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Copyright
by
Trent William Hanna
2003
The Dissertation Committee for Trent William Hanna
certifies that this is the approved version of the following dissertation:
Concerto for Piano and Orchestra
Committee:
_________________________________Kevin Puts, Supervisor
_________________________________Donald Grantham
_________________________________Russell Pinkston
_________________________________Stefan Kostka
_________________________________Thomas J. O’Hare
Concerto for Piano and Orchestra
by
Trent William Hanna, B.M., M.M.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Musical Arts
The University of Texas at Austin
May 2003
Dedication
My Concerto for Piano and Orchestra is dedicated to
Dr. John Paul, my piano instructor for many years.
v
Acknowledgements
First, I would like to thank the members of my committee: Kevin Puts,
Donald Grantham, Russell Pinkston, Stefan Kostka, and Thomas O’Hare. I am
grateful for their patience.
I would also like to thank Dan Welcher, with whom I studied composition
for two years before starting my dissertation.
It would be impossible to mention everyone who has supported me
through this dissertation and degree. I would, however, like to thank Beth Hiser,
David Diers, David Patton, Kyle Kindred, and Mike Vernusky for their friendship
these past few years. Thanks also to Tanya Weisheit, who not only helped with
proofreading the paper, but also showed her support in so many caring ways.
Thanks to my grandfather and my father, Ray and John Hanna. They both
received Doctoral degrees and I am proud to continue in this tradition.
I would also like to thank my daughter, Symphony, for being as patient as
a seven-year-old can be when her father is too busy completing his dissertation to
play with her every waking hour (as I would prefer).
Most importantly, thank you to my mother and step dad, Victoria and Joe
Courtney, for their constant, loving support throughout my educational pursuits.
vi
Concerto for Piano and Orchestra
Publication No._____________
Trent William Hanna, D.M.A.
The University of Texas at Austin, 2003
Supervisor: Kevin Puts
The Concerto for Piano and Orchestra is a one-movement work.
Symmetry and simple mathematical formulae are used throughout the work as
compositional tools and structural framework. This dissertation consists of the
full score for the concerto, followed by a discussion of the formal, symmetrical,
thematic, and mathematical elements of the music.
vii
Table of Contents
MUSICAL SCORE: CONCERTO FOR PIANO AND ORCHESTRA 1
ANALYSIS 62
Introduction .......................................................................................................62
Analysis .............................................................................................................65
Opening Section ........................................................................................65
Rehearsal Letters A Through C .................................................................72
Rehearsal Letter A ...........................................................................72
Rehearsal Letter B ...........................................................................74
Rehearsal Letter C ...........................................................................76
Rehearsal Letter D.....................................................................................78
Rehearsal Letters E and F ..........................................................................85
Rehearsal Letter E ...........................................................................85
Rehearsal Letter F ............................................................................86
Rehearsal Letter G.....................................................................................88
Rehearsal Letters H Through K .................................................................94
Rehearsal Letter H ...........................................................................94
Rehearsal Letter I .............................................................................94
Rehearsal Letter J ............................................................................94
Rehearsal Letter K ...........................................................................94
Rehearsal Letters L Through P ..................................................................96
Rehearsal Letter L ...........................................................................96
Rehearsal Letter M ..........................................................................97
Rehearsal Letter N ...........................................................................98
viii
Rehearsal Letter O ...........................................................................98
Rehearsal Letter P ............................................................................98
Closing Section .........................................................................................99
Rehearsal Letter Q ...........................................................................99
Rehearsal Letter R .........................................................................100
Rehearsal Letter S ..........................................................................100
Rehearsal Letter T .........................................................................100
Rehearsal Letter U .........................................................................101
Rehearsal Letter V ..........................................................................101
Rehearsal Letter W .........................................................................101
Rehearsal Letter X ..........................................................................102
Rehearsal Letter Y ..........................................................................102
Rehearsal Letter Z ..........................................................................104
Rehearsal Letter AA ......................................................................105
Rehearsal Letter BB........................................................................105
Rehearsal Letter CC........................................................................106
Rehearsal Letter DD .......................................................................107
Conclusion ......................................................................................................108
Vita ................................................................................................................109
With Intensity h = 72
With Intensity h = 72
© 2003 Trent Hanna
Trent Hanna (ASCAP)
Concerto for Piano and Orchestra
poco rit.
poco rit.
Piccolo
1Flute
2
1Oboe
2
English Horn
1Clarinet in Bb 2
3
1Bassoon
2
Contrabassoon
1Horn in F
2
3Horn in F
4
1Trumpet in Bb 2
3
1Tenor Trombone
2
Bass Trombone
Tuba
5 Timpani
Percussion 1
Percussion 2
Percussion 3
Percussion 4
Harp
Piano
Violin I
Violin II
Viola
Violoncello
Contrabass
Score in C
MMLOMMMM
ff
cadenza
1
h = 60
Pno.
fff
molto accel.
Perc. 2
Perc. 4
Pno.
Tam-Tam
pp
Bass Drum
pp
p p
p
f
2
2Powerful q = 112
Powerful q = 112
Senza misura
Senza misura
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
Cl. 1
2Cl.
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
1Tpt. 2
3
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
ff
ff
ff
ff
ff
ff
ff
p
stagger breathing
ff
p
stagger breathing
ff
ff
ff
ff
p
stagger breathing
ff
p
stagger breathing
ff
p
stagger breathing
ff
p
stagger breathing
f
p
Anvil
ff
f
Brake Drumff
f
cresc.
(cadenza)
f
5 56
7
div.
ff
div.
ff
div.
ff
ff
div.
p
( )
ff
div.
p
( )
3
1Bsn.
2
Cbsn.
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 2
Perc. 4
Pno.
Vc.
Cb.
Tam-Tam
pp
Bass Drum
pp
cresc.
1212 12 12
ff
9 9
9 9
4
5
(q = 112)
(q = 112)
Senza misura
Senza misura
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
Cl. 1
2Cl.
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
1Tpt. 2
3
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
ff
ff
ff
ff
ff
ff
ff
p
ff
p
ff
ff
ff
ff
p
ff
p
ff
p
ff
p
f
p
ff
f
ff
f
(cadenza)
f cresc.
3 3
6
fff
3 3
66
6
ff
ff
ff
unis.
ff
div.
p
( )
unis.
ff
div.
p
( )
5
8
As before q = 112
As before q = 112
accel.
accel.
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 2
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mf
ff
36 6
a2
mf
ff
36 6
a2
mf
mf
mf
ff
3
mf
ff
3
a3
mf
mf
mf
ff
3
mp
2
cresc.
ff
3
mp
cresc.
ff
3
mf
mf
a2
mf
ff
3
mf
mf
a2
mf
ff
3
a3
mfff
3
mf
mf
mf ff
3
mp
cresc.
mf ff
3
mp
cresc.
ff
3
mp
cresc.
f
3
Tom Toms
f
3
f
3
mf
mf
mf
ff
3
6
6
cresc.
3
mf
mf
mf
div.
ff
3
mf
mf
mf
div.
ff
3
mf
mf
mf
div.
ff
3
unis.
mp
cresc.
div.
ff
3
mp
cresc.
ff
3
6
13
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 2
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
3 cresc.
fff
3 5
3 cresc.
fff
3 5
3 cresc.
fff
3 5
3
cresc.
fff
3 5
3 cresc.
fff
3 5
3 cresc.
fff
3 5
3
cresc.
fff
3 5
3
cresc.
fff
3 5
3
cresc.
fff
3 5
3 cresc.
fff
fp
3 5
3 cresc.
fff
fp
3 5
3
cresc.
fff
fp
3 5
3
cresc.
fff
3 5
3 cresc.
fff
D-Eb, A-Gb, B-Bb3 5
3
cresc.
fff
Play toms randomly
as fast as possible 3 5
3
cresc.
fff
3 5
fff
5 6
cresc.
3
5 6
3 cresc.
fff
3 5
3 cresc.
fff
3 5
3 cresc.
fff
3 5
3
cresc.
fff
3 5
3
cresc.
fff
3 5
7
A
A
Sedate q = 72
Sedate q = 72
17
Picc.
Ob. 1
Cl. 1
Cbsn.
Hn. 1
Perc. 1
Hp.
Vln. I
Vln. II
Vla.
Vc.
Cb.
p
p
p
ppp
p
p
p
ppp
p
p
p
ppp
p
p
p
ppp
con sord.p
p
p
ppp
Triangle
p
mp
F#
B§
solo
p
four players
p
1/2 section
p
( )
solo
p
four players
p
1/2 sectionp
ppp
solo
p
four players
p
1/2 section
p
ppp
solo
p four players
p 1/2 section
p
ppp
solo
p three players
p
1/2 section
p
( )
(a tempo)
(a tempo)
26
molto rall.
molto rall.
As before q = 72
As before q = 72
Cl. 1
Hn. 1
Hn. 3
Tbn. 1
Tbn. 2
Vln. I
Vln. II
Vc.
Cb.
mp
solo
mp
cresc.
mf
mp
con sord.
p
con sord.p
con sord.p
con sord.p
pp
ppp
tutti
pp
pp
tutti
pp
ppp
8
34
B
B
poco accel.
poco accel.
Calm q = 104
Calm q = 104
Cl. 1
Hn. 1
Hn. 3
Tbn. 1
Tbn. 2
Pno.
