Yahoo Tuning Groups Ultimate Backup tuning My last message

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NOTA BENE
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This is my last message on regular basis on this List. I taked this
decision with a certain degree of sadness but certainly not with angry or
resentment. Since it could be easily misinterpreted as resulting of
apparent conflict with Paul, I would like to express clearly that my
problem was not at this level.

I posted my first message the first day of august and I ceased from writing
on my website the august 20. I see I was unconsciously waiting for
something impossible. With efforts it is possible to write ideas in English
(or in German, or ...) but it was not (to me) truly possible to feel
belonging at a new community. First, I escaped too much nuances. Secondly I
did'nt write what I wanted. I was very restricted by what I was able to
write in a reasonable amount of time. Moreover I didn't find how to keep
focus on what I would find pleasure to write. It's only on that base I
lived frustration in my dicussions with Paul. I saw mountain of questions
growing which seemed to me delaying the moment where I could find pleasure
to expose my ideas. There is no reason to blame here.

I had decided to leave completely the List to return more easily to my
searching and writing rituals but at the precise moment having a last look
on messages, I saw, in last ones, those of Margo Schulter (#15038) "A
gentle introduction to neo-Gothic progressions". I changed my mind. I shall
certainly scan the List from time to time hoping to find such pearls. If I
underline here a coincidence having found an ultimate pleasure at decision
time, it is not to minimize other moments or sources of pleasure. Otherwise
I would not have stayed until now.

My decision is to write now in French. Maybe I shall post announces of new
papers if I think someone could be interested. Besides I shall seek for
someone to translate some papers. Finally, I keep the same URL and address.

Now my last answers in discussion about COHERENCE, then a SPECIAL HALLOWEEN
and a final CHALLENGE.

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HEURISTIC APPROACH OF COHERENCE Last Part
------------------------------------------

[ I shall answer only few questions. I apologize for
leaving without answers a long list of questions I
keeped on a file. ]

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Quoting 14367 :

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

Paul wrote (14371) :

<< 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. >>

(1) About "musico-historically", I try to translate a precedent Ricoeur
quotation.

<< For historicism, to understand is to find anterior forms,
sources, evolution direction. With structuralism, that are
arrangements, systematic organizations in a given state
that are first intelligible. Ferdinand de Saussure begin
to introduce this reversing with distinction between
language and speech. >>

(2) For sure, people didn't need Saussure before begining to speak, to make
grammars or to define style rules.

(3) The 13 tones are used to reveal hidden properties about scales used
since millenaries. Besides, choosing one of the 16 proper(*) modes of these
13 tones, like the well-known 1 9/8 5/4 4/3 3/2 5/3 15/8, there exist a
family of gammiers (having same tonal generator) in which this mode is a
proper mode. There exist also other gammiers or gammoids like the Indian
one (1 3 9 27 45 81 243) in which this particular mode is also a proper
mode (meaning a contiguous path across the octave in a specific set). To
sum up, this 7-tone scale has not only one possibility to be framed like
the 13-tone gammier #9 structure (that was only heuristically built here).

(4) However, with the tools, I can assert that the minimal coherent set of
tones in which this contiguous path is possible is precisely the gammier #9
for which the matrix (1 3 5 9 15) define not only the tones but all the
triplets of the composition law.

(5) Open question : "Could mode confining (about pitches) be the simplest
(conscious) way to approximately obtain (unconscious) algebraic structure
confining (about intervals between pitches)?"

(*) "proper" == "propre" in sense of property of the set as a whole.

--------

Quoting 14367 :

<< 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. >>

Paul wrote (14371) :

<< Can you explain this? >>

(1) Explanation about what is an isomorphism like here

G/2S --> Z/n

Let F be a bijection (or a "biunivoque" application) of a set A structured
with the composition law (*) on a set B structured with the composition law
(+). We write simply

F : A --> B

Let x and y be generic elements in A. The bijection F is an isomorphism if

F(x * y) = F(x) + F(y)

(2) Explanation about the symbols used. In G/2S --> Z/n, G and Z are
structured sets and 2, S and n represent equivalence relations. Now, the
details.

Z == the set of all relative integers {... -2, -1 , 0, 1, 2 ...}

Z/n == additive cyclic group of integers modulo n like
Z/5 == {0, 1, 2 ,3 ,4} with addition modulo 5

G == a set of intervals with gammoid structure (examples and main
property of gammoid structure forward)

G/2 == set of tones (interval classes by octave equivalence)
corresponding to the set G

Examples :

G/2 = {1, 10/9, 9/8, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3,
16/9, 9/5} for the chinese gammier

G/2 = {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} for our 13-tone gammier

G contains the corresponding intervals in all octaves.

