Martijn Hooning ABOUT HARMONIC ANALYSIS

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Martijn Hooning
ABOUT HARMONIC ANALYSIS
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degrees and functions; main degrees and secondary degrees
tonic, dominant and subdominant / main degrees
secondary degrees
closes, (final) cadences
embellishing tones (passing tone, neighbouring tone, suspension,
anticipation)
harmonic rhythm
motifs, bar groups, and harmonic structure
hemiola and syncopation
structural and non-structural harmony / faux-bourdon / pedal point
sequences
modulation, secondary functions / degrees
examples of: harmonic reduction, outer-voice reduction
2009, February / March
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51 ?
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This text is mainly about harmony and harmonic analysis - though other aspects of analysis are
mentioned too, along the line. It is possible (probable even) that you are informed about some topics
in this text. I am mentioning these things mainly for the sake of some completeness.
Since around 1600, in most compositions chords are used: triads and seventh chords. In other
words: the harmony plays an important role. In a harmonic analysis observations can be made about
the relations between chords, and about their place in the key(s). It may also be important to look
for relations between the harmony and other aspects of a composition, like the melody or melodies,
the phrases, the form, and the meter and rhythm.
Some tones in tonal music are part of the melody, but not of the harmony; these tones are called
embellishment tones.1 And in most tonal music the harmony is layered, meaning that not all chords
are equally 'important': we can distuingish between structural and non-structural chords.2
When making harmonic analyses you could use several techniques, such as:
- analysis of the degrees: the harmony of a composition is described using Roman numerals for the
scale degrees;3 the inversions of the chords can be added. Instead of Roman numerals sometines
chord symbols are used.4
- harmonic reduction: the score is 're-written' in such a way, that only, or mainly, the harmony is
visible. Of course, this technique may be combined with an analysis of the scale degrees.5
- harmonic/melodic reduction: usually the score is 're-written' in such a way, that only, or mainly,
the (most important) tones of the bass and the upper voice are shown. An analysis of the scale
degrees may be added.
When making, or reading and understanding, an harmonic analysis it is necessary to be aware of the
meaning of the chords, mentioned in the analysis. Sometimes the meaning of a chord, or a group of
chords, is clear through description with a Roman (scale degree) numeral(s), sometimes it makes
sense, or turns out to be necessary, to add further explanation or interpretation to a scale degrees
analysis (or to a reduction).
degrees and functions; main degrees and secondary degrees
Every tone of a minor or major key has its own character, and therefore plays specific roles in a
piece of music (and doesn't play certain other roles). A quite obvious example is the leading tone in
the major and harmonic minor keys: usually it resolves to the root of the key. Another - maybe less
obvious - example is the sixth in the major and (harmonic) minor keys: this tone often leads down,
to the fifth of the key. In tonal music, such 'roles' are determined by the audible relation with the
root of the key: the presence of this root creates the possibility for a certain tone to play specific
roles.
For triads and seventh chords applies the same, more or less. For instance: a chord that contains the
leading tone, logically is followed by a chord in which the root of the key plays an important role.
The role of a chord often can be described even more precisely than the role of a single tone: the
combination of three or four tones (and therefore: the combination of several 'roles') clarifies the
status of the chord they are forming.
1 Further explanation: see the chapter embellishing tones (passing tone, neighbouring tone, suspension, anticipation
2 Further explanation: see the chapter structural and non-structural harmony / faux-bourdon / pedal point. We can
also speak of structural and non-structural harmony.
3 For an explanation see the section degrees and functions; main degrees and secondary degrees below.
4 The use of chord symbols is common practice when describing Jazz music, for instance. Chord symbols are also
sometimes practical in the description of music of the twentieth century.
5 Some examples of harmonic reduction you can find in my text “An example of sonata-rondo: Beethoven, sonata in
c minor op. 13 (Pathetique), third movement...” (from page 4).
3
On every tone of the major and minor keys a chord (triad or a seventh chord) can be placed.6 Such a
chord on a tone of a key is called: degree. There are seven degrees, as many as there are tones in the
key. Degrees can be 'labelled', using Roman numerals. For example: III means: the triad on the third
tone of the key, IV7 means: the seventh chord on the fourth tone, I6 means the triad on the first
tone, in first inversion.
Degrees differ of course, because they are on different tones of the key (and thus consist of different
tones), but they differ above all because of their different sound:
1. in a major key the triads on I, IV and V are major ; the triads on II, III and VI are minor; the
only diminished triad is on VII.
2. in a minor key the triads on I and IV are minor; the triad on V normally is major - because
of the use of the (artificial) leading tone in minor keys7 In the so-called natural minor key
the triad on V is minor. On III8 and VI stand major triads, and on II and VII usually
diminished triads. The use of the leading tone is causing the diminished triad on VII: in the
natural minor key the triad on VII is major.
The minor key is more complex than the major key: in fact a minor key consists of nine tones, as
(often) the seventh and (less often) the sixth tone may be raised:9
1
2
3
4
5
6
6#
7
7#
Therefore, in the minor key in principle more different chords are possible than in the major key.
A schematic overview of the degrees (triads) in the major and (harmonic) minor keys:
(Maj = major min =
minor dim = diminished)
major: [VI] [VII] I
II III IV V
VI VII
[min] [dim] Maj min min Maj Maj min dim
minor: I
min
As a music example:
II
dim
III IV V VI VII [I] [II]
Maj min Maj Maj dim [
min [dim]
]
example 1
In the above schedule is visible that the chords in a major and its relative minor key almost
completely correspond: the minor key is a minor third below its relative major key, and consists
basically of the same tones. For example, an E minor triad can be used as II in D major, but also as
6 Chords might alos be placed on some raised or lowerd tones. I am disregarding these possiblities here.
7 The so-called harmonic minor scale
8 In a minor key, when the leading tone is used, III appears as an augmented triad. III as an augmented triad is much
less common than III as a major triad - and for that rason only the major triad is mentioned in the schedule.
9 It is common practice to speak of harmonic minor (minor scale with raised 7), and of melodic minor (minor scale,
using raised 6 and 7 when going up, and using 'natural' 6 and 7 when going down). In practice we observe 'mixed
use' of these scales though. Therefore they can be considered as merely theoretical 'constructions'.
4
IV in B minor; a D major triad can be used as I in D major, but also as III in B minor, etc. As
normally the leading tone is used in V and VII in a minor key, these degrees differ from III and V in
the relative major key (in example 1: A# ). The degrees of a major key correspond exactly with the
degrees of its 'natural' minor relative:
(Maj = major min =
minor dim =
diminished)
majeur:
[VI] [VII] I
II
[min] [dim] Maj min
natuurlijk mineur: I
min
example 2
As a music example:
II
dim
III
IV
Maj min
III
min
IV
Maj
V
VI VII
Maj min dim
V
min
VI
Maj
VII [I]
[II]
Maj [min] [dim]
Degrees can also be used in the form of seventh chords. Below is a list of the most common seventh
chords in major and (harmonic) minor keys:
(Maj = major min = minor
dim = diminished
dom = dominant seventh
chord
hfdim= half diminished)
As a music example:
majeur:
[VI7] [VII7] I7
[min] [hfdim] Maj
harm. mineur: I7
min
II7
III7
hfdim Maj
II7 III7 IV7 V7 VI7
min min Maj dom min
VII7
hfdim
IV7 V7 VI7 VII7 [I7]
[II7]
min dom Maj dim [min] [hfdim]
example 3
The term function is used to describe which role(s) a chord plays in a piece of music. We
distinguish three functions: tonic, dominant and subdominant (abbreviated: T, S and D). Each
chord in a piece of music has one of these three functions. Some chords always have the same
function, others may have different functions, and can therefore play different roles, depending on
the context.
5
The seven degrees of the major and minor keys can be divided in main degrees and secondary
degrees. Main degrees are functionally clearer than secondary degrees. The main degrees are: I, IV
en V ; the function of I is: tonic ( I can be considered as the 'central chord' in a tonal piece, because
it stands on the root of the key); the function of IV is subdominant, and the function of V is:
dominant.
