===       ===     =============        ====
             ===       ===           ==            ==   ==
            == ==    ====           ==           ==      =
           ==   ==== ===           ==           ==      ==
          ==     ==  ==           ==            =      ==
         ==         ==           ==             ==   == 
        ==         ==           ==               ====
       M U S I C          T H E O R Y         O N L I N E
                     A Publication of the
                   Society for Music Theory
          Copyright (c) 1994 Society for Music Theory
| Volume 0, Number 10   September, 1994    ISSN:  1067-3040   |
  All queries to: mto-editor@husc.harvard.edu
AUTHOR: Renwick, William, J.
TITLE: "A Subject of Four Notes":  William Crotch's Experiment in
Motivic Saturation
KEYWORDS: fugue, counterpoint, grundgestalt, cell
William J. Renwick
McMaster University
Department of Music
Hamilton, Ontario, CANADA L8S 4M2
ABSTRACT: This paper discusses the circumstances that drew
William Crotch into a highly specific and systematic
compositional procedure, and describes how the severe material
and procedural restrictions of his *Introduction and Fugue on a
Subject of Four Notes, no others being used all thro' the
movement* (ca. 1830) nevertheless permit the creation of a
successful musical work on both intellectual and artistic
scales.  The analytical focus of the article is the manner in
which Crotch establishes and deploys a varied content through a
coherent and developmental design despite the self-imposed
ACCOMPANYING FILES: mto.94.0.10.renwick1.gif (music, p. 1)
                    mto.94.0.10.renwick2.gif (music, p. 2)
                    mto.94.0.10.renwick3.gif (music, p. 3)
                    mto.94.0.10.renwick4.gif (music, p. 4)
                    mto.94.0.10.renwick5.gif (Examples 1-7)
                    mto.94.0.10.renwick6.gif (Example 9)
NOTE: Readers who wish may retrieve high quality EPS files of the
composition, as well as a MIDI performance by calling the
following files from mcmail.mcmaster.ca via ftp:
[Editorial note: To retrieve the .eps files, ftp to mcmail.mcmaster.ca,
and login anonymously, using "anonymous" when prompted for a loginID, and
using your full email address when prompted for a password.  Once
connected, change directories to pub/brahms (type: cd pub/brahms). Set
FTP for a binary transfer (at the ftp> prompt, type: bin).  Then, use the
'get' command to retrieve the files (e.g.: get renwick1.eps).  Use the
'get' command to retrieve each file individually, or use the 'mget' command
to retrieve all at once.]
[1] Throughout the ages, composers have delighted in the
challenges afforded by severe pre-compositional restrictions,
whether structural limitations, as in the ancient practice of
canon and cancrizans, favoured by Josquin and Ockeghem among
others, material limitations such as the exhaustive and
systematic treatment of a single theme, as demonstrated in Bach's
*Art of Fugue*, or procedural limitations, as in the serial music
of Webern, Babbitt and Boulez.  *Mikrokosmos* is a wonderful
illustration of how Bartok's fertile imagination gained
inspiration from a wide variety of limitations, tonal, rhythmic,
and technical.  Indeed, pre-compositional restrictions are often
fundamental in establishing the necessary limitations that
balance the dynamic, growing forces that together determine the
proportions, form, and qualities of a composition.(1)  This paper
discusses the circumstances that drew William Crotch into a
highly specific and systematic compositional procedure, and
describes how the self-imposed compositional material and
procedural restrictions of his *Introduction and Fugue on a
Subject of Four Notes, no others being used all thro' the
movement* permit the creation of a successful musical work both
on the intellectual and artistic scales.  As is evident from the
title, and from a cursory observation of the composition, the
entire fugue is constructed from statements of the theme and
nothing else.(2)    
1. This concept is considered at some length in Siegmund Levarie
and Ernst Levy, *Musical Morphology: a Discourse and a
Dictionary* (Kent, Ohio: The Kent State University Press, 1983),
pp. 11-37. 
2. See the first four GIF files listed at the beginning of the
[2] William Crotch (1775-1847) embraced probably the all-time ebb
in English musical composition, from the late Georgian era
through the first part of the nineteenth century, a transitional
era caught between conflicting values: after the decline of the
ancient regime of Handel and J.C. Bach, but before Mendelssohn
had provided a focal point around which the new spirit of the
romantic movement could flourish in Victorian England.  The most
precocious musical prodigy in recorded history, Crotch rose
meteorically to the highest ranks of the musical establishmenton
the basis of his astonishing talents, but never attained
comparable heights as a composer.  He was playing public concerts
at the age of two, touring from three to nine, assistant to Dr.
Randall, Professor of Music at Cambridge at twelve, Organist of
Oxford Cathedral at fourteen, Bachelor of Music at nineteen, and
Professor of Music at Oxford at age twenty-one.(3)  His music in
general possesses all the technical requisites of greatness, but
the expression of an inspired imagination rarely shines through. 
