00100				CHAPTER VII
00200	 
00300	 
00400	 
00500				MODULATION
00600	 
00700	 
00800		In this book the term "modulation" has been reserved for
00900	use in connection with the movement of basic tonics on the highest
01000	level.  The term is not meant to define the process of moving to a
01100	new tonic; modulation will refer to the state of having moved to a
01200	new basic tonic.  (For example, we will speak of music as having
01300	modulated, but we will not say that music is modulating.)  The means
01400	of getting to a new tonic or the duration of the process will not be the
01500	most important influences on the decision as to whether or not a
01600	modulation has taken place.  Rather, it is how the new tonic, once
01700	attained, is used that is of the greatest importance.
01800	
01900		The new tonic must become, in relative terms, truly a basic
02000	tonic.  Thus modulation might occur in any formal scheme, just so long
02100	as a change of the basic tonic is involved.  The uses of a new tonic
02200	that constitute modulation cover a wide variety of situations and will
02300	often be open to question.  Probably the clearest modulations include
02400	new melodic material of considerable length -- perhaps even in a new
02500	tempo and/or meter.  There is every conceivable gradation between such
02600	cases and those where the modulation borders on being merely an
02700	elaborated cadence or extended tonicization.
02800	
02900		Reference to some of Mozart's Piano Sonatas may help to
03000	clarify these points.  Very many Mozart movements are in some kind
03100	of sonata form which usually includes very clear modulations of basic
03200	tonics.  The first movement (variations) of the Sonata in A, K.300i(331),
03300	contains neither modulations nor varied control tonics.  The Finale 
03400	(Alla turca) has one simple modulation and return.  This section of 32
03500	bars (repeated) in the tonic on vi (f#) presents new, contrasting
03600	material.  The movement to a temporary tonic on A about two-thirds of
03700	the way through this section by no means represents a return to the
03800	original key on the highest level.  This bit in A is surrounded by, 
03900	and completely dependent on, f#.  The modulation back to A is effected
04000	by the return of previously heard material associated with that key.
04100	The Menuetto, however, has more complex tonic movements and is in what
04200	might be called (and this is true of many menuets, scherzos, etc.) a
04300	miniature sonata form.  In this movement, at measure 11, entirely 
04400	new material is presented in the dominant key.
04500	
04600	****Example 71 ******
04700	
04800	Figure 71
04900	
05000	
05100	
05200	
05300		The above is certainly a borderline case.  It would not be
05400	considered as a modulation except for the independence of bars 11 to
05500	18, which form a unit with importance almost equal to that of the
05600	first ten bars.  What makes this a dubious example of modulation 
05700	is the lack of harmonic variety in the E section.  There are no new
05800	temporary tonics; in fact, there is only one chord that does not
05900	function as I or V.  An only slightly broader view would be necessary
06000	to call the movement to E a change of control tonics in the level
06100	of the middle ground.
06200	
06300		After the double bar, we have a brief development section 
06400	freely based on the material of bars 5 to 10.  Since there is no
06500	local reference to A at the double bar, the following progressions
06600	are shown as dependent on a basic tonic of e.  Under this basic
06700	tonic are control tonics of b (V) and a (iv), this latter becoming 
06800	major and then the basic tonic upon the return of the original material
06900	(measure 31).
07000	
07100		The status of the functions after the double bar (measure 19)
07200	naturally are somewhat different when the repeat is taken.  In fact,
07300	the three notes, E-F#-G, can easily be heard as the extension of the
07400	final A chord, so that it becomes the dominant function of D.  But
07500	then D's vi (b) may be immediately heard in direct relation to the
07600	basic tonic, A.  We must not forget, however, that when music is 
07700	repeated literally, the functions of the opening measures, the second
07800	time through, take on a dual aspect.  At once we realize we are taking
07900	a fresh start (or continuing once again from earlier material) and, at
08000	the same time, we are continuing on from the final part of the section.
08100	Only briefly do repeats influence our comprehension of the functional
08200	details.  But the influence on the background elements can be more
08300	far-reaching.  It is enough that the diagram above shows only the
08400	differences caused by the repeat.
08500	
08600		The movement to the D of the Trio constitutes a modulation of
08700	a different sort.  This Trio is actually a separate, self-contained
08800	piece.  This is usually the case in this type of work, even when a
08900	passage of retransition to the Menuetto might be added at the end of
09000	the Trio.  Within this Mozart Trio it is best to consider that there
09100	are no real modulations.  The main reason for taking this position
09200	is that even though the dominant (A) becomes a strong control tonic,
09300	the passage using this key presents no important independent material
09400	but merely repeats two cadential formulas.
09500	
09600	**** Example 72.  Mozart, Sonata in A, K.300i(331), 2nd movement Trio.