Vc.
ppp
p
ppp
senza sord
p
ppp
senza sord
p
ppp
senza sord
p
ppp
senza sord
mp
p
mp
p
mp
p
pp
div. mp
41
Hn. 1
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
senza sord
mf
p
mp p mp mf
mp
p
mp
p
33 3
3
7
mp
tutti
mf
tutti
mf
tutti
p
div.
mf
ppp
p
div.
mf
tuttimp
ppp
9
47
Hn. 1
Tbn. 1
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mf
unis.
52
Picc.
1Fl.
2
Tbn. 1
Timp.
Perc. 2
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mf
a2
mf
f
mf
Bb-B§
Suspended Cymbal
ppp
mf
f
mp
9
96
6
f
decresc. mf
f
decresc.
( ) div.
mf
f
decresc.
( ) mf
f
decresc.
( ) mf
f
decresc.
( ) mf
10
57
Picc.
1Fl.
2
Eng. Hn.
1Cl.
2
Bsn. 1
Timp.
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
ppp
(no rit.)
ppp
mp
ppp
mf
dim.
mp
ppp
mp
ppp
mp
MLLOLLML
mp
glis
s.
p
p
pp
(no rit.)
4:3
4:3
ppp
mf
decresc.
mp
ppp
decresc. mp
decresc. mp
decresc. mp
11
C
C
Steady and Relaxed q = 120
Steady and Relaxed q = 120
62
Eng. Hn.
Bsn. 1
Timp.
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
p
ppp
p
ppp
p
l.v. Bb - B, Gb - A, Eb - F
bell tree
p
mp
glis
s.
pp
cresc.
mp
ppp
ppp
ppp
p
ppp
ppp
p
ppp
ppp
p
ppp
12
70
Fl. 1
Fl. 2
1Ob.
2
Cl. 1
Cl. 2
Cl. 3
1Bsn.
2
Cbsn.
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mf
f
3
3
mf
f
3
3
mp
mf
1.
f
mp
f
mp
mf
f
f
mp
1.
mf
f
f
mp
mp
mf
f
3 3
3
3
mp
f
mp
f
mp
f
mp
f
mp
pizz. arco
f
13
76
Fl. 1
Fl. 2
1Ob.
2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
1Bsn.
2
Cbsn.
Timp.
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mp
mp
3
3 3 3
mp
mp
33 3
mp
1.mf
mp
3 3
mp
mp
mp
mp 3
3
3 3
mp
mf
3
mp 3
3
3 3 3
1.
mp
mp
mp
mp
mp
LMLONNML
mp
mf
mp
3 3 3 3 3 3
3
3 3 3 3 3
mp
div.
mp
div.
mp
div.
mp
div.
pizz arco
14
81
Picc.
Fl. 1
Fl. 2
1Ob.
2
Eng. Hn.
Cl. 1
Cl. 2
1Bsn.
2
Cbsn.
Timp.
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mf f
mf
7
cresc.
mf f
mf
6
cresc.
mf
cresc.
f
p
cresc.
f
p
cresc.
mf
f
mf
mf
cresc.
mf
cresc.
f
p
1.
mf
cresc.
f
p
mf
F - E
f
decresc.
cresc.
f
decresc.
7
6
cresc.
f
cresc. unis.
f
cresc.
unis.
f
cresc.
unis.
f
cresc.
f
15
87
Picc.
Fl. 1
Fl. 2
Eng. Hn.
Cl. 1
Cl. 2
Bsn. 1
Bsn. 2
Cbsn.
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mf
5
mp
3
mf
f
5
f
p
3
mp
3 mp
3
mf
mf
3 mp
3
3 mf
f
3
f
mf
3 p 3
3
3
3
mp
p
3
3
3 3
3
mp
p
3
3
3 3
3
bell tree
mf
MNMOMNMM
p
3
cresc.
3 3
3
3 3 3 3
5
3 3
3
3
3
3 3 3
5
decresc.
unis.
ppp
decresc.
ppp
decresc.
ppp
ppp
pizzdecresc.
decresc.
ppp
16
D
D
Shimmering (q = 120)
Shimmering (q = 120)
92
Picc.
Fl. 1
Fl. 2
Cl. 1
Cl. 2
Cl. 3
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
ff
decresc. poco a poco (to m. 96)
f
ff
decresc. poco a poco (to m. 96)
f
ff
decresc. poco a poco (to m. 96)
f
ff
decresc. poco a poco (to m. 111)
ff
decresc. poco a poco (to m. 111)
ff
decresc. poco a poco (to m. 111)
Triangle
f
decresc. poco a poco (to m. 121)
Vibraphone
ff
decresc. poco a poco (to m. 119)
Vibraphone
ff
decresc. poco a poco (to m. 121)
Glockenspiel
f
l.v.
decresc. poco a poco (to m. 121)
ff
decresc. poco a poco (to m. 121)
C§
C#
ff
decresc. poco a poco (to m. 120)
1/2
ff
decresc. poco a poco (to m. 124)
div.
ff
decresc. poco a poco (to m. 124)
div.
ff
decresc. poco a poco (to m. 124)
ff
1/2decresc. poco a poco (to m. 124)
17
97
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
f
decresc. poco a poco (to m. 116)
f
f
decresc. poco a poco (to m. 116)
decresc. poco a poco (to m. 116)
( f )
( f )
( f )
( f )
( f )
( f )
( f )
( f )
C§
C#
( f )
div.
( f )
1/2
( f )
1/2
( f )
div.
( f )
18
102
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
( mf )
( mf )
( mf )
( mf )
( mf )
( mf )
( mf )
( mf )
( mf )
( mf )
( mf )
C§
C#
( mf )
( mf )
( mf )
( mf )
( mf )
19
107
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
( mp )
( mp )
( mp )
( mp )
p
( mp )
p
( mp )
p
( mp )
( mp )
( mp )
( mp )
( mp )
C§
C#
( mp )
1/2
( mp )
div.
( mp )
div.
( mp )
1/2
( mp )
20
112
Ob. 1
Ob. 2
Eng. Hn.
Bsn. 1
Bsn. 2
Cbsn.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
( p )
pp
( p )
pp
( p )
pp
p
decresc. poco a poco (to m. 120)
p
decresc. poco a poco (to m. 120)
p
decresc. poco a poco (to m. 120)
( p )
( p )
( p )
( p )
( p )
C§
C#
( p )
div.
( p )
1/2
( p )
1/2
( p )
div.
( p )
21
117
rit.
rit.
q = 60
q = 60
rit.
rit.
Eng. Hn.
Bsn. 1
Bsn. 2
Cbsn.
Tbn. 1
Tbn. 2
B. Tbn.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
mp
solo
( pp )
ppp
( pp )
ppp
( pp )
ppp
con sord
pp
decresc. poco a poco
ppp
con sord
pp
decresc. poco a poco
ppp
con sord
pp
decresc. poco a poco
ppp
ppp
( pp )
ppp
( pp )
( pp )
ppp
motor on
( pp )
ppp
ppp
( pp )
ppp
( pp )
ppp
ppp
ppp
div.
tutti div.
1/2
ppp
1/2 div.
1/2
div.
ppp
EPeaceful q = 60
126
Eng. Hn.
Pno.
p
solo
con.
22
136
Pno.
Vc.
slowly decelerate trill
mp
3
3
solo
pp
144
poco rall.
poco rall.
poco rall.
1Fl.
2
1Cl.
2
Pno.
Vla.
Vc.
mp
mf
pp
3
3
3 3
mp
mf
pp
3 3 3 3 3 3 3
3
3 3
3
3 3
33
3
1/2 mf
pp
1/2 mf
pp
As before q = 60
As before q = 60
148
(q = h.)
(q = h.)
F
F
Regal h. = 60
Regal h. = 60
Hn. 1
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
con sord.
p
p
mf
9 mp
f
6 6 6 6 6 9
9
6 6 6 6
69
tutti
p
sul pont. mp
mp
naturale
sul pont.
p
mp
mp
pizz.
tutti
sul pont.
p
mp
mp
pizz.
tutti
p
sul pont.
mp
pizz.
mp
pizz.
mp
23
158
Eng. Hn.
Hn. 1
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mp
ppp
mp
mf
171
Fl. 1
Eng. Hn.
Bsn. 1
Timp.
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mf
ppp
mf
ppp
mp
mf
ppp
p
mf
f
f
f
f
ff
3
f
sim.
arcof
sim. arco
f sim.
arco
f
sim.
pizz.
f
arco
24
181
Fl. 1
Fl. 2
Cl. 1
Cl. 2
Perc. 3
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mp
cresc.
3
mp
cresc.
3
mp
cresc.
3
mp
cresc.
3
Vibraphonemf
mf
cresc.
mp
dim.
mp
dim.
pp
mp
pizz. pp
mp
dim.
pp
pizz.
mp
pp
192
accel.
accel.
Picc.
Fl. 1
Fl. 2
Cl. 1
Cl. 2
1Bsn.
2
Cbsn.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mf
f
ppp
mf
f
ppp
mf
f
ppp
mf
f
ppp
mf
a2
mf
mf
mf
mf
3 3 3 3
3 3
3 3
mf
mf
arco
mf
mf
mf
mf
arco mf
mf
25
G
G
q = 192
q = 192
200
Picc.
1Fl.
2
Ob. 1
Ob. 2
Cl. 1
Cl. 2
Bsn. 1
Bsn. 2
Cbsn.
1Hn.
2
3Hn.
4
Timp.
