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.CENTER		
CHAPTER II
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.CENTER			
DIATONIC FUNCTIONS
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.INDENT 6
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.BEGIN VERBATIM

Scales and Tonality
.END
	A particular tonality is defined by a few essential interval
relations in any succession of tones.  Paradoxically, in the music
here to be dealt with, the tonal center itself is not a note
that need figure in any of the essential intervals.  In fact, an
unheard note on the tonal center may achieve its role through a
kind of musical default, wherein all other possibilities are
ruled out.  First of all it must be realized that in tonal music
the minor mode has no separate existence, but represents merely
a fairly consistently applied group of alterations -- flattings --
of certain parts of the major mode.  These alterable parts of the
major scale are the 7th, 6th, 3rd, and even the 2nd (most often
as the root of the "Neapolitan" chord or as a non-chord auxiliary).
.begin verbatim

Example 1
.end
.SKIP 6
	The remaining notes, the 5th, 4th and 1st, can never be
altered, as functional tones,$$See page zzz regarding the exception
of the Augmented Sixth Chord.$ without causing at least a tendency
to shift tonal centers.  (It should be noted that chromatically
%2raising%1 any note of the %2major%1 scale causes some tendency to shift tonality.
See following chapters.)  However, the presence of a potential 1st,
4th and 5th may still be tonally inconclusive without
the appearance of the leading tone -- major 7th of the scale.
In minor keys especially, the lowered 7th may often be
heard, but -- in the broadest sense -- almost always as a
descending auxiliary tone.
.begin verbatim

Example 2
.end
.skip  11
	When the %4F%*7th degree appears as a chord tone and is not, in
some sense, passing downward toward the 5th, the tonal center
tends to shift.
.begin verbatim

Example 3
.end
.skip 11
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	Continuing this approach, we find that the %2tritone%1 (augmented
4th or diminished 5th) may be thought of as the tonality-defining
interval, since its presence between the 4th and 7th is
unique in relation to all the other intervals found between the
degrees of a %2major%1 scale.
.begin verbatim


Example 4
.end
.skip 10
	However, one more note must be involved so that we may be
made aware of the particular role of each part of the tritone.
Since, in a %2major key%1, one or the other of the parts of a tritone
must be the unalterable 4th of the scale, the whole step above it
%2must%1 be the equally unalterable 5th.
.begin verbatim


Example 5
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.skip 7
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	It follows that when a note is heard a half-step above either
part of a tritone, it must be the tonic.
.begin verbatim

Example 6
.end
.skip 5
	Thus, in the major mode, we have two groups of three notes,
either of which may suffice to define a tonal center -- the 1st,
4th, and 7th, or the 5th, 4th, and 7th.  It must be noted that
these intervals need not occur between %2adjacent%1 tones only.
Other less critical notes may separate these scale degrees 
within a melodic unit.  An important thing to remember is that the
tritones formed as the result of the alterations (flattings) that
create the %2minor%1 scale do not serve the same purpose of tonic
definition.  Of course, composers often take advantage of this
obvious possibility of ambiguity in order to shift tonics.  The
"altered" notes in the original key become "unaltered" notes in
the new key, etc.
.begin verbatim

Example 7
.end
.skip 6
	The next example shows how the various temporary tonics of
a somewhat chromatic line might be ascertained.  Among the notes
heard in measure 17 and the first two beats of measure 18, two
tritone relations may be found -- A%4F%1-D and F-B.

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.begin verbatim

Example 8.  Mozart, Sonata in F, K.189e (280), first movement.
.end
.skip 12
	It  is  clear  that  A%4F%1-D cannot be the critical tritone, since
the E%4N%1, a whole step above D, rules  out E%4F%1 as tonic.  The context of  the
passage makes it doubtful that A%4F%1 could be interpreted as G%4S%1, because  the
G%4N%1 is a  normal goal  for the  flatted sixth  in C  Major-minor.  No  such
problems come with the interpretation of (G)-F-B-(C) as the tonic-defining
intervals, and so C is  the tonic at that point.   It should be seen  that
there is no reason to indicate the tritone relation between notes such  as
the E and B%4F%1 of measure 18.  The E%4N%1 becomes altered to E%4F%1 and unless there
is some reason to call the E%4F%1  now D%4S%1, there is no tonic-defining  element
present, but rather a move into the minor mode.  The same line of  thought
will apply to the rest of the passage, due to its sequential nature.


