GARCÍA, Miguel V. Harmonic Analysis in Practice _ a Critical Review of the Labels Employed to...

8/12/2019 GARCÍA, Miguel V. Harmonic Analysis in Practice _ a Critical Review of the Labels Employed to Describe Harmony i…

http://slidepdf.com/reader/full/garcia-miguel-v-harmonic-analysis-in-practice-a-critical-review-of-the 1/89

Harmonic analysis in practice: A critical review of the labels

employed to describe harmony in common practice music

Miguel Vicente Garc’a



2

Contents

¥ Introduction

Contextualization. What is Òcommon practiceÓ? Harmonic function vs. Linear 4function. Functional harmony.

General considerations for the elaboration of a ÒvocabularyÓ of labels 7

¥ The system

1. Figured bass and roman numerals: a paradoxical association 8

2. Diatonic harmony 21

a) Major and minor scales. b) Inversions.c) Seventh chords.

d) Inversions of seventh chords

3. Chromatic harmony 29

a) Secondary functions. Tonicization

b) Modulation

c) Altered chords- Mixture

- Augmented sixth chords. Conventions and problems- The quest for a systematic approach: Louis & ThuilleÕs

Harmoniehlere

- Terminology matters I

- The Dutch connexion: E. MulderÕs Harmonie

- Terminology matters II

- Later discussion

- Terminology matters III

- Symbol vs. Letter - Altered chords in use

-

Enharmonic possibilities- Further implications of altered chords

4. Ninth chords 70

5. The limits of the system. 72

a) Non-tertian constructions.

b) Overcoming functions I. Sequential patterns

c) Modal Contexts

d) Overcoming functions II



4

Introduction

Harmonic analysis is a subject that every musician, not only those focused on the theory

of music but also singers and instrumentalists, will face at some stage of their education,

and it should become a tool for better understanding and interpretation of the music. But

students often face a problem that discourages their efforts: every book and everyteacher seem to have a different system to approach the subject, even using different

terms for analogous situations. Considering the undeniable and understandable pre-eminence for the main subject in both time and efforts of the students (and study plans),

it seems that every effort to facilitate the learning (and application) of the side subjects

seems justified.

In the case of harmonic analysis, the system we use to label the chords is not as trivial

as it is sometimes regarded. In Thinking about harmony, D. Damschroder states that

Òsome matters, such as Arabic versus Roman numerals, or all-capital Roman numerals

versus a mix of capital and small Roman numerals are of little significanceÓ, in the

whole process of understanding a harmonic progression.1 That may be true for the

professional theorist, but from the point of view of the student, who is building up newknowledge it may be crucial to have one concrete system.Many aspects have to be considered. First of all the repertoire that is the subject of

analysis. In this paper we will focus exclusively on the so called ÒCommon practiceÓ

repertoire. This somewhat vague term is used to designate the music extending roughly

from the early eighteenth century to the beginning of the twentieth century. Although

this huge span of time embraces different style periods, namely late-Baroque, Classical

and Romantic, we can find strong characteristics that justify the idea of considering all

of them under the name of Common practice, and consequently analyze them with the

same tools. These strong features are the consistent organization of chords insuperposed thirds, the harmonic direction and hierarchical organization of the chords

within a key (represented by the pre-eminence of the tonic, the relationship of thedominant and the subdominant with the tonic, and the relevance of fifth relationships in

general) and the use of specific voice leading procedures such us the resolution of

disonances and leading tones, the preference for stepwise movement and the avoidance

of parallel perfect consonances.

The choice for such a long period has a lot of influence on the way we are dealing with

the subject. It seems that most textbooks and methods on harmony (if not all of them),

being more practice-based (writing exercises) or analytical, start from simple diatonic

harmony and they progress to more complex situations, getting in trouble (or justavoiding) to continue with the same system of labelling when dealing with more

chromatic music (Wagner, Franck...) and for this motive every author seems to look forhis own way to solve this problem, resulting in an unconnected (with simpler examples

and with other text books) and sometimes even incoherent array of chord names and

symbols. For that reason in this paper we will look forward from the beginning to the

more complex situations, always within common practice, so some of the choices we

make in early steps in diatonic music are already conditioned by our vision on

chromatic music, with the aim of making one coherent system that will apply to both

harmonic contexts.

We also make a choice for an essentially vertical approach for understanding the

harmony. With this choice we do not disregard the benefits of a horizontal or linear

analysis, but on the contrary we see it as a complementary tool. Nevertheless we think

1 Damschroder, D., Thinking about harmony, p. 218.



5

that common practice harmony can be convincingly described in this way so that every

vertical sonority can be precisely identified and classified (therefore understood) at local

level. This detailed description of the foreground is only one step in the complete

description of the harmonic content of a piece, that may be completed by the description

of the background harmonic events.

Several aspects of this vertical approach are sometimes criticized:

1. The focus on individual chords prevent the student from recognizing the overallshape and large scale structures. ÒIf you want to get a more detailed understanding of

the musicÕs harmonic structure, then you have to consider its linear patterns, and you

cannot do this if you reduce everything to harmonic symbolsÓ.2

This risk is certainly true but can be successfully avoided by a conventionalized

knowledge3 of tonal progressions. Linear patterns can be easily associated to harmonic

progressions. This includes cadences and other standard progressions, such us passing

progressions (I-V -I6) and sequences.

2. Vertical approach is a mechanical procedure, consisting only in the identification

of a chord within a key.

Our approach to harmonic analysis, as it will be explained in the following pages,

combines two different concepts: Stufentheorie (scale step theory) and Funktionstheorie (function theory). The combination of these two approaches makes us regard the scale

degrees in a very concrete manner that restricts the possible harmonic situations. Theserestrictions are given by the functional relationships between the chords within a

tonality. For that the analyst has to take choices that involve his own way of hearing the

harmonies, that is interpreting them. As we will also see, sometimes several

interpretations may be possible, thus proving that this kind of analysis is far from being

mechanical.

3. Such an analysis does not tell about the function of the chords. Vertical approach

Òcan be misleading, for it ignores the origin, behavior and function of some of thechords.Ó4

This reproach comes often from the more linear approaches. Aldwell & Schachter and

Gauldin textbooks are representatives of the linear approach. But at this point we

should ask: What do we mean when talking about functions? This is a crucial issue that

is often overlooked on harmony textbooks. Both Aldwell & Schachter and Gauldinrefer to functions derived from Schenker's structural analysis, who differentiates

between harmonic or tonal functions and contrapuntal functions of chords. Wallace

Berry gives a concise definition of both functions:

ÒMelodic and harmonic functions are of two kinds: the first of these has to do with position,

identity, and hierarchic status in each of the system components of the particular tonality in

question (tonal function); the second is the role of the event in the melodic-harmonic linear

stream (linear function), also hierarchically defined with respect to a given level of

reference. Linear function is the relation of an event to the structural (relatively ÒessentialÓ)

2 Cook, N., A guide to musical analysis, p. 25

3

The term Conventional Knowledge will be used to designate the basic knowledge that is needed for atask and it is assumed to be part of the student previous knowledge.4 Aldwell, E. and Schachter, C., Harmony and voice leading , p. 55.



6

linear frame or basis, or its auxiliary (subsidiary, elaborative, embellishing, prolonging)

relation to an event of higher order.Ó5

Without questioning the validity of the reasoning of the linear function theory, westrongly believe that it overlooks one crucial issue: every chord has an inherent

harmonic function (even when this function is not fulfilled), namely Tonic, Dominant or

Subdominant.6 The idea of harmonic function results from the judgment that some

chords and tonal combinations sound and behave alike within a tonal center. We are

dealing then with what D. Harrison calls perceptual impressions.7 Although the

concepts of Tonic, Dominant and Subdominant were already in use since Rameau, it

was Hugo Riemann (1849 Ð1919) who fully developed a theory of harmonic functions.

Riemann also developed his own way of labeling according to his theories, in which heuses T, D, and S to designate the function of every chord, adding different symbols to

that letters to differentiate the chords that could realize that function (see example 3.14)If we agree on the functional understanding of the progressions proposed by Riemann,

why do not we keep his way of labeling then? In our opinion working with scale

degrees simplifies the notation of the chords, and facilitates the location of the chordswithin a key. This choice will imply that the information of function is not given in the

labeling of the chord itself (although our choice for a concrete label is certainly

conditioned by the function of the chord) and the function is only known by association

of concrete chords (and progressions) with functions.

Since our approach is functional, from now on when we refer to functions we will meanharmonic functions (otherwise it will be specified). Back to the critic, in any case

(harmonic or linear) functions in our system are not specified in the labels, but onlyunderstood within the context, again relying on the conventional knowledge.

The sources we will be using range from general music theory textbooks to specific

textbooks on harmony and analysis. The authorÕs own analytical experience will to

some extent condition his view on the subject but the intention is to be as critical withoneÕs practices as with the ones explained in the different sources. The application of

the procedures in musical examples from the repertoire will be the final test for they

validity.

Our final choices may not be perfect and definitive but hopefully will help to open the

discussion about the specific topics here considered.

What started as a study of the different ways of labeling harmony in harmonic analysis

will eventually become a whole revision of our way of hearing, analyzing and

understanding harmony.

5 Berry, W., Structural functions in music, p. 29

6

To make it more confusing Aldwell & Schachter and Gauldin use also this terms, but only to designatethe chords built on the I, V, and IV degrees respectively.7 Harrison, D., Harmonic function in tonal music, p. 36



7

General considerations for the elaboration of a ÒvocabularyÓ of labels

The criteria for elaborating a standard labeling system are diverse and complex. Each

case must be considered both in the general context and separately, because for the

different choices we make, it is possible that the specific weight of the criteria varies

from one point to the other.The objective is to describe harmonies in such a way that is clear and coherent. By

ÒclearÓ we mean that the signs used, maybe contrarily to the music that they are tryingto describe, are in themselves unequivocal and are not open to ambiguous interpretation.

By ÒcoherentÓ we mean that the first choices we will make will affect to the successive

ones, and also coherent in the sense that the labeling reflects the way in which we hear

and understand harmonic procedures.

In addition, we strive for a theory that is based on the music itself. The system will be

deducted from the music examples, not being a purely theoretical abstraction.

Another aim for our proposal is to be specific and generalized. Once delimited the scope

of the music that can be successfully subjected to such an analysis, in this case common

practice music, the system should strive to be as general as possible so as to describeany possible harmonic situation within that context. In practice this last purpose seemsutopian, as music is not a mathematical science but art, and in the second place because

we work a posteriori, deducing the procedures after the music itself and the determined

practices (of a particular author for example) may not lead to any generalization and

may be subjected to a specific stylistic gesture. In practical terms one of the goals is that

the description of an harmonic progression is valid for any key, so it is generalized.

With this we mean that the labels describe the relationships within a key, the prevailing

key, that is used as a system of reference and as far as the relationships between the

chords are the same, the same labels will be valid independently of which is thegoverning key (this point will be relevant on the discussion on figured bass in the first

chapter).Tradition will unquestionably have an important influence in our choices. It always will

be a reference to look at, with both respect and criticism. Nevertheless we think it would

be very pretentious (and ingenuous) to try to impose some radical changes in the way

theorists have been working for over two centuries. Experience tells that even good

ideas find difficulties to be accepted by the mainstream body of analysts-teachers-

theorists.

Finally, we will strive for simplicity and economy of means. The motto will be ÒDo not

use two symbols if one can be explicit enoughÓ. The goal is to avoid analysis overcrowded of symbols, that may diminish the clarity rather than improve the quality of

the description. A restrained selection of symbols and/or letters will make this practicemore accessible, especially for students.

When all these criteria are respected, the didactic usefulness of the procedures and its

value as an efficient tool for musical analysis will be assured.



8

1. Roman numerals and figured bass: A paradoxical association

The idea of using numerals to describe harmony consists in assigning a number from

one to seven to each of the triads that can be built on each note of a given diatonic scale.

