melodic transformation in george garzone's triadic

SMT | Nov 7, 2014

Melodic Transformation in George Garzone’s

Triadic Chromatic Approach

Jonathan De Souza

SMT | Nov 7, 2014

Jazz, Math, andBasket Weaving

Jonathan De Souza

SMT | Nov 7, 2014

sti

SMT | Nov 7, 2014

sti

“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 actively, like a singer, player, or composer, thinking: ‘I am at s; what characteristic transformation do I perform to arrive at t?’” (Lewin 1987, xxxi)

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Jazz

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George GarzonePhoto © R. Cifarelli

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George Garzone, “Have You Met Miss Jones,” Fours and Twos (1995)(Transcription adapted from Lorentz 2008, 116)

G Ab E Am

F F#

A Bb

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“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

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D6, a dihedral groupS3, a symmetry group

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!"#$%&''&()*+$#,-./'#0$)''1023-2#$2"#$)./3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%&7$-*4$2"#$/3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%'78$91#$2&$2"#$5"3&.-2)5$*-213#$&%$2")0$5&*5#/2:$/'#-0#$*&2#$2"-2$)%$-*$#,-./'#$"-0$*&$2).#$0)+*-213#:$-''$*&2#0$-3#$*-213-'$1*'#00$2"#3#$)0$-$0"-3/$&3$%'-28$$$$

$$$

$$;&$%-3:$-''$2"#$#,-./'#0$"-<#$4#-'2$()2"$.-=&3$23)-408$$>'#-0#$?*&($2"-2$2"#$@-*4&.$!3)-4)5$A//3&-5"$-//')#0$2&$.)*&3:$4).)*)0"#4$-*4$-1+.#*2#4$23)-40$-0$(#''8

!"&()*+$%'$,$-./)+.$0+12/3$

!"&()*+$%&$,$4()./)+.$0+12/3$$

- 3 -

(a) A melody constructed according to the triadic chromatic approach, with Garzone’sannotations

!"#$%&''&()*+$#,-./'#0$)''1023-2#$2"#$)./3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%&7$-*4$2"#$/3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%'78$91#$2&$2"#$5"3&.-2)5$*-213#$&%$2")0$5&*5#/2:$/'#-0#$*&2#$2"-2$)%$-*$#,-./'#$"-0$*&$2).#$0)+*-213#:$-''$*&2#0$-3#$*-213-'$1*'#00$2"#3#$)0$-$0"-3/$&3$%'-28$$$$

$$$

$$;&$%-3:$-''$2"#$#,-./'#0$"-<#$4#-'2$()2"$.-=&3$23)-408$$>'#-0#$?*&($2"-2$2"#$@-*4&.$!3)-4)5$A//3&-5"$-//')#0$2&$.)*&3:$4).)*)0"#4$-*4$-1+.#*2#4$23)-40$-0$(#''8

!"&()*+$%'$,$-./)+.$0+12/3$

!"&()*+$%&$,$4()./)+.$0+12/3$$

- 3 -

(b) An “improper” sequence of triads that does not follow Garzone’s approach

Figure 1: Illustrations of the triadic chromatic approach (Garzone 2008, 3)

Figure 2: Network of three-note contour segments connected by rotation (r)and inversion (I), isomorphic to the group of rotations and reflections of anequilateral triangle, D3

2

135 351 513

531 315 153

Network of triadic rotations and flips(1 = root, 3 = chordal third, 5 = chordal fifth)

SMT | Nov 7, 2014

!"#$%&''&()*+$#,-./'#0$)''1023-2#$2"#$)./3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%&7$-*4$2"#$/3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%'78$91#$2&$2"#$5"3&.-2)5$*-213#$&%$2")0$5&*5#/2:$/'#-0#$*&2#$2"-2$)%$-*$#,-./'#$"-0$*&$2).#$0)+*-213#:$-''$*&2#0$-3#$*-213-'$1*'#00$2"#3#$)0$-$0"-3/$&3$%'-28$$$$

$$$

$$;&$%-3:$-''$2"#$#,-./'#0$"-<#$4#-'2$()2"$.-=&3$23)-408$$>'#-0#$?*&($2"-2$2"#$@-*4&.$!3)-4)5$A//3&-5"$-//')#0$2&$.)*&3:$4).)*)0"#4$-*4$-1+.#*2#4$23)-40$-0$(#''8