	The preceding  discussion  has  been  in  terms of a succession of
tones -- a melodic  line.  The same  things prove to  be true when  dealing
with the interval relations in chord progressions, since chords, in  tonal
music, may be  thought of  as simultaneous vertical  occurrences of  scale
parts.  Every  chord  progression is  inextricably  bound up  with  linear
implications;  chords  grow   out  of  the   verticalization  of   melodic
combinations and melodies  are directed  so as  to serve  the purposes  of
particular chord progressions.  Since music never exists in a static form,
the  linear  impulse  --  the  impulse  to  move  forward  --  is   always
predominant.   However,  early   in  the  history   of  tonal  music   the
relationships of the vertical elements  became so conventional that  their
abstract manifestations were clear to all.  Strike any dominant 7th  chord
before even the  most untrained listeners  and they will  be able to  sing
back the notes  of the expected  tonic chord, but  without any  particular
regard for voice leading between the two chords.  Thus in tonal music  the
simple melodic impulse must share primacy with the impulse of the harmonic
progression.

.CENTER		        
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.FILL
.BEGIN VERBATIM

Harmonic Functions
.END

	Harmonic functions are the basic relationships between the
chords -- the relationships of the %2roots%1.  It is not inferred  that the
roots are to be heard as a kind of imagined bass line.  Nevertheless, the
roots might be called the generating tones of each chord, in that the
relationships of the roots to the tonic note usually give fair representation of
each chord's function.  Chords seem to be less stable -- that is,
they tend to move on to other chords -- when the 3rd, 5th, or 7th appears
in the bass.  In fact, the simple %4A%1 position (with the 5th in the
bass) most often has no independent existence.$$See section on substitute
functions, page 16.$  Usually the upper two
notes of such a chord act as appoggiaturas or neighboring notes to
a following chord.


.begin verbatim

Example 9
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.COMMENT skip 7 ;
	Another frequent usage of the %4A%1 chord occurs when the bass,
and perhaps other notes, move in a stepwise manner.  Aside from chords
moving in complete parallelism, this passing %4A%1 chord is the main
formation that can be called a "contrapuntal" chord rather consistently.


.begin verbatim

Example 10
.end
.skip 9
	Of course chords in other positions may be found in similar
usage, but the basic acoustical situation caused by the interval
of the 4th with the bass has apparently led composers away from a freer
use of the %4A%1 chord.
It must be remembered that there will always be some exceptions
to the general rules given here.  One prominent exception in regard to
the use of %4A%1 chords occurs when the lowest voice is serving in a
primarily melodic role.  In this sort of situation the lowest voice may
sometimes work as a kind of "misplaced" soprano, with the true bass line
being only implied.  Also it is not really rare to find a V chord  in the
%4A%1 position, but then the bass will usually move in a stepwise manner.
(See next example.)
In some rather chromatic situations it may be best to give the
%4A%1 position functional status equal to that of other positions.  This may occur
when the usual resolution of the %4A%1 chord does not appear or when 
the chord acts as a pivot to a new tonic.
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.begin verbatim
Example 11
.end
.skip 8
	In speaking of functions, the cardinal principle to remember
is that it is only the context that determines the function of each chord.
When the context remains uninfluenced by the appearance of
chromaticism,$$Since the normal usage of the minor mode includes much
chromaticism, the following discussion will generally refer to
situations in the major.$
 the functions are usually self-evident; the labels
applied by numbering the scale degrees of the roots usually give
indication of the functions.


