ARTS & HUMANITIES

Bilateral Keyboard Symmetry in the Music of Einojuahni Rautavaara

Brandon Paul School of Music

20th-century Finnish composer Einojuhani Rautavaara’s fascina- tion with symmetry played an integral role in his compositional evolution. Inherent in 12-to-the-octave composition, pitch sym- metry has been exploited by many twentieth-century composers. One type of visual symmetry—bilateral keyboard symmetry—in- fluences Rautavaara’s compositions in both piano and non-piano settings. Bilateral keyboard symmetry organizes the piano keys into a symmetrical plane creating axes at pitch classes {2} and {8}. Every note consequently has a complement in mirrored re- spect to the axes (i.e., {7} complements {9} and {1} complements {3}) which can then be used to generate harmonies and melodies. These harmonies and melodies coincide with tonal and aesthetic choices in the music of Rautavaara yielding triadic relationships such as D-major complementing G-minor. This non-transposing type of symmetry is not based upon auditory organization of pitch- es, but of visual organization of piano keys and is used with mir- rored physiological motions of the hands at the piano. This system can also be found in many non-piano pieces such as Ballad for Harp and Strings and Sonata for Solo Cello no. 1, which feature sections resembling bilateral keyboard symmetry. Using a novel system of analytic notation, bilateral keyboard symmetry can be examined on local and global levels.

Introduction

This research focuses on a method of composition that is based upon the physical construct of the piano and organizing the piano keys into symmetrical groups. Many composers, including Persichetti and Chopin, have understood the symmetrical relationships of keys on the piano. However, Einojuhani Rautavaara has taken this compositional system a step further to create new musical forms. Through studying the majority of Rautavaara’s scores, I have come up with a sys- About the Author tem to analyze Rautavaara’s music and determine that the symmetry of the piano Brandon Paul is currently keys has influenced Rautavaara’s compositions for all types and groups of instru- researching the social func- ments. I clearly define the technical aspects of my analysis in the paper; only a few tions of listening to nostalgia- musically technical terms need to be defined. Pitch classes, abbreviated pcs, refer inducing autobiographical songs. He is interested in emo- to numbers that represent traditional note names: (0=C, 1=C#/Db, 2=D, 3=D#/Eb, tions that are perceived and 4=E, 5=F, 6=F#/Gb, 7=G, 8=G#/Ab, 9=A, 10=A#/Bb, and 11=B). Pitch classes are experienced by listeners. He intends to graduate in Spring placed in brackets, such as {2}. Interval classes refer to the number of semitones 2010 with a degree in Musi- that separate one pitch class from the next. Dominant and tonic relationships, cal Theory, and then to go on discussed in the Tonality section, deal with a relationship between two chords to earn a PhD in Psychology, examining emotions and other where the dominant chord is built on pitch class {7} from the tonic chord built on effects arising from listening to pitch class {0} and is a common chord movement in tonal music. Advanced theory auditory stimuli such as music and speech. His undergradu- knowledge is not necessary to understand the relationships I discuss; my argument ate advisors have been David comes from a primarily visual and spatial paradigm. Huron and Gregory Proctor. 93 ARTS & HUMANITIES acteristic melodies and harmonies. Natural exploitations of exploitations Natural harmonies. and melodies acteristic char are texture symmetric the through permeating Often choices. aesthetic and tonal defined relatively,clearly with to createcomplexsymmetrical relationships. mediate and deferred realization can be used in conjunction simultaneity.in Im not simply or statement initial an after played are classes pitch more or one complementsof when occurs realization Deferred complements. their and pitches select contains hand right the with played triad C-minor a and hand left the in played triad A-major an example, For complements. their with simultaneously played are classes pitch of collection or class pitch a when occurs realization Immediate realization. deferred and realization immediate and atonalsenses. tonal both in applications residual for allows This Bartók. of music or the Webern in find would one as system, pitch pitch symmetry, which is unlike a transposable symmetrical of system non-transposing a is plane. result piano The a on based upon the visual organization of keys in physical space is referredtoas complements {10}, and {7} {6} complements {9}. This system {11}, complements {5} {3}, complements {1} {4}, complements {0} complement: reflective a has class pitch remaining each {8}, and {2} pitch-classes of axes around symmetry.reflective groups of symmetrical the Examining axes certain allows keys piano of arrangement The sition. compo of system generative a become has keys black and white of relationship the and piano the organizationat sual Rautavaara’s oeuvre; the symmetry of physical space by vi Rautavaara). 