the twelve-tone technique and beyond: overview and...

NACUSA Presentation

The Twelve-Tone Techniqueand Beyond:

Overview and Applications

October 6, 2014

Gershon Wolfe, PhD

Structure of Talk

● Tonal behavior● Atonality● Tonal Atonality

– Composing music

● Twelve-Tone Technique● Examples● Future Work

Tonal Music

● Major and minor tonality– Established tonic or central chord

– Each triad has a tonal function wrt the tonic

– The basic harmonic functions are the I and V

– Tension is created by moving from consonance(stable) to dissonance (instability)

● Dissonance implies tension (distance) against the tonic● Resolution, progression from dissonance to consonance● Resolution uses counterpoint

Consonance and Dissonance

● Consonances– Perfect: Unison, octaves, P4, P5

– Imperfect: M3, m6, m3, M6

● Dissonances– M2, M2, m7, M7, tritone

Harmonics

Note Fundamental 1st overtone 2nd overtone 3rd overtone 4th overtone 5th overtone

C 261.62 523.24 784.86 1046.48 1308.10 1569.72

E 329.62 659.24 988.86 1318.48 1648.10 1977.72

G 399.99 783.98 1175.97 1567.96 1959.95 2351.94

Note Fundamental 1st overtone 2nd overtone 3rd overtone 4th overtone 5th overtone

C 261.62 523.24 784.86 1046.48 1308.10 1569.72

Eb 311.13 622.25 933.38 1244.50 1555.63 1866.75

G 399.99 783.98 1175.97 1567.96 1959.95 2351.94

C Major

C Minor

Dissonance

● Dissonance has the need to resolve toconsonance.

● Dissonance is culturally conditioned● Dissonance is complement of consonance● The buildup and resolution of dissonance is

what makes music beautiful and expressive.

Definitions

● Consonance– Large number of aligning harmonics

● Dissonance– Close but non-aligning harmonics

Harmony

● Governed by chords– Usually consonant by design, some more

consonant than others

– Any note not within harmony is dissonant● Steps: less dissonant● Leaps: more dissonant● Resolution: usually by steps● Dissonance on strong beats have more emphasis

20th Century Music

● Triadic structure does not have to generate atonal structure

● Non-triadic harmonic formations can function asreference elements

● The twelve tone complex does not preclude theexistence of tonal centers

Musical Phrases in Tonal Music

● Move from consonance to dissonance and back● Notes spaced over several octaves function in

the same way as if they were placed in oneoctave

● Tonal music assumes that chord within thescale have harmonic implication and function

● Secondary tonal centers are established bycadences– Two or more chords that end a section

Mid 18th Century

● Began using Neapolitan, French, and Italiansixth chords

● These chords temporarily suspend a sense ofmode– Changing between major and minor voice for the

tonic chord

– Made listener unsure of whether the music was inmajor or minor

– Increase of the use of notes that were not part ofthe seven notes, a chromatic chord. Mahler andStrauss

● Chromaticism

Neapolitan Chord

● It's not the quality of the chord, it's therelationship to tonality that gives it its dissonant-like sound.

C Major

I: C E G

IV: F A C

V7:G B D F

I: C E G

N6:Dd F Ab → F Ab Db

Neapolitan in Action

Regular

Neapolitan

Neapolitan

Regular

Atonality

● Lack of a tonal center, or key● In the Western tradition and is not tonal● Arnold Schoenberg, Alban Burg, Anton Webern● Music without tonal center had been written

previously, Franz Liszt 1885● The use of ambiguous chords● Harmonic relationships hardly functioned at all● The twelve-tone technique (1923) was taken as

the inspiration for serialism

Tonality in Atonal Music

● Fo a piece to be tonal– Must have functional harmony

● Dominants (leads to tonic)● Subdominants● Tonics

– Voice leading● How dissonance is treated

● In atonal music, Roman numerals do not apply

Tonality in Atonal Music

● Atonal music can have pitch-class centers– Webern's String Quartet, Op. 5 --> ref. to C#

– Bartok's String Quartet, No. 2 --> 4-19 (0148)

– Stravinsky's Symphony in C --> triads CEG EGB● Supports either B or C in melody

– Schoenberg's Piano Piece, Op. 11 --> Key E, F#, G● All are correct.

