00100		Following is the well-known introduction to the second movement
00200	of Dvorak's New World Symphony.  The notation here differs
00300	enharmonically from the original.
00400	
00500	Example 67
00600	
00700	
00800	
00900	
01000	Figure 67
01100	
01200	
01300	
01400		Even though the movement proper is in Db, the key at the 
01500	beginning, as listed on the highest level, is E.  This represents a
01600	continuation of the e of the first movement.  The tritone relation 
01700	between two major chords (E and Bb) is usually heard as V-bII or
01800	vice versa, the former being used in this case because of the positions
01900	of the two chords,  V5-3 bII6, and because of the subsequent harmonic
02000	developments.  By the time we reach the third measure we are sure that
02100	the E chord was working as a dominant to A.
02200	
02300	Example 68a
02400	
02500	
02600	
02700	
02800	
02900		However, the Db chord which broke the continuity is most simply
03000	heard as dominant to A's sixth degree, f#.  It must be noted that the 
03100	absence of emphasis in the first two bars makes immediate recognition of
03200	the functions of all but the first tonic E very difficult.  Until more
03300	is heard any chord may be dominant, subdominant, or tonic, etc.  When
03400	the phrase is complete -- and thus develops points of emphasis -- these
03500	distantly related chords are retrospectively assigned functions in terms
03600	of the end results.  The f# chord serves as VI to A (Example 68a);
03700	f# is tonicized by the progression of the fourth to the sixth chord
03800	(Example 68b); and the f# chord is IVbb3 (gb) to Db (Example 68c).
03900	A has been tonicized by the first three chords, but the A chord also 
04000	serves as bbVI (Bbb) to Db (Example 68c).
04100	
04200	Example 68b
04300	
04400	
04500	
04600	Example 68c
04700	
04800	
04900	
05000		The later appearances of this material in the movement are
05100	somewhat simpler because they both begin and end in Db.
05200	
05300	Example 69
05400	
05500	
05600	
05700	Figure 69
05800	
05900	
06000	
06100	
06200		In the twentieth of Beethoven's Diabelli Variations, Op.120,
06300	the listener is frequently kept in doubt as to whether or not the
06400	chromaticism is functional.  The slow, regular rhythm and the constant
06500	use of free imitation makes it difficult to specify a functional or
06600	non-functional role for every note.  Notes such as the C# of bar 4,
06700	which had virtually no functional significance earlier in the work
06800	(see Variation I, bars 4 and 6), now share equal status, timewise,
06900	with some of the most clearly functional notes.  Another factor which
07000	creates doubt is the occasional use of notes in the upper voices
07100	which may have double significance as both dissonant pedal points and
07200	true chord tones (see bars 5, 7, etc.).  The voice leading is
07300	extremely carefully worked out -- every chromatic change has its
07400	logical linear result. However, the harmonic implications of the
07500	combinations of the lines may be experienced on their own plane.  Of
07600	course, it must never be forgotten that particular harmonic functions
07700	never have a real existence apart from the total music which produces
07800	them.  The alternating functions in this variation certainly grow
07900	out of a mode of thought that considers all musical factors
08000	simultaneously.
08100	
08200	Example 70.  Beethoven, Diabelli Variations, Op.120
08300	
08400	
08500	
08600	
08700	
08800		From the broadest point of view, all the altered notes in 
08900	bars 4-8 and 17-20 may be taken as non-functional elaborations of the
09000	dominant.  (The same is true for bars 21-24, but with a different
09100	tonic.)  In bars 2 and 3 the bass is melodic, so there need be no
09200	question of a 6-4 chord.
09300	
09400	Figure 70a
09500	
09600	
09700	
09800		As the variation continues, we are faced with progressions
09900	which do not respond so well to this approach.  The chords of each
10000	dotted half-note seem to acquire much more separate significance.
10100	With this in mind, we might make an alternative (but not necessarily
10200	"better") analysis of the above discussed measures.  Let us consider
10300	that Ab=G# in bar 4 and that the Gn of bars 4, 5, and 7 (and 17, 18,
10400	and 19) is an intermittent pedal point within the d minor chord.  Now
10500	the elaborations are shown as functional under ii (d), concluding
10600	in each case with a C half-cadence.