The origins of this analytical practice can be traced back to the first half of the

eighteenth century (J. F. Lampe: A Plain and Compendious Method of TeachingThorough Bass, 1737). Different authors made use of numerals to describe harmonies

throughout the eigthteenth and nineteenth century, using either Roman (Schršter,Vogler, Weber) or Arabic numerals (Fšrster, Jelensperger).8 It is remarkable that in

these early stages the numerals could refer either to the lowest sounding voice or to the

actual root of the chord, depending on the author. Georg Joseph Vogler in

Tonwissenschaft und Tonse[t]zkunst (1776) seems to be the first in associating Roman

numeral to chordal roots, a practice that will eventually prevail.9 The use of Roman

numerals became standarized because of a practical reason: Arabic numerals were used

already for something else, and that could lead to confusion. The arabic numerals had

been in use since the begining of the seventeeth century, to indicate to the performers of

a bass line which tones could be added on top of this bass so as to complete theharmony. This practice is known as thorough bass. There was the possibility ofalterating the notes by means of sharps and flats, and this was indicated in different

ways. For simplification of use, some conventions were accepted in the course of the

years (as the use of only 6 for , or no figures for ). The information provided by those

numbers is purely descriptive and does not involve any kind of interpretation, only toadd the notes on the corresponding intervals. Besides, as Nicholas Cook points out,

figured bass Òdoes not categorize chords as such at all. It does not distinguish chordsfrom ÒnonchordsÓ- formations resulting, say, from passing notesÓ.10

Ex. 1.1 Notation and realization of figured bass. In b) the absence of numbers implies , and thealteration affects to the third if it is not specified otherwise.

Very often the note that is in the bass happens not to be the root of the chord. ThomasCampion (1567-1620) had already foreshadowed the principle of chordal inversion.

11

The Roman numerals are associated to chordal roots, whatever the lowest sounding

voice is, and they describe the fundamental progressions of the harmony, an idea that

was developed by Rameau with his concept of basse fondamentale. In harmonic

analysis it was necessary to make a distinction between chords in root position and

8 Detailed information about Eighteenth and Nineteenth century treatises on analysis can be found in

Damschroeder, D. Thinking about Harmony: Historical Perspectives on Analysis, Cambridge University

Press, 2008.9 Nevertheless it is possible to find isolated examples in 20

th century literature in which the Roman

numeral refers to the bass, like in the TraitŽ de lÕHarmonie (3 vols, 1923-6) by Charles Koechlin10 Cook, N., A guide to musical analysis, Oxford Universoty Press, Oxford, 1987, p. 17

11 Campion, Th., The new way of making four parts in counterpoint (ca. 1613)



9

inverted chords that shared the same root. For this reason figured bass numbers were

added to the Roman numerals so as to make explicit the differences between chords

with the root, the third or the fifth in the bass. So adapting the convention from figured

bass practice, the Roman numeral alone represented the root position (root on the bass),

the 6 for first inversion (meaning the third of the chord in the bass, and on top of that

note a third and a sixth, that actually is the root) and for the second inversion(meaning the fifth of the chord in the bass, and on top of that note a fourth Ðactual rootÐ

and a sixth). If we carefully examine what is the information that each element (the

Roman numeral and the Arabic numerals) provide we can observe that what the figured

bass is telling us is only descriptive, and that our nowadays immediate assimilation of 6

for first inversion and for second inversion is only associative, and has been accepted

by convention.12 This means that the standard labeling mixes two procedures that have

not only different origin but also a contrasting idiosincracy.

Roman numerals always refer to the root of the chord, without considering if it is the

lowest sounding voice or not, and are depending on the tonal context (prevailing key).

A chord formed by the notes c-e-g can be I in C Major, V in F Major, VI in e minor, etc.

This means that when we assign a Roman numeral to a chord we are not only describing

it but interpreting it within a tonal context. This implies that roman numerals work in ageneric way, meaning that a determined harmonic progression, for instance I-IV-V-I

will be notated in the same way independently of which one is the tonic key.

Ex. 1.2 Same chord progression (I-IV-V-I) in different keys.

The governing key is stated at the left of the Roman numerals, followed by a colon.

On the contrary, figured bass does not interpret the context, but neutraly identifies thenotes sounding at a precise moment. Moreover bass figuring relates always to the

lowest sounding voice, be this the root or not, and it is not absolute but relative, that is it

may be different to describe the same kind of chord in two different keys. For instance,

the need for a leading tone in a minor key will make that a dominant chord in the key

of a minor (in root position) will be notated with a sharp (to make the g sharp leading

tone), but in f minor the same functioning chord will be notated with a natural sign (so

as to rise the e flat a half step).13

12 In fact other strategies that a priori seem more direct have been proposed. Jacques Chailly uses a

system in which a triangle above the Roman numeral denotes root position and a number of dots above

the roman numeral indicate the inversion (TraitŽ historique de analyse harmonique, Alphonse Leduc et

Cie, Paris, 1977, p. 5) . Some English harmony textbooks from the beginning of the twentieth century

onwards, for instance E. ProutÕs Analytical key to the exercises in Harmony, its theory and practice

(1903), employ the letters b and c to refer to first and second inversions respectively. None of these

alternative systems has been strong enough to substitute the one originated from figured bass.13 In Baroque, # often means Òraising a diatonic scale tone a semitoneÓ, regardless of its diatonic

notation.



10

Ex. 1. 3 Different figured bass symbols within the same context: alteration of the seventh

degree in order to have a leading tone

This paradoxical situation becomes sometimes problematic, especially when dealing

with non-harmonic or ornamental tones, such us suspensions, neighbor and passing

tones.

When analysing harmonies we must therefore make a choice of which notes are part of

the structural chords and which ones are ornamental. Sometimes there are more than

one possible interpretations.

Ex. 1.4 Two interpretations of the cadential chord.

The analysis of bar two presents two possible interpretations. The upper analysis

interprets the chord in the first beat as a second inversion of the tonic triad, while the

second interpretation sees the whole bar as a dominant in which the 6th

and the 4th

function as ornamental tones, being suspensions to the chord tones. Both interpretations

can be understood. The second interpretation has the benefit of recognize the quality of

the non-harmonic tones e and c, but could be misleading, as if we consider the label V

alone we will interpret it as a second inversion of a dominant triad, so the 5 th would be

in the bass.

Ex. 1.5 Equivocalness in the interpretation of label V



11

What is different in these examples is the status of the sixth and the fourth. They are

structural chord notes in b), but in a) they are non-harmonic tones that resolve to

structural tones, the fifth and the third of the chord.

This dilemma has been an issue considered by many theorists. Those with a linear

conception of harmony (Aldwell & Schachter, Gauldin) will choose for

V interpretation adducing that a label such us I does not tell about the Òorigin, behavior and function of the chord.Ó

14

Ex 1.6 Notation of the cadential in a) Gauldin (p. 213) and b) Aldwell & Schachter (p. 183)

On the other hand, by using the I we avoid this ambiguous situation of having one

symbol for two completely different chords. A similar problem is shown in Ex. 1.7.

Ex 1.7 Equivocalness in the interpretation of label V6

The horizontal line between two arabic numerals prevents us from the misunderstandingof the chord as a dominant in first inversion15. This line denotes voice leading, implying

that one ornamental tone moves to a structural tone (root, third or fifth of the chord) orvice versa. This practice has also some disadvantages. In first place, if we look to

figured bass practice, the horizontal line is used with a different meaning, in fact

opposite. It does not mean that a voice moves but on the contrary it means that a note

stays still while other notes in the chord move.

A moving voice will be indicated by two successive numbers, without line in between.

This practice in itself can lead to confusion, as R. Hunt warns:

14

Aldwel, E. and Schachter, C., Harmony and voice leading , 4th

ed., Schrimer, Boston, 2011, p. 55.15 A variant can be seen in the example of Gauldin, in which the figures stand above an horizontal

bracket.



12

ÒThe student may find it difficult at first to distinguish between figured bass indicating

chords and that indicating suspensions. It should be remembered that, when two figures are

found next to each other under the same bass note, and the second figure is a consonance

(3rd, 4th, 5th, 6th or 8th), the first of the two will almost always indicate a suspension (...)

The number 7 in the second bar of ex. 26 (a) seems to suggest a [dominant]16

seventh in

root position; in 26 (b) Ðfirst minim of second barÐ that figuring is that associated with thefirst inversion of a [dominant] seventh; and the figuring in 26(c) suggest the third inversion

of that chord Ð but figures that follow in each case correct that wrong impression.Ó17

Ex. 1.8 Hunt, R.: A second harmony book , p.29

Problems also arise when the bass changes before the resolution of the ornamental tone.

Ex. 1.9 Change of bass note before the resolution of suspensions

The 9 8 suspensions are transformed into 9 3 and and 9 6 suspensions, because of the

change in the bass.

Ex. 1.10 Example of suspension over a changing bass in Gauldin (p. 214)

16

This is probably a mistake. As we can see in the example, the chord implied by the figure 7 alonewould not be a dominant seventh, but a minor seventh.17

Hunt, R.: A second harmony book , London, Herbert Jenkins Ltd., 1965, p. 29-30



15

Ex. 1.15 C. Franck, Violin sonata, I, b. 6-10

Louis & Thuille call situations like the one in example 1.15 Òconceptual dissonancesÓ19,

in which we deal with Òchords that are consonant when taken out of musical context,

but which are applied occasionally in a way which invites their interpretation as

harmonic dissonances.Ó20

Another aspect for consideration in the use of roman numerals is the alternative use ofcapital letters and lower case letters to make distinctions about specific chord quality.

An early example of this practice can be found on Gottfried WeberÕs Versuch einer

geordneten Theorie der Tonse[t]zkunst.21

This treatise starts with a extensive chapter on what Weber calls General musical

instruction, in which he thoroughly describes the tonal system, intervals, inversions,

chords, etc. In p. 166 he presents his method of designating the fundamental harmonies:

ÒFor designating a major three-fold harmony, we will use a large German letter. A large

German C, e. g. shall denote the major three-fold chord of c; C#, D, E , &c.

To designate the harmony of the minor three-fold chord, we will take small German letters,

as, e. g. a, c, c#, d, eb, &c. Which will accordingly stand as representatives of the minor

three-fold chords of a, c, c#, &c.

To denote the diminished three-fold chord, we will use the same small letters with a little

cypher prefixed, as, e.g. ¼b, ¼c, ¼d, ¼g#, ¼e , &c.

In order to represent the principal four-fold chord, inasmuch as it consists of a major three-

fold chord with a minor seventh, we will use a large letter with the figure 7 annexed to it; e.

g. G7, C

7 C#

7, B

7, A

7, E

7, &c.

For the minor four-fold chord, (with minor third, major fifth, and minor seventh), we will,

for similar reasons, employ a small letter with a figure 7, e. g. a7, c

7, c#

7, d

7, &c.

For the four-fold chord with minor fifth, we will use a small letter with a cypher (¼) and the

figure 7, thus: ¼b7, ¼c

7, ¼d

7, ¼g#

7, ¼f#

7, &c.

Finally, for the four-fold chord with major seventh, we will employ a large letter and a

figure 7 with a stroke through it (7); as, e. g. C722

, C#7F7,B7, &c.Ó

It is relevant how Weber explains that this system of designation was not of general use,

although he proudly points out that its validity and practical use was confirmed Òby the

19 Riemann calls situations like this Scheinconsonanzen (Òpseudo-consonancesÓ).

20 Louis, R. and Thuille, L. Harmonielehre, Stuttgart, Ernst Klett Verlag, [1907],

41913 trans. R. I.

Swchartz, PhD diss., Washington University, 1982, p. 65.21

Weber,G., Versuch einer geordneten Theorie der Tonse[t]zkunst , Mainz, Chott, 1817-21, 3rd

ed., trans.

by J. F. Warner as Theory of musical composition, boston, Wilkins and Carter, 1842-46, augmented by

John Bishop, London, Cocks, 1851.22 In this examples the 7 does not appear as superscript, like the 7 in the other seventh chords.