The role of the fifth in the harmony of tonal music probably is the main reason that I, IV and V can
be described (and heard) as main degrees:
example 4 in G major
- The fifth can be considered as the 'simplest' interval (after the
10
octave, but an octave is hardly heard is an interval). When two
chords with roots a fifth apart are connected, this connection is
perceived as 'strong'. (We could also say: as 'easy
understandable': every other connection of two chords is more
complex.) I and V are a fifth apart, as are IV an I:
- A diatonic key (a key, consisting of five whole tones and two semitones) basically is a row of
perfect fiths.11The major and the 'natural' minor keys both are diatonic keys. In other words: major
and minor keys consist of a part of the circle of fifth (namely half of it, plus one tone):
example 5
It is therefore logical that chords on tones just one fifth apart from the tonic (in both directions) have
special importance. Chords on tones that are further away from the tonic (measured in fifths) have
weaker relations with the tonic, which results in less functional clarity. For instance: a triad on D in
the key of A minor is just one fifth apart from the tonic, and has clearly dominant function; a triad
on F on the other hand is four fifth apart from the tonic, and therefore functionally much less clear.
In the next chapter I try to specify the meaning of the terms tonic, subdominant and dominant.
10 After the octave, the fifth is the most consonant interval. This is visible in the structure of the overtone series.
11 Next to major and minor keys, the so-called “church modes” (or better, and shorter: “modes”) are diatonic as well.
6
tonic, dominant and subdominant / main degrees
The functions: tonic and dominant are relatively easy understandable. The tonic can be regarded as
resting point, or starting point. The first degree in particular is clear: the vast majority of tonal
music ends on I, most pieces also start on I. In other places in a composition a conclusion of a
phrase on I mostly sounds as a resting point as well
The dominant tends to 'lead to' this resting point. Thereby the following points are important:
1. A chord with dominant function always contains a leading tone (= the seventh of the scale, a minor
second below the root). The presence of the leading tone can be regarded as 'minimum condition' to
which a chord must meet to be able to act as a dominant. The leading tone has a melodic function: it
leads into the root of the key, in the same voice. (in other words: it resolves to the root:: 7^ —> 1^).
In minor keys, in all dominant chords the (artificial) leading tone must be used, as the seventh of the
'natural minor' scale is a major second below the root, and therefore not useable as leading tone. See
example 6a.
2. The main degree V contains the leading tone (= the third of V ), and moreover the fifth of the key.
This fifth has a tendency to fall a fifth down (or jump a fourth up) to the root of the key. This
tendency can be called: the harmonic aspect of the dominant function.
example 6
d.
So, in the triad on V, the melodic function of the leading tone
is combined with the harmonic function of the dominant; see
examples 6c. en 6d.
The relation V – I can be understood as: the dominat creates
tension, which is put to rest in the tonic.
The subdominant is less clear than the tonic or the dominant (several theorists even deny the
existence of the subdominant ...). Since Rameau12 introduced the term “subdominant”, a wide range
of explanations have been given. I more or less follow the explanation of Schoenberg::13
Two chords standing a fifth apart are strongly related, because they have a common tone (through
the distance of a fifth). This applies to V and I, but also to VI and II, and so on. Schoenberg calls
this the 'harmonic tie'14 between two chords (see the dotted lines in example 6d). Logically, every
triad is harmonically connected with two other triads in this way: with the triads a fifth above, and a
fifth below. For instance: I is harmonically 'tied to' V (the fifth of I is the same tone as the root of
V), and to IV (the root of I is the same tone as the fifth of IV).
The relations between I and IV, and between V and I are similar: I could be considered as a 'kind of
dominant' to IV (see example 4): I may 'lead to' IV, as V is leading to I - because of the distance of
the fifth, the 'harmonic tie'. The function of IV, and of the subdominant in general, could then be
described as: 'leading away from the tonic, moving away from the centre' (centrifugal).The
dominant is 'moving to the centre' (and therefore can be called: centripetal). It is then logical to
assume that after I – IV (moving away from the tonic) V – I should follow (recomfirming the tonic)
12 Rameau: Nouveau Système de Musique Théorique (1726)
13 Arnold Schoenberg: Harmonielehre (1911)
14 German: harmonischer Band
7
When we look at these basic
harmonic relations in this way, the
relation between IV and V actually
is almost a 'non-relation': these
chords have no common tone (no
harmonic 'tie'), as they do not stand
a fifth apart. In minor keys there is
a melodic relation though: the third
of IV (6^ of the key) normally
leads to 5^ - see example 7.
example 7
It is also possible to describe the function 'subdominant' in a different way:
Especially in the Classical era, the harmony at the beginning of pieces, or at the beginning of
sections of a form, often consists of only tonic and dominant chords. We then have the impression
that the harmony does not really move yet, and that some harmonic development is needed, before
the music can aim at some kind of harmonic 'conclusion' (a final cadence, for instance). The
subdominant often appears after moving forth and back from tonic to dominant for some time (in
themes the first subdominant often appears at the beginning of the consequent). The subdominant
then can be seen as starting point of a harmonic 'conclusion'. In a way this is logical: as soon as the
subdominant appears, all three fuctions have been used, and the music then will move towards a
cadence.
This phenomenon is clearly visable in the first theme of Mozart's piano sonata in G major (see
example 35): in the first four bars only tonic and dominant harmonies are used. The subdominant
appears in bar 5, and from here the music is aiming at the full cadence in bar 8 (or bar 10). In the
first theme of Mozart's piano sonata in A minor (see example 12) we observe a similar procedure: in
the first four bars only I and V7 are used (on top of a tonic pedal point); the consequent (from bar 5)
starts from a tonic harmony, and in bar 6 the harmony is 'put into motion' by IV.
secondary degrees
The relation betweeen the three main degrees can be understood from the distance of the fifth
(between IV and I, and I and V); the relation between the main degrees and the secondary degrees
can be understood from the distance of the third. Triads with roots a third apart have two common
tones; therefore their sound is somewhat similar (though not all triads a third apart are sounding
equally 'similar'). Probably this is the reason that chords with roots a third apart are functionally
related. The three main degrees each are functionally related to two secondary degrees, standing a
third higher and a third lower. IV for example is functinally related to II, and to VI.
As visable in example 8, there are a few ‘overlaps’ (which is logical: three main degrees relate to
six secondary degrees – but a key contains only seven degrees!): III and VI are a third apart from
two main degrees with different functions. In major keys both chords are functionally unclear - the
8
context has to clarify their function. The same applies to VI in minor keys - but not to III: when no
leading tone is used in this chord, it has tonic function; when the leading tone is used, it
automatically has dominant function .
In minor keys, when talking about harmony, we normally can assume the harmonic minor key: in
tonal music the seventh of the key often is raised (leading tone). You should realize though that
scales like 'natural', 'melodic' and 'harmonic' minor are basically artefacts: in most tonal music they
are mixed. The choice to use the leading tone (and/or the raised sixth tone) nearly always depends
on melodic direction: the ‘high’ 6^ and 7^ ‘cause’ ascending motion, the ‘low’ 6^ and 7^ ‘cause’
descending motion. Else than V and VII, III in minor keys seldom contains the leading tone:15
example 8
I will not give detailed description of the relations between secondary degrees and main degrees
here. Just a few observations:
- Secondary degree mostly are ‘weaker representatives’ of a funtion than the main degree with the
same function: they stand further apart from the root of the key (measured in fifths).
III and VI are the 'weakest' degrees - the function of these degrees is not clear beforehand (but: III in
minor keys is functionally clear, see example 8!)
- VII is rather clearly a dominant: the chord does not contain the fifth of the key (this fifth
characterizes V, and the relation between V and I), but: the leading tone in VII makes that it leads to
a tonic chord. In other words: VII is lacking the harmonic dominant function, but the melodic
function ( 7^ —> 1^ ) makes clear enough where VII is 'aiming at'.
- II is very clearly a subdominant. The relation between I and II is less clear than between I and IV
(as they are not a fifth apart); but the relation between II and V is more compelling than IV – V: II
and V are a fifth apart (see example 9). This distance between II and V is without any doubt an
important reasons that II is used very often instead of IV:
example 9
II i instead of IV, before the dominant
- Secondary degrees often are used as substitutes of main degrees. A secondary degree may also be
used after the main degree with the same function, causing the function of the main degree to last
longer; in this case we can speak of prolongation16 of a function. See example 10.
15 So, III in minor normally has tonic function, and seldom dominant function.
16 This term is used also for other 'extension techniques'. Compare page 30: voice leading chord within a prolongation
of a function.
9
- The funcions of VI and of III depend of the context. In the chord progression V - VI, VI takes the
place of I (because we expect I after V - substitutution ); VI in this situation is a (weak) tonic. In
the chord progression I - VI - V – I , the function of VI is: subdominant (it takes the place of IV or
II). In the chord progression I - III - IV – V, III is a tonic, because it is prolonging I (and thats why,
in minor keys, in this progression the use of the leading tone in III is not possible..) See again
example 10.