Perc. 2
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
f
ff
5:6
f
pp
f
pp
f
ff
5:6
pp
f
ff
5:6
pp
pp
f
ff
5:6
pp
f
ff
5:6
f
ff
5:6
f
ff
5:6
mf
ff
5:6 F-A, A-C, B-D
Suspended Cymbal ff
dampen
Glockenspiel
f
ff
5:6
f
f
ff
5:6 pp
f
ff
5:6
pp
f
f
5:6 pp
arco
f
ff
ff
5:6 pp
26
210
Picc.
Ob. 1
Ob. 2
Cl. 1
Cl. 2
Bsn. 1
Bsn. 2
Perc. 4
Vln. I
Vln. II
Vc.
Cb.
p cresc. poco a poco
mp
mp
p
pp
pp
p
pp
p
p
pp
pp
p
pp
p
p
Glockenspiel
p
p
p
p
p
mp
27
220
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Cl. 1
Cl. 2
Bsn. 1
Bsn. 2
Tpt. 1
Tpt. 2
Tbn. 1
Tbn. 2
B. Tbn.
Perc. 4
Vln. I
Vln. II
Vc.
Cb.
mf
mf
mf
mf
mf
mf
mp
mp
mf
mp
mf
mf
mp
mp
mf
mp
mf
mf
con sord.mf
mf
con sord.
mf
mf
senza sord.
mp
mp
mf
senza sord.
mp
mf
senza sord.
mp
mp
mf
mp
mf
mp
mf
mp
mf
mp
mf
mf
28
229
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Cl. 1
Cl. 2
Bsn. 1
Bsn. 2
Cbsn.
Tpt. 1
Tpt. 2
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 4
Pno.
Vln. I
Vln. II
Vc.
Cb.
f
f
mf
f
f
f
mf
f
f
f
f
f
decresc.
f
f
mf
f
f
f
f
mf
f
f
ff
mf
f
f
f
mf
f
f
f
f
f
f
f
f
f
f
ff
ff
f
f
mf
f
f
ppp
f
ppp
f
ppp
f
ppp
29
238
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Cl. 1
Cl. 2
Bsn. 1
Bsn. 2
Cbsn.
Tpt. 1
Tpt. 2
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 2
Perc. 4
Pno.
Vln. I
mf
p
f
mf
mp
f
mf
mp
p
p
mp
p
mp
f
mp
f
mf
mp
f
mf
mp
mf
Tom Toms
ff
mp
cresc.
ff
30
247
Fl. 1
Fl. 2
Cbsn.
Tpt. 1
Tpt. 2
Tba.
Timp.
Perc. 2
Pno.
p
p
ppp
p
p
p
ppp
254H rit.
rit.
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Cl. 1
Cl. 2
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Pno.
ff
ff
f
ff
f
ff
f
f
ff
ff
D-Db, C-B, E-Eb, G-C
Crotales
ff
ff
Suspended Cymbal
ff
Bass Drum
ff
fff
31
264
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Cl. 1
Cl. 2
Cl. 3
mf
mp
mf
mf
mp
mp
mf
p
p
I Senza misura q = 80
274
poco accel.
Cl. 1
Cl. 2
Cl. 3
Pno.
ppp
ppp
ppp
p
cresc.
274
(q = 160) molto rit.
Pno.
32
275
J Brash q = 100
accel.
accel.
Cbsn.
Hn. 1
Hn. 2
Hn. 3
Hn. 4
Tpt. 1
Tpt. 2
Tpt. 3
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Perc. 3
Pno.
f
f
7:8
f
3
3
3 3
f
5:4
3
f
senza sord.
f
3 5
senza sord. f
3 3
senza sord.f
3 3
f
f
f
senza sord.
f
f
Ratchet
ff
K
q = 160284
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Perc. 4
Pno.
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
Slap Stick
ff
33
289
(no decresc.)
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Pno.
3
3
3 3 3 3
3
3
3 3 3 3 3 3
3
3
3 3 3 3
33 3 3 3
3 3 3 3 3
3 3 3 3
3
3
3 3 3 3 3 3
3
33 3 3 3
3
3 3 3 3 3 3
294
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Pno.
fp
( )
fp
( )
fp
( )
3
fp
( )
3
3
fp
( ) 3
3
3
fp
( )
fp
( )
fp
( )
3
fp
( )
3 chromatic cluster 5 6 7
34
L
L
Pulsating q = 120
Pulsating q = 120
298
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
1Bsn.
2
Cbsn.
1Tpt. 2
3
Tbn. 2
B. Tbn.
Tba.
Perc. 4
Pno.
Vc.
Cb.
fff
fff
fff
fff
fff
fff
fff
fff
fff
f
p
f
p
harmon mute
f
Use q. duration to open
3
con sord.
f
p
con sord.
f
p
con sord.
f
p
Bass Drum
mf
fff
gliss.
5 55
con
gliss.
5 5
55 5 5 5 5 5 5 5
pizz
f
(non div.) pizz
f
305
Pno.
55
55 5 5 5
5
5
5 5 5 5 5 55
313
Pno.
5 5
5
5 5 5 5
5
5 5 5 5 6
35
320
Pno.
ff
55
55
55
56 6 5
55
55
56 6 5
5 5
327
Pno.
5
55
55
55
55
5 6
331
Pno.
5
5 5 5 5 5 5 5 5 5
5 5 5 5
M
M
336
1Tbn.
2
B. Tbn.
Pno.
Vc.
Cb.
f
a2 3
f
3
5 5
3 5 5 5 5 5 5 3 5 5 5 5
5 5 3 5 5 5 5 5 5 3 5 5 5 5
f
3
f
3
36
342
1Fl.
2
1Cl. 2
3
1Tbn.
2
B. Tbn.
Pno.
Vln. I
Vln. II
Vc.
Cb.
a2
f
3
a3
f
3
senza sord.
f
3
f
senza sord.
3
5
5
5
5
5
5
5
4:3
5
4:3
3
4:3
5
4:3 4:3
5
4:3 4:33:2 3:2
4:3 4:33:2 3:2
4:3 6
5 5 3 5 5 5 5 5 5 5 5
6
f
3
f
3
f
3
f
3
347
1Fl.
2
1Cl. 2
3
Hn. 1
Hn. 3
Tpt. 1
1Tbn.
2
B. Tbn.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
3
f
3
f
f
senza sord.
f
f
3
f
3
5
5
5
5
5
5
4:3
3:2 3:2
4:3 6 3 4:33:2
3:2
4:3 6 3 4:33:2
3:2
4:3 6
5 5
6 35 5
6 35 5
6
f
3
f
3
div.
f
f
3
f
3
37
350
1Fl.
2
1Cl. 2
3
1Hn.
2
3Hn.
4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
3
f
3
f
f
1.2. senza sord.
f
f
3
3
f
3
3
5
5
5
5
6 4:3
3:2 3:2
4:3 6 6 4:33:2 3:2
4:3 6
6
5 5
6 6
5 5
6
f
3
f
3
f
f
3
3
f
3
3
38
352
Fl. 1
Fl. 2
1Cl. 2
3
1Hn.
2
3Hn.
4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
5 5
5 5
5 5
5 5
5 5
a3 senza sord.
f
5 5
5 5
5 5
senza sord.
f
5 5
5
5
5
5
6 4:33:2
3:2
4:3 6 6 4:33:2
3:2
4:3 6 6
6
5 5
6 6
5 5
6 6
5
5
5
5
5
39
molto rit.
molto rit.
354
Picc.
Fl. 1
Fl. 2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f cresc.
3 3 3 3 3 3
cresc.
3 3 3 3 3 3
cresc.
3 3 3 3 3 3
f cresc. 3 3 3 3 3 3
f
cresc.
cresc.
f
cresc.
f
cresc.
cresc.
cresc.
cresc.
cresc.
cresc.
cresc.
5
5
5
4:33:2
3:2
4:3 6 6 4:33:2
3:2
4:3
5 5
6 6
5 5
div a3
cresc.
div a3
cresc.
cresc.
div cresc.
div cresc.
5
40
N
N
Powerful q = 112
Powerful q = 112
355
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 3
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
ff
a2
ff
a2
ff
ff
a3
ff
ff
ff
ff
ff
ff
ff
ff
ff
f
Anvil
ff
Brake Drum
ff
Bass Drum
ff
5
5
ff
3:24:3
3:2 4:35 6
7
55
tutti ff
div.
tutti
ff
div.
ff
ff
ff
41
358
molto rit.
molto rit.
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Cl.
2
Cl. 3
1Bsn.
2
Cbsn.
1Hn.
2
Hn. 1
3Hn.
4
1Tpt. 2
3
Tpt. 1
Tpt. 2
Tpt. 3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
sfp
sfp
sfp
sfp
sfp
sfp
sfp
sfp
sfp
ppp
ppp
Suspended Cymbal
cresc. LMLOLMMLf
gliss.
ff
8:6
12 20
tutti
div.
tutti
ppp
gliss.
tutti div.
tutti
ppp
gliss.
tutti
ppp
gliss.
tutti
ppp
gliss.
42
O
O
Soaring q = 80
Soaring q = 80
361
rit.
rit.
Timp.
Perc. 2
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
l.v.Db-D, Eb-A
ff
l.v.
ff
gliss
.
fff
ff
gliss.div.
fff
ff
f
ff
gliss. fff
ff
f
ff
gliss. fff
ff
f
ff
fff
ff
f
tutti
ff
fff
ff
f
P
P
Senza misuraca. 20"
ca. 20"
369
Q
Q
Driving q = 160
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
gliss. Sul G
one by one (on conductor's cue)players gliss to D4 taking 2"-4".
gliss. p
gliss freely - unsteady rhythmq = 120-200 - bow as necessary
gliss. Sul G gliss.
p
gliss.
p
gliss.
