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.SKIP 2
.CENTER		
CHAPTER VI
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.CENTER			
ALTERNATING PROGRESSIONS AND INDIRECT RELATIONSHIPS
.SKIP 1
.INDENT 6
.FILL
.ADJUST
.SELECT 1
 
 
	The simplest form of alternating progression is that in which
a particular chord is approached from two points of view with regard 
to its harmonic function.  In most cases there is not the slightest
doubt as to the primary function (usually dominant) of the final chord.

.CENTER
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.BEGIN VERBATIM
Example 68







Figure 68





.END
.FILL INDENT 6

	Here the delayed resolution of the D7th chord to %2its%1 tonic, ↓_G_↓,
may be shown as above.  The primarily dominant function of the G chord 
is ensured by the immediately preceding subdominant (II%5H%1)
function.  Thus the G major chord is tonicized by the second chord and
"dominantized" by the third chord, the last relationship in the
series being the most influential.  Here, once more, the voice-leading
or contrapuntal element becomes relevant and, from a broad point of
view, the bass notes F%4S%1, A%4F%1, and G form a kind of cambiata figure
which minimizes the effect of the D7th chord as a dominant.$$ %1Many
books on harmony would call this D7 chord simply II%4S3%1.  This seems like
an oversimplification.  A dividing line must be drawn somewhere in these
matters; in this book the sharped 4th and 5th degrees of the scale will only
be admitted as "functional" when they are part of augmented sixth or
augmented fifth chords.$  In detail,
however, the above example %2does%1 offer us the musical effect
indicated by the analysis.

	Similar progressions occur in the %2Mazurka No.38%1, Op.59,#3,
of Chopin (measures 16-24):

.begin verbatim

Example 69a
.end
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.skip 12
.FILL INDENT 6

	As the progressions under study become more complex, the
following method for clarifying the relationships may be used.  The
horizontal bars connecting various parts of the Chordal Reduction
serve the purpose of showing which of the harmonies may be directly
related to one another.  %2On any given level, the horizontal bar
connects  chords which all serve the same temporary tonic%1.  It
should be noted that sometimes a chord of substitute function will
be found on a given tonic's bar even though the chord is at the same
time more clearly related to another tonic.  These Chordal Reductions
are not to be confused with the diagrams outlining Tonic Guide Tones.
Tonic Guide Tones will be used solely as a means of clarifying
relationships among the %2tonics%1 involved in any progression.  (See page 57.)

.begin verbatim

Figure 69

Chordal Reduction
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.skip 13



.FILL INDENT 6
	Here the control tonic of III (↓_A_↓) has been established.  The
section in ↓_A_↓ is far too brief to warrant the use of the term
"modulation".  In measure 17 the G and the clearly implied C%4S%1 form
a tritone which is most readily heard as dominant to ↓_D_↓.  However,
measure 18 first offers us a 7th chord on B -- this heard as dominant
to ↓_E_↓.  Then follows a d minor chord, the logical result of measure 17.
In measure 19 is heard an E7th chord (over a tonic A pedal) which is
a substitute for the simple tonic expected after the B7th chord.  At
measure 21 things become a little more complex.  The last chord contains
an A%4S%1, which could be interpreted merely as a chromatic passing note,
but which certainly does tend to tonicize the following B (as the
sixth degree, normally a minor tonic, of ↓_D_↓), which then functions as it
did earlier.  At the end of measure 22, the tonic expected after the
A7th chord of four beats back is replaced by the II%5I%1 of ↓_A_↓.

	Since the successive functions in such music sometimes bear
only a secondary relation to each other, very skillful voice leading,
as in the Chopin example, is necessary in order that the progression
remains clear.  From one point of view, there is no doubt at all that
this passage can be interpreted as a chromatic "working out" of the
A chord.
.begin verbatim

Example 69b.  (Reduction of Example 69a.  The doubling of the
		highest part at the octave below is omitted.)
.end
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.skip 5




.FILL INDENT 6
	However, the particular form the chromaticism takes in this
Mazurka also gives the unmistakable impression of a series of varied
harmonic functions.
.SKIP 1

	For more examples of chords which have both contrapuntal and
harmonic significance we will turn to the opening of Schubert's
%2Quintet in C%1, Op.163.
.next page
.begin verbatim
Example and Figure 70.  Schubert, String Quintet in C, Op.163
		(Reduction of measures 1-26.)
.END
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.SKIP 15

.FILL INDENT 6

	Complete diminished 7th chords (at bars 4 and 13) are often
used non-functionally, as multiple neighboring notes.  Taking this
point of view, Figure 70a treats most of the altered chords as
purely contrapuntal.  Thus the harmonic movement remains quite
simple and without any new tonicization until the end of the excerpt.
This is very likely the way the music is heard (and intended to be heard).


