Analysis of Twentieth-Century
Music Using the Fourier Transform
Jason Yust, Boston University
Music Theory Society of New York State
Binghamton, April 11, 2015
A copy of this talk is available at
people.bu.edu/jyust/
Outline
I. Characteristic Functions and DFT
1. Characteristic functions and pc-distributions
2. DFT
3. Properties of DFT
4. Phase spaces
II. Example: Webern Op. 5 no. 4
1. Previous analyses: Forte, Perle, Burkhart
2. DFT magnitudes and harmonic universes
3. Phase Space (2, 3)
III. Scale Theory, Subset relations, and Phase Space 5
1. Example: Debussy, Prelude I/3 “Le vent dans la plaine”
2. Example: Satie, “Idylle” (Avant dernières pensées 1)
3. Example: Stravinsky, Piece for String Quartet no. 1
IV. Feldman, Palais de Mari
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
I. Characteristic Functions
and DFT
1. Characteristic functions and pc-distributions
2. DFT components
3. Properties of DFT
4. Phase spaces
Discrete Fourier Transform on Pcsets
Lewin, David (1959). “Re: Intervallic Relations between
Two Collections of Notes,” JMT 3/2.
——— (2001). “Special Cases of the Interval Function
between Pitch Class Sets X and Y.” JMT 45/1.
Quinn, Ian (2006–2007). “General Equal-Tempered
Harmony,” Perspectives of New Music 44/2–45/1.
Amiot, Emmanuel (2013). “The Torii of Phases.”
Proceedings of the International Conference for
Mathematics and Computation in Music, Montreal,
2013 (Springer).
Yust, Jason (2015). “Schubert’s Harmonic Language
and Fourier Phase Spaces.” JMT 59/1.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Characteristic Function of a Pcset
And using
values,
can
The
characteristic
function
of
a pcset
ispc-vector
a 12-place
vector
By
allowing
othernon-integer
integer
values,
the the
characteristic
function
describe
pc-distributions
with
1s
for
each
pc
and 0s elsewhere:
can
also
describe
pc-multisets
0, 0,
1, 0,
( 1,
2, 0, 0.5,
0, 0.25,
0, 1,
C
Jason Yust
C#
D
E∫
E
F
F#
1, 0, 0.25,
0, 0.5,
0, 0 )
G
G#
A
Analysis of Twentieth-Century Music using DFT
B∫
B
MTSNYS 4/12/2015
DFT Components
=
C major triad
Component 1:
magnitude = 0.043
Jason Yust
+
Component 2:
magnitude = 0.083
Analysis of Twentieth-Century Music using DFT
+
...
MTSNYS 4/12/2015
DFT Components
=
C major triad
Component 3:
magnitude = 0.186
Jason Yust
+
Component 4:
magnitude = 0.144
Analysis of Twentieth-Century Music using DFT
+
...
MTSNYS 4/12/2015
DFT Components
=
C major triad
Component 5:
magnitude = 0.161
Jason Yust
+
Component 6:
magnitude = 0.083
Analysis of Twentieth-Century Music using DFT
+
...
MTSNYS 4/12/2015
DFT Components
The DFT is a change of basis from a sum of pc spikes
to a sum of discretized periodic (perfectly even) curves.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
DFT Components
Quinn’s generic prototypes are pcsets that maximize a given
component. Subsets and supersets of the prototypes are the
best representatives of each component
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
DFT Components
Notation
fn
The nth DFT component
|fn|
The magnitude of the nth component
|fn|2
Squared magnitude
n
The phase (0 ≤ n ≤ 2p) of the nth component
Phn
Phase normalized to pc-values: (0 ≤ Phn ≤ 12)
 (|f1|2, Ph1), (|f2|2, Ph2), (|f3|2, Ph3), (|f4|2, Ph4), (|f5|2, Ph5), (|f6|2, Ph6) 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Phase spaces: One dimensional
One-dimensional phase spaces are Quinn’s Fourier balances,
superimposed n-cycles created by multiplying the pc-circle by n.