09700	
09800	Figure 72
09900	
10000	
10100	
10200		In the foregoing diagram note that A is first shown as the
10300	control tonic at the point at which the various factors of the music
10400	reach A as a goal.  What follows is an extension; e appears as the
10500	control tonic over its dominant chord because the "head" of the opening
10600	material is there restated.  C appears as the control tonic immediately
10700	upon presentation of the new material in that key.
10800	
10900		The e control tonic at measure 17 becomes ii of D rather than
11000	remaining as v of A, because A never is heard as anything more than the
11100	the elaborated dominant of D.  The repeat becomes important here too.
11200	After once having heard the A7 chord at measure 16 proceed to the D
11300	chord when the first section of the Trio is repeated, the subsequent
11400	movement to the B7 chord sounds like a sudden digression to the ii (e)
11500	area.  This situation is only slightly different from that found at the
11600	first double bar in the Menuetto.  There, the independence of the
11700	material in the dominant key, and especially the absence of a 7th chord
11800	just before the repeat, are the factors which put this music on the 
11900	other side of the admittedly academic dividing line.
12000	
12100	                       -------------------
12200	
12300		The next movement to be considered, the Rondeau en Polonaise
12400	from Mozart's Sonata in D, K.205b(284), contains a simple modulation
12500	and some firmly established control tonics.  In the following analysis
12600	some sections are condensed to save space.
12700	
12800	Figure 73a.  Mozart, Sonata in D, K.205b(284), Rondeau en Polonaise
12900	
13000	
13100	
13200	
13300		There is a cadence on a new tonic of E at bar 25.  E is shown 
13400	here as a control tonic rather than a modulation because of the 
13500	relative brevity of its use after the cadence (six bars out of the
13600	first thirty of the piece), and because its repetitious character
13700	gives it the sound of an extended cadence formula.  The new tonic at
13800	bar 47 (f#) is completely independent and so is shown as a modulation,
13900	despite its brevity.
14000	
14100		Treatment of the following music in D is open to varying
14200	opinion.  It has been listed as a new control tonic on D (VI in f#)
14300	for two reasons:  first, the music present at bar 53 has already
14400	been heard as continuation, and then transition material (see bar 17);
14500	second, no clear reference is made back to the original basic tonic
14600	with material that was in that key (A), or with new material, until
14700	bar 70.  The first control tonic at bar 53 is shown as VI/I (***)
14800	rather than simply VI, so that the latter one at bar 61, VI/V, may
14900	appear on the same horizontal line.  This way the lower two levels of
15000	the middle ground at bars 53-64 are the exact parallel of 17-28.  It
15100	is especially important that the diagram makes clear the similarities
15200	of such parallel passages.
15300	
15400		If the movement to D is taken as a modulation, the diagram
15500	appears simpler, but then perhaps some of the subtlety of this work's
15600	large harmonic structure is missed.
15700	
15800	Figure 73b (upper levels only)
15900	
16000	
16100	
16200	
16300		The analysis given in Figure 73a is preferred.
16400	
16500	               -------------------------
16600	
16700		In both the first and last movements of Mozart's Sonata in
16800	F, K.300k(332), we find what is commonly called a transition section
16900	linking the two main parts of the exposition.  In each case the
17000	material of the transition follows a full close in F, is new, and
17100	starts out clearly in d.  However, d proves to be only a step on the
17200	way to C.  Such transitions should be analyzed under the basic tonic
17300	of departure, the new basic tonic appearing only when a clear goal
17400	is reached.
17500	
17600	Figure 74a.  Mozart, Sonata in F, K.300k(332), first movement.
17700	
17800	
17900	
18000	
18100	
18200	Figure 74b.  Third movement.
18300	
18400	
18500	
18600	
18700	
18800		The second pivot involving tonics might have appeared as early
18900	as bar 40, thus bringing out the role of d as ii of C.  However, C as
19000	a control tonic should not appear before bar 41, where a varied form
19100	of the material of bar 36 is presented (see Figure 74b).
19200	
19300	Figure 74c
19400	
19500	
19600	
19700	
19800		In both movements above, the material of the transitions was 
19900	of moderate duration and had no especial thematic connection with
20000	what followed.  Notice that C in its first tonic appearance in the
20100	first movement was minor, but when it became the basic tonic, it then
20200	was major. In the third movement this process is reversed. However,
20300	since the music changes to major not long after, the minor indication
20400	might show only in the middle ground.  For example:
20500	
20600	
20700	
20800	
20900	                          -----------------
21000	
21100		In the first movement of the Sonata in C, K.189d(279), the
21200	first basic tonic area closes with a half-cadence.  The following
21300	new material, while not in the dominant key, does lead quickly and
21400	without break to the dominant.  Other decisive factors here are that
21500	the last two quarter-notes (D's) of bar 16 (which belong to the new
21600	material) are heard as a continuation of the G chord which closed the
21700	first section, and that the beginning of the phrase that contains the 
21800	reappearance of the G chord, now as I (bar 20), is elided with the
21900	end of the previous phrase.
