MULTIPLE TIME-SCALES

IN ADÈS’S RINGS

DANIEL FOX

INTRODUCTION

Tterms, be it “the magneticabout

HOMAS ADÈS OFTEN SPEAKS his music in scientific or physical
forces of notes” or the importance of
“responding to temperature.”1 He claims that “Berlioz has brought us
into the Modern world of relativity and uncertainty.” 2 Adès’s music
often conveys the sense of a natural process unfolding. Simple, identical
processes unfolding at varying rates create expansions and contractions
of harmonies in pitch space. Despite a sometimes overwhelming level
of detail, one feels the physical causality behind the movement of
notes. The aim of this paper is to argue that a scientific understanding
of an extended physical metaphor unifies the processes at work in
Rings, the first movement of Adès’s violin concerto Concentric Paths
(2005).3 The metaphor of celestial motion is offered to us on the cover
of the score and in the composer’s program notes.
 Multiple Time-Scales in Adès’s Rings 29

COLOR, INSTRUMENT, AND PROCESS

Adès has spoken of Kurtag and Ligeti as “inventors of colour and

instruments—not in the sense of actual musical instruments, but an
‘instrument’ being a complex of timbre and interval and harmony and
rhythm, which could be an implement that you could use. . . .”4
Dominic Wells points us to the “relationship between the opening of
Arcadiana (Adès’s first string quartet, 1994) and that of Ligeti’s Vio-
lin Concerto, which also begins with open fifths. . . .”5 Adès’s violin
concerto also opens with an ostinato of open fifths (see Examples 1A
and 1B). However, Ligeti’s piano Etude No. 6 Automne à Varsovie
offers a more extensive model for the musical processes in Rings.
Example 2 shows the opening bars (re-notated for analytical purposes)
of Automne à Varsovie. The octave Ebs form a perfect crystal. The Fb in
m. 2 introduces an impurity into the crystal, and the rest of the piece
details the reaction of the crystal to this impurity: the disturbance
propagates through the crystal, deforming it into a quasi-crystal. 6 The
final bars of Automne à Varsovie present the final collapse of the crystal
structure. All that is left is the chromatic descent that was initiated by
the impurity. We will find in Rings an elaboration on this process.
Rings opens with energetically alternating D6s and G4s in the solo
violin, supported thinly in the orchestra (see Example 3).7 In the
second measure, when Flute 2 tries to leap down a perfect fifth from
D6 to G5, instead of a perfect twelfth to G4, there is too much energy
in the system, and it overshoots slightly to F#5. Flute 1 leaps from D6

EXAMPLE 1A: THE OPENING MEASURES OF LIGETI’S VIOLIN CONCERTO

LIGETI KONZERT FOR VIOLIN AND ORCHESTRA COPYRIGHT © 1990, 1992 BY SCHOTT MUSIC GMBH & CO. KG.
ALL RIGHTS RESERVED. USED BY PERMISSION OF EUROPEAN AMERICAN MUSIC DISTRIBUTORS COMPANY, SOLE U.S. AND
CANADIAN AGENT FOR SCHOTT MUSIC GMBH & CO. KG.

EXAMPLE 1B: THE OPENING MEASURES OF RINGS

30 Perspectives of New Music

EXAMPLE 2: THE OPENING MEASURES OF LIGETI’S AUTOMNE À VARSOVIE

LIGETI ETUDES POUR PIANO, BOOK 1 COPYRIGHT © 1985 BY SCHOTT MUSIC GMBH & CO. KG. ALL RIGHTS RESERVED.
USED BY PERMISSION OF EUROPEAN AMERICAN MUSIC DISTRIBUTORS COMPANY, SOLE U.S. AND CANADIAN AGENT FOR
SCHOTT MUSIC GMBH & CO. KG.

EXAMPLE 3: THE OPENING MEASURES OF RINGS INTRODUCE THE

TETRACHORD [67E2]

to B4 and thus the tetrachord [67E2] 4-20(0158) is introduced. At

m. 3 the soloist begins a rapid arpeggiation of this [67E2] chord
within the span of G4 to D6 (see Example 4). 8 This pitch set, a GMaj7
chord, remains stable for mm. 4 and 5.
Similar to the introduction of the Fb in m. 2 of Ligeti’s Automne à
Varsovie, when Flute 2 introduces the F#5 in m. 2 of Rings, it creates
instability in the pitch-class G; G develops a tendency to decay to F#.
In m. 6 the instability in G–F # spreads: the D6 in the solo part slips to
C#6 (this is reinforced in the orchestra). None of the pitches in [67E2]
 Multiple Time-Scales in Adès’s Rings
EXAMPLE 4: MM. 4–9 OF RINGS. THE PITCH SET [67E2] 4-20(0158) IS TRANSFORMED BY T−1
TO THE PITCH SET [56T1] 4-20(0158) THROUGH THE INTERMEDIATE SET [1ET6] 4-14(0237)