206; (Sivuoja-Gunaratnam 2002 year palindromic the of symmetry in the word Arabescata as well as the celebration inherent the including composition, his influences also try his first serial period of composition. Extramusical symme through material for generation of source primary the was work Prevariata (Sivuoja-Gunaratnam 77). Pitch symmetry orchestral the in as well as “Arabescata” 4 no. Symphony stage of his evolution as a composer, evidenced in the every serial in found are structures symmetrical Formal cissus. Nar piece piano solo driven symmetrically the including works, recent most his into up evolved have symmetry of uses various Preludes, piece Symmetrical piano Three solo 1950 the symmetry with Beginning products. of compositional his into amount vast a incorporates composer, ish Background Paul /BilateralKeyboardSymmetry Rautavaara incorporates bilateral keyboard symmetry keyboard bilateral incorporates Rautavaara Bilateral keyboard symmetry has two main operations: is symmetry keyboard bilateral of reflectivity The One unique type of symmetry is prominent throughout Finn 20th-century prominent Rautavaara, Einojuhani bilateral keyboard symmetry . ------unl are quently compositional techniquestretchesoveraspanof50years. portrays bilateral keyboard symmetry. The influence of this that collection pitch a around based completely almost is 1 8 no. Symphony as such works several in found be can Evidence well. as music non-piano generate to means a as composition, Rautavaara has used this piano-related system vital inmuchofRautavaara’s music. is ideas neo-tonal or tonal with symmetry keyboard lateral bi of juxtaposition The minor. D-flat and major A-flat as about the relationship between C major and A minor as well relationships with D acting as a symmetrical axis. The same may be said tonic-and-subdominant or dom inant-and-tonic its to due favored particularly is minor G menting comple major D of relationship symmetrical The music. tonal conventions are usually a central facet in Rautavaara’s Figure 1 a of layout pitch the to proportionately formation metrical sym a of register relative the displays notation The tion. sec a of component measures) two or one (typically local a indicate Brackets section. a of part aggregate or section a of structures aggregate represent braces of confines the in Graphics purple. colored are collection aggregate an in hands left and right both by played Pitches hand. right the structures, blue and hand left the represent structures red tion and subsequent operations. Collections emphasized by realiza deferred and immediate show to pitches the plays dis system this keyboard, a of layout the Resembling tion. nota graphic this of sample a shows 1 Figure symmetry. keyboard bilateral of relationships global and local show Graphic Notation which thesetechniqueswereemployed. during period active most the of core the at lie and system try, yet these pieces exemplify all qualities found within the symme keyboard bilateral of principles the feature works the and Sermon” Fischer”, die und “Christus , a piece for solo cello. The cello. solo for piece a , The musical works that will be referenced most fre most referenced be will that works musical The Exploiting bilateral keyboard symmetry as a method of A unique system of graphic notation will be used to used be will notation graphic of system unique A . SampleGraphic Notation. in Cnet n. 1 no. Concerto Piano “The Journey” and the and Journey” “The Three Symmetrical PreludesThreeSymmetrical Piano Sonata no. 2 no. Sonata Piano Ballad for Harp and Strings and Harp for Ballad , in Snt n. 1 no. Sonata Piano Nirvana Dharma, Nirvana Cello Sonata no. Sonata Cello . Many other Many . “The Fire “The ------ARTS & HUMANITIES 94 - - - - - “The , Vol. I , Vol. Piano Sonata Oculus Piano Sonata no. 2 M. 66-67 “Christus und die Fischer”, mm. 61–68 feature two- An example of one-handed interval expansion and Two handed interval expansion and contraction in M. 63-64 contraction may be found in the class