Sul A gliss.
p
gliss.Sul G gliss.
p
43
374
Timp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
mp
ppp
ppp
ppp
ppp
pizz
f
pizz
f
380
Timp.
Pno.
Vc.
Cb.
385R
R
1Ob.
2
1Cl.
2
Perc. 1
Pno.
Vln. I
Vln. II
Vla.
Vc.
mf
a2
a2
mf
Marimba
ff
cresc.
mf
f
pizz.
f
mf
f
mf
f
mf f
44
389
1Ob.
2
1Cl.
2
Perc. 1
Pno.
Vln. I
Vln. II
mf
pizz.
f
393
S
S
Picc.
Fl. 1
1Ob.
2
1Cl.
2
Perc. 1
Pno.
Vln. I
Vln. II
Vla.
Vc.
mf f
6
mf
f
6
f
6
f
6
f
f
6
6
6
arco
f
6
arco
f
6
f
6
f
6
397
Hp.
Pno.
Vc.
Cb.
mp
6 6 6 6 6 6 6
arco
mp
arco
mp
45
401
T
T
1Fl.
2
Timp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
mf
mf
f
mf
f
f
f
mf
arco
f
mf
f
mf
f
3 3 3 3
404
Picc.
1Fl.
2
1Cl.
2
Timp.
Perc. 2
Perc. 3
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
1.
f
f
a2
f
Xylophone
f
Vibraphone
f
7
7
f
f
46
U
U
408
Picc.
1Fl.
2
1Ob.
2
Cl. 1
Cl. 2
1Bsn.
2
Tbn. 1
Pno.
Vln. I
Vln. II
Vla.
a2
mfmf
a2
mf
mf
3 3 3 3
a2
mf
mf
ff
mf
mf
mf
3 3 3 3
413
Picc.
1Fl.
2
1Ob.
2
Cl. 1
Cl. 2
1Bsn.
2
Tbn. 1
Timp.
Perc. 2
Vln. I
Vln. II
Vla.
mf
3 3 3 3 3 3 3 3 3 3 3 3 3 3
mf
3 3 3 3 3 3 3 3 3 3 3 3 3 3
gliss.
f
mf
Xylophone
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
47
V
V
418
Picc.
1Fl.
2
1Ob.
2
Cl. 1
Cl. 2
1Bsn.
2
Tbn. 1
Timp.
Perc. 2
Pno.
Vln. I
Vln. II
Vla.
f
f
f
f
f
f
A-Gb
f
423
W
W
1Cl. 2
3
1Hn.
2
3Hn.
4
1Tpt. 2
3
Pno.
Vln. I
Vln. II
Vla.
a3
f
con sord.
f
con sord.
f
a3 con sord.f
ff
f
f
f
48
428
1Cl. 2
3
1Hn.
2
3Hn.
4
1Tpt. 2
3
Pno.
Vln. I
Vln. II
Vla.
432
X
X
1Cl. 2
3
1Hn.
2
3Hn.
4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
senza sord.
ff
ff
senza sord.
ff
ff
a2
ff
ff
ff
ff
ff
ff
ff
Crotales
ff
Bass Drum ff
ff
4:3 4:3 4:3 4:3
f
49
4361
Hn. 2
3Hn.
4
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
50
441Y
Y
1Bsn.
2
Cbsn.
Hn. 1
Hn. 2
Hn. 3
Hn. 4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
a2
ff
ff
ff
ff
ff
ff
senza sord ff
a3
a2ff
ff
ff
f
Suspended Cymbal
ppp
ff
l.v.
Chimesff
Chimesff
Bass Drum
f
cresc.
fff
glis
s.gl
iss.
ff
51
447
(x = x)
(x = x)
Z
Z
Eng. Hn.
1Bsn.
2
Cbsn.
Hn. 1
Hn. 2
Hn. 3
Hn. 4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
ppp
ppp
ppp
ppp
ppp
ppp
ppp
ppp
ppp
ppp
ppp
B-Bb, A-F, C-Ab
Marimba
f
Snare Drum
f
ppp
ppp
ppp
ppp
ppp
ppp
52
454
Ob. 1
Eng. Hn.
Cl. 2
Bsn. 1
Bsn. 2
Tpt. 2
Tbn. 2
Perc. 1
Vln. II
Vc.
f
f
f
f
f
f
f
f
53
461
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Bsn. 1
Bsn. 2
Cbsn.
Hn. 1
Hn. 2
Hn. 3
Hn. 4
Tpt. 1
Tpt. 2
Tpt. 3
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
f
p
Crotalesff
Tam-tam
p
Chimes
ff
Glockenspiel
ff
f
f
f
f
f
f
54
467
(no rit.)
(no rit.)
Picc.
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Bsn. 1
Bsn. 2
Cbsn.
Hn. 1
Hn. 2
Hn. 3
Hn. 4
Tpt. 1
Tpt. 2
Tpt. 3
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
55
AA
AA
Triumphant q = 80
Triumphant q = 80
471
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
Tpt. 1
Tpt. 2
Tpt. 3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
ff
f
p
ff
ff
11 11 11 11
ff
ff
ff
ff
ff
56
473
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
Tpt. 1
Tpt. 2
Tpt. 3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
f
1111
11 11
57
475
molto rit.
molto rit.
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
Tpt. 1
Tpt. 2
Tpt. 3
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 2
Perc. 3
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
cresc.
3
cresc.
3
cresc.
cresc.
cresc.
cresc.
cresc.
cresc.
cresc.
cresc.
cresc.
3
cresc. 3
cresc.
ff
3 D-Db, F-Eb
Chimes
ff
3
Chimes
ff
3
3
9
9 9
3
1212
12
cresc.
3
cresc.
3
cresc.
3
cresc.
cresc.
58
BB
Relentless q = 160478
Pno.
mf
cresc.
482
CC
CC
Picc.
1Fl.
2
Ob. 1
Cl. 1
Cl. 2
Cl. 3
Bsn. 1
Cbsn.
Hn. 1
1Tpt. 2
3
Tbn. 1
Tbn. 2
B. Tbn.
Timp.
Perc. 1
Perc. 2
Perc. 4
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
ff
6 6 6
ff
6 6 6
ff
6 6 6
ff
fp
6 6 6
ff
6 6 6
ff
6 6 6
fp
fp
fp
ff
gliss.
ff
gliss.
ff
gliss.
ff gliss.
fp
mp
Flexatone
ff
gliss. gliss. Tam-Tam
p
Slapstickff
LMLOLNLL ff
fff
ff
sul A
ff
gliss. (l.h. pizz)
sul A
ff
gliss. (l.h. pizz)
sul A
ff
gliss.
(l.h. pizz)
sul A
ff
gliss.
(l.h. pizz) arco
div a4
p
59
486
Fl. 1
Fl. 2
Ob. 1
Ob. 2
Eng. Hn.
Cl. 1
Cl. 2
Cl. 3
Bsn. 1
Bsn. 2
Cbsn.
Hn. 1
Hn. 2
Hn. 3
Hn. 4
Tpt. 1
Tpt. 2
Tpt. 3
Tbn. 1
Tbn. 2
B. Tbn.
Tba.
Timp.
Perc. 2
Hp.
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
fp
arcodiv a4
p
arco div a4
p
arco div a4
p
( )
div a4
p
( )
( )
60
DD
DD
Chaotic q = 144
Chaotic q = 144
490
Picc.
1Fl.
2
1Ob.
2
Eng. Hn.
1Cl. 2
3
1Bsn.
2
Cbsn.
1Hn.
2
3Hn.
4
1Tpt. 2
3
1Tbn.
2
B. Tbn.
Tba.
Timp.
Perc. 1
Perc. 2
Perc. 3
Perc. 4
Pno.
Vln. I
Vln. II
Vla.
Vc.
Cb.
fff
3
fff
3
fff
3
fff
3
fff
3
fff
a2
3 3 3
fff
3 3 3
fff
3
fff
3
fff
3
fff
a2
3 3 3
fff
3 33
fff
3 3 3
Db-D§ ff
fff
3
3 3
Anvil
ff
3
l.v. Tom-toms
ff
fff
6
3 6
Brake Drum
ff
dampen tam-tam
3
Bass Drumff
3
3 3
palm cluster - white notes
forearm cluster - chromatic
sfffz
6
3
6
fff
3
non div.
fff
3
non div.
fff
3
non div.
fff
3
non div.
fff
3
non div.fff
3
non div.fff
3
non div.fff
3
3 3
non div.fff
3
33
non div.fff
3
33
non div.fff
March 2003
3
33
61
62
INTRODUCTION
I have long had the desire to compose a piano concerto. As a piano performance
major during my undergraduate and masters years I competed in several piano concerto
competitions and was fortunate enough to win a few of them. This, in time, led to the
opportunity of performing piano concerti with several orchestras. All this experience
helped me gain an appreciation for well-written “competition-winning” piano concerti.
My experience playing Gershwin’s Rhapsody in Blue and Concerto in F, Corigliano’s
Concerto for Piano and Orchestra, Ravel’s Piano Concerto in G Major, and Ginastera’s
Concerto for Piano and Orchestra left wonderful impressions on my mind, sparking the
aspiration to compose a piano concerto of my own some day.