31
32 Perspectives of New Music

remain stable for very long. It is as if each pitch is an atom in a crystal

lattice, but there is so much heat that the atoms are jostled out of
place. As in Automne à Varsovie, a process of chromatic descent is
initiated. However, the characters of the two works differ significantly.
Ligeti’s process feels as if it is describable by laws of physics; Adès’s
process feels more biological, an association put forth by the composer
himself: “The moment I put a note down on paper it starts to slide
around the page. And the writhing that I could see when I look at a
note under the microscope, you would see with any living thing.” 9
This instability, this tendency for a pitch to decay down a semitone, is
played out over the next 85 measures, taking us through about two-
thirds of the movement. Fighting against this tendency is the less
frequent leap up a major seventh, to which we will return.
The chromatic descents lead to a harmonic progression when
applied to the tetrachord [67E2]. The progression filled out by the
soloist and reinforced in fragments by the orchestra is diagramed in
Example 4. A T−1 transformation is applied to [67E2] to produce
[56T1] by m. 7, though a transition region mediates, because each
pitch loses stability and slides down by a semitone at a different rate. It
is as if each pitch is moving at a time-scale that differs slightly from
that of its neighbors. This variation in decay rate deforms the set-class
but then allows it to regain integrity. The different rates at which
pitches descend lead to suspensions reminiscent of common practice
counterpoint. In fact the concerto has something of a Baroque feel,
due, in part, to the initiation of a “perpetual motion machine.” 10
Taruskin observed that Adès possesses an “omnivorous range of
reference,”11 and to understand the soloist’s material it is useful to
turn to the Presto from J. S. Bach’s Sonata for Solo Violin in G minor.

THE PRESTO FROM J. S. BACH’S SONATA

FOR SOLO VIOLIN IN G MINOR

J.S. Bach’s Presto from the G minor Sonata for Solo Violin serves as a
model in four ways: 1) It has a near ceaseless perpetual motion texture;
2) It uses arpeggiated seventh chords sliding by (modal) step; 3) It
uses sequenced suspensions; and 4) It displays regular interval class
motion. The perpetual motion texture of unwavering sixteenth notes
in the Presto is briefly suspended at the end of each of the two repeated
sections. In Rings the soloist alternates between the driving sixteenths
and melodic lines that unfold on a slower timescale and, largely, in a
different register; but there is not a measure of Rings in which the six-
teenth note tattoo is not carried by some instrument (in the literal
 Multiple Time-Scales in Adès’s Rings 33

sense or in Adès’s broader sense of a “complex of timbre and interval

and harmony and rhythm”).
In comparing Rings and the Presto, it is clear that the former
undergoes chromatic transformations such as T−1, whereas the latter is
more rigidly confined to a mode at any given time and undergoes
modal transformations. To emphasize the relationship between
passages of Rings and the Presto, it will be useful to regard sections of
the Presto to be confined to a given minor mode and to measure
intervals from within that mode, rather than from within the broader
chromatic context. When the mode is G natural minor we will use
integer notation in which

G=0, A=1, Bb=2, C=3, D=4, Eb=5, F=6

and we will calculate using arithmetic modulo 7. For example, mm. 9–

11 of the Presto consist of “V” shaped arpeggiations of the chords
Cm7, BbMaj7, Aø77 (see Example 5A). Using modal integer notation,
the tetrachords can be realized as [3502], [2461], and [1350]. Each is
obtained from the last using a T−1 transformation, where the step of −1
is regarded within the mode.12 Measures 67–69 have the same sequence
but placed in the mode of C natural minor (see Example 5B). Another
sequence of seventh chords is found in mm. 17–24 (see Example 5C).
There the arpeggiation pattern leads to the (G natural minor pitch-class)
sequence [5, 2, 5, 0, 5, 0, 3, 0, 3, 5, 3, 5]. This is projected through a
T1 transformation twice. The third transformation is only approximate,
producing an FMaj chord without a seventh in mm. 23–24, as opposed
to the seventh chords that have appeared in mm. 17–22. In Rings Adès
generalizes13 these transformations into a chromatic post-tonal context
and uses a gradual deformation process to interpolate from one
seventh-chord to the next (see Example 4).
The suspensions in Rings that result from the uneven rate of decay
of each pitch (see mm. 6–7 in Example 4) also have a clear precedent
in the Presto (see Example 6). Measures 70–73 consist of eleventh
chords rooted on a cycle of (rising) fourths. The C in the G dominant
seventh chord of m. 70 is a suspension from the ii ø77 (of C minor)
from the previous measure and it is resolved to Bn (with a bounce!).
The F of m. 71 is a suspension from m. 70 (with octave displacement)
and is resolved to Eb (again with a bounce). One can trace a line from
B4 in m. 70 to Bb4 at the end of m. 71, to A4 in m. 72, and Ab4 in
m. 73. Measures 70–73 consist of a sequence of 4-3 suspensions of
these seventh chords.
34 Perspectives of New Music

EXAMPLE 5A: BACH PRESTO MM. 9–11

EXAMPLE 5B: BACH PRESTO MM. 67–69

EXAMPLE 5C: BACH PRESTO MM. 17–23

EXAMPLE 6: BACH PRESTO MM. 70–73

 Multiple Time-Scales in Adès’s Rings 35

The Presto serves as a model for the initial details of the solo violin
material in Rings. It also provides a model for the closing material of
the soloist. Beginning on the seventh sixteenth-note of m. 121 of
Rings, the penultimate measure, the soloist fills out Hex 2,3 with the
PC sequence

[E, 6, 7, 2, 3, T; E, 6, 2!, 7!, 3, T; E, 6, 7, 2, 3, T].