.begin verbatim


Example 12
.end
.skip 10
	The two main problems that must be faced in diatonic
situations have to do with substitute functions and the distinction
between chord tones and consonant non-chord tones.  Once the
concept of substitute functions is understood, the differentiation
between chord and non-chord tones should offer little difficulty
(until we reach music written near the end of the tonal
era).  When doubt occurs, the simplest choice (that is, the one
closest to the most common progression) is usually the right one.
.CENTER		        
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.FILL
.BEGIN VERBATIM

 
Substitute Functions
.END
	Substitute functions occur in those situations where the notes
of one chord are found in a context that indicates the function of
another chord.  The intuitive use of substitute functions may have
grown out of a need for a kind of process of abbreviation that
could be applied to both harmonic and melodic situations.  Usually the
dominant function is involved.  The VII chord often carries a
dominant feeling and can be considered as an abbreviated V%47%1.  The small
%2s%1 following a Roman numeral gives indication of substitute
function.

.begin verbatim

Example 13
.end
.skip 8
	When the 13th of an incomplete V chord leaps directly to the
tonic a 3rd below, we may often consider it to be an abbreviation of a
three-note scale movement.$$Concerning these "frozen" accessory tones
see Roger Sessions, Harmonic Practice, pp.224-231.$

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.begin verbatim

Example 14
.end
.skip 6
	Abbreviations depend on their context for specificity and so
are somewhat more subtle than their sources.  The VII%4F7%1 chord is
almost always heard as an abbreviated V%4F9%1.  But in this abbreviated
form it can function as the dominant to any of four different tonics,
depending only on where the composer leads it.

.begin verbatim

Example 15
.end
.skip  9
	Similarly, even the II triad in the minor may sometimes be
heard as the upper three notes of an incomplete V%4F9%1.

.begin verbatim

Example 16
.end
.skip 7
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	When the diminished 7th chord appears in its 2nd inversion
(VII%4D%1) and moves to I in root position, the strong bass
movement from 4 to 1 gives the impression of a plagal cadence.  In
such a case VII seems to be substituting for IV.

.begin verbatim

Example 17
.end
.skip 8
	Likewise, the VII%4F7%1 may gain something of the subdominant
function when it moves from its 3rd inversion to the root position of
the dominant or tonic.

.begin verbatim

Example 18
.end
.skip 8
	Another important substitution for the dominant function is
the tonic %4A%1 chord.  This usually is heard as an abbreviation
of the conventional formula, I%4A%1-V.  Quite often a composer will
bring the music to the I%4A%1 chord, but head off in another
direction before taking the time for the resolution to the V.  The best
example of this comes at the beginning of the Classic concerto
cadenza.  In some rare instances the music will continue directly to the tonic
in root position or 1st inversion.  (See next example.)

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.begin verbatim
Example 19.  Beethoven, Sonata in d, Op.31,#2, first movement.
.end
.skip 11
	The case of the III chord of the major implying the V%4Z%1 or
I%47%1 presents a somewhat different problem.  In completely diatonic
situations the III chord seems "stronger" than only the VII.
Probably this is partly due to the identification of its root and 3rd
with the I chord and its 3rd and 5th with the V chord.  The existence
of this triad as a true III function (i.e., not as a substitute
for V or I) is fairly rare outside of sequential patterns with 4th or
5th related roots (see Example 12).  When the root of III, appearing
in a melodic role, moves to I, III frequently carries a
dominant function.

.begin verbatim

Example 20
.end
.next page
	The III as a substitute continuation of I is sometimes found
in the I-III-IV progression.
.begin verbatim


Examples 21a, 21b
.end
.skip 9
	In Examples 21a and 21b the functions involved are clearly
similar, since the B sounds as a passing note, en route to A, in both
cases.  However, with the same upper parts and a new bass line
(Examples 22a and 22b) the substitute character of the III tends
to disappear.  Now, due to the strength of the 5th descent in the
bass, three functions might be heard instead of two.  The  tied-over C
can be heard as a suspension with the following B as its resolution.
.begin verbatim

 
Examples 22a, 22b
.end
.skip 9
.next page     
	The very special effect of the %2ascending%1 3rd progression
may be seen as an occasional characteristic of Brahms' work.
.BEGIN VERBATIM