Ph3
Ph2
Ph1
Ph4
Ph5
Ph6
N.B. counter-clockwise orientation
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Phase spaces: One dimensional
The position of the pcset in the phase space is the
circular average of the individual pcs
{CEG}
{CEG}
Ph3
Jason Yust
Ph5
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Phase spaces: One dimensional
Opposite phases cancel one another out.
Therefore pcsets can have undefined phases.
C
{CD} has undefined Ph3,
|f3| = 0
Ph3
D
Jason Yust
×
This is a kind of
Generalized Complementation:
Complements balance one another
in all phase spaces.
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Phase spaces: One dimensional
An analytical proto-methodology:
Each Fourier component measures an independent
musical quality: (1) chromaticism, (2) quartal harmony,
(3) triadic harmony, (4) octatonicism,
(5) diatonicism, (6) whole-tone balance.
Distances in phase spaces indicate:
• Relatedness of harmonies on the given dimension
• Whether the harmonies reinforce one another or
weaken one another on the given dimension
when combined.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Phase Spaces: Two dimensional
A two-dimensional phase space tracks the phases of
two components, and is topologically a torus.
Amiot (2013) and Yust (2015) use Ph3–5-space to
describe tonal harmony.
from
Amiot,
MCM 2013
proceedings
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Phase Spaces: Two dimensional
Amiot (2013) and Yust (2015) use Ph3–5-space to describe
tonal harmony
Pcs, consonant
dyads and triads,
and Tonnetz
in Ph3–5-space,
from Yust (2015)
(JMT 59/1)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
2. Webern, Satz for String
Quartet, Op. 5, no. 4
1. Previous analyses: Forte, Perle, Burkhardt
2. Fourier magnitudes and harmonic
universes
3. Phase space (2, 3)
Webern, Op. 5 no. 4
Principal section
Contrasting section
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4
“Perhaps the most interesting formal feature of this miniature . . .
is the structural disjunction which takes place as a result of the
‘incomparable’ relation. . . . The middle section (bars 7–9) . . .
has its own set of sets, all interrelated by inclusion, but
incomparable with respect to all the sets of 4 notes or more
in the outside sections.”
“The middle section [emphasizes IC4] and the sections bounding
it . . . emphasize IC6”
Forte (1964), 176
Jason Yust
Analysis of Twentieth-Century Music using DFT
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Webern, Op. 5 no. 4
Other authors focus on underlying continuities:
Perle (1991) observes that the inversion of V
about E–F# (T10I(V)) is a subset of F.
Burkhart (1980) views A'+A'' = T4(01236789)
as a source set for the entire piece.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4
Principal section: Inclusion relations
A  A'' B
= A'
A
A'
A''
A'
Jason Yust
T1(B)
C
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4
Principal section: DFT magnitudes
Set
Set Class
B
(0156)
A
(0167)
AB
(012567)
A  A'
(012678)
A  A'  A'' (01236789)
A + A'
(01225627)
A + A' + A'' (012223672829)
C
(012567)
DFT magnitudes^2








1,
0,
1,
0,
0,
0,
0,
2,
9,
12,
13,
16,
12,
36,
48,
12,
4,
0,
1,
0,
0,
0,
0,
2,
1, 1, o 
4, 0, 0 
1, 1, 1 
0, 0, 4 
4, 0, 0 
4, 0, 0 
0, 0, 0 
0, 2, 0 
All
important
sets have
markedly
high
|f2|
B also
has a strong
|f3|,abut
for other
sets
(ones involving A),|f3| is zero
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4
Principal section: DFT magnitudes
Set
V
m. 6 Flyaway
F
F+E
E
D
mm.