If we swap the pitch classes [2] and [7], decorated with exclamation
marks, then the whole sequence follows the unordered interval class
pattern of 5, 1, 5, 1. . . . Such regularity is obscured by wild octave dis-
placement (see Example 7).
A similarly simple sequence disguised by octave displacement is
found in mm. 43–46 of the Presto (see Example 8). If we use D natural
minor pitch-class space (D=0, E=1, F=2, G=3, A=4, Bb=5, C=6) then
the sequence is [2, 0, 5, 3, 1, 6], [0, 5, 3, 1, 6, 4], [5, 3, 1, 6!, 4, 2].
Within each six-note segment the line moves by the (modal) interval
class sequence 5, 5 . . . and between each six-note segment the motion
is by a (modal) T−2 transform. The one exception is the (modal) PC 6
marked with an exclamation in the third segment. The altered pitch C #
endows this third sequence with a dominant function within D minor
that ejects the music from the sequence. Both Rings and the Presto

EXAMPLE 7: THE CLOSING MEASURES IN WHICH

THE SOLOIST MOVES AS IF ON A STRANGE ATTRACTOR

EXAMPLE 8: BACH PRESTO MM. 43–45

36 Perspectives of New Music

explore simple sequences infused with a dizzying energy through

octave displacement and tireless rhythmic insistence.
In the last measures of Rings the set Hex2,3 functions like a strange
attractor.14 In the mathematical subject of dynamical systems theory, a
strange attractor is a subset of the space of all possible states (known as
the phase space) to which the system converges, but on which it moves
chaotically (but deterministically). Leading up to m. 119 in Rings, the
soloist saturates the aggregate before it is sucked into the Hex 2,3 subset
of chromatic space. The passagework gives the impression of chaotic
motion, even though it moves with an underlying regularity (or
determinism?) in terms of interval class. The analogy is made more
congenial by the fact that one of the first chaotic dynamical systems to
be studied was the N-body system of planetary motion.15

CONCENTRIC PATHS LEAD TO MULTIPLE TIME-SCALES

In his review of the British premiere of Concentric Paths, Paul Griffiths

observed that, “People on concentric paths may be going in the same
direction, and perhaps can touch each other, but they can never link
up—or get away from one another—and they will always come back to
the same place.”16 He goes on to quote Adès as depicting the first
movement as “fast, with sheets of unstable harmony in different orbits.”
The imagery of “different orbits” is reinforced by the image of a “Map
of the Earth and Planetary Orbits” from The Celestial Atlas by Andreas
Cellarius (1661), which appears on the cover of the score. Our large-
scale sense of time is derived in part from the changing of the seasons,
whose rates of change in turn arise from the frequency of the Earth’s
orbit about the sun. The slowly shifting harmonies in Rings come and
go like the seasons and, like the seasons, are often muddled up.
The analogy between planetary orbits and the unfolding of the
music in Rings serves as a conceptual motivic seed that can unify the
structure of the music. One of Johannes Kepler’s great accomplish-
ments was to formulate the relationship between the period of a
planet’s orbit and its distance from the sun. Using astronomical units
to hide distracting constants, he found that T3=a2 where “T” is the
period of the planet’s elliptical orbit and “a” is the semi-major axis of
the ellipse. This law implies that objects in different orbits move at
different speeds and that the more distant orbits lead to slower speeds.17
In Rings, and particularly in the part of the violin solo, we find both
the multiple time- scales of different orbits and the tendency of outer
orbits to move more slowly if we associate the distant orbits with the
 Multiple Time-Scales in Adès’s Rings 37

high register. The solo violin spends most of its time swirling around
at the rate of sixteenth-notes, but when it launches into an outer orbit
of the upper register it slows down dramatically (see Example 9). The
soloist looks down from its great height on the voices of the orchestra
as they wind through their closer (lower register) and faster orbits.
Different instruments (either in the literal sense, or in Adès’s use of
the term to describe “a complex of timbre and interval and harmony
and rhythm”) travel on different orbits, so they move in and out of
alignment with one another, as Griffiths emphasized. Example 10
shows an example of this from the opening of the piece. Through m.
12 the soloist and orchestra are aligned. Beginning in m. 13, differing
rates of harmonic change begin to be felt, with the orchestra lagging
behind the soloist. The re-alignment coincides with the first climax of
the piece in m. 20.
The multiple time-scales present in the work operate at two levels.
The first is at the microscopic level: Pitches have a tendency to decay
down a semitone, but the rate at which this happens varies. This leads
to contrapuntal resolutions of suspensions (e.g. B4→A#4=Bb4 in mm.
6–7, Example 4). The second is at a larger scale: Different instruments
orbit at different speeds. This causes the pitch content to spread across
the aggregate and then to contract. In mm. 18–20 the harmony is
stretched out to fill the aggregate except that it is missing PC 9. In m.
20 the instruments realign on an explosive Hex 2,3 chord, and remain
aligned in PC space for the next few measures. The missing PC 9 is
delivered shortly after as part of a [T]–[9] suspension-resolution as the
soloist’s melody unfolds more slowly in a distant orbit. Thus the
suspensions at the microscopic level are composed out in a suspension
of the completion of the aggregate. In fact, there is yet another type of
suspension structuring the piece to which we will return when
discussing the large-scale structure.

TE VS. T-1 AND THE CHROMATIC SUSPENSION CABLES

The chromatic descent in the upper voices found in Ligeti’s Automne

à Varsovie is a prominent feature of a number of Adès’s compositions.
For example, multiple chromatic descents weave through Summa, the
first movement of Traced Overhead.18 The descending chromatic lines
function like suspension cables: Rather than building harmonies upon
a bass line, the harmonies hang from the cables like a bridge or an
alpine cable car as they descend through chromatic space. These chro-
matic descents are often palpably audible.
 38
Perspectives of New Music
EXAMPLE 9: RINGS MM. 20–23: THE SOLOIST IS LAUNCHED INTO AN OUTER ORBIT
 Multiple Time-Scales in Adès’s Rings 39

