This handout adapts some slides from the presentation and includes a select bibliography

128

128

?

&#n#

n

b ?

? ##

? bn

bbb

An B-chord with coloring agent. Or an S-chord with Dflat as coloring agent. Or aT-chord with two coloring agents (E and C). Or BS1s1

?

i[6] i[5] i[4] i[3] i[2] i[1]

& # ## n

?n

&

Taruskin IV:226 (Scriabin aggregates)

# n # n n n”“

? # b

&

M3 upper slurs, m3 lower slurs, p4 under beam

œ

?&

œb

œ# œ# œ

wwwwww~~w

œ# œ# œ œœ œ œn œn

~~w œb œ# œ œn œb

œœœœ‚‚

œ

œ œbœ œb œ œb

œb œ

œb œn œ# œœ œœ œ# œn œ œ œ#œ œ

œ œ œ œn œbœ œ

œn œœ œœ œœ œœ œ œ œ#œ œ

œ œ œ œnœ œ

œ œ# œn œ# œ œ œn œ#

œ œ

œ œ#

=

=

=

=

=

?œ œb

œ œb œ œbœb œ

Key to Symbols in the following chord analyses: i[6] i[5] i[4] i[3] i[2] i[1]

&#n#

n

b ?

? ##

? bn

bbb

An B-chord with coloring agent. Or an S-chord with Dflat as coloring agent. Or aT-chord with two coloring agents (E and C). Or BS1s1

?

i[6] i[5] i[4] i[3] i[2] i[1]

& # ## n

?n

&

Taruskin IV:226 (Scriabin aggregates)

# n # n n n”“

? # b

&

M3 upper slurs, m3 lower slurs, p4 under beam

œ

?&

œb

œ#

wwwwww~~w

œ# œ# œ œœ œ œn œn

~~w œb œ# œ œn œb

œœœœ‚‚

œ

œ œbœ œb œ œb

œb œ

œb œn œ# œœ œœ œ# œn œ œ œ#œ œ

œ œ œ œn œbœ œ

œn œœ œœ œœ œœ œ œ œ#œ œ

œ œ œ œnœ œ

œ œ# œn œ# œ œ œn œ#

œ œ

œ œ#

=

=

=

=

=

&

Taruskin IV, 194

”“

?

&#n#

n

b ?

? ##

? bn

bbb

An B-chord with coloring agent. Or an S-chord with Dflat as coloring agent. Or aT-chord with two coloring agents (E and C). Or BS1s1

?

i[6] i[5] i[4] i[3] i[2] i[1]

œ œbœ œ œn œn

œ# œ# œ œ

œ# œ# œ

wwwwww~~w

œ# œ# œ œœ œ œn œn

~~w œb œ# œ œn œb

œœœœ‚‚

œ

œ œb œ œbœb œ

=

=

Two sketches for aggregate chords in Scriabin’s Acte préalable (after Taruskin, IV: 226)

Berg Altenberg Lieder #3, analysis of �nal aggregate chord.

Proko�ev, Sonata for �/vln and piano, op. 94. Analysis of opening period.

(Note “values” don’t pertain to proximity relationship but to tonal hierarchy, not discussed in this paper.)

i[x] = a harmonic interval of x semitones

Daniel HarrisonSMT Milwaukee, Nov. 7, 2014

Extending Harmony to Extended Chords

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‰œJœ

‰œJ œ ‰ œ

J œ Œ œ œ# œ œœ œ œœ

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¢

°

¢

Violin

Piccolo

Flute

Oboe

Violin I

Trombone

VioloncelloContrebass

ppp

ppp pp ppp

c

c

c

c

c

c

c

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∑

&##### ∑ ∑ ∑ ∑

&#####

n

∑ ∑ ∑ ∑

&##### ∑ ∑ ∑ ∑ ∑

&##### ∑ ∑ ∑

?##### ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

?##### ∑ ∑ pizz. ∑ ∑

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˙

Ó

w

w ˙ ww ww ˙

ÓÓ

˙

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ww

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ÓÓ

˙

ÓÓ

ww

w˙ ˙n www ∑

www˙˙ Ó

˙˙

˙˙n

wwww ∑

wwww ∑

˙˙ Ó

˙˙ Ó

wwwn wwwn

Œœn œn œn Œ

œn œn œn Óœn

Œ Óœn

Œœn

Œœ

Œœn

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p dolente dim.