- It makes some sense that the 'row of thirds' in example 8 starts on II and ends on VII. It is a nice
idea that the tonic is standing in the middle (and it belongs in the centre, of course..) More important
is that II en VII, though they are a third apart, are hardly connected - in spite of the two common
tones. II is related to IV (and: a third apart from IV), and always a subdominant; VII is related to V
(and: a third apart from V), stands on the leading tone, and therefore always has dominant function.
This functional difference rules out any direct relation between II and VII.
example 10 prolongation and substitution, using secondary degrees
- We can deduce from example 8 (or example 5) that secondary degrees - like the main degrees - are a
fifth apart from each other. Relations like V - I have equivalents among the secondary degrees:
we could argue (simplifying matters a little) that the progression V - I can be 'imitated' by secondary
degrees.17 For example: between VI and II, or III and VI a dominant-tonic-relation can be suggested
- as it were on a lower level: the second chord is no 'real' tonic, and the first chord no real dominant
(often there is no leading tone in this chord). Because of the distance of a fifth the progression is
nevertheless similar to V – I . In such cases we could speak of ‘figurative' or 'improper'
dominants. As the major and minor keys may be described as a row of fifths (see example 2), every
degree can act as ‘quasi-dominant’ of another degree. Eventually, all degrees can be put in a row of
fifths:
I
IV
III
II
VII
VI
V
I
This order of all scale degrees is very common in tonal music. The last three chords then act as
final, complete cadence :18
17 Maybe even I – IV is an 'imitation' of V – I ?
18 Complete cadence: see page 14.
10
example 11
all degrees, in a row of fifths
(diatonic sequence with descending fifths)
This chord succession sometimes is called a sequence with (diatonic) dominants; it is more
correct to call it a diatonic sequence with descending fifths: 'descending fifths' because of the
distances between the chords, 'sequence' because a pattern is constantly repeated. Sometimes it is
simply called: row of (descending) fifths.We normally tend to hear the chords in this sequence 'two
by two', and then (often through the melody) perceive a sequential pattern with descending
seconds.19
In diatonic sequences with descending fifths in minor keys the leading tone is only used in the V,
preceding the final tonic: VII en III contain no leading tone: they use the 'low' seventh of the key.
We could speak of 'natural' VII and 'natural' III here, or adopt the common Dutch (and German)
practice and call them eolian (abbreviated:VIIeol , IIIeol).
Example 11 also shows that II takes a special place among the secondary degrees: it is, as the 'last
fifth before V' , particularly suitable as substitute for IV in a final cadence.
Some of the above mentioned points are visable in example 12. The first four bars of this theme
move forth and back between tonic and dominant (on top of a pedal point 20 in the bass). In fact the
harmony starts to move not earlier than in bar 5, the first bar of the consequent. Bars 5-9 contain a
complete sequence with falling fifths (see the harmonic analysis in the example). From II6 in bar 8
we can speak of a final cadence - within the sequence with falling fifths..21 The leading tone (G#) is
only used at the end of the sequence, in V7:
example 12 Mozart, sonate in A minor, K 310, beginning of the first movement
19 Because: two descending fifth result in a descend of a second. Bars 5-9 of Mozart's Piano Sonata in A minor are a
good example of this procedure (see example 12): we hear sequences per bar in the melody, falling a second. The
underlying chirds form a sequence with descending fifths.
20 See the chapter structural and non-structural harmony / faux-bourdon / pedal point
21 See page 12.
11
Closes, (final) cadences
Cadences (or: closes) are an important means to structure music: they mark for example the end of
musical phrase,s or of sections of a musical form.
Below a schematic overview of the different types of closes/cadences. I am using two different sets
of terms, referring to different phenomena (though there is some overlap):
The following terms describe in which way we reach a final tonic:
1. authentic cadence: an ending, in which the final tonic is preceded by a dominant (mostly: V).
see example 13a.
2. plagal cadence: an ending, in which the final tonic is preceded by a subdominant (mostly: IV
or II). see example 13b.
The following terms describe on which function we are ending:
1. full cadence: an ending on I, the tonic. Normally this final I is in root position. We speak of a
perfect full cadence when the the bass and the highest voice both end on the root of the key
(= the root of I ). When the highest voice ends on the third or the fifth of the key, the cadence
sounds less 'stable'. We then speak of an imperfect full cadence. See example 13c.
2. semicadence (or: half cadence): a 'temporary' ending on the dominant, normally V ; the usual
place of a semicadence is halfway a musical phrase (typically: halfway a period). As a rule, V
in a semicadence is in root position, but endings on 6-, 6/5-, 4/3- of 2-positions are also
possible. See examples 13c en 13d.
3. deceptive cadence: in an ending, the final chord after the dominant is not I (though we are
expecting I ), but another degree.22 Common procedure: the leading tone (in the dominant
chord) is resolving to the root of the key, though this root is part of a 'wrong chord' (often VI,
less often IV6, and sometimes still another chord). A deceptive cadence often is used to achieve
prolongation of a musical phrase: after the decepetive cadence a new attempt has to be made to
reach the final tonic. See examples 13e en 13f.
example 13
cadences
22 Sometimes the German term Trugschluß (trügen=to deceive) is used. In French (and Spanish, Portugese etc.) the
deceptive cadence cadence is called a broken cadence: cadence rompue, cadenca rota etc.
12
Deceptive cadences can cause prolongation of a phrase, because a 'proper ending' is avoided.
Plagal cadences often are prolongation in itself, because they are following after a full cadence. See
example 14. This use of a plagal cadence is, for example, frequently found in the 'Amen' at the end
of a section of a mass:
example 14
plagal cadence as prolongation
An example of an Amen, linked to a plagal cadence - the example is from 1869:23
example 15
Bruckner, final bars of the Ave
Maria From the motets)
Three centuries earlier (1567), in a composition by Palestrina:
23 See also the beginning of the second movement of Beethoven's Third Symphony (example 19), bar 8.
13
example 16
Palestrina, Missia Papae
Marcelli, final bars of the
Kyrie
In fact, the term cadence is used in different ways; apart from describing certain types of endings
(like described above: full cadence, deceptive cadence etc.), it is also used in a more general way, to
indicate that some endpoint or cesura is reached. And it is also used to describe certain groups of
chords.
- Cadence, in the most literal sense of the word means: a closure (on a certain scale degree - often the
tonic is meant; the harmony 'falls to' the tonic, cadere=to fall).24 The word 'cadence' can be used in
sentences like: "After the modulation we end with a cadence in G major”. In this context, we could
also speak of a 'cadential point', to indicate the precise point where the cadence is completed (in a
closure V – I this point is the I ).25We can speak of a complete cadence when an ending contains all
three functions, for example: IV V I of II V I. See example 17:
example 17 cadence
- Another meaning of 'cadence' is: a series of chords, 'defining' a key. Mostly we need at least three
chords for that purpose: combination of for example the C major and F major triads can be
interpreted in different ways: for example as V – I in F major, or as I – IV in C major. It is quite
obvious that many different combinations are possible – sone combinations though are of particular
interest, as they often occur, and therefore can be regarded as a kind of 'fixed formulas':
24 The cadences in example 13 could also be called 'closures' in this sense – only the deceptive cadence is a bit
problematic, because an actual ending is avoided through of a deceptive cadence.
25 I don't believe cadential point really is part of existing English terminology though..
14
1. the above mentioned complete cadence: this cadence is often shown with a tonic at the
beginning (which is not always the case in ´real music´...), and then consists of the functions
tonic-subdominant-dominant-tonic, for example expressed through the degrees I – IV – V – I
or I – II – V – I .
2. the so-called extended cadence: I – VI – IV – II – V – I . Here, we can see VI as
prolongation of I , and II as prolongation of IV.
1. the deceptive cadence, and what is happening after it: after a deceptive cadence normally
follows either an authentic, or a plagal cadence; for example: I – S – V – VI – S – D – T , or
I – S – V – VI – S – T
example 18
'default'
cadences
Abbreviations:
T=tonic,
S=subdominant,
D=dominant
15
embellishing tones (passing tone, neighbouring tone, suspension, anticipation)
In most compositions, not all tones are part of the harmony. Next to the chords used in a piece,
mostly horizonral lines play an important role. Those lines hardly ever are moving in the same pace
as the underlying harmony (mostly they move faster, sometimes slower). Imagine for example a
stepwise moving melody above a single chord, or a melody over a slower moving chord succession:
example 19
Beethoven, Third
Symphony,
beginning of the
second movement
The first three bars of the above example contain just a single harmony: the tonic triad, consisting of
the tones C, Eb and G. The tones B, D and F in the first bar are not part of the harmony (as they are
not part of the tonic triad). The same applies, in bar 5, for the tones Eb and G in the first violin and
for C and Eb in the double bass: the underlying harmony, VII7, consists of B, D, F and Ab. Such
tones are called: embellishing tones (or: ornamental tones). We are looking then from the
harmony, and note that some tones in a (melodic) line are part of the harmony, and some are not.