MEASURE SOLOIST PC SET ORCHESTRA PC SET

1–5 [67E2] [67E2]
6 [6TE1] [6TE1]
7–8 [56T1], [56T1E] [56T1], [56T1E]
9–10 [9T025] Aligned [9T025]
10–11 [489E] [489E]
11–12 [348E] [348E]
12 [2378T] [2378T]
13–14 [1269] [237T]
15 [1568] Separating [2679]
16–17 [6801] [1269]
17 [0158] [0158] and [78013]
18 [0357] Reconnecting [0158], [780], [570], [78T023]
18–20 [E02357] [E0247], [E0457], [237T]
20 [2367TE] [2367TE]
21–23 [9T123] [E2367] and [6TE1]
24 [56TE1] Aligned [56TE]
25 [56T] [56T1]
26 [378] [03578]

EXAMPLE 10: MULTIPLE TIME-SCALES LEAD TO SEPARATION AND

REALIGNMENT BETWEEN THE SOLO VIOLIN AND THE ORCHESTRA, CAUSING
THE PITCH CONTENT OF THE HARMONIES TO EXPAND AND CONTRACT

Beginning in m. 6 of Rings, the highest sounding pitch is treated to

a chromatic descent at the pace of about one semitone every two to
three measures (see Example 11). The chromatic suspension cables
grow to enormous length, though the continuity is sometimes
obscured as they move through inner voices. Example 12 traces one
such chromatic descent. It is remarkable that this chromatic line winds
continuously through the orchestra over such a long span of time. In
fact, one can trace this chromatic suspension cable all the way to m.
113 if one allows for occasional octave leaps, which normally occur at
climactic moments, as if high-energy explosions launch the pitches into
a higher register. The line descends with hardly a break (though with
 40
Perspectives of New Music
EXAMPLE 11: IN RINGS SUSPENSION CABLES OF CHROMATIC DESCENTS
PROVIDE LARGE-SCALE STRUCTURE

Violin Concerto—Concentric Paths by Thomas Adès © Copyright 2010 by Faber Music Ltd., London.
 Multiple Time-Scales in Adès’s Rings
EXAMPLE 11 (CONT.)

41
Violin Concerto—Concentric Paths by Thomas Adès © Copyright 2010 by Faber Music Ltd., London.
 42
Perspectives of New Music
EXAMPLE 11 (CONT.)

Violin Concerto—Concentric Paths by Thomas Adès © Copyright 2010 by Faber Music Ltd., London.
 Multiple Time-Scales in Adès’s Rings 43

MEAS. PITCH INSTRUMENT MEAS. PITCH INSTRUMENT

# # vlns. 1 & 2,
4 G5, F 5 solo vln. 28 A4–G 4
clar. 1
7 b
G 5, F5 solo vln. 29–31 A4–Bb4 ob. 2, vln. 1
9 F5 solo vln. 21–33 A4–G#4 vln. 1, clars.
F5–E5, E5–D#5,
10–11 solo vln. 34–36 Ab4–G4 clars.
Eb5–D5
12–13 D5 solo vln. 37 G4–F#4 vln. 1, clars.

# solo vln.,
14 D5–C 5 solo vln. 38 F#4–F4
clars., ob.

# b solo vln.,
15–17 C 5 (D 5) 39 F4–E4 solo vln.
vlns. 1 & 2
solo vln., #
18–20 C5 40 E4–D 4 solo vln.
vln. 2

b solo vln.,
21 B4 vln. 2, winds 41–42 E 4–D4
vln. 1

# solo vln.,
23 B4–A 4 vln. 2 43 D4–C4, C4
clar. 1, vln. 2

b vlns. 1 & 2, vlns. 1 & 2,

24–25 B 4 44 C4–B3
solo vln. clar. 2

b b fl. 1, clar. 1,
26 B 4–A4 vlns. 1 & 2 45 B 5
vln. 2, solo vln.
27 A4 vlns. 1 & 2

EXAMPLE 12: A CHROMATIC DESCENT LISTED ACCORDING

TO MEASURE, PITCH, AND INSTRUMENT

occasional octave leaps) from m. 4 until m. 90 (three measures before

Rehearsal 9), where it vanishes for a measure. The chromatic cable
breaks off with a B5 in the viola. In m. 91 the clarinet and solo violin
have Bb4 which initiates a chromatic line that now ascends until m.
110, then descends again briefly until m. 113 (nine measures from the
end of the movement), at which point the solo violin begins to rage
wildly through Hex2,3 (see Example 7). It would appear that during
mm. 4–91 almost every pitch in the score is succeeded only a short
time later by a pitch (not just pitch class) a semitone lower.
Countering the decay of chromatic descent is the tendency to leap by
a major seventh. The first important occurrence of this is the soloist’s
44 Perspectives of New Music

leap from Bb5 to A6 across the seam of mm. 12–13. The highest
pitches in the soloist’s arpeggios have been sliding down by semitone
at the rate of approximately one semitone every two to three measures.
But the highest line touches B b5 for only the last sixteenth note of
m. 12 before it is catapulted eleven semitones to A6, as if that Bb5 was
infused with an explosive energy. Building towards this leap the soloist
crescendos up to forte and then to fortissimo on the A6, the highest dy-
namics yet. After the leap, the slurred bowing in groups of two to thir-
teen notes give way to bowing each sixteenth-note individually. In PC
space the harmony continues to sink chromatically from m. 12 to m. 13.
But the leap of eleven semitones provokes a pivot in pitch space around
the D5: the G3 of m. 12 is reflected up to A6 in m. 13, each a perfect
twelfth away from D5. This suggests interpreting the transformation
from [237T] at the end of m. 12 to [1269] in m.13 as being enacted
not by sliding down by TE but by inverting by T4I. The existence of
two distinct transformations sending [237T] to [1269] is equivalent to
the fact that T5I is a symmetry of [237T]. Throughout Rings one finds
the contrast of descending chromatic lines and explosive upward
leaps.19 Of course the transition to a higher register also prolongs the
chromatic descent which otherwise would be cut off by the lower
limits of the violin’s register.