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≈≈ œœnr≈≈ œœœr

≈≈ œœœnr≈≈œœœnr≈≈œœœnr≈≈œœœnr

≈≈ œœœœnr

≈≈œœœœnr

≈≈œœœœnr

≈≈œœœœnr

≈≈œœR

≈≈œœR

≈≈ œœR≈≈ œœœR

≈≈ œœœœnR

≈≈œœœœR

≈≈œœœœR

≈≈œœœœR

fiœjœ ™J

œ œœœ®œKrœ œ®œ

Krœ œ# œ

œ ≈ œ œn œ≈ œ œœ œ œ

œœœœœœœœœœœœœœœœœœœœœœœœœœœ œœœ œœœœœœ œœœ œœœœ#œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ œœœ

œœœœœœ

≈ ≈≈œ#r œ ™jœ œœœJ

œj≈≈≈

œœœœœœœœœœœœ#œœœœœœœœJ ≈≈≈œ

œR

≈≈ œœœR

≈≈œœœR

≈≈œœœR

≈≈œœœR

Noteworthy chord registers Strauss, Also Sprach Zarathustra, ending Beethoven, op. 110, II (two passages)

A characteristic passage of mixed tertian and fourth-chords.Hindemith, Symphonie Mathis der Maler, II, opening.

Harrison, p. 2

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&

& B

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025

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œ œœ

œ œb œb œ œœ

œ œœb

œ œ œn

œ œœœ

œ œœœ œ œœ

œ œ œœœ

œ œœ œ œ

œ

œ œ œ œ œ œ œ œœ

œ œ œ

œ œ œ œ œ œ œ œœ

œ œ œ

œ œ œ œ œ œ œ œ œ œ œ œb

œ œ œ œ œ œ œ œ œ œ# œ œ

2

T Adjacent pitches are separated by i3 or i4.

A Adjacent pitches are separated by i3, or i4, or i5.

B Adjacent pitches are separated by i3 or i4 or i5, and one or more i6.

Sn An A- or B-relation with n i2s.

sn An A- or B-relation with n i1s.

37

Proximity Relations

Built-out palette of triads according to TABSs relations

Harrison, p. 3

A

B

C

D

i ii iii iv V VI

A development of Harrison 1988 in re: invertible counterpoint. Demonstrated on the 24 arrangements of a model tetrachord.

Columns show conjugations: cyclic rearrangements of chord tones. Conjugation № I demonstrates the conventional theory of chord inversion.

Rows show positions of chord inversion (alphabetic labels after Day 1845). Each chord shown is the closed-position (most compact) exemplar of its conjugation.

& # ## n

?n

&

Taruskin IV:226 (Scriabin aggregates)

# n # n n n”“

? # b

&

M3 upper slurs, m3 lower slurs, p4 under beam

œ

?&

œb

œ œb

œb œn œ# œœ œœ œ# œn œ œ œ#œ œ

œ œ œ œn œbœ œ

œn œœ œœ œœ œœ œ œ œ#œ œ

œ œ œ œnœ œ

œ œ# œn œ# œ œ œn œ#

œ œ

œ œ#

=

=

=

Morris 1987: the Five Basic Spacing-Typesand their application to the model arrangements above

1. Overtone: larger intervals, lower pitches Conjugation III.2. [Undertone]: smaller intervals, lower pitches “ ” “ ” IV.3. [Even]: similarly-sized intervals, all pitches “ ” “ ” I, VI.4. Focused: smaller intervals, central pitches “ ” “ ” II.5. [Polar]: larger intervals, central pitches “ ” “ ” V.

Proximate pitches dispersed by invertible counterpointa tetrachordal example

Harrison, p. 4

Bernard, Jonathan W. 1997. "Chord, Collection, and Set in Twentieth-Century Theory." In Music Theory in Concept and Practice, edited by J. M. Baker, D. W. Beach and J. W. Bernard. Rochester: University of Rochester Press.

———. 2003. "Zones of Impingement: Two Movements from Bartok's Music for Strings, Percussion, and Celesta." Music Theory Spectrum 25.1: 3–34.