22000	
22100	Figure 75a.  Mozart, Sonata in C, K.189d(279), first movement.
22200	
22300	
22400	
22500	
22600		In the recapitulation this area should be analyzed a little
22700	differently, because a control tonic of G is established just before the
22800	new material appears.  The C recurs as the control tonic immediately
22900	upon presentation of the material of the second key area (at *).
23000	
23100	Figure 75b
23200	
23300	
23400	
23500	
23600	
23700	                     -------------------
23800	
23900		A great extension of the situation just discussed is found in
24000	Schubert's Quintet in C, Op.163, first movement.  The full exposition
24100	clearly ends in G (followed by a single transitional 7th chord), but
24200	the first key heard in the exposition's second group is Eb.  The second
24300	group is preceded by an eleven-bar passage over a G pedal note.
24400	Because of the absence of any simple V-I progression in G in these 
24500	eleven bars, G manages to maintain its role as dominant to the basic
24600	tonic of C.  It would be quite normal for the music to continue at 
24700	this point with G being heard immediately in both the roles of control
24800	tonic and basic tonic.  However, when the second 'cello descends
24900	chromatically from G at bar 59, we may well expect the goal to be a
25000	C chord rather than the Eb chord, since the last heard G chord still
25100	had a dominant quality relative to the control tonic C.  (G had
25200	been tonicized, but only as the fifth degree of C.  Even the eleven-bar
25300	prolongation of G fails to give it the status of a control tonic.)
25400	
25500		Once Eb is reached, the total texture leaves no doubt that
25600	we are in an entirely new realm.  The next tonal movements again
25700	raise a question regarding the location of the control tonic.  C, as
25800	the tonic VI of Eb, is touched upon at bar 71, and once more G sounds
25900	as a tonic at bar 79 -- but still mainly as a secondary element.  The
26000	whole process is repeated and then finally, at bar 100, G assumes a
26100	primary tonic role.  Nevertheless, the whole second section of the
26200	exposition may be analyzed under a single basic tonic, because the
26300	unified character of the thematic material gives the G goal of all the
26400	harmonic movement of the section strong predominance.  The upper levels
26500	of a condensed analysis of this music follow.
26600	
26700	Figure 76.  Schubert, String Quintet in C, Op.163, first movement.
26800	
26900	
27000	
27100	
27200	
27300	
27400	
27500	
27600		The important thing in an analysis is to show indications
27700	which parallel the actual progress of the music, not only on the
27800	small, or point-to-point level, but also on the structurally higher
27900	section-to-section level.  In the diagram above, when Eb first appears
28000	it is directly related to C as bIII.  The new pλematic material,
28100	however, gives Eb a stronger role as the bVI relation to G.  It is
28200	true that there is no way of knowing this on first hearing.  The
28300	large structure of a work is fully grasped only upon reflection after
28400	the music is ended.  This in no way undermines the affective quality
28500	of harmonic "surprises" during subsequent hearings, whether they be
28600	found in the functional details or in the movements of the tonics.
28700	On the contrary, the affect of the music can only begin to be fully
28800	appreciated when the relationship of all the parts to the whole is
28900	sensed -- even if only intuitively.
29000	
29100		Continuing with the above analysis, at bar 71 C appears as 
29200	VI of Eb.  There is no direct connection between the role of this
29300	tonic C and the tonic of the movement's opening.  The full cadence on 
29400	G establishes a control tonic on the dominant of C (bar 79).  After the
29500	entire progression is repeated, the control tonic on G continues and
29600	thus becomes a factor on the same level as Eb.  When this happens, a
29700	pivotal movement puts G's tonic function on the uppermost
29800	level of the middle ground.
29900	
30000	
30100	Exercises for Chapter VII
30200	
30300		Analyze the following.  Some of the listed works need not
30400	be thought of as involving true modulation.
30500	
30550	
30600	From the Sonatas of Mozart:
30700	
30800	  1. Sonata in C, K.189d(279), second movement up to the double bar.
30900	  2.   "    in F, K.189e(280), finale (Presto) up to the double bar.
31000	  3.   "    in Eb, K.189g(282), Menuetto I, all.
31100	  4.   "     "  "     "    "  , Menuetto II, all.
31200	  5.   "    in a, K.300d(310), first movement up to the double bar.
31300	  6.   "    in Bb, K315c(333), finale (Allegretto) first 36 bars.
31400	  7.   "    in D, K.576, first movement up to the double bar.
31500	
31550	
31600	From the Mazurkas of Chopin:
31700	
31800	  8. Mazurka No.2 in e, Op.17,#2, all.
31900	  9.    "    No.14 in g, Op.24,#1, all.
32000	 10.    "    No.22 in g#, Op.33,#1, all.
32100	 11.    "    No.27 in e, Op.41,#2, all.
32200	 12.    "    No.34 in C, Op.56,#2, all.