CLIMACTIC ALIGNMENTS

Pitch class [T] plays a structural role throughout the work—it is a har-
binger of the climactic alignments. In this sense, Rings is a [T]-centric
work or, using a term introduced by the composer himself, we might
call Bb a “fetish note.” At the end of m. 20 (see Example 9), at the first
climactic chord, the soloist launches up to B b7, holding it for seven six-
teenth notes and initiating a phrase where each note is held for seven
sixteenth notes. The accented notes of this outer orbit melody are Bb7
and A7. This pitch pair satisfies three projected expectations: 1) The
constant tendency for pitches, or in this case pitch classes, to decay by
semitone; 2) In the sequence of highest notes, Bb was essentially skipped
over and now it is filled in, achieving a sort of gap-filling with respect
to temporal expectations; 3) The dynamic increases to triple-forte with
the arrival of Bb7, continuing the association of Bb with high-energies
that was initiated in m. 12. (See the discussion in the last section for
more detail.)
The PC [T] continues to be pivotal. The pitch classes of the soloist’s
melody, beginning in m. 20 (see Example 9), are [T, 2, 3, 9, 2, 1].
After this the soloist returns to a faster orbit for mm. 24–25 where it
 Multiple Time-Scales in Adès’s Rings 45

once again happens upon a Bb (Bb4) that launches it into an outer

orbit.20 Although the resolution/decay of Bb4 to A4 occurs in m. 26
as part of a chord that stretches from G3 to Eb7, what we are more
aware of is that the soloist left off with a B♭4 in m. 25 and is launched
up to Eb7 on the climactic alignment on the downbeat of m. 26. Pitch-
class [T] has similar functions at m. 30 and m. 45. We will return to
the significance of the pair Bb–Eb shortly.
When the music resets around m. 92, a B b is present at the begin-
ning of the chromatic ascent that balances the previous descent. At the
first big climax after the reset, in m. 110, the soloist lands on B6 but
soon arrives on Bb6 with a tenuto mark, once again adding weight to
this pitch class, even though its role is not as central as in earlier
climaxes. The piece ends on a Hex2,3 chord with the soloist landing on
Bb3. Thus PC [T] plays a dominant role in the early climaxes and in
the final chord, where it also appears in the oboes, clarinets, and
trombones—it is the PC most doubled in the final chord.
These instances of Bb do not exhaust its significance. We will return
to its large-scale structural role after a broader discussion of the climaxes.
The alignment of celestial bodies has immediate physical conse-
quences. When the Earth-Moon-Sun system forms a syzygy it causes
Moonquakes and leads to the stronger spring tides21 on Earth. In Rings,
the alignment of the orbiting instruments also leads to a spring tide,
manifested in periodic climaxes of explosive power (see Example 13).
In Rings the periodic swells to climactic chords release tension. They
function not so much like harmonic cadences from tonal music, but
like explosive releases of energy resulting from the alignment of the
various orbiting instruments. Example 13 catalogs the harmonies that
occur at these climaxes. While Hex2,3 obviously plays an important
role, it is more difficult to understand the logic of the harmonies at the
other climaxes. One possibility is that we should view these harmonies
as snapshots of a pitch process. Sometimes the snapshot captures the
pitch structure clearly, but other times a single still frame is misleading,
and an analysis tracing the paths of individual voices (moving with
their own sense of time) would be necessary to find the cause for a
harmony to sound at a given climax. What is common to all of these
climaxes is that they catapult the soloist, and sometimes other
instruments as well, into outer orbits. The slower pace of the outer
orbits provides a sense of repose after a climax—the mad rush
continues on but we are now observing it from a distant orbit and so
are less affected by it.
46 Perspectives of New Music

MEAS. PITCH-CLASS SET PRIME FORM SOLOIST–EJECTION

20 Hex2,3 (014589) 6-20 b
B 7
26 [9T02357] (013468T) 7-34 Eb 7
28 [4568T0] (012468) 6-22 D7→B 7 b
34 [35790] (02469) 5-34
38 [6T1] (037) 3-11 b
B 7→B3
45 [35679T] (013457) 6-Z10 Ab4 . . . Gb6
59 [E0134578] (01245689) 8-19 C6→G6
62 [369TE] (01258) 5-Z38 b
E6→B 7
110 [234679TE] (0124578) 8-20 b
E 6→B6→G7
114 [13569T] (014579) 6-31 F6→Db8
115 [7E2] (037) 3-11 F#7→B7
116 [T23] (015) 3-4 Bb3→G4
117 [7T2] (037) 3-11 D7→Db8
118 [T23] (015) 3-4 D4→Eb5
119 [27] and [67E] (015) 3-4 D6→Eb7
120 Hex2,3 (014589) 6-20 D7→B7, B5→E 7 b
121–22 Hex2,3 (014589) 6-20 G6→B7

