. DEVICE XGP    
.spacing 10*5 mills;
.EVERY HEADING(,{PAGE},)
.AREA TEXT LINES 4 TO 40
.FONT 1 "METL"
.font 2 "METLI"
.font 3 "METS"
.font 4 "MUS[HHA,LCS]"
.PORTION MAIN;
.PLACE TEXT;
.COMPACT
.PAGE←39
.NEXT PAGE
.<< Put in a footnote. >>
.
.COUNT FOOTNOTE INLINE FROM 1 TO 999 IN PAGE PRINTING ⊂"*****"[1 TO FOOTNOTE]⊃
.<<	(IF THISDEVICE = "XGP" THEN "%51%*" ELSE "[1]");>>
.
.FOOTSEP ← "__________";
.AT "$$" ENTRY "$"
.	⊂
.	NEXT FOOTNOTE;
.	FOOTNOTE!;
.	SEND FOOT
.		⊂
.		BEGIN "NEXT FOOTNOTE"
.		SELECT 1;
.		SINGLE SPACE
.		SPACING 0 MILLS
.		INDENT 0,0,0;
.		(FOOTNOTE! & " ");
ENTRY
.		END "NEXT FOOTNOTE";
.		⊃;
.	⊃;
.
.TURN ON "%↓_↑↓[&]"
.SKIP 2
.CENTER		
CHAPTER IV
.SKIP 2
.CENTER			
FUNCTIONS OF THE MINOR
.SKIP 1
.INDENT 6
.FILL
.ADJUST
.SELECT 1
.BEGIN VERBATIM

Major-minor Interchangeability
.END
 
 
	As stated earlier, minor scales will be treated as existing
as the result of a fairly consistent set of alterations of the
parallel major scales (see page 8 and following).  While minor
scales may not have originated in this way, it is probable that
their use in tonal music may best be approached from this point
of view.  Two alterations suffice to give the complete impression
of minor harmony -- the flatted third and the flatted sixth.  In
some cases, even the sixth need not be altered.  However, if this
scale degree is lingered upon, the minor effect will be weakened
unless it has been flatted.  Conversely, the %4F%16 may be found in
proximity to the %4N%13 without completely destroying the major
quality.
 

	When the third or sixth of a scale is flatted and appears as
the third of a chord, there is most often no change in the function
of the chord.  The flatted seventh of the scale %2can%1 change the
function of the V chord, especially if it is used in some
ascending manner.  It is rare that a real V%4F3%1 chord is heard in a
position of structural importance.  When it exists, the V%4F3%1 usually
loses its "dominant" function, unless the composer has made a special
effort to impress upon the listener that the "leading tone" is
to be a whole step below the tonic note.  However, it is only by
examining the whole of a phrase that one can ascertain whether a
chord containing the flatted seventh of the scale is functioning
exclusively in the original minor key or rather dually in that key
and the relative major, or some other closely related key.  It
should be mentioned that the "closeness" of any two keys to one
another depends largely on just how much their scales have in
common, or sometimes %2potentially%1 have in common.  (By this is
meant, the conventional ascending and descending forms of minor scales
must %2both%1 be considered.)  Thus ↓_C_↓ is a little "closer" to ↓_G_↓ than
it is to ↓_a_↓, because only one note (the 4th) of the ↓_C_↓ scale
must be altered to produce the ↓_G_↓ scale, but two notes (the 4th and 5th)
must be altered to produce the ascending form of the ↓_a_↓ scale.
 
.begin verbatim

Example 43
.end
.skip 4
 
 
	On the other hand, ↓_C_↓ is somewhat "closer" to ↓_a_↓ than a is to
↓_e_↓ or ↓_E_↓.  After an ↓_a_↓ tonic is established, ↓_C_↓ may be tonicized merely
by means of elements already found in the conventional
descending form of the ↓_a_↓ scale; whereas the ↓_e_↓ or ↓_E_↓ scale requires
at least D%4S%1 for its establishment.
 
.begin verbatim

Example 44
.end
.skip 5
 
 
	In music where there is much major-minor alteration, almost
any keys may be directly related.  As to the problem of "close" or
"far", the absolute number of chromatic changes required to establish
a new key will not be so important as the question of just what
role the altered notes play in each of the scales involved.  The
problem of the larger relationships between keys will be dealt with
in subsequent chapters.
 
