is obviously defined.

Some
> tones can't be divided by lower tones in the set.

What exactly does that mean?

> Excepted {1}, these tones
> are the prime tones of the set, and a contiguous path has only
steps pick
> out among the prime tones.

Prime tones? Don't get it.

Here we have only {16/15, 10/9, 9/8} as
> contiguous steps.

You've lost me.

>There exist a
> matrix analog to Partch's diamonds for each COHERENT context. But
> generators are not limited by the concept of N-limit.

Well, Partch derived the seven-tone scale
1/1 6/5 5/4 3/2 4/3 8/5 5/3
just the same way you did (if I didn't
misintepret you) and called it the 5-limit
diamond. These are the tones that are
consonant (within the 5-limit) against 1/1. All
of Partch's diamonds come from choosing
certain minimum level of consonance against
the 1/1, as defined by odd-limit. A use of
the Partch diamond construction without the
odd-limit, hence without a minimum
consonance requirement against the 1/1,
would require a whole new raison d'etre.

Not only this matrix
> gives the ratios, but the algebric composition law on these ratios
is
> totally defined by this matrix.

I'd like to understand what you mean by
this.
>
> 15 1 9 5
>
> 15 1 16/15 6/5 4/3
>
> 1 15/8 1 9/8 5/4
>
> 9 5/3 16/9 1 10/9
>
> 5 3/2 8/5 9/5 1

That's neat, and I'm sure Partch would have
enjoyed seeing that work out too, but I
don't see the principle behind it. Is there one,
or is the set 15 1 9 5 simply reverse-
engineered to produce your pre-determined
13-tone set?

>
>
> 9/8 15/8
> / \ / \
> / 5/4 5/3 \
> / / \ / \ \
> 1 - 10/9 4/3 - 3/2 9/5 - 2
> \ \ / \ / /
> \ 6/5 8/5 /
> \ / \ /
> 16/15 16/9
>
Do I understand that this shows the 16
possible contiguous paths (whatever that
means)? Does this display some relationship
with the 4-by-4 matrix above it?
>
> 1 5 3 15 9
>
> 1 1 5/4 3/2 15/8 9/4
>
> 5 4/5 1 6/5 3/2 9/5
>
> 3 2/3 5/6 1 5/4 3/2
>
> 15 8/15 2/3 4/5 1 6/5
>
> 9 4/9 5/9 2/3 6/5 1

Okay, so the set 1 3 5 9 15 produces the
same octave-reduced scale as the set 1 5 9 15
when the Partch diamond construction is
used. That's comforting, but I still don't
know what the principle behind it.

If you'd like to learn my approach to
understanding the "coherence" of the
diatonic scale, in the evolutionarily successive
contexts of pure melody (ancient Greece and
Gregorian Chant), triadic harmony
(Renaissance), and tonal music (Baroque-
Romantic), read the first part of my paper
(_Tuning, Tonality, and Twenty-Two Tone
Temperament_) -- read the stuff _before_
where I introduce decatonic scales in 22-tET
(as a possible future direction).

Also see my _Gentle Introduction to Fokker
Periodicity Blocks_ for a much more direct
way (than what you seem to be presenting
above) of coming to strict-JI diatonic scales
like your "Aristoxenian" and "Zarlinean"
examples (odd names because both men
seemed to be dissatisfied with strict JI) using
essentially only these two assumptions:

(1) The tones form 5-limit consonances with
_one another_.

(2) The process of constructing more and
more tones through successive 5-limit
consonances stops when the melodic interval
between any two tones becomes too small.

Please forgive me for, and help me correct
any misunderstandings I may have
committed above. I'll make an effort to be
open to and absorb any further material you
wish to show me. I hope you are able to look
at my two papers above and I'll try to
elaborate (in French, if necessary) on any
elements that are unclear to you.

🔗Pierre Lamothe <plamothe@aei.ca>

----------------------------------
HEURISTIC OF COHERENCE DISCUSSION
On Paul H. Erlich message # 14331
Answer - Part II
----------------------------------

Quoting message 14259 (14261) :

<< After {1} and {5/4, 4/3, 3/2, 5/3} we see minimaly that
the threesteps in the series are forcely soon implied. >>

Paul wrote (14331) :

<< I don't understand this. Why do the first five
tones [..snip..] get chosen only based on
their concordance against the 1/1, and not based
on their concordance with one another,
while subsequent tones are not chosen based
on their concordance with 1/1 but only based on
their concordance with other tones? >>

(1) I've soon begin to answer in 14333 message.