Often the melody is the voice that is moving faster than the underlying harmony; sometimes another
voice (a middle voice, or the bass) is moving faster. So: embellishing tones can occur in all voices.
There are four types of embellishing tones: passing tone, neighbouring tone, suspension and
anticipation. And within these four types we can distinguish between different forms. Below a
schematic overview; at the left examples with only two voices, at the right similar examples with
four voices.
16
- passing tones stand on weak beats, and connect different tones; they are not part of the underlying
harmony, whereas the tones before and after are part of the harmony:
example 20 passing tone
When a passing tone causes a
dissonant, we can speak of a
dissonant passing tone; when a
passing tone is not causing a
dissonant we call it consonant
passing tone.
p = passing tone
diss. = dissonant
cons. = consonant
Sometimes two notes form a kind of
'passing tone group': see the first
bars of both examples: -->
Sometimes a passing tone is
standing on a (relatively) strong
beat, giving the impression that it is rel. strong p. = relatively strong
coming 'too late' (in the second bars passing tone
of the examples the passing tone
could stand on the second half of the
first beat as well).In such situations
we can speak of relatively strong
passing tones.26
- Neighbouring tones also stand on weak beats; in principle they connect two identical tones; they
are not part of the underlying harmony, whereas the tones before and after are part of the harmony:
example 21 neighbouring tone
We can distinguish between upper
neighbouring tones and lower
neighbouring tones (or: upper
neighbour / lower neighbour).
Like passing tones, neighbouring
tones may cause a dissonant - which
means that we can speak of
dissonant and consonant
neighbouring tones (neighbours):
n = neighbouring tone
A neighbouring tone may be
incomplete: it may be reached by
leap - in fact it is then reached from
another tone of the chord; or it
might be left by leap - in fact it is
then leaping to another tone of the
chord.27
26 This is certainly not a standard English term though...
27 In Dutch (and German) these neighbours are described as 'freely entering' and 'escaping' respectively. There is no
appropriate English description for this distinction.
17
- Like passing and neighbouring tones, anticipations stand on weak beats. Anticipations are tones
that 'appear too early'; they are not part of the underlying harmony, but part of the harmony on the
following strong beat. Anticipations mainly occur in closures, where then almost always the root of
the key is antocipated, in the upper voice.28 The anticipation appears then on top of a dominant
harmony, just before the final tonic:
example 22 anticipation
Anticipations always cause a
dissonant:
a = anticipation
A small example, the final bar of the Air from Bach's Second
French Suite. The penultimate tone in the upper voice (C) is an
anticipation, forming a dissonant with the bass (the interval
between bass and upper voice is a dissonantn fourth). On the
third beat the two voices form a consonant again (octave):
- Different from all other embellishing tones, suspensions stand on strong beats; they are 'delaying' a
tone of a chord29 Suspensions are mostly dissonant, and resolve into a consonant on a weak beat.
We distuingish two types of suspensions:
–
prepared suspension: a consonant tone on a weak beat is reapeated, or tied to, a tone on the
next strong beat; on the strong beat this tone is forming a (dissonant) suspension;
–
unprepared suspension: the suspension is reached by step or leap - there is no tone repetion (or
tie) from the preceding weak beat:
example 23 vertraging30
Prepared suspension: imagine the
'procedure' as:
preparation
consonant
suspension
dissonant
resolution
consonant
s = suspension
Unprepared suspension:
28 But: compare example 32!
29 In a way suspensions are exactly the opposite of anticipations: a suspension causes that a tone (which is part of a
chord) is appearing 'too late'; an anticipation is a tone that is appearing 'too early' itself.
30 Compare example 32.
18
Sometimes a suspension is left by leap. In that case it is in fact not resolving – though probably
another voice takes over the resolution. In example 32, bar 10, Bb in the upper voice should resolve
into Ab, but is leaping up to Eb instead. The same happens in bar 14 (also in the upper voice).
Sometimes a suspension is not really dissonant, though we perceive a dissonant - because it is not
part of the underlying harmony, though not forming a dissonant with this harmony. This is called:
consonant suspension. As there is no factual dissonance, we can speak of
“Auffassungsdissonanz”:31
example 24
Schubert,
Piano Sonata
in A major
D64,
beginning of
the second
movement
I believe we hear the harmony in the first and fifth bar of this movement as I, with consonant - suspensions 6-5 in the soprano (the fifth of I is suspended by the sixth). But
objectively, on the first beat of the first bar stands VI in first inversion...
With the term syncopated dissonance we describe the situation that the preparation of a
suspension is tied to the suspension, while the other voices move:
example 25
syncopated dissonance
This type of suspension is already very common in music of the Renaissance, usually in closures.
There we always see the following pattern:
preparation
consonant
suspension
dissonant
resolution
imperfect
consonant
close
perfect consonant
(mostly octave)
In the following example of Josquin des Prez three of these 'patterns' are indicated:
31 Meaning: 'dissonance by perception' or 'dissonance by interpretation'. There is no adequate English translation for
this German term.
19
example 26
Josquin des Prez, Missa
da Pacem, Crucifixus
p = passing tone
n = neighbouring tone
s = suspension
20
harmonic rhythm
The number of chords per bar, and the exact placement of these chords in the meter, form the
harmonic rhythm. The speed of the harmonic rhythm at a distinct moment in a piece of course is
interdependent with other aspects of a composition, like the structure of the phrase, the musical
tension at that moment, and so on.
In the first theme of the Mozart's piano sonata in G major (see example 35), the harmonic rhythm
in bars 1-7 is per bar: all these bars contain just a single harmony. 32 From bar 8, the harmonic
rhythm is acellerating: every two beats show a change of chord.33 Such accelerations cause increase
of harmonic tension. In most four-part chorales the harmonic rhythm is relatively simple: the
melody and the harmony move at the same, or almost the same, pace. Only when for instance a
suspension and its resolution is taking two beats, or when other embellishing tones are used in the
soprano, the harmonic rhythm is moving slower than the melody. Faster notes in other voices
likewise often are embellishing tones. Some examples:
example 27
Bach:
beginnings of
chorales Es
is Genug and
Nun danket
alle Gott
In most instrumental music, and a quite a lot of non-chorale vocal music, we encounter more
complex patterns than in the above chorales: usually melodies move faster than the underlying
harmonic rhythm: normally multiple tones of a melody are combined with one chord; melodies
often are (partly) consisting of tones of the underlying chord(s) (whether or not in combination
with passing tones, suspensions, etc.). Especially in fast movements melodies tend to move much
faster than the underlying harmony. See for example the beginning of Schumann's Rhenish
Symphony: example 28 shows the melody, the chords, and the harmonic rhythm; the second system
contains a harmonic reduction of the same bars:
32 The situation in bar 7 actually is no more than a change of inversion: V4/3 becomes V6
33 This accelerating harmonic rhythm is connected with the hemiola in these bars. See page 29.
21
example 28 Schumann, Rhenish Symfony, beginning of the first movement,
melody and harmony./harmonic rhythm. And a harmonic reduction:
The same fragment, complete score:
22
Example 29 shows the harmonic rhythm of the first 13 bars of the Graduale of Bruckner:
example 29 Bruckner, Graduale Christus factus est pro nobis (from the motets) Rhythm of the melody and harmonic
rhythm
The complete score of the beginning of this piece:
23
motifs, bar groups, and harmonic structure
Closes, cadences and harmonic rhythm hardly can be considered seperated from the role they play
in musical structures: logically (for example), semicadences usually stand at the end of the first half
of a phrase, full cadences at the end (and the phrase then can be called a period). A semicadence at
the end of a phrase rather suggests: 'now some development will follow'. A semicadence at the end
of a section (often lasting for some time) suggests: 'hereafter a new section will start' . And so on...
Examples 30 and 31 show various closes, and various forms of harmonic rhythm. The examples
also show that final chords usually stand on strong beats; normally these strong beats are first
beats.
example 30 Mozart, Violin Sonata in E minor, beginning of the first , begin van het eerste deel
24
Bars 1-8 of the above example form a period (semicadence in bar 4, full cadence in bar 8). At the
end of the consequent (bars 7/8) the harmonic rhythm is accellerating. After a 'connection' (bars 912), the first period is repeated (from bar 13, with some changes..). This example shows that there is
no need to write actual vertical chords to make clear that harmony is present: despite the unisono of
the first 12 bars we clearly hear harmonic functions. Sometimes though, such implied harmony is
unclear: given the harmonization in bars 13-20, it is quite uncertain whether bars 5/6 really have
'subdominant character'.