Today only a handful of his compositions are heard with any
3. The standard account of Crotch's life and works is Jonathan
Rennert's *William Crotch*. (Lavenham, Suffolk: Terence Dalton,
[3] As Professor of Music, Crotch promoted a systematic approach
to musical aesthetics derivative of contemporary English theories
of the pictorial arts.(4)  Crotch was, after all, a highly gifted
landscape artist and a close friend of Constable.  Essentially
music could be considered according to three categories, sublime,
beautiful, and ornamental.  Sublime comprised the lofty sacred
works of Bach and Handel, while "pure sublime" distinguished the
*prima pratica* of Palestrina, Byrd, and Gibbons.(5)  Beautiful
music encompassed the expressive style of the baroque,
exemplified by the instrumental music of Corelli and by opera
seria.  Much of Handel combines the sublime and the beautiful. 
The ornamental included music of an eccentric nature, whether in
rhythm, harmony, modulation, etc.; music designed to provoke an
immediate reaction from the listener, but not necessarily to have
lasting value.  Haydn was judged by Crotch as a composer who
succeeded in combining the beautiful and the ornamental.
4. Edmund Burke and Joshua Reynolds both developed systematic
approaches to aesthetics.  See Burke, *A Philosophical Enquiry
into the Origins of Our Ideas of the Sublime and Beautiful*
(1757), ed. J.T. Boulton (Oxford: Oxford University Press, 1958);
Reynolds, *Discourses* (1769-97) (New Haven: Yale University
Press, 1975).
5. Rennert, p. 43.
[4] Crotch was an academic, and he became increasingly interested
in the theoretical and speculative aspects of the art at a time
when personal expression was growing in importance.(6)  As Crotch
progressed towards a more severe and pious philosophy in his
later years, his work increasingly assumed charachteristics of
his sublime aesthetic.  Fugue and chant became the focus of his
interest in music, and, like Webern, Crotch became increasingly
interested in miniature composition.  He excelled at the simplest
forms and devoted much of his energy to writing hymn tunes and
Anglican chants--simple melodies of ten or twenty notes
harmonized in chorale style.
6. In his adjudication of Samuel Sebastian Wesley's brilliant
anthem *The Wilderness* for the Gresham prize, Crotch's caustic
criticisms pointed to just those aspects which mark Wesley's
uniquely personal and original style and which give it a lasting
value.  Ironically, it is by and large Wesley's music, not
Crotch's, that remains the more performed and more highly valued
[5] According to Rennert, Crotch's view was that "the way to re-
introduce the pure sublime into modern music was to study `the
mysteries of canon, fugue and imitation', and then imitate the
sixteenth-century sacred style in original compositions.  . . .
sublimity `does not strike and surprise, dazzle and amuse, soothe
and delight', as does the ornamental.  Instead it `elevates and
expands the mind'".(7)  Regarding the *Well-tempered Clavier*,
Crotch noted: "The fugue in the major key of E (*Specimens of
Music*, Vol. iii. p. 1.) is perhaps the best of the whole. "(8)
(See Example 7 (mto.94.0.10.renwick5.gif).  Interestingly,
Crotch's *Fugue on a Subject of Four Notes* resembles Bach's
fugue in its copious use of stretti and diminutions.  We might
note that it was similar strict procedures that characterized to
a high degree the compositional techniques of the second Viennese
school.  Indeed, one is tempted to expand the aesthetic argument
to the point at which pure mathematics, as demonstrated through
the serial procedures of Babbitt and Boulez for example, becomes
the new "sublime" style of the 20th century, as it primarily
gratifies the mind rather than the senses.  The *Introduction and
Fugue on a Subject of Four Notes* is itself an exercise in
reduction to minimal content.
7. Rennert, p. 47.
8. *Substance*, p. 101
[6] The *Introduction and Fugue on A Subject of Four Notes* does
not appear in any of the usual bibliographies, such as the
*Catalogue of Printed Music in the British Library to 1980* (9)
or *The New Grove*, or in works lists such as that in Rennert's
biography.  Apparently the work has up to now been unknown.  The
source for my copy, and the only known source at present, is
bound in a volume in the extensive collection of mainly English
organ music which was purchased in London en masse by the
University of Alberta as the basis for the establishment of its
music library in 1940.(10)
9. *Catalogue of Printed Music in the British Library to 1980*,
62 vols. (London: K.G. Saur, 1981-87).
10. The catalogue number is M10 C65 I6 R88.  It appears that a
large portion of the material in this collection, which comprises
bound volumes of compositions originally published separately, at
one time belonged to Edward John Hopkins (1818-1901), organist of
the Temple Church in London from 1843-1898.
[7] Although the composition is undated, we can establish the
limits within which it must have been published on the basis of
the extensive title:
*Introduction & Fugue/On a Subject of 4. Notes, no others being
used all thro' the movement/Composed for the Piano Forte or
Organ,/by/Wm. Crotch. Mus Doc./Professor of Music in the
University of Oxford & Principal of the Royal Academy of Music
London./London, Pubd. at the Royal Harmonic Institution, Argyll
Rooms, 246 Regent street by Welsh & Hawes,/Music Sellers by
special appointment to his Majesty, Their Royal Highnesses the
Dukes of York,/Clarence, Sussex & the Duchess of Kent, & to be
had at all the Music Warehouses in the United Kingdom.*
While Crotch held the post of Professor of Music at Oxford
University from 1797 until his death in 1847, he was Principal of
the Royal Academy of Music only from 1822 to midsummer, 1832.(11) 