 
.CENTER		        
_______________
.FILL
.adjust
	When we truly have a group of functions based on the natural
minor scale (i.e., third, sixth, and seventh degrees flatted both ascending
and descending), we may say that the harmony formed therefrom
is %2modal%1.  Following is an example of modal and tonal treatments
of the same melodic fragment.
 
.begin verbatim
 
Examples 45a,b.  Bach, Chorale 28
.end
.skip 9
 
	The analysis of the first phrase is as follows:
 
.begin verbatim

Figure 45a
.end
.skip 5
	The %4N%13 in the third chord must be indicated, since it is a
deviation from the norm.  The %4S%13 is shown in the second bar, since
the V now becomes a true tonal dominant.  (The second chord may be
analyzed either as II or IV; in any case, the function is
subdominant.) 
The last phrase gives:$$Note that the auxiliary tone E 
(instead of E%4S%1)weakens the tonic effect of the f%4S%1 chord.  
It seems to refer back to the modal V of the first phrase.$
.begin verbatim
 
Figure 45b1
.end
.skip 5
 
	We hear much more harmonic activity in this phrase, so our
diagram becomes more complicated.  It might be argued that the
last eighth note of bar 1 does not form a new function.  There is in
the alto, however, a very distinctive leap of a 4th across the
bar line which tends to make us hear the B as a chord note.  A
broader interpretation of this passage, based on the stepwise
movement of the outer voices at this point, would lead us
to the omission of both the last eighth of bar 1 and the first
eighth of bar 2 as structural chords.  Thus:
 
.begin verbatim
 
Figure 45b2
.end
.skip 6
 
 
	Either of these two analyses of the last phrase is acceptable,
the choice depending on how much detail is desired.  The important
thing is that the details be comprehended.
 
	Many examples of modality may be found in 19th-century music,
especially in the works of Chopin (see Mazurka #15, Op.24, #2,
second section, Lydian mode; Mazurka #26, Op.41, #1, Phrygian
mode.)  These self-conscious exoticisms might be called "modal
tonality" (as opposed to major-minor tonality) and are most often presented
as clear deviations from the norm.  The functions of the various
modally-formed chords are no different, but these chords can serve as
direct pivots to tonal areas which might be more distant from an ordinary
major-minor tonality.
.SKIP 1
.CENTER		        
_______________
.FILL
.adjust
.NEXT PAGE
	It will be seen that any degree of either a major or minor
scale may be tonicized.  Usually the new tonics will be major or
minor, depending on the normal forms of the various triads in the
original tonic.  When the original key is major, II, III and VI
will most often be tonicized as minor keys, while IV and V will
be major.  VII will rarely be tonicized, since it presents a
diminished chord in the home key.  (The chord built on the
leading tone of C major is B, D, F%4N%1.)  On the other
hand, VII frequently appears as a tonic goal when the first key
is minor.  (The chord built on the seventh tone of the c minor
scale is B%4F%1, D, F.)  With a minor scale we have III, VI,
and VII forming major tonics, and IV and V minor tonics.  Since the
chord on the second degree of the minor scale is normally diminished,
II now becomes rare as a tonic, whereas the major tonic on %4F%1II,
the Phrygian second, is often used.  (See Beethoven, %2Sonata #23 in f%1,
Op.57, opening bars; %2Sonata #29 in B%4F%1, Op.106, third movement.)
In actual practice, especially in the music of the 19th century, the
major and minor elements are freely interchangeable.

 
	%2From this point on, the distinction between major and
minor tonics in the upper two levels of an analysis will be indicated
by the use of upper and lower case Roman numerals respectively%1.  The
functions %2within%1 the tonics (the lowest level of an analysis) will
continue to be shown by upper case numerals.  Non-functional alterations,
such as the "tierce de Picardie" (I%4S3%1 in the minor), may be shown in
the usual fashion.

 
	However, when a temporary tonic appears as major after the
preceding music leads one to expect a minor tonic, certain problems
are sometimes raised.  Often such chords are experienced as dominant
functions to chords more closely related to the scale of the
original key, as in bar 8 following (see also Chapter V).
 
.NEXT PAGE
.BEGIN VERBATIM
Example 46.  Schubert, Meeres Stille, Op.3, #2
.END
.SKIP 14
 
.BEGIN VERBATIM

Figure 46
.END
.SKIP 9
 
 
 
	On the next page is a complete table of indications for the possible
temporary tonics that might be found in %2direct%1 relation to a basic
tonic of ↓_C_↓ major.  Add the appropriate accidentals when dealing
with other basic tonics.
 