In the following example, from Schubert's Moment Musical No. 2, the harmonic rhythm is more
complex than in the above Mozart-example. At first sight (see the degrees under the notes in
example 31) the harmonic rhythm at the beginning of this fragment seems as follows: one harmony
in bar 18, one harmony in bar 19, three harmonies in bar 20. But, when we look at the functions of
these chords, we can rightly claim that 'little or nothing' is happening in these bars: I is the most
important chord in all three bars, the other chords are 'encircling' the tonic, and are dependent of it:
they are neighbouring chords.34 In fact the harmony starts to move not earlier than at the end of bar
21, as the bass moves to a new tone (and the harmony changes from I to I6). In the consequent
(bars 24-31) we observe something similar: in bars 24-26 the harmony is much more 'static' than
from bar 27: bar 24 contains one harmony; the next harmony is lasting two bars (bars 25/26).
From bar 27 the harmony moves faster: we mostly see two harmonies per bar. In bars 32-35, the
transition to the next section of the piece,35 the harmony moves slower again:bars 32 and 33 contain
one harmony per bar, and in bars 34/35 one harmony is sustained:
example 31
Schubert,
moment musical
Nr. 2 in Ab
major, bars 1835
34 See also from page 30.
35 This next section is not displayed in example 31
25
hemiola and syncopation
The rhythm in a piece may of course move at the same pace is the meter; more common is
interaction between the metric pulse, and rhythmic structures within the pulse. When rhythmic
patterns contradict the meter, this is mostly caused by a hemiola or a syncopation.
Syncopations, caused by ties in one or more voices, combined with suspension in one voice or
several voices, are very common (the so-called syncopated dissonance)36 Such suspensions are all
over the place in example 32: in the first section of this piece (Allegretto) the ties cause several
dissonant suspensions on first beats, in the upper voice. This is clearly audable in bars 9-16, 27/28
and 31-33, especially because the same material has been presented before without suspensions
(compare for example bars 1-8 with 9-16). The upper voice also contains suspensions on the first
beats of bars 18, 20, 23 and 24. From bar 37 (the Trio) we encounter anticipations instead of
suspensions. 37
example 32
Beethoven, Piano Sonate in C# ninor Op. 27 Nr. 1, Allegretto
36 See page 19/20, examples 25 and 26.
37 The anticipations in the Trio in example 32 are quite of a quite uncommon kind. in het Trio in example 121 behoren
niet tot de meest voorkomende. Anticipation of the root of the key in the top voice is much more usual - see page 18.
26
Especially from the Classicism sometimes syncopations in all voices (at the same time) are used.
These syncopations cause a shift of metrical accent to a tone (mostly) before or (sometimes) after
the syncopation. In extreme situations the listener may believe that the bar itself is shifted one or
several beat(s) - until a 'normal' first beat appears (see example 33); or he may believe that
temporarily the meter is changed (see example 34):
example 33
Beethoven, Piano Sonata Op. 109, fragment from the first movement
27
The concluding motif
(last beat of bar 24 and first beat of bar 25) is repeated several times, until bar 29. Through
the tie between the two tones of this motif:
a long-lasting syncapted rhythm is created (from bar 29). We
perceive a shift of meter, and at some moment we don't hear
anymore, but:
instead.
In bar 35 the meter is readjusted - but even thereafter syncopations are permanently present (see the top voice from bar
37, and the bass from bar 41).
example 34
Brahms, fragment
from the first
movement of the
Violin Conceerto Op.
77
The solo violin part suggests three 5/4 bars, instead of five 3/4 bars. The bass is making the situation even more
confusing, as it is suggesting still another meter, namely a succession of 4/4, 5/4 en 3/4 bars. The middle voices, though
'organized' in 3/4, cause confusion through syncopations, more than helping to clarify the 3/4 meter.
28
We speak of hemiola when a shift of metric accent is causing the impression of a temporary change
of meter. This may happen for example, when in triple meter, through a shift of metrical accent, two
groups of three beats are changed to three groups of two beats.This is quite common in endings,
especially in Baroque music - we could then speak of a 'closing hemiola'38 The following example,
the first theme of the first movement from Mozart´s piano sonata in G major, contains such a
´closing hemiola': at the end of the theme (bars 8-10, and again in bars 14-16) two 3/4 bars actually
are changed to a single 3/2 bar. We perceive a hemiola here because of the placement of the chords
in the meter (on first, third and second beats), and because of the angular points in the melody (it
changes direction on the third beat of bar 9, and agin on the second beat of bar 10):
example 35
Mozart, Piano Sonata in G major, beginning of the first movement (first theme)
Bars 1-4: antecedent, mainly determined by
the tonic triad (the dominant chords in bars 2
and 3 are neighbouring chords, meaning they
are harmonically 'weak').
Bars 5-8: actual consequent (the first beat of bar 8 could have been
the end of a clear final cadence.. ). Extension of motif 1, together
with melodic and rhythmical compaction, causes increase of musical
tension: motif 1 has a feminine ending now, and is repeated after just
one bar. The appearance of the first subdominant of the piece, in bar
5, adds to the increase of harmonic tension.
Bars 8/9: extension of the consequent through
a hemiola, in combination with
a clear complete authentic cadence T S D T .
Bar10 contains the ending that could have
been in bar 8 already. In bars 11-16 the whole
consequent (with extension) is repeated.
Dynamic accents may also go against the meter to a certain extent: when accents are given on weak
beats, they stand against the meter – though there is no syncopation or hemiola involved.Especially
Beethoven likes the use of such 'antimetric' dynamic accents.
38 This is not an English term, I fear - rather a literal translation of the Dutch slothemiool, or German: Schlußhemiole.
29
structural and non-structural harmony / faux-bourdon / pedal point
In tonal music, mostly the chord succession is not an independent phenomenon. It normally is
connected with, s idependent of, or even: arises from horizontal phenomena, in the sense that (more
or less) independent voices together form the harmony. The term voice leading then describes the
course of these indivual voices. In homophonic music usually the outer voices (top voice and bass)
bear more importance than the middle voices: normally the upper voice contains the melody, and
the bass is the most important voice for the harmony.
We have seen about one aspect of the relation between harmony and voice leading already:
embellishing tones.39 The use of embellishing tones is adding complexity to the relation between
horizontal (the voices) and vertical (the harmony), as the harmony - and the harmonic rhythm40 - is
then moving slower than the individual voices.
We have seen above that the distance of a fifth can be regarded as primary harmonic relationship41;
this holds true for the relation between chords, and also for - at a larger scale - for the relation
between keys.42 Distance of a second, on the contrary, can be regarded as the most important
melodic relation: in the voice leading seconds usually are the most common interval. Melodic
movement in secunds often arises from the use of embellishing tones. Embellishing tones may
create a chord, quasi 'by accident', because several voices move in seconds at the same time: such
chords may be seen as resulting from passing tones, neighbouring tones, suspensions, and/or
anticipations. Therefore they may be called: voice leading chords. Voice leading chords are
harmonically and functionally dependent on their environment, and often occur within a
prolongation43 of a function. See example 36:
- The first three chords may be labelled: I V6/4 I6 ; V6/4 is 'caused by' voice leading: it arises from
a passing tone in the bass, and therefore is not independent. In other words: the tonic 'controls' the
first three beats; V6/4 appears within a tonic prolongation.
- The chord on the third beat of bar 2 may be labelled: I6/4 ; this chord is not independent, as the
tones A and C act as suspensions; the harmonic function of both the third and fourth beat of bar 2 is
dominant.
- The first three chords of bar 3 may be labelled: I IV6/4 I ; IV6/4 is 'caused by' voice leading: it
arises from two neighbouring tones (in alto and soprano). The tonic 'controls' the first three beats;
IV6/4 appears within a tonic prolongation.
- The chords in bar 4 may be labelled: I6/4 IV6 I6/4 V; IV6 is not independent, as it arises from a
neighbouring tone in the bass. I6/4 is also not independent; it arises from two suspensions (in
soprano and tenor). The dominant 'controls' the whole bar.
In general we can say that
- succession of functions often is at an even slower pace than succession of chords (as several
chords with the same function may follow each other44, or chords with the same function may be
connected through a voice leading chord)
In other words:
- Often melodies contain more tones than the number of chords (degrees); and likewise often there
are more chords than independent functions.
39
40
41
42
43
44
See from page 16.