Other works of Crotch issued by the same publishers, Welsh &
Hawes at the Royal Harmonic Institution, span the identical
period, 1822-1832.(12)   Thus, we can be confident in dating the
composition to the period 1822-1832.(13)  Only in the case that
Crotch retained the title Principal of the Royal Academy of Music
after his resignation, can we date the composition in the period
11. Among his duties at Oxford was the unlikely one of
adjudicating his own composition exercise for the Mus. Doc. in
1799.  Not surprisingly, he judged it satisfactory.
12. Rennert suggests 1820 as an approximate date for the Royal
Harmonic Institution's reprint of Crotch's popular anthem
"Methinks I hear the full celestial choir".  See Rennert,
*William Crotch*, p. 101.
13. Crotch's other published fugal works focus on the year 1825.
[8] Secondary evidence that narrows the date even further, but
which is perhaps less concrete, is the fact that this composition
does *not* appear in the list of Crotch's published works as
appended to the *Substance of several courses of lectures on
music*, "to be had at the Royal Harmonic Institution, Regent
Street", first published in 1831.(14)  Thus, barring the
accidental omission of this piece from the list, we can only date
its publication as 1831-32.
14. *Substance of several courses of lectures on music, read in
the University of Oxford, & in the Metropolis* (London: A. & R.
Spottiswoode, 1831).  The works-list appears on pages 171-173.
[9] As previously noted, the pre-compositional basis of the fugue
is the limitation of the content to statements, transpositions,
inversions, augmentations and diminutions of a single four-note
motive.  Only at the final cadence of the composition is this
pattern broken, as in the tradition of strict canon, in order to
form a satisfactory perfect authentic cadence.  One wonders
whether Crotch's theme was inspired by the fugue-theme from the
Finale of Mozart's Symphony No. 41 (see Example 6, see
mto.x.y.renwick5.gif).  After all, according to Bennar Rainbow, 
"For William Crotch the musical world of his youth had been
divided into two opposing factions--`the one despising the
trifling melodies of the opera, and the other the barbarous and
mechanical structure of the fugue.' (*Substance*, pp. 148-149) 
His mature view of the matter was that during his lifetime those
factions were at length reconciled when Mozart, the greatest of
all modern composers', demonstrated by his effective use of
counterpoint that `science could no longer be held in ridicule'.
(*Substance*, p. 149)."(15)
15. Rainbow, Introduction to *Substance*, p. ix.
[10] The introduction is a simple affair, the purpose of which is
merely to propose the subject and establish the tonality.  The
tempo indication is by Crotch's standard method, the length of a
free-swinging pendulum. Sixteen inches is about 104 beats per
minute.  However, this should indicate the tempo of the quarter
note, not the eighth note, as the music suggests.  Curiously no
pendulum designation appears for the fugue.
[11] Crotch himself provides a motivic analysis of the fugue
using carats of four sizes, as shown at the bottom of the first
page of the fugue (see mto.94.0.10.renwick2.gif).  Note that
retrogrades are never used.  The reason for this should quickly
become clear, for the introduction of retrogrades would confuse
the compositional plan, by admitting virtually any pattern that
utilizes stepwise motion.  Also, statements of the motive are
never conjoined-that is no note is used twice, as the final note
of one statement and the initial note of the next-except in one
location, the connection of m. 52 to m. 53.  Finally, what is
perhaps obvious but is mentioned here as a significant
distinction compared to atonal and serial techniques, statements
of the motive are always presented melodically, that is within
the context of a single part of the four- or five-part texture.
[12] One might think that Crotch's own analysis provides a fairly
complete explanation of the piece, but it really provides only
one level of motivic analysis.  It shows very clearly the
derivation of all the notes, but it does not really address how
this plethora of statements of the motive, some 290 in total,
combine to form a convincing artistic whole.  In this paper I
attempt to provide a second level of motivic analysis that
addresses this problem directly.  There are therefore really two
levels of motivic structure; the basic one shown by Crotch, and
the more complex one which relates to issues of structure and
[13] For the purposes of this discussion, it is useful to make a
distinction between the motive, the basic idea upon which the
piece is based, and the fugue-subject, a particular form of the
motive employed at a particular transposition and developed in a
formalized manner.  It is not improper, I believe, to consider
the motive as a true *Grundgestalt*, for Crotch's utilization of
the motive follows very well the principles of transformation and
development that are implied by Schoenberg's concept.  Indeed,
this is the means by which Crotch is able to create variety and
coherence within the formal pre-compositional constraints.  The
motive is a rising third followed by a descending step, presented
in equal note-values.
