CHAPTER VII
 
 
 
			MODULATION
 
 
	In this book the term "modulation" has been reserved for
use in connection with the movement of basic tonics on the highest
level.  The term is not meant to define the process of moving to a
new tonic; modulation will refer to the state of having moved to a
new basic tonic.  (For example, we will speak of music as having
modulated, but we will not say that music is modulating.)  The means
of getting to a new tonic or the duration of the process will not be the
most important influences on the decision as to whether or not a
modulation has taken place.  Rather, it is how the new tonic, once
attained, is used that is of the greatest importance.

	The new tonic must become, in relative terms, truly a basic
tonic.  Thus modulation might occur in any formal scheme, just so long
as a change of the basic tonic is involved.  The uses of a new tonic
that constitute modulation cover a wide variety of situations and will
often be open to question.  Probably the clearest modulations include
new melodic material of considerable length -- perhaps even in a new
tempo and/or meter.  There is every conceivable gradation between such
cases and those where the modulation borders on being merely an
elaborated cadence or extended tonicization.

	Reference to some of Mozart's Piano Sonatas may help to
clarify these points.  Very many Mozart movements are in some kind
of sonata form which usually includes very clear modulations of basic
tonics.  The first movement (variations) of the Sonata in A, K.300i(331),
contains neither modulations nor varied control tonics.  The Finale 
(Alla turca) has one simple modulation and return.  This section of 32
bars (repeated) in the tonic on vi (f#) presents new, contrasting
material.  The movement to a temporary tonic on A about two-thirds of
the way through this section by no means represents a return to the
original key on the highest level.  This bit in A is surrounded by, 
and completely dependent on, f#.  The modulation back to A is effected
by the return of previously heard material associated with that key.
The Menuetto, however, has more complex tonic movements and is in what
might be called (and this is true of many menuets, scherzos, etc.) a
miniature sonata form.  In this movement, at measure 11, entirely 
new material is presented in the dominant key.

****Example 71 ******

Figure 71




	The above is certainly a borderline case.  It would not be
considered as a modulation except for the independence of bars 11 to
18, which form a unit with importance almost equal to that of the
first ten bars.  What makes this a dubious example of modulation 
is the lack of harmonic variety in the E section.  There are no new
temporary tonics; in fact, there is only one chord that does not
function as I or V.  An only slightly broader view would be necessary
to call the movement to E a change of control tonics in the level
of the middle ground.

	After the double bar, we have a brief development section 
freely based on the material of bars 5 to 10.  Since there is no
local reference to A at the double bar, the following progressions
are shown as dependent on a basic tonic of e.  Under this basic
tonic are control tonics of b (V) and a (iv), this latter becoming 
major and then the basic tonic upon the return of the original material
(measure 31).

	The status of the functions after the double bar (measure 19)
naturally are somewhat different when the repeat is taken.  In fact,
the three notes, E-F#-G, can easily be heard as the extension of the
final A chord, so that it becomes the dominant function of D.  But
then D's vi (b) may be immediately heard in direct relation to the
basic tonic, A.  We must not forget, however, that when music is 
repeated literally, the functions of the opening measures, the second
time through, take on a dual aspect.  At once we realize we are taking
a fresh start (or continuing once again from earlier material) and, at
the same time, we are continuing on from the final part of the section.
Only briefly do repeats influence our comprehension of the functional
details.  But the influence on the background elements can be more
far-reaching.  It is enough that the diagram above shows only the
differences caused by the repeat.

	The movement to the D of the Trio constitutes a modulation of
a different sort.  This Trio is actually a separate, self-contained
piece.  This is usually the case in this type of work, even when a
passage of retransition to the Menuetto might be added at the end of
the Trio.  Within this Mozart Trio it is best to consider that there
are no real modulations.  The main reason for taking this position
is that even though the dominant (A) becomes a strong control tonic,
the passage using this key presents no important independent material
but merely repeats two cadential formulas.

