Melodic Transformation in George Garzone’s Triadic Chromatic Approach Jonathan De Souza
SMT | Nov 7, 2014
Jazz, Math, and Basket Weaving Jonathan De Souza
SMT | Nov 7, 2014
Jazz, Math, and Basket Weaving Jonathan De Souza
SMT | Nov 7, 2014
i
s
t
SMT | Nov 7, 2014
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“Instead of regarding the i-arrow […] as a measurement of extension between points s and t observed passively ‘out there’ in a Cartesian res extensa, one can regard the situation situation actively acti vely,, like li ke a singer, singer, player play er,, or composer, composer, thinking: ‘I am at s; what characteristic transformation transformation do I perform to arrive at t?’” (Lewin 1987, xxxi)
SMT | Nov 7, 2014
Jazz SMT | Nov 7, 2014
Photo © R. Cifarelli
SMT | Nov 7, 2014
G
F
A
E
Am
F
A
B
George Garzone, “Have You Met Miss Jones,” Fours and Twos (1995) (Transcription adapted from Lorentz 2008, 116)
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SMT | Nov 7, 2014
“Basic Principles 1. Triads MUST be connected with a half-step in between 2. The same inversion CANNOT be repeated back to back” (Garzone 2008, 1)
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Math SMT | Nov 7, 2014
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D6 , a dihedral group S3 , a symmetry group SMT | Nov 7, 2014
135
351
513
531
315
153
Network of triadic rotations and flips (1 = root, 3 = chordal third, 5 = chordal fifth) SMT | Nov 7, 2014
Network of rotations and flips for a three-note contour segment (0 = lowest note, 1 = middle note, 2 = highest note) SMT | Nov 7, 2014
012
120
201
210
102
021
root position
&adg &dga &gad
&gda &dag &agd
1st inversion
&dgq &gqd &qdg
&qgd &gdq &dqg
2nd inversion
&gqe &qeg &egq
&eqg &qge &geq
Table of melodic permutations used in the triadic chromatic approach SMT | Nov 7, 2014
Illustration of the triadic chromatic approach (Garzone 2008, 3)
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(r0 , +1) (r2I, 0)
(I, +1)
(I, 0)
(I, 0)
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“the drunkard’s walk”
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0.5
...-4
0.5
-3 0.5
0.5
0.5
-2 0.5
0.5
-1 0.5
0.5
0 0.5
0.5
1 0.5
0.5
2
0.5
3
0.5
4... 0.5
“the drunkard’s walk” Markov chain SMT | Nov 7, 2014
D
A F
G
C A
B
E
E
F D
B
A F
C
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C
B
C /D
B
D E
A
E
A G
F
F
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X
=
C C 0 C 0 D 0 E 0.167 E 0.167 F 0.167 F 0 G 0.167 A 0.167 A 0.167 B 0 B
0
C 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0
D 0 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167
E 0.167 0 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167
E 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167 0 0.167
F 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167 0
F 0 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167
G 0.167 0 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167
A 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0 0 0.167
A 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0 0
B 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0
B 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0
Transition probability matrix for a random walk on the Tonnetz, corresponding to “within-triad” movement
SMT | Nov 7, 2014
Y
=
C C 0 C 0.5 D 0 E 0 E 0 F 0 F 0 G 0 A 0 A 0 B 0 B
C D E E F F G A A B B 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0.5 0.5 0 0 0 0 0 0 0 0 0 0.5 0
Transition probability matrix for a random walk on the pc clockface, corresponding to “between-triad” movement SMT | Nov 7, 2014
XY
=
Z
=
C C 0 C 0 D 0.083 E 0.083 E 0.167 F 0.083 F 0.167 G 0.083 0.167 A A 0.083 B 0.083 B
0
C 0 0 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083
D 0.083 0 0 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083
E 0.083 0.083 0 0 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167
E 0.167 0.083 0.083 0 0 0 0.083 0.083 0.167 0.083 0.167 0.083
F 0.083 0.167 0.083 0.083 0 0 0 0.083 0.083 0.167 0.083 0.167
F 0.167 0.083 0.167 0.083 0.083 0 0 0 0.083 0.083 0.167 0.083
G 0.083 0.167 0.083 0.167 0.083 0.083 0 0 0 0.083 0.083 0.167
A 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0 0 0 0.083 0.083
A 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0 0 0 0.083
B 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0 0 0
B 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0 0
Transition probability matrix combining both random walks, corresponding to a complete “step” of Garzone’s approach
SMT | Nov 7, 2014
Z
2
=
C C 0.125 C 0.097 D 0.104 E 0.069 E 0.076 F 0.056 F 0.069 G 0.056 A 0.076 A 0.069 B 0.104 B