[14] This motive is among the smallest conceivable motives upon
which such a composition can be based.  The continual stepwise
motion which this motive exhibits is a necessary condition to
permit sufficient freedom in the construction of contrapuntal
combinations.  That is, linear motion is capable of
interpretation as passing and neighbouring dissonance, whereas
leaps generally imply only consonance, and are therefore
comparatively limiting.(16)  Three notes in a step-wise
relationship are not enough to establish a unique contrapuntal
motive.  If such a three-note motive were wedge-shaped, it would
be incapable of asserting retrograde formations.  Likewise, a
uni-directional motive--a scale-segment--is incapable of
asserting any distinction between retrograde and inversion.(17)  
These cells, however, combines a leap and a step in the opposite
direction, a possibility not readily accommodated by the
requirement of stepwise motion here.  I will designate the motive
as any statement of any permutation of the four-note group,
whether transposed, inverted, augmented or diminished--note that
Crotch never uses the retrograde.(18)  Thus the motive is the
general idea behind the composition, whereas the subject and
various countersubjects that I will identify are specific
formations of the motive.
16. Frescobaldi's "Recercar Ottavo, Obligo di non uscir mai di
grado", from  *Recercare et Canzoni*, 1615, is a fascinating
specimen of music in the strict style constructed with absolutely
no stepwise motion, and therefore with no dissonance.
17. The trichords in Webern's Concerto for Nine Instruments, Op.
24, represent the ultimate reduction of material.
18. Crotch was no doubt aware of the potential for retrograde
constructions, however.  His double chant in G major, is based
entirely on retrograde, in the form A, B, A(r), B(r).
[15] The subject itself is the simplest possible expression of
the motive, beginning on the tonic in half-notes: f-g-a-g.  Once
the motive is expressed in actual notes, it assumes a tonal
aspect.  In essence the subject represents a rise from 1^ to 2^,
implying a basic harmonic motion I-V (see Example 1,
mto.94.0.10.renwick5.gif).  The third note, then, can be taken as
a reaching-over which forms a harmonic consonance with the f,
while the second note forms a passing tone.  It is most unusual
that a fugue-subject end on a weak beat, as this one does, but
this is an important condition for the counterpoint, as will
become clear.  The subject, then, is an incomplete linear motion,
suggesting a continuation with either f or a.  F would close the
thought, resolving the motion back to 1^, while a would complete
a rising third-progression, in which the g of m. 2 would operate
as a passing note.  In the music, m. 3 provides both notes. 
While most fugue subjects express a complete tonal motion, in
this case the lack of such completion is actually necessary to
allow any degree of freedom in the construction of progressions. 
These tonal interpretations of the subject are borne out by the
*Introduction*, which in 3/4 time places the harmonic motion and
metric emphasis on the first and last notes of the subject.  In
the *Introduction*, the rising linear implication is realized
through a sequential ascent.  Further statements of the motive as
half-notes, beginning on various scale degrees, will be
considered transposed repetitions of the subject.  The *answer*,
naturally, denotes the subject transposed to the fifth of the
operative key.
[16] It is the distinction between motive and subject made here
that allows the possibility of creating out of the motive one or
more countersubjects that simultaneously provide contrast and
unity in the composition.  Crotch creates three recognizable
countersubjects, each based on a different of species of strict
counterpoint (see Examples 2-4, mto.94.0.10.renwick5.gif). 
Countersubject 1 (CS1) denotes the quarter-note form of the
motive (first species) normally beginning on the second quarter,
and normally using the inversion, i.e. a-g-f-g in m. 3.  Note
that the *rectus* form of CS1 uses the inversion of the motive. 
The *inversus*, CS1i, uses the prime form of the motive.  This
distinction, while somewhat awkward, is by no means trivial or
artificial, for it is central in establishing a diverse content
from the motive.  At the quarter-note rate, two statements of the
motive are necessary to provide sufficient length to accompany
the subject, and to create a full statement of CS1.  In this new
rhythmic form, the final note of the motive occurs on a strong
beat, allowing its tonal importance to be expressed.  CS2 denotes
the syncopated form of the motive in half notes (fourth species)
normally also using the inverted form of the motive, and capable
of a combination *per arsin et thesin* (at the distance of one
quarter-note) with the subject.  CS3 denotes the eighth-note form
of the motive (third species) beginning on the weak part of the
beat, and normally using the rectus form, as if to provide
further distinction to CS1.  However, CS3 is not systematically
deployed.  Rather, its function is to provide linking and
harmonizing material and textural contrast to the more systematic
utilizations of S, CS1, and CS2.  In three instances the motive
appears in whole notes (see Example 5).  Due to their location
and role in the formal process, they can be considered simply as
augmentations of the subject.  
[17] In terms of the present analysis, Crotch's analytic labels
identify all the statements of the motive.  His Subject inverted
in diminution, I define as CS1; his subject in double diminution,
I define as CS3.  CS2 is of course is the subject inverted, but
on the off-beats.  Therefore it is not distinguished in Crotch's
[18] It should become clear that the ultimate success of this
composition is dependant entirely on Crotch's very careful
establishment of a range of well defined and contrasting thematic
materials from the single motive.  The subject, in its normal and
augmented forms, follows the metrical plan, always beginning on a
strong beat, while the diminution and double diminution of CS1
and CS2 provide an *arsis* characteristic.  This dichotomy is
extremely important to the contrapuntal working of the
composition, and it is the success of these definitions that
allows a larger coherence to pervade the music.  The challenge
then is to organize the materials in such a way that a sense of
form, contrast, development, and unity can be established.  It is
just as much the means by which Crotch limits his materials, as
the means by which he exploits the range of possibilities, that
gives coherence and sense to the composition.