outer limit. This motion is demonstrated in Figure 2. This motion is demonstrated class outer limit. volves the intervals between the lowest and highest pitch class of a formation. In music, Rautavaara’s this motion is almost always concurrent with immediately-realized sym like Much symmetry. deferred with combine may but metry moves, pitch one as contraction, and expansion one-handed its complement will move accordingly. In no. 1 sym initial The contraction. and expansion interval handed metrical structure of pcs {7 10 0 4 6 9} expand to larger symmetrical structures as m. 68 approaches. The graphic Fire Sermon”. The right-handed material in mm. involves an 95–96 initial symmetrical structure built on pcs {7} Pcs sur higher. octave an {2} and 9}) ({7 {2} axis rounding expands interval outer the as static remain 95 m. in {9} and symmetri new a 96 m. In {4}. and {0} to {2} and {2} from {4} and {0} 6}. 4 0 {10 pcs involving stated is structure cal remain static as the interval expands from {10} and {6} to {8} and {8}, then to {6} and {10}. The pitches are, at this point inverted, and they have reached the general interval------bilateral key “The Fire Sermon,” I, mm. 95–96. Graphic Analysis of Right Hand Material. Analysis of Right Hand I, mm. 95–96. Graphic “The Fire Sermon,” “Christus und die Fischer,” I, mm. 61–66. Graphic analysis. “Christus und die Fischer,” Piano Sonata no. 2 Piano Sonata no. 1 and not to any formal or pitch-space sym M. 65-66 discussed henceforth will refer to . Rautavaara, M. 61-62 . Rautavaara, Inherent in bilateral keyboard symmetry is the lateral Figure 2 class 17 is generally the extent to which one-handed expan one-handed which to extent the generally is 17 class sion reaches. ate complement depending on deferred or immediate sym the hand one in contracts and expands interval the As metry. pitch-classes will eventually invert. The limit of interval- ic increases or decreases its interval by inward or outward As more. or semitone by complements successive to motion appropri the reach will pitch second the changes, pitch one either one hand or in both hands. stat In not is that one-handed interval initial an interval contraction, and expansion motion across the plane of piano keys. The first basic mo tion involves the expansion and contraction of intervals in Principles of Motion Symmetry symmetry board stated. unless explicitly metry, a given octave will be the next adjacent octave or higher. (pc[s]) classes pitch of terms in discussed be will Structures and referenced by tonal pitch names when appropriate. piano. In other words, any repeating octave to the right of Figure 3 95 ARTS & HUMANITIES in Snt n. 2 no. Sonata Piano the in as such works, several in demanded explicitly often are keys black and white of Clusters works. piano many in features these utilizes Rautavaara keys. white the to son compari in keys black of elevation relative the especially piano, the of feature another yet are keys black and keys symmetry.keyboard bilateral White of principles basic the to contribute space its of awareness the and piano the of qualities Physical themselves. keys piano the of ganization or of use the through exemplified further is style sitional eral keyboardsymmetry. bilat maintaining still while differentoctaves in collection 12-pitch the through moves structure the yet 6), Figure in seen (as intervals through change not does structure metric sym The motion. concurrent and reflective two-handed of Piano Concerto no. 1 metry. An example of reflective motion may be found in the sym keyboard bilateral in feature a as exists such, as and, contract. The distance between the hands therefore changes or expand complements when inherent is hands the of tion the pitch-classes and octave placement may. Reflective mo but change, not may intervals general the works, many In try is not always by expansion and contraction of intervals. metrical pcscomplementsof{7}and{9}(Figure5). 100, the intervals have expanded and have also added sym m. in starting recapitulation the During 4. Figure in cally tion built on pcs {0 2 4 6 8 10}. The material is seen forma graphi symmetrical a on based material opening features work. a of system. the with coinciding melody the Note 68. m. to formation trans structure’s initial the presents 3 Figure in notation Figure 5 Paul /BilateralKeyboardSymmetry Figure 4 The visual and spatial aspect of Rautavaara’scompo of aspect spatial and visual The symme keyboard bilateral two-handed of motion The Interval expansion can also occur at the structural level . Rautavaara, . Rautavaara, Narcissus Te ie emn. n te works, other In Sermon”. Fire “The Narcissus Narcissus , Op. 45. Mm. 83–103 contain a string , a solo piano piece written in 2002, in written piece piano solo a , , mm.100–103.GraphicNotation. , mm.1–4.GraphicNotation. ------Notation oforiginal symmetricstructure. 