10700	
10800	Figure 70b
10900	
11000	
11100	
11200	
11300		For the sake of simplicity, the first diagram (Figure 70a)
11400	will be used in the complete analysis.
11500	
11600		The motive heard in bars 17-20 is continued in bars 21-24 as
11700	an elaboration of a diminished 7th chord.  There is little basis for
11800	settling on any one tonic, although d or f would seem to be indicated
11900	by the notation.  For the final analysis, f will be chosen because
12000	the tonics involved in the last part of the variation all stand in
12100	simple relation to that tonic.
12200	
12300		All the subtleties of the alternating progressions at bars
12400	9-13 and 25-28 would be difficult to express.  In the first phrase
12500	the approach to d as a temporary tonic seems clear;  and that this
12600	d should become major and the dominant to G is quite conventional.
12700	The unexpected appearance of the C chord at bars 11 and 12 has the
12800	effect of prematurely wrenching away the tonic role of d.  Because
12900	of the alien surroundings, this C chord is most immediately heard
13000	as a sudden return to the I which might have been expected after
13100	the clear V of bar 8.  Then the C chord works as a quite normal
13200	pivot to a full cadence on V (G), which may be listed briefly as
13300	a new control tonic at bar 15.
13400	
13500	Figure 70c
13600	
13700	
13800	
13900	
14000		The second phrase including alternating progressions is more
14100	difficult.  At measure 25 the bb6-4 eventually assumes three important
14200	functions.  One of these, II of Ab, does not become apparent until
14300	bar 27, however.  The Eb7 chord of measures 27 and 28 might have been 
14400	listed only as the G6 function of g, since there are no other chords
14500	following which bear direct relationship to it.  The earlier bb6-4
14600	chord does form, however, a conventional preparation for the function
14700	of Eb as the dominant of Ab.  Thus, these two functions of the Eb7
14800	chord can be shown in alternation with the G6 of bb.
14900	
15000	Figure 70d
15100	
15200	
15300	
15400	
15500	
15600		It may be noticed that after the first few bars in this
15700	Beethoven variation, very few simple tonic functions (i.e., not
15800	substitutes for tonics) are heard.  Only the tonics at bars 15-16
15900	and at the end are decisive.  In the last 12 bars, even most of the
16000	dominant functions are presented by substitutes.  It will be found
16100	that most of the harmonic vagueness of later tonal music is 
16200	attributable to a low percentage of simple tonic (or even dominant)
16300	functions within rapidly changing temporary tonic areas.
16400	
16500		Despite the complications, our analysis will show the
16600	simplicity of the largest harmonic design in this variation.  All the
16700	alternating progressions, substitute functions, and what have you,
16800	serve only as very special elaboration of the various temporary tonics,
16900	which in turn contribute to the definition of the control tonics
17000	I V I iv I.
17100	
17200	Figure 70e.  The complete variation.
17300	
17400	
17500	
17600	
17700	
17800	
17900	
18000	
18100		In this chapter many new elements have been added to the
18200	analytical diagram.  One must always be on guard against overloading
18300	the analysis with non-essential detail.  When new, and possibly
18400	confusing indications are added to the basic diagram, great care
18500	should be taken that the music under consideration really warrants
18600	their use.  Especially for the performer, the potential danger of
18700	too much attention to detail in analysis is probably greater than
18800	the danger of over-simplification.
18900	
19000	      ---------------------
19100	
19200	Exercises for Chapter VI
19300	
19400		Analyze the following works.  Although a great deal of
19500	chromaticism is involved in these excerpts, not all of them
19600	contain alternating progressions.
19700	
19800	1.  Chopin, Mazurka  1, in f#, Op.6, #1, first 16 bars.
19900	2.     "       "     6, in a,  Op.7,  #2, first 32 bars.
20000	3.     "       "    11, in e,  Op.17, #2, first 24 bars.
20100	4.     "       "    49, in f,  Op.68, #4, all.
20200	5.     "    Prelude in e,  Op.28, #4, all.
20300	6.     "       "    in f#, Op.28, #8, all.
20400	7.     "       "    in f,  Op.28, #18, all.
20500	8.     "       "    in g,  Op.28, #22, all.