(2) I recall it's an arbitrary starting point. There exist a very large
space of coherent structures (gammiers). Here is an heuristical way to
build one of them without mathematical axioms.

(3) The values {1/1, 5/4, 4/3, 3/2, 5/3} are not chosen simply for their
actual relation to 1/1 but mainly for the potential to have between them
great concordance. We have here, reordered, the Maj9 chord

2/3__5/6__1/1__5/4__3/2 == 8:10:12:15:18

which contains much subchords like 2 major triads 4:5:6, 1 minor triad
10:12:15, Maj7 and min7 tetrads 8:10:12:15 and 10:12:15:18, etc.

(4) Once free use principle is applied to these 5 tones, the subsequent
tones are not chosen based on their concordance with these 5 tones but they
are implied as difference tones between them, without regard to their
position in dissonance curve. They are simply there, without direct choice
(choice is free use).

It's why I talk about diversity in kind of tones in a coherent set. (I
refer to comparativity profiles in witdth and sonance dimensions). It could
be particularly incoherent to applied free use principle to difference
tones like {16/15, 10/9, 9/8}. These tones act, in the chosen set, above
all as width. When they are used in chords, we have seen, with the example
of choice between 9/8 and 10/9, that their proper sonance qualities are not
determinant : concordance discrimination arise from difference tones
sonance qualities. What they need is coherent qualities like having common
factors with other tones of the set.

--------

Paul wrote (14331) :

<< It seems you need to justify your choice of the
number five (or seven) as a delimiter between the
two stages of scale formation. It certainly would
seem hard to justify based on considerations of
the history of musical practice. (In case anyone's
confused, the five note scale 1/1 5/4 4/3 3/2 5/3
and the seven note scale 1/1 6/5 5/4 4/3 3/2 8/5 5/3
have nothing to do with the pentatonic and heptatonic
scales of historical importance -- neither Pierre nor
I are suggesting that they do). >>

(1) You're right saying there is no consideration of the history of musical
practive. Practice implies tuning problems which are not in the scope of
this discussion. I add also that the << five note scale >> and the << seven
note scale >> are not, here, scales at all. The five notes are a chord and
the seven notes are a simple subset of the 13.

(2) The number 5 is not a theoric delimiter. The 13 tones generated are the
set of tones corresponding to gammier 9 generated by the chord (1 3 5 9
15). The "chordic dimension" here is 5. By convention let us write 5H. The
first 5H corresponds to gammier 4 generated by (1 3 5 9 11). The first 4H
corresponds to gammier 1 generated by (1 3 5 9). The first 6H corresponds
to gammier 11 generated by (1 3 5 7 9 15), etc. And I add, to show that
large number may be used, that Indian gammoid has dimension 7H generated by
(1 3 9 27 45 81 243). There is no theoric upper limit. (I would think,
however, that confusion may arise using too long chordic generator, but
that concerns only constitutive harmony and not vertical harmonic extension.)

--------

Quoting 14259 (14261) :

<< And we understand immediately that the comparativity
profile of these intervals are very different. The steps
{16/15, 9/8, 10/9} appears mostly as Width than as Sonance.
And here, not only the tones can be ordered, but 16/15
appears easily in Width as half of 9/8 or 10/9. The
approximative Width relation 4-5-7-9 (12-tET) for the best
consonances is less immediate than 1-2 for the steps. >>

Paul wrote (14331) :

<< I'm having trouble understanding the implications
of Width-Sonance Comparativity (?). >>

(1) I think I've partially answered at (4) of the first question.

(2) I would like to say that I never used this expression before. I
believed that it could only help me to explain.

(3) I seek to say that all tones in a coherent set don't play an equivalent
role. For unison 1/1 it's obvious. In a simple approach (without theoric
tools) to constitute a coherent set we could focus on width or sonance.
Focusing on width would mean look first at tones having "width profile"
like contiguous steps. We would have to discover how they combine and see
after resulting harmony. Other way is easier. Focusing on sonance means
look first at type of chords in view starting with tones having "sonance
profile" and then obtain other tones by simple matrix of difference tones.

