# Some Remarks on the Use of Riemann Transformations

## Henry Klumpenhouwer

KEYWORDS: Riemann, harmonic dualism, Wagner

ABSTRACT: The essay examines Lewin’s and Hyer’s
use of transformations derived from Riemann’s
work, and suggests a number of revisions. An
expanded collection of transformations is
introduced and used in an short analysis of
excerpts from Wagner’s *Ring*.

Copyright © 1994 Society for Music Theory

[1] In this paper I suggest revisions to an
analytical methodology developed in Lewin 1987,
1992 and Hyer 1989. Their work draws upon
certain themes in Riemann”s writings in order to
construct procedures for investigating harmonic
relationships in Wagner, procedures that also
engage transformational approaches to analysis.
My revisions also refer explicitly to Riemann”s
writing, but they refer to an earlier stage in
his career, and in particular to his *Skizze
einer Neuen Methode der Harmonielehre* (Riemann
1880).

**I. Chord Models and Chord Transformations**

[2] Hyer (1989) develops a group of transformations on triads from a collection of four relations generally associated with Riemann”s work: parallel, relative, leittonwechsel, and dominant. Lewin (1987, 1992) begins with a somewhat larger number of basic transformations. A few originate in Riemann (parallel, relative, leittonwechsel). Others (mediant, submediant) are adapted from other theorists. And still others (slide relation) are constructed by Lewin himself.

[3] **Example 1** investigates how these transformations interact with the triads or klangs they
relate. Example 1a is a network of three arrows
labelled with Riemann transformations, and three
nodes containing klangs. The relationship
between C major and E minor is interpreted as
leittonwechsel (L), defined by Lewin as a
transformation on triads that maintains the two
pitches that form a minor third (mod 8ve), and
moves the remaining pitch one semitone so as to
form another triad (1992, 49). The network
interprets the relationship between E minor and
G major as an instance of the relative
transformation (R), which maintains the two
pitches that form a major third (mod 8ve), and
moves the remaining pitch a whole tone so as to
form another triad. The arrow that extends from
C major to the G major is labelled with the
subdominant transformation (S), which transposes
a klang by a perfect fifth up. The network
suggests the following functional equality: L
followed by R yields S.

[4] Aside from node content, the network given in example 1b is very similar to the network in example 1a. Both networks have leittonwechsel and relative labels on corresponding arrows. But while the arrow on example 1a that extends from C major to G major is labelled with the subdominant transformation (S), the corresponding arrow on example 1b (extending from A minor to D minor) is labelled with the dominant transformation (D), which transposes a klang up a perfect fifth down. The network in example 1b, contrary to the network in example 1a, suggests that L followed by R yields D.

[5] The difference bothers me because the
difference (and other similar differences in the
system) is brought about by separating dualist
chord models from dualist transformations
(leittonwechsel, relative) and then using those
transformations in conjunction with functions or
relations (dominant, subdominant, submediant,
mediant) derived from fundamental-bass
conceptions of chord structure. My point here
is not that Hyer and Lewin have been unfaithful
to the wishes of Riemann. My point is that
technical properties of leittonwechsel and
relative transformations—in particular the
property that they are their own inverses—
induce what amounts to a dualist relationship
between major and minor chords. For instance,
as one examines the values and arguments on the
relevant function table of the leittonwechsel
transformation, one sees that in general the
transformation changes the modality of a major
triad to its parallel minor and transposes a
major third up, while it changes the modality of
a minor triad to its parallel major and
transposes a major third *down*. The reversal
of direction that accompanies the reversal in
modality is the imprint of dualist thinking.
The imprint is indelible: the only way to avoid
the dualism is to redefine the transformation so
that its function table reverses the modality of
a triad and transposes it a certain directed
interval regardless of modality. But that would
simply turn leittonwechsel either into Lewin”s
SUBM (submediant) transformation or into his M
(mediant) transformation, depending on the
chosen directed interval.