Example 23.  Brahms, Romance, Op.118,#5
.end
.skip 9 
	Any two-note  chord  (or  even  a  single  note)  can  be  a
substitute function whose numeral designation depends entirely on the
context.  In two-part tonal counterpoint the clearly  implied  roots
of  chords  are  very  often omitted.  When doubt does arise in such
music,  usually  the  analysis  which  shows   the   simplest,   most
conventional  progression,  or  one  which  is part of an established
pattern or sequence, is the right one.
.begin verbatim
 
Example 24
.end
.skip 6
	Quite subtle is the relationship between II, especially
II%4B%1, and IV.  It is clear that when the II in root
position moves to V its function will not be confused with that
of IV.  However, in cadential formulas the sense of II%4B%1 or II%46%1
is certainly heard as subdominant -- with the root of II being heard
usually as an anticipation, added 6th, or passing tone.
.next page
.BEGIN VERBATIM
Example 25
.end
.skip 7
	In cases as above, where the bass (and often the soprano too)
moves stepwise, the problem of harmony versus counterpoint presents one
of its most perplexing facets.  In Example 25 it is completely
possible to conceive of the entire II%4B%1 as a passing chord
between the I%46%1 and the V.  It is argued by some that all harmonies
that occur over stepwise bass movement (excepting those that come
at the high or low point of such movement) have no functional significance.
If this view is accepted, some distinction (though not a hard an fast
one) must be made between the "structural" and "contrapuntal"
chords of a progression.  In situations including chromatic alteration
the problem is often simplified, since in most cases the alterations
serve to lead our attention to the main chords.
 
 
.CENTER		        
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.FILL
 
 
	The following examples serve to illustrate that the context
is the chief consideration in designating the relative importance of
chords.
.begin verbatim
 
Example 26.  Mozart, Sonata in A, K.300i (331)
.end
.skip 10
 
	Heard alone, the above is a simple ↓_D_↓: IV-V-I progression;
but heard in its continuing context (Example 27) the I chord
retrospectively becomes a pseudo %4A%1 chord.  Because of the use of the
A pedal at the starred (*) points and the fact that each of the two bar
groups presents the A chord on the strong beats, the F%4S%1 and D of the
D chord, even when moved to the bass, are heard as auxiliaries to E and
C%4S%1 of the A chord.
.begin verbatim
 
Example 27
.end
.skip 13
 
	Despite the two-part presentation, the essential movement of the
following passage from Bach's Partita #5 is clearly heard as a
descending series of chords; VI-V-IV-III-II.  Needless to
say, the progression found in the details (VI-II | V%47%1-I | IV-VII |
III%47%1-VI | II) does exist.  But the context causes us to hear the
alternate chords (II,I,VII,VI) as subsidiary to the other
chords.  The fact that VI-V-IV-III-II hardly constitutes an
exciting harmonic progression does not alter our hearing of this
passage.  It must be understood that, in the largest sense, this passage
gives us a movement from VI to II.  In relative terms, the main
harmonies in between are passing chords, each elaborated by chords
at the 4th above.  However, the %2particular%1 effect of the
example depends on the functions of all the harmonic details -- VI-II
V%47%1-I -- etc.
.NEXT PAGE
Example 28.  Bach, Partita #5, Praeambulum
.skip 14
 
	Undoubtedly the internal structure of this passage may be
heard on many different levels.  The number of levels may vary but,
generally, the use of three in an analysis will suffice to give a clear
picture of the way in which a piece can be heard.  The primary level of this
sort of analysis will indicate the key of the particular area of the
piece under consideration.  The secondary level will show the
larger harmonic motions within the area, and the tertiary level will give
virtually all the harmonic details.  The middle level will offer the
greatest problems, since it is listening at this level (i.e., not
concentrating on every detail and, at the same time, maintaining
a harmonic awareness of more than just the end points) that the most
variety occurs.  In returning to the example from the Partita we see that
the first level may be considered for now as ↓_C_↓: I.  Because of the clear
stepwise descent of the bass from first beat to first beat of each bar,
which is concluded only on the low D, we can say that the movement on
the second level is simply VI-II (or alternatively, because of the
patterns of the upper part, the passage may be heard in two-bar groups
 -- VI-I, IV-VI, II).  The third level offers -- without
going into the details of minute chord inversions -- the analysis given
under the example.  Putting the three levels together we have:
.begin verbatim
 