--------

That closes Part II of the answer. Part I was message 14333. Other parts
are following. Sorry, I can't write more rapidly in English.

Pierre Lamothe

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

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

> (3) The values {1/1, 5/4, 4/3, 3/2, 5/3} are not chosen simply for
their
> actual relation to 1/1 but mainly for the potential to have between
them
> great concordance. We have here, reordered, the Maj9 chord
>
> 2/3__5/6__1/1__5/4__3/2 == 8:10:12:15:18
>
> which contains much subchords like 2 major triads 4:5:6, 1 minor
triad
> 10:12:15, Maj7 and min7 tetrads 8:10:12:15 and 10:12:15:18, etc.

I would lattice this as

5/3-------5/4
/ \ / \
/ \ / \
/ \ / \
/ \ / \
2/3-------1/1-------3/2

>
> (4) Once free use principle is applied to these 5 tones, the
subsequent
> tones are not chosen based on their concordance with these 5 tones
but they
> are implied as difference tones between them,

Implied as difference tones? I don't get it. The only difference
tones
you can get out of this scale is 2/3 and its harmonics.
>
> --------
>
> Paul wrote (14331) :
>
> << It seems you need to justify your choice of the
> number five (or seven) as a delimiter between the
> two stages of scale formation. It certainly would
> seem hard to justify based on considerations of
> the history of musical practice. (In case anyone's
> confused, the five note scale 1/1 5/4 4/3 3/2 5/3
> and the seven note scale 1/1 6/5 5/4 4/3 3/2 8/5 5/3
> have nothing to do with the pentatonic and heptatonic
> scales of historical importance -- neither Pierre nor
> I are suggesting that they do). >>
>
>
> (1) You're right saying there is no consideration of the history of
musical
> practive. Practice implies tuning problems which are not in the
scope of
> this discussion.

Practice implies many musical features which also appear not to be in
the scope of this discussion -- see below.

> (2) The number 5 is not a theoric delimiter. The 13 tones generated
are the
> set of tones corresponding to gammier 9 generated by the chord (1 3
5 9
> 15). The "chordic dimension" here is 5. By convention let us write
5H. The
> first 5H corresponds to gammier 4 generated by (1 3 5 9 11).

What happened to 7?

What are these chordic generators, in terms of musical structure? Or
is musical structure outside the scope of this discussion?

🔗Pierre Lamothe <plamothe@aei.ca>

----------------------------------
HEURISTIC OF COHERENCE DISCUSSION
On Paul H. Erlich message # 14359
----------------------------------

Quoting message 14354 :

<< (4) Once free use principle is applied to these
5 tones, the subsequent tones are not chosen based
on their concordance with these 5 tones but they
are implied as difference tones between them >>

Paul wrote (14359) :

<< Implied as difference tones? I don't get it. The only
difference tones you can get out of this scale is 2/3
and its harmonics. >>

(1) Sorry for the confusion. With the term "difference tones", I wanted to
refer to width difference of log (ratios). That corresponds to division of
ratios.

(2) Since there is only one composition law in this semi-algebra, the term
"difference" was used in his generic sense englobing all types of
composition law. We need a generic term when the law is not defined as
such, talking only about axioms. I use "difference matrix" in this sense,
values being ratios, cents or whatever.

(3) However, I see that it was here a source of confusion for all. I'll try
to be more vigilant

--------

Quoting message 14354 :

<< We have here, reordered, the Maj9 chord

2/3__5/6__1/1__5/4__3/2 == 8:10:12:15:18

which contains much subchords like 2 major
triads 4:5:6, 1 minor triad 10:12:15, Maj7 and
min7 tetrads 8:10:12:15 and 10:12:15:18, etc. >>

Paul wrote (14359) :

<< I would lattice this as

5/3-------5/4
/ \ / \
/ \ / \
/ \ / \
/ \ / \
2/3-------1/1-------3/2 >>

(1) Mathematically speaking, the following graph A is a chordoid. Chordoid
theory don't apply only in music. Why it is a chordoid? (Forgeting details)
it's because there exist a chordic generator C for this graph A

0---O---O---O
/ \ / \ / \ / \
O---0---0---O---O
\ / \ / \ / \ /
O---0---0---O

And the chordic generator C is

0---O
/ \ / \
O---0---0

How? Refering to origin, the chord C may be translated to take all the
positions where one of his node occupies origin (without rotation).