C 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.097 0.104
D 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069
E 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076
E 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056
F 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069
F 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056
G 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076
A 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069
A 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104
B 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097
B 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125
Transition probability matrix for two complete “steps” of Garzone’s random triadic approach
SMT | Nov 7, 2014
Z5
=
C 0.082 0.082 0.082 0.083 0.083 0.084 0.084 0.085 0.084 0.084 0.083 0.082 0.083
C C 0.082 C 0.082 D 0.083 E 0.083 E 0.084 F 0.084 F 0.085 G 0.084 A 0.084 A 0.083 B 0.083 B
D 0.083 0.082 0.082 0.082 0.083 0.083 0.084 0.084 0.085 0.084 0.084 0.083
E 0.083 0.083 0.082 0.082 0.082 0.083 0.083 0.084 0.084 0.085 0.084 0.084
E 0.084 0.083 0.083 0.082 0.082 0.082 0.083 0.083 0.084 0.084 0.085 0.084
F 0.084 0.084 0.083 0.083 0.082 0.082 0.082 0.083 0.083 0.084 0.084 0.085
F 0.085 0.084 0.084 0.083 0.083 0.082 0.082 0.082 0.083 0.083 0.084 0.084
G 0.084 0.085 0.084 0.084 0.083 0.083 0.082 0.082 0.082 0.083 0.083 0.084
A 0.084 0.084 0.085 0.084 0.084 0.083 0.083 0.082 0.082 0.082 0.083 0.083
A 0.083 0.084 0.084 0.085 0.084 0.084 0.083 0.083 0.082 0.082 0.082 0.083
B 0.083 0.083 0.084 0.084 0.085 0.084 0.084 0.083 0.083 0.082 0.082 0.082
B 0.082 0.083 0.083 0.084 0.084 0.085 0.084 0.084 0.083 0.083 0.082 0.082
Transition probability matrix for five complete “steps” of Garzone’s random triadic approach
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Weaving SMT | Nov 7, 2014
SMT | Nov 7, 2014
SMT | Nov 7, 2014
“According to the standard view, the form pre-exists in the maker’s mind, and is simply impressed upon the material. Now I do not deny that the basket-maker may begin work with a pretty clear idea of the form she wishes to create. The actual, concrete form of the basket, however, does not issue from the idea. It rather comes into being through the gradual unfolding of that field of forces set up through the active and sensuous engagement of practitioner and material.
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“Effectively, the form of the basket emerges through a pattern of skilled movement , and it is the rhythmic repetition of that movement that gives rise to the regularity of form.” (Ingold 2000, 342)
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“The notion of making […] defines an activity purely in terms of its capacity to yield a certain object, whereas weaving focuses on the character of the process by which that object comes into existence. To emphasise making is to regard the object as the expression of an idea; to emphasise weaving is to regard it as the embodiment of a rhythmic movement.” (Ingold 2000, 346)
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SMT | Nov 7, 2014
D
A F
G
C A
B
E
E
F D
B
A F
C
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“Even if you are following all the technical rules it’s important to remember that it’s not mathematics, as George says.” —Ben Britton
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“As a former student of George’s I think after 20-30 years of playing he’s figured out a way to explain what he does naturally. I don’t think when he’s playing he’s thinking ‘up a major 3rd here, play a diminished triad, down a minor second, play a major triad up, etc.’ “He’s just doin’ his thing...” —Greg Sinibaldi
SMT | Nov 7, 2014
“I cannot , will not and won’t even try to apply TCA directly (‘note-for-note’) in my improvisations. “I don’t see the point in doing so. I personally work hard at the TCA simply to open my ears and fingers to different sounds and new possibilities. “...and honestly, I don't think even George Garzone himself applies the concepts when he blows! It’s more of a practice tool than a literal way of playing.” —Marc-André Seguin SMT | Nov 7, 2014
“The goal is to get this into your subconscious.” —George Garzone (Downbeat January 2009, 99)
SMT | Nov 7, 2014
“If I am at s and wish to get to t, what characteristic gesture […] should I perform in order to arrive there?” (Lewin 1987, 159)
SMT | Nov 7, 2014
i
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“If I am at s and wish to get to t, what characteristic gesture […] should I perform in order to arrive there?” (Lewin 1987, 159)
SMT | Nov 7, 2014
i
s
t
SMT | Nov 7, 2014