[19] It follows that since all the material of the piece is
created from statements of the motive, invertible counterpoint
will play an especially important role in this fugue.  It would
be an interesting exercise to explore all of the possible
contrapuntal combinations of rectus, inversus, diminution and
augmentation, and stretto, and it is likely that Crotch worked up
initial sketches for the composition in that manner, considering
his general propensity towards contrapuntal combinations.  Due in
particular to the stepwise nature of the motive, the number of
contrapuntal combinations available is extremely large.  An
exhaustive listing of possibilities is unnecessary; certainly the
composition itself illustrates the range of possibilities
[20] In order to explain the manner in which the various thematic
transformations combine to form a coherent whole, I will adopt
for the following discussion a narrative-analysis that explains
how the various parts relate to one another.  Readers should
follow the score (mto.94.0.10.renwick2.gif through
mto.94.0.10.renwick4.gif) as well as the chart of sections given
below as Example 8. 
Example 8: Design of the Fugue.
Mm.    content                                          tonality
1-10:  exposition: S, CS1, CS2, CS3                         I
10-15: additional entries in stretto
16-17: episode leading to V                                 (V)
18-26: entry in V, introduction of CS1i, leading to III     III
26-37: S and CS2 in chromatic forms, leading to VI;         VI
       introduction of Si     
37-39: stretto on S and Si with CS3; leading to III/VI
39-46: stretto on S with CS3, returning to VI
46-50: S+CS1+CS3; CS1 in chromatic form, leading to V/VI
50-52: episode, leading to II                               II
52-60: S and Si combinations, leading to diminution of same
61:    link to 
62-65: diminutions in contrary motion leading to IV         IV
66-78: Augmentation, in combination with CS1 and CS3
78-85: episode leading to V                                 V
85-88: pedal point supporting diminutions
88-93: pedal point supporting stretto of S
93-97: stretto leading to final cadence                     I
[21] The subject (S) occurs in the tenor.  At the entry of the
answer in the alto, Crotch establishes the crucial distinction of
materials as described previously by creating CS1 out of a chain
of diminished inverted motive-statements, and beginning on the
second quarter, so that the final note of each statement
coincides with a metric strong point.  Therefore the
accompaniment to the answer portrays a contrasting melodic
profile and a contrasting rhythm--this is essentially the means
by which Crotch is able to create structural distinctions among
the motivic statements, and thus to permit the development of a
form with some depth.  In the first part of the piece, the
subject occurs only in rectus, while the CS1 statements use only
the inversus form of the motive.  The bridge before the third
entry is created out of a statement of CS1 in the alto,
accompanied in parallel thirds and tenths in the tenor. 
Construction of a bridge in this manner, out of the
countersubject, is a venerable method that gives the semblance of
canon at the outset of the piece.  Here too, the continued use of
CS1 provides for a contrast to the music of the following entry,
without introducing new material.
[22] At the statement of the subject in the soprano in m. 7--
below the alto--Crotch does not continue with CS1, but instead
introduces CS2, the suspension figure.  Continued use of only S
and CS1 throughout the exposition would have proven tedious at a
very early point in the fugue.  Again, to contrast with the
thematic character of the subject, CS2 uses the inverted form of
the motive.  At the fourth entry, the answer in the bass, CS3
appears for the first time.  As well, CS2 appears again in the
soprano.  Thus the exposition systematically fulfils the formal
objective of stating all of the thematic materials, S, CS1, CS2,
and CS3, while projecting a continual growth of rhythmic activity
and textural density.
[23] Apparently, mm. 10-15 serve essentially as a development of
the subject, exploring a variety of ways in which the subject can
interact with itself in stretto, however, using only the rectus
form.  Rather than setting the exposition off from later entries,
the first stretto overlaps with the end of the fourth entry.
During this passage, the only other accompanimental material is
CS3, first presented in rectus form, then in inversus (CS3i), and
then in a variety of rectus and inversus forms.  The introduction
of CS3i is quite important here in allowing the necessary
flexibility of the accompanimental parts as they harmonize the
stretti.  In mm. 16-18, CS3 and CS3i are played out in relation
to one another, providing a brief episode in V that separates the
stretto passage from what follows.
[24] At m. 18, S in V (accompanied by CS2 in the soprano)
provides the basis for a modulation to III as VI of V.  Here Si
appears for the first time, in the tenor.  However, this
statement of Si should not be considered a significant event in
the motivic development of the piece, but only a convenient
accompanimental inner voice.  Only later does Si appear with a
significant formal connotation.  Following S in the bass at m.
18, the bass continues by reintroducing CS1 against A in the
alto.  This simple reintroduction of CS1 prepares for the next
significant transformation; in m. 24, CS1 appears for the first
time in inversion against itself: CS1 in the treble, and CS1i in
the bass.  This new formation is followed immediately by its own
inversion:  CS1 in bass accompanied by CS1i in soprano (itself
harmonised in parallel sixths with the alto on CS1i).
[25] A PAC on III at m. 27 (A minor), created from free
statements of CS3, heralds a new passage in the upper register,
constructed of alternations of S and Si, in each instance
accompanied by CS2.  This passage, therefore develops the idea of
stretto per arsin et thesin as it projects an intense
chromaticism through a broad descending sweep in mm. 27-37,
leading to VI (D minor).  The statements of CS2 beginning in mm.