Anticipating, I give also example of G/2S for the chinese gammier

G/2S = {{1}, {10/9, 9/8, 6/5}, {5/4, 4/3}, {3/2, 8/5},
{5/3, 16/9, 9/5}} --> {0, 1, 2, 3, 4} = Z/5

(3) Explanation about quotients like Z/n. Let S be a set with an algebraic
structure and R be an equivalence relation on S. Then S/R is the set of
classes corresponding to the partition of S by the equivalence relation R.
If there is a morphism between S and S/R (with the standard law of
composition on classes), then the equivalence relation is called a congruence.

(4) Explanation about the three equivalence relations used to construct the
structured sets of classes G/2, G/S, G/2S and Z/n.

( x = y mod n ) additive equivalence meaning there exist k
in Z such that x = y + n*k

( x = y mod 2 ) multiplicative equivalence meaning there exist
k in z such that x = y * 2^k

( x = y mod S ) modal equivalence meaning there exist k in
srutal matrix S such that x = y * k

The srutal matrix S associated with a gammoid is the matrix generated by
the tonal generator of this gammoid.

Examples :

For the chinese gammier (#1), the tonal generator (minimal steps)
is (10/9 9/8 6/5), then the matrix S is

| 10/9 9/8 6/5
-----|-------------------
10/9 | 1 81/80 27/25
|
9/8 | 80/81 1 16/15
|
6/5 | 25/27 15/16 1

Two tones x and y in the chinese gammier separated by an element
k in S belongs to the same tone class (degree).

For the gammier #9, the tonal generator is (16/15 10/9 9/8)
and the corresponding matrix S has, with the unison 1, this set
of modal srutis {81/80, 80/81, 25/24, 24/25, 135/128, 128/135}.

(5) Now, global explanation about the isomorphism G/2S --> Z/n. If G/2 has
the gammoid structure, then there exist a unique epimorphism F of G/2 on
Z/n where ker(F) = {1}, meaning unison class is reduced to {1}. This
epimorphism is decomposable like that

G/2 --> G/2S --> Z/n

where there are a canonical epimorphism of tones G/2 on tone classes G/2S
and an isomorphism of tone classes G/2S on the cyclic group Z/n.

(6) It is very strange for me that nobody seems to have remarked how the
group Z/7 has modeled the western music and notation. Is it for III + III =
V hides the underlying 2 + 2 = 4?

(7) As APPENDICE at this answer I list the three first gammiers (and more)
having partition in degrees between 5 and 12.

RANK GENERATOR ONE EXAMPLE AMONG PROPER MODES

[5 DEGREES]

1 (1 3 5 9) 1 9/8 5/4 3/2 5/3 2 Chinese
2 (1 3 7 9) 1 8/7 9/7 3/2 7/4 2 Slendro
3 (1 3 9 11) 1 12/11 4/3 3/2 18/11 2 Pelog
------------------------------------------------------------------
?? (1 3 9 81) 1 9/8 81/64 3/2 27/16 2 Ling Lun

[6 DEGREES]

7 (1 3 5 7 15) 1 7/6 5/4 3/2 7/4 15/8 2
13 (3 5 9 11 15) 1 5/4 15/11 3/2 5/3 11/6 2
16 (1 3 5 13 15) 1 6/5 13/10 3/2 13/8 15/8 2
---------------------------------------------------------------
89 (forward) 1 7/6 4/3 7/5 3/2 7/4 2 Blues

[7 DEGREES]

4 (1 3 5 9 11) 1 11/10 11/9 4/3 3/2 18/11 20/11 2 Rast
6 (1 3 7 9 13) 1 9/8 9/7 18/13 3/2 13/8 13/7 2 Pelog
9 (1 3 5 9 15) 1 9/8 5/4 4/3 3/2 5/3 15/8 2 Zarlino
-------------------------------------------------------------------

?? (1 3 9 27 243) 1 9/8 81/64 4/3 3/2 27/16 243/128 2 Pythagore

[8 DEGREES]

17 (1 3 5 7 13 15) 1 13/12 7/6 5/4 3/2 13/8 7/4 15/8 2
24 (3 5 7 9 11 13 15) 1 13/12 7/6 5/4 7/5 3/2 5/3 11/6 2
28 (1 3 5 9 11 17) 1 17/16 9/8 5/4 11/8 3/2 5/3 11/6 2