● Centric effects play a crucial role in shapingatonal music

Set Class Table

http://lulu.esm.rochester.edu/rdm/pdflib/set-class.table.pdf

Centric Effects

● Centricity comes form the use of stablereferential collections– Diatonic 7-35 (013568T) Most of Western tonal

music

– Octatonic 8-28 (0134679T) Highly symmetrical wrttransposition and inversion

– Whole-tone 6-35 (02468T) highly symmetric,contains only two distinct members

● Mixing these together: diatonic and octatonic

Diatonic Collections

● Used without the functional harmony or voiceleading of tonal music

● Stravinsky's Petrushka, uses only diatoniccollections– G-Mixolydian G A B C D E F G first eight measures

– A-Dorian A B C D E F# G A

– Allows for a change in centricity

● Tonal music: triads are divided up vertically ● Atonal music: triads, with other diatonic subsets

(harmonies) → 4-23 (0257) and 3-9 (027)

Stravinsky's Rake's Progress

● Uses A B E and A D E, two forms of 3-9 (027)and together they form 4-23 (0257), one ofStravinsky's favorite sets.– The passages are centered around A, and on the

perfect 5th A-E, but the fifth is not filled with thirds, itis filled with seconds and fourths. This created thecharacteristic Stravinsky sound.

– The music may be diatonic, but is not triadic ortonal.

Octatonic Scale

● Favored by Bartok and Stravinsky● Has only three forms● Subsets incluse

– 3-2 (013)

– 3-11 (037) major/minor triad

– 4-3 (0134)

– 4-10 (0235) tetra chord

– 4-26 (0358) minor 7th

– 4-27 (0258) dom or half-dim 7th

– 4-28 (0369) diminished 7th

Octatonic Scale

● Bartok's Mikrokosmos, No. 101, diminished 5th

– Combines 4-10 (0235) with T(6) → Complete OCT

– 3-7 (025), then shifts back to 4-10

● Stravinsky's Petrushka, Second Tableau– Two Major triads a tritone apart → incomplete OCT

● Messian's Quartet for the End of Time, 3rd mov– Uses 3-2 (013) by T(3) and T(6)

Whole-Tone Scale

● Contains only even intervals● Berg's Four Songs, Op. 2

– Uses set class 4-25 (0268)● Maps onto itself twice under transposition and inversion● Has only six distinct members and the music cycles

through all six forms

– 3-8 (026)

● Stravinsky's Requiem Canticles uses whole-tone harmonies

Diatonic/Octatonic Interactions

● Octatonic is rich in triads– Four Major and minor triads, and others

● Stravinsky's Symphony of Psalms– Uses octatonic, on E: E F G Ab Bb B C# D

– Set class 4-3 (0134) is contained in this orderingand was stated by Stravinsky as the basis idea forthe entire piece.

– $-3 (0134) Interval vector <212100>

Inversional Axis

● An axis of symmetry can function as a pitch-class center → play a centric role in the piece

● Inversionally symmetrical sets maps onto itselfunder T

nI

● Schoenberg's Orchestra Pieces (3rd), Op. 16 issymmetrical around the axis E-Bb, using thefollowing chord:

1 2 1 4

G# A B C E

1 4 4 1

B C E G# A

Axis of Inversion

C

F#

D#

E

F

C#

D

B

G

G#

A

A#

This is symmetry of pitch class, not of pitch. Pitch class E plays a central role in the piece.

(0 4 8 9 E)

Twelve-Tone Pieces

● Tonal music is communal, from composer tocomposer a large amount of musical material isshared

● In twelve-tone music, little is shared from pieceto piece (i.e. no two pieces use the sameseries). The series shapes the music.

● The series uses four different orderings– Prime: Original set or series

– Retrograde: Reversal of prime

– Inverse: Inversion of the pitch

– Retrograde inverse: Thr retrograde of the inverse

Twelve-Tone Row

I0 I2 I4 I6 I8 I10 I1 I3 I5 I7 I9 I11

P0 C D E F# G# A# C# D# F G A B R0

P10 A# C D E F# G# B C# D# F G A R10

P8 G# A# C D E F# A B C# D# F G R8

P6 F# G# A# C D E G A B C# D# F R6

P4 E F# G# A# C D F G A B C# D# R4

P2 D E F# G# A# C D# F G A B C# R2

P11 B C# D# F G A C D E F# G# A# R11

P9 A B C# D# F G A# C D E F# G# R9

P7 G A B C# D# F G# A# C D E F# R7

P5 F G A B C# D# F# G# A# C D E R5

P3 D# F G A B C# E F# G# A# C D R3

P1 C# D# F G A B D E F# G# A# C R1

RI0 RI2 RI4 RI6 RI8 RI10 RI1 RI3 RI5 RI7 RI9 RI11

P and IR

● P and IR are very similar, they have the sameintervals in reverse order. Upside down andbackwards versions of one another.

● Comparing R, RI have complementary intervalsin the same order

● For any series, there are 48 forms. All areclosely related in terms of pitch class andintervals.

● Most twelve tone pieces use fewer than 48forms.

Prime Form

● A form that represents the most fundamentalrepresentation of the set classes

● If sets have the same prime form they willsound alike, contain the same number ofpitches, and intervals.