EXAMPLE 13: CLIMACTIC ALIGNMENTS AND THE RESULTING EJECTIONS

OF THE SOLOIST INTO AN OUTER ORBIT

HEXATONIC COMPLETION

Ligeti’s Automne à Varsovie has a central section of calm involving a

slow line with an immense octave-displaced tritone doubling. The
octave displacement imparts the sense of a vast space. Beginning with
the climax at m. 62 in Rings, the soloist climbs (or is flung?) higher
and higher (passing through a scintillating violin-piccolo duet) until it
comes to rest on an E7 (harmonic). At m. 86 it becomes clear that we
have made a return to the opening of the work: the soloist has rapid
perfect twelfths in a similar register as the opening measures. However,
it is not a circle we have made, but a spiral:22 The soloist has F#–B
instead of D–G and the low B0 in the double bass creates the sense of
having achieved a great height. In m. 93 the winds introduce impuri-
ties to the F#–B crystal and the soloist follows, now sweeping out
 Multiple Time-Scales in Adès’s Rings 47

[237T] (0158) with its racing arpeggios instead of the [67E2] (0158)
at the opening of the work. This time the harmonies are transformed
by T1 instead of the T−1 process that guided the first two-thirds of the
work. The movement has something of a U-shape: harmonies sink by
T−1 until about m. 60; there is a calm repose from mm. 86–92; and
then the harmonies transform by T1 from m. 92 until close to the end
of the work. The spiral nature arises from the TE catapults: the bottom
of the U is perceived from a great height.
The large-scale structure composes out the set Hex2,3 (see Example
14). Recall the opening twelfths D–G and the subsequent F#–B from
Flute 2 in m. 3. With hindsight we see that the Bb–Eb needed to
complete the hexatonic set is withheld. Taking a larger perspective, the
opening descending twelfth is D–G and the return to this texture at m.
86 uses F#–B. What is missing, to complete the hexatonic set, is the
Bb–Eb. Of course we hear much of Bb and Eb throughout the work, but
a decisive statement of these two pitch classes as a descending twelfth
is withheld. The finale completes the set Hex2,3 and includes the Bb–Eb
in a sequence of descending twelfths. Griffiths hears the ending as “a
new rotating machine, of loud stuttering chords from the full orchestra”
that “cuts the soloist off.”23 The “rotating machine” he refers to is
presumably the repeating sequence of descending perfect twelfths, D–
G, F#–B, Bb–Eb, that fill out Hex2,3 (see Example 15). Parsing the
hexatonic scale into a T4-cycle of perfect fifths also appears at the
opening of Arcadiana.24 There the progression is stated completely, as
opposed to Rings, where a clear statement of the complete cycle of
perfect fifths is withheld until the end of the movement. In the last
measures of Rings the orchestra cycles through this sequence of
twelfths as the soloist thrashes wildly through Hex2,3. The completion
of the cycle of fifths creates a rounded form, though a spiral one rather
than a circular one.

EXAMPLE 14: LARGE SCALE HEXATONIC COMPLETION

48 Perspectives of New Music

EXAMPLE 15: THE HEXATONIC “ROTATING MACHINE”

The Bb–Eb has played a less obvious structural role even before the
finale. The soloist’s outer orbit melody of m. 20 begins on Eb and the
slightly more heliophilic orbit of m. 26 begins on B b, thus completing
the hexatonic sequence of fifths from the opening measures. This type
of relation appears in a somewhat diluted form in the pair of m. 30 and
35, and again in m. 62. One sees the gradual alignment of orbits,
climaxing in the explosive triple alignment of the hexatonic fifths, first
in the initial climax of Rings (m. 20) and finally in the terminal chord
of the movement.
Although Adès’s interest in hexatonic sets has been thoroughly
investigated by Roeder,25 I am not aware of an exposition in the litera-
ture discussing the sort of movement-length hexatonic completion
described above. Rings is built around a movement-long composing-
out of the T4-cycles of perfect fifths that appear in various movements
of Arcadiana.
 Multiple Time-Scales in Adès’s Rings 49

CONCLUSION

Adès states that in his work, “The form is a result of the inner music.” 26
Though one might also interpret this in a more philosophical light, in
Rings the microscopic details do shape the form of the music: the ini-
tial decay of a note descending a minor second initiates a process
which directs the structure of the movement. The existence of multiple
time-scales causes the processes to diverge from one another and then
realign, like orbiting planets. Adès’s music is filled, sometimes saturated,
with the unfolding of simple processes. Roeder analyzes examples of
expanding pitch-interval processes, decreasing durational processes, and
multi-dimensional pitch transformation processes;27 he also uses physical
processes (e.g. “the inexorable trickle of raindrops down a window-
pane”) as metaphors for Adès’s music.28 Joshua Banks Mailman claims
that “processes in music often depict narratively significant natural forces
or physical processes. . . .”29 The association of musical processes with
physical processes is, in part, what makes effective the use of physical
analogies to characterize aspects of the music of Adès and Ligeti.
The initial decay of a note descending a minor second is pursued
over the first two-thirds of Rings. The prolongation of such a simple
pattern over that time-scale is a procedure more often found in
minimalist process music. In particular, the unwinding of the
chromatic suspension-cables that underlies the structure of Rings is
reminiscent of the process used in Alvin Lucier’s Crossings (1984). In
Crossings, a pure sine tone modulates in frequency from the infrasonic
to the ultrasonic over sixteen minutes. Using a divided orchestra to
hocket, the pitches of a rising chromatic scale are introduced just
higher than the sine tone, creating accelerations and de-accelerations
of beats. In Rings the descending or ascending chromatic cables that
thread the orchestra parallel the modulating sine tone in Crossings; the
oscillations between neighboring chromatic tones in Rings are
analogous to the orchestral pitches and resulting beats of Crossings.
Both pieces are concerned with alignments and both acknowledge the
significance of the spatial location of the source of music through a
pervasive use of hocketing. Furthermore, Rings and Crossings both
probe the extremes of register.30
Adès’s music reflects the current “mental climate”31 in that it engages
with musical process just as we are all prompted to engage with the
notion of physical process through a scientific understanding of the
world. Such an association is suggested by the composer’s own
common use of scientific analogies, a few of which were quoted at the
beginning of this article. The use of the physical process of planetary
motion as a metaphor for the structure of Rings is analogous to but
50 Perspectives of New Music