	As we come to know the piece rather well, especially as
regards the violoncellos' repetition of the opening theme at bar 33
(see Example 71), we may feel that the various altered chords in the
first part also carry a potential of having functional implications.
Any given listener may not always hear this example in the same way, even
if he has memorized it.  While the main purpose of our analyses is to
show how the harmony of a piece is most simply heard, we should also
try to provide for other, less simple possibilities that may be
inherent in the music.


	Figure 70b shows the implications of the various altered chords.
The diminished chord at bar 13 is shown as ↓_d_↓: Vs of V (A%4F%1=G%4S%1) simply
because of its being in a position analogous to that of the first 
diminished chord, bar 4.  Bar 13 has, of course, at the same time the 
role of Vs of ↓_C_↓.  This is demonstrated by the indications below the
second analysis.  Thereby the symmetry of the first and second
phrases is clearly shown.  At (*) three functions are given for the
d minor harmony.  They represent, 1) its relation to the previous
function, 2) its present function, as defined by a new statement of
the main motive, and 3) its relation to the following function. Such
triple indications should be used sparingly, since they could easily
obscure the most important elements of a progression.

	Perhaps Schubert wished to prove that these altered chords
could have harmonic functions, since his ↓_varied_↓ restatement of the
opening gives the following progression.

.begin verbatim

Example 71.  Schubert, Quintet in C, Op.163 
             (Reduction of bars 33-46)






Figure 71
.end
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.SKIP 2
.FILL INDENT 6
	Now, the first diminished chord is certainly working as a
tonicizing agent to ↓_G_↓.  The three chords found in measures 40 to 44
give the effect of a deceptive progression in ↓_a_↓, the final chord
of the group being also subdominant in terms of what follows.  Note
that no pivotal tonic is used when the control tonic returns to ↓_I_↓.
It is felt that the time spent under the control tonic of ↓_ii_↓ (↓_d_↓) is
of such relatively brief duration that C could scarcely be heard as
a tonicized VII.  However, some other piece with a different
%2presentation%1 of a similar progression might be analyzed rather
differently.


	In the %2March to the Scaffold%1 from Berlioz' %2Fantastic Symphony%1
occurs a passage that has puzzled many musicians.  Its dramatic effect
is highly dynamic and specific, but its musical justification has been
difficult.  The problem arises from the alternation of the tonic ↓_g_↓
minor harmony and the D%4F%1 chord.  It might seem that the closest link
between these tritone-related minor and major chords would be found at
the fourth above the tonic, or ↓_c_↓ minor.  Thus the relationship could
be shown:
.BEGIN VERBATIM

Figure 72
.END

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.SKIP 2


.FILL INDENT 6
	However, the composer does not seem to take this view of the
relationship (nor does he treat D%4F%1 in this manner when it appears a
few moments earlier).  The alternation of chords continues until D%4F%1
is almost heard as a new and functionally unrelated tonic.  But, as the 
drive of the passage subsides, it progresses to E%4F%17 and thence to a
tutti chord, at which time the previous happenings may be reassessed.
The analysis following the example represents the %2result%1 of this
reassessment.  There is inadequate basis for choice until the end
of the passage.
.next page
.begin verbatim
Example and Figure 73.  Berlioz, Fantastic Symphony 
				(Reduced from the fourth movement)
.end
.CENTER
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.skip 16
.fill indent 6
	Below the main body of the analysis is shown the separate
movement of each of the two alternating progressions.  The one in ↓_g_↓ is 
quite simple; in fact, the whole g chord acts as a pedal until the
augmented sixth chord leads to the dominant.  The progression begun
by the D%4F%1 chord is listed under an ↓_A%4F%1_↓ tonic because, in view of the
ultimate hearing of the E%4F%17 (enharmonic), ↓_A%4F%1_↓ is the most direct
connecting link to ↓_g_↓, even though the I
of ↓_A%4F%1_↓ is not used.  The
functions listed in parentheses might be interpreted as chromatic
elaboration of the D%4F%1 chord (see Example 70).  Thus the overall
progress from the D%4F%1 chord is likewise simple -- ↓_A%4F%1_↓: IV...V,
wherein the V serves doubly as the G6 of ↓_g_↓.  Notice that in all this
the possibility of treating the D%4F%1 chord as a %4F%1V or %4S%1IV relationship
in ↓_g_↓ is avoided.  A true direct relation of the tritone between two
tonics tends to destroy both of them.
Such a relationship would merely constitute a juxtaposition of keys
whose tonic chords were incompatible except as they both served as
subsidiaries to a third tonic (see page 46).