● Sort-of the same as how all major chords areequivalent in tonal music.

Calculating Prime Form

● P = C E G# G F● P = 0 4 8 7 5

● PN = 8 7 5 4 0 Most compacted to the left

● PNI

= 4 5 7 8 0

● Prime = 0 1 3 4 8– 5z-17 (01348) ~ 5z-37 (03458)

● Z-related sets have same Interval vector– ICV = <212320>

Prime Form

● Considered to be the simplest version of thepitch class set, an ordered set or series (i.e.Serial Composition)

● Once you know the prime form, look it up in atable of prime forms to get its interval vectorand related pitch class sets

● Use the prime from to better understand,control, and manipulate the harmonies in yourmusic

Composing Atonal Music

● Choose pitches that do not imply tonality– Reverse the rules of the common period

● The use of dissonant counterpoint (atonalcounterpoint)– First species is all dissonant

– Consonances are resolved through skips, not steps

● George Perle: Structural coherence is achievedthrough intervallic cells, a fixed intervalliccontent as a chord or melodic figure

● Allen Forte: developed the theory behind atonalmusic

Composing Atonal Music

● Tonal music: chords belong to the same scale● Atonal music: different operations on the chords

are defined● Notes are represented on a circle● Two main operations: transposition and

inversion

Invariants

● Notes that stay identical after transformation● No difference is made between the octave in

which the note is played.– All D#s are equivalent no matter the octave in which

they occur in.

Structure to Atonal Music

Equivalent chords● Invariants● Z-related pairs● Identical subsets● All give a continuity to the piece and sort-of

makes up for the lack of atonality by defining anew equivalence relationship between chords

Twelve-Tone Technique

● Invented by Arnold Schoenberg around 1921● An intervallic cell (tone row) that contains all

twelve tones of the chromatic scale and treateach semi-tone with equal importance– As opposed to treating the tonic and dominant as

bing more important

Tone Row

● The fundamental of the TTT is the tone row– An ordered arrangement of the chromatic scale

● Four postulates of the tone row– A specific ordering of the 12 notes without regard to

octave placement

– No note is repeated in the row

– The row undergoes intervalic-preservingtransformations (P, R, I, RI)

– The row in any of its transformations may begin onany degree of the chromatic scale

● Transpositions are indicated by an integer (0-11)

Tone Row

P

R

I

RI

Tone Row Analysis: P, RI

● Notice, for P and RI, there are three sets of two-pitch sequences are the same

P

RI

Transformations

● Every row has up to 48 different row forms– Some have fewer due to symmetry, a result of a

derived row leading to invariance

● Ascending chromatic scale, as a tone row, hasinvariance

– RI = P and I = R, only 24 forms of the P0-row are

available

● Typically, the RI has three points where thesequence of two pitches are identical to the P-row

Properties of Transformation

● Basic tone row = P0 ans give 12 pitches, there

are 12! tone rows (479,001,600), although469,022,400 of these are transformations ofother rows

● There are 9,979,200 truly unique twelve-tonerows possible

Derived Rows

● Transforming segments of the chromatic scaleto yield a complete set of 12 tones– Trichords (except dim-triad 0,3,6)

– Tetrachords (except interval class 4, M3 betweenand tow elements)

– Hexachords

● Anton Webern often used derived rows– P, RI, R, I

● Invariance is a side effect of a derived row

Webern's Concerto, Op. 24

B Bb D Eb G F# G# E F C C# A

Let B = 0 then the row becomes 0 11 3 4 8 7 9 5 6 1 2 10

The third trichord, 9 5 6, is the first trichord, 0 11 3,backwards, 3 11 0, and transposed by 6, 3+6, 11+6,0+6 = 9 5 6 mod 12

Combinatoriality

● A row and one of its transformations combine toform a pair of aggregates

● Schoenberg often combined P0 and I

5 to create

two aggregates between the first hexachord ofeach, and the second hexachord of each,respectively.

● Op. 24 is all combinatorial, P0 being

hexachordally combinatorial with P6, R

0, I

5, and

RI11

Tone Row: 0 11 3 4 8 7 9 5 6 1 2 10I0 I11 I3 I4 I8 I7 I9 I5 I6 I1 I2 I10

P0 0 11 3 4 8 7 9 5 6 1 2 10 R0

P1 1 0 4 5 9 8 10 6 7 2 3 11 R1

P9 9 8 0 1 5 4 6 2 3 10 11 7 R9

P8 8 7 11 0 4 3 5 1 2 9 10 6 R8

P4 4 3 7 8 0 11 1 9 10 5 6 2 R4

P5 5 4 8 9 1 0 2 10 11 6 7 3 R5

P3 3 2 6 7 11 10 0 8 9 4 5 1 R3

P7 7 6 10 11 3 2 4 0 1 8 9 5 R7

P6 6 5 9 10 2 1 3 11 0 7 8 4 R6

P11 11 10 2 3 7 6 8 4 5 0 1 9 R11

P10 10 9 1 2 6 5 7 3 4 11 0 8 R10

P2 2 1 5 6 10 9 11 7 8 3 4 0 R2


P0 and P

6

P0 and R

0 (T2)