Nevetheless it appears as a superscript in further appearances, when used relating to roman numerals.



16

circumstance that, immediately after the appearance of the first volume of my theory,

other writers adopted them and appropriated them to themselves.Ó23

In order to generalize the system, Weber proposes to apply it in the same way to

Roman numerals:

Ò...instead of employing the German letters, we will use the Roman numerals to denote thedegrees of the scale on which chords have their fundamental tones; and in the place of large

letters, we will use large numerals, while in the place of the small letters we will use small

numerals, and we will mark these numerals with the characters7,7, and ¼, just as we did the

German letters.Ó24

This clever arrangement could be questionable regarding the possible qualities of the

triads and seventh chords described. In a lengthy remark Weber exposes his reasons for

his choices:

ÒMen dispute, and contend and quarrel, on the question, how many fundamental harmonies

there are (...) The question here cannot be, how many genera there probably are, but only,

into how many the species admit of being most conveniently arranged, in order to bring thegreatest possible number of species agreeing with one another in the largest number of

common characteristics under the smallest possible number of principal clases.Ó25

Weber continues naming some examples from other theorists who describe other chordsas the superfluous (augmented), the major-diminished or the double diminished triad,

and a series of seventh, ninth and thirteenth chords. Weber concludes:

Ò... my seven fundamental harmonies are perfectly adequate to the explanation of all the

harmonic combinations occurring in music, and it certainly is better to make enough out of

a little, than out of much.Ó26

The seven fundamental harmonies proposed by Weber are thus:

- Triads (three-fold chords): Major, minor, diminished- Seventh chords ( four-fold chords): Principal (dominant seventh), minor four-

fold chord (minor seventh chord), four-fold chord with diminished 5 (half

diminished seventh chord), major four-fold chord (major seventh chord).27

The discussion about which chords should be considered as self-standing harmonies is

still open nowadays and that choice will notably condition the way we understand, and

therefore label, harmonic progressions.

In common practice, chords are built in superposed thirds. There is a general consensusin harmony books and music theory books about the self-standing status of the

following chords:

23 Weber, p. 167

24 Weber, p. 286

25

Weber, p. 16526 Ibid.

27 The same chords were already described by Kirnberger.



17

Ex 1.16 Triads and seventh chords in Aldwell & SchachterÕs Harmony and voice leading

What all these chordal formations have in common is that all can appear within a

diatonic context, considering the major scale and all three forms of the minor scale

(natural, harmonic and melodic). Nevertheless common practice music literature abounds in examples of chordal

formations that do not fit in any of these categories.

Ex. 1.17 J. S. Bach, B minor Mass, Credo: Crucifixus (vocal parts only)



19

Weber, as well as other theorists of his time, was aware of these chordal formations but

tried to explain them in relation to their fundamental chords. Thus Weber would explain

the chord consisting of a diminished triad and a diminished seventh (diminished seventh

chord) as being a principal four-fold chord (dominant seventh) with an added minor

ninth (that he calls Òindependent ninthÓ, so as to distinguish them from suspension

ninths) in which the fundamental tone is omited. The reasons for this radical choice(designate as root a tone that is not actually sounding) are at first just obviated:

ÒInasmuch as the minor ninth, in such a chord, stands at a distance of a diminished seventh

from the proper third, it not unfrequently obtains in musical usage the appellation

diminished seventh-chord or chord of the diminished seventh. We will willingly adopt this

name in our technical language, though we must not forget, meanwhile, that what is here

called a seventh is not properly such (not a fundamental seventh), but is strictly a minor

ninth, which is here denominated a seventh, merely on the ground that, being reckoned

from the base tone onward, it is the seventh tone.Ó30

Later in text Weber explains that the relationship between a dominant chord and a triad

a fourth higher (or fifth lower) was akin to the relationship between the diminishedseventh chord and a triad a half step higher (remark p. 206). This similar relationship

was later explained by Riemann and his theory on harmonic functions. Actually

Riemann notation shows clear connection with WeberÕs reasoning but seems necessary

to bear in mind the remark that Diether de la Motte makes in his Harmonielehre to that

practice: he considers that the diminished chords can not be derived from a ninth chord

unless the ninth chord exists as such, and this will not happen in baroque and classical

music. For this reason he proposes de label sDv , acknowledging then the autonomous

status of the chord, as well as its functional duality.Weber will employ similar deductions for the explanation of the other chordal

formations that did not belong to his fundamental harmonies.

In the following example we see how the chord marked with an arrow is identified as asecond degree in e minor, assuming the nonexistent f# as root. Weber sees the chordal

formation c-e-g-a# as a modified half diminished chord, in which the third is raised anda ninth is added.

Ex. 1.19 Weber, The theory of musical composition, vol. 1, p. 344

This chord is in fact one of the chords described by Willemze, the double diminished

seventh chord. Accepting it as a fundamental chord prevents us from assuming a

fictional tone as root.

30 Weber, p. 198



20

In spite of the essential differences in origin and meaning between roman numerals and

figured bass they have often been combined for harmonic analysis. In the search for a

refined method we will have to take into account the status of the tones (chord tones or

ornamental tones) and the variety of chordal formations that in practice seems to be

bigger than the usually described chord types.



21

2. Diatonic harmony

a) Major and minor scales

By diatonic harmony we mean that the chords used in the progresions only use the notes

of a diatonic scale. The diatonic scales used in common practice music will be mainly,and almost exclusively, the major scale and the minor scale.

On each step of the major scale we can build triads, by adding the diatonic third andfifth on each degree, giving place to the following:

The triads are identified according to the place that the lowest of its tones (root) has

within the scale. As we saw in the previous chapter, Roman numerals have been chosen

for this purpose. Besides, each degree is also known with one name:

I: Tonic

II: Supertonic

III: Mediant

IV: Subdominant

V: Dominant

VI: Submediant

VII: Leading-tone triad

The quality of the triads in the major scale is diverse. Thus the I, IV and V triads are

major (major third and perfect fifth). The II, III and VI triads are minor (minor third and perfect fifth), and the VII triad is diminished (minor third and diminished fifth).31 These

qualities remain invariable whatever the key is, provided that it is a major key.

Common practice music is characterized by the different relationships between the

triads in a key. These relationships are explained by the theory of harmonic functions,

that was thoroughly described by Hugo Riemann. According to this theory every chord

has a function within its context in a progression. There are only three possible

functions: Tonic, Subdominant and Dominant. The designation of the functionscoincides with the name of the triads that most strongly fulfill those functions: I, IV and

V respectively.

32

In few words we could describe the tonic as representing stability and rest. Dominant

represents tension against the tonic, and a need of resolve this tension by going to the

31Althoug there is a certain tendency, specially in American texts, to use lower case numerals for the

minor triads as Weber proposed, I will stick to the use of capitals only. As we saw in the previous chapter

the duality expressed by the capital/lower case roman numerals excludes the possibility of notating chords

containing a diminished third.32

Refering to both triads and functions with the same terms may lead to ambiguous interpretations. Etyan

Agmon in his article ÒFunctional harmony revisited. A prototype-theoretical approachÓ proposes that the

terms Tonic, Dominant and Subdominant Òare used exclusively to refer to functions, that is categories of

chords; reference to individual chords is made by Roman numerals. II, for example represents an abstractcategory (namely the subdominant), which is represented also by IV; at the same time IV is of course the

more prototypical representative of that category.Ó (Music Theory Spectrum, p. 204)



22

tonic. The subdominant represent another kind of tension (more ÒvagueÓ), that may be

resolved directly to the tonic, or to the tonic via the dominant.

In a general way each function can be seen as a category, in which different chords can

be put.Ex. 2.1 Diatonic chords grouped according their function

Functions Tonic Dominant Subdominant

Chords I, VI, III V, VII, III IV, II, VI, (VII)

When dealing with the minor mode we find that there are three different versions of this

scale: the natural minor, the harmonic minor and the melodic minor.

The natural minor scale employs the same pitches as its major relative (in the example,

the c natural minor scale uses the same pitches as its relative E major). The fact that the

seventh degree is a whole tone distance to the first degree33 (as opposed to what

happens in major, where the leading tone is a half step apart from the tonic)

makes thatit lacks the strong tendency to resolve to the tonic that is so characterisitc of functional

harmony. For this reason, the seventh degree is rose a half step in the harmonic minor.

This alteration will make that the distance between 6 and 7 becomes the unmelodic

interval of an augmented second. To avoid this melodic gap, the so called melodic

minor scale will add an ascending alteration to the sixth degree, so that the augmented

second is avoided. As the seventh degree only needs to be a leading tone when it

proceeds to 1, when descending it can be kept in its original form, and consequently sodoes 6, so descending the melodic minor will be exactly as the natural minor.

The specific importance of each of these three versions of the minor scale on strictly

harmonic grounds is uneven. There is a clear preference for the use of the seventh as

leading tone instead of subtonic. Nevertheless on harmonic grounds the use of the sixthdegree of the natural scale seems to be more often used than the raised form, that will be

used whenever the melodic considerations described above apply and occasionally for

coloristic purposes (see ÒMixture ChordsÓ below).

How will this affect the task of harmonic analysis?

The scale we choose as primary for the minor mode will condition our way of labelling.As V will have for the majority of cases a major third, that is the seventh of the scale

will be leading tone and usually the root of VII will be a half step apart from the tonic, it

seems wise to take these (the ones derived from the harmonic scale) as the primary

33 In this case the seventh degree is known as Subtonic, to differentiate it form the seventh degree that lies

a half step under the tonic, that is called Leading-tone.



24

Ex. 2.3 Labelling of chords built on the seventh degree as subtonic and as leading-tone in Aldwell E.,

Schachter, C., Harmony and voice leading , 4th

ed. Boston: Schirmer, 2011, p. 276-7

The procedure employed in c), this is adding the alteration that negates the leading-tone

to the roman numeral, presents to my view one important disadvantage: the alteration

symbol will change in some keys to a flat (for example in c minor: VII) or even to a

sharp! (in g# minor: #VII meaning a lowered seventh degree). Since one of the premises

we considered necessary for the elaboration of a vocabulary of labels for analysis was

that it should be valid independently of the key, we discourage this practice. In ouropinion, as the meaning of the Roman numeral stands independently of the key, so

should do the alterations that modify this numeral. By this we mean that VII will stand

for a lowered seventh degree (subtonic) independently from the alteration that is neededin a specific key.

Ex. 2.4

The apparent contradiction of using an alteration to indicate a diatonic degree is justified by the fact that the leading-tone is more frequently used both in VII and V than

the subtonic to the point of being the a priori expected 7, and therefore subtonic is seen

as a deviation of the rule.

Once accepted the harmonic scale as primary framework we notice how on the third

degree we would get an augmented triad (a minor: c-e-g#). As Aldwell & Schachter

point out, this augmented triad Òis more visible in harmony books than audible in realmusic. The basic form of III in composition Ð a major rather than an augmented triad Ð

is the one derived from the natural form of minor.Ó36

In that case, as in the fifth degreewith minor third, it seems necessary to specify that the chord is using the subtonic

instead of the leading tone. This can be done in different ways:

36 Aldwell & Schachter, p. 49-50.



25

- By using figured bass symbols, e. g. III»

- Specifying the scale to what the chord belongs, e. g. IIInat, or IIIaeol (aeol stands for

aeolian, the mode corresponding with the natural minor scale).

- Specifying the quality of the chord.

Figured bass symbols again will result in differences in notation according to the key.

Looking forward, in chromatic harmony the procedure of specifying the scale from

where chords can be deducted will result unsuccessful, so we will choose for specifying

the quality of the chords (for example #IV double diminished). For that reason it will be

coherent to do so also in diatonic harmony. The labelling for diminished (¼) and

augmented (+) is relatively standardized (Gauldin, Benward & Saker, Blatter). For

minor triads those authors use lower case, but Aldwell & Schachter usually do not make

any distinction.37 Since we declined to use lower capitals, we should find another way

to show the minor triads. In this case we could borrow a symbol used in jazz notation.For minor chords they use different labels, such us: D-, Dm, Dmi. Considering that we

are using symbols for diminished and augmented, it seems then logical to use the - symbol for minor.