Before I began working on the concerto, I knew from my competition experience
how exciting and rewarding a powerful ending can be for both the performer and
audience. Because I wanted to create such an ending in my concerto, my focus began at
the end. For years I had been playing a percussive “ditty” on the piano that had promise
as closing material. I am also very interested in symmetry as a compositional tool and
decided to use it as an underlying factor in the concerto. This fit in well with my plans,
for the little “ditty” possesses symmetry of its own, establishing the core concept for the
ending of the concerto. From this foundation, I began conceptualizing the rest of the
piece.
I became interested in symmetry in the Fall of 2001 while working on my first
string quartet (Tetrasomia). Although I was already employing symmetrical ideas in this
piece, I became further interested in their use when I began studying Bartok’s Fourth
String Quartet and learned of the importance of symmetry therein. Like my string
quartet, this piano concerto is symmetrically centered around the pitch D4. This works
63
out well on the piano, for not only are the pitches symmetrical in interval, but are visually
symmetrical as well.
Example No. 1
Although symmetry is an important element throughout the concerto, it does not
dominate the entire piece. Whereas I use symmetrical sets with varying degrees of
consonance, I also “break free” of this symmetry from time to time. My approach to
symmetry involves a focus on tonal or modal symmetrical sounds in opposition to the
more chromatic “dissonant-sounding” possibilities, which symmetry easily allows. It is
perhaps true that during the act of playing simultaneous random sounds at the keyboard,
dissonant harmonies are more often encountered than consonant ones. If symmetry is the
only guideline for this same practice, much of the same result would occur. My goal was
to use symmetrical pitch sets and to search for “pleasant” harmonies and melodies, and
contrast them with the more dissonant possibilities. As a result, symmetry replaces the
use of a key, so instead of a Concerto in D, for example, I have composed a Concerto in
Symmetry, in which D acts as the pitch center.
Another concept I focus on in my concerto is the use of proportions, contractions
and expansions of both meter and rhythm. Throughout the concerto there are many
64
instances of time signatures expanding or contracting the length of a measure, and
additions or subtractions governing the number of chord or pitch repetitions. The first
measure of the concerto (the introductory piano cadenza) introduces some of the
symmetrical and numerical concepts found throughout the concerto. While most of the
numerical processes in this concerto, mostly of addition and subtraction by one, are
somewhat simple, the likelihood of the audience hearing (or at least “feeling”) these
simpler deviations is far greater than if I used more involved mathematical formulae to
create rhythmic variation and development.
The orchestration consists of a standard size orchestra. The wind section includes
piccolo and two flutes, two oboes and English horn, three clarinets, two bassoons, and
contrabassoon. The brass section employs four horns, three trumpets, two tenor
trombones, bass trombone, and tuba. The percussion involves the use of four players
whose instrumentation includes glockenspiel, xylophone, vibraphone, marimba, two sets
of chimes, crotales, bass drum, tam-tam, snare drum, suspended cymbal, tom toms, slap
stick, flexatone, anvil, brake drum, ratchet, and bell tree. The instrumentation also
includes five timpani, harp, standard strings, and, of course, piano.
I have always had a fondness for single-movement works and therefore decided to
compose this piano concerto in one movement. Usually I have some sort of
programmatic notion in my compositions, but this piano concerto breaks from that
practice.
65
ANALYSIS
OPENING SECTION
As I have stated, most of the piece is symmetrical around D, and more specifically
D4. The concerto begins with a piano cadenza of intense force, commanding the
listener’s attention. The piano introduction, before the entrance of the orchestra, can be
divided into three main parts. The first part begins with A0, the lowest possible note on
the piano, followed by G7, the symmetrical counterpart of A0. After the pitch A1 is
added in the left hand, the right hand mirrors the left with G6. Following this, another
octave is added, and both hands are employed to accomplish the addition of symmetrical
pitches in both registers. The upper register’s symmetrical imitation of the lower register
continues as more and more pitches are added until both hands are playing five-note
chords in the extreme registers of the piano (See Example No. 2 below). As a means of
creating even greater forcefulness and intensity, grace-note chords are systematically
added to the beginning of these chordal outbursts. The lower register has two
unanswered instances of this before the upper register responds with its own grace-note
chord. Then the systematic repetition continues as both registers add two and three
grace-note chords, respectively. Up to this point the only pitches involved have been the
white notes excluding B and F – the pentatonic collection. With these two pitches
omitted, the two most dissonant intervals – the tritone and the minor second (and major
seventh) – have been avoided.
66
Example No. 2
The second part (first system, p. 2) introduces the previously mentioned
dissonance. With the grace-note chord count rising to four, the hands separate, and both
registers resound together for the first time. However, the grace-note chords are no
longer foreshadowing the same chord, but are announcing the arrival of symmetrical
dissonance. The left hand moves up to B0 and Bb1 while the right hand goes down to F#6
and F7. Pitches B and F are now introduced as well as the black-note counterparts (Bb
and F#). More dissonant intervals (the tritone and the minor second) and the black-note
“realm” are introduced simultaneously.
67
Example No. 3
Following this swift transitional chord is a black-note contrasting gesture to C#4
and D#4 (Example No. 3). This occurs two more times, with the only difference being an
additional grace-note chord before each. After the third utterance, the arrival at C#4 and
D#4 is not sustained as both hands rise together in diminished-seventh arpeggios a major
second apart, ending at the extreme high register of the piano; A#7 and C8 (the
enharmonic equivalent of a major second).
Example No. 4
In the introduction’s third part, the music returns to the middle register of the
keyboard. No longer employing C#4 and D#4 as an intervallic center, it is expanded
68
outward by a minor second to C4 and E4. Once again pitches are symmetrically added to
each hand. This time, however, the addition of pitches occurs in an outward direction
simultaneous in both hands and involves more dissonance than the pentatonic gestures at
the outset of the piece.
Example No. 5
These expanding gestures arrive on a chord that contains the previously avoided pitches
B and F, as well as the E and C from whence they came. Each chord reaches a total of
five notes before beginning again on the same two pitches. This chord is repeated four
times before an extra pitch is added to each hand. A written-out accelerando continues
as the chord is repeated five times before the final pitch is added to each hand. A bass
drum and tam-tam roll join in as the repetition of this chord accelerates and builds
intensity into the introduction of the full orchestra.
69
The orchestra makes a powerful entrance and is symmetrical as well. After the
initial Ds on the downbeat in the low register, the axis moves up an octave and a tritone
for the accented chordal response occurring on beat 3. This is in keeping with a familial
axis, D and Ab sharing the same symmetry.
Example No. 6
The low Ds sound again, followed by two chords, which now converge on the D axis by
semitone. The low Ds respond in the next measure by expanding by semitones to a
sustained Eb and Db dyad as the piano continues with symmetrical virtuosic cadenza
material.
Example No. 7
70
After this display the orchestra part returns with material similar to its entrance.
However, this time the accented chords spread outward by semitone, while the low Ds
expand even further to a sustained tritone (B and F), over which the piano performs a
rather difficult octave section. The next full orchestral entrance imitates a previous part
of the piano introduction (2nd system, p. 2) before reaching a repeated chord with a
written-out accelerando, much like before. This repeated chord, however, accelerates at
three levels.
Example No. 8
The most obvious of these is the actual accelerando indicated in m. 11. The second is the
written-out accelerando which involves the orchestra beginning with half notes, followed
by half-note triplets. This is imitated twice in diminution, followed by four sixteenths
and five sixteenths, respectively, before reaching the measure of fermata (m. 15). While
this is going on, the piano part follows a different pattern. Each two beats that pass
involve the piano part playing one additional utterance of a chord. The hands are spread
at the extreme registers like at the beginning of the concerto, but move inward every two
beats to a different, symmetrical chord. Also, pitches are added much like before,
reaching five pitches per hand by m. 13. In the final measure before the fermata (m. 14),
the repeated chords are replaced by trilled chords that ascend together into the high
71
register of the piano, reaching a highly dissonant, raucous trill as the orchestra also makes
its frenzied arrival at the measure of fermata. A measure of rest follows to allow the
listener to reflect on what just took place, and to imagine what might possibly lie ahead.
72
REHEARSAL LETTERS A THROUGH C
Rehearsal Letter A
As a means of relief, the following section (Rehearsal Letter A, beginning in m.
17) is quite subdued. The orchestration begins with woodwind quintet, triangle, harp,
and solo strings. D4 is the first pitch sounded, followed by C3 and E5. The next two
pitches, however, do not follow the usual scheme of symmetry, introducing the first
example of what I term “combined symmetry.” This involves a group of notes that do
not form symmetry by themselves, but create symmetry when combined with a nearby
group of notes. In mm. 18-19, Bb1 and F6 are introduced, but are not symmetrical to D4.
In m. 20 D4 returns, followed by E5 and C3. This time, however, F#6 and B1 replace
their symmetrical counterparts. These two pairs of pitches are symmetrical to each other.
Example No. 9
I first experimented with this concept of combined symmetry in my string quartet.
In it, as a scale is being repeated up and back from A3 to D5, the pitches being employed
alternate between those of B-natural minor and Bb major. The B and C# offset the Eb and
F, while the Bb and C do the same to the E and F#. Together the two scales form a
symmetrical bond.
73
Example No. 10
In the concerto, I use this concept as a way to incorporate some tonal elements while
maintaining a symmetrical basis.
The main musical theme (Theme 1) of the entire concerto is introduced in mm.
26-28 by the solo clarinet. While the outer pitches (G1 and A6 - an inversion of the
beginning pitches of the concerto) are sustained from the previous section, the clarinet
begins the main theme on the primary axis of D4. This is a symmetrical theme, using
each “white-key” pitch class. The first three measures of this melody contain the
thematic material that is used throughout the concerto.
