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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.
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& w w# w wb wn w w# w w# w wb wnC Cis D Es E F Fis G Gis A Bes B
Absolutes for singing symmetric harmoniesAlex
& cVoice œ œ œ# œ# œb œ œD E Fis Gis Bes C D,
1. Whole-tone scaleA. Whole-tone scale #1 œ œ œb œ# œ# œ œ
D C Bes Gis Fis E D!
œ œ# œ# œb œ œ œetc.
Starting on different degrees
&4
œ œ œ# œb œ œ œA B Cis Es F G A...
B. Whole-tone scale #2
œ œ# œb œ œ# œC. Different orderingsi. Augmented triads œ œb œ# œ œ# œ
&7
œ œ œ# œ# œb œ œii. Zig-zag
œ œb œ œ# œ œ œb œiii. Tritones
œ œœ# œœœ# œœœœbD. Sustaining exercises
œœœœœ œœœœœœ ˙˙˙˙˙˙
Symmetry ExercisesAlexander Ness
& 44 .. .. .. ..Voice
Lent e»ª™wœ. œ. œb . œ# . œ.
-œ# . œb . œ.^
!
doucement sonore
œ œ œ œb œ# œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ3 3 3 3 3 3 3 3
& .. ..3
œ œ .œ# œb ˙ œ œb œ# œ œ œ .œ œ œ .œ œ œ .œ œ .œ œ3 3
œ .œ œ .œb œn œ .œ œ .œ œ jœ"Cloches à travers les feuilles"
(Debussy's Images, Book 2, p. 43)
& 42
Modéré e»••(Dans un rhythme sans rigueur et caressant)
très doux
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Œ œœ# -
pœœ œœ# œœ œœb . .. .œœ#
œœ# œœb# Œ œœ# -
pœœ œœ# œœ œœb œœ#
più pœœ œœb# œœb œœ ˙ œœ Œ
&13
Œ ‰ œ#! très souple
œb œ œ# œ# œœ œœb#rœœ " œ!
œ œ œ œ# œœ œœb#rœœ " œ œ œ œ# œœ# œœ ..œœb œœ œœ# œœ œœ
3
jœœb# œœ#
"Voiles" (Debussy's Preludes, Book 1, p. 3)
& 4218
œAndante q»¡º•
p dolce
œ# œ œ# œ œ# œ œ œ# œ# œ œ# œ œ# œ œ# ˙
&24
œ#Pœ# œb œ œb œ# œ# œ œ# œ# œb œ# œ œ œ œ# œ# œ œ# ˙#
"Whole-tone Scale"(Bartók's Mikrokosmos, Book 5, p. 32)
Singing the Whole-Tone ScaleAlexander Ness
&(non legato, martelé)
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œ#ƒ
Décidé, vigoureux, granitique, un peu vif (q»¡¶§ env.)
œ œb - œ œ# œ# œ œb œ# œ œb œn œb œ ˙# œ œb œ- œn œb œ œ# œ# œ
&34
œ# œ œ# œ œb - .œ- ˙# - œ(octatonic)
œ œb œ œ œ# œ# œb œb œ œ œ
&36
œ-œ œb œ. œ- œ-
œ œb œ œ œ# œ œb œ œ# œ œ .œb - -"Danse de la fureur, pour les sept trompettes"
(Messiaen, Quartour pour la Fin du Temps, 23)
2 Singing the Whole-Tone Scale
& 23Voice œ œ œ œ œ# œb œn œ# ˙2. Octatonic scaleA. Octatonic scale #1, first mode (whole step on bottom)
large tetrachords(whole-half-whole)
œ œ œ œ œb œ œ# œ# ˙B. Octatonic scale #2
œ œ# œ œ œb œ œ# œb ˙nC. Octatonic scale #3
& c4
œ# œ œ œ œ œ# œb œn ˙#D. Octatonic scale #1, second mode (half step on bottom)
small tetrachords(half-whole-half)
˙˙˙# ˙˙#nbE. Other orderingsi. two diminished-seventh chords œœœœœn#b
œœœœœbn#bœœœœœ#n#b
œœœœœ#b
ii. interlocking major triads/dominant-seventh chords/dominant-ninth chords
&7 œœœœœ
bb œœœœœbbbb œœœœœ
bbbbbœœœœœ#bthe same, spelled tonally
˙bb ˙˙˙n#
iii. two minor-seventh chords a tritone apart(see final cadence of "From the Island of Bali" in Mikrokosmos V) œœœœb œœœœ#n œœœœ#b# œœœœbb#
iv. interlocking minor triads/half-diminished seventh chords
˙˙˙˙b# ˙˙˙˙#n
v. two "Bartok chords" a minor third apart(see "From the Island..." or melodies from Ainsi la nuit)
Symmetry Exercises: OctatonicAlexander Ness
&
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œ œ œb œ# œ œ œ œ#
!
œ œ œ œ œ œ# œ# œbetc.
!
œœœœ# œœœœ#nb œœœœ#nn œœœœ#nbTry this with open voicings too.
!
œœœœ# œœœœ#bn œœœœnn# œœœœnb#
!
&
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5 œœ#b œœ œœ œœ#
œœb œœb œœ#b œœ#b
œœb œœ#n œœ## œœnbetc.
œœ œœ œœb œœ
œœ œœ## œœb œœn œœb œœn œœb œœetc.
œœ œœbb œœ# œœn œœb œœ## œœnn œœbb
œœ- œœb - œœ- œœ-etc.œœ# - œœ# - œœ#b - œœ#n -
Octatonic SequencesAlex
& wwb ww ww
&4 wwb wwb# ww#
&7
ww#
&9
wwb
&11
wwb
&13
ww
Octatonic worksheetYour name: _________________
Write the octatonic scale with these notes:
Find two octatonic scales with these notes:
& c 43 23 89 86Voice ˙# ˙#
3. Other symmetrical scalesA. Two scales made up of augmented triadsi. six-note scale made of two adjacent augmented triads
œ œ# œ œ œ# œ3 transpositions (starting on C, C#, D)2 modes (half step or minor third on bottom)
˙# ˙# ˙b#ii. nine-note scale made of three augmented triads
œ œ# œ œ œ œ# œ# œ œb3 transpositions (starting on C, C#, D)3 modes (h-h-w, h-w-h, w-h-h)
& 86 c5
œ œb œn œ# œ œbB. Several scales with 6 transpositionsi. six-note scale with three-note chromatic clusters
œ œ œ# œ œ# œb œ œ#ii. eight-note scale with two major tetrachords
& 8107
œ œ œ œ# œ# œb œn œiii. weird eight-note scale
œ œb œn œ œ# œ œb œniv. eight-note scale with four-note chromatic clusters
œ œb œn œ œ# œ# œ œb œn œv. ten-note scale with five-note chromatic clusters
Symmetry Exercises Con'd.Alexander Ness
&
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œœœ# œœœ## œœœbb œœœnn
Down by minor thirds:major triads
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
&
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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)
&
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w w w# w# wb w
!
wb w
!
w wb wn w# w# w wb w
!
&
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w# wb w w w w
!
w w# wb
!
w# w wb w w# w w wb wn
!
&
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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
&
&
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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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