(2) Then, If you apply that to your lattice you'll span the 13 tones I name
gammier 9. Why gammier rather than chordoid? For there exist supplemental
properties like abelian group isomorphism between (factorization) class of
tones G/2S and cyclic integers Z/7. The paths in its factorization lattice
are the musical modes.

--------

Quoting message 14354 :

<< (2) The number 5 is not a theoric delimiter.
The 13 tones generated are the set of tones
corresponding to gammier 9 generated by the
chord (1 3 5 9 15). The "chordic dimension"
here is 5. By convention let us write 5H. The
first 5H corresponds to gammier 4 generated
by (1 3 5 9 11). >>

Paul wrote (14359) :

<< What happened to 7? >>

(1) I never talk about 7 tones as starting point. It was your hypothesis in
message 14331 when you wrote :

<< If you then allowed octave-equivalence, you have a total
of seven tones per octave (this seems to help lead to what
you do below -- let me know if it is a misinterpretation). >>

I repeat. It was a misinterpretation.

(2) The 7 degrees arising in gammier 9 are a macrotonal property of the 13
tones took as a whole. If we remove 10/9 and 9/5, we have the 11 tones of
gammier 8 generated by (1 3 9 15) having 12 pentatonic modes similar to
japanese scale. If, rather, 16/15 and 15/8 are removed, then we have the 11
tones of gammier 1 generated by (1 3 5 9) having 12 pentatonic modes
similar to chinese anhemitonic scale. It's probably the most important
thing to understand at this moment : distinction between microtonal and
macrotonal properties. I can elaborate if it's not clear.

--------

Paul wrote (14359) :

<< Practice implies many musical features which
also appear not to be in the scope of this
discussion -- see below. >>

and below :

<< What are these chordic generators, in terms
of musical structure? Or is musical structure
outside the scope of this discussion? >>

(1) Structure is what I talk about. The term coherence refer directly to
algebraic structures in math manipulation and paradigmatic structures to
explain categoric perception and tonal intelligibility.

(2) What is not in the scope of the discussion, for much reasons are both
tuning problematic and musical artistic structures. For the art, it's
clear, I'm not musician. For the tuning problematic there are many reasons.
First it's soon very confused with this restricted scope. Second, I'm very
interested by the question but I'm not prepared for that. A part of my
research has for goal a renewal of the tuning problematic using new
"lutherie" concepts. I need to work with a musician to spot missing musical
ones. Third, the language barrier.

(3) For the chordic generator, I hope the graphic chordic generator example
was enlightening. Since we are yet in heuristic way, I won't use heavy
maths definition and I'll give only example. With gammier 9, (1 3 5 9 15)
is the minimal harmonic generator building the matrix giving our 13 tones.
The 5 tones used as starting tones are a chordic generator of the same
matrix. It's a chord and using these 5 tones is equivalent to use (1 3 5 9
15) for calculation.

Now, there exist in the matrix, five lines and five columns. These 10
objects are chords : five Maj9 chords and five min9 chords. If we take any
of these 10 distinct chords, we generate the same matrix of 13 tones. That
are all chordic generators of gammier 9. There exist also 8 chords
corresponding to generator (1 5 9 15) giving also the 13 tones but the
composition law on these 13 tones is not completely explicited by the
4-dimension matrix. By example, it is not explicited by the matrix that
there exist 9/8 interval between 4/3 and 3/2. These two intervals appears
in opposite corners of the matrix. In basic theory that refer to a LARGE
CHORDOID rather than CHORDOID.

(4) I add that in gammoid (and gammier) the matrix has toroidal structure
and tones class can be aligned on diagonals in matrix and circles in torus.

(5) Perhaps have you a definition for musical structure not covered by
these explanations. If it is the case, could you explicit your definition?

--------

Pierre Lamothe

what is missingunderstand musical constraint I needthe transposition
question,s clearis first taking account of transposition problem with fixed
sounds instruments and

🔗Pierre Lamothe <plamothe@aei.ca>

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

I wrote,

<< Implied as difference tones? I don't get it. The only
difference tones you can get out of this scale is 2/3
and its harmonics. >>

Pierre wrote,

>(1) Sorry for the confusion. With the term "difference tones", I wanted to
>refer to width difference of log (ratios). That corresponds to division of
>ratios.