Example No. 11
74
The second part of the melody (mm. 52-56) does not return until Rehearsal Letter AA (m.
471), an intense climatic section of the work. The melody in its entirety radiates a sense
of hope as the clarinet searches high and low before eventually locating its resting spot an
octave below its beginning pitch.
Rehearsal Letter B
At Rehearsal Letter B (m. 36) the secondary theme (Theme 2) is introduced by
the cello section, and the piano makes its first entrance since the cadenza. The piano part
acts as an accompaniment, while its pitch content is based on the primary theme.
Example No. 12
The left hand echoes each pitch of the right hand except for the final pitch. At
this moment the left hand begins the theme and the right hand imitates, continuing the
pattern. The cellos are divisi: the upper part has the melody while the lower is in
symmetry. By m. 42 the violins and basses relieve the cellos due to the range
considerations caused by the outward expansion of the two lines. The piano part has a
small written-out accelerando propelling into m. 46, at which point the string section
expands into symmetrical six-note chords. Here, the right hand is relegated to continue a
75
somewhat imitative version of the accompaniment performed previously by both hands.
However, as the musical line reaches its valleys and peaks, the intervals between the
imitated pitches are reduced each time by one step, resulting in fourths by the middle of
m. 51. Also, the piano part avoids playing any Ds throughout this section, and the
intervals are altered where necessary to avoid this pitch.
Example No. 13
The piano part throughout Rehearsal Letter B begins with all white notes, but pitches are
altered in order to conform to the symmetrical harmony being produced in the strings. I
call this “altered symmetry.” The concept of symmetry is still the overriding factor, but
rules are bent in order to maintain pitch consistency between the piano and orchestral
parts.
76
Rehearsal Letter C
Rehearsal Letter C (m. 62) begins with the piano, in an accompanying role,
emerging from the chord being sustained from the previous section. The tempo is
slightly faster and the alternating meters of 3/4 and 7/8 add a gentle bounce to the
section’s beginning. Both the right hand and left hand parts in the piano use combined
symmetry. The piano part alone introduces the accompaniment, but soon fades out as the
string section takes over. Beginning in m. 70, three aspects control the piano part. The
piano begins with a transposed retrograde version of the primary theme in octaves (A).
This is followed by a rising dissonant “push” (B -- a variation on the third part of the
piano’s introduction). A transitional flourish of sixteenth notes in stepwise, parallel
quintal harmonies (C) returns the piano part to the main theme, no longer in retrograde
and harmonizing itself in the left hand at the interval of a major second (m. 77).
Example No. 14
77
Another stepwise sixteenth note flourish follows, this time in quartal harmony (m. 81).
Now, in m. 82, the piano part is displayed in thirds and, in m. 89, fourths and fifths.
Interval expansion is the overriding factor of the piano part in this section, much like the
earlier interval contraction in mm. 46-51. Throughout this section, I also wanted to do
something uncommon in piano concertos; coloring the solo line with a woodwind texture.
As the melodic line reaches different levels and intensities, the orchestral timbre changes
ever so slightly. As the section nears its end, the strings fade out as the music rises up in
the woodwinds and piano to the upper register. This is where the next section begins.
78
REHEARSAL LETTER D
Beginning with a bright, shimmering orchestration and diminishing in both range
and intensity throughout its length, Rehearsal Letter D (m. 92) employs several methods
of organization to control its shape and flow.
The section’s main framework consists of two three-note chords of quintal
construction in each measure; the lower chord ascends by semitone as the upper chord
descends by the same interval. Since the distance between the two sets of pitches is four
semitones, the chords exchange positions after five measures. I have numbered the
voices of the chords from bottom to top (1 to 6, respectively) for analysis.
Example No. 15
The pattern of beats per measure throughout the section is as follows:
Example No. 16
Mm. Beats per measure92-96 5 4 3 2 197-102 4 5 3 2 1102-106 3 4 5 2 1107-111 2 3 4 5 1112-116 1 2 3 4 5
79
In the first five measures of the section the high descending quintal chord group
consists of piccolo and flutes 1 and 2 (mm. 92-96), while the lower group begins with the
clarinets. Following this, the descending chord group drops down an octave, and
clarinets replace the flutes and piccolo. The pattern continues as shown below (Example
No. 17).
Example No. 17
The strings at Rehearsal Letter D are called upon to play high harmonics
following the same chordal pattern described above. However, like the woodwinds, the
strings do not maintain the same position in each five-measure grouping. In fact, the
strings never repeat the same five-bar pattern, but maintain a structural symmetry as
shown below (Example No. 18).
80
Example No. 18
The piano part in this section uses only four of the six notes from the two chords,
producing two-note tremolos in each hand. Each five-bar section in the piano part uses a
different arrangement of four of the six pitches, each pitch following the same semitone
ascent or descent as in the orchestral part. In the piano part, each pitch position is used a
total of six times. As the section nears the end in mm. 117-120, a different arrangement
of four of the six pitches changes each measure.
Example No. 19
mm. 92-6 97-101 102-6 107-11 112-6 117 118 119 1206 6 6 6 6 6 = 65 5 5 5 5 5 = 64 4 4 4 4 4 = 63 3 3 3 3 3 = 6
2 2 2 2 2 2 = 61 1 1 1 1 1 = 6
81
Example No. 20
The glockenspiel takes the place of one woodwind instrument per measure (an
octave higher), creating both a coloristic effect for this section and a chance for the
instrumentalists to breathe. The glockenspiel part changes position each measure,
starting in position 1 and continuing in opposite numerical order (i.e. 6, 5, 4, etc.).
Example No. 21
82
In this section I was also searching for a defining, methodical embellishment in
the piano part, and my decision was to employ grace notes. The grace notes follow their
own numerical layout, while pitch-choice is defined as follows. Each five-bar repetition
of the five-chord exchange pattern involves some repetition of pitches identical in both
the ascending and descending lines. These pitches are A, B, and Bb, and are excluded as
grace notes, while E is also excluded as it is the centrally introduced, non-repetitive pitch.
The remaining pitches in order (based on the ascending pattern) are G, D, Ab, Eb, F, C,
F#, and C#.
Example No. 22
I avoided using grace notes in every measure for the sake of variety and ease.
Each measure of 1/4 is denied grace notes due to the short length of these measures,
while the first measure in each five-measure grouping is without embellishment as well,
due in part to its following the short measure of 1/4. This leaves fifteen measures to be
accounted for. The first of these fifteen measures contains one grace note (m. 93),
followed by two in the second, and so on, until the middle of the fifteen measures is
reached, which has the highest grace note count of eight (m. 104). Then each following
measure reduces the total by one, returning to one grace note by the final measure
involved (m. 116). The placement of the grace notes in the right and left hand is as
follows: (the vertical lines indicate bar lines and the numbers indicate the order of the
grace-note’s appearances)
83
Example No. 23
The pitch order of the grace notes (G, D, Ab, Eb, F, C, F#, C#) is not maintained,
but is altered much like the time signatures’ arrangement in this section is altered. While
the time signatures, grouped in five-bar sets, make a complete reversal after five cycles,
the order of the grace-note pitches follows the same pattern, but the number eight
controls the formula. The chart below demonstrates how each eight-note group involves
the use of a new pitch in the hierarchy. In other words, as each new row emerges, the
closest pitch to the beginning that has not yet been the first pitch becomes the first pitch
for the next row, while the remaining pitches maintain their order. This formula works
out so that no matter how many pitches (or time signatures) are being accounted for, it
takes that same number of times for the order of the pitches (or time signatures) to be
reversed.
Example No. 24
84
A somber English horn solo marks the end of this section (mm. 120-127) and introduces
the main theme (Theme 3) for the upcoming slow section.
85
REHEARSAL LETTERS E AND F
Rehearsal Letter E
The slow section of the concerto (Rehearsal Letter E, m. 126) is the most “tonal”
sounding of all the sections. The left hand remains symmetrical throughout most of the
section, starting with the first two pitches of the concerto – G and A. This repeated
accompaniment figure, beginning with a dyad consisting of G3 and A3 followed by D4,
expands outward; G3 and A4 expands to F3 and B5, while D5 expands to C5 and E5.
The right hand has the melody (Theme 3), which is actually a transformation of the main
theme reduced to just five pitches.
Example No. 25
The piano part becomes slightly more involved beginning in m. 136 as a solo
cello is called upon to perform a rendition of Theme 2. In mm. 144-145, the piano part
maintains octave-equivalent symmetry offset by differing rhythms in each hand, while
the flutes imitate the right hand a beat later.
86
Example No. 26
Two dissonant symmetrical flourishes expand outward (mm. 148-149), catapulting the
music into virtual chaos.
Rehearsal Letter F
This is interrupted by a light, innocent waltz in the strings, introduced at
Rehearsal Letter F (m. 150). The melody in the first violins is a transformed version of
Theme 3, while the accompaniment involves pizzicato in the rest of the section. The
phrases are five, four, three, and two measures long, respectively, and each contains eight
notes that maintain a similar shape. During these measures, the theme attempts to return
to its original pitch level (Bb, C, D, E, F) as each of the phrase’s beginning pitches is a
minor second higher than the one before.
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Example No. 27
The piano makes its entrance in m. 164 and sounds as if it will be returning to the original
pitch level of Theme 3. This, however, is not the case as the piano part begins on Db just
as the first violins had, but eventually does return to the D combined symmetry by m.