.NEXT PAGE
.BEGIN VERBATIM
Figure 47
.END
.SKIP 12
 
 
	Group 1 lists the tonic chords available from the resources
of the unaltered ↓_C_↓ scale.  The second group includes tonic chords
that are available as the result of flatting any of the original
scale degrees except the 1st, 4th and 5th; hence, ↓_c_↓ minor.  (The
%4F%1vii, ↓_b_↓%4F%1, as a tonic is rare, but see the opening of Mozart's
%2Quartet in C%1, K.465.)  Group 3 is set apart because it includes the
less common temporary tonics, all of which involve either flatting or
sharping of the 1st, 4th or 5th of the original scale in order to
produce the new tonic chord.  These chords, even when surely approached
as tonics, often prove to be working as dominant or subdominant from the
largest point of view.$$This type of progression will be treated somewhat
differently in the section on added levels of tonicization in Chapter V.$
.BEGIN VERBATIM

Example 48
.end
.skip 6
.next page
.begin verbatim
Figure 48
.end
.skip 6
 
	It will be noted that there is %2no%1 temporary tonic listed
in Figure 47 that has as its %2root%1 a sharping of any part of the original
tonic major scale or a flatting of the 1st, 4th or 5th degrees.
It is very difficult to make a case for the existence of such a
direct relationship of tonics.  (See Chapter VI, Alternating
Progressions.)
 
.CENTER		        
_______________
.BEGIN VERBATIM


Augmented Sixth Chords
.END
.FILL
.INDENT 6
.adjust
	It is now clear that chromatic alteration in a chord very often
changes its function.  This will most often be true when the
alteration involves the 1st, 4th, 5th or 7th degree of the scale,
or when it occurs within a phrase rather than at its end.  As stated
before, every judgment must be based on the whole of any particular
context under consideration.
 

	The alterations that go into making chords of the augmented
sixth (A6) give us a special case.  Such chords evolved as separate
entities from the practice of "freezing" the chromatic passing tone
between the 4th and 5th of the scale.  When these chords are used in
their conventional manner and within a single tonality, they give
us an exception to the rule that the 4th of a scale, as a chord tone,
may not be altered without causing a shift of tonics.  The
traditional names for the three most common augmented sixth chords
are as follows:
.next page
.BEGIN VERBATIM
Example 49
.END
.SKIP 5
 
 
 
	Strictly speaking, they should be indicated thus for a major key such
as ↓_C_↓:
.SKIP 5
It will generally
suffice to refer to them all in the abbreviated manner, A6.  When
they actually are used as augmented sixth chords, they almost always have
primarily contrapuntal significance, the outer voices expanding by
half step movement.
.BEGIN VERBATIM
 
Example 50
.END
.SKIP 5
 
 
	However, since they are so often used as a means of tonicizing
new areas (see Example 33), it is usually advisable to make their
presence clear in an analysis.  The augmented sixth interval creates a
situation wherein the dominant tends to be tonicized (Example 50, F%4S%1
leads to G) and at the same time a tonic role for the dominant
is weakened by the effect of presenting its 2nd as flatted (Ab to G).
The actual result is that added strength is given to the %2dominant role%1
of the final "resolution" of the A6 chord, since the flatted note is
most easily taken as the %4F%16 of the original tonic.  Thus, augmented
sixth chords will be said to function as part of a tonic a major 3rd
%2above%1 the %2lower%1 member of the augmented sixth interval (see Example zzz).
 

	Because the G6 chord presents the identical acoustical
situation as an ordinary dominant 7th chord (i.e., major 3rd,
minor 3rd, minor 3rd), its enharmonic form will often be used as a
means to move rapidly to a relatively distant tonic.
.begin verbatim

Example 51
.end
.skip 5
 
 
 
.begin verbatim
 
Figure 51
.end
.skip 5
 
 
	Even without recourse to enharmonic change, the F6 chord may work
as the V%4G%1 structure of the original key's dominant
(see Example zzz.)  One notable use of the V%4G%1 occurs at
the end of Schubert's %2Quintet in C%1, Op.163.
 