38

In this case we will have the following labels:

Ex. 2.5 Labelling of triads in the minor mode.

Of course this symbols sould be applied also to the triads in the major mode, in the

same way:Ex. 2.6 Labeling of triads in the major mode.

b) Inversions

We will stick here to the most usual way of labeling the inversions, that is with figured

bass notation. As we saw in the last chapter, this method will be unequivocal once we

understand and differenciate the chord tones, from eventual ornamental tones.

Thus the Roman numeral by itself will denote root position. The first inversion (third of

the chord in the bass) will be labelled with a 6 in the right of the Roman numeral (as an

37 One of the few spots where they do so is in minor chords on the fifth degree. They use the minus sign

before the roman numeral, e. g. ÐV. As the standard practice for augmented and diminished symbols is to

use them after the roman numeral, we propose that the symbol for minor is used in the same position.38

This is also the choice of Mark Levine in his book The Jazz Theory Book . He also mentions that the

minus (-) sign can also be equivalent in some contexts to the flat ( ) sign (C7 9

= C7-9

), but maybe to avoid

confusion, he recommends to stick to the flat symbol in such cases.



26

abbreviation for ) and the second inversion (fifth on the bass) with a 6 and a 4. When

only one figure is used it will be written as a superscript, following the most usual practice. When several figures are needed they will appear from the top in descending

order.Ex. 2.7 Labelling of inverted triads.

c) Seventh chords

By adding a note a third higher on top of a triad we get a seventh chord. The interval of

a seventh between the bass and the higher note is dissonant and it will require special

treatment.

Ex. 2.8 Seventh chords in major and minor

As we did with the triads, we will proceed to identify the seventh chords with the degree

in which they are constructed, plus the identification of the kind of seventh chord they

form.We will start with the chord on the fifth degree in major and harmonic minor. It was

properly called the principal four-fold chord by Weber, and it is certainly the most

commonly used seventh chord. Today it is known as dominant seventh chord and it is

unanimously labelled V7. In this case the minor seventh (given by the diatonic) scale is

added to a major triad. So it does in VII in minor and the label VII7 can be

consequently used.39

The other major triads we find (I, IV in major and III and VI in minor) will add adiatonic major seventh, thus forming major seventh chords on those degrees. For thesecases (and obviating for a moment the fact that some of these chords are more likely

than others to appear in common-practice music) there is not a standard way of

labelling, but most authors again make no distinction with the label for a dominant

seventh and would use the label IV7 for both a chord with minor or major seventh,

arguing that the context will indicate what the label stands for.

39 Nevertheless an alternative label for this chord will be suggested when it will be considered in relation

with its harmonic function (secondary dominant for III in minor). See the chapter Secondary functions.



27

Ex. 2.9 Same notation for two different seventh chords

in Gauldin, Harmonic practice in tonal music, p. 225

I think that if we choose for stating the quality of the chord in the label, we must be

consistent and avoid ambiguities like this.The solution may come again from a label borrowed from jazz for the major seventh

chord: a superscript triangle ( ) at the right of the Roman numeral, plus the seventh

indication of seventh chord (for example IV 7).

For the minor seventh chord we can use the same symbol as for the minor triad (-), plus

a seventh denoting the seventh chord (II-7).

Diminished and half-diminished seventh chords have already standard symbols that are

generally accepted and used: for the diminished seventh chord we take the ¼ , used forthe diminished triad and add a seventh. For the half-diminished seventh the standard

notation is ¿ plus the superscript seven.The remaining two chord qualities, minor-major (II and VI in minor) and augmented

seventh chord (III in harmonic minor), hardly ever will be used as self-standing chords.

In the minor-major seventh chord the seventh is usually an ornamental tone, resolving

to the tonic. Nevertheless, in case we find a minor-major self-standing chord, it could be

unambiguously labeled in this way: I- 7, the minor symbol standing for the minor triad

and the standing for the major seventh. In the same way, an eventual augmented

seventh chord could be labelled III+ 7, the plus standing for the augmented triad and the

standing for the major seventh.

With this limited array of symbols we can deal with all the diatonic triads and seventh

chords.

Although the symbols and¿ directly imply seventh chords (they are not used for

triads), it seems coherent to add the figure seven also to them.



28

Ex. 2.10 Labelling of seventh chords

d) Inversions of seventh chords

Here again we consider the usual practice derived from figured bass as an efficientmethod for identifying the different inversions of the seventh chords. Thus, the root

position will be denoted with the superscript seven. The first inversion will be labelled

(standing for 6/5/3), the second inversion (standing for 6/4/3) and the last inversion 2

(standing for 6/4/2).40

Ex. 2.11 Labelling of inverted seventh chords

40

Some authors use (Aldwell & Schachter argument that this notation is more often used than 2), butthe use of the shorter version is preferred by a simple matter of economy of means: the 2 itself

unambiguously denotes the third inversion of a seventh chord.



29

3. Chromatic harmony

a) Secondary functions. Tonicization.

One of the most important concepts in RiemannÕs theory of harmony is the idea of

secondary or applied functions.This theory allows us to understand certain chords that use non-diatonic pitches. It can

be explained in the following way: any major or minor diatonic triad other than thetonic may be preceded by a chord whose function is understood in relation to the chord

that it precedes, without losing the feeling of the main key. For a short moment the

diatonic chord that follows the non-diatonic chord (in relation to the main key)

functions as a local I. We say that this chord is momentarily tonicized.

Ex. 3.1 Beethoven, Piano sonata op. 13 n.8, II

The 8 first bars of the Beethoven example are clearly in A-flat major. However, in this

fragment we find three pitches that do not belong to the main key. While the e on bar 4

can be understood as an ornamental tone, namely a chromatic passing tone between the

e and the f, in the other two altered notes of this fragment the case is different. The

alteration from d to d on bar three results on a dominant chord on b , that is followed

by a E chord, V in the key of A-flat. Locally the relationship of the B

7

chord and theE chord is V-I, but the feeling of A-flat as governing key is still strong. Similarly in bar

6, the a produces a dominant chord on f that is followed by a B minor chord (II- in A )

keeping the local V-I relationship, without losing the feeling of A major, the key that is

confirmed by an authentic cadence in bar 8.

The principle of secondary functions is widely accepted as explanation for that kind ofchromaticism. The ways of notating it are based on two different procedures:

The first one is to notate the altered chord as a modified diatonic degree. This can be

done by a horizontal line across the Roman numeral (Schoenberg) or by figured bass

signs applied to a diatonic degree (Schenker). Figured bass has the already mentioned



30

inconvenience of changing according to the different keys, and the method used by

Schonberg does not give any information about the way in what the chord is modified.

In addition none of those labels make reference to the concrete function of the chord.

Ex. 3.2 Secondary dominant notated as a modified diatonic degree in Schenker, Harmony, p. 64

Ex. 3.3 Secondary dominants in Schšnberg, Structural functions of harmony, p.19

The second procedure labels the secondary functions in relation to the degree that they

temporarily tonicizy. It is in this way that most present day standard text books

approach this situation. While there may be differences in the concrete way of labelling,

all of them seems clear and unequivocal. Still we can find small nuances that entail

every approach:

V of V(Piston), V/V, (Gauldin). This approach is very clear. The only disadvantage

could be the redundance, because the chord that is tonicized appears frequently after thesecondary function. This is avoided in the following way:V!V (Aldwell & Schachter)

or (V)V (derived from Riemann, using numerals instead of letters, see example infootnote 41 below). This last approach is especially practical for extended tonicizations

(more than one secondary chord, see ex. 3.10 below).

Ex. 3.4 Alternative notations for secondary dominants



31

Very often we find sequential chains of secondary dominants like the following:

Ex. 3.5 Chain of secondary dominants

From my point of view in this case the use of the arrows has the advantage of showing

the relationship between two consecutive chords, i. e. the B7 chord functions as a

dominant for an e chord, that itself is modified to become E7, and function as dominant

to a. The fact that at the same time that the dominant is resolved, the chord is modified(but keeping the same root) does not cancel the sense of resolution of the first secondarydominant in the next chord, although on the background we have the following

progression:

Ex. 3.6 Progression in the background of a chain of secondary dominants

Taking into account both the chains of secondary dominants and the extended

tonicizations the use of brackets and arrow seems the most suitable system. The

addition of the arrow is convenient because of two reasons. Linear based analysis use

brackets with a different meaning (ornamental chords within a fundamental

progression) so the arrow makes explicit that we are not using that procedure. Besides

sometimes the secondary function can only be explained in relation with the chord that

precedes it and not with the chord that follows. In such cases an inverted arrow will beuseful to show that circumstance.

Ex. 3.7 Back relating secondary dominant

Every chord that can function as a dominant can be used as a secondary dominant, and

they are consequently labelled in relation to the chord that they tonicize.



33

Nevertheless they describe another possible situation, that more that one chord (usually

a subdominant and a dominant) are used for a temporarily tonicization of a diatonic

degree. This is called extended tonicization.

In example 3.10 we see in bar four a D7 and a E7 chords in first inversion that can be

seen as IV and V , relating to the following a minor chord, VI in C major. This is

repeated sequentially, a third lower resulting in IV and V to a F chord, IV in C major.

The dominant pedal starting at bar 7 confirms that the fragment should be considered as

a whole in C major, with the mentioned tonicizations.

Ex. 3.10 Bach, Kleine prelude BWV 924, mm.1-8

As any other dominant, a secondary dominant can sometimes not resolve to its local

tonic, but resolve deceptively. One of the most common situations is that the dominant

is followed by a chord whose root is a second higher, producing a local V-VI

progression.

Ex. 3.11 Deceptive progression quoted in Gauldin, Harmonic practice in tonal music, p. 368. The notation used by Gauldin does not show the close relationship between the B

7 chord and the C major

chord. Both chords can be understood as a V-VI progression in e minor (VI of the main key, G major)



34

Ex. 3.12 Schubert, Quartet n. 14, I, mm. 13-21

In the Schubert example we see a variant of the V-VI progression in which the VI

appears in first inversion. Although the E chord in bar 17 and the F chord in bar 18 can

both be understood in d minor ( II and III respectively) the second interpretation

proposed seems to be more telling of the meaning of these progressions. Square

brackets denote a chord that is implied but finally is not coming. This notation was

proposed by Riemann (see ex. 3.14 below).

Some authors, like Aldwell & Schachter and Laitz & Bartlette, have a different vision

about similar chord progressions. Aldwell & Schachter talk about apparent applied

chords, and show the following example (3.13) to indicate that:

ÒRather frequently a chord that appears to be V (or V7) of VI moves to IV rather to VI. The

progression is a varied form of I-(III)-IV, as you can see by comparing Example 26-26 with

Example 16-10a, a quotation from the beginning of that song. As we have seen before the

bass of a seventh chord may move by ascending 2nd to accommodate the resolution of the

seventh. The B7

chord on beat 2 of bar 14 (Example 26-26) represents simply an alteredform of III

7 moving up a step to IV. Only if the larger context strongly suggest E minor as a

temporary center (and here it does not) should we think of such a progression as a deceptive

resolution of an applied dominant (V7 of VI moving deceptively to VI of IV).Ó

42

Ex. 3.13 Aldwell E., Schachter, C., Harmony and voice leading , p. 483

Aldwell and Schachter finally point out that the III# chord is an example of secondary

mixture. Here we deal not only a with different way of labeling, but with a different

way of understanding harmonic progressions. Aldwell & Schachter prioritize the

relevance of a preconceived (and indeed present earlier in the piece) tonal succession I-

42

Aldwell E., Schachter, C., Harmony and voice leading , p. 482



35

III-IV, over the functional meaning of the chords. From a functional point of view I

think that the strong sense of direction that a dominant chord has (and especially when it

has its seventh) is so strong that it can not be neglected as such even when it is not

confirmed by the expected resolution. In this kind of harmonic analysis it is very

important to consider the expectations of the listener. In ex. 3.13 the expectation of an e

minor chord is undeniable, as it is undeniable that when the B7

is followed by the Cchord we hear a local V-VI progression. Indeed this is a variant of the III-IV

progression that we hear at the beginning of the piece, but this does not mean that thefunctional meaning of the latter should be overlooked. We could also say that the

functional value of a secondary dominant is intrinsic, and it is not dependent on the

resolution or not of that dominant. We strongly believe in this property as a very

important point in our understanding of common practice harmony. Within a context of

functional harmony the expectations of the listener (be the expectations intrinsic by the

music itself or created by education or/and previous experience) are a decisive factor to

understand harmonic progressions.