[6] I have not faulted Hyer and Lewin with
having misrepresented Riemann”s work because I
do not count that as a fault, and because there
is a sense in which an uncomfortable
juxtaposition of dualist and non-dualist
thinking is a prominent characteristic of
Riemann, particularly of his later work: the
well-known contemporary critique of Riemann in
Belinfante 1904 makes this very point. In fact,
it could be fairly argued that in what follows
it is *I* who will be misrepresenting Riemann.
My own distortions (as far as I can tell)
involve projecting a particular ideational
structure on Riemann”s transformations (as they
appear in the *Skizze einer Neuen Methode der
Harmonielehre*), and then appropriating the
transformations for analytical purposes other
than those discussed by Riemann, who seems most
interested in using transformations to chart out
the topographies of his *Dur-Moll* and *Moll-
Dur* systems. Following Lewin and Hyer, I will
be interested in extending the use of Riemann”s
transformations to seek out repetitions of
harmonic patterns, which are then presented as
motivic.

[7] Accordingly, my proposed revision of the analytical methodology involves firstly turning away from non-dualist transformations adapted from dominant, subdominant, mediant, and submediant relations, and secondly identifying in Riemann 1880 a less conflicted group of transformations.

[8] Since the transformations I shall be
discussing *do* compel a dualist conception of
triads, it will be worthwhile to establish at
least the dominant features of such a view. But
because Riemann”s own explanation of chord
dualism is premised on the existence of harmonic
undertones, I shall use an alternative yet
functionally equivalent explanation, developed
from Hauptmann”s discussion of major and minor
triads (1853, 25–35).

[9] Hauptmann assigns three “functions” to
pitches that constitute major and minor triads:
unity (*Einheit*); duality or separation (*Zweiheit*); union (*Verbindung*). The three functions are labelled respectively I, II, and III
for reference, and are assigned to triad members
according to two rules:

1) I and II form a perfect fifth (mod 8ve);

2) I and III form a major third (mod 8ve).

The rules stipulate that only the pitch that
acts as I or the *Einheit* participates in both
the perfect fifth (mod 8ve) and the major third
(mod 8ve) relationships.

[10] **Example 2** distributes the three symbols
I, II, III among the pitches that form a major
triad. Example 2a calculates which two pitches
may be assigned I and II under the first rule
above. Since the formalism asserts only that I
and II form a perfect fifth, a perfect fourth,
or one of their compounds, we cannot determine
the assignment of function labels beyond what is
shown in the example: F is either I or II;

[11] **Examples 3a**, **3b**, and **3c** carry out on a
minor triad the procedures for assigning the
functions I, II, and III. Example 3a calculates
which pitches may be assigned I and II under the
first rule above. Since the formalism asserts
only that I and II form a perfect fifth, a
perfect fourth, or one of their compounds, we
cannot determine the assignment of function
label beyond what is shown in the example: F is
either I or II;

[12] Comparing the assignment of function
labels in minor triads and major triads,
Hauptmann analyzes the constitutive perfect
fifths and major thirds as intervals directed
upwards: in major klangs the two intervals
extend respectively from I to II and from I to
III; in minor klangs the two intervals extend
respectively from II to I and from III to I. In
major klangs, says Hauptmann, the pitch that
acts as I *has* a perfect fifth and major third,
while in minor klangs the pitch that acts as
I *is* a perfect fifth and major third.

[13] Under Hauptmann”s explanation, a dualist model organizes aural sensations in roughly the following way: When listening to a triad, pick out a major third or its inversion, and pick out a perfect fifth or its inversion; when you do, you will become aware that one pitch in the triad member is overdetermined, and thereby seems more prominent than the others. By way of contrast, a fundamental bass model organizes sensations in roughly the following way: When listening to a triad, reorganize it so that it takes up the smallest registral space and so that only thirds and fifths are formed; assign prominence to the lowest pitch and take note of the quality of the third between that pitch and the next lowest. And a figured bass model organizes aural sensations in roughly the following way: When listening to a triad, concentrate on the lowest-sounding pitch, and assign it prominence; imagine a third and a fifth above the lowest pitch (the qualities of which are determined by a contextual diatonic collection); pitches that do not lie a diatonic third or fifth above the prominent pitch are momentarily displacing the pitches that do. The degree to which figured bass and fundamental bass protocols have been hypostatized by theorists (to the extent that many will claim that one or the other defines more or less the response end of a reflex arc) is something worth worrying about.