!"&()*+$%'$,$-./)+.$0+12/3$

!"&()*+$%&$,$4()./)+.$0+12/3$$

- 3 -


!"#$%&''&()*+$#,-./'#0$)''1023-2#$2"#$)./3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%&7$-*4$2"#$/3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%'78$91#$2&$2"#$5"3&.-2)5$*-213#$&%$2")0$5&*5#/2:$/'#-0#$*&2#$2"-2$)%$-*$#,-./'#$"-0$*&$2).#$0)+*-213#:$-''$*&2#0$-3#$*-213-'$1*'#00$2"#3#$)0$-$0"-3/$&3$%'-28$$$$

$$$

$$;&$%-3:$-''$2"#$#,-./'#0$"-<#$4#-'2$()2"$.-=&3$23)-408$$>'#-0#$?*&($2"-2$2"#$@-*4&.$!3)-4)5$A//3&-5"$-//')#0$2&$.)*&3:$4).)*)0"#4$-*4$-1+.#*2#4$23)-40$-0$(#''8

!"&()*+$%'$,$-./)+.$0+12/3$

!"&()*+$%&$,$4()./)+.$0+12/3$$

- 3 -




2

Network of rotations and flipsfor a three-note contour segment

(0 = lowest note, 1 = middle note, 2 = highest note)

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012 120 201 210 102 021

root position &adg &dga &gad &gda &dag &agd1st inversion &dgq &gqd &qdg &qgd &gdq &dqg2nd inversion &gqe &qeg &egq &eqg &qge &geq

Figure 3: Table of melodic permutations used in the triadic chromatic approach

X =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

C C♯ D E♭ E F F♯ G A♭ A B♭ BC 0 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0C♯ 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0D 0 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167E♭ 0.167 0 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167E 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167 0 0.167F 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167 0F♯ 0 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167G 0.167 0 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167A♭ 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0 0 0.167A 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0 0B♭ 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0B 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Figure 4: Transition probability matrix for a random walk on the Tonnetz, cor-responding to “within-triad” movement in the triadic chromatic approach

3

Table of melodic permutations used in the triadic chromatic approach

SMT | Nov 7, 2014

!"#$%&''&()*+$#,-./'#0$)''1023-2#$2"#$)./3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%&7$-*4$2"#$/3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%'78$91#$2&$2"#$5"3&.-2)5$*-213#$&%$2")0$5&*5#/2:$/'#-0#$*&2#$2"-2$)%$-*$#,-./'#$"-0$*&$2).#$0)+*-213#:$-''$*&2#0$-3#$*-213-'$1*'#00$2"#3#$)0$-$0"-3/$&3$%'-28$$$$

$$$

$$;&$%-3:$-''$2"#$#,-./'#0$"-<#$4#-'2$()2"$.-=&3$23)-408$$>'#-0#$?*&($2"-2$2"#$@-*4&.$!3)-4)5$A//3&-5"$-//')#0$2&$.)*&3:$4).)*)0"#4$-*4$-1+.#*2#4$23)-40$-0$(#''8