As we see in the ex. 3.14 Riemann also acknowledges secondary dominants resolving

deceptively. He notes between square brackets the chord that is implied by thesecondary dominant, but is not coming, because it is substituted by another chord.

Ex. 3.14 Riemann, Harmony simplified or The theory of the tonal functions of chords, p.130

b) Modulation

Generally speaking we define modulation as a change of key. This change of keyusually becomes explicit by the use of chromatic pitches, out of the diatonic collection

of the initial key. This use of non-diatonic pitches is also happening in tonicizations, but

in those cases the initial tonal center is not completely lost, as they are relatively short

and the music proceeds in the same initial key after the chromatic alteration. But if this

tonicization is longer, the feeling of the initial key may be lost, and we would have amodulation. Since the sensation of a change of key may vary from one listener to another,

in some situations the line that separates tonicization from modulation might be not very

precise. Besides some authors make distinctions between transient modulations and

extended modulations. The elements that condition our way of hearing the change of tonal center are mainly

two: the way the transition from one tonality to the other is achieved and the way the

music continues afterwards.

Concerning the first element, three types of modulations are usually described in

textbooks: diatonic, chromatic and enharmonic.

A given chord can occur in more than one key as a diatonic formation. This feature is

used in the diatonic modulation. In example 3.15 we see how the g minor chord in the



36

second half of measure 11 is understood as VI in B major, but as the music continues it

is reinterpreted as II in F major. This key is subsequently confirmed by a imperfect

authentic cadence V -I. The unequivocal continuation of the piece in F major

(alternating dominant and tonic until bar 23, where eventually the second theme will

appear in the expected key of F, V of B ) confirms us that the C7 chord of bar 12

introduces a long tonicization, and thus a modulation. The chord that is belonging to

both the initial and the new key is called pivot chord.

Ex. 3.15 Mozart, Sonata in B Major, kv 333, I, mm. 9-15

There is a standardized practice to notate this kind of modulations in which the new key

is notated at a different line, so in the pivot chord the two identities of the chord arenotated one below the other.43 In these cases Gauldin recommends to indicate the goal

of the modulations according to their scale degree and triadic function and not

according to the designation of keys by name of the tonic. He argues that Òin closed

tonal forms, which are those that begin and end in the same key, we should still

consider these areas [modulations] as extended tonicizations of diatonic triads. Whenwe do so we never lose sight of their function in terms of the overall tonal scheme of the

entire movement.Ó44

This statement seems far to general. Consider for example the first movement of

BeethovenÕs Symphony n. 3 in E Major, ÒEroicaÓ (1804). It starts and ends in E

Major, so it has a closed tonal form. Nevertheless halfway the development (bar 284) anew theme is introduced in e minor, a key that is by no means a tonicization of a

diatonic triad, but actually a modification of a chromatic degree ( II).45 Besides if we

relate all the modulations to the initial key we lose the relation between the consecutive

keys. For this reason we think that it is better to label the key of the modulations with

the letter of the key. In this way we can more easily check the relationship both to the

initial key and the key immediately prior to the modulation.

43 Mulder adds the symbol Ò=Ó (probably borrowed from Riemann), and other authors enclose the two

identities of the pivot chord within a square but we think that the notation is already unequivocal without

any of these markings.44 Gauldin, Harmonic practice in tonal music, p. 368

45 The key of e minor is the result of a series of modulations in ascending fifths: f Ðc Ð gÐdÐe.



37

Why is it important to know both relationships? LetÕs put an example. Imagine a piece

in C Major in which we find a section (in the development, for instance) in F# Major,

the furthest key, in terms of the circle of fifths. In such a case it will make a big

difference if that key is achieved by means of intermediate keys following the circle of

fifths (for example C, G, D, A, E, B and finally reaching F#) or by means of a direct

enharmonic modulation.

The second type of modulation described in textbooks is the chromatic modulation. Wefind different views on this subject.

According to Aldwell & Schachter Òthe term chromatic modulation is often used to

describe a key change in which the pivot is a chromatically altered chord in one or both

of the keys.Ó46 This view is shared among others by Riemann , Louis & Thuille and

Kostka & Payne. A chromatically altered chord is a chord in which one (or several) of

his tones does not belong to the diatonic scale, because it is modified by an accidental

sign. This kind of chromatic alterations can result in different kind of altered chords,

such as secondary dominants and secondary subdominants.

Ex. 3.16 Pivot chord in chromatic modulation in Kostka, S., Payne, D., Tonal harmony, p. 318

On the other hand other authors, like Gauldin and Benward & Saker, do not

acknowledge the presence of a pivot chord.

Ex. 3.17 Chromatic modulation in Gauldin, Harmonic practice in tonal music, p. 262

In fact, Benward & Saker call Common-chord modulation to what we defined as

diatonic modulation, thus making explicit that there is not a common chord in a

chromatic modulation. In their opinion Òa chromatic modulation occurs at the point of a

chromatic progression (a progression that involves the chromatic inflection of one or

more tones). The letter name remains the same in a chromatic progressionÑfor

46 Aldwell E., Schachter, C., Harmony and voice leading , p. 551



38

example, in the following Bach chorale [ex. 3.18]. At chord 2, the tenor is D, but in

chord 3, the D becomes D-sharp.Ó47

Ex. 3.18 Chromatic modulation in Benwards, B., Saker, M., Music in theory and practice, p. 317

In our view the existence of a common chord in both GauldinÕs and Benward & SakerÕs

examples is undeniable. In GauldinÕs example (3.17) the first chord of the second bar (a

G major chord) can be understood as V in C major, and the previous chord can be

interpreted then as a secondary dominant to the dominant. The G chord in bar two is

then reinterpreted as I in G major, so it is the actual pivot chord. We find a similar

situation in Benward & SakerÕs example (in this case the e minor chord is the pivot

chord: it is VI in G and I in e). Actually in both cases we deal with diatonicmodulations.

Moreover, if we go back also to the Kostka & PayneÕs example (3.16), we could also

interpret the D chord in bar 15 as V in G and I in D (being the previous chord again a

secondary dominant for the dominant). If we understand in this way the progression, the pivot chord is diatonic in both keys, so it is a diatonic modulation. This interpretation is

confirmed in bars 18-19, where we find an analogous progression, actually a half

cadence, in G: II-6-V /V-V.

But then, what is a chromatic modulation? We agree with Aldwell & SchachterÕs

definition, but it should be precised that the chord after the pivot chord must not be

diatonic chord in the initial key (this is what happens in the Kostka & PayneÕs example)otherwise the altered chord can always be understood as a secondary dominant or

subdominant, and the feeling of the modulation is delayed at least until the next chord.

47 Benwards, B., Saker, M., Music in theory and practice, p. 316



39

Ex. 3.19 Chromatic modulation in Schubert, Quartet n. 14, I, mm 233-241

The closer the keys are in terms of the circle of fifths, the more possibilities are that amodulation is diatonic, because close related keys share several diatonic triads. Still, it

is possible to make chromatic modulations to keys that are only two fifths away.

Ex. 3.20 Chromatic modulation to a key two fifths away

The next example, taken from the development of the first movement of BrahmsÕ fourth

symphony shows a series of chromatic modulations in which the pivot chord is chromatic in

both the initial and the new key.

Ex. 3.21 Brahms, Symphony n. 4, I

b¬:(V7)! V7 II¿

V2 (V7)! [VIB¬ M]

g#/ a¬: IV- V

7II

¿ V

2 (V

7)![VIA¬ M] c: (V

7)! V

7 (V )![VIC M]

d: (V )! V7

(V )![VID M]

e: (V )! V7 II¿

This example also poses a new question: Can we speak of a modulation when there is not acadence and not even a tonic? Traditionally textbooks make distinctions between temporary



40

(or embellishing) tonicizations, transient (or intermediate) modulations and extended (or

true) modulations. It seems that there is agreement about the concept of temporary

tonicization, that implies that the sense of the main key is not lost. Regarding the

difference between transient and extended modulations there is no universal agreement

among the different authors. While according to Blatter Òa true modulation

requires, in the minds of most theorists, both the existence of a pivot chord and acomplete (authentic) cadence in the new keyÓ48, Gauldin states that if a tonal shift is not

confirmed by a continuation in the new key, even when this key have been confirmed

by a complete cadence, we are dealing with a transient modulation. The next phrase

may either immediately go back to the original tonic or move to yet another key, as he

shows in the next example

Ex. 3.22 Gauldin, Harmonic practice in tonal music, p. 379

In BlatterÕs view, the move to d minor will be considered a real modulation (the d minor

chord in bar 194 would be the pivot chord C: II-6

= d: I-6

), but the shift to a minor will

be a transient modulation. GauldinÕs approach has the advantage of showing the large

scale structure: an extended section on the supertonic region and a smaller section on

the submediant region to finally go back to the initial key.

In any case the subtle differenciation between temporary tonicizations, transient

modulations and extended modulations seems to some extent irrelevant when trying to

48 Blatter, A., Revisiting music theory, p. 135.



41

Òquantify something that admits of innumerable shades of grayÓ using StellÕs words.49

A more useful differentiation would be between those keys that have structural

importance and those which does not. These structural differentiations are not given by

the foreground harmonic analysis (the one we are dealing with) and therefore ask for

supplementary comentary.

The last procedure for modulation is the enharmonic modulation in which the pivot

chord needs to be respelled to be understood in both the original and the new key.

Ex. 3.23 Enharmonic modulation

Two notes in diminished chord on bar 4 (C: VII¼7) are enharmonically respelled so that

the resulting chord is understandable in the new key as VII¼ . The altered chords that

will be discussed in next chapter will add more possibilities for enharmonic changes.

49 Stell, J. T., The flat-7th scale degree in tonal music, p. 151



42

c) Altered chords

The explanation and classification of altered chords is probably one of the most

controversial topics in the theory of tonal harmony.

Traditionally, textbooks deal with altered chord by making different categories:

Mixture, Neapolitan and Augmented sixth chords, usually adding another Ðratherambiguous- category for other chromatic chords (Aldwell & Schachter) or

embellishing chromatic chords (Gauldin). LetÕs have a look to each of this categories.

Mixture

In the broad sense the concept of mixture, as the use of chords in a key that belong to

the parallel key (using chords from c minor in C major or vice versa), is commonly

accepted by theorists but their application in practice may differ quite a lot.

In the next example Gauldin (p. 394) compares the scale steps of the major and minor

scales.Ex. 3.24 Gauldin, Harmonic practice in tonal music, p. 394

giving place to the following mixture chords in the major mode:


Gauldin assumes that »6 and »7 already belong to the melodic minor scale, and the

chords that use those notes are not borrowed from major. Besides he considers that theuse of a major tonic at the end of a minor piece (Picardy third) can hardly be regarded as

an example of mixture.In addition he talks about the use of III in major, explaining that Òthis chord is

ÒborrowedÓ from the dominant harmony of the relative minor key.Ó50

In that case wewould be dealing with a completely different type of mixture:


50 Gauldin, p. 402



44

Ex. 3.27 Embellishing chromatic chords according to Gauldin, Harmonic practice in tonal music, p.502

In this classification Gauldin rejects again any functional interpretation by emphasizing

the quality Ònot applied chordsÓ in example 19-A55

(some of those chords Ðthe E-major,

A- major and D-Major in the key of C MajorÐ correspond to the concept of secondarymixture of Aldwell & Schachter). He justifies his vision with the following example

(3.28), adducing that the altered chord (VI) stands in a chromaticized 3rd relationship to

the functional harmonies that surround it:

Ex. 3.28 Gauldin, Harmonic practice in tonal music, p.503

GauldinÕs remark obviates once more the functional significance of the chord. The A

major chord on bar 2 can be interpreted as a secondary dominant to the second degree.