See from page 21.
See page 6, and page 10/11.
As for example the relations between the keys in forms like sonata form, rondo, large ternaries etc.
Prolongation: see also page 9.
See page 9/10.
30
example 36
voice leading
chords (chords,
arising from
melodic
movement and /
or prolongation of
a function.
Most voice leading chords are inversions (of triads or seventh chords): root positions are 'stronger'
(more stable) than inversion, and therefore functionally clearer. And voice leading chords should
preferably be functionally 'weak'.
Voice leading chords are similar to embellishing tones: like embellishing tones are not part of the
underlying harmony, voice leading chords are not part of what is called: the structural harmony.
As with embellishing tones, we can distinguish different types of voice leading chords (see example
36):
- passing chord: a chord on a weak beat that connects two other chords on stronger beats; passing
chords mostly are based on a passing tone in the bass.
- neighbouring chord: a chord on a weak beat that connects two other chords on stronger beats; a
neighbouring chord is based on a neigbouring tone in the bass, or on one or more neighbouring
tones in other voice(s).
- suspension chord: a chord on a strong beat that is perceived as suspension for the following chord
on a weak beat; suspension chords mostly are based on suspensions in other voices than the bass.
There is a link with harmonic rhythm:45 passing and neighbouring chords are on weak beats, and are
perceived as 'not very important' in the succession of harmonies, and in the harmonic rhythm.
Suspension chords stand on strong beats; they are 'not very important' though in the sense that
not the suspension chord, but its resolution belongs to the structural harmony. Some examples:
In bar 1 of the slow movement of Beethoven's String Quartet Op. 18 Nr. 1 (see example 37) VII6 is
used as a passing chord between I and I6: it clearly arises from the combination of a neighbouring
tone (in the second violin) and two passing tones (in viola and violoncello). I6 prolongs I in root
position, which is the only structural harmony in bar 1. In bar 2 we likewise find a passing harmony
between I6 and I: V4/3. The first change in the structural harmony is at the beginning of bar 3, as V
is introduced.
In fact, until the beginning of bar 4, we are dealing with three harmonic layers:
- the structural harmony, consisting of I – V – I
- prolongation of I through I6
- passing harmonies within the prolonged I: VII6 en V4/3
45 See also page 21-24.
31
example 37
Beethoven, String Quartet Op. 18 Nr. 1, beginning of the second movement
From bar 4 similar things are happening, though it might seem a little more 'abstract': from bar 4 the
bass is moving stepwise up; I is at the beginning of this stepwise motion (in bar 4), I6 at the end (in
bar 7). It is possible to consider all chords in between as passing chords: though they are more
independent than VII6 and V4/3 in bars 1 and 2, they appear within a tonic prolongation:
- VI6 ecan be regarded as a neighbouring chord 'coming from' I (see the second violin)
- II7 can be regarded as a suspension chord for VII7 (see the first violin)
- VII7 / VII6 / VII7 can be interpreted as actually a single chord, which is forming a passing chord
between I en I6
When we interprete the whole passage in this way, the first structural harmony after I in bar 1 is a
subdominant: II6/5 in bar 8. G# in the violoncello in bar 8 can be interpreted as a chromatic passing
tone, which means that the chord on this tone: (VII7) can be heard as a passing chord. Then the next
structural harmony is V at the beginning of bar 9.
This Beethoven-example makes clear that melody, harmony and structural harmony each may move
at their own pace. The reduction below only shows the structural harmony (what I regard as the
structural harmony..) of this fragment: see the degrees in the example (open noteheads refer to
structural harmonies):
32
example 38
Beethoven, String Quartet Op. 18 Nr. 1,
beginning of the second movement –
reduction of the harmony
The harmonic rhythm in the first bars of Brahms' song
“Traun, Bogen und Pfeil..” is recognizable as:
On the first and second beats of bars 3 and 4 stand two different chords (see the degrees in the
example). But in both cases we find only a single structural harmony: I6/4 in both bars arises from
two suspensions (Bb and G on the first beats), and therefore is not independent, but has to be
regarded as a suspension chord for V on the second beats. This means that it does not have tonic,
but dominant function, along with V46 Actually, the appearance of I6/4 does not interfere with the
harmonic rhythm in quarter notes and half notes, as the harmonic function does not change from the
first to the second beat:
example 39
Brahms, begin
van het lied
“Traun, Bogen
und Pfeil..”,
Op. 33
In bars 7 and 8 is a 'real' change of harmonic rhythm: here we find three harmonies per bar. Again,
we could ask the question whether the chords form structural harmonies: I tend to interprete the first
inversions in bars 6-8 (IV6 and V6) as 'weak'. To a certain extent a shift of meter is taking place
here, as the harmony causes somewhat emphasizes the second beats (as there are root positions of I
on the second beats). From bar 9 we return to the initial harmonic rhythm, in quarter notes and half
notes. In bar 11, (VII7) can be regarded as a (chromatic) passing chord.
In the first bars of example 31 (see page 25) neighbouring chords are used; both IV6/4 and II2 are
46 Compare with example 35, where , in bars 9 and 15, likewise I6/4 is appearing as suspension chord for V. The same
holds true for bars 3 and 7 in example 32. See also example 36.
33
voice leading chords, arising from neighbouring notes on top of F# in the bass. Both chords do not
have subdominant function; actually the tonic triad is 'encircled' through a 'double neigbour' (or:
'double neighbouring tone'):
example 40
double neighbour, and: Schubert,
Moment musical
Nr. 2 in Ab major, bar 18-21,
reduction. See example 31
for the score.
Example 40 (and bars 4-6 of example 37) show that it is possible to put more than a single passing
or neigbouring chord between two structural harmonies. Example 36 shows a common possibility
of 'layered' voice leading chords: in bar 4 of this example a suspension chord, I6/4 , is prolonged by
a neighbouring chord!
In a so-called fauxbourdon also several voice leading chords stand between structural harmonies. A
fauxbourdon consists of a 'chain' of (stepwise moving) parallel sixth chords. These sixth chords
hardly have any harmonic meaning: in fact fauxbourdon is a melodic connection between two
structural harmonies. This is evident in bars 45/46 (and 47/48) of the first movement of Mozart's
piano sonata in G major: a 'chain' of sixth chords form a coinnection between I and I6 ; it hardly
makes sense to 'label' the individual sixth chords as seperate degrees:
example 41 Mozart, piano sonata in G major, first movement, final bars of the exposiotion
Actually fauxbourdon (French for false bass) is an 'invention' from the fifteenth century, as a
technique of harmonisation, in which a melody (cantus firmus) in the top voice is accomponied by
two other parts, a perfect fourth (middle voice) and a sixth (bass) below the top voice; all three
voices move parallel - though at the beginning and end of phrases the lowest voice forms an octave
with the top voice. Starting point and end of phrases therefore sound stable; the faux bourdon
34
between these stable points essentially is instable, passing. 47 The accompanying voices sometimes
have embellishments. Like in the above Mozart-example, in the fifteenth century usually just a
small part of a composition employs fauxbourdon technique. An example from a composition by
Dufay:
example 42
Dufay, beginning
of Ave maris
stella, in faux
bourdon (ca.
1440)
Since in the course of the sixteenth century the parallel fifth became more and more obsolete (and
finally was banned from music, in the middle of the sixteenth century..), parallel motion between
two, or more than two, voices is only possible in thirds and sixths. Or in thirds and sixths, in other
words, in fauxbourdon. Therefore it is quite logical that precisely this - actually medieval harmonisation technique did survive.
The fauxbourdon in example 41 (Mozart) is relatively simple. Sometimes a fauxbourdon is 'hidden',
for example when other inversions stand between the sixth chords of the fauxbourdon. This is the
case at the beginning of the first movement of Schumann's Rhenish Symphony 48: the sixth chords
do not always follow each other - whereas in the background nevertheless a fauxbourdon is
perceivable : I – VI6 – V6 – IV6 – III6 (etc.). In Bruckner's Graduale Christus factus est pro nobis49,
the harmony from bar 7 is employing a fauxbourdon - a rather complex one, because of the voice
crossings and suspensions, and especially because of the chromaticism inthis passage. In the
background a fauxbourdon - with 7-6 suspensions on the first and third beats - is audible though:
example 43 Bruckner, Graduale
Christus factus est pro nobis, bar
7-11 (reduction).
On the first and third beats the
sixth chords are suspended by a
seventh.