.next page
	Following is the well-known introduction to the second movement
of Dvorak's %2New World Symphony%1.  The notation here differs
enharmonically from the original.
.BEGIN VERBATIM

Example and Figure 74
.END
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.skip 18


.fill indent 6
	Even though the movement proper is in ↓_D_↓%4F%1, the key at the 
beginning, as listed on the highest level, is ↓_E_↓.  This represents a
continuation of the ↓_e_↓ of the first movement.  The tritone relation 
between two major chords (E and B%4F%1) is usually heard as V-%4F%1II or
vice versa, the former being used in this case because of the positions
of the two chords,  V%4H%1-%4F%1II%46%1, and because of the subsequent harmonic
developments.  By the time we reach the third measure we are sure that
the E chord was working as a dominant to ↓_A_↓.

.BEGIN VERBATIM

Example 75a
.END
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.skip 4

.fill indent 6



	However, the D%4F%1 chord which broke the continuity is most simply
heard as dominant to ↓_A_↓'s sixth degree, ↓_f_↓%4S%1.  It must be noted that the 
absence of emphasis in the first two bars makes immediate recognition of
the functions of all but the first tonic ↓_E_↓ very difficult.  Until more
is heard any chord may be dominant, subdominant, or tonic, etc.  When
the phrase is complete -- and thus has developed points of emphasis -- these
distantly related chords are retrospectively assigned functions in terms
of the end results.  The f%4S%1 chord serves as VI to ↓_A_↓ (Example 75a);
↓_f_↓%4S%1 is tonicized by the progression from the fourth to the sixth chord
(Example 75c); and the f%4S%1 chord is IV%4FF3%1 (g%4F%1) to ↓_D_↓%4F%1 (Example 75b).
↓_A_↓ has been tonicized by the first three chords, but the A chord also 
serves as %4FF%1VI (B%4FF%1) to ↓_D_↓%4F%1 (Example 75b).

.BEGIN VERBATIM

Example 75b
            







Example 75c
.end
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.fill indent 6

	The later appearances of this material in the movement are
somewhat simpler because they both begin and end in ↓_D_↓%4F%1.

.begin verbatim

Example and Figure 76
.end
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.skip 11

.fill indent 6
	In the twentieth of Beethoven's %2Diabelli Variations%1, Op.120,
the listener is frequently kept in doubt as to whether or not the
chromaticism is functional.  The slow, regular rhythm and the constant
use of free imitation makes it difficult to specify a functional or
non-functional role for every note.  Notes such as the C%4S%1 of bar 4,
which had virtually no functional significance earlier in the work
(see Variation I, bars 4 and 6), now share equal status, timewise,
with some of the most clearly functional notes.  Another factor which
creates doubt is the occasional use of notes in the upper voices
which may have double significance as both dissonant pedal points and
true chord tones (see bars 5, 7, etc.).  The voice leading is
extremely carefully worked out -- every chromatic change has its
logical linear result.  However, the harmonic implications of the
combinations of the lines may be experienced on their own plane.  Of
course, it must never be forgotten that particular harmonic functions
never have a real existence apart from the total music which produces
them.  The alternating functions in this variation certainly grow
out of a mode of thought that considers all musical factors
simultaneously.
.next page
.begin verbatim
Example 77.  Beethoven, Diabelli Variations, Op.120
.END
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.FILL INDENT 6
	From the broadest point of view, all the altered notes in 
bars 4-8 and 17-20 may be taken as non-functional elaborations of the
dominant.  (The same is true for bars 21-24, but with a different
tonic.)  In bars 2 and 3 the bass is melodic, so there need be no
question of a %4A%1 chord.
.begin verbatim

Figure 77a
.END
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.SKIP 4
.FILL INDENT 6

	As the variation continues, we are faced with progressions
which do not respond so well to this approach.  The chords of each
dotted half-note seem to acquire much more separate significance.
With this in mind, we might make an alternative (but not necessarily
"better") analysis of the above discussed measures.  Let us consider
that A%4F%1=G%4S%1 in bar 4 and that the G%4N%1 of bars 4, 5, and 7 (and 17, 18,
and 19) is an intermittent pedal point within the d minor chord.  Now
the elaborations are shown as functional under ii (↓_d_↓), concluding
in each case with a C half-cadence.
.BEGIN VERBATIM