31 and 36 introduce a new form of the motive, consisting entirely
of semitonal motion.
[26] Mm. 37-45 exploit a new configuration; S and Si working
against sequential figures composed of CS3 and CS3i.  At mm. 46-
49, CS1 is added to the mix, but now it too is presented in
semitonal motion.  Following another episode (mm. 50-52) based on
forms of CS3 and CS3i, and leading to II (G minor), S is combined
directly with Si, at the tenth, octave, and sixth in succession,
while CS3 and CS3i provided the continual running motion in the
inner parts.  It is at this juncture that the end of the
statements of CS3 and CS3i intersect with the beginning of the
statements of S and Si.  The musical effect of this occurrence is
elision. The following passage then prepares a climax at mm. 59-
60 composed of CS1 and CS1i combined in five parts, but placed
dramatically for the first time on the down beat.  Its context as
an accelerated repetition of the music of the previous measures
makes explicit that these new forms of CS1 and CS1i arise out of
the process of diminution of S and Si.  Therefore this passage,
which must be taken as harmonically and formally the point of
furthest remove, is a synthesis of S and CS1, which were
originally presented antithetically in the exposition.
[27] A rhetorical rest and reduction to two parts provide
connections to the following passage, mm. 62-65, composed of
combinations of CS1 and CS1i, now beginning on beat 4, and
expanding through the registers.(19)  The sudden contrast at this
point, and ultimate reduction to eighth notes is as close as
Crotch comes in this composition to an example of the ornamental
style, in ths case as an inserted parody.  However, it also
serves to reestablish the anacrustic nature of the following
statements of CS1 and CS1i.  The resolution of this passage to IV
(B-flat) at m. 66 coincides with S augmented in whole notes in
the bass.  In a sense, IV substitutes for I in a kind of
subdominant recapitulation at this point.  The following eleven
measures exploit S in augmentation, traversing the registers from
bass to soprano in the manner of the D-sharp minor fugue of
*Well-tempered Clavier I*, accompanied by CS1 and CS3-CS3i, in a
very fine triple-counterpoint.  At m. 78 the subject, inverted
and augmented, occurs in the alto part, but this should really be
understood as an augmentation of CS2, since its function is as a
counterpoint for a statement of S in augmentation.
19. The sudden contrast at this point, and ultimate reduction to
eighth notes is as close as Crotch comes in this composition to
an example of the ornamental style, since the sudden reduction
can easily be heard as an inserted parody.  However, it also
provides the means for reestablishing the anacrustic nature of
the following statements of CS1 and CS1i.
[28] Mm. 80-84 provide for a suitable episodic transition to the
pedal point on V as more and more instances of CS3 and CS3i
develop a tension that resolves on V at m. 85.  The pedal point
begins with a descending 6/3 series of CS3i that spans the fifth
progression g2-c2 and reduces the tension of the previous
measures.  At the final instant in m. 87, CS3i appears in the
intervallic diminution of semitones, representing conceptually
the ultimate reduction of the motive, both intervallically and
durationally.  We can state this idea in set-theoretical terms as
a systematic reduction from 0-2-4 at the outset to 0-1-3 and its
inversion, 0-2-3 in the minor-key sections, and finally 0-1-2,
the chromatic form.  This passage also echoes the music of the
pedal point at the end of the Introduction, providing a
satisfying connection of the two movements, and strengthening the
imperative to resolve what was unresolved at the end of the
[29] The pedal point itself is fully integrated into the motivic
structure.  Its beginning is the final note of a statement of
CS3, and later the same note becomes the first note of the final
stretto series.
[30] At m. 94, and for the two following measures, the motivic
structure finally breaks--as is truly a necessity, for there is
no means of creating a traditional PAC out of the strictly
stepwise motion of the motive.  The bass becomes a functioning
root progression, 1-5-1, and the upper voices provide the
traditional falling cadence.  The upper voice itself presents S,
after which only one additional note, f1 provides the conclusion
with the absolute minimum of digression from the motivic
[31] Just as important to the motivic structure as the systematic
combination of motive forms is the avoidance of certain
combinations.  We have already noted the complete absence of
retrogrades, explained by their lack of provision for any really
new material in comparison to their potential for increased
confusion in the identity of motivic forms.  Also, Crotch never
combines CS1 and CS2.  CS2i never appears, just as Si rarely
appears, except as a direct contrary motion accompaniment to S. 
CS1i occurs only as an inversus accompaniment for CS1 (rectus). 
The only exception is in m. 70, where it accompanies S in
augmentation and can therefore be construed structurally as CS3
in augmentation.  Si in Augmentation only occurs in arsin et
thesin form, relating it specifically as an augmentation of CS2. 
S in augmentation never occurs in combination with S or Si,
probably since that would call into question whether the focus of
the passage is S or the augmentation of S, and whether S was
really CSi in augmentation.  The fatal mistake which any novice
would have made in such a composition would have been
overemployment of possibilities.  It is Crotch's selective use of
materials that allows him to build a coherent form.