distinct from the use of a narrative as a metaphor for music; it diverts

our focus away from a perception of agency within the music and
towards a perception of causality within the music. Mailman uses
Lucier’s Crossings to illustrate the narrative possibilities of minimalist
process music.32 He suggests that minimalist process music implies a
sense of agency outside of the time frame of the music and provides an
opportunity for the listener to engage in “ad hoc imaginative play.”33
In Rings Adès composes both the “process” of the chromatic suspen-
sion cables and the “imaginative play” of arpeggios and melodic lines.
Using the opening of Adès’s Piano Quintet, Emma Gallon illustrates
how a process guided by pitch continuity may override conventional
expectations of harmonic material within sonata form.34 She sees the
processes within the music as estranging the traditional sense of agency
often ascribed to the themes of sonata form. Gallon makes the striking
argument that, in the Piano Quintet, the agent is “time.”35 Both
Gallon and Mailman find that processes within music distance the
sense of agency from its traditional locus within the music.
A sense of stepping outside of the music has been invoked in other
ways with regard to Adès’s work. Roeder poetically interprets the end
of Arcadiana as the transfiguration of “mortal linearity” into “eternal
periodicities.”36 That moment would also seem to be an example of
what Taruskin called “a serene overview of the preceding music, as if
from a great height.”37 That serene moment arrives in the middle of
Rings when, once again, there is a sense of looking down from a great
height, or of looking inward from a distant orbit. The central place-
ment of this calm moment follows the form Ligeti constructed in
Automne à Varsovie.38 The explosive end of Rings contrasts sharply
with the serene ending of Arcadiana, but Rings also undergoes a
convergence to “periodicities.” In m. 115 of Rings the orchestra initi-
ates the T4-cycle of descending twelfths that fill out hexatonic space.
The soloist saturates the aggregate before the orchestra captures it in a
hexatonic trap where the soloist thrashes wildly in pitch space, but with
regular periodicity in pitch-class space. Whereas Automne à Varsovie
terminates with a saturation of chromatic space, Rings concludes with
the soloist saturating the hexatonic space that is constructed in
cadential gestures by the orchestra. Adès has suggested that all of
Ligeti’s pieces are tending toward “the heat death of the universe.” If
one is limited to the twelve chromatic tones, then saturating the
aggregate reaches towards a type of total disorder. Adès distinguishes
himself from Ligeti by claiming that his own pieces escape from the
“vanishing point” of “total entropy.”39 While Rings grapples with
dissipative effects that push towards disorder, it is rescued from the
disorder of the aggregate by a high-energy hexatonic (strange) attractor.
 Multiple Time-Scales in Adès’s Rings 51

ACKNOWLEDGEMENTS

Violin Concerto—Concentric Paths by Thomas Adès © Copyright

2010 by Faber Music Ltd., London. Excerpts from the printed
score by kind permission of the publishers.

Ligeti Etudes pour piano, Book 1 copyright © 1985 by Schott

Music GmbH & Co. KG. All rights reserved. Used by permission of
European American Music Distributors Company, sole U.S. and
Canadian agent for Schott Music GmbH & Co. KG.

Ligeti Konzert for violin and orchestra copyright © 1990, 1992 by

Schott Music GmbH & Co. KG. All rights reserved. Used by per-
mission of European American Music Distributors Company, sole
U.S. and Canadian agent for Schott Music GmbH & Co. KG.
52 Perspectives of New Music

NOTES

1. Thomas Adès and Tom Service, Thomas Adès: Full of Noises, Con-
versations with Tom Service (New York: Farrar, Straus and Giroux,
2012), 3, 10.
2. Ibid., 88–89.
3. Thomas Adès, Concentric Paths (London: Faber, 2010).
4. Adès and Service, Thomas Adès: Full of Noises, 138.
5. Dominic Wells, “Plural Styles, Personal Style: The Music of
Thomas Adès,” Tempo 66 (2009), 7.
6. This process brings to mind Beethoven’s Opus 111. In m. 11 of
the first movement of the sonata the repeated G note is like a crys-
tal, and the introduction of the Ab is the impurity—it causes a wave
of disturbance to propagate downwards. Later this walk down from
the b6th degree to the tonic is developed into a theme.
7. Wells says that the F# “infects” the pitch material. See Wells, “Plu-
ral Styles,” 12.
8. I will use “T” and “E” for the respective pitch classes Bb and B.
9. Adès and Service, Thomas Adès: Full of Noises, 25.
10. Paul Griffiths observed that Adès “. . . initiates a perpetual motion
machine in running semiquavers and at the same time introduces
what Mahler might have called a ‘nature sound,’ a raw acoustic
fact gleaming from the whole first movement’s horizon.” Paul
Griffiths, “Violin Concerto,” The Times Literary Supplement, 16
September 2005; p. 20; Issue 5345.
11. Richard Taruskin, “A Surrealist Composer Comes To the Rescue
of Modernism,” The New York Times, 5 December 1999,
http://www.nytimes.com/1999/12/05/arts/a-surrealist-com-
poser-comes-to-the-rescue-of-modernism.html.
12. This means that not all steps of “size 1” within the mode contain
the same number of equally tempered chromatic semitones.
13. This is an example of generalization in the sense defined by Straus
in “Remaking the Past,” p. 17.
14. “Strange attractor,” accessed 2 July 2013, http://www.encyclo
pediaofmath.org/index.php?title=Strange_attractor&oldid=26959.
 Multiple Time-Scales in Adès’s Rings 53