[9 DEGREES]

20 (1 3 7 9 13 15) 1 9/8 7/6 5/4 4/3 3/2 13/8 7/4 15/8 2
22 (1 3 5 7 11 13 15) 1 13/12 7/6 5/4 11/8 3/2 13/8 7/4 15/8 2
46 (1 3 5 9 11 19) 1 9/8 19/16 5/4 11/8 3/2 19/12 5/3 11/6 2

[10 DEGREES]

34 (1 3 5 9 15 17) 1 17/16 9/8 5/4 4/3 17/12
3/2 5/3 30/17 15/8 2
40 (1 3 7 9 13 15 17) 1 17/16 9/8 6/5 9/7 18/13
3/2 13/8 7/4 15/8 2
42 (1 3 5 7 9 13 15 17) 1 17/16 9/8 5/4 4/3 17/12
3/2 13/8 7/4 15/8 2
------------------------------------------------------
95 (1 3 5 7 9 15 21) 1 21/20 9/8 5/4 21/16 7/5
3/2 5/3 7/4 15/8 2

[11 DEGREES]

170 (1 3 5 9 11 19 21) 1 9/8 19/16 5/4 21/16 11/8
3/2 19/12 5/3 7/4 21/11 2
174 (1 3 5 7 9 11 19 21) 1 12/11 8/7 6/5 24/19 4/3
3/2 18/11 12/7 9/5 36/19 2
265 (1 3 5 9 11 19 23) 1 9/8 19/16 5/4 11/8 23/16
3/2 19/12 5/3 11/6 23/12 2

[12 DEGREES]

63 (1 3 5 9 15 17 19) 1 17/16 9/8 19/16 5/4 4/3 17/12
3/2 8/5 17/10 9/5 19/10 2
188 (1 5 7 9 15 19 21) 1 21/20 9/8 19/16 5/4 21/16 7/5
3/2 8/5 12/7 9/5 19/10 2
189 (1 3 5 7 9 15 19 21) 1 15/14 9/8 7/6 5/4 4/3 7/5
3/2 19/12 5/3 7/4 15/8 2

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[I return now to a precedent message where my comment leaved too much space
for misinterpretation and finally confusion.]

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 >>

Then, what was my imprecise comment :

<< 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). >>

And Paul wrote (14371) :

<< Yes, Dan Stearns and others were taking many different
"C"s and drawing the corresponding "A"s about a year ago
on this list. >>

(1) I wanted to "forget the details" and I tried to show how the
translation process where the graph generator nodes occupy successively the
origin point (representing unison) is equivalent to matrix generation by an
odd chord.

(2) The graphs A and C are not really chordoid and its generator but only
parts of them. I apologize for confusion resulting with drawings presented
on this list about a year ago.

(3) In the generator C used here, only thirds and fifth are represented.
The true graphic chordoid generator corresponding to C has an interval
between all its 5 nodes. I can't draw (in this post) all the vertices.
There exit 6 x 2 + 1 = 13 intervals, each having some emplacements. (Unison
is represented by loop at each node.)

(4) What is very important to deeply understand is the fact that graph
chordoid A has not an interval (vertice) between all his nodes like graph
generator C. The vertices of graph chordoid A are generated by vertices of
the true graph C. Thus there exist only 13 vertice types in graph A, those
collected in graph C. And the 13 vertice types correspond to the 13 nodes
of graph A. Each "vectorial" vertice type corresponds to the unique vertice
between node 1 and one of the 13 nodes.

-----------------
SPECIAL HALLOWEEN
-----------------

I met yesterday an old witch who showed to me how to give "halloween
colors" to diatonic scales. She said you put a "colored" third x in the
weird "hat"

-1 x 0 0 0
0 -1 3/2x 0 0
"hat" = 0 0 -1 x 0
0 0 0 -1 3/2x
8/9 0 0 0 -1

then you hit the "hat" with a magic wand giving "hat inversion" and then
you obtain (a rabbit? ... no) a gammier similar to Zarlino gammier where
major triad appears on I, IV and V degrees in the major mode. She picked
few thirds in the Stern-Brocot tree and took out from the inversed "hat"
their structured contexts. She called these colored "scales bunches" :
"Zarlino", "Ptolemy", "Hindemith", "Schulter", etc. I verified and
effectively with the major third x = 5/4 the matrix inversion gave the
Zarlino diamond I wanted to approach heuristically in precedent messages.
["Have you said COHERENCE?"]