[14] In the course of this paper we shall
represent klangs as ordered pairs. The first
element defines Hauptmann”s I-function or
*Einheit* (*not* the *Grundton*). The second
element defines the klang”s modality. “+”
represents a major (or over or super) klang, or
a “positive” *Einheit* Hauptmann calls it. “-”
represents a minor (or under or sub) klang, or a
“negative” *Einheit*. The klangs in examples 2c
and 3c are respectively represented as (

**II. Schritts and Wechsels**

[15] Having established the relevant chord
model, we can now investigate appropriate
transformations. We begin by examining
Riemann”s Schritts, the first of two classes of
transformations he presents in *Skizze einer
Neuen Methode der Harmonielehre*. Quintschritt
(abbr. Q) transposes a klang by the directed
interval that extends from I to II. In the case
of (C,+), where C functions as I and G as II,
the relevant interval is a perfect fifth up.
Accordingly, Q(C,+) = (G,+). In the case of
(C,-), where C functions as I and F as II, the
relevant interval is a perfect fifth down.
Accordingly, Q(C,-) = (F,-), i.e., a

[16] Gegenquintschritt (abbr. -Q) transposes a klang by the directed interval that extends from II to I. In the case of (C,+), where G functions as II and C as I, the relevant interval is a perfect fifth down. Accordingly, -Q(C,+) = (F,+). In the case of (C,-), where F functions as II and C as I, the relevant interval is a perfect fifth up. Accordingly, -Q(C,-) = (G,-). Q and -Q are inverse-related transformations.

[17] Terzschritt (abbr. T) transposes a klang
by the directed interval that extends from I to
III. In the case of (C,+), where C functions as
I and E functions as III, the relevant interval
is a major third up. Accordingly, T(C,+) =
(E,+). In the case of (C,-), where C functions
as I and

[18] Leittonschritt (abbr. L) transposes a
klang by the directed interval yielded by the
composing the relevant directed intervals of Q
and T. In the case of (C,+), the composition of
Q (a perfect fifth up) and T (a major third up)
produces a major seventh up. Accordingly,
L(C,+) = (B,+). In the case of (C,-), the
composition of Q (a perfect fifth down) and T (a
major third down) produces a major seventh down.
Accordingly, L(C,-) = (

[19] Ganztonschritt (abbr. G) transposes a
klang by twice the directed interval that
extends from I to II. Hence, G = Q2. In the
case of (C,+), where C functions as I and G as
II, the relevant interval is two perfect fifths
up, viz., a (compounded) whole tone up.
Accordingly, G(C,+) = (D,+). In the case of
(C,-), where C functions as I and F as II, the
relevant interval is two perfect fifths down,
viz., a (compounded) whole-tone down.
Accordingly, G(C,-) = (

[20] Kleinterzschritt (abbr. K) transposes a
klang by the directed interval that extends from
II to III. In the case of (C,+), where G
functions as II and E functions as III, the
relevant interval is a minor third down.
Accordingly, K(C,+) = (A,+). In the case of
(C,-), where F functions as II and

[21] Riemann only discusses these six schritts. Following the pattern established by the pair quintschritt/gegenquintschritt, we can define four more schritts: gegenterzschritt (abbr.-T), gegenleittonschritt (abbr. -L), gegenganztonschritt (abbr. -G), and gegenkleinterzschritt (abbr. -K), which will be inversely-related to terzschritt, leittonschritt, ganztonschritt, and kleinterzschritt respectively. Two others, which we shall call identity (abbr. I) and tritonusschritt (abbr. R) are generated by demands of group structure. Appendix I lists all twelve schritts, and provides paradigms for each. Those not explicitly defined by Riemann are given in angle brackets.

[22] The names of the transformations help us
remember their particular effect: an X-schritt
transposes by the *magnitude* X (where X is a
German interval designation) in the direction
determined by the operand: major (or over)
klangs extend X upwards; minor (or under) klangs
extend X downwards. Gegen-X-schritt may be
taken either to reverse the direction of X, or
to replace the magnitude X with its inverse.
Accordingly, X-schritt and gegen-X-schritt are
inverses. Kleinterzschritt and its gegen-form
are the only schritts that do not follow the
pattern: one needs provisionally to take
kleinterz as signifying a major sixth.

[23] Schritts combine according to the following rule:

Xschritt*Yschritt = (X+Y)schritt

In other words, summing the directed intervals of composed schritts (and reducing any compounds) yields the appropriate product schritt. For example, Composing K and T in the context of major klangs yields -L, since a major sixth up (the directed interval by which K transposes major klangs) and a major third up (the directed interval by which T transposes major klangs) sum to a minor ninth up, a compound of a minor second up (the relevant directed interval by which -L, the inverse of L, transposes major klangs). Composing K and T in the context of minor klangs still yields -L, since a major sixth down (the directed interval by which K transposes minor klangs) and a major third down (the directed interval by which T transposes minor klangs) sum to a minor ninth down, a compound of a minor second down (the relevant directed interval by which -L, the inverse of L, transposes minor klangs).