Figure 28
.end
.skip 7
 
.BEGIN VERBATIM
	1.  The key is established (although the I chord 
		occupies no prominent position).
	2.  The main movement within the key is from 
		VI to II (or the alternative given above).
	3.  The details of harmonic elaboration are given.
.end
.CENTER		        
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.FILL
.skip 1
 
	For a somewhat more complicated example we turn to a passage from
Schubert's %2Piano Sonata in B%4F%1 (fourth movement, measure 86).
.begin verbatim
 
Example 29
.end
.skip 17
	Here the first level gives us ↓_F_↓: I.  In cases where the
second level becomes quite subtle, it is often advisable to bypass it
and return later from the third level.  The third level offers no more
problems than did the Bach example.  Here the progression is:
.begin verbatim
 
Figure 29a
.end
.skip 3
	Although this passage certainly does not sound "weak", all
the direct progressions in the first four bars are by 2nds and 3rds.  If
we apply the concept of substitute functions to the 3rd relations,
the situation becomes more clear.  Thus III may serve for I and VI
may serve for IV.  The II%4A%1 is non-functional, since its upper
two notes resolve stepwise to the final VI.  This may be indicated:
.begin verbatim
 
Figure 29b
.end
.skip 3
	However, the descending figure of two notes in the soprano in
each of the last four bars draws one's attention to the last note
%2and%1 chord in each bar.  The first beat in each of these bars may be
considered a kind of harmonized appoggiatura, even though the bass does
not always move in a stepwise manner.  Now:
.begin verbatim
 
Figure 29c
.end
.skip 3
becomes our second level.  The three levels together form:
.NEXT PAGE      
 
Figure 29d
.skip 7 
	In the foregoing discussion, the word "levels" is used in
full realization that its use in this connection could become confused with
its use in the overlapping and somewhat similar realm of pure harmonic
functions.  Decisions concerning the broader structural levels
heard in a piece will generally be much less specific than those
concerning the levels of harmonic function.  In the former
category, the totality of a piece must be dealt with; in the latter,
one must deal primarily with an abstraction of the relations between
the vertical occurrences of a piece -- i.e., harmony.  It is
believed that the development of consistent terminology for the discussion
of harmonic functions must supercede the total analysis of
music.  In the following chapters, "levels" will be
used with regard to harmonic functions.
.CENTER		        
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.FILL
 
 
.next page
.begin verbatim
 
Exercises for Chapter II
 
I.	Write out the following passages.  Show each tritone
	relationship and indicate the half or whole steps on 
	either side of the tritone.  From this, ascertain the 
	various temporary tonics involved.  Follow the form 
	used in Example 8 on page 12.
 
	From the Mozart Sonatas:
 
.end
.ONCE NO FILL INDENT 0 
	1.  Sonata in B%4F%1, K.189f(281), first movement, bar 48 to
.begin verbatim
			first note of bar 54 (both hands).
 
	2.  Sonata in a, K.300d(310), last movement, bars 52-59
		(right hand only).
 
	3.  Phantasie (nr.4) in c, K.475, six bars, starting
		from 12 before Andantino (both hands).
 
	4.  Sonata in C, K.545, first movement, bars 29-42 
		(right hand only).
 
II.	Analyze the following diatonic passages.  Show all 
	substitute functions and "passing" or "contrapuntal" 
	chords.
 
	From the 371 Chorales of Bach:
 
	1.  Chorale 1, first 10 bars
	2.  Chorale 5, first 6 bars
	3.  Chorale 14, first 6 bars
	4.  Chorale 125, first 4 bars
	5.  Chorale 209, last 5 bars
 
	From the Mozart Sonatas:
 
	6.  Sonata in G, K.189h(283), first movement, bars 1-10
	7.  Sonata in G, K.189h(283), second movement, bars 1-4
	8.  Sonata in C, K.545, third movement, last 10 bars
	9.  Sonata in D, K.284c(311), second movement, bars 1-11
.end