Figure 77b
.END
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.SKIP 7
.FILL INDENT 6

	For the sake of simplicity, the first diagram (Figure 77a)
will be used in the complete analysis.
.next page
.FILL INDENT 6
	The motive heard in bars 17-20 is continued in bars 21-24 as
an elaboration of a diminished 7th chord.  There is little basis for
settling on any one tonic, although ↓_d_↓ or ↓_f_↓ would seem to be indicated
by the notation.  For the final analysis, ↓_f_↓ will be chosen because
the tonics involved in the last part of the variation all stand in
simple relation to that tonic.

	All the subtleties of the alternating progressions at bars
9-13 and 25-28 would be difficult to express.  In the first phrase
the approach to ↓_d_↓ as a temporary tonic seems clear;  and that this
↓_d_↓ should become major and the dominant to ↓_G_↓ is quite conventional.
The unexpected appearance of the C chord at bars 11 and 12 has the
effect of prematurely wrenching away the tonic role of d.  Because
of the alien surroundings, this C chord is most immediately heard
as a sudden return to the I which might have been expected after
the clear V of bar 8.  Then the C chord works as a quite normal
pivot to a full cadence on V (↓_G_↓), which may be listed briefly as
a new control tonic at bar 15.
.BEGIN VERBATIM

Figure 77c
.END
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.next page
.FILL INDENT 6
	The second phrase including alternating progressions is more
difficult.  At measure 25 the b%4FA%1 eventually assumes three important
functions.  One of these, II of A%4F%1, does not become apparent until
bar 27, however.  The E%4F7%1 chord of measures 27 and 28 might have been 
listed only as the G6 function of ↓_g_↓, since there are no other chords
following which bear direct relationship to it.  The earlier b%4FA%1
chord does form, however, a conventional preparation for the function
of E%4F%1 as the dominant of ↓_A%4F%1_↓.  Thus, these two functions of the E%4F7%1
chord can be shown in alternation with the G6 of ↓_b%4F%1_↓.

.BEGIN VERBATIM

Figure 77d

.END
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.FILL INDENT 6

	It may be noticed that after the first few bars in this
Beethoven variation, very few %2simple%1 tonic functions (i.e., not
substitutes for tonics) are heard.  Only the tonics at bars 15-16
and at the end are decisive.  In the last 12 bars, even most of the
dominant functions are presented by substitutes.  It will be found
that most of the harmonic vagueness of later tonal music is 
attributable to a low percentage of simple tonic (or even dominant)
functions within rapidly changing temporary tonic areas.

	Despite the complications, our analysis will show the
simplicity of the largest harmonic design in this variation.  All the
alternating progressions, substitute functions, and what have you,
serve only as very special elaboration of the various temporary tonics,
which in turn contribute to the definition of the control tonics
↓_I_↓ ↓_V_↓ ↓_I_↓ ↓_iv_↓ ↓_I_↓.
.BEGIN VERBATIM

Figure 77e.  The complete variation.
.END
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.NEXT PAGE
.FILL INDENT 6
	In this chapter many new elements have been added to the
analytical diagram.  One must always be on guard against overloading
the analysis with non-essential detail.  When new, and possibly
confusing indications are added to the basic diagram, great care
should be taken that the music under consideration really warrants
their use.  Especially for the performer, the potential danger of
too much attention to detail in analysis is probably greater than
the danger of over-simplification.
.CENTER		        
_______________
.FILL
.adjust
.begin verbatim

Exercises for Chapter VI
.END
.FILL INDENT 6
	Analyze the following works.  Although a great deal of
chromaticism is involved in these excerpts, not all of them
contain alternating progressions.
.BEGIN VERBATIM

.end
.ONCE NO FILL INDENT 0 
	1.  Chopin, Mazurka #1, in f%4S%1, Op.6, #1, first 16 bars.
.BEGIN VERBATIM

	2.  Chopin, Mazurka  #6, in a,  Op.7,  #2, first 32 bars.

	3.  Chopin, Mazurka #11, in e,  Op.17, #2, first 24 bars.

	4.  Chopin, Mazurka #49, in f,  Op.68, #4, all.

	5.  Chopin, Prelude in e,  Op.28, #4, all.

.end
.ONCE NO FILL INDENT 0 
	6.  Chopin, Prelude in f%4S%1, Op.28, #8, all.
.begin verbatim

	7.  Chopin, Prelude in f,  Op.28, #18, all.
.END