[24] Since schritts are quantities that combine under simple addition, the group is commutative: in general X*Y = Y*X.

[25] But schritts only relate klangs of the same genus. Wechsels, the second class of transformations presented in Riemann 1880, relate klangs of opposing genera.

[26] Seitenwechsel (abbr. W) inverts a klang
about I (or about I and I). The transformation
reflexively maps the major klang and the minor
klang whose I-functions are executed by the same
pitch. In other words, it exchanges “positive”
and “negative” forms of the same *Einheit*, so
that W(C,+) = (C,-), and W(C,-) = (C,+).
Seitenwechsel appears in Goetschius 1917 (114)
as his “stride relation,” defined as “a perfect
fifth downward from any major keynote, and
upward from any minor keynote, with a change in
mode.”

[27] The remaining wechsels may be construed
as the composition of some schritt and
seitenwechsel. Terzwechsel (abbr. TW)
transposes a klang by the directed interval that
extends from its I to its III, that is, applies
terzschritt, and then inverts the result around
its I, that is, applies seitenwechsel. One
might then understand the designation
“terzwechsel” as an elision of
“terzschrittseitenwechsel.” Terz(schritt) only
affects the first element of an order pair;
(seiten)wechsel only affects the second element.
In the case of (C,+), TW(C,+) = W(E+) = (E,-);
in the case of (C,-), TW(C,-) = W(

[28] Leittonwechsel (abbr. LW) is the
composition of leittonschritt (L) and
seitenwechsel (W). In the case of (C,+),
LW(C,+) = W(B,+) = (B,-); in the case of (C,-),
LW(C,-) = W(

[29] Riemann defines three more wechsels: quintwechsel (abbr. QW), ganztonwechsel (abbr. GW), and tritonuswechsel (abbr. RW). Quintwechsel is functionally equivalent to Lewin”s and Hyer”s parallel transformation (abbr. P), and may alternatively defined as inversion of klang about I and II. Six more wechsels are constructed by extending the idea of composing gegenquintschritt, kleinterzschritt, gegenkleinterzschritt, gegenleittonschritt, gegenterzschritt, and gegenganztonschritt in turn with seitenwechsel. Appendix II lists all twelve wechsels, and provides paradigms for each. Those not defined explicitly by Riemann are given in angle brackets. Appendix III provides rules and relevant examples for combining the twenty-four schritts and wechsels given in the first two appendices. The group is simply-transitive: in particular, for any two klangs j and k, there exists a unique member of the schritt-wechsel group S such that S(j)=k. The analytical use of groups with the property are considered in Lewin 1987.

**III. Analysis**

**Example 4**. Valhalla Theme I

(click to enlarge)

**Example 5**. Valhalla Theme II

(click to enlarge)

**Example 6**. Valhalla Theme I (after Lewin’s Figure 3b)

(click to enlarge)

**Example 7**. Valhalla Theme II (after Lewin’s Figure 4b)

(click to enlarge)

[30] **Examples 4** and **5** investigate some
possibilities of schritts and wechsels by using
them as arrow labels on network analyses of two
excerpts from Wagner”s *Ring*. Both networks
are developed from analyses recently presented
in Lewin 1992: **Example 6** is based on Lewin”s
figure 3b, which studies the harmonic structure
of the Valhalla theme as it appears in *Das
Rheingold* sc.2; **Example 7** is based on Lewin”s
figure 4b, which studies the harmonic structure
of the same theme as it appears in *Die
Walkuere*, act II, sc. 2. The networks are not
exactly as they appear in the cited figures in
Lewin 1992. The networks that appear in
examples 6 and 7 were designed to have the same
node-and-arrow configurations, viz., to contain
the same number of nodes and arrows, with
corresponding arrows extending in the same
direction, while Lewin”s figures were
constructed to serve other analytical purposes
that did not require identical node-and-arrow
configurations. In any case the divergences
from Lewin”s figures are slight and the networks
in examples 6 and 7 are well-formed according to
standards of the relevant methodologies.