-1 6/5 0 0 0 1 5/4 3/2 15/8 9/4
0 -1 5/4 0 0 8/5 1 6/5 3/2 9/5
0 0 -1 6/5 0 -- inversion --> 4/3 5/3 1 5/4 3/2
0 0 0 -1 5/4 16/15 4/3 8/5 1 6/5
8/9 0 0 0 -1 8/9 10/9 4/3 5/3 1

This diamond has the generator (1 3 5 9 15). With other thirds it were also
gammier structures. With x = 7/6 we find the following gammier (and the
same gammier with x = 9/7 but transposed).

-1 7/6 0 0 0 1 7/6 3/2 7/4 9/4
0 -1 9/7 0 0 12/7 1 9/7 3/2 27/14
0 0 -1 7/6 0 -- inversion --> 4/3 14/9 1 7/6 3/2
0 0 0 -1 9/7 8/7 4/3 12/7 1 9/7
8/9 0 0 0 -1 8/9 28/27 4/3 14/9 1

Paul, probably Harry Partch would be appreciated also to meet this old
witch. I list the tries now ordered by gammier generator.

THIRD TRIAD DIAMOND EXAMPLE AMONG PROPER MODES

5/4 4:5:6 (1 3 5 9 15) 1 9/8 5/4 4/3 3/2 5/3 2

9/7 6:7:9 (3 7 9 21 27) 1 28/27 7/6 4/3 3/2 14/9 7/4 2

13/10 10:13:15 (5 13 15 39 45) 1 9/8 13/10 4/3 3/2 26/15 39/20 2

19/16 16:19:24 (1 3 9 19 57) 1 9/8 24/19 4/3 3/2 32/19 36/19 2

17/14 14:17:21 (7 17 21 51 63) 1 9/8 17/14 4/3 3/2 34/21 51/28 2

11/9 18:22:27 (9 11 27 33 81) 1 9/8 11/9 4/3 3/2 18/11 16/9 2

14/11 22:28:33 (7 11 21 33 99) 1 9/8 14/11 4/3 3/2 56/33 21/11 2

16/13 26:32:39 (1 3 9 39 117) 1 9/8 39/32 4/3 3/2 13/8 117/64 2

Maybe, I shall draw eventually all the corresponding lattices in my website.

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CHALLENGE
---------

My first message on the List was a question titled CHALLENGE. I would like
to end this last message by two such challenging questions on Hexany and
Blues scale. Since I leave, I not intend to give solutions. Thus why asking
these questions?

I was here to write about solutions, but maybe need for solutions is less
important than need for questions. Giving solutions may be like giving
fishes while giving question may be like giving a fishing rod.

-----On Hexany-----

Let be a diamond generated by (a b c d) where the letters can be (1 3 5 7)
to facilitate the understanding. We can represent this matrix in 3D space
by a symmetric figure where 1/1 is the center of a sphere and all other
ratios are on the surface. If we delete 1/1 and trace an arc between all
ratios being on same horizontal or vertical lines in the matrix then we
have what I name a "peripheral accordance graph" for the structure
generated by (a b c d) as chord. (What is deleted with 1/1 is "neutral
accordance graph".)

If you want to know easily how appears the spatial distribution of the
ratios, try to arrange symmetricaly (in 3D) four identical circles around
the same center. The 12 points where circles are crossing two by two
correspond to positions of ratios and all parts of circles between these
points correspond to vertices between ratios defined by lines and columns
in the matrix.

Now, in figure with vertices as lines, we find, in the spatial surface, 8
triangles and 6 squares. The 8 triangles correspond to the 8 chordic
sub-generators of 3 elements. With (1 3 5 7) we have (1 3 5) (1 3 7) (1 5
7) (3 5 7) in both harmonic and dual "flavor".

What little geometric transformation (keeping triangles but cutting edges)
of this structure may result precisely to hexany figure?

[Hint : in the transformation, the six peripheric triangles (chords) are
changing their "flavor"]

More generally, what underlying (musical) sense of CPS may be suggested here?

What are the two distinct ordered structures used in all CPS design? In
what distinct musical spaces these ordered structures have sense?

-----On the Blues scale-----

Find (v w x y z) the minimal odd generator of the Blues scale

1 7/6 4/3 7/5 3/2 7/4 2

corresponding to the gammier 89. Its lattice is there

http://www.aei.ca/~plamothe/pix/tan-blues.gif

Pierre

My last message

🔗Pierre Lamothe <plamothe@aei.ca>

🔗D.Stearns <STEARNS@CAPECOD.NET>

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

🔗D.Stearns <STEARNS@CAPECOD.NET>