[32] In *Free Composition*, Schenker argued against special
contrapuntal devices for their own sake, as artificial, non-
organic means of creation.(20)  Crotch's work is artificial in
the sense that the restrictions that Crotch imposed potentially
hamper the free development of tonal materials and voice-leading
progressions.  But on the other hand, the entire composition is
predicated on such constraints.  And Crotch achieves an integral
balance of motive and structure in that it is entirely the motive
that determines the possibilities and limitations of the piece. 
Example 9 (mto.94.0.10.renwick6.gif), a middleground structural
analysis of the fugue, illustrates some of the significant tonal
events, and in particular attempts to demonstrate how the mass of
motivic detail resolves into a coherent tonal picture.  To be
brief about the contents of Example 9, The fugue actually
displays by and large a classic tonal structure for fugue, as it
projects a single broad tonal progression throughout its length. 
However, in some of the details the peculiarities of the
composition show through.  The bass line in particular displays
somewhat less of the tonal clarity associated with a typical
Bachian fugue.  See for example, the passage from mm. 37 to 44. 
On the other hand, the middleground level of structure provides
strong fifth progressions in the bass that are not possible on
the conjunct surface.  See for example the cadential progressions
leading into mm. 18 and 27.
20. Heinrich Schenker, *Free Composition* (New York: Longman,
1979), p. 78.
[33] The upper voice projects a very clear *Urlinie* from 3^--
that is very clear within the genre of fugue, which by nature is
problematic--expanded by the upper neighbour, b-flat, and
consistently broadened by coupling of lower and upper octaves of
the principal upper voice notes, in a manner reminiscent of
Bach's Fugue 1 in C Major, *Well-tempered Clavier II*.  In fact,
this registral coupling is one of the principal means by which
the fugue attains a sense of controlled forward direction towards
the ultimate conclusion.
[34] Considering the simplicity of the motive and the excellent
control of tonal structure which the fugue displays, it comes as
something of a surprise that the fugue does not contain any
hidden statements of the motive that can be considered
significant.  One can, for example, take the principal notes of
the *Urlinie* in the upper register, a2-b-flat2-a2-g2, as a giant
augmentation of the motive in retrograde, but it is not easy to
support such an interpretation as having any significance other
than the realization of the imperatives of the tonal system.  It
is only a result of the postulates of the *Urlinie* as linear,
and in this case embellished by an upper neighbour.  Rather, in
this piece, it seems that the proliferation of surface instances
of the motive in four distinct magnitudes actually works against
the perception of hidden motives at higher levels.  In contrast,
the free treatment of the motive in the Introduction seems
naturally to lead to motivic expansions, such as the bl-al-g-
sharpl-al form of the upper line in mm. 8-12.
[35] In a work such as this one, where the compositional
parameter of monothematicism is so rigidly predetermined, a major
problem is the differentiation of primary or expository material
from secondary, transitional, or developmental material, in order
that the composition can generate a classical sense of rhetorical
logic and formal proportion.  It is the creative solution to this
problem that shows Crotch's mastery.
[36] In *Serial Composition and Atonality*, George Perle referred
thus to Webern's, *Concerto for Nine Instruments*, Op. 24: 
"Every compositional element is thus ultimately derived from a
single microcosmic detail . . . "(21)  This statement admirably
describes this fugue composed about 100 years earlier!  Of
course, where Crotch had the problem of creating a traditional
tonal coherence, Webern had a different problem, that of
developing a unique coherence and organization of elements. 
Ironically, it is partially his premature musical development
that caused Crotch to develop an aesthetic of music that was
outdated by the time he reached his maturity, and he was unable
to retool the aesthetic of his formative years to reflect a new
set of musical values.  Not until the twentieth century did such
academic composition again become appreciated for its special
21. George Perle, *Serial Composition and Atonality*, fifth
edition, (Berkeley: University of California Press, 1981), p. 79.
[37] In his lectures on musical aesthetics, Crotch stated:  "If a
fugue be a barbarous invention, unworthy of an ingenious
composer, let it be omitted altogether.  But I should rather say,
it is one of the most interesting of all kinds of movement,
resembling the discourse of a great orator, who, having chosen
his theme, states it clearly; examines it in all its bearings;
views it through all its changes and varieties of aspect; and, in
conclusion, by recapitulating the whole, stamps his argument
strongly on the mind." (*Substance*, p. 101)  At the outset of
these same lectures, he stakes out his ground with a general
principle:  "Music is both an art and a science."(22)  There is
no better testament of his commitment to this principle than his
*Introduction and Fugue on a Subject of Four Notes*. 
22. Crotch, *Substance*, p. 1
A: THE MUSIC: (note that in the GIF and EPS editions stem
directions have been normalized)
  m. 1: the tempo indication should refer to the quarter note,
not the eighth note.
  m. 15, beat 3: e-flat2 was e-natural2.
  m. 26, lower staff: natural missing on b.
  m. 28, upper staff beats 2-4: the University of Alberta print
is illegible and has been inked over by hand.  However, the
motivic context makes the reconstruction plain.
  m. 29, upper staff: tie missing to the first g2.
  m. 32, upper staff: sharp missing on f1.
  m. 32, lower staff: whole rest missing.
  m. 49, upper staff: natural missing on d2.
  m. 96, upper staff: dot missing on c1.
