Symmetry Class Summary Alex July 24, 2006 Here’s what we did in class: • We learned about symmetrical scales. – I started with the whole-tone and octatonic scales. – I moved on to six- and nine-note scales of interlocking augmented chords, and the eight-note scale that’s a whole-tone scale plus another tritone. – I ended with the six-, eight- and ten-note “barbell” scales of chro- matic clusters, and explored a relationship between minor seconds and perfect fifths (the “switcharoo”). • We did a lot of ear-training exercises. – We tuned symmetrical scales in the squash court. – We sang canons. – We sang a little bit of Messiaen’s Quartet for the End of Time (sixth movement). – I had you listen to seventh chords in different inversions, and tried to pick out the root. • We listened to a lot of late-nineteenth- and twentieth-century music: – Debussy: selections from Pr´ eludes and Images – Bart´ok: selectionsfrom Mikrokosmos, the fifth string quartet, and Music for Strings, Percussion and Celesta 1
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Symmetry Class Summary
Alex
July 24, 2006
Here’s what we did in class:
• We learned about symmetrical scales.
– I started with the whole-tone and octatonic scales.
– I moved on to six- and nine-note scales of interlocking augmentedchords, and the eight-note scale that’s a whole-tone scale plusanother tritone.
– I ended with the six-, eight- and ten-note “barbell” scales of chro-matic clusters, and explored a relationship between minor secondsand perfect fifths (the “switcharoo”).
• We did a lot of ear-training exercises.
– We tuned symmetrical scales in the squash court.
– We sang canons.
– We sang a little bit of Messiaen’s Quartet for the End of Time(sixth movement).
– I had you listen to seventh chords in di!erent inversions, and triedto pick out the root.
• We listened to a lot of late-nineteenth- and twentieth-century music:
– Debussy: selections from Preludes and Images
– Bartok: selections from Mikrokosmos, the fifth string quartet, andMusic for Strings, Percussion and Celesta
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– Ravel: a selection from Miroirs
– Messiaen: Quartet for the End of Time
– Dutilleux: Ainsi la nuit (string quartet)
– Ligeti: selections from the piano studies
– Wagner: prelude to Tristan and Isolde (on CD)
– Webern: first movement of the Symphonie
– Recordings of Hindustani ragas marwa, shree, and todi
• We analyzed many of the pieces we heard.
• We studied basic serial (twelve-tone row) techniques.
– I showed you how to make a twelve-tone row matrix.
– We looked for row-forms in Dallapiccola’s “Quartina”.
• In the fourth week, we talked a little about bitonality in Western andIndian music, and touched on symmetrical harmonies in quarter-tonetemperament.
All we have left is to prepare a demo!
• We’ll review some of the activities from the first two weeks of Walden,such as singing scales, chords, and canons.
• You’ll write some songs and melodies in di!erent symmetrical scales.
• We’ll try to find a way to make musical analysis exciting for an audienceto watch.
2
& w w# w wb wn w w# w w# w wb wnC Cis D Es E F Fis G Gis A Bes B
Triad progressions that fit in the octatonic scale
œ œ œb œ
œœœ#
œNotice that we cycle through the inversions of the triad, and end up in a different place than we start.
œœœb œœœn œœœ## œœœbbbDown by minor thirds:minor triads
œ œ œ# œb
œœœb
œYou can also go up.
&
?
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œœœbb œœœn œœœ#Up by major thirds:major triads
Triad progressions that fit in the six-note scale with alternating minor seconds and minor thirds ("oriental" scale)
œb œ œ
œœœbbœb
Here, the triad ends where it started.
œœœ œœœ## œœœnbUp by major thirds:minor triads
œ œ# œ
œœœœ
&
?
44
44
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œœœœb œœœœ# œœœœ### œœœœbbbSeventh-chord progressions that fit into the octatonic scale
œ œ œ# œb
œœœœbn
œ
Symmetrical Relationships with Tonal ChordsAlex
Things to think about:1. Common tones2. Voice-leading patterns (expanding or contracting intervals?)3. Diminished triad relationships within seventh chords
Minor seventh,Half-diminished seventh,Major seventh,Dominant ninth
(to be continued)
&
?
44
44
w w w# w# wb w
!
wb w
!
w wb wn w# w# w wb w
!
&
?
4
w# wb w w w w
!
w w# wb
!
w# w wb w w# w w wb wn
!
&
?
7
œœœœb œœœœn# œœœœbbb œœœœbnn
œ œ œb œ
œœœœb œœœœn# œœœœ### œœœœnnb
œ œ# œ# œ
œœœœbbbb œœœœn#n œœœœbnn œœœœbbbbœb œ œ œb
Whole-tone scales with bonus notes
Alex
Whole-tone scale plus a tritone (symmetrical at the tritone)
This has four different modes (inc. Ligeti ostinato)...try them all!
Whole-tone scale plus an augmented triad (symmetrical at the major third)
This has three different modes. If you cycle through dominant, minor, or half-diminished seventh chords by major thirds, you use the notes in this scale (see below).
Examples:
&
&
w w# w
!
w# w w#
!
w wb wn w
!
w# w wb wn
!
wb wn w w# w
!
wb wn w w# w
!
&
&
7
w w# w wn wb w w w w# wb w# w
w w# wn w wb w w# w w w w# wb
&
&
8
w w# w
w w w
w# w w#
w# w# w#
w wb wn w
w w w w
w# w wb wn
w# wb wb w
wb wn w w# w
wb
w wb wn w w#
"Barbells" (chromatic cluster scales)Alex
Six-note scale (small barbells)
All these scales are symmetrical at the tritone.
Eight-note scale(medium barbells)
Ten-note scale(large barbells)
The minor-second/perfect-fifth switcharoo
The minor second and perfect fifth (inverted: major seventh and perfect fourth) are the only intervals that cycle through all twelve pitches (why?). A weird and interesting consequence is that a twelve-tone row turns into a different twelve-tone row if you substitute perfect fifths for minor seconds:
The same substitution with the barbells gives you the same scale, rearranged:
(What's kind of a scale do the missing notes make?)
(This is the chromatic scale with a missing tritone.)
(Exercise: fill in)
Notice that groups of notes stay the same. Why?Hint: visualize the circle of fifths.
& w w w w w w wsa re ga ma pa dha ni
wb wb w# wb wbre ga ma dha ni
&3
w w w w w w w# wb wb wb wb w#sa pa re dha ga ni ma re dha ga ni ma
&4
œ œ w w w w w œ œ w w w w wœ œ w w w# w w œ œ w w wb w w#
&8
œ œ w w# wb w wbœ œ w wb wb w# wb œ œ w# wb wb wb wb
œ œ wb wb w wb wb
&12
œ œ wb wb w wb w œ œ wb w w wb w œ œ wb w w w w œ œ w w w w w
Bitonality in Hindustani ScalesAlex
"Shuddh" (natural) notes
Accidentals: "komal" (flat) and "tivra" (sharp)
(sa and pa are always natural)
The twelve notes arranged as a circle of fifths Aug. 3rd!
Scales with augmented 3rds:
Marwa
Shree Todi (one more)
Diatonic scales
Diatonic scales
Row for Webern Symphony Op. 21
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Music engraving by LilyPond 2.8.5 — www.lilypond.org
RowforDallapiccola'sQuartina
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Music engraving by LilyPond 2.8.5 — www.lilypond.org