7–9
Set Class
DFT magnitudes^2
(0157)
(0123679)
 0.27, 7, 2, 1, 3.73, 4 
 1, 7, 2, 7, 1, 1 
(01468)
(01224568t)
(0347)
(048)
 0.27, 3,
 3, 3,
 2, 0,
 0, 0,
5,
9,
8,
9,
1, 3.73, 9 
3, 3, 9 
4, 2, 0 
0, 0, 9 
In the contrasting section, |f3| is most prominent
Meas. 6 is transitional: f2 still dominates but is dampened.
In the middle section f2 is weakened by E and D.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4
Principal section: DFT magnitudes
Set
V
m. 6 Flyaway
mm.
7–9
F
F+E
E
D
Set Class
DFT magnitudes^2
(0157)
(0123679)
 0.27, 7, 2, 1, 3.73, 4 
 1, 7, 2, 7, 1, 1 
(01468)
(01224568t)
(0347)
(048)
 0.27, 3,
 3, 3,
 2, 0,
 0, 0,
5,
9,
8,
9,
1, 3.73, 9 
3, 3, 9 
4, 2, 0 
0, 0, 9 
Set V is very similar to F, the differences being that
F loses the strong |f2| and brings |f3| and |f6| to the fore
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4
Principal section: DFT magnitudes
Set
Set Class
B
(0156)
DFT magnitudes^2
 1, 9, 4, 1, 1, o 
B combines the harmonic properties of the
principal section (high |f2|), and the middle
section (high |f3|). It therefore acts as a kind
of motivic seed, setting up the space of
activity for the whole piece.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4: Phase space
 Ph3 
Important
sets are
concentrated
in one region
of phase
space, in the
vicinity of pcs
C, B, and E
 Ph2 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4: Phase space
You can also
subtract a set
by adding its
negation.
 Ph3 
Sums of
equallyweighted sets
directly
across from
one another
(phase difference of 6)
annihilate the
component.
AB – {E}
=A
Flyaway + {F}
= AA'A''
B + {Bb, F#}
= AA''
V + {Bb, F}
= AA''
 Ph2 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4: Phase space
 Ph3 
The activity
of the entire
piece occurs
in a region
defined by the
Ph2 values of
A, A', and A'',
with the
exception of
the last
“Flyaway.”
A'
A
A''
 Ph2 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4: Phase space
The middle
section lives
in a region
constrained
in the Ph3
dimension by
the values of
D and E.
D
E
 Ph3 
The two parts
of the first
section lie
just outside
this region,
on either side.
 Ph2 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Webern, Op. 5 no. 4: Phase space
 Ph3 
Note how V
lies in the
center of this
scheme, and
has the same
Ph3 value as
E, another
manifestation
of its role as
mediator
(reinforcing
Perle’s
analysis).
 Ph2 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
3. Scale Theory, Subset
Relations, and Ph5-Space
1. Debussy, “Le vent dans la plaine”
2. Satie, “Idylle”
3. Stravinsky, Piece for String Quartet no. 1
Debussy, Le vent dans la plaine
Meas. 1–4:
6∫ diatonic
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Debussy, Le vent dans la plaine
Meas. 19–23:
7∫ acoustic
0# (=12∫) diatonic
or 1# (=11∫) acoustic?
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Debussy, Le vent dans la plaine
N.B.: No C
!!
from Tymoczko, “Scalar Networks and Debussy” JMT 48/2 (2004)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Debussy, Le vent dans la plaine
“This measure involves a complex set of scalar affiliations. The
first three beats of m. 21 involve only white notes, suggesting
D dorian. However, the fourth beat of the measure adds a D∫,
which, taken together with the earlier music, implies G
acoustic.
—Tymoczko, “Scalar Networks and Debussy” JMT (2004)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Ph5 as circle of fifths balance
2∫ Diat.
1∫ Diat.
0# Diat.
1# Diat.
2# Diat.
Phase Space {5}
( = Circle of Fifths)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Ph5 as circle of fifths balance
2∫ Diat.
1∫ Diat.
0# Acous.
0# Diat.
1# Diat.
2# Diat.