**** Example 72.  Mozart, Sonata in A, K.300i(331), 2nd movement Trio.

Figure 72



	In the foregoing diagram note that A is first shown as the
control tonic at the point at which the various factors of the music
reach A as a goal.  What follows is an extension; e appears as the
control tonic over its dominant chord because the "head" of the opening
material is there restated.  C appears as the control tonic immediately
upon presentation of the new material in that key.

	The e control tonic at measure 17 becomes ii of D rather than
remaining as v of A, because A never is heard as anything more than the
the elaborated dominant of D.  The repeat becomes important here too.
After once having heard the A7 chord at measure 16 proceed to the D
chord when the first section of the Trio is repeated, the subsequent
movement to the B7 chord sounds like a sudden digression to the ii (e)
area.  This situation is only slightly different from that found at the
first double bar in the Menuetto.  There, the independence of the
material in the dominant key, and especially the absence of a 7th chord
just before the repeat, are the factors which put this music on the 
other side of the admittedly academic dividing line.

                       -------------------

	The next movement to be considered, the Rondeau en Polonaise
from Mozart's Sonata in D, K.205b(284), contains a simple modulation
and some firmly established control tonics.  In the following analysis
some sections are condensed to save space.

Figure 73a.  Mozart, Sonata in D, K.205b(284), Rondeau en Polonaise




	There is a cadence on a new tonic of E at bar 25.  E is shown 
here as a control tonic rather than a modulation because of the 
relative brevity of its use after the cadence (six bars out of the
first thirty of the piece), and because its repetitious character
gives it the sound of an extended cadence formula.  The new tonic at
bar 47 (f#) is completely independent and so is shown as a modulation,
despite its brevity.

	Treatment of the following music in D is open to varying
opinion.  It has been listed as a new control tonic on D (VI in f#)
for two reasons:  first, the music present at bar 53 has already
been heard as continuation, and then transition material (see bar 17);
second, no clear reference is made back to the original basic tonic
with material that was in that key (A), or with new material, until
bar 70.  The first control tonic at bar 53 is shown as VI/I (***)
rather than simply VI, so that the latter one at bar 61, VI/V, may
appear on the same horizontal line.  This way the lower two levels of
the middle ground at bars 53-64 are the exact parallel of 17-28.  It
is especially important that the diagram makes clear the similarities
of such parallel passages.

	If the movement to D is taken as a modulation, the diagram
appears simpler, but then perhaps some of the subtlety of this work's
large harmonic structure is missed.

Figure 73b (upper levels only)




	The analysis given in Figure 73a is preferred.

               -------------------------

	In both the first and last movements of Mozart's Sonata in
F, K.300k(332), we find what is commonly called a transition section
linking the two main parts of the exposition.  In each case the
material of the transition follows a full close in F, is new, and
starts out clearly in d.  However, d proves to be only a step on the
way to C.  Such transitions should be analyzed under the basic tonic
of departure, the new basic tonic appearing only when a clear goal
is reached.

Figure 74a.  Mozart, Sonata in F, K.300k(332), first movement.





Figure 74b.  Third movement.





	The second pivot involving tonics might have appeared as early
as bar 40, thus bringing out the role of d as ii of C.  However, C as
a control tonic should not appear before bar 41, where a varied form
of the material of bar 36 is presented (see Figure 74b).

Figure 74c




	In both movements above, the material of the transitions was 
of moderate duration and had no especial thematic connection with
what followed.  Notice that C in its first tonic appearance in the
first movement was minor, but when it became the basic tonic, it then
was major. In the third movement this process is reversed. However,
since the music changes to major not long after, the minor indication
might show only in the middle ground.  For example:




                          -----------------

	In the first movement of the Sonata in C, K.189d(279), the
first basic tonic area closes with a half-cadence.  The following
new material, while not in the dominant key, does lead quickly and
without break to the dominant.  Other decisive factors here are that
the last two quarter-notes (D's) of bar 16 (which belong to the new
material) are heard as a continuation of the G chord which closed the
first section, and that the beginning of the phrase that contains the 
reappearance of the G chord, now as I (bar 20), is elided with the
end of the previous phrase.