>(2) Since there is only one composition law in this semi-algebra, the term
>"difference" was used in his generic sense englobing all types of
>composition law. We need a generic term when the law is not defined as
>such, talking only about axioms. I use "difference matrix" in this sense,
>values being ratios, cents or whatever.

I think I get it now! The 2/3 5/6 1/1 5/4 3/2 is simply rewritten as 1 5 3
15 9 and the "Partch diamond" (i.e. the cross-product of the set with its
inverse) is calculated, giving the 13 ratios. But what motivates this? The
13 ratios represent the _intervals_ between the 5 _pitches_ in {2/3 5/6 1/1
5/4 3/2}. What justifies one to take these 13 ratios and reinterpret them as
_pitches_?

>1) Mathematically speaking, the following graph A is a chordoid. Chordoid
>theory don't apply only in music. Why it is a chordoid? (Forgeting details)
>it's because there exist a chordic generator C for this graph A

> 0---O---O---O
> / \ / \ / \ / \
> O---0---0---O---O
> \ / \ / \ / \ /
> O---0---0---O

>And the chordic generator C is

> 0---O
> / \ / \
> O---0---0

>How? Refering to origin, the chord C may be translated to take all the
>positions where one of his node occupies origin (without rotation).

Yes, Dan Stearns and others were taking many different "C"s and drawing the
corresponding "A"s about a year ago on this list. You reminded me of one
thing: in addition to the chordiod being interpretable as the set of
intervals in the chordic generator, it's also interpretable as the set of
_pitches_ created by constucting all modes of the chordic generator on a
single root. So is that your justification for reinterpreting the intervals
as pitches? Are you suggesting that all the modes of five-tone "scale" need
to be constructed from a single "tonic"? Why?

>(2) Then, If you apply that to your lattice you'll span the 13 tones I name
>gammier 9. Why gammier rather than chordoid? For there exist supplemental
>properties like abelian group isomorphism between (factorization) class of
>tones G/2S and cyclic integers Z/7. The paths in its factorization lattice
>are the musical modes.

Can you explain this?

Pierre wrote,

><< (2) The number 5 is not a theoric delimiter.
> The 13 tones generated are the set of tones
> corresponding to gammier 9 generated by the
> chord (1 3 5 9 15). The "chordic dimension"
> here is 5. By convention let us write 5H. The
> first 5H corresponds to gammier 4 generated
> by (1 3 5 9 11). >>

I wrote,

<< What happened to 7? >>

>(1) I never talk about 7 tones as starting point.

I meant, why did you skip the number 7 in (1 3 5 9 11)?

>(2) The 7 degrees arising in gammier 9 are a macrotonal property of the 13
>tones took as a whole.

I understand that, but it seems that you have to construct a 13-tone
chordoid scale _before_ you can get your 7-tone scale. That is what I think
I'll have a hard time with. I'd argue that the diatonic scale is
musico-historically and logically prior to the 13-tone chordoid.

>Structure is what I talk about. The term coherence refer directly to
>algebraic structures in math manipulation and paradigmatic structures to
>explain categoric perception and tonal intelligibility.

I'll like to see more about how you "explain categoric perception and tonal
intelligibility".

>Now, there exist in the matrix, five lines and five columns. These 10
>objects are chords : five Maj9 chords and five min9 chords.

Yes -- the Maj9 chords are the five transpositions of the original five
tones, and and min9 chords are each created by following where one of the
original five tones moves during the five transpositions.

>These two intervals appears
>in opposite corners of the matrix.

I don't immediately see what of practical importance the position in the
matrix represents. At this stage, I must say your theory seems highly
abstract with a preponderance of notions that correspond to no evident
elements of musical reality. I seek to invoke William of Ockham and his
razor -- "hypotheses should not be multiplied beyond necessity". In my
_Gentle Introduction to Periodicity Blocks_ I give a derivation of the
various JI diatonic scales (and, in connected documents, of pentatonic
scales, chromatic scales, and the sruti scale of India) with far fewer
abstract notions -- and what abstract notions there are are immediately
motivated by a mathematical formulation of a real musical problem that a
musician may have faced. In my paper _Tuning, Tonality, and Twenty-Two Tone
Temperament_, I give psychological and psychoacoustical rules for what
constitutes "diatonicity" and so far my friends on the list and I have found
only very few "generalized diatonic" scales -- this specialness is a more
convincing explanation to me of the scale's musical importance than its
derivability from a mathematical framework that seems utterly contrived
(though I eagerly await further understanding which I hope will dissuade me
of this opinion).