!"&()*+$%'$,$-./)+.$0+12/3$

!"&()*+$%&$,$4()./)+.$0+12/3$$

- 3 -


!"#$%&''&()*+$#,-./'#0$)''1023-2#$2"#$)./3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%&7$-*4$2"#$/3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%'78$91#$2&$2"#$5"3&.-2)5$*-213#$&%$2")0$5&*5#/2:$/'#-0#$*&2#$2"-2$)%$-*$#,-./'#$"-0$*&$2).#$0)+*-213#:$-''$*&2#0$-3#$*-213-'$1*'#00$2"#3#$)0$-$0"-3/$&3$%'-28$$$$

$$$

$$;&$%-3:$-''$2"#$#,-./'#0$"-<#$4#-'2$()2"$.-=&3$23)-408$$>'#-0#$?*&($2"-2$2"#$@-*4&.$!3)-4)5$A//3&-5"$-//')#0$2&$.)*&3:$4).)*)0"#4$-*4$-1+.#*2#4$23)-40$-0$(#''8

!"&()*+$%'$,$-./)+.$0+12/3$

!"&()*+$%&$,$4()./)+.$0+12/3$$

- 3 -




2

Illustration of the triadic chromatic approach(Garzone 2008, 3)

SMT | Nov 7, 2014

!"#$%&''&()*+$#,-./'#0$)''1023-2#$2"#$)./3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%&7$-*4$2"#$/3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%'78$91#$2&$2"#$5"3&.-2)5$*-213#$&%$2")0$5&*5#/2:$/'#-0#$*&2#$2"-2$)%$-*$#,-./'#$"-0$*&$2).#$0)+*-213#:$-''$*&2#0$-3#$*-213-'$1*'#00$2"#3#$)0$-$0"-3/$&3$%'-28$$$$

$$$

$$;&$%-3:$-''$2"#$#,-./'#0$"-<#$4#-'2$()2"$.-=&3$23)-408$$>'#-0#$?*&($2"-2$2"#$@-*4&.$!3)-4)5$A//3&-5"$-//')#0$2&$.)*&3:$4).)*)0"#4$-*4$-1+.#*2#4$23)-40$-0$(#''8

!"&()*+$%'$,$-./)+.$0+12/3$

!"&()*+$%&$,$4()./)+.$0+12/3$$

- 3 -


!"#$%&''&()*+$#,-./'#0$)''1023-2#$2"#$)./3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%&7$-*4$2"#$/3&/#3$.#2"&4$&%$2"#$3-*4&.$23)-4)5$-//3&-5"$6!"#$%'78$91#$2&$2"#$5"3&.-2)5$*-213#$&%$2")0$5&*5#/2:$/'#-0#$*&2#$2"-2$)%$-*$#,-./'#$"-0$*&$2).#$0)+*-213#:$-''$*&2#0$-3#$*-213-'$1*'#00$2"#3#$)0$-$0"-3/$&3$%'-28$$$$

$$$

$$;&$%-3:$-''$2"#$#,-./'#0$"-<#$4#-'2$()2"$.-=&3$23)-408$$>'#-0#$?*&($2"-2$2"#$@-*4&.$!3)-4)5$A//3&-5"$-//')#0$2&$.)*&3:$4).)*)0"#4$-*4$-1+.#*2#4$23)-40$-0$(#''8

!"&()*+$%'$,$-./)+.$0+12/3$

!"&()*+$%&$,$4()./)+.$0+12/3$$

- 3 -




2

(r0, +1) (I, +1)

(I, 0) (I, 0)(r2I, 0)

SMT | Nov 7, 2014

“the drunkard’s walk”

SMT | Nov 7, 2014

“the drunkard’s walk”

...-4 -3 -2 -1 0 1 2 3 4...0.5

0.5

0.5 0.5 0.50.5 0.5 0.50.5

0.5 0.5 0.5 0.50.5 0.5 0.5

Markov chain

SMT | Nov 7, 2014

GCF D A

BbEb FAbDb

BEA F# C #

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G

C

F

D

A

Bb

Eb

F

Ab

DbB

E

A

F#

C #/

SMT | Nov 7, 2014

012 120 201 210 102 021

root position &adg &dga &gad &gda &dag &agd1st inversion &dgq &gqd &qdg &qgd &gdq &dqg2nd inversion &gqe &qeg &egq &eqg &qge &geq

