CHAPTER II

		DIATONIC FUNCTIONS

Scales and Tonality

	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 appled 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.)

		Example 1









	The remaining notes, the 5th, 4th and 1st, can never be
altered, as functional tones,* without causing at least a tendency
to shift tonal centers.  (It should be noted that chromatically
raising any note of the major 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.

		Example 2





						Auxiliary tones will
						often be marked X.








	When the b7th degree appears as a chord tone and is not, in
some sense, passing downward toward the 5th, the tonal center
tends to shift.

		Example 3













	Continuing this approach, we find that the tritone (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 major scale.

		Example 4















	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 major key, one or the other of the parts of a tritone
must be the unalterable 4th of the scale, the whole step above it
must be the equally unalterable 5th.

		Example 5










	It follows that when a note is heard a half-step above either
part of a tritone, it must be the tonic.

		Example 6








	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 adjacent 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 minor 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 dey become "unaltered" notes in
the new key, etc.

		Example 7












	The next example shows how the various temporary tonics of
a some what chromatic line might be ascertained.  Among the notes
heard in measure 17 and first two beats of measure 18, two
tritone relations may be found -- Ab-D and F-B.

  Example 8.  Mozart, Sonata in F, K.189e (280), first movement.





















	It  is  clear  that the Ab-D cannot be the critical tritone, since
the E!, a whole step above D, rules  out Eb as tonic.  The context of  the
passage makes it doubtful that Ab could be interpreted as G#, because  the
G! 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 Bb of measure 18.  The E! becomes altered to Eb and unless there
is some reason to call the Eb  now D#, 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 tru 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.

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

Harmonic Functions


	Harmonic functions are the basic relationships between the
chords -- the relationships of the roots.  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 their
relationships 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 6-4 position (with the 5th in the
bass) most often has no independent existence.*  Usually the upper two
notes of such a chord act as appoggiaturas or neighboring notes to
a following chord.


Example 7

					Note that this is the opposite
					situation from that wherein a
					sustained bass pedal point may
					have little or no influence on
					the details of harmonic function.





	Another frequent usage of the 6-4 chord occurs when the bass,
and perhaps other notes, move in a stepwise manner.  Aside from chords
moving in complete parallelism, this passing 6-4 chord is the main
formation that can be called a "contrapuntal" chord rather consistently.


Example 8


					In a detaled analysis, the
					numerals for chords that are felt
					to be passing or non-functional
					should appear in parentheses.






	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 to the 6-4 chord.
	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*, the functions are usually self-evident; the labels
applied by numbering the scale degrees of the roots usually give
indication of the functions.


Example 9











	The two main problems that must be faced in diatonic
situations have to do with substitute functions and the distinction
between chord toes 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.

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

Substitute Functions

	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 V7.  The small
s following a Roman numeral gives indication of substitute
function.

Example 10
				In a very detailed analysis, the designation
				of a chord according to its makeup should
				appear in parentheses below the indication
				of its function.




	When the 13th of an incomplete V chord leaps directly to the
tonic a 3rd below, we may often consider it to be an abbrevation of a
three-note scale movement.*


Example 11










	Abbreveations depend on their context for specificity and so
are somewhat more subtle than their sources.  The VII b7 chord is
almost always heard as an abbreviated V b9.  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.

Example 12










	Similarly, even the II triad in the minor may sometimes be
heard as the upper three notes of an incomplete V b9.

Example 13













	When the diminished 7th chord appears in its 2nd inversion
(VII 4-b3) 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.

Example 14













	Likewise, the VII b7 may gain something of the subdominant
function when it moves from its 3rd inversion to the root position of
the dominant or tonic.

Example 15













	Another important substitution for the dominant function is
the tonic 6-4 chord.  This usually is heard as an abbreviation
of the conventional formula I 6-4, V.  Quite often a composer will
bring the music to the I 4-6 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 rare instances, the music continues directly to the tonic
in root position or 1st inversion.

Example 16











	The case of the III chord of the major implying the V13 or
I7 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 9).  When the root of III, appearing
in a melodic role, move to I, III frequently carries a
dominant function.

Example 17