174. Following this, the outward expanding flourishes return with woodwind
reinforcement, no longer deceiving the listener as they send the music into controlled
chaos.
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REHEARSAL LETTER G
At the beginning of Rehearsal Letter G (m. 200), the harmony from Rehearsal
Letter E returns with pitches G and A inverted. The music is no longer subdued (as
before), but brash, demanding attention. The dyad of A and G is sounded twice at the
dotted half note, followed by the F-B dyad three times in half notes. This pattern of
addition by one dyad continues, with four and then five dyads presented, each evenly
spaced within six beats.
Example No. 28
This section involves eight different repetitive melodic lines, each a
transformation of one of the first three themes. One line at a time is introduced and
continues as a new one is added. However, the number of measures before the
introduction of a new line lessens by one each time. As soon as all eight of the lines have
reached fruition, they each are removed from the texture, one by one, starting with the
chordal framework in the strings. The elimination of lines, however, is spaced by beats
rather than measures. The basses are the first instruments eliminated following one
measure after the introduction of the timpani part (the last of the eight lines being
successively added). Following this, the cellos exit two beats after the basses. Then
three beats pass before the third part (Vln. II) is eliminated. This pattern continues until
every line is gone except the piano and the percussion.
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Example No. 29
Mm. 8 7 6 5 4 3 2 1
Ob. 1.2, Vla.Cl. 1.2., Bsn. 1.2.
Picc., Glock.Tbn. 1.2., B. Tbn.
Fl. 1.2., Tpt. 1.2.Piano (Theme from Waltz)
Cbsn., Tba.Timp.
Beats 1 2 3 4 5 6 7 8 9 10Cb.
Vc.Vln. II
Vln. IOb. 1.2.
Cl. 1.2., Bsn. 1.2.Picc., Glock.
Tbn. 1.2., B. TbnFl. 1.2., Tpt. 1.2
Cbsn., Tba.
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Example No. 30
Example No. 31
Group Note Count Note Value Duration (beats)8 5 Eighth-note 2 1/27 5 Eighth-note 2 1/26 10 Eighth-note 55 11 Quarter-note 114 5 * 7 1/24 5 r.h. ** 9
4 l.h. 62 12 Dotted quarter 11 1/21 5 Half-note *** 10
* Durations change and follow same pattern used with alteration of time signatures at Rehearsal Letter D.** See below.*** Pitches change every half-note value, but rhythmic values get systematically added with eachrepetition.
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The piano part of Rehearsal Letter G (beginning in m. 229) consists of two main
elements. The melody is based on the theme from the waltz section and is presented in
octaves in the right hand. While this repeated melody consists of five notes, a
symmetrical pattern contained inside the octave melody embodies a four-note pattern.
This same four-note pattern is used in the left-hand accompaniment as well. However it
neither synchronizes rhythmically with the right hand representation nor is it presented in
the same order.
Example No. 32
The melody at the onset lasts for three bars and is repeated twice. The following two
times, the melody is reduced to dotted quarters (m. 235), now totaling only five measures
compared to the previous six. The next two repetitions of the melody reduce the rhythm
of the right hand to alternating quarter and eighth notes (m. 240), decreasing the number
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of bars once again by one. This is followed by a representation of the melody in eighth-
notes (m. 245), repeated six times in order to complete a cycle, synchronizing the
beginning of the measure with the beginning of the pitch pattern.
Example No. 33
During this, the left hand accompaniment changes in accordance with the melody,
but follows its own patterns. First the left-hand rhythm resembles that of the
accompanimental rhythms towards the end of Rehearsal Letter F. The four-note pattern
(C#, D#, C, E) cycles through three times in the left hand as opposed to two in the right.
As the rhythm in the right hand reverts to dotted quarter notes (m. 235), the
accompaniment avoids coinciding with the melody. As the right hand melody reverts to
the use of quarters and eighths (m. 240), the left-hand pitches are each held for the
duration of seven eighth notes. When the melody is finally reduced to eighth notes, the
left hand’s octaves are held for the duration of five eighth notes (mm. 245-249). This
finally culminates in both hands playing alternating sixteenth notes, repeating the melodic
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pattern six times (once again) in order to synchronize the cyclic pattern with the bar line.
Then both hands, in octaves, descend into the low register of the piano, avoiding the
target G# until abruptly landing on it with the support of the bombastic percussion. This
is quite an important pitch in the piece as it is the other axis of symmetry (Ab).
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REHEARSAL LETTERS H THROUGH K
Rehearsal Letter H
As the G#s in the piano and crotales diminish, the upper woodwinds emulate the
piano’s descent with a brittle rendition of their own in quarter note durations, alternating
between the five-note symmetrical pattern around both D and Ab, until finally resting on
D5.
Rehearsal Letter I
An elision in the piano begins a slow solo interlude displaying Theme 1 in all
twelve keys (m. 274). A crescendo and accelerando throughout Rehearsal Letter I propel
the symmetrical motives downward to low Abs.
Rehearsal Letter J
Following this, the brass section mimics the material from the previous section,
but whereas the piano was moving deeper and deeper into the lower register, the brass
section rises out of the depths before reaching a climatic dissonant symmetrical chord (m.
283). Also, the theme moves further and further from symmetry and becomes
rhythmically more complex.
Rehearsal Letter K
As if reverberating from the biting brass chord, a frantic woodwind section
appears (Rehearsal Letter K, m. 284) that foreshadows the closing section’s material.
Nine woodwind instruments are used in this section and are divided into three groups by
classification (piccolo and flutes, clarinets, and oboes and English horn). Each group has
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one instrument beginning on either pitch D, Eb, or C#, and every instrument follows the
pitch order of C#, D, Eb, D. The woodwinds have six beats of sixteenth notes using these
pitches before the piano makes its entrance. The piano part consists of the same pitches
as well. The woodwinds return with just five beats before the piano answers with three
beats. This pattern is repeated once more before the forces join together. At this
moment, the piano’s minor seconds are expanded to major, then minor to major thirds,
and so on, until the pianist is dared to attempt such an erratic spread as two minor
sevenths in each hand (mm. 290-293). This is all going on as the woodwinds are
seemingly running out of “rhythmic” gas. But that is not the case. They are just waiting
for their moment to trill – and reintroduce a strong brass downbeat as the piano
dissonantly reaches a cluster and glissando.
Example No. 34
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REHEARSAL LETTERS L THROUGH P
Rehearsal Letter L
The middle cadenza (Rehearsal Letter L, m. 298) begins with a thunderous cluster
and a rising glissando in the right hand, quickly answered by a descending glissando in
the left. A quintal arpeggiation in the left hand begins at the same time as the cluster and
becomes the underlining force throughout the cadenza. An upward arpeggiation of the
quintal chord consisting of pitches D3, A3, and E4 is followed by the symmetrical
equivalent (C4, G4, and D5). This arpeggiation lasts for seven measures before a new
arpeggiated chord and its symmetrical equivalent takes place. This pattern continues two
more times, one measure in length shorter each time. The notes being displayed outside
of the arpeggiations are also symmetrical and their rhythmic placement is proportional as
well.
Example No. 35
Starting in m. 320, Theme 3 is now placed in the left hand in octaves as the right
hand arpeggiates both of the quintal harmonies from the first measure of Rehearsal Letter
D (m. 92). The left hand, when not playing the melody, supports the right hand with
arpeggiations of its own. As the melody ends, the right hand arpeggiation returns to
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arpeggiating the beginning harmony from Rehearsal Letter D. When the two hands
arpeggiate simultaneously, the left hand has B, F#, and C# as the right hand has G, D, and
A, and vice versa.
Rehearsal Letter M
The orchestra makes its return at Rehearsal Letter M (m. 336) with Theme 3
beginning on F#. After the second entrance of this group, woodwinds and violins answer
beginning on Eb, the five pitches as a group being symmetrical to the five in the first
group. A third group makes an entrance after each has begun again. This group
represents the possibilities of combined symmetry between Bb major and B minor
simultaneously.
Example No. 36
In mm. 352-354 more pitches get added, producing a cluster-like chord, yet still
maintaining its symmetry. The piano part throughout this orchestral conglomeration
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continues its arpeggiations of the previously mentioned quintal chords as additional
pitches get added, producing a much more dissonant effect in the piano part as well.
Rehearsal Letter N
The material at Rehearsal Letter N (m. 355) represents an inverted version of the
orchestral entrance at the beginning of the concerto. The low Ds are replaced by high Ds
and the clangorous chords are now produced in the low register of the orchestra. This
section is much shorter than the beginning’s version because of the absence of a cadenza.
The piano, however, still performs some rather virtuosic passagework, but in much more
condensed forms. M. 357 displays borrowed material from m. 7 of the beginning
cadenza, but now the octaves alternate between both hands and ascend. In m. 360 the
material is borrowed from the end of m. 4, while the rhythm is altered to accommodate 8,
12, 16, and 20 pitches, respectively, in each beat.
Rehearsal Letter O
The next section is a rich, lush, shortened version of Rehearsal Letter B
(Rehearsal Letter O, m. 361) and remains symmetrical throughout. The section
culminates in the strings reaching a Db major seventh chord three times.
Rehearsal Letter P
As the third chord is held, each player is to slowly glissando, one by one, towards
D4 on the conductor’s command (m. 369). As each player reaches this pitch, he is to
glissando inward and outward from it within a semitone. This sets up the final section of
the work.