Figure 3: Table of melodic permutations used in the triadic chromatic approach

X =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

C C♯ D E♭ E F F♯ G A♭ A B♭ BC 0 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0C♯ 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0D 0 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167E♭ 0.167 0 0 0 0 0 0.167 0.167 0.167 0 0.167 0.167E 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167 0 0.167F 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167 0F♯ 0 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167 0.167G 0.167 0 0.167 0.167 0.167 0 0 0 0 0 0.167 0.167A♭ 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0 0 0.167A 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0 0B♭ 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0 0B 0 0 0.167 0.167 0.167 0 0.167 0.167 0.167 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Figure 4: Transition probability matrix for a random walk on the Tonnetz, cor-responding to “within-triad” movement in the triadic chromatic approach

3

Transition probability matrix for a random walk on the Tonnetz, corresponding to “within-triad” movement

SMT | Nov 7, 2014

Y =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

C C♯ D E♭ E F F♯ G A♭ A B♭ BC 0 0.5 0 0 0 0 0 0 0 0 0 0.5C♯ 0.5 0 0.5 0 0 0 0 0 0 0 0 0D 0 0.5 0 0.5 0 0 0 0 0 0 0 0E♭ 0 0 0.5 0 0.5 0 0 0 0 0 0 0E 0 0 0 0.5 0 0.5 0 0 0 0 0 0F 0 0 0 0 0.5 0 0.5 0 0 0 0 0F♯ 0 0 0 0 0 0.5 0 0.5 0 0 0 0G 0 0 0 0 0 0 0.5 0 0.5 0 0 0A♭ 0 0 0 0 0 0 0 0.5 0 0.5 0 0A 0 0 0 0 0 0 0 0 0.5 0 0.5 0B♭ 0 0 0 0 0 0 0 0 0 0.5 0 0.5B 0.5 0 0 0 0 0 0 0 0 0 0.5 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Figure 5: Transition probability matrix for a random walk on the pitch-class cy-cle, corresponding to “between-triad” movement in the triadic chromatic ap-proach

XY = Z =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

C C♯ D E♭ E F F♯ G A♭ A B♭ BC 0 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0C♯ 0 0 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083D 0.083 0 0 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083E♭ 0.083 0.083 0 0 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167E 0.167 0.083 0.083 0 0 0 0.083 0.083 0.167 0.083 0.167 0.083F 0.083 0.167 0.083 0.083 0 0 0 0.083 0.083 0.167 0.083 0.167F♯ 0.167 0.083 0.167 0.083 0.083 0 0 0 0.083 0.083 0.167 0.083G 0.083 0.167 0.083 0.167 0.083 0.083 0 0 0 0.083 0.083 0.167A♭ 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0 0 0 0.083 0.083A 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0 0 0 0.083B♭ 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0 0 0B 0 0.083 0.083 0.167 0.083 0.167 0.083 0.167 0.083 0.083 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Figure 6: Transition probability matrix combining both random walks, corre-sponding to a complete “step” of the triadic chromatic approach

Z2 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

C C♯ D E♭ E F F♯ G A♭ A B♭ BC 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097C♯ 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.104D 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069E♭ 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076E 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069 0.056F 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056 0.069F♯ 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076 0.056G 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069 0.076A♭ 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104 0.069A 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097 0.104B♭ 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125 0.097B 0.097 0.104 0.069 0.076 0.056 0.069 0.056 0.076 0.069 0.104 0.097 0.125

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Figure 7: Transition probability matrix for two complete “steps” of the triadicchromatic approach, with non-zero probabilities for every pitch-class regard-less of the starting note

4

Transition probability matrix for a random walk on the pc clockface, corresponding to “between-triad” movement

SMT | Nov 7, 2014

Y =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝


⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


XY = Z =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝


⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


Z2 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝


⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


4

Transition probability matrix combining both random walks, corresponding to a complete “step” of Garzone’s approach