This expectation is deceived by the following chord, when the voice leading c-c# is not

resolved, but changed again into a c natural.

Concerning the chords on example 3.27 B, Gauldin explains that Òthese sonorities are

sometimes called double mixture chords since they represent a two step borrowing from

the minor mode, first the chord itself and then its minor versionÓ 56 and he adds thefollowing example:

55 Strangely enough, when Gauldin speaks about the major tonic in minor, he does not regard it as a

mixture chord but acknowledges it as a secondary dominant to iv (besides its use as Picardy third at theend of a composition).56

Gauldin, Harmonic practice in tonal music, p. 504



45


Although GauldinÕs text acknowledges that the tonicization of iii is achieved by a

sequence of the previous tonicization ( II) he chooses for a different notation of this

tonicization, that states the relationship of individual chords to the initial (and

eventually also final) tonic. The problem is that the label iii does not reflect the actual

(temporary) tonic function of that chord. In fact we could say that from a functional

point of view Ò iiiÓ is meaningless. We cannot find a direct dominant or subdominant

relationship to the main key. Earlier in the book when talking about mixture chords in

the minor mode Gauldin states that Òthe use of the minor mediant and submediant is

restricted to key centres rather than individual harmoniesÓ.57 We could extrapolate such

affirmation also to major keys. In that case it seems much more convenient the labelling

proposed in example 3.30. We are focusing then in local tonal functions. It implies thatthe functional meaning of every chord is understood, but it is also true that being more

telling, this labeling is incomplete and only tells a part of what happens in the fragment.

The reference to the special feature of this modulation, which is going to a modified

degree and change the mode of this degree, needs to be pointed out. That information iscertainly relevant when describing in text the overall shape of the piece, something we

need to do after a chord-by-chord analysis, to have a general picture of the piece. We

are thus making a distinction between local relationships (not only chord by chord, but

considering tonal centres), that usually have a complete functional explanation reflected

in our labels, and large scale events that may reflect distant harmonic relationships, like

in this example, that we could describe in a complementary text.

57 Gauldin, Harmonic practice in tonal music, p. 402



47

3. Dorisches Moll

Concerning the terminology the authors explain:

ÒThe term ÒDorian minorÓ is selected for the minor with a major sixth, and ÒAeolian

minorÓ for the minor with a minor seventh. Both terms refer to the Dorian and Aeolianchurch modes. Intervals of the major sixth and minor seventh are characteristic of the

Dorian and Aeolian modes, respectively. This choice of terms is easily understood because

the name ÒDorian sixthÓ was always applied to the minor [this is clearly a misprint in the

translated version, it should say major: Ò...als der Name Òdorische SextÓ fŸr die gro§e Sext

in Moll...Ó] sixth in minor.Ó58

The English translator of the text has chosen the term Òmelodic minorÓ for the authorÕs

term ÒDorian minorÓ, Òharmonic minorÓ for Òordinary minorÓ, and Ònatural minorÓ for

ÒAeolian minorÓ. We will follow the translator terms, for considering that the reference

to church modes may be misleading: the use of the major sixth in minor does not

necessarily give modal character to the music, but on the contrary it is generally

produced by melodic considerations (avoid the augmented second interval between 6and »7) that are in fact product of enhancing tonal relationships: the addition of a

leading tone in minor, to have a functional dominant-tonic relationship.

The authors explain that the harmonic minor is neither a pure nor a natural mode, but a

combination of both major and minor ( Es ist ein aus Moll und Dur gemischtes

Geschlecht (Durmoll)). The term Durmoll refers then to the use of an element from

major (Dur), specifically the leading tone, in minor (Moll). Nevertheless, in the Englishversion the term is translated as minor-major , meaning a minor scale that borrows

elements from major (maybe the influence of the graphic showed in ex. 3.32 Ðp. 147 of

the original versionÐ , actually dealing with the melodic scale, that shows the lower

tetrachord of minor, and the upper tetrachord of major, makes the translator choose for

minor-major )Ex.3.32

Confusion can arise when later the term Molldur is introduced, refering to the use ofminor elements in major, that it is consequently translated major-minor. To avoid

confusion we will refer to the original German terms. According to Louis & Thuille the

only minor element that is borrowed to major is the 6 of the key. The use of this note in

IV will enhance the subdominant character of this chord (Òa chord with the major third

can fulfill a subdominant function, but accomplishes it far better if the chord contains a

minor thirdÓ59), being counterpart to the »7 in minor, that not only enhances but actually

gives V its dominant character.

58 Louis, R., Thuille, L. Harmonielehre, 4th

ed, Stuttgart: Ernst Klett Verlag, 1907. Trans., by Schwartz,

R. I., An annotated english translation of ÒHarmonielehreÓ of Rudolf Louis and Ludwig Thuille, Ph. D.diss. Washington University, 1982, p. 181 59

Ibid. p. 194



49

progression as a standard harmonic pattern will make us interpret the complete

succession as subdominant-dominant-tonic.

Concerning labelling, and once stated that the functional meaning of the mixture chords

is not compromised by the alterations, we can easily continue to apply the method we

applied so far, in which the information given by the Roman numeral is completed by

the indication of the quality of the chord. Once again the important thing is to have aclear picture of the qualities of the chords in each of the scales.

Ex. 3.33 Chord quality changes produced by mixture in major:

Ex. 3.34 Chord quality changes produced by mixture in minor:

Although these lists shows all the possible chords, we should point out that some ofthem are uncommon, such us the minor-major seventh chord on IV in major, or the

augmented seventh chord in III in minor.

Augmented sixth chords. Conventions and problems.

Within the broad (and rather loose) group of altered chords, a particular series of chords

have been always given special attention. Those are the chords containing the interval of

an augmented sixth, and on broad terms they are traditionally known as augmented sixth

chords. Considering this group of chords as a distinct category means to change thecriteria that are followed to describe a chord: instead of ordering the pitches insuperposed thirds and accept the lower of those pitches as the root of the chord, that is

subsequently described in relation to the key, when we characterize a group as

augmented sixth chords what we do is to accept the existence of a single interval as the

decisive feature, and this will bring several problems, as we are going to see.

Three chords are usually listed in textbooks as the paradigms of augmented sixth

chords. They are named Italian [augmented] sixth, French [augmented] sixth and

German [augmented] sixth chords, abbreviated as It.6, Fr. and Ger. (Gauldin,

Aldwell & Schachter) or It+6

, Fr +6

, Ger +6

(Kostka & Payne)

These terms are almost unanimously used in textbooks in English. The term ÒItalian

sixthÓ can be traced back to the eighteen century. John Holden in An essay towards a Rational system of Music (1770) states in the comments on one musical example that



51

origin and its relationship to the phrygian cadence.65

Gauldin even justifies that

Òbecause of their linear derivation, using functional Roman numerals to label these

chords hardly seems appropriateÓ.66

Aldwell & Schachter affirm that the augmented 6th

has essentially a passing function.

From a functional point of view, interpretations that only consider linear arguments are

insufficient and again refer to only very concrete situations. It might be true that theseconcrete situations are the most often found in music literature, but by taking them as a

model we commit what is called the Ògenetic fallacyÓ67

. This becomes evident insituations when trying to describe other dispositions (different note in the bass) of these

chords. Then authors start to speak about inversions of a (augmented) sixth chord. In the

first place we could wonder if that makes any sense at all, but furthermore we find that

some of the inversions even lack the characteristic interval of the augmented sixth!68 In

practice this results in the introduction of new designations like diminished third chord ,

to refer to the setting of a German sixth chord, in which the #4 is in the bass (see ex.

1.17 from BachÕs b minor mass). This procedure for denoting a chord by one of its

intervals is also used by Gauldin to refer to the doubly augmented fourth chord, for the

respelled use of the Ger in major.

The labels for such chords are also diverse: Gauldin chooses for Ger¼3, whereas Aldwell

& Schachter use Roman numerals plus continuo figures. For other inversions of the

German chord Gauldin sticks to the label Ger, adding figures for inversions (not for

accidentals).Another arguable aspect is that when the chord is not identified by its quality and root

every different situation will ask for a new distinct name. In ex. 3.36 Gauldin proposes a

new nickname (as arbitrary as the other geographical names), when a chord with the

same structure of the German chord, but rooted in another degree, resolves to the tonic

directly.

65 In a tonal context, a phrygian cadence is usually interpreted as a half cadence IV

-6-V.

66 Gauldin, Harmonic practice in tonal music, p. 424

67 The genetic fallacy or fallacy of origins is a fallacy of irrelevance where a conclusion is suggested

based solely on somethingÕs origin rather than its current meaning or context.68

This fact was already pointed out in the nineteenth century by Adolf Bernhard Marx (1795-1866), who

criticizes the use of a designation (augmented sixth chord) that applies only in the context of inversion:

ÒThe chord d -g-b-f boasted formerly of a high-sounding name. Just because the superfluous sixth, d -b attracted

principal attention, it was called the chord of the superfluous sixth, or some similar name; and not satisfied with

this, the same name was given to another chord, d -f-a -b, which had altogether a different origin, and thus, sext -

chords, terz-quart chords, and quint-sext chords were all thrown into one category. This name is not only

superfluous, but it is absolutely unsystematical and confusing.Ó [ Die Lehre von der musicalischen Komposition

(Saroni) p. 261]

Aldwell & Schachter also refer to this subject affirming that although the expression Òinverted augmented

sixthÓ is not literally correct, because the augmented sixths are not root position chords, Òthe expression

nonetheless conveys a truth. The linear forces that govern the origin and resolution of this family ofchords are usually most effectively expressed when the bass and an upper voice form an augmented

sixthÓ (p. 580)



52


Aldwell & Schachter also propose a new category of chords (Common tone augmented

sixth chords) for similar examples.

As we mentioned earlier the music literature will show many different types of

chromatic chords that share some characteristics with the geographical chords, but that

lack a specific name. The designation of a chord with the name of a piece in which it

first appeared or has an important role (the actual criteria is not clear) such as the socalled Tristan chord or Till EulenspiegelÕs chord, seems to be just papering over the

cracks; what kind of analytical information are they giving?

Ex. 3.38 Some altered chords and their alternative labels in different textbooks

Key: a a a d C C a a F Gauldin It6

GerGer¼3

Ger Ger

(AA4th)

ÒAmericanÓ?

Sixth FrÒTristanÓ?

Sixth ÒTillÓ?

Sixth

Aldwell

/Schachter

It63

Ger#IV7 Ger 64+

3

Common

Tone A6Fr

Weber ¼II7

Kostka/

Payne

It+6 Ger +6 Ger +6 Ger +6 Fr +6+6 +6

Piston It. Ger Swiss Fr

To summarize we could say that the choice of the augmented sixth interval as the

essential feature for defining a category of chords and the use of geographical

nicknames, result in a group of labels that refer to similar chords in a somehow

unpredictable manner, that are often neither informative of the quality of the chord, nor

telling about their situation within a given tonality. As a consequence of this, the only

way to learn them would be then to memorize one by one each of them!

The quest for a systematic approach: Louis & ThuilleÕs Harmoniehlere

The chapter on altered chords on Louis & ThuilleÕs Harmoniehlere presents an original

approach, that excludes preconceived prototypes (as the geographical chords) as the

basis for understanding such chords.