>(5) Perhaps have you a definition for musical structure not covered by
>these explanations. If it is the case, could you explicit your definition?

Musical structure -- how music is constructed from the scale. Melody,
harmony, tonality as we know them -- how do they arise? How do their
development influence the further development of scales? Perhaps we should
leave this aside. Though I concern myself with those "musical structure"
issues in _Tuning, Tonality, and Twenty-Two Tone Temperament_, I don't in
the _Gentle Introduction to Periodicity Blocks_. There the concern is simply
harmonic connectedness and categorical clarity (which you mentioned above).
Perhaps you could do me the favor of looking at

http://www.ixpres.com/interval/td/erlich/intropblock1.htm
http://www.ixpres.com/interval/td/erlich/intropblock2.htm
http://www.ixpres.com/interval/td/erlich/intropblockex.htm
http://www.ixpres.com/interval/td/erlich/intropblock3.htm

and then

http://www.egroups.com/message/tuning/5245

and then, if time permits,

http://www.ixpres.com/interval/td/erlich/srutipblock.htm
http://www.ixpres.com/interval/td/erlich/ramospblock.htm
http://www.ixpres.com/interval/td/erlich/partchpblock.htm

Perhaps (seems unlikely, but who knows?) it will turn out that in some
sense, your theory is mathematically equivalent to periodicity-block theory.
But periodicity-block theory would remain the better _musical_ theory since
it _explains_ with a minimum of abstract notions. For an example of
mathematically equivalent theories with different explanatory powers,
consider Einstein's development of special relativity. Before Einstein,
Lorentz and Fitzgerald has essentially written down all the equations of
special relativity. So what was Einstein's great achievement? A few still
maintain, "nothing", but the vast majority of physicists see Einstein's
brilliance in the _interpretation_ he gave to these equations -- in terms of
time and space being relative to one's frame of reference, with all frames
of reference obeying the same laws of physics. In fact, this interpretation
proved so powerful that the equations themselves could be derived from it
(and from the assumption that the speed of light is the same in all
reference frames) -- rather than derived from considerations of a seeming
natural "conspiracy" to hide one's state of motion relative to the medium of
electromagnetic waves. When Wien recommended that Lorentz and Einstein be
jointly awarded a Nobel Prize, he stated:

"While Lorentz must be considered as the first to have found the
mathematical content of the relativity principle, Einstein succeeded in
reducing it to a simple principle. One should therefore assess the merits of
both investigators as being comparable... "

So, Pierre, unless I'm missing large portions of your theories, they seem to
be in need of underlying _musical_ principles. I would like to see if I can
be the Einstein to your Lorentz (not comparing myself to Einstein in any
way). Or maybe even help show that Fokker serves as the Einstein to your
Lorentz (Fokker did happen to work with Einstein!).

🔗Pierre Lamothe <plamothe@aei.ca>

[Nota Bene]

Questions come now from messages 14331, 14359
and 14371. So the message 14367 answering
questions in message 14359 will be considered
in sequence as Part III of the answers.

----------------------------------------
HEURISTIC APPROACH OF COHERENCE Part IV
----------------------------------------

In 14371 Paul wrote :

<< So, Pierre, unless I'm missing large portions of your
theories, they seem to be in need of underlying _musical_
principles. >>

(1) Even if you're missing large portions, you're wright to talk about a
need of musical comprehension, and that, not only because I know almost
nothing about music but more deeply because it belongs by principle to
structural paradigm. However, for an eventual serious discussion about
relations between structural explanation and musical understanding, I would
have to insert this question in the vast philosophical debate between "to
explain" and "to understand". Paul Ricoeur shows with great luminosity the
fecundity of a fine dialectic between the two irreducible cognitive modes
in his article "Entre herméneutique et sémiotique", 1990.

--------

In 14371 Paul wrote :

<< my friends on the list and I have found only very few
"generalized diatonic" scales -- this specialness is
a more convincing explanation to me of the scale's musical
importance than its derivability from a mathematical
framework that seems utterly contrived. >>

(1) I don't understand. It seems here you didn't ask much to be convinced.