Sometimes, we encounter situations very similar to fauxbourdon, where actually no 'real'
fauxbourdon is at place. This is especially the case when a bass and an upper voice, moving in
parallel sixths, sometimes form sixth chords together with the middle voice(s). A good example is
the beginning of the Allegretto of Beethoven's piano sonata Op. 27 nr. 150. The parallel motion in
sixths in the first two bars of this piece causes instability, whereas the cadence in bars 3/4 leads to
stability. Bars 1/2 are similar to a fauxbourdon, mainly because of the parallel motion in sixths. See
example 44: Beethoven's original notes are at the left; at the right a 'reconstruction' of the faux
bourdon resembling these notes:
47 The stable sounds at the beginning and the end consist of perfect consonants; ths sixth chords in between are less
stable because they contain imperfect consonants.
48 See example 28.
49 See example 29.
50 See example 32.
35
example 44 Beethoven,
Allegretto from piano sonatae
Op. 27 Nr. 1 (initial bars, and
'reconstruction' of the faux
bourdon)
As with a faux bourdon, a pedal point causes what we could call 'layering' in harmony. A pedal
point is a sustained tone in the bass, indicating the structural harmony at that moment. Mostly this
bass tone is the root of V or I . The higher voices move 'freely' on top of the sustained bass, which
may lead to dissonances with the bass: the harmony is as it were split in two layers, where the
higher voices form a chord succession, (quasi) independent from the pedal point. In the final bars of
the first Prelude of Bach's Welltempered Piano we see successive pedal points on V and on I. It is
clearly audible that the harmony on top of both pedal points is more or less independent:
example 45 Bach, welltempered piano, first book, final bars of the first prelude
Pedal points on V often act as a kind of long 'announcement ' of the (return of) the tonic. This
happens often at the end of the development section in sonata forms51, and sometimes before the the
second theme (hence: at the end of the transition). The pedal tone must not be actually sustained: it
is sufficient when the bass returns to this tone often - the listener then will be able to 'think through'
the pedal point himself.. See the following example, the end of the development section of the first
movement of Mozart's horn quintet: Bb is not sustained; we still clearly hear a pedal point on the
dominant:52
51 This is called: dominant pedal point, see my text: “Where is the beginning of the second theme?...”, page 5.
52 In this pedal point, V is alternating with I6/4. This I6/4 has dominant function, as it appears within a
prolongation of the dominant, as neighbouring chord.
36
example 46 Mozart, horn quintet K 407, first movement, end of the development and beginning of the recapitulation
Alternating chords sometimes function as pedal points as well, especially when V is ´encircled´. In
the harmonic rhythm, V then is stronger than the other degree(s). The following example contains
the final bars of the development section, and the beginning of the recapitulation of Mozart's piano
sonata in A minor. Until the end of the development section, every V is on a strong bar, whereas
every I stands on a weak bar. The remaining chords are on even weaker beats. Until the
recapitulation, V is structurally the most important harmony; all other chords 'aim at' V:53
example 47 Mozart, piano sonata in A minor, first movement, end of the development and beginning of the recapitulation
53 You can find the other bars of this development section in example 59.
37
Pedal points on I often can be found at the end of pieces, or at the end of a major section of a form.
Such pedal points emphasize the final I, and intensify the concluding character of the preceding
final cadence. Moreover, often the higher voices have a second final cadence on top of the pedal
point. See example 45: the final tonic in bar 32 is prolonged four bars through a pedal point; the
higher voices meanwhile form an 'independent', cadensing chord succession.54
Pedal points at the beginning of pieces - often on I - are quite common as well. On top of such tonic
pedal points the higher voices then form a cadence (for example I IV V I ), or form alternating
tonic and dominant chords, or tonic and subdominant chords - quasi independent from the bass. See
example 12 (Mozart, piano sonata in A minor, beginning of the first movement): in the first four
bars of this piece we see the chords I V7 I V7 on top of a tonic pedal point.. Some pedal points
are extremely long, and 'control' large parts of a piece (even a few pieces exist that are entirely
based on a pedal point!).55
It is not unusual to combine pedal point and fauxbourdon: then, on top of a sustained bass, at
least three voices move in parallel sixth chords. This is happening in the following example, the end
of the development section of Beethoven's piano sonata in C minor, Op. 10 nr. 1. V forms the
starting point and the end of the fauxbourdon (see bars 158, 162 and 166). At the end (see bars
167/168) the right hand is moving from the fauxbourdon via I6/4 and V7 to I at the beginning of
the recapitulation:
example 48
Beethoven, piano
sonate in C minor,
Op. 10 nr. 1, first
movement, end of the
development secion
and beginning of the
recapitulation
54 Compare with example 41: the last 3 bars of the exposition of this sonata movement contain a tonic pedal point very
similar to the one in example 44 (Bach).
55 To mention just a few quite extreme examples:
- the beginning of Brahms' First Symphony first is on a long tonic pedal point, followed by a long dominant pedal
point..
- the development section of Brahms' Third Violin Sonata in D minor contains an 'extreme' dominant pedal point:
the whole development is on a pedal point on A!
- the whole second movement of Ravel's Gaspard de la Nuit (Le Gibet) is on a pedal point.
- the whole of Debussy's Second Piano Prelude is based on a pedal point.
38
sequences
The general meaning of sequence is: repetition on a different pitch, in the same voice(s). A sequence
essentially differs from imitation: imitation is a repeat in another voice (on the same, or on a
different pitch). Compare example 49 with example 50: example 49 contains a sequence; example
50 also contains a sequence (bar 2 is a sequence of bar 1), but an imitation as well (the lower voice
imitates the upper voice).
Example 49 shows that I am using, next to the term sequence, the term “model” - as a means to
indicate which notes are put into sequence later. Some theorists use the term sequence to describe
the combination of model and sequence(s) as well - I try to avoid that though, as I believe it is quite
misleading.56
We can distinguish between different types of sequences:
- in a melodic sequence the melody moves to a different pitch, but the harmony does not move along
with the melody. In the example below bars 5-8 are a melodic sequence of bars 1-4 (the sequence is
not entirely literal, but in principle the melody moves up a second). The harmony is not put into
sequence though:
example 49
Mozart, Piano
Concerto Nr. 20
in D minor K
466, beginning of
the first
movement
(reduction)
In fact the main bass tones
form a sequence in itself. The
motion in this bass-sequence is
contrary to the soprano.
56 Maybe one reason not to adapt this practice is the literal meaning of “sequence”: “what follows”, sequitur= to
follow.
39
example 50
Bach, beginning
of the two-part
invention in C
major: sequence
and imitation
Likewise, in the following example we see some melodic sequences:
example 51 Bach, beginning of the two-part invention in Eb major
- in a harmonic sequence a chord succession is moved up or down: a complete group of chords is
repeated on a different pitch. Strictly speaking, the melody should not move together with the
harmony - and it is my belief that this situation is not very common; normally the melody moves
together with the harmony. An example of a harmonic sequence (I fabricated it myself):
example 52
harmonic
sequence
- In a harmonic/melodic sequence both melody and harmony are moved.57 We may stay in the same
key - and then in the sequence preferebly diatonic chords are used. We may then speak of a nonmodulating sequence, and, when only diatonic chords are used, furthermore of a diatonic
sequence.58 It is also possible that we leave the key through moving the chord pattern. For example:
when the chord succession I IV V in C major is repeated, as a sequence, as I IV V in A minor,
then V in A minor is no scale degree in C major: we have modulated to the key of A minor. In this
case we can speak of a modulating sequence. Some examples:
example 53 Beethoven, Piano
Sonata in E major Op. 109,
beginning of the first movement
57 This for example normally happens in a sequence of falling fifths. See page 11.
58 Diatonic sequences of falling fifths are a good example; see page 11, examples 11 and 12.
40
Example 53 contains a variant of the so-called “Pachelbel-sequence”, named after Johann
Pachelbel's Kanon und Gigue in D-Dur für drei Violinen und Basso Continuo (canon and gigue
for three violins and thorough bass). Pachelbel (see example 54) wrote a three-part canon on top
of a two-bar pattern in the bass, which is constantly repeated (=basso ostinato); the bass consists
of the following tones:
1 – 5 – 6 – 3 –
4 – 1 –
4 – 5 – 1 .
|_model_| |_sequence 1_| |_sequence 2 _|
The bass has to be harmonized, as we are dealing with a basso continuo. As there are no figures
in the score, we can assume root positions have to be played on the basstones; this results in the
degrees:
I – V – VI – III – IV – I – IV – V – I .