Phase Space {5}
( = Circle of Fifths)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Ph5 as circle of fifths balance
2∫ Diat.
1∫ Diat.
0# Acous.
0# Diat.
Single-semitone
voice leadings
1# Diat.
2# Diat.
Phase Space {5}
( = Circle of Fifths)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Debussy, Le vent dans la plaine
Measure 21
(“White notes”)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Debussy, Le vent dans la plaine
{DFGAB}
G acoustic
Meas. 1–4
Meas.
19–20
6∫ diatonic
7∫ acoustic
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Satie
Satie: “Idylle” (Avant Dernières Pensées no. 1, 1915)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Satie
Satie: “Idylle” (Avant Dernières Pensées no. 1, 1915)
Ostinato (ABCD)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Satie
Satie: “Idylle” (Avant Dernières Pensées no. 1, 1915)
Ostinato (ABCD)
Subset of two possible
diatonic collections
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Satie
Satie: “Idylle” (Avant Dernières Pensées no. 1, 1915)
Ostinato (ABCD)
Bois secs:
Ph5 = 5.5
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Satie
Satie: “Idylle” (Avant Dernières Pensées no. 1, 1915)
Arbres:
Ph5 = 0
Ostinato (ABCD):
Ph5 = 2.5
Bois secs:
Ph5 = 5.5
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Satie
Satie: “Idylle” (Avant Dernières Pensées no. 1, 1915)
Mon cœur:
Ph5 = 11
(Diatonic)
(Harmonic minor)
Jason Yust
Le soleil: Ph5 = 5.5
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Satie
Satie: “Idylle” (Avant Dernières Pensées no. 1, 1915)
“What do I see?
The brook is all wet;
Semicolon: go sharpward
Period: go
flatward
But my heart is very small.
Semicolon:
The trees look like great misshapen combs;
go sharpward
and the sun, like a beehive, has golden rays. Period: go flatward
And the wood is all dry and flammable as a switch.
But my heart shivers with fright.
Return
The moon has blurred with its neighbors;
and the brook is soaked to the bones.”
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Satie
Satie: “Idylle” (Avant Dernières Pensées no. 1, 1915)
Mon cœur:
Ph5 = 11
Ostinato (ABCD):
Ph5 = 2.5
Arbres:
Ph5 = 0
Bois secs:
Ph5 = 5.5
Jason Yust
La lune
Ph5 = 3.67
Le soleil: Ph5 = 5.5
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Stravinsky
Stravinsky: Three Pieces for String Quartet (1914)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Stravinsky
Stravinsky: Three Pieces for String Quartet (1914)
The pcsets in each part have relatively large values for
component 1 because they are clustered in a narrow range,
component 5 because they are diatonic subsets, and
component 4 (vln2 and cello) as octatonic subsets.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Stravinsky
Stravinsky: Three Pieces for String Quartet (1914)
GABC: Vln1, Ph5=2.1
CD∫E∫:
Cello,
Ph5 = 9.2
C#D#EF#: Vln2, Ph5 = 6.5
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Stravinsky
 Ph5 
Stravinsky: Three Pieces for String Quartet (1914)
Narrow 4 spread
Wide 5 spread
 Ph4 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Scale Theory and Subsets: Stravinsky
Stravinsky: Three Pieces for String Quartet (1914)
Oct2,3
 Ph5 
Oct0,1
Narrow 4 spread
DFT magnitudes for the
sum of the three pcsets:
 8.5, 3, 5, 19, 1.5, 9 
 Ph4 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Feldman, Palais de Mari
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Features of the piece:
• Composed in 1986, Feldman’s last work for solo piano.
• Long but sparse: the 9-page score takes ca. 25 minutes to
play.
• Made up of discrete gestures, frequently repeated and
varied (often in subtle ways).
• Pedal is held continuously throughout most of the piece.
This blurs the distinction between successive and
simultaneous sounds.
• Extended segmentational analysis in Hanninen, A Theory of
Music Analysis (2012).