P0 and I

5

P0 and RI

11

Partitioning and Mosaics

● Create segments from set or aggregate throughregister difference

● Cross-partition, a two dimensional configurationof pitch class, a reordering of the verticaltrichords while keeping the pitch classes in theircolumns

● Mosaic is a partition that divides the aggregateinto segments of equal size

Mosaics and Symmetry

Time

Pitc

h

Symmetry Relations

P = 0 11 3

R = 9 5 6

RI= 4 8 7

I = 1 2 10

B Bb D Eb G F# G# E F C C# A

Cross Partition

● Arrange pitch classes into a rectangular design– Vertical columns (harmony): derived from adjacent

segments of row

– Horizontal columns (melody): a result of theplacement of vertical columns and may contain non-adjacencies

62 43 34 26

Cross Partition

1 2 3 4 5 6 7 8 9 10 11

34 partition

0 3 6 9 0 5 6 111 4 7 10 2 3 7 102 5 8 11 1 4 8 9

Cross partitions were used by Schoenberg Op. 33Klavierstuck, also used by Alban Berg andLuigi Dallapicolla.

Serialism

● A method of composition uses a series ofvalues to manipulate elements of music.However, serialism itself is not a method ofcomposition.

● Placing pitches into series, the tone row.● Music constructed according to permutations of

a group

Summary

● The rules of the twelve-tone technique shouldbe broken

● Stravinsky used cyclic permutations● Berg incorporated tonal elements

– Lyric Suite for string quartet, only the 1st and 6th movements use twelve-tone technique, the 2nd and4th movements don't.

– 1st movement: P0 is divided into two hexachords, the

second a retrograde of the first. Each half containssix of the seven notes of the diatonic scale

Berg's Lyric Suite

● Berg also reorders the six note groups toproduce two derived rows– The first group in scalar form

– The second group as a sequence of perfect fifths

● The diatonic aspect of the Lyric Suite is verycharacteristic of Berg, and thus creating tonalassociations in atonal music.

● In this way, Berg was very different thanSchoenberg and Webern.

Constructing Piece1P

9 = A B D F G A# F# G# E C C# D#

I9 I11 I2 I5 I7 I10 I6 I8 I4 I0 I1 I3

P9 A B D F G A# F# G# E C C# D# R9

P7 G A C D# F G# E F# D A# B C# R7

P4 E F# A C D F C# D# B G G# A# R4

P1 C# D# F# A B D A# C G# E F G R1

P11 B C# E G A C G# A# F# D D# F R11

P8 G# A# C# E F# A F G D# B C D R8

P0 C D F G# A# C# A B G D# E F# R0

P10 A# C D# F# G# B G A F C# D E R10

P2 D E G A# C D# B C# A F F# G# R2

P6 F# G# B D E G D# F C# A A# C R6

P5 F G A# C# D# F# D E C G# A B R5

P3 D# F G# B C# E C D A# F# G A R3


Constructing Piece1

● Take and P9 and add RI

9 to the lower register.

These two series are related to each other byT

6I

● This piece will only use P9 and RI

9 and their

retrogrades, R9 and I

9.

● Don't just state the series verbatim, a goodcomposer will use a bit of creativity and musicalprowess.

Constructing Piece2

5z-17 ( 01348) P = E G# C B A

I4 I8 I0 I11 I9

P4 E G# C B A R4

P0 C E G# G F R0

P8 G# C E D# C# R8

P9 A C# F E D R9

P11 B D# G F# E R11

RI4 RI8 RI0 RI11 RI9

I4 I8 I5 I3 I0

P4 E G# F D# C R4

P0 C E C# B G# R0

P3 D# G E D B R3

P5 F A F# E C# R5

P8 G# C A G E R8

RI4 RI8 RI5 RI3 RI0

5z-37 (03458) P= E G# F D# C

Quasi-Sonata Form

Exposition Development Recapitulation Coda

Maj: I V V V Modulations V I I I I

Theme 1 tr Theme 2 c Lots of Freedom Theme 1 tr Theme 2 c C

Tonal Atonal

Exposition: Establishes tonic ---> modulation to new key Modulation via interval permutation.Axis-System andhexachord combinatoriality

Development: Transpose Exposition Transpose Exposition

Recapitulation:

the twelve-tone technique and beyond: overview and...

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