Their theory is based on the observation of how chromaticism can result in two different

situations: modulation69 or altered chords:

69 Modulation in Louis & ThuilleÕs text must be understood in the wider sense of the word, referring also

to transient tonicizations ( passing digression in Louis & ThuilleÕs words).



53

ÒChromatic tones are either components of independent chords or non-chord tones

appearing in incidental harmonic formations. If the tone of an independent chord is

chromatically altered, either the relationship to the tonic is maintained Ðin which case we

are still in the sphere of influence of the tonic-or a modulation is introduced(...) If no

modulation is introduced, we obtain an Òaltered chordÓ by means of chromatic alteration -

i.e. an Òaltered chordÓ in the true and narrow sense of the word- provided that the resulting

harmony is not possible at all in any key as a diatonic formation.Ó70

Louis & Thuille also noticed that the chromatic tones belonging to altered chords

always worked as leading tones for one of the tones of a tonic triad:

ÒEvery chromatic alteration of independent chords Ðnot producing a modulation- owes its

creation to a desire for a leading tone, i. e., the desire to be able to reach a tonally important

scale degree not with a whole-step movement originally presented in the scale, but rather

with the smallest movement applied in our music- the half step.Ó71

From these two premises they then proceed to systematically describe all possible

chords produced when applying one determined artificial leading tone.

Ex. 3.39 Possible leading tones to tonic triad in major and minor (Diatonic leading tones within brackets)

The influence of each artificial leading tone is discussed in relation to major, minor and

the modes obtained from mixture. As we can see in ex. 3.39, #2 can only be used in

major and 4 can only be used in minor. Moreover the authors remark that the use of #4

in major does not produce altered chords, because Òthe entire key is shifted to the

ÒrightÓ Ðso to speak- i.e. in the sense of the circle of fifthsÓ72

, meaning that all the

chords produced can be diatonically understood in the key of the dominant.

Ex. 3.39 The #4 as shift to the dominant region. Louis, R., Thuille, L. Harmonielehre (trans.) p. 270

LetÕs now have a look to the altered chords resulting by applying #4 in minor and

Molldur :

70

Louis, R., Thuille, L. Harmonielehre (trans.) p. 26771 Ibid.

72 Ibid. p. 269



54

Ex. 3.40

In both examples chords 5 and 7 are not altered chords in the narrow sense of the word,

because they are diatonic in E major. Regarding all the other chords Louis & Thuille

point out a common element: the diminished third between the raised fourth and natural

sixth scale degree. The authors here explain that this interval is enharmonically identical

to the major second and even when the interval is inverted the enharmonic

equivocalness persists (in this case between the augmented sixth and the minorseventh). Nevertheless the tendency to resolve outward from the augmented sixth to the

octave is Òmuch more easily perceivedÓ than the resolution of the diminished third into

the unison (or diminished tenth into the octave). This would explain the reservations to

the use (and recognition) of positions of these chords containing a diminished third.

Louis & Thuille make a summary of all inversions not containing the diminished third

interval but later they also shortly describe the use of the diminished third position:

ÒThe introduction of the diminished third (instead of the augmented sixth) can most easily

result in a passing tone, such that the interval receives quasi-melodic character (...) In more

recent music the diminished third occurs frequently, and it is created by the enharmonic

change of the minor second. The following situation, however might also occur:

Just as the augmented sixth naturally resolves to the octave, so does the diminished third to

the unison.Ó73

In fact the reference to more recent music seems to be unnecessary. As we already saw

in BachÕs mass example (ex. 1.17) the diminished third had been employed since long

ago. Mark Ellis goes even further asserting that ÒBach makes very frequent use of the

chord in ÒrootÓ position (with the sharp note in the bass)Ó, remarking later that Òin

broad terms (both geographically and chronologically) this is extremely unusual:

occasional examples have been identified in the music of Buxtehude, Vivaldi and

Pergolesi, but the sharp inversion chord as a distinct stylistic feature is otherwise found

only in Kuhnau.Ó74

73 Ibid. p. 279-280

74 Ellis, M. A chord in time , p. 135



55

Terminology matters I

Back to Louis & Thuille, in their description of the chords containing the augmented

sixth they mention that the so-called Òaugmented sixth chordÓ is obtained from the

altered subdominant triad in minor and Molldur , Òotherwise known as the Òdoubly-

diminishedÓ triad [doppelt verminderten Dreiklang ].Ó75

This statement is crucial because it reveals the acknowledgment of such formations as self-standing chords.

Actually the term doppelt verminderten Dreiklang appears already in A. B. MarxÕs Die Lehre von der musicalischen Komposition (1837-47).76 This designation addresses

directly the intervalic structure of the chord: a diminished third and a diminished fifth. 77

Louis & Thuille talk about the Òdoubly-diminishedÓ seventh chord [doppelt

verminderten Septaccord ] in which a diminished seventh is added to the doubly-

diminished triad. Further they comment the so-called Òhard-diminished triadÓ

[hartverminderten Dreiklang ]. As R. I. Schwartz points out in his English translation of

Louis & ThuilleÕs book, the term hartverminderten stems from hart : major and

verminderten: diminished. It is curious though that he chooses for ÒhardÓ as translation.

The term Òhard-diminishedÓ in English seems more to address a (highly subjective)aural quality of the chord. The literal translation would be Òmajor-diminished triadÓ.Again the intervalic structure is used to designate a kind of chord that does not fit in any

of the standard categories (major, minor, agumented, diminished and half-diminished).

Although Louis & Thuille only acknowledge these two terms (doppelt verminderten, for

the triad and the seventh chord; and hartverminderten, only for the triad), terms that as

we saw were already in use, this procedure will result very useful to address all altered

chords lacking an specific name.

Concerning the function of the chords affected by the upward alteration of the fourthscale degree the authors state that the formations belonging to the fifth and seventh

scale degrees have a dominant function, and all the others have subdominant function.Subdominants can be therefore be followed by a dominant or the tonic.

Louis & Thuille continue by thoroughly describing the other possible alterations, i.e. the

upward alteration of the second degree (only in major) and the downward alteration of

75 Louis, R., Thuille, L. Harmonielehre (trans.), p. 274

76 A posthumous edition by H. Riemann was translated into English by H. S. Saroni as Theory and

practice of musical composition. There the term doppelt verminderten Drieklang is translated as double-

diminished triad (p. 261). [Probably double diminished triad is more accurate, as opposed to doubly

diminished referred to an interval (doubly diminished fourth) meaning that the interval is twicediminished (from perfect we have to make it smaller a half tone twice). In the case of the triad it does not

contain any doubly diminished interval but two diminished intervals (the third and the fifth)]. N. B. The

term double diminished chord is also used in Jazz theory referring to the eight-note chord including all

the notes of the diminished scale. It is constructed by playing two fully diminished seventh chords in

which the root note of each chord are a major ninth apart from each other.

77 According to Damschroder earlier similar approaches appear in Marpurg [verminderte verminderte]

and Rey [ Accord de Quinte et Tierce diminuŽes] in Exposition ŽlŽmentaire de lÕharmonie (1807)



56

the second and fourth (only in minor) scale degrees. Some of their remarks deserve

special attention. They mention that the chords resulting from the use of #2 in minor

also produce the interval of a diminished third or augmented sixth, with the natural

fourth degree. Furthermore they point out that chords constructed on the raised second

and seventh degrees of the major scale sound exactly as the chords produced by the #4

in the relative minor key adding that both the resolution and the tonal relationships aredifferent.

A similar parallelism (although not involving relative keys) can be found between thetriads and seventh chords on the fifth and seventh scale degrees in minor or Molldur

with an altered 2, and the chords obtained by the use of #4 into the minor harmonies of

the second and fourth degrees. This dual interpretation results in functional multiplicity

of the altered chords. In Louis & ThuilleÕ words:

ÒWe can conclude that all of these chords have either a dominant or a subdomiant function;

b-d#-f ( and b-d#-f-a) is either a II (IV) in the key of a minor (or major) or a V in e minor

(or major), and d#-f-a (and d#-f-a-c) is either a IV in the key of a minor (or major) or a VII

(=V) in e minor (or major).Ó78

Ex. 3.41 Equivocalness produced by different alterations in different keys

To complete their theory about altered chords the authors explain the reasons for only

accepting artificial leading tones for the tones of the tonic triad as part of the altered

chords.

ÒAscending leading tones to other scale degrees are created in the following manner:

1. To obtain a leading-tone to the fifth above the dominant (second scale degree), the tonic

itself must be altered. If this alteration occurs in an independent chord, the resulting

formation loses its harmonic relationship to the tonic (...)

2. There already exists an ascending leading-tone to the subdominant in major. Its

introduction into minor would create a passing digression to the key of the subdominant.

3. An ascending leading tone to the third above the subdominant (sixth scale degree)

already exists in minor and major-minor. Its introduction into pure major would create a passing digression to the key of the relative minor (or even to the key of the subdominant).

4. The minor seventh above the tonic is ordinarily applied to reach the third above the

dominant with a half-step movement (i. e., the seventh scale degree). This is true even if

the voice containing the seventh ascends. This seventh already exists as a diatonic form in

the descending melodic minor scale but makes much more sense as an augmented sixth in

major. Nevertheless, if the chromatically raised sixth scale degree is introduced as a

component of an independent chord , the only applicable formations would be those which

decisively result in a passing digression to a foreign key(...)

Downward alteration of scale degrees other than the second and fourth degrees only occurs

in incidental passing harmonic formations, but not in independent chords. The reasons for

this are listed below:

78 Louis, R., Thuille, L. Harmonielehre (trans.), p. 289



57

1. A descending half step movement from the tonic to the seventh scale degree

(dominant third) is diatonic not only in major but also in minor.

2. The chromatically lowered seventh scale degree in minor belongs to the key

itself. It always creates a passing digression to the key of the subdominant (or its relative

minor) if the tone is introduced into major.

3. The sixth scale degree (subdominant third) exists in two forms not only in

minor but also in major, i. e. as a minor sixth and a major sixth (provided minor-major isconsidered to be equally as valid as pure major).

4. Downward alteration of the dominant always functions in a modulatory fashion

in major and minor. It always makes more sense to fill in the whole-step between the

dominant and subdominant with a chromatically raised subdominant than with a

chromatically lowered dominant, even if the voice containing the alteration descends.

5. Finally, the minor third introduced into major with an independent chord would

always make it completely impossible to related the chord to a major tonic.Ó79

Finally they discuss the combination of several alterations in one chord, stating that the possibilities are reduced to the use of two alterations in a single chord, giving place to

new chord configurations.

Ex. 3.42 Louis, R., Thuille, L. Harmonielehre (trans.), p. 301

Occasionally the use of two alterations in a chord will question the statement that an

altered chord in the true sense of the word is not possible at all in any key as a diatonicformation. In example 250 f) the sharp alteration of the second and fourth degrees in

major produces a diminished chord, that could be interpreted as VII¼7 in e minor.

Instead, its resolution to C chord makes that we understand such progression as a plagal

cadence in C major, then the #2 and #4 must be understood as alterations.Also the possibility of combining an upward and downward alteration of one and the

same scale degree is mentioned.

79 Louis, R., Thuille, L. Harmonielehre (trans.), p. 286,301



58

Ex. 3.43 Louis, R., Thuille, L. Harmonielehre (trans.), p.303

Ex. 3.44 Upward and downward alteration of one and the same scale degree in SchoenbergÕs

Kammersymphonie op. 9 (reduction bars 3-4)

In spite of the very precise description that the authors make of the altered chords theirsystem to label them in the musical examples is very rudimentary. Certainly it was not

their main concern to supply explicit labels to each new chord. They just show the scaledegree in which the chord is based, and between brackets the primary triad (I; IV or V)

that it replaces (see ex. 3.42). Neither explicit mention of chords built in altered degrees

(like #2 or #4) nor indications of chord inversion are added.