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Features of the piece:
• Long sections on the piece tend to dwell on a limited set of
gestures, giving the piece a sense of trajectory that is
nonetheless non-teleological.
• Composed around the same time as his Second String
Quartet, which Feldman described as “a dialectic of sorts
between such elements as . . . chromaticism/consonance.”
• “Reverse Development”: Gestures often appear before the
idea from which they are derived, replacing a process of
development with one of revelation.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
The initial gesture (Hanninen set A) stages a
chromatic–diatonic conflict
The first three notes are highly diatonic,
but the final E introduces a concentrated chromaticism.
Hanninen: “The contrast between harmonies rich in ics 2 and 5,
versus those rich in ic1, resonates throughout the piece.”
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
The initial gesture (Hanninen set A) stages a
chromatic–diatonic conflict
(A∫E∫)
(FE)
The gesture can also be divided by part into
a fourth and a minor second.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
ic2
ic3
 Ph5 
Ph1,5-space
ic2 and ic3
are the
highestmagnitude
ics in the
space and
are
balanced
between f1
and f5
 Ph1 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
(FE)
 Ph5 
Ph1,5-space
Dyads from
the upper
and lower
voices of
m. 1—
ic1 and ic5
occupy the
same
positions in (A∫E∫)
the space.
 Ph1 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Ph1,5-space
The upper
and lower
voice dyads
are close
in Ph1 and
distant in
Ph5.
 Ph5 
N.B.: They are
also imbalanced
(bottom: high f1;
top: high f5).
(3.71, 0.35)
(5.71, 2.35)
(0.31, 3.75)
 Ph1 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
One of Feldman’s basic harmonic techniques is
Transpositional Combination
m. 20:
m. 41:
T1
(CD)
(B∫F)
T2
(Hanninen set C)
Jason Yust
ic1*ic2
ic5*ic2
(Hanninen set E/d)
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Convolution Theorem: Transpositional combination (with
doublings retained) is the same as multiplying DFT
magnitudes and adding the phases.
(t7)
× (02)
=(t079)
m. 20:
 (0.27, 4.5), (3, 3), (2, 7.5), (1, 6), (3.73, 10.5), (0, –) 
(B∫F)
T2
= (t7) ×(02)
Jason Yust
× 
(3, 11), (1, 10), (0, –), (1, 2),
(3, 1),
(4, 0) 
=  (0.8, 3.5), (3, 1), (0, –), (1, 8), (11.2, 11.5), (0, –) 
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Transposition of entire gestures by semitone
reinforces component 1 and cancels out component 5
(0235) = 2*3 is balanced
between f1 and f5, but
multiplying by (01)
weakens f5 in favor of f1
Jason Yust
(46)
× (03)
 (3, 7)1, (3, 5)5 
×  (2, 10.5)1, (2, 10.5)5 
= (4679)
=  (6, 5.5)1, (6, 3.5)5 
(4679)
× (01)
 (6, 5.5)1,
(6, 3.5)5 
×  (3.73, 11.5)1, (0.27, 9.5)5 
= (4567289) =  (22.4, 5)1,
Analysis of Twentieth-Century Music using DFT
(1.6, 8)5 
MTSNYS 4/12/2015
Feldman, Palais de Mari
This idea is repeated
frequently throughout the
last part of the piece.
RH
This variant of the idea
makes it evident that both
are products of (013) with
ic5 or ic1:
LH
RH
LH
(Hanninen G/a)
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Hanninen:
“ The arrival of G/a287–88 . . . is the
keystone in a remarkable confluence of
events. First, it defines the center of subset
G/a, and also of set G. Second, it forms a
bridge to set A, recalling and rearranging
intervals and key pcs of set A. Third, it forms
a second, and stronger, bridge to set C.”