example 54 Pachelbel, Kanon
und Gigue in D-Dur für drei
Violinen und Basso Continuo,
beginning
In the beginning of Beethoven's sonata Op. 109 (example 53) the bass is descending stepwise,
causing the appearance of alternating root positions and first inversions:
I – V6 / VI – III6 / IV – I6 . Like in the Pachelbel-example, this sequence is completely
diatonic
Example 55 contains modulating sequences, which is not very astonishing when we bear in mind
that these bars form the beginning of a development section:
41
example 55 Schubert, Symfony Nr. 5 in Bb major, beginning of the development section: modulating sequence
42
modulation, secondary functions/degrees
In most tonal compositions we encounter more than just a single key. The most important key, the
main key or home key, is 'the key of the piece' (mostly indicated by the key signature). Almost all
tonal pieces start and end in their main key. Some other keys have specific 'names'. Some other keys
also have a specific ´name'. For instance, the key a fifth above the home key is called: dominant
key; the key a fifth below the home key is called: subdominant key; keys with the same key
signature are called: parallel keys (in major: the minor key a minor third below, in minor: the major
key a minor third above the main key). Major and minor keys with the same root (eg C major and A
minor) are called: relative keys.59
Transitions from one key to another are called: modulations. As a rule, we can assume a
modulation when the new key actually is confirmed - often through a cadence in the new key.60
When a new key only is 'suggested', or ´touched´, without clear confirmation, we better do not
speak of modulation, but of (temporary) tonicization.61
Chromaticism within the key (when we are not leaving the key, in spite of the use of chormatic
tones) mostly is caused by secondary functions: a chord, or a succession of chords is as it were
'borrowed' from another key, whereas no modulation (or even: tonicization) is taking place. For
example: when we use the chord D-F#-A-C (dominant seventh chord) in the key of C major, this
does not necessarily mean that we are modulating: the chord may be used within the key of C
major, with F# as chromaticism. I will come back to this topic.
In modulations often a pivot chord is used: a chord, forming a scale degree in both keys, and used
as the centralmpoint in the modulation. Pivot chords can be labelled in both the 'old' and the 'new'
keys with Roman numerals.62 We could imagine the 'old' key as the 'past' of such a chord, and the
new key as its 'future'. Therefore, the new key normally is not yet audible in the moment the pivot
chord appears: we, as listeners, indeed don't know yet that a modualtion is taking place (as we still
hear the chord in the 'old' key). Only later we realize which chord did form the pivot chord in the
modulation, so: where precisely the modulation took place. A little example:
example 56 modulation (to the dominant key), with pivot chord
It is customary (in analyses, and in harmony) to distinguish between three types of modulation:
59 It is useful to know that in Dutch (and German, etc.) relative keys are called parallel. And parallel keys are called:
gelijknamig (German: gleichnamig), which means: 'having the same name'.
60 See pages 14 and 15.
61 This is for instance common in fugues: during the second entrance of the subject (comes / answer ) a modualtion to
the dominant key is initiated. But, as we stay in the dominant key for a very short time (because we modulate back
to the home key immediately after the second entrance), we better speak of tonicization of the dominant.
62 This is (also) visable) in examples 39 and 52: at some points I used double explanations (different scale degree
numbers, in different keys, one written below the other).
43
1. in diatonic modulations a chord is used that is forming a diatonic scale degree in both keys; in
other words: a diatonic pivot chord is used: a chord, forming a scale degree in the 'old' key, and at
the same time forming another scale degree in the 'new' key. The modulations in example 56 are
diatonic: the F# minor chord in bar 4 is VI in A major, and appears to become II in E major. And the
E major chord in bar 8 is I in E major, but as soon as the repeat is beginning it turns out that we can
hear it as V in A major as well.
In example 55 something similar happens: the keys Db major, Bb minor and Gb major are
connected through diatonic pivot chords; in both modulations VI changes to I in the new key (see
bars 122 and 126).
It is obvious that diatonic modulations mainly occur when keys are not very far apart (meaning:
their key signatures are not too distant), as in the modulation a chord, forming a scale degree in both
keys, is used. The two keys, involved in the modualtion, therefore must have at least three common
tones.
2. in enharmonic modulations likewise a pivot chords is used, but in this case the pivot chord may
not form a diatonic scale degree in one of the keys involved. In the pivot chord one or several
tone(s) are enharmonized (for example: A# is written as Bb). This is called: enharmonization; in
the modulation we find an enharmonic pivot chord.
In enharmonic modulations, always partial enharmonization is used: no more than two tones of a
triad, or three tones of a seventh chord are enharmonized. In a modulation it doesn't make sense to
enharmonize all tones of a chord (complete enharmonization), as then really only the notation is
changed: when, for example, the chord F#-A#-C#-E is notated as Gb-Bb-Db-Fb, essentially
nothing has changed, as the F# chord forms V7 in B, and the Gb chord in Cb. And in fact B and Cb
are the same key!
Through (partial) enharmonization in a chord we get
– an identic chord, but: in another inversion, or
– another chord, with identical sound
Not all chords are suitable for enharmonization. Probably the most important are:
– diminished seventh chord
– augmented triad
63
– 'hard-diminished' seventh chord
when enharmonization is used in one of these chords, the inversion changes (and along with that:
the key in which the chord forms a - diatonic or non-diatonic - scale degree)
– dominant seventh chord; this chord may be enharmonized to a doublediminished seventh
chord.64
These are by far not the only possibilities (I am leaving lots of possibilities out here...). An example,
showing the chords mentioned above:
example 57 enharmoniszations in diminished, doublediminished and 'hard-diminished' seventh chords, and in the
augmented triad.
63 This is not an English term, but literal translation of the Dutch 'hardverminderd' (or German: hartvermindert). The
chord consists of major third, diminished fifth and minor seventh. The second inversion is known in English as
'French augmented'.
64 This term is not very comon in English (though it is used). The chord consists of diminished third, diminished fifth
and diminished seventh. The first inversion is known in English as 'German augmented'.
44
For harmonic analysis, it is important to know that composers often don't notate enharmonizations.
In other words: in enharmonic pivot chords often a single notation is chosen; normally for instance
an A# is not slurred to a Bb. That enharmonic modulation is at issue then has to be derived from the
context. In the following example though, Beethoven has been kind enough to indicate the
enharmonization precisely:
example 58 Beethoven, Sonatra in C minor, Op. 13 “Pathetique”, first movement, beginning of the development section
The following example shows the major part of the development section of Mozart's Piano Sonata
in A minor65. As can be expected in a development, several modulations are taking place. The
development starts in C major (the relative major key of the main key A minor), and modulates to F
major after a few bars. In bars 56 and 57 V in F major (dominant seventh chord on C) is
enharmonized to a doublediminished seventh chord. This enharmonization causes the modulation
from F major to E minor:
example 59 Mozart, Piano Sonate first movement, development section until dominant pedal point
65 You can find the last bars of this development in example 47; the beginning of the exposition is shown in example
12.
45
46
3. we speak of chromatic modulation when
– a chromatic step is used (for example C-C# of B-Bb) and/or
– a chord is used that does not form a scale degree in the 'old' key, but does form a scale degree in
the new key.
Chromatic modulations normally do not contain a pivot chord: through the chromaticism we
immediately leave the 'old' key, and reach the new one; the connection with the old key is suddenly
lost. For example: when we are modulating from C major to D minor via the chromatic step C-C#,
then the tone C belongs to the key of C major, and C# to D minor.
At a slightly larger distance though, it often is possible to find a relationship between the two keys:
in the following example there is no pivot chord at the actual 'modulation point': the tone F in bar
50 belongs to the key of Bb major, the tone F# in bar 51 belongs to the key of G minor. But, as soon
as the G minor chord is reached, it is possible to perceive it (also) as VI in Bb major. Situations like
this - diatonic 'connection' in the background of chromatic modulations - are very common.
example 60 Schubert, Symfony Nr. 5 in Bb major, first movement, fragment of the transition
The modulations in bars 58, 62 and 66 of the development section of Mozart's piano sonata in A
minor (see example 59) are to be considered as chromatic,66 even when no chromatic steps are
visable. The reason is that we every time expect a minor triad as resolutions of the preceding
dominant - but a major triad is appearing instead (which subsequently proves to be dominant in a
new key ). In bars 58-70 we encounter a chord succession of falling fifths (Bmajor – Emajor –
Amajor – dminor); because the first three chords appear to be dominant for the following chord we
can speak of a chromatic sequence of falling fifths (or: chromatic sequence of dominants). This
is followed by a (much faster) diatonic sequence of falling fifths s67 in bars 70-73.
66 Apart from the enharmonic modulation in bars 56/57, see page. 45. Compare also example 31, bars 32/33: here the
modulation is taking place via a chromatic step (in the bass).
47
67 See also page 11.
48

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