SMT | Nov 7, 2014

Y =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝


⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


XY = Z =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝


⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


Z2 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝


⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


4

Transition probability matrix for two complete “steps” of Garzone’s random triadic approach

SMT | Nov 7, 2014

Z5 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝


⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Figure 8: Transition probability matrix for five complete “steps” of the triadicchromatic approach, starting to converge on the Markov chain’s stationary dis-tribution

Selected BibliographyClampitt, David. 1998. “Alternative Interpretations of Some Measures from

Parsifal.” Journal of Music Theory 42: 321–34.Cohn, Richard. 2012. Audacious Euphony: Chromaticism and the Triad’s Second

Nature. New York and Oxford: Oxford University Press.Garzone, George. 2008. The Music of George Garzone and the Triadic Chromatic

Approach. Savannah, GA: JodyJazz.———. 2009. “Basics of the Triadic Chromatic Approach.” Downbeat 76/5:

58–59.Gollin, Edward. 2000. “Representations of Space and Conceptions of Distance

in Transformational Music Theories.” Ph.D. dissertation, Harvard Univer-sity.

Harrison, Daniel. 1988. “Some Group Properties of Triple Counterpoint andTheir Influence on Compositions by J. S. Bach.” Journal of Music Theory 32:23–49.

Ingold, Tim. 2000. The Perception of the Environment: Essays on Livelihood,Dwelling, and Skill. London and New York: Routledge.

Lewin, David. 1987. Generalized Musical Intervals and Transformations. NewHaven: Yale University Press.

Lorentz, Jonathan. 2008. “The Improvisational Process of Saxophonist GeorgeGarzone with Analysis of Selected Jazz Solos from 1995–1999.” Ph.D. dis-sertation, New York University.

Rockwell, Joti. 2009. “Banjo Transformations and Bluegrass Rhythm.” Journalof Music Theory 53: 137–62.

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Transition probability matrix for five complete “steps” of Garzone’s random triadic approach

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Weaving

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Tim Ingold

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“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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GCF D A

BbEb FAbDb

BEA 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

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“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

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“The goal is to get this into your subconscious.”

—George Garzone (Downbeat January 2009, 99)

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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)

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sti

“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)

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sti

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Z5 =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝


⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

Figure 8: Transition probability matrix for five complete “steps” of the triadicchromatic approach, starting to converge on the Markov chain’s stationary dis-tribution

Selected BibliographyClampitt, David. 1998. “Alternative Interpretations of Some Measures from

Parsifal.” Journal of Music Theory 42: 321–34.Cohn, Richard. 2012. Audacious Euphony: Chromaticism and the Triad’s Second

Nature. New York and Oxford: Oxford University Press.Garzone, George. 2008. The Music of George Garzone and the Triadic Chromatic

Approach. Savannah, GA: JodyJazz.———. 2009. “Basics of the Triadic Chromatic Approach.” Downbeat 76/5:

58–59.Gollin, Edward. 2000. “Representations of Space and Conceptions of Distance

in Transformational Music Theories.” Ph.D. dissertation, Harvard Univer-sity.

Harrison, Daniel. 1988. “Some Group Properties of Triple Counterpoint andTheir Influence on Compositions by J. S. Bach.” Journal of Music Theory 32:23–49.

Ingold, Tim. 2000. The Perception of the Environment: Essays on Livelihood,Dwelling, and Skill. London and New York: Routledge.

Lewin, David. 1987. Generalized Musical Intervals and Transformations. NewHaven: Yale University Press.

Lorentz, Jonathan. 2008. “The Improvisational Process of Saxophonist GeorgeGarzone with Analysis of Selected Jazz Solos from 1995–1999.” Ph.D. dis-sertation, New York University.

Rockwell, Joti. 2009. “Banjo Transformations and Bluegrass Rhythm.” Journalof Music Theory 53: 137–62.

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melodic transformation in george garzone's triadic

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