|————Phrygian tetrachord—————| |—————Ionian tetrachord—————-——| Ex. 1: Overtones and “undertones”: extrapolated from Arthur von Öttingen, Harmoniesystem in dualer Entwicklung (1866). “The tones of the major triad are common components of the tonic fundamental; those of the minor triad have a common phonic overtone.” TECHNOLOGIES OF SYMMETRY IN BRAHMS’S “DOUBLE” CONCERTO Roger Moseley, Cornell University [email protected]Ex. 2a: The symmetrical relationship between Ionian and Phrygian tetrachords in the formation of Ionian (major) and Phrygian scales Ex. 2b: Riemann’s “minor-major” or “harmonic major” scale Tonic = D (D + ) Ex. 2c: “Major-minor” or melodic minor scale Phonic = A (°A); °G when inverted |————Ionian tetrachord—————| |————Phrygian tetrachord————| |————Phrygian tetrachord————| |————Ionian tetrachord————-——| |————Phrygian tetrachord—————| |—————Phrygian tetrachord——————| |—————Ionian tetrachord—————-——| |————Phrygian tetrachord—————| |————Ionian tetrachord————-——| |————Ionian tetrachord————-——| |————Ionian tetrachord—————| |————Phrygian tetrachord—————|
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TETRACHORDS, HEXATONIC MOTION, AND MODAL … · Ex. 3: Tonic and phonic triads and their respective Gegenklänge Ex. 4: Brahms/Heine, “Der Tod, das ist die kühle Nacht,” op.
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|————Phrygian tetrachord—————|
|—————Ionian tetrachord—————-——|
Ex. 1: Overtones and “undertones”: extrapolated from Arthur von Öttingen, Harmoniesystem in dualer Entwicklung (1866).“The tones of the major triad are common components of the tonic fundamental; those of the minor triad have a common
phonic overtone.”
TECHNOLOGIES OF SYMMETRY INBRAHMS’S “DOUBLE” CONCERTO
Ex. 6: Brahms, “Double” Concerto, I: 268-93Performed by Julia Fischer (violin), Daniel Müller-Schott (cello), and the Netherlands
Philharmonic Orchestra Amsterdam, conducted by Yakov Kreizberg
TETRACHORDS, HEXATONIC MOTION, ANDMODAL SHIFTS IN THE FIRST MOVEMENT
[———Phrygian tetrachord————] + [——Ionian tetrachord———] = A melodic minor (or conceivably E harmonic major)
(mm. 1-4)
A flat+ A flat/G#- E+ E-
C+ C- A flat+ A flat/G#-
E7 = V7
continued...
(mm. 268-79;neo-Riemannian
annotations)
diatonic (circle of fifths) hexatonic
melody:
key: °E (in Riemannian terms)
Sunday, February 21, 2010
(as described by Richard Cohn)
Ex. 7: Joachim, “Abendglocken,” op. 5 no. 2, central episodePerformed by Florin Paul (violin) and Birgitta Wollenweber (piano)
Ex. 9: Brahms, “Double” Concerto, I: 396-417
Geg. = Gegenklang
Geg. Geg. Geg. Geg.
STINGS IN THE TAIL? GEGENKLÄNGE IN THECODA OF THE FIRST MOVEMENT
Geg. Geg.
Geg. Geg.
(mm. 396-400;Riemannian annotations)
(mm. 409-17)
C+ °E
°A
F+ °A
Sunday, February 21, 2010
alla zoppa rhythms (characteristic of the style hongrois)
Ex. 8: The symmetrical relationship between Joachim’s “Gisela” and “Frei aber einsam” mottos,which outline an Ionian and Phrygian tetrachord respectively