--------

Quoting 14367 :

<< These two intervals appears in opposite corners of the
matrix. >>

Paul wrote (14371) :

<< I don't immediately see what of practical importance the
position in the matrix represents. At this stage, I must
say your theory seems highly abstract with a preponderance
of notions that correspond to no evident elements of musical
reality. I seek to invoke William of Ockham and his razor --
"hypotheses should not be multiplied beyond necessity". >>

(1) I admit I've contributed to confusion here giving a detail about tools.

(2) How could you judge of any preponderance about a theory not yet
exposed? Theme of discussion is heuristic approach of coherence. I would
like to talk about the coherence notion but, for now, I have to answer not
only to questions about what I've written but also to general comments
about what I've not written.

(3) Since detail about tools is far from hypothesis, I ask to me why Ockham
is convoked.

--------

Quoting 14367 :

<< Structure is what I talk about. The term coherence refer
directly to algebraic structures in math manipulation
and paradigmatic structures to explain categoric perception
and tonal intelligibility. >>

Paul wrote (14371):

<< I'll like to see more about how you "explain categoric
perception and tonal intelligibility". >>

"L'étonnant de la perception, c'est que nous ne
cessons de déchiffrer un sens qui s'enlève sur
l'opacité de la présence brute et muette, sans
pour autant s'arracher jamais à la limitation
d'une perspective, sans jamais renier l'inhérence
de la conscience à un point de vue."

P. Ricoeur, Hommage à Merleau-Ponty, 1961

(1) Categoric perception refers to quick integration of perceptions toward
sense decoding using predeterminate categories.

(2) The expression "categoric perception" underlines its distinction with
pure "sensitive perception". Grosso modo, as correlated to acoustical
parameter a sound appears like a sensitive perception and as correlated to
psychic structure where it has a predeterminated place it appears like a
categoric perception. Men and women have distinct pitchs and formants for
the vowels. When a vowel "o" appears in a conversation its quick perception
is categoric and organized immediately in the speech perception. But
someone can be attentive to difference between female and male "o" and then
its perceptions regress toward sensitive ones.

(3) The expression "tonal intelligibility" corresponds to "intelligibilité
tonale" where the term "tonale" denote class of intervals by octave
equivalence. So it corresponds to a relational property rather than a
sensible property. Acoustics tends to reduce interval at a sensible
property like pitch for which the sensation is correlated to the frequency
parameter. But an interval is not the sensible "correlat" of one parameter.
First, it corresponds to a relation between two frequencies, such relation
being not tied to a particular region of pitchs. Second, there exist
minimaly two associated perceptive fields, width and sonance, informing
about intervals. In this context, intelligibility refer to the possibility
of a rationnal (though inconscious) organization of these relations in such
manner than quick categoric perception can be obtained facilitating
immediate insertion in musical woof.

(4) As mathematical, the gammier theory is purely axiomatic and as such,
like all mathematical theories, it's a developped tautology. Technical
rather than musical knowledge is required to derive tools. However, as it
appears that gammier structures have morphology similar to long-lasting
widespread musical scales like anhemitonic, by example, a speculative
theory in developpement is superposed. That starts with justification with
only two general principles for the axioms chaining. Main goals are to
constitute a metalanguage helping musical theorists to define more
rigourously their concepts and to prepare technical tools permitting
perhaps a renewal of relations with technology (complexity mastering to
merge with new interfaces).

(5) First step derive from "How sound may act as a musical sign?" I've soon
talk about possibilities to keep relations in acoustical channel. I said
that intensity intervals varied with translation in pitchs and intensities
spectra. So it is impossible to maintain algebraic structures other than
groups based on set of oppositions. But continuous ordered structures are
maintained and continous variations are largely used by music in a proper
intelligibility segment, driving particularly emotions.

Only frequency and duration intervals are elementary invariants. Timbres
are, in a certain manner, complex invariants (but not elementary) so their
relations are hardly structurable otherwise than oppositions and they are
mainly used for textures and voices distinction. Actual focus on timbres
(mass work, spectral microstructures, morphing, etc.) permits to extend
possibilities, but algebraic structuration is out of reach. It's not
fortuitous that acuity and duration signs constitute majority of signs in a
traditional partition.