.begin verbatim

Example 52
.end
.skip 7
 
 
 
	In this particular case it might be inferred (perhaps
fancifully) that this chord is used to point in the subdominant
direction, a tradition of long standing for endings.  Thus:
 
.skip 2
.CENTER			
might imply
.INDENT 6
.FILL  
.adjust
.skip 3
 
 
	For a stronger example, this time employing the G6, we turn to
Schubert's posthumous %2Sonata in A%1.  Here, there is little possibility
of confusion with an altered V, since there is no root of V present.
 
.begin verbatim

Example 53.  Schubert, Sonata in A
.end
.skip 7 
 
 
	Of the two following possible analyses, the second is better.
In the first, we assume that since the F is omitted at (*), an
E, though not present, might be implied.  (Even so, this would give the
sound of the subdominant's F6.)  In the second, we more
reasonably assume that the F persists in the memory and that the context
does not lead us to interpret the chord B%4F%1-D-F-G%4S%1 as an inversion
of ↓_A_↓: VII%4G%1.
.BEGIN VERBATIM

Figure 53
.END
.NEXT PAGE
 
	It should be pointed out again that no chord may be evaluated
until all its parts have been heard.  However, no set rules can be
given for this; as usual, the context must be the determining factor.
In the Schubert %2Sonata%1, the B%4F%1 chord fills a measure and a quarter
before the appearance of the G%4S%1.  Thus the relatively simple progression
of I to %4F%1II has time to be accepted before the function is altered by
the addition of a fourth pitch.  Such particulars are certainly open to
varying interpretations.  In both of these Schubert pieces, such ambiguous
procedures are possible because of the extremely firm establishment of
the main tonic in the preceding passages.

 
	A case with some similarity is presented in the %2Prelude #23
in F%1 of Chopin, where an E%4F%1 is found in the final arpeggiated
chord.  This E%4F%1 is usually heard merely as an intensification of
the already present 7th partial of the bass note's overtone series.
However, it is undeniable that some tendency toward the subdominant,
a tonal area which is almost completely neglected in the piece, is
heard.  At the end of a piece devoted to almost nothing more than
figuration on I and V, this E%4F%1 comes as a welcome bit of
fantasy.
.begin verbatim

Example 54.  Chopin, Prelude, Op.28, #4
.end
.skip 7 
.next page
	And then there is the standard "Blues" progression:
 
.begin verbatim

Figure 51
.end
.skip 7
 
 
	Many variants of the main A6 chords may be found in later
tonal music.  But once they recognized for what they are, they should
cause no difficulties in an analysis.  Similarly, there are many
chromatic variants of the dominant, subdominant (the A6 is one!), and
other functions, except for the I function.$$But remember that 
there is always the %2potential%1 of a change of
function when any chromaticism is introduced.$  Various notes may be
%2added%1 to tonic function, and the root may even be omitted, but if
the root or 5th are altered, the tonic function becomes dubious.

 
	Chromaticism has always caused composers trouble with
notation.  The augmented sixth and diminished seventh chords in
particular have raised problems, because of the ambiguity of
the situations in which they are often found.  Composers cannot be
totally consistent in notating such chords when enharmonic
equivalence is involved.  It will always be necessary to determine
the function or functions of chromatic chords by studying their
surroundings rather than by taking their notation at face value.
 
.CENTER		        
_______________
.FILL
.next page
.begin verbatim

Exercises for Chapter IV

.end
	Analyze the following works.  Show the distinctions between
major and minor tonics through the use of upper and lower case
numerals on the two highest levels of the analyses.
.begin verbatim

 
From the 371 Chorales of Bach:*

	1.  Chorale  11, up to the 3/4.
	2.  Chorale  37, all.
	3.  Chorale  47
	4.  Chorale  94
	5.  Chorale  223
 

From the Sonatas of Mozart:
 
	6.  Sonata in D, K.205b (284), last movement, Variation VII.
 
	7.  Sonata in a, K.300d (310), first movement, 24 bars
		starting at recapitulation.  (In cases where there are
		many suspensions and appoggiaturas it is usually best
		to write out a chordal reduction of the passage before
		working out the analysis.)
 
	8.  Sonata in C, K.300h (330), second movement, first 20 bars.
 

From the Preludes, Op.28, of Chopin:
 
        9.  Prelude  1, in C
       10.  Prelude  6, in b
       11.  Prelude 20, in c

_____________
* Note that all bass lines in the Bach Chorales are probably
intended to be doubled at the octave below.  Hence, even if the
notation of the bass part crosses above the tenor part, the bass 
part remains the true bass.
.end
.NEXT PAGE