6 Figure Piano Concertono.1 in as clusters of white and black keys are written explicitly, such ymty ltr scin cnrs imdae symmetry immediate contrast sections B later symmetry, While the B sections in the preludes do not feature deferred symmetry.immediate the breaks that section B a by lowed preludes, an A section three featuring immediate the symmetry is of fol each For movement. each for sections B and A distinct creates symmetry immediately-realized of ing break and establishment the piece, Withinoeuvre. this his ludes Formal Construction inner layerofarpeggiation. The hand and arm clusters move reflectively around a thick the in exists also clusters note black and white with coinciding motion tive Fiddlers 6; Op. Icons, in found be may Examples structures. musical between movement the controls hands the between space physical of traction con and expansion as Rau motion Reflective in music. tavaara’s feature prominent a also is symmetry keyboard Physical motion without the requirement of bilateral of requirement the without motion Physical Rautavaara’s 1949 piano piece is the first instance of bilateral keyboard symmetry in Piano Sonata no. 1 no. Sonata Piano Rautavaara, . , Op. 1; and 1; Op. , Piano Sonata No. 2 No. Sonata Piano Etudes . in Cnet n. 1 no. Concerto Piano Piano Concerto no. 1 no. Concerto Piano “Christus und die Fischer” and Fischer” die und “Christus , Op. 42. reflec Op. of , example An Three Symmetrical Pre “The Fire Sermon”. Fire “The I m 8. Graphic 83. m. I, , , Op. 45; Op. , The ------ARTS & HUMANITIES 96 ------Piano , Vol. I , Vol. Oculus , Op. 45. The first The 45. Op. , , Op. 45, the qualities of sonorities are sonorities of qualities the 45, Op. , Piano Concerto no. 1 no. Concerto Piano . “Christus und die Fischer”, I, mm. 27–31: a Narcissus The juxtaposition of bilateral keyboard symmetry One of the most obvious features of triadic harmonies as well, only vice-versa. One of the most the in prominent occurs amples ex movement opens with an arpeggiation of a D-major/minor triad in the left Acting hand. under the domain of deferred symmetry, a G-major/minor triad enters in m. 7. Figure 9 shows this relationship. By using major/minor triads in the 1 no. Concerto Piano preserved. If D-major was the first chord, G-minor must have been the second, thus changing the quality of the so level later in the work. This is similar to the result of result the to similar is This work. the in later level Sonata no. 1 mul These structure. create to axis the over flipped section 2002 pia also be found in the can of symmetry layers tiple no piece and Tonality Bilateral Keyboard Symmetry with tonal or neo-tonal ideas is central tional to method of the Rautavaara. Triadic composi relationships; tonic, dominant and predominant functions; modality; and poly tonality are just a few areas of exploitation derived from Rautavaara’s Throughout ideas. both of use cooperative the works, the coincidence occurs most frequently from1969 to 1973 when bilateral keyboard symmetry is most active. Harmonic and melodic ideas alike can be transformed and developed through this system. relationship the is symmetry keyboard bilateral in operating between triads built upon G and D. The D-major triad is complemented by a G-minor triad; D-minor and G-major 27–31 confirms the symmetry. The material in the B sec tion is also flipped over the axis in mm. 32–41. This dem onstrates an immediately-realized and deferred symmetry section that has been immediately realized on a structural - - - - , I, mm. 1–2. “Christus und die “Christus und die Fischer,” I, aggregate mm. 1–5, 15–16 and 27–31. I, aggregate mm. “Christus und die Fischer,” “Christus und die Fischer” features Fischer” die und “Christus Piano Sonata no. 1 Piano Sonata no. 1 Three Symmetrical Preludes Three Symmetrical . Rautavaara, . Rautavaara, Piano Sonata no. 1 no. Sonata Piano A monophonic melody leads into the realization of the of realization the into leads melody monophonic A

The initial material of Three Symmetrical Preludes fo Preludes Symmetrical Three of material initial The Figure 8 A’ section—starting in A’ m. 27. Graphic notation (Figure 8) showing the aggregate outer voices of mm. 1–5 and mm. deferred A section of A deferred Fischer”. Completely flipped over the axis, a combination the create sections A first the of 15–16 mm. and 1–4 mm. of important for later realization. material starting an octave Using higher. a combination of immediately-realized and deferred symmetry, this B sec tion features a repeating pc {1} in the right hand, which is and an A-major triad in the right move by expansion into continues pattern The reversed. but complements, same the section with the until right A the handed restatement of the able bilateral keyboard The symmetry. following interlude section in measures 6–8 is built on tertian harmonies that are immediately realized. A C-minor triad in the left hand beginning of the sonata, a descending melody underneath a underneath melody descending a sonata, the of beginning notice no displays first at {10} pc on built collection quintal symmetry. Moreover, quintal harmonies are combined with combined are harmonies quintal Moreover, symmetry. deferred symmetry and tertian harmonies are joined with immediate symmetry to define each At section the further. much like the goal of the A section. A much like the goal of the contrasting sections defined by immediate and deferred order to articulate the left-hand accompaniment to the mel the to accompaniment left-hand the articulate to order The termination ody. of this section on pcs {1 2 3} further emphasizes the {2} axis and its proximal complements, order to contrast the initial statement. Rautavaara utilizes the inherent symmetry of 12-to-the-octave composition in cuses on a collection surrounding the pc {2} axis operating operating axis {2} pc the surrounding collection a on cuses under immediately-realized symmetry graphically seen in Figure The 7. B section breaks the immediate symmetry in with deferred symmetry as in Piano Sonata no. 1 “Christus und die Fischer”. Figure 7 97 ARTS & HUMANITIES ymty n o-in msc s the is music non-piano in symmetry are realized after 15 measures with complements {9 4 6 7}. 9} 10 0 {7 pcs solo, soprano the follows that music the In for example, contains a melodic passage for soprano voice. piece vocal The piano. the of plane rical yet the material often is transformable through the symmet no music. Most material is not generated by these methods, eral keyboard symmetry lies in the translation into non-pia Applications inNon-PianoMusic of sonorities. collection his into incorporates Rautavaara that harmonies triadic of use creative a for allows symmetry deferred This complements. of collection its by followed is always that flat, and so on. The right hand features a single-line melody D- by A-flat F-sharp, by followed E-flat triad, E-minor an by succeeded is triad F-major arpeggiated an hand, left the In 1–5. mm. in succession in occur complements their and Piano Sonata No. 2 the In works. many in apparent technique another is ments of theprelude.SeeFigure10forgraphicnotation. resolves back to the C-major dyad. This is repeated in m. 20 and AF suggesting a dominant-9th B, chord. The 9th chord G, then notes the with measures, the of two beat on curs oc relationship dominant AE. and C notes the contain 4 third prelude from Nirvana Dharma Light”, in as such music non-piano in symmetry keyboard bilateral of application the in ploited ex further are G and D between relationships nority.The ship. 9 Figure Paul /BilateralKeyboardSymmetry One of the strongest examples of bilateral keyboard bilateral of examples strongest the of One An interesting implication of Rautavaara’s use of bilat comple their to them deferring and triads Building the in implied is relationship tonic-dominant Another Symphony no. 8 no. Symphony . Rautavaara, . G Major/Minor . Three Symmetrical Preludes “The Fire Sermon” II, a string of triads Piano Concerto no. 1 no. Concerto Piano “The Journey” and the vocal work vocal the and Journey” “The Symphony no. 7 no. Symphony D Major/Minor el Snt n. 1 no. Sonata Cello , I. D and G relation G and D I. , Nirvana Dharma Nirvana . Mm. 3 and “Angel of “Angel ------. , 3–4. GraphicNotation. 10 Figure tion to include pcs {5} and {11}. The trend continues to continues trend The {11}. and {5} pcs include to tion opera expansion an through presented are 10} 9 7 {6 pcs metry occurs in m. 15 where a collection of complementary sym of appearance symmetry.first keyboard The bilateral of collection a resembles harp, the for especially material, Strings and on pcs{7}and{9}movingoutto{6}{10}. starting occurs process same The outward. continuing and complements of pcs {1} and {3} expanding to {0} and {4} symmetric involves 41 Measure sonata. this in occurs also contraction and expansion interval handed One 11. Figure metry. Graphic notation displays the aggregate collection in sym deferred resembles distinctly material the measures, five opening the Examining operations. symmetrical cal typi demonstrate movements second and first entire The Graphic Analysis. 12 Figure Graphic Notation. Figure 11 Another example for discussion is the . Rautavaara, . Rautavaara, . . Rautavaara, . . Bound in a symmetrical form of ABA, The B The of ABA, form symmetrical a in Bound . Cello Sonata no. 1 Three Symmetrical Preludes Symmetrical Three Ballad for Harp and Strings and Harp for Ballad C Major G9 Chord , I, mm. 1–5. Aggregate Ballad for Harp , mm. 34–35. mm. , , no. 3, mm. 3, no. , - - - - ARTS & HUMANITIES 98 - - , Vol. I , Vol. Einojuhani . Finland: . Edited by Oculus . Edited by Fazer Musik. Fazer by Edited . . Edited by Fennica Gehrman. Nirvana Dharma Nirvana Ballad for Harp and Strings Narcissus Magdeburg: Otto-von-Guerike-Universität (1999). Rautavaara’s First Serial Period 1997. Tiedeakatemia, Suomalainen Fazer, Warner/Chappell Music. Finland, 1973. Espoo, Helsinki, Finland, 1979. Helsinki, Finland, 2002. [4] ---. Narrating with Twelve Tones. [9] ---. ---. [10] [11]---. . - - Topics, Topics, Thorough Thorough analysis of Rautavaara’s music has defini “Christus und die , Op 46. Edited by “The Fire Sermon.” choices allows a deaf system to be musical to a deaf system allows choices Conclusion tively shown the relationship of a composition symmetrical applied style to his of music. Whereas Rautavaara’s symmetry is explicit, a system of analysis from visually- oriented perspective can lead to composers who use future a piano as a tool for composition. analysis The for all layout and construction of the piano or keyboard, as well as placing our hands to the instrument, has some form of the determine can research Future composition. in influence If compositions. into piano the from relationship of amount re and abstract, mathematical, the of outside think theorists the of paradigm a then theory, traditional of aspects lational human body’s physical relationship with instruments can its evolution. shed light on describing music and ------Music. Espoo, Finland, 1969. Piano Sonata no. 1 Cello Sonata no. 1 Piano Sonata no. 2 Edited by Fazer, Warner/Chappell Music. Warner/Chappell Edited by Fazer, Espoo, Finland, 1970. Texts, Texts, Tensions: Essays in Music Theory by Breitkopf & Hartel, Leipzig, Germany, 1969. Fazer, Warner/Chappell Music. Finland, 1969. Espoo, Fischer.” Edited by Fazer, Warner/Chap pell (2002). Accessed 16 November 2008. (2002). sicus’ or an Intertextual Perspective on the Oeuvre of Einojuhani Rautavaara,” , the appearance [8] ---. [6] ---. [7] ---. [3] Sivuoja-Gunaratnam, Anne. “’Narcissus Mu - - . Juva: W. . Juva: W. , Op. 45. Edited Cello Cello Sonata no. 1 . Edited by Fazer, Three Symmetrical Omakuva . http://www.fennicageh . Edited by Fazer, Warner/ Fazer, by Edited . . Edited by Fazer, Warner/ . Edited by Fazer, Warner/Chap Highlights The Fiddlers, Op. 1 Piano Concerto no. 1 Icons, Op. 6 Etudes, Op. 42 Op. Etudes, Chappell Music. Espoo, Finland, 1969. Warner/Chappell Music. Finland, 1952. Music. Finland, Warner/Chappell Chappell Music. Espoo, Finland, 1955. Preludes pell Music. Espoo, Finland, 1950. Year,” Year,” rman.fi/highlights/hl12_rautavaara.pdf Söderström, 1989. These examples reveal that bilateral keyboard sym [5] ---. [4] ---. [4] [3] ---. [2] ---. Referenced Scores [1] Rautavaara, Einojuhani. [2] ---. “Some Reflections on a Symmetrical [1] Rautavaara, Einojuhani. References is inherently a deaf system. That is to say, aural pitch orga is inherently That a is deaf to system. say, struc these of generation the for responsible not is nization tures. The guidance of tonal, aesthetic and conventional mance can be just as rewarding as the experience of listen ing to the interaction of sound and vision. Moreover, it is essential to keep in mind that bilateral keyboard symmetry melody to a global symmetrical structure. Physical motion spatial the of result direct a is plane piano lateral the across and symmetrical organization of the keys and its perfor for orchestrations of all types. The sonorous possibilities of a visual system can suggest everything from a two-note ure 12 shows the expansion and contraction. the expansion ure 12 shows metry is not just a means for composing piano music, but 6 7 9 As 10 in 11}. the of one-handed expansion and contraction is found in mm. Fig collection. main the from classes pitch the using 34–35 include include pcs {1} and {3}. The main collection that demon strates symmetry in the final expansion includes pcs {1 3 5