B: ANALYTICAL MARKS: In some cases an inversus is marked as a
rectus and vice versa.  In other cases an analytical mark is
  m. 5, upper staff: SDi is marked as SD.
  m. 6, upper staff: SDi is marked as SD.
  m. 13, upper staff: SDDi is marked as SDD.
  m. 13, lower staff: SDD is marked as SDDi.
  m. 14, lower staff: SDD and SDD are not marked.
  m. 17, upper staff: SDD is indicated on the first eighth, not
the second.
  m. 22, upper staff, SDD is missing.
  m. 41, upper staff, SDD on the second half is marked as SDDi.
  m. 44, lower staff, SDD on the second half is marked as SDDi.
  m. 49, lower staff, SDD is omitted on the second half, lower
  m. 51, upper staff, SDD is missing on the second half, lower
  m. 68, upper staff, SDDi on the second half is marked as SDD.
  m. 74, upper staff, SDD on the first half is marked as SDDi.
  m. 75, upper staff, SDD is missing.
  m. 75, lower staff, SDi is missing.
  m. 83, upper staff, SDD is missing for lower part, second half.
  m. 83, lower staff, SDDi is missing for lower part, second
  m. 88, lower staff, S is missing.
  m. 91, upper staff, SDDi is missing--twice--for lower part.
The University of Alberta copy contains inked-in suggestions in
the fugue only for performance on the organ with pedals as
indicated below.  It is most unlikely that these annotations
reflect Crotch's manner of organ performance, for there is no
recorded mention of him using the organ pedals.  The markings
likely date from the mid-nineteenth century, the period during
which the pedal organ quickly gained popularity in England
following Mendelssohn's important visit of 1829. 
  m. 9: "Ped:" marked at the bass entry of S.
  m. 15: "Man:" marked at the f2.
  m. 18: "Ped:" marked at the bass entry of S.
  m. 25: "Ped 8ves:" marked at the F.
  m. 33: "Ped:" marked at the bass entry of S.
  m. 38: "Man:" marked at the tenor entry of Si on d1.
  m. 40: "Ped 8ves:" marked at the bass entry of S on F.
  m. 42: "Ped:" marked at the bass entry of SDD on F.
  m. 48: "Ped:" marked at the g-sharp.
  m. 51: "Ped[:]" marked at the d.
  m. 66: "Ped:" marked at the bass entry of S-augmented.
  m. 71: "Ped:" marked at the e.
  m. 85: "Ped 8ves:" marked at the C that begins the pedal-point.
In addition, from mm. 88-91 and mm. 94-95, a retained C has ben
added, indicating that the pedals would retain the pedal-point C
throughout these passages.
Burke, Edmund.  *A Philosophical Enquiry into the Origins of Our
Ideas of the Sublime and Beautiful* (1757), ed. J.T. Boulton. 
Oxford: Oxford University Press, 1958. 
Crotch, William.  *Substance of Several Courses of Lectures on
Music*.  London: Longman, Rees, Orme, Brown, and Green, 1831. 
R/with an introduction by Benarr Rainbow.  Clarabricken, Ireland:
Boethius Press, 1986.
Crotch, William.  *Specimens of Various Styles of Music*.
Birchall, 1808, 2nd. Ed. Royal Harmonic Institution, c. 1822.,
3rd. Edition, with corrections and additions.  Cramer, Addison
and Beale, c. 1845.
Siegmund Levarie and Ernst Levy.  *Musical Morphology: a
Discourse and a Dictionary*.  Kent, Ohio: The Kent State
University Press, 1983.
Perle, George.  *Serial Composition and Atonality*.  Fifth
edition.  Berkeley: University of California Press, 1981.
Rennert, Jonathan.  *William Crotch*.  Lavenham, Suffolk: Terence
Dalton, 1975.
Reynolds, Joshua.  *Discourses* (1769-97).  New Haven: Yale
University Press, 1975.
Schenker, Heinrich.  *Free Composition*.  2 Vols.  New York:
Longman, 1979.
Copyright Statement
[1] Music Theory Online (MTO) as a whole is Copyright (c) 1994,
all rights reserved, by the Society for Music Theory, which is
the owner of the journal.  Copyrights for individual items 
published in (MTO) are held by their authors.  Items appearing in 
MTO may be saved and stored in electronic or paper form, and may be 
shared among individuals for purposes of scholarly research or 
discussion, but may *not* be republished in any form, electronic or 
print, without prior, written permission from the author(s), and 
advance notification of the editors of MTO.
[2] Any redistributed form of items published in MTO must
include the following information in a form appropriate to
the medium in which the items are to appear:
	This item appeared in Music Theory Online
	It was authored by [FULL NAME, EMAIL ADDRESS],
	with whose written permission it is reprinted 
[3] Libraries may archive issues of MTO in electronic or paper 
form for public access so long as each issue is stored in its 
entirety, and no access fee is charged.  Exceptions to these 
requirements must be approved in writing by the editors of MTO, 
who will act in accordance with the decisions of the Society for 
Music Theory.