In the above diagram, the abrupt skip down to a third
level of tonics (which resolves to a I on the second level)
accurately parallels the effect of the example.  Notice the problems 
created by the use of a string of diminished chords over a basically
chromatic line.  Such situations may be analyzed as
substitutes for the "circle of fifths" progression, but generally it
is best to consider all but the first and last chords of the string
as parallel-moving passing chords (see Bach, Partita #1 in Bb, Gigue;
Mozart, Fantasy in C, K.394).  However, in 19th century music,
details of voice leading (especially the melodic skips) often tend
to give functional importance to the inner chords of strings of
diminished chords (see Chopin, Etude in E, Op.10,#3, measure 38,
etc.; Liszt, Transcendental Etude #6, "Mazeppa", beginning).

	Most often the return to the main tonic will not be made by 
the same route as was the departure.  When this happens, a pivotal
tonic may appear in the middle area of the analysis (see below at
the sign @).

Example 58






Figure 58






Example 59. Mozart, Sonata in D, K.576, 2nd movement.





	The exact location of the pivotal tonic will usually depend on
the association of the particular means of presenting a given harmonic
function with earlier presentation of the same function.  In other words,
the return to the chord of departure, even with the same function on the
lowest level, can be inconclusive until we also return to the same
melodic and rhythmic presentation of that function.  In Example 53 the f#
chord at (*) has a tonic function, but f# as a tonic of the middle area
is still subsidiary to another tonic (D) until the return of the 
original material (see Example 59).

Figure 59











	The middle area of the first analysis is preferred at (@),
because the tonic chord of f# is completely consistent with the scale
of D, and there are no other factors that might deter one from
hearing the passage as a stepwise progression of tonics (I, ii, iii).
If, for any reason, it seems undesirable to show a pivot in cases where
two or more levels of tonics are involved, the wavy vertical line ( )
might be used to show the lack of continuity along any given level
(see Figure 64).  However, it rarely happens that a logical connection
between tonics cannot be made.


Control Tonics

	An extension of the previous line of thought will serve well
when considering the following analysis of music typical of the work
of Bach.

Example 60.  Bach, Invention No.4 in d.



Figure 60.  Bach, Invention No.4 in d (bars 7-38).







	The form of this work is defined mainly by a group of
cadential formulas.  The strength of the points of definition is 
greatly enhanced by the fact that each cadence offers us a new tonal
center.  However, due to the exclusive use of a single set of motives
and the brevity of the piece, modulation (as it is known in the main
body of tonal music) does not occur.  The larger harmonic movement
of this piece is certainly heard as a series of tonics, whose relations
to the main, beginning tonic are at least as strong as their
relations to one another.  Thus, even though a appears first as iii of
F, it is finally established as an equal of F (in relation to d)
by the occurrence of the well-prepared cadence.  Please note, however, 
that a's role as something more than a transient tonic of the F
scale is not too definite until the point of cadence.  To emphasize these
varying levels of relationships among the tonics, we must start at the
beginning of our diagram with two levels in the middle ground, so 
that when a temporary tonic is somehow given greater status and 
becomes what may be called a control tonic this change will
appear in a consistent form.  In the above analysis note that on the 
highest level of tonicization the progress from i to III to v coincides
exactly with the full cadences.  However, F and a are first tonicized
well before the cadences (this shows on the second level of
tonicization).  The numerals of the control tonics will be
underlined.

	The use of strong cadences is not the only method which can
give a temporary tonic the status of a control tonic.  Another favorite
method is the clear presentation, in some new tonic, of
material (either old or new) of relatively great importance; e.g., in
the Bach Invention, the return to the original d is defined by the 
return of the opening statement, rather than by a full cadence.  The
presentation should be long enough so that the new tonic may be heard
as independent beyond the extent of simple tonicization.  The 
functions under a control tonic established in this manner need not
include a strong cadence.  The music might then progress smoothly to
yet another control tonic.  If the important factors of the music seem
to warrant its independence, a new control tonic might even appear as
a lower level of tonicization, below an old control tonic (see next
chapter, Schubert example).

	Much music can best be analyzed in terms of series of control 
tonics, the term "modulation" being reserved for a shift of the
basic tonic.  No set rule can be given as to when it is advisable to
use an extra, higher level of indications in the middle area of the
analytical diagram.  Any variant of the basic diagram is acceptable
just so long as it contributes to the understanding of the music under
consideration.

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

	It is very evident that as the complexity of harmony increases
there is usually a similar increase in the ambiguity of individual
harmonic functions as they relate to the whole.  This is, of course,
the reason for the particular effect of late "Romantic" and "Impressionistic"
harmony, and also the source of the ultimate breakdown of functional
tonality as a meaningful system.  When every harmonic function becomes
ambiguous, then something other than a tonal center (as defined by
harmonic functions) must take over as the prime cohesive force (see last
chapter).  It is significant that in music of the Classical era the 
more complex harmonic progressions are most often relegated to
developmental passages -- passages that depend greatly on motivic
continuity.  In such instances, it is common to hear a series of tonics
which seem to relate only to their immediate surroundings.  (It is
possible for even the original tonic of a developmental section to
reappear for a considerable time without its having any direct
functional relationship with its earlier presentation.)  The
analysis of an extended developmental passage will show continually
added levels in the middle ground, the return to a single level being
conditioned by the reappearance of familiar, stable material in its
proper key.  This traditionally happens at the point of recapitulation
in the sonata form.*

***************************************************************
*  *Among the numerous exceptions to the above statements are the
* first movement recapitulations of Mozart's Sonatas in D, K.284c(311)
*  and in C, K.545.  In the first of these, the recapitulation begins with
*  the material of the second key area, but now heard in D instead of
*  A.  The recapitulation of the C sonata offers the original
*  opening material, but in the subdominant, F.
********************************************************************

Exercises for Chapter V


Analyze the following works:

From the 371 Chorales of Bach:

	1.  Chorale 21
	2.  Chorale 92
	3.  Chorale 200
	4.  Chorale 231
	5.  Chorale 279

From the Two-Part Inventions of Bach:

	6.  Invention 11, in g
	7.  Invention 12, in A
	8.  Invention 13, in a

From the Well-Tempered Clavier, First Book, Bach:

        9.  Fugue I, in C
       10.  Fugue X, in e

From the Preludes, Op.28, of Chopin:

       11.  Prelude 5, in D
       13.  Prelude 22, in g