Figure 75a.  Mozart, Sonata in C, K.189d(279), first movement.




	In the recapitulation this area should be analyzed a little
differently, because a control tonic of G is established just before the
new material appears.  The C recurs as the control tonic immediately
upon presentation of the material of the second key area (at *).

Figure 75b





                     -------------------

	A great extension of the situation just discussed is found in
Schubert's Quintet in C, Op.163, first movement.  The full exposition
clearly ends in G (followed by a single transitional 7th chord), but
the first key heard in the exposition's second group is Eb.  The second
group is preceded by an eleven-bar passage over a G pedal note.
Because of the absence of any simple V-I progression in G in these 
eleven bars, G manages to maintain its role as dominant to the basic
tonic of C.  It would be quite normal for the music to continue at 
this point with G being heard immediately in both the roles of control
tonic and basic tonic.  However, when the second 'cello descends
chromatically from G at bar 59, we may well expect the goal to be a
C chord rather than the Eb chord, since the last heard G chord still
had a dominant quality relative to the control tonic C.  (G had
been tonicized, but only as the fifth degree of C.  Even the eleven-bar
prolongation of G fails to give it the status of a control tonic.)

	Once Eb is reached, the total texture leaves no doubt that
we are in an entirely new realm.  The next tonal movements again
raise a question regarding the location of the control tonic.  C, as
the tonic VI of Eb, is touched upon at bar 71, and once more G sounds
as a tonic at bar 79 -- but still mainly as a secondary element.  The
whole process is repeated and then finally, at bar 100, G assumes a
primary tonic role.  Nevertheless, the whole second section of the
exposition may be analyzed under a single basic tonic, because the
unified character of the thematic material gives the G goal of all the
harmonic movement of the section strong predominance.  The upper levels
of a condensed analysis of this music follow.

Figure 76.  Schubert, String Quintet in C, Op.163, first movement.








	The important thing in an analysis is to show indications
which parallel the actual progress of the music, not only on the
small, or point-to-point level, but also on the structurally higher
section-to-section level.  In the diagram above, when Eb first appears
it is directly related to C as bIII.  The new thematic material,
however, gives Eb a stronger role as the bVI relation to G.  It is
true that there is no way of knowing this on first hearing.  The
large structure of a work is fully grasped only upon reflection after
the music is ended.  This in no way undermines the affective quality
of harmonic "surprises" during subsequent hearings, whether they be
found in the functional details or in the movements of the tonics.
On the contrary, the affect of the music can only begin to be fully
appreciated when the relationship of all the parts to the whole is
sensed -- even if only intuitively.

	Continuing with the above analysis, at bar 71 C appears as 
VI of Eb.  There is no direct connection between the role of this
tonic C and the tonic of the movement's opening.  The full cadence on 
G establishes a control tonic on the dominant of C (bar 79).  After the
entire progression is repeated, the control tonic on G continues and
thus becomes a factor on the same level as Eb.  When this happens, a
pivotal movement puts G's tonic function on the uppermost
level of the middle ground.


Exercises for Chapter VII

	Analyze the following.  Some of the listed works need not
be thought of as involving true modulation.


From the Sonatas of Mozart:

  1. Sonata in C, K.189d(279), second movement up to the double bar.
  2.   "    in F, K.189e(280), finale (Presto) up to the double bar.
  3.   "    in Eb, K.189g(282), Menuetto I, all.
  4.   "     "  "     "    "  , Menuetto II, all.
  5.   "    in a, K.300d(310), first movement up to the double bar.
  6.   "    in Bb, K315c(333), finale (Allegretto) first 36 bars.
  7.   "    in D, K.576, first movement up to the double bar.


From the Mazurkas of Chopin:

  8. Mazurka No.2 in e, Op.17,#2, all.
  9.    "    No.14 in g, Op.24,#1, all.
 10.    "    No.22 in g#, Op.33,#1, all.
 11.    "    No.27 in e, Op.41,#2, all.
 12.    "    No.34 in C, Op.56,#2, all.