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CLOSING SECTION
Rehearsal Letter Q
The closing section of the concerto begins at Rehearsal Letter Q (m. 370). Two
new rhythmic transformations of the main theme control most of this section. The first
(Theme 4) is presented by the piano beginning in m. 377 and is based on combined
symmetry expanding outward, while the rhythm is non-retrogradable. Theme 5 is
introduced in m. 387 by the oboes and clarinets and follows the same rhythmic pattern set
up by Theme 4. It uses the same pitches as Theme 1, but differs in that it is includes an
elided retrograde version.
Example No. 37
The piano part is rather percussive throughout the final section, based on the little
“ditty” mentioned earlier. The pianist is called upon to play this section much as a mallet
player would play a marimba or xylophone part, except the pianist uses his index fingers
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instead of mallets. It begins with a rapid succession of repeated D4s. Once again the
focus is on the combined symmetry between Bb-F and B-F#.
Rehearsal Letter R
At Rehearsal Letter R (m. 387) the marimba replaces the piano and continues the
repeated sixteenth notes, while the repeated pitches now are G and A, sounding
simultaneously. Oboes and clarinets, as mentioned above, are called upon to play Theme
5 with the axis moved up to A4. After four measures, the piano joins in, filling the
melody in with quartal arpeggiations.
Rehearsal Letter S
As the piano part regains the repeated sixteenth notes (m. 396), the axis returns to
D4. The quartal concept remains in the piano part, first punctuating downbeats, then
strong beats, and finally taking over every beat.
Three measures of transition move the axis of symmetry back up to A4, but this
time the marimba is not called upon, as the piano part remains to perform the repeated
sixteenth notes, punctuated with arpeggiations.
Rehearsal Letter T
This time the punctuation is on the first beat of each measure, getting closer
together as each measure contracts by a beat (mm. 403-406).
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Rehearsal Letter U
At Rehearsal Letter U (m. 408) the orchestra takes over the melody for the first
time with the D4 axis. Four rhythmic transformations of Theme 5 are introduced at two-
measure intervals. Only the first and third synchronize at the bar line.
Example No. 38
Rehearsal Letter V
Rehearsal Letter V (m. 418) begins with an axial transposition up a whole step to
B4. The pianist is no longer using just his index fingers, as octaves are used to play the
moving outer parts of Theme 4.
Rehearsal Letter W
As the orchestra mimics the previous eight bars of the piano (Rehearsal Letter V),
it is the piano’s turn to show off by playing some bravura sixteenth note passages, two
octaves apart (m. 426).
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Rehearsal Letter X
Rehearsal Letter X (m. 432) has a return to the D axis. While the material here is
similar to the initial entrance of the orchestra (m. 2), the previously symmetrical chords in
the orchestra are replaced by only piano and crotales. Furthermore, the chords no longer
contain all the pitches from before, but are replaced with the harmony from the first three
measures of Rehearsal Letter A. The second chord uses the pitches from mm. 20-22,
while the third chord continues this pattern, using the same pitches as in mm. 23-25.
Here, the crotales sound the chord once as the piano plays it four times. This chord is
repeated again with an additional symmetrical pitch in both hands and is played only
three times. This pattern of contraction happens once more before a sixteenth note octave
sweep downward in the piano, landing on octave Ds in both hands, followed by a
powerful glissando upward to high Ds in octaves, at which the brass makes its entrance
(Rehearsal Letter Y).
Rehearsal Letter Y
Here Theme 1 is split into its two symmetrical halves (m. 442), resulting in one
ascending and one descending line. This is actually not the first occurrence, for Theme 4
was also split in this way at the previous Rehearsal Letter. However, this is more
transparent as both lines sustain the pitches throughout the section. The chimes is the
only instrument playing both halves of the theme, but it features the theme with the upper
part of the melody as the lowest part and the lowest part as the highest.
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Example No. 39
Theme 1 is announced four times, and each time the rhythm gets more compact.
Example No. 40
104
Rehearsal Letter Z
At Rehearsal Letter Z (m. 450) the meter suddenly changes to 12/16 as the brass
fade out. The accented rhythm is no longer syncopated, but synchronized with the beat.
In this section, I wanted Theme 1 introduced in a systematic way. The only parameters
being altered for each introduction of the theme are register and rhythmic placement. I
used nine separate introductions of the theme throughout this section. The organization
of the theme’s entrances concerning register is simple. There are three divisions of
register - middle, high, and low – and three groups introduced in order using each of
these register placements. This results in nine occurrences of Theme 1. I use three
possible orders of register placement that contain no repetition of placement in either of
the groups; middle-high-low, high-low-middle, and low-middle-high. My desire was to
start out in the middle register and end in the high register, so the order above worked out
best. The first group employs one instrument for each of the registers. The second and
third groups use a division of the string section, plus two and four instruments for each of
the registers, respectively.
Example No. 41
Also, each register placement avoids beginning on the same beat subdivision. Since there
are twelve possible subdivisions of the beat in a measure of 12/16, this is formulated in
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such a way that each register placement begins on a different subdivision of the beat,
which leaves three remaining, equally spaced subdivisions of the beat unused. These are
the beats on which the pitched percussion parts (chimes, crotales, bells) heroically
attempt to introduce the upcoming new focal point of F#.
Example No. 42
Rehearsal Letter AA
In my opinion, Rehearsal Letter AA (m. 471) is the climactic section of the entire
concerto. A real test for the listener occurs, as the music of this section has only been
heard once before. At Rehearsal Letter A, the clarinet introduced the main theme of the
concerto, which was only the beginning part of the melody. This section uses the
remainder of this melody, performed here with full orchestral forces as the piano
arpeggiates rapidly in the upper register.
Rehearsal Letter BB
At Rehearsal Letter BB (m. 478), the sixteenth-note material returns in the piano,
but the material continues to expand outward, much like it did in both hands right before
the beginning of the middle cadenza (mm. 290-294). Following this, one measure of
erratic, unforeseen material occurs (m. 484), as if shaking the foundations of the piece.
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Violins, violas, and cellos grace this measure with one long downward glissando. The
woodwinds rapidly ascend and descend, while trumpets, flexatone, and slapstick
highlight their apex. Trombones enter on the third beat with their own ascending
glissando, their accented arrival marking the beginning of Rehearsal Letter CC.
Rehearsal Letter CC
With the foundation now shaken, new material seems to pop out of nowhere
(Rehearsal Letter CC, m. 485), the only organizing factor being symmetry. As the right
hand part of the piano arpeggiates an A minor seventh chord involving a repeated pattern
of five sixteenth notes, the left hand uses quarter notes to play each of the five black
notes, alternating fifths and fourths in these two measures.
Example No. 43
As the piano part continues to ascend, the orchestra acts like an amplified sustain pedal
for the piano. The strings enter on the white notes as each pattern of five sixteenth notes
begins, while the rest of the orchestra sustains the black notes as they are introduced,
culminating in a rather intense rush of energy.
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Rehearsal Letter DD
At Rehearsal Letter DD (m. 490) one measure of primal-sounding clusters in the
piano accompanied by bass drum, anvil, and brake drum takes place before the full
orchestra joins in. The rhythm is the same as that of the string section beginning in m.
207 of Rehearsal Letter G. The following measure has every instrument playing in a
powerful range, repeating the same rhythm. The winds have eleven of the twelve
possible tones (excluding D) in both the upper and lower divisions of the sections while
maintaining symmetry. Both violins I and II are divisi with triple stops, involving all
twelve tones, while violas, cellos, and basses are divisi with double stops, also producing
all twelve tones. One lone instrument, the English horn, is relegated to bravely holding
D4 for the entire measure while the surrounding bedlam ensues. The final measure
begins with all instruments remaining on their previous pitch (minus the D4 from the
English horn) being repeated as eighth-note triplets before dropping the upper register to
brashly close in the lower register where the piece began. The entire work ends on D in a
dramatic forceful flash.
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Conclusion
As I think back to when I first started composing the concerto, it is hard to believe
that I am sitting here, typing a concluding paragraph. I knew that eventually the end
would come, but some say compositions are never really completed. I dare to disagree.
Though the concerto will likely need minor changes here and there, I believe in looking
towards the future and not dwelling on the past. I do not plan on doing revision after
revision of any of my works, including this concerto. Instead, I look forward to the next
time I am sitting in front of a blank sheet of manuscript paper, wondering what will fill
the pages as I begin a new composition.
The concerto is roughly 16 minutes long and was completed on March 20, 2003.
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Vita
Trent William Hanna was born on Williams A.F.B. in Williamson County,
Arizona, on April 30, 1969, the son of John William Hanna and Victoria Lynn Hanna.
Upon graduation from Bryan High School, Bryan, Texas, in 1987, he entered Sam
Houston State University in Huntsville, Texas. He received his Bachelor of Music
degree in theory, composition, and piano performance in 1992, and his Master of Music
degree in piano performance in 1995. During the following two years he was employed
as an adjunct faculty member at Sam Houston State University and taught private and
class piano, fundamentals of music, and conducted both of the jazz ensembles. In August
2000 he entered the Graduate School of the University of Texas at Austin. Trent has
received awards in both composition and piano performance including first place in the
2001 SCI/ASCAP Student Composition Competition, Region VI, in the 1994 TMTA
Collegiate Competition, in the Corpus Christ International Young Artists 24th Concerto
Competition, in the 1995 Sorantin Young Artist International Concerto Competition, and
in the 1996 Texas Young Artists Concerto Competition.
Permanent Address: 1308 Esther Blvd., Bryan, Texas 77802
This dissertation was typed by the author.