The Dutch connexion: E. MulderÕs Harmonie

In spite of the success of Louis & ThuilleÕs Harmonielehre in Germany (ten editions

appeared between 1907 and 1933) its influence abroad was relatively limited. An

English transaltion was not available until 1982. Nevertheless it had a very big

influence in the Netherlands, specifically in E. MulderÕs Harmonie (1947). As wealready saw, Mulder borrows the Molldur and Durmoll concepts (with subtle

differences), but also all the theory on altered chords. In the second part of the first

volume Ð Theorie/Analyse Ð he makes a distinction between concentrische alteratie

(literally Òconcentric alterationsÓ) and excentrische alteratie (Òeccentric alterationsÓ).

Under concentrische alteratie he explains the alterations that produce leading tones to

the tones of the tonic triad, i. e. the alterations that produce altered chords in the true

sense of the word, in Louis & ThuilleÕs words, following the same system. He calls

excentrische alteratie the alterations that produce a change of the tonal center. These are

the alterations appearing in secondary dominants and subdominants (wisseldominant and wisselsubdominant 80

).Concerning terminology, Mulder borrows the terms doppelt vermindert and

hartvermindert , translating them as dubbelverminderd and hardverminderd . Unlike

Louis & Thuille, Mulder does uses the term hardverminderd for a seventh chord that

has a major third, a diminished fifth (thus a hardverminderde triad) and a minor

seventh. This is an important fact because he assumes a minor seventh as the expected

seventh to a hardverminderde triad. This is indeed happening in the seventh chord on

the second degree in minor with the ascending alteration in the fourth degree (d-f#-a -c

80

Mulder employs these terms for any secondary dominant or subdominant. However the termswechseldominante and wechselsubdominante, used by Riemann, have a more restricted use, denoting only

the secondary dominant for the dominant and secondary subdominant for the subdominant respectively.



59

in c minor) or in the seventh chord on the fifth degree with the descending alteration of

the second degree (g-b-d -f in C). This is in fact similar to the expectation of a minor

seventh in a minor seventh chord and a major seventh in a major seventh chord.

Nonetheless the combination on several alterations in a chord can produce a chord

consisting on a hardverminderde triad and a diminished seventh, for instance whenusing the in the 2 and 4 in the seventh chord on the fifth degree (g-b-d -f in C).

Mulder does not give a particular name to these chords. For labelling the examples heuses a combination of roman numerals with figured bass symbols (sometimes adding

extra information concerning mixture: ÒmolldurÓ/ÓdurmollÓ) and letters for concrete

chords like the secondary dominants and subdominants (W.D. and W.S. standing for

wisseldominant and wisselsubdominant ) and the cadential suspension I64 (D6

4). Several

critics can be made to this approach. In first place, the already discussed handicap

related to the use of figured bass, that implies different label according to the different

keys. It is remarkable that figured bass remarks every alteration on the chord, but when

the root itself is altered no indication is given in the label (as it was the case in Louis &Thuille). Besides, the use of W.D and W. S. as labels for the secondary dominants and

subdominants is insufficient because lacks any specification about the type of chord (inthe case of a secondary dominant for instance it could be a dominant seventh chord, a

diminished triad or a diminished seventh chord).

Ex. 3.45 Mulder, Harmonie, vol. 1, p. 75 and 89



60

Although hardly innovative in comparison with Louis & Thuille, MulderÕs book

importance relies on its position as reference textbook for several generations of music

theorists in The Netherlands.

Terminology matters II

Over the years new terms had been added in Dutch music theory to dessignate altered

chords other than the hardverminderde and the dubbelverminderde. Theo Willemze inhis Algemeene Muziekleer (1964) describes three new chords: the overmatig

dominantseptiemakkoord , the dubbelverminderd-klein septiemakkoord and the klein-

verminderd septiemakkoord . When possible Willemze derives these chords from scales:

thus the klein-verminderd septiemakkoord is found on the seventh degree of the gypsy

minor scale and on the third degree of the gypsy major scale. This chord can be

explained also in terms of altered chord, as the seventh degree of the minor scale, with

the upwards alterations of the second degree or in the third degree in major with the

downwards alteration of the second scale degree.

The overmatig dominantseptiemakkoord and the dubbelverminderd-klein septiemakkoord do not appear in any scale and can only by explained by the use ofaltered scale degrees.

As we can see the terms klein-verminderd septiemakkoord and dubbelverminderd-klein

septiemakkoord are in fact descriptions of the chord structure81

: the klein-verminderd

septiemakkoord consists of a minor triad (klein) and a diminished seventh (verminderd )

and the dubbelverminderd-klein septiemakkoord consist of a double-diminished triad

(dubbelverminderd ) and minor seventh (klein). Following this logic, and mainly within

the circle of the music theory department in the Koninklijk Conservatorium in The

Hague, new names were adopted to describe other chordal formations resulting ofaltered scale degrees. Thus we can find the hardverminderd-verminderd septiemakkoord

and the groot-verminderd septiemakkoord . A problem is found when the triad resultingfrom the altered degrees has not a particular name. This happens on the chords built on

the seventh degree with the downward alteration of the fourth scale degree. This

alteration produces a doubly-diminished fifth in relation with the bass. In those cases we

will need to name independently all three characteristics of the chords: the third, the

fifth and the seventh giving place to names like verminderd-dubbelverminderd-

verminderd septiemeakkoord and klein-dubbelverminderd-verminderd septiemeakkoord .

The advantage of this systematic approach is that it can be used for any chord and being

itself descriptive, does not involves the memorization of a long list of chord names orintervalic patterns. In my opinion this is so far the most complete, systematic and

comprehensive approach to chromatic harmony.

Later discussion

More recent articles related to the topic of altered chords like D. HarrisonÕs Supplement

to the theory of augmented-sixth chords and Ch. J. SmithÕs The functional extravagance

of chromatic chords, present approaches that consider some of Louis & ThuilleÕs ideas

(obviating some others), but finally they fail to give a full embracing theory on

description and labelling. In HarrisonÕs article, who quotes Louis & ThuilleÕs list of

augmented sixth chords, the fact of considering the augmented-sixth interval as the

81

This procedure was already used to coin the term hartverminderte Dreiklang , with the difference thatin this case the name describes the quality of the third and the fifth, and here, for the seventh chords the

name refers to the quality of the fifth and the seventh.



61

distinctive quality of the chords make him struggle with the problems we already

discussed and leads him to very generic conclusions: ÒThe intervalic constitution of

augmented-sixth chords need not be specified or prescribed since it is the augmented-

sixth interval that gives the chord its powers; the remaining notes accompany the

generating interval and provide it with different sonorous shadings.Ó82 For designating

chords he proposes a dualistic approach in which a chord is named in relation withanother chord because Òinversional intervalic relationshipsÓ giving place to names like

the ÒdualÓ German augmented-sixth chord.

Ex. 3.46 Explanation of the ÒDualÓ German augmented-sixth chord in Harrison, D., ÒSupplement to the

theory of augmented-sixth chords.Ó p. 184

Smith proposes a procedure similar to Louis & ThuilleÕs to derivate the altered chords

by altering scale degrees, but in a more restricted way. In first instance he only

considers the upward and downward alteration of the second degree (#2 and 2) and

only within dominant functioning chords. Later he changes his method and builds

dominants and half-diminished sevenths on different degrees of the scale.

The labels he employs are highly confusing : Òvii¯7Ó is used for a chord with a major

third (example a-2) and for a chord with a diminished third (example b-2). Besides he

obviates two important features that go together in common practice music: chordconstruction (tertian structure) and voice leading. Take for instance the chord on

example d-2: it is labelled as a half-diminished chord on the flattened sixth degree, a in

C. Nevertheless if a is the root, the chord should be spelled a -c -e -g . Considering the

note of resolution the e can not be understood as such because it is resolving to c, so it

is in fact a d. The proposal of understanding the a as g# to keep the tertian structure is

also not possible, because it is resolving to g natural. Again taking in consideration the

note of resolution, the most plausible chord configuration is b-d-f#-a , that would result

from the chord on the seventh scale degree with the upward alteration of the fourthdegree in minor.

82 Harrison, D., ÒSupplement to the theory of augmented-sixth chords.Ó Music Theory Spectrum Vol. 17,

No. 2 (Autumn, 1995), p. 185



63

Terminology matters III

Considering that the Louis & Thuille system, and the Dutch way of naming altered

chords is to our understanding the best way to understand and label chromatic harmony,

we find that this approach could be exported into the English-speaking world. To realize

this we need a translation of the terms used in these systems. For this purpose we willuse as reference the Terminorum musicae index sept linguis redactus (Polyglot

dictionary of musical terms)83

and the compendium Muziektermen Engels- Nederlands/Nederlands-Engels by Kees van Hage84.

For most of the terms there is no problem, as a direct translation exists in English:

GERMAN DUTCH ENGLISH

vermindert verminderd diminished

klein klein minor

ŸbermŠ§ig overmatig augmented

Hart/ gro§ groot major

Other terms may require some comment and discussion:

Doppelt vermindert (Ger.) /dubbelverminderd (Dutch): As we saw, probably the best

translation would be Òdouble-diminishedÓ. This term has only appeared so far as literal

translation for the German term85, but what we propose here is to coin it as a self-

standing name in English. Therefore it would have then the same definite status as

major, minor, diminished, etc. when referring to a triad (diminished third anddiminished fifth) or a seventh chord (diminished third, diminished fifth and diminished

seventh).Even more problematic is the term hartvermindert (Ger.). Already the Dutch translation

adopted by Mulder is somewhat misleading (hardverminderd ), as it is the Englishtranslation proposed by Schwartz (hard-diminished, see discussion above). On the other

hand we find that coining a concrete term, in absence of a better one we would chose for

hard-diminished, can have some advantages. By choosing this term, in fact we coin a

new proper name (the descriptive character of the original German term is lost), and

again it gives a definite status to this chordal formation (triad with major third and

diminished fifth, or seventh chord with major third, diminished fifth and minorseventh86). Although one alternative would be the literal translation, Òmajor-

83

Leuchtmann, H. et al., Terminorum musicae index sept linguis redactus, 2nd

ed., BŠrenreiter, 198084

van Hage, K, Muziektermen Engels-Nederlands/Nederlands-Engels,

http://wrvh.home.xs4all.nl/kvhage/muziektermen/85

The Terminorum musicae index sept linguis redactus only recognizes the term doppelt vermindert as

related to interval, where it is consequently translated as Òdoubly diminishedÓ86

The structure of the hard-diminished seventh chord can also be defined as a dominant seventh with a

lowered fifth. Indeed in many situations the function of this chord can be understood as a dominant for

the next chord a fourth higher. In example a) below we find a hard-diminished seventh chord on the fifth

degree of C major resolving to the tonic. In b) we see a hard-diminished seventh chord on the second

degree moving to the fifth degree, that is a fifth lower. It can thus understood as a secondary V hard-

diminished to the g chord, but it can be also simply understood as an altered second degree in C (with

upward alteration of 4, and the use of 6 resulting from mixture).



64

diminishedÓ we think that it could also lead to confusion in the following context: letÕs

assume we use the term Òmajor-diminishedÓ for a triad, but also for a seventh chord,

then we would have the Òmajor-diminished seventh chordÓ, but this term could also

refer to a chord with a major triad and a diminished seventh.

It would be still possible to make explicit the difference of those two chords by stating

all three intervals in each case (major-diminished-minor and major-minor-diminished) but to keep it more concise seems to be more uselful the other approach, that is accept

the term hard-diminished as self-standing, and use only the three term label for thosechords in which the triad does not belong to any standard category.

Sometimes the actual meaning of the hard-diminished chord can be at least ambiguous, as it happens in

the ending bars of the theme in the fourth movement of BrahmsÕ fourth symphony. Here the key signature

is e minor, so the chord in the seventh bar can be understood as a V hard-diminished, nevertheless the use

of the Picardian third in the following tonic makes that it can be heard as a dominant for an a chord, that

actