Yale University Department of Music
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SOME ASPECTS OF
THREE-DIMENSIONAL
TONNETZE
EdwardGollin
This paper explores how a three-dimensional(3-D) Tonnetzenables
an interesting spatial representationof tetrachordsand the contextual
transformationsamongthem. The geometryof the Tonnetz,a functionof
its set-class structure,emphasizes certain operations within a larger
group of transformationsbased on their common-toneretentionproperties.
After reviewing some featuresof the traditionaltwo-dimensional(2D) Tonnetz,we will explorefeaturesof a 3-D Tonnetzbased on the familiar dominant-seventh/Tristan
tetrachord[0258]. We will then generalize
the 3-D Tonnetzto accommodaterelationsamong membersof any tetrachordclass. Lastly,we will investigatethe relationof a Tonnetz'sgeometry to the group structureof the transformsrelatingits elements.
The Two-dimensional Tonnetz
The traditionalTonnetz(as manifest in the nineteenth-centurywritings and theoriesof Oettingen,Riemann,et al.) is an arrayof pitches on
a (potentiallyinfinite)Euclideanplane.' Figure la illustratesa region of
a traditional2-D Tonnetz.We may summarize several of its essential
aspects as follows:
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1) Pitches are arrangedalong two independentaxes (not necessarily
orthogonal), arrangedintervallicallyby perfect fifth on one axis
and by majorthirdon the other (See Figure lb).2
The
2)
triangularregions whose vertices are defined by some given
pitch and those eithera perfectfifth andmajorthirdabove or a perfect fifth andmajorthirdbelow correspondto some triad,eitherthe
Ober- or Unterklangrespectivelyof that given pitch.
The
3)
edges of the triangularregions correspondto the intervals
within the triad.Only two of these lie along axes of the array;the
thirdedge may be definedas a secondaryfeatureresultingfrom the
combinationof the primaryaxes.
4) The dual structureof major and minor triads is visually manifest
withinthe Tonnetz.Trianglesrepresentingmajortriadsareoriented
oppositely from those representingminortriads.
5) Triangles/triadsadjacentto any given triangle/triadsharetwo common tones if they share a common edge, or one common tone if
they sharea common vertex.
Three particularoperations acting on triads, L, P and R (for Leittonwechsel or Leading-toneexchange, Parallel,and Relative) which maximize common-tone retention,occupy a privileged position on the Tonnetz: they are representableas 'edge-flips' among adjacent triangles/
triads.3Arrowsin Figure la illustratethe mappingsof L, P and R about
a C-majortriad.
Severalnineteenth-centuryassumptionsunderlyingthe traditional2-D
Tonnetzare not obligatory.For instance,if one assumes an arrangement
of equally-temperedpitch classes (ratherthanjust-intonedpitches), the
Tonnetzwould be situatednot in an infinite Cartesianplane, but on the
closed, unboundedsurfaceof a torus.4 RichardCohn (1997) and David
Lewin (1996) have demonstratedthatthe Tonnetzneed not representrelations merely among triads, but may do so among members of any trichordalset class; Cohn (1997) has furthershownthe extensionof the Tonnetz in chromaticcardinalitiesother than c= 12.5 Further,one need not
adhereto the restrictionof representingtrichordsin two dimensions,but
may extend the Tonnetzin three dimensions to accommodaterelations
among tetrachords.6
An [0258] Tonnetz
Figure2 illustratesa portionof a 3-D Tonnetz,consistingnot of points
on a plane but of points (representingpitch classes) in a space lattice.
Whereastwo axes were sufficientto locate points in the 2-D Tonnetz,the
additionaldegree of freedom in three dimensionsrequiresan additional
axis to describe the location of all points. I have chosen axes of a nonorthogonalcoordinatesystem in Figure 2 (interaxialangles are all 60?),
196
Lattice = espaço "em rede".
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D
G#
C#
F#
B
A
Ab
r
L.VL
G
C
F
Db
B
E
1R
Bb
D#
Eb
D
Bb
(a) A region of traditional2-D Tonnetzand threecontextualinversions
(sharinga common edge) with a C-majortriad
/M3
axis
P5axis
(b) The axis system of the traditional2-D Tonnetz
Figure 1
and have labeled these a, b, and c (avoidinglabels x, y, and z to distinguish my axes from those of the Cartesiancoordinatesystem). Axes are
labeled accordingto a right-handedconvention(rightthumbpoints in a
positive directionalong the a-axis, right index finger in a positive direction along the b-axis, middle finger in a positive directionalong the caxis). Points within the lattice are arrangedat unit distances in positive
and negative directionsalong the axes from all other points. The regular
arrangementof points in this Tonnetzconstitutes one of two uniform
ways of filling space with spheres-crystallographers refer to this arrangementas cubic closest packing (ccp).7
Throughoutthis discussion, we will assume equal temperament.Doing so induces a modular geometry to the 3-D Tonnetz-the Tonnetz
occupies the closed, unboundedvolume of a hyper-torusin 4-dimensional space. Units in a positive directionalong the a-axis correspondto
the musical intervalof 4 semitones,units in a positive directionalong the
b-axis correspondto the intervalof 7 semitones, and units in a positive
directionalong the c-axis correspondto the intervalof 10 semitones.Any
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ACE
c-axis
b-axis
a-axD
F#is
a-axis
Figure 2. A region within an [0258] Tonnetz
point in the lattice may be locatedin referenceto any otherpointin terms
of units traversedalong each of the axes, and consequently,if the pitch
class of the firstpoint is known, one can determinethe pitch class of the
second. For instance, in Figure 2, a point one positive unit along the baxis from C (=0) is 0+7 = G. A point one positive unit along the a-axis,
one positive unit along the b-axis, and one negativeunit along the c-axis
from C is (1.4) + (1-7) + (-1.10) = 1 = C#.
The 60* angle is chosen because it is the angle formed at the edges of
a regulartetrahedronmeeting at a vertex. Just as triangularareas of the
2-D Tonnetzcorrespondto triadicelements (or trichordalelements,in the
case of a generalized2-D Tonnetz),tetrahedralvolumes in our 3-D Tonnetz correspondto its tetrachordalelements.In Figure2, one can observe
that any point, along with the points lying one positive unit along the a-,
b-, and c-axes, describe the vertices of an upward-pointingtetrahedron
and thatthe pitch classes representedby these points constitutea 'dominant-seventh'chord. Similarly,any point and the points lying one negative unit along a-, b-, and c-axes describe the vertices of a downward
pointing tetrahedron,and the pitch classes representedby these vertices
correspondto some Tristanor half-diminishedseventhchord.Analogous
to the invertedtriangles/triadsof the 2-D Tonnetz,the dual structuresof
Tristanand dominant-seventhchordsare visually manifestas oppositely
orientedtetrahedrain the 3-D Tonnetz(i.e., pyramidswith peaks pointed
upward,versus those with peaks pointeddownward).
Before I discussrelationsamongtetrachordalelementsin our Tonnetz,
it will be helpfulto develop a contextualnotationfor identifyingthe pitch
elements of the tetrachords.For this, we can adapta contextualnotation
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developed by Moritz Hauptmannto identify the tones of a triad(Hauptmann [1853]1991).
Figure 3a presentsHauptmann'sdesignationfor the tones of a major
triad,and one of his versions for designatingthe tones of a minor triad.
Roman numeralI designates the Einheit of a triad, that chord tone to
which the others refer by means of 'directly intelligible intervals' (for
Hauptmann,these are the octave, perfect fifth and majorthird).Roman
numeralII refers to the tone in the triadthat lies a perfect fifth from the
Einheit,romannumeralIIIto the tone thatlies a majorthirdfromthe Einheit. The differencebetween majorand minor for Hauptmannis that in
the case of major,the Einheitlies a perfectfifth and a majorthirdbelow
the otherchordtones (i.e. it has a perfect fifth and majorthird),whereas
in minor the Einheit lies a perfect fifth and majorthird above the other
chordtones (i.e., it is a perfectfifth and a majorthirdto the tones below).
One need not accept Hauptmann'sacousticalbiases nor the dialectical connotationsinherentin his reference to II and III as Zweiheit and
Verbindungin order to adopt a similar system of tokens that identifies
tones based on their intervallicenvironmentwithin tetrachordsof some
Tn/TnI class. In Figure 3b, I illustrate a neo-Hauptmanniansystem
designating two forms of set class [0258]: a 'dominant-seventh'chord
[C,E,G,Bb],and a 'Tristan'chord [C#,E,G,B].I designate C and B as
'Einheiten' of their respective chords-altering Hauptmann'ssymbology, I label these with a lower case romannumerali. 'Einheit'here designates a chord tone that lies four semitones, seven semitones, and ten
semitones from the other chord tones. In (C,E,G,Bb), 'Einheit' is the
familiarroot of the chord;in (C#,E,G,B),B is the 'dualroot'-the upper
tone in a traditionalthird-stacking.Romannumeralsii, iii and iv referto
those pcs thatlie four, seven andten semitonesrespectivelyfromthe Einheit, either above (in the nominally 'major'dominant-seventh)or below
(in the nominally 'minor'Tristanchord). Our initial choice of 'Einheit'
was arbitrary-any tone could be the basis to which the others refer.
What is importantis that the system of tokens offers a dual system of
namingelements of any asymmetricaltetrachordclass.8
Our neo-Hauptmannianlabels offer a means of identifying the contextual transformsamong tetrachordswithin our lattice, in particular,
inversions between tetrachordssharing common tones. Figure 4 illustrates the common-tonemappingsabout a nexus tetrahedronrepresenting the dominantseventhchord [C, E, G, Bb].Eachmappingself-inverts:
the directionof mappingwithin the Tonnetzis dependentupon whether
the operation acts on a 'major' dominant seventh or 'minor' Tristan
chord. Thus each mappingmay be viewed either as a mappingfrom the
dominantseventh chord [C, E, G, Bb] to an adjacentTristanchord, or
conversely a mappingfrom some Tristanchord to [C, E, G, Bb].
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I
IIl
C
II
G
~ffY
II
F
P5
tl?
II
ab
I
C
?
(a) Hauptmann'sdesignationof triadicchordtones
i
C
ii
E
iii
G
iv
Bb
iv
C#
0
10I
iii
E
ii
G
i
B
(b) a neo-Hauptmanniansystem of tokens identifyingchordtones in set class [0258]
iv
iv
ii
iii
iii
ii
i
i
(u,v,w)
(u,v,w-1)
(u,v,w)(u+l,v,w)(u,v+l,w)
(u,v,w+l)
(u,v-l,w)(u-l,v,w)
(c) neo-Hauptmanniantokens appliedto generalizedtetrachordsin the 3-D Tonnetz
Figure 3
!!!
Figure 4a isolates the six transformsthatmap or exchange tetrahedra
sharinga common edge. The exchangeis representedspatiallyas a 'flip'
of the two tetrahedraabout that common edge. For instance, the operation Ii exchanges tetrahedraabout their common edge bounded by
chordalelements i and ii. In Figure4a, the Iii operationflips the central,
upward-pointing[C, E, G, Bb] tetrachordabout the C-E edge, and
exchanges it with the downward-pointing[E, C, A, F#] tetrachordjust
below and to the front of the illustration.The operation Ii similarly
'flips' two tetrahedraabouttheircommonelements ii and iii. In Figure4,
Iliiexchangesthe central[C, E, G, Bb]tetrachord(flippingit aboutthe GE edge) with the downward-pointing[B, G, E, C#]tetrachordto the lower
right in the illustration.Each 'edge-flip'maintainsat least the two common tones thatconstitutethe common tetrahedraledge. In the case of Iii
in this Tonnetz,mapped [0258] tetrachordssharethreecommon tones.
Figure 4b isolates the four transformsthat map or exchange tetrahedra sharinga common vertex.The exchangeis representedspatiallyby a
'flip' of the two tetrahedraabout that common vertex. For example, the
operationIIexchanges two tetrachordsthat sharea common 'Einheit.'In
Figure 4b, Ii exchanges the central [C, E, G, Bb] tetrachordwith the
downward-pointing[C,Ab,F, D] tetrachordbelow andto the forwardleft
in the illustration.Each 'vertex-flip'maintainsat least one common tone
(the commonvertex).However,I"land I'vmap [0258] tetrachordssharing
two common tones, and maps [0258] tetrachordssharingthree.
Iii
We can observe the following
degeneracyamong transformsin this
equal-temperedTonnetz:Il = Iv and
Iv. The degeneracyarises from
Ii= set class [0258], and from our
the symmetries inherent in our chosen
impositionof equal temperament.Specifically,the equivalenceof I"and
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D
c
c-axis
a-axis
b-axis
a-axis
AL
a-axs
-,?<
LL
al
Figure4a. Six 'edge-flips'abouta nexus tetrachord,(C,E,G,B?),
within an [0258] Tonnetz
iv
c-axis
D
b-axis
$
a-axis
A
F
E
E
A
C#
A#
Figure4b. Four 'vertex-flips'abouta nexus tetrachord(C,E,G,Bb),
within an [0258] Tonnetz
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(u-,v,w+
(u,v,w+l)
(uv-
(
(u-1,v+1,w
c-axis
ufv+1w ,v+l,w)
uv(u+l,v,w)
(u-1,v,w)
1,W)w
XiS(u-,vb-axis
u+l,v-l,w)
u+,v-,w)
(u,v-l,w)
a-axis
uw-l
+,w-
(u,v,w-1)
1
(u+1,v,w-1)
Figure 5. A region of generalizedTonnetzaboutthe point (u,v,w)
iv occurs because the intervalfrom ii to iv (6 semitones) spans half the
and
cardinality of our chromatic universe. The equivalence of
Iiv
within an
results from the symmetry of the [036] trichordembeddedIii
in
the
the
modular
of
[0258].9Moreover,
geometry
hyper-toroidalspace
in which our equal-temperedTonnetzresides, lattice points identical in
pitch are also identical in location. Thus, (X)Ii and (X)I', where X is
some [0258] tetrachordin this Tonnetz,are the same tetrachordin our
curvedmodularspace.'0
A generalized 3-D Tonnetz
Figure5 illustratesa region of lattice points in a generalized3-D Tonnetz adjacentto a point (u,v,w).The triple(u,v,w) indicatesa pointu units
(in a positive direction)along the a-axis, v units along the b-axis and w
units along the c-axis from an origin point (0,0,0). An equivalentfamily
of points could be constructedaroundany point in the space. If we select
coefficients a, 3, and y that correspondto intervalsin some universeof
cardinalityN, and posit these as unit lengths along the a-, b-, and c-axes
respectively,and if we fix the origin point as 0 in an integer notational
system, then the identity of any point (u,v,w) is given by au + 3v + yw
(mod N). Forinstance,fixing (0,0,0) = C = 0, and settinga, 3, andyequal
to 1, 4 and 8 semitonesrespectivelyin a mod 12 pitch-classspace, a point
(2,-3,5) would representthe pitch class (1.2)+(4--3)+(8-5) = 2-12+40 =
30 mod 12 = 6 = F#in an [0148] Tonnetz.
Figure 3c illustrateshow our neo-Hauptmanniantokens may be affixed to points in the generalized3-D Tonnetz.If we let ii referto a pitch
class thatlies one unit along the a-axis from some Einheit, let iii referto
a pitch class one unit along the b-axis, and let iv referto a pitch class one
unit along the c-axis (in eithera positive or negativedirection),thengiven
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some point in the lattice (u,v,w) as 'Einheit,'we may define a 'major'
form of any tetrachordas the tetrahedroncomprisingthe four vertices
(u,v,w), (u+1,v,w), (u,v+1,w), and (u,v,w+1). A tetrahedronrepresenting
a 'minor' form of a tetrachordclass built from the same Einheit would
consist of the four vertices (u,v,w), (u-1,v,w), (u,v-l,w) and (u,v,w-1).
Analogously to the [0258] Tonnetz,contextual transformationsin
the generalized 3-D Tonnetzmay be representedas edge- or vertexflips among adjacenttetrahedrons.For instance, Iii is an edge-flip that
inverts a tetrachordabout its 'Einheit' and its element ii, mapping
to [(u+1,v,w),(u,v,w),(u+1,v-1,w),
[(u,v,w),(u+1,v,w),(u,v+1,w),(u,v,w+1)]
(u+1,v,w-1)] and vice versa.
I have been cautiousto specify the Tonnetzin which a particulartetrahedron-invertingoperationoccurs, since our neo-Hauptmannianlabels
(and consequentlyour inversionlabels) are tied to the intervalliclayout
of our axes. Were we to explore relationsin an [0258] Tonnetzwith differentlyarrangedaxes (for examplepositing units along the a-, b-, and caxes of 3, 6 and 8 semitones respectively), the same operationsof our
original [0258] Tonnetzwould obtain, but with different 'I' labels. The
same is not the case if one considers operationswithin Tonnetzeof different set classes: the family of common-tone preserving, contextual
inversionsof one set class is in general a differentfamily of contextual
inversionsfrom any other.
Group Structure of Transforms within the Tonnetz
If one exhaustivelycomposes the L, P, and R transformsin the triadic
2-D Tonnetzof Figure 1, the resulting closed group of operationsis a
dihedralgroupknown as the Schritt/Wechselgroup (S/W)." The group,
acting on harmonic triads, consists of 12 mode-preservingoperations
(Schritte)and 12 mode-invertingoperations(Wechsel).The symbols So,
will here indicate 12 Schrittethatmap a triadto the mode,
S1,S2,
SIlwhose root lies 0, 1, 2,..., 11 semitones away in the direcidentical
.... triad
tion of chord 'generation'(up in the case of majortriads,down in the case
of minor).12For example, (E major)S3=(G major),but (E minor)S3=(C#
min.). Wo,W1,W2etc. are Wechselthatmap mode-invertedtriadswhose
dual roots lie 0, 1, 2, etc., semitones apartin the directionof chord 'generation.'Thus in the triadic2-D Tonnetz,L equals WI1,P equals W7 and
R equals W4.13
Analogously, if we exhaustivelycompose the edge- and vertex-flipping transformsin our [0258] Tonnetz,the resultinggroupof operations
is identicalin structure(or isomorphic)to the S/W group acting on harmonic triads. The isomorphic S/W group acting on [0258] tetrachords
mapsmode-identicalandmode-invertedtetrachordsbasedon the directed
intervalsamongtheir 'Einheiten.'Thus in our [0258] Tonnetz,IIis equiv203
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alent to Wo,Iii is equivalentto W4, etc.14 Note thatif we alterour system
of labels (i.e., select some othertone as 'Einheit')the Wechselassociated
with each particularcontextualinversionare all changed by some constant.The groupsresultingfrom the relabelingof tetrahedralelements in
the Tonnetzare thereforeautomorphicimages of one another.
The exhaustivecompositionof the contextualinversionsof any asymmetricaltetrachordal(or trichordal)set class will similarlyyield an S/W
group isomorphic to the S/W groups acting on triads or [0258] tetrachords (or else isomorphic to a dihedralsubgroupthereof-contextual
inversions among members of [0248] for instance yield only the even
Schritteand Wechselof the largergroup).
This remarkableresult, that the same group of operationsunderlies
Tonnetzeof such drasticallydifferentgeometries (and even of different
dimensions), suggests that the Tonnetzis an entity independentof the
S/W group.It also suggests an interestingrelationbetween the structure
of the Tonnetzand the transformationsamong its members.Elementsof
mathematicalgroups are largely indifferentto the music-analyticalsituations upon which they are called into service in transformationaltheories. In the context of the S/W groupqua group,any Wechselis basically
like any other (all of order 2, etc.). Yet one could hardlyclaim that the
musical effect of the Relativetransform(W4)actingon a triad(e.g., mapmin.) is at all the same as that of the Gegenkleinterzping C
wechselmaj.E--A
min.). Thus we may view the
(W3, e.g. mapping C
maj.--Ab
that allows one to make distincTonnetzas an independentframework
tions among the (otherwise indifferent)transformsthat underlieits elements. It allows one to posit analyticalmeaningor value to certainfamilies of transformsfrom the largergroup based on the distancebetween
elements mappedby those transformswithin the Tonnetz,whose geometry in turn is determinedcontextuallyby the intervallic structureof its
chordal elements. Distance in the Tonnetz,moreover,is a function of
common-toneretention.Hence, the function of the Tonnetzin neo-Riemanniantheory is not unlike that of Lewin's INJ function:it provides a
measureof the progressiveversus internalquality of a contextualfunction acting on a 'Klang' of a trichordalor tetrachordalset class.'"
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NOTES
1. For a historyof the Tonnetzandits evolutionfromEulerthroughRiemann,see
Mooney1996.A studythatdiscussesthe Tonnetzandits relationto Funktionstheorie in Riemannandhis successorsis Imig 1970.
2. Thenineteenth-century
basisof intervalselectionwasacoustical:theintervalscorrespondedto whatRiemann(alongwithnumeroustheoristsbeforehim)believed
to be the only intervals(alongwith theoctave)givenby nature.
3. RichardCohn (1997) has recentlyexploredthe Tonnetzrepresentation
of these
operations,notingtheirpotentialfor common-toneretentionas well as theirparsimoniousvoice-leading.Theabbreviations
derivefromBrianHyer1989and1995
(101-38).
4. Cohn 1997cites Lubin1974as the firstrecognitionof the toroidalstructureof an
equallytemperedTonnetz.Hyer'sFigure3 (1995, 119) presentsa Klangnetz,the
geometricdual of a Tonnetz(see Douthettand Steinbach1998), in pitch-class
space,whichsimilarlyinducesa toroidalstructure.
5. Lewin'shexagonalgraphof a 3-valuedCohnfunction(Lewin 1998; 1996, 189),
is, like Hyer's,a dualof an [013] Tonnetz.
6. Otherwritingsthatextendthe Tonnetzinto threedimensionsincludeVogel 1993
andLindleyandTurner-Smith
1993.
7. It is so called becausethe unitcells of the lattice(the smallestunitsthatcan fill
spaceby translationalone)areface centeredcubes (latticepointsexist at the six
verticesandat the centerof eachcubeface).
8. HenryKlumpenhouwer
(1991, chap.2) exploresa similarsystemof contextual
labeling.
9. Theembedded[036]is alsothereasonthatIIi!andIiRmaptetrachords
in our[0258]
Tonnetzwith threecommontones.It is interestingto observethatnone of the 29
in a chromaticuniverseof cardinality12 arewithoutdegenertetrachord-classes
atecommon-tone-preserving
contextualtransforms:
in c= 12 either
anytetrachord
(a) containsan ic 6 dyad,(b) is symmetricalor embedsa symmetricaltrichordal
subset,or (c) satisfiesconditions(a) and(b).
10. Thiswouldnot be truein a just-intonedversionof ourTonnetz,where
(C,E,G,B,)
# (C,E,G,Bb)Itv= (Ab,Fb,Db,Bb).
Sucha Tonnetzwould,
howIii= (G#,E,C#,A#)
an
infinite
Euclidean
See
123
and
ever,occupy
ff.,
space. Vogel1993,
Lindleyand
Turner-Smith1993, 66-68, for discussionsof unequallytempered[0258] Tonnetze.
11. Riemannexpoundshis system of Schritteand Wechselas directedroot-interval
relations among triads in Skizze einer neuen Methode der Harmonielehre (Rie-
mann1880). Klumpenhouwer
1994 reinterprets
andextendsRiemann'sSchritte
andWechselintoa groupof operationsactingon theset of majorandminortriads.
12.Thisis a vestigeof Riemann'sbeliefthatmajortriadswereacousticallygenerated
fromtherootup (viaovertones),andthatminortriadswereacousticallygenerated
fromtheirupper,dualrootdown(via undertones).
13. OnecanderivethecompleteS/W groupfromtheL andR transformsalone.Since
LR = W11W4= S7is an elementof order12, it may be combinedwith L or R or
any Wechsel(all areof order2) to generatethe completegroup.
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14. Analogously,we canproducegeneratorsforour[0258]S/Wgroup.lifiIii = WIIW4
= S7is anelementof order12,whichmaythenbe combinedwithanyedge-orvertex-flipto generatethecompletegroup.
15.INJis discussedin Lewin 1987, 123ff.
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