Blättler, Damian Joseph. 2013. A Voicing-Centered Approach to Additive Harmony in Music in France, 1889–1940. Ph.D. Dissertation, Music, Yale University.

Brown, Stephen C. 2003. "Dual Interval Space in Twentieth-Century Music." Music Theory Spectrum 25.1: 35–57.Chapman, Alan. 1981. "Some Intervallic Aspects of Pitch-Class Set Relations." Journal of Music Theory: 275-290.Chrisman, Richard. 1977. "Describing Structural Aspects of Pitch-Sets Using Successive Interval Arrays." Journal of Music

Theory 21.1: 1–28.Day, Alfred. 1845. A Treatise on Harmony. London: Cramer, Beale, and co. Goehr, Alexander. 1975. "The Theoretical Writings of Arnold Schoenberg." Perspectives of New Music: 3-16.Harrison, Daniel. 1988. "Some Group Properties of Triple Counterpoint and Their Influence on Compositions by J. S.

Bach." Journal of Music Theory 32.1: 23-49.———. 2011. "Three Short Essays on Neo-Riemannian Theory." In The Oxford Handbook of Neo-Riemannian Music Theories,

edited by E. Gollin and A. Rehding. New York: Oxford University Press.Hindemith, Paul. 1937 [1942]. Craft of Musical Composition. Translated by A. Mendel. Vol. 1 (Theoretical Part). New York:

Associated Music Publishers. Original edition, Unterweisung im Tonsatz (Mainz: Schott, 1937). Hohenegger, M. 2002. "The Banned 'Verklarte Nacht' - Ninth Chords in the Fourth Inversion for the First Time in

Schoenberg?" Osterreichische Musikzeitschrift 57.2: 32-38.Krenek, Ernst. 1940. Studies in Counterpoint. New York: G. Schirmer. Lansky, Paul. 1975. "Pitch-Class Consciousness." Perspectives of New Music 13.2: 30–56.Lewin, David. 1987. "On the 9th-Chord-in 4th-Inversion From'verklarte Nacht'." Journal of the Arnold Schoenberg Institute

10.1: 45-64.Martin, Henry. 2000. "Seven Steps to Heaven: A Species Approach to Twentieth-Century Analysis and Composition."

Perspectives of new music 38.1: 129-168.McGowan, James John. 2005. Dynamic Consonance in Selected Piano Performances of Tonal Jazz. Ph.D. Dissertation,

Theory, University of Rochester, Rochester.Morris, Robert. 1987. Composition with Pitch-Classes : A Theory of Compositional Design. New Haven: Yale University Press. ———. 1995. "Equivalence and Similarity in Pitch and Their Interaction with Pcset Theory." Journal of Music Theory 39.2:

207–244.Persichetti, Vincent. 1961. Twentieth-Century Harmony : Creative Aspects and Practice. [1st ed. New York: W.W. Norton. Ricci, Adam. 2003. What's Wrong with the Minor Ninth?: Chord Roots and Extensions. Paper read at Society for Music

Theory, at Madison, WI.Robison, Brian. 1994. "Modifying Interval-Class Vectors of Large Collections to Reflect Registral Proximity among

Pitches." Music Theory Online 0.10. http://mto.societymusictheory.org/issues/mto.94.0.10/mto.94.0.10.robison.art.Samplaski, Arthur G. 2000. A Comparison of Perceived Chord Similarity and Predictions of Selected Twentieth-Century

Chord-Classification Schemes, Using Multidimensional Scaling and Cluster Analysis, Music Theory, Indiana University.

Schoenberg, Arnold. 1912 [1978]. Theory of Harmony. Translated by R. E. Carter. Berkeley: University of California Press. Schubert, Peter. 1993. "'A New Epoch of Polyphonic Style': Schoenberg on Chords and Lines." Music Analysis: 289-319.Taruskin, Richard. 2005. The Oxford History of Western Music. 6 vols. Oxford ; New York: Oxford University Press. Tymoczko, Dmitri. 2011. A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. New York:

Oxford University Press. Ulehla, Ludmilla. 1966. Contemporary Harmony: Romanticism through the Twelve-Tone Row. New York: The Free Press. Wason, Robert W. 1984. Viennese Harmonic Theory : From Albrechtsberger to Schenker and Schoenberg. Ann Arbor, Mich.: UMI

Research Press.

Harrison, p. 5