15. Florin Diacu and Philip Holmes, Celestial Encounters: The Origins of
Chaos and Stability (Princeton: Princeton University Press, 1996).
16. Griffiths, “Violin Concerto,” 20.
17. It requires slight calculation to reach this conclusion. That is
because the distance traveled also increases with the size of the orbit
so that it might be that the two factors balance out and the average
speed is constant for all orbits. That this is not the case can be seen
for circular orbits using a bit of algebra: For a circular orbit, a is
the radius of the circle. An object traversing the circumference of a
circle of radius a travels a distance of 2πa during the period T.
Thus the average speed S (given by the ratio of the distance traveled
and the time needed to travel that distance) is S=2πa/T=2πa/
a3/2=2πa−1/2. The average speed decreases as the inverse square
root of the radius of the orbit. Due to the rotational symmetry of
the circular orbit, the average speed is also the instantaneous speed.
18. John Roeder, “Co-operating Continuities in the music of Thomas
Adès,” Music Analysis 25/1–2, 135.
19. A “fuzzy” inversion might be taking place in the soloist’s material
in mm. 41–42. The sc in each measure is (01568). Set [2378T]
transforms to [9T235] via T 7. It is interesting to note that
T5I[2378T]=[79T23], which differs from the desired result only
by the presence of [7] in place of [5]. Thus there appears to be an
approximate pivot around the [23]-axis, similar to the pivot
around [2] found in mm. 12–13. Note that T5I[2378]=[239T].
20. This B♭4 can be traced back chromatically to the B4–A#4 of the
orchestral violins in m. 24, which in turn traces back to the C5
carried by Vln. 2 in mm. 18–20 (before the cataclysmic alignment)
and which was also carried by the soloist in m. 18.
21. “Spring” here refers to the “jump” in the size of the tides, not the
season. These spring tides occur approximately twice per month.
22. The composer has offered, “In my music it’s very often spiral
rather than circular.” Adès and Service, Full of Noises, 8.
23. Griffiths, “Violin Concerto,” 20.
24. See Examples 3a and 3b in Roeder, “Co-operating Continuities,”
128. Dominic Wells characterizes such pitch structures as a “5+2”
progression and provides many examples of them in Adès’s work.
See Wells, “Plural Styles, Personal Style: The Music of Thomas
Adès,” 12.
54 Perspectives of New Music

25. Roeder, “Co-operating Continuities,” 128. Also see John Roeder,

“A Transformational Space Structuring the Counterpoint in Adès’s
‘Auf dem Wasser zu singen,’” Music Theory Online 15/1 (2009),
2.3.
26. Adès and Service, Thomas Adès: Full of Noises, 131.
27. Roeder, “Co-operating Continuities,” 125–27.
28. Roeder, “Co-operating Continuities,” 135–37.
29. Joshua Banks Mailman, “Agency, Determinism, Focal Time
Frames, and Processive Minimalist Music,” in Music and Narra-
tive Since 1900, ed. Michael L. Klein and Nicholas Reyland,
(Bloomington: Indiana University Press, 2013), 140.
30. Adès is famous for his use of the outer limits of register. For other
examples of his use of extreme upper registers, see the music of
Ariel in The Tempest or the music of the cello in Lieux Retrouvés.
For low register extremes, see the music of the Duke and Hotel
Manager in Powder Her Face, or listen to his basso profundo voice
in an interview.
31. J. W. N. Sullivan, Beethoven: His Spiritual Development,” (New
York: Vintage Books, 1927), 7.
32. Mailman, “Agency, Determinism, Focal Time Frames, and Proces-
sive Minimalist Music,” 137.
33. Mailman, “Agency, Determinism, Focal Time Frames, and Proces-
sive Minimalist Music,” 141.
34. Emma Gallon, “Narrativities in the Music of Thomas Adès,” in
Music and Narrative Since 1900, ed. Michael L. Klein and
Nicholas Reyland (Bloomington: Indiana University Press, 2013),
216.
35. Emma Gallon, “Narrativities in the Music of Thomas Adès,” 223.
36. Roeder, “Co-operating Continuities,” 147.
37. Taruskin, “A Surrealist Composer Comes To the Rescue of Mod-
ernism.”
38. A precursor of this calm center is also found in the aria “Fancy,
fancy being rich” from Adès’s first opera, Powder Her Face (1995).
39. Adès and Service, Thomas Adès: Full of Noises, 139.
 Multiple Time-Scales in Adès’s Rings 55

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Diacu, Florin, and Philip Holmes. 1996. Celestial Encounters: The Ori-
gins of Chaos and Stability. Princeton: Princeton University Press.
Gallon, Emma. 2013. “Narrativities in the Music of Thomas Adès: The
Piano Quintet and ‘Brahms.’” In Music and Narrative since 1900,
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Griffiths, Paul. 2005. “Violin Concerto.” The Times Literary Supple-
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Encyclopedia of Mathematics. “Strange attractor.” Accessed 2 July
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Roeder, John. 2006. “Co-operating Continuities in the Music of
Thomas Adès.” Music Analysis 25/1–2: 121–154.
———. 2009. “A Transformational Space Structuring the Counterpoint
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Sullivan, J. W. N. 1927 Beethoven: His Spiritual Development. New
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http://www.nytimes.com/1999/12/05/arts/a-surrealist-composer
-comes-to-the-rescue-of-modernism.html
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