(6) Second step derive from "What musical functionality may be associated
with algebraic structuration of pitch and duration intervals?" It would be
trivial to simply talk here about categoric perception and fast decoding
possibilities. Questions are "What is the real advantage for music?" or yet
"What wouldn't be obtained without algebraic structuration?" The main
concept here is the rapid perception of similarities. Life implies dynamic
identities or, in other words, keeping identity thru transformations.
Musical appreciation implies not only sonorities and textures perception
but also pattern simililarities recognizing and global arrangement. Even if
there exist maybe much possibilities to define similarities class (like
fuzzy class) it seems, in first approximation, that algebraic class concept
allows a simple comprehensive distinction between paradigmatic and
syntagmatic axis.
It would be hard to give more details here, so I prepared in French a non
musical example showing how similarity may imply algebraic structure.

http://www.aei.ca/~plamothe/similar.htm

-Pierre

-----------------------------------------------------------------

APPENDICE

(1) Quelques réflexions de Paul Ricoeur tirées de "Structure et
herméneutique" 1963) concernant la légitimité du paradigme
structural.

<< Pour l'historicisme, comprendre, c'est trouver la genèse, la
forme antérieure, les sources, le sens de l'évolution. Avec le
structuralisme, ce sont les arrangements, les organisations
systématiques dans un état donné qui sont d'abord intelligibles.
Ferdinand de Saussure commence à introduire ce renversement en
distinguant la langue et la parole. >>

<< Mais voici le troisième principe [..] Il a sutout été dégagé par
les phonologues, mais il est déjà présent dans l'opposition
saussurienne entre la langue et la parole ; les lois linguistiques
désignent un niveau inconscient et en ce sens non réflexif, non
historique de l'esprit [..] c'est un inconscient kantien plutöt que
freudien, un inconscient catégoriel, combinatoire ; c'est un ordre
fini ou le finitisme de l'ordre mais tel qu'il s'ignore. Je dis
inconscient kantien, mais par égard seulement pour son organisation,
car il s'agit bien plutôt d'un système catégoriel sans référence à
un sujet pensant. >>

<< L'entreprise structuraliste me paraît donc parfaitement légitime
et à l'abri de toute critique aussi longtemps qu'elle garde la
concience de ses conditions de validité, et donc de ses limites. >>

<< Quelle est la place d'une "théorie générale des rapports" dans une
théorie générale du sens? >>

--------

(2) Passage de "La méthode de Solesme" (Dom Joseph Gayard) sur
la distinction des niveaux sensible et intelligible
concernant le rythme.

<< S'appuyant sur la distinction fondamentale entre, d'une part
les qualités matérielles, physiques, des sons : mélodie,
intensité, durée, produite par une disposition particulière
des vibrations sonores elles-mêmes (nombre, amplitude ou durée),
et absolument indépendantes les unes des autres, et, d'autre
part, les relations qui naissent forcément des différentiations
ainsi produites entre les sons successifs relativement aigus
ou graves, forts ou faibles, brefs ou longs, Dom Maquereau a
été amené à conclure, selon les lois de la plus rigoureuse
logique, que le rythme ne s'identifie à aucun de ces trois
ordres matériels, mais qu'il consiste, lui, dans l'organisation
du mouvement né de ces variations. IL N'EST PAS, COMME LES
AUTRES ÉLEMENTS, UNE QUALITÉ PHYSIQUE, MATÉRIELLE ; SA
PERCEPTION EST UNE OPÉRATION PRINCIPALEMENT INTELLECTUELLE,
D'UN ORDRE ABSOLUMENT SUPERIEUR, QUI DOMINE ET GOUVERNE TOUS
LES AUTRES. SON RÔLE ESSENTIEL, SA FONCTION PROPRE, EST
PRÉCISÉMENT DE SE SAISIR DE CES RELATIONS, DE LES PRÉCISER,
DE LES ORDONNER, DE LES HIÉRARCHISER, DE MANIÈRE À LES FAIRE
TOUTES CONCOURIR À L'UNITÉ, CETTE UNITÉ SANS LAQUELLE IL N'EST
PAS D'OEUVRE D'ART. Ce n'est plus un élément de division, mais
de synthèse, de groupement, de fusion. >>