[31] The nodes in examples 6 and 7 are filled
with triads construed according to the
fundamental bass model, and are represented by
the relevant fundamental bass and an indication
of the third”s quality. The arrow labels are
Lewin”s designations: L represents
leittonwechsel; P, parallel; S, subdominant; D,
dominant. LP represents the composition of L
and P. Comparing the left sides of example 6
and 7, we see that they are identical except for
node content, and are thus strongly isographic
structures. The right sides of examples 6 and 7
are not identical: while the node and arrow
configurations are the same, the corresponding
arrow labels are different. The transformations
L and LP on the left side of both networks
invoke dualism on the relevant nodes. (P is
neutral with respect to chord model.) Traveling
to the right side of each network, one is
constrained by the transformations S and D to
adopt a non-dualist conception of major and
minor chords, to shift from Riemann to Rameau.
Accordingly the

[32] Example 4 adapts example 6 to conform to
the present methodology: harmonies are
represented by the dualist ordered pair format,
and arrows are relabelled with the appropriate
schritt or wechsel. Only leittonwechsel appears
on both example 6 (as L) and example 4 (as LW).
Quintwechsel (QW) replaces parallel (P), and
terzschritt (T)—which uses as its paradigm the
directed interval the extends from I (

[33] Example 5 carries out similar changes on example 7. Comparing examples 4 and 5, one sees that the corresponding arrow labels are strictly identical. The shift from the analysis in examples 6 and 7 to the analysis in examples 4 and 5 allow us then to assert a strong isography between the harmonic relations in two Valhalla themes. With respect to node content, one could assert that the network in example 4 is a positive or major form of the network in example 5.

[34] I will not pretend that schritt and wechsel transformations are the only way to establish a link between the two Valhalla themes. One could replace L and LP on example 6 by Lewin”s submediant transformation (SUBM) and the composite transformation SUBMP respectively, and replace L and LP on example 7 by Lewin”s mediant transformation (M) and the composite transformation MP respectively, thereby shifting the networks entirely into the discourse of fundamental bass theory. Corresponding arrows on example 6 and 7 would then be labelled by inverse-related (Rameau-) transformations.

**APPENDIX I
Schritts**

(C,+) → (C,+); (C,-) → (C,-)

<Gegenleittonschritt> [-L]

(C,+) → (

Ganztonschritt [G]

(C,+) → (D,+); (C,-) → (

<Gegenkleinterzschritt> [-K]

(C,+) → (

Terzschritt [T]

(C,+) → (E,+); (C,-) → (

Gegenquintschritt [-Q]

(C,+) → (F,+); (C,-) → (G,-)

<Tritonusschritt> [R]

(C,+) → (

Quintschritt [Q]

(C,+) → (G,+); (C,-) → (F,-)

<Gegenterzschritt> [-T]

(C,+) → (

Kleinterzschritt [K]

(C,+) → (A,+); (C,-) → (

<Gegenganztonschritt> [-G]

(C,+) → (

Leittonschritt [L]

(C,+) → (B,+); (C,-) → (

**APPENDIX IIWechsels**

(C,+)↔(C,-)

<Gegenleittonwechsel> [-LW]

(C,+)↔(

Ganztonwechsel [GW]

(C,+)↔(D,-)

<Gegenkleinterzwechsel> [-KW]

(C,+)→(

Terzwechsel [TW]

(C,+)→(E,-)

<Gegenquintwechsel> [-QW]

(C,+)→(F,-)

Tritonuswechsel [RW]

(C,+)→(

Quintwechsel [QW]

(C,+)→(G,-)

<Gegenterzwechsel> [-TW]

(C,+)→(

<Kleinterzwechsel> [-KW]

(C,+)→(A,-)

<Gegenganztonwechsel> [-GW]

(C,+)→(

Leittonwechsel [LW]

(C,+)→(B,-)

**APPENDIX IIIRules for Combining Schritts and Wechsels**

Xschritt * Yschritt = (X*Y)schritt

Q*L = Q+L = R

Xschritt * Ywechsel = (X*Y)wechsel

Q*LW = (Q+L)W

Xwechsel * Yschritt = (X*-Y)wechsel

QW*L = (Q-L)W = -TW

Xwechsel * Ywechsel = (X*-Y)schritt

QW*LW = (Q-L) = -T

Henry Klumpenhouwer

University of Alberta

Edmonton, Alberta T6G 2C9

userklum@mts.ucs.ualberta.ca

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