Gesture from set A:
Gesture from set C:
20
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Ph1,5-space
Feldman, Palais de Mari
The (0235)
and G–A∫
subsets are
spread out in
Ph1 and close
in Ph5
 Ph5 
The individual
chords are
more spread
out in Ph5,
meaning they
have
stronger f5
(0.51, 7.55)
(61, 65)
(1.51, 8.55)
(3.71, 0.35)
(2.31, 5.75)
 Ph1 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Ph1,5-space
 Ph5 
The (013)s of
the variant
gesture are
spread out in
Ph5 and close
in Ph1,
making the
sum strongly
chromatic.
(5.71, 2.35)
(21.41, 0.65)
(5.71, 2.35)
 Ph1 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
One important gesture reveals how Feldman “cripples”
symmetries by asymmetrically dividing a symmetric entity
 (3.73, 5.5)1, (0.27, 3.5)5 
(F#GG#D)
 (5.73, 1.3)1, (2.27, 2.8)5 
+ (AB∫BCE)
 (4, 3)1, (4, 3)5 
= (F#GG#AB∫BCDE)
The symmetric source chord is balanced between f1 and f5,
but it is split into more heavily chromatic chords.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Feldman, Palais de Mari
Ph1,5-space
(5.71, 2.35) (41, 45)
(3.71, 0.35)
 Ph5 
The Ph1
spread shows
how more
chromatic
chords are
extracted
from a
balanced
parent chord.
 Ph1 
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Summary
• DFT is a change of basis applied to the domain of pcdistributions
• Each DFT component measures a musically
interpretable quality relating to a type of evenness.
• DFT magnitudes can replace much of pcset-theory’s
use of interval content to relate harmonic entities.
• The fifth Fourier component measures
diatonicity, and provides a more systematic
approach to reconciling subsets and supersets
with scale theory.
• Distances in phase space indicate the extent to which
sets reinforce or oppose one another in a given
dimension.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Bibliography
Ames, Paula Kopstick. “Piano (1977).” In The Music of Morton Feldman, ed. T. Delio (Westport,
Conn.: Greenwood Press), 99–146.
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Music 1: 157–72.
———. 2013. “The Torii of Phases.” Proceedings of the International Conference for
Mathematics and Computation in Music, Montreal, 2013, ed. J. Yust, J. Wild, and J.A.
Burgoyne (Heidelberg: Springer).
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Journal of Mathematics and Music 5/3, 149–69.
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V, ed. Salzer (New York: Columbia University Press) 317–34.
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332.
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Science 320: 346–8.
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(Westport, Conn.: Greenwood Press), 39–70.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Bibliography
Feldman, Morton. 2000. Give my Regards to Eighth Street: Collected Writings of Morton
Feldman, ed. B.H. Friedman. Cambridge, Mass.: Exact Change.
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———. 2011. “Spelled Heptachords.” In Mathematics and Computation in Music: 3rd
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Emmanuel Amiot, Jean Bresson, and John Mandereau, 84–97. Heidelberg: Springer.
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Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Bibliography
Lewin, David. 1959. “Re: Intervallic Relations between Two Collections of Notes.” Journal of
Music Theory 3: 298–301.
———. 2001. Special Cases of the Interval Function between Pitch-Class Sets X and Y. Journal of
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———. 2007. Generalized Musical Intervals and Transformations, 2nd Edition. New York:
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———. 2008. “Set-Class Similarity, Voice Leading, and the Fourier Transform.” Journal of Music
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Yust, Jason. 2015a. “Restoring the Structural Status of Keys through DFT Phase Space.”
Proceedings of the International Congress for Music and Mathematics (forthcoming).
———. 2015b. “Schubert’s Harmonic Language and Fourier Phase Space.” Journal of Music
Theory 59/1, 121–181.
Jason Yust
Analysis of Twentieth-Century Music using DFT
MTSNYS 4/12/2015
Analysis of Twentieth-Century
Music Using the Fourier Transform
Jason Yust, Boston University
Music Theory Society of New York State
Binghamton, April 11, 2015
A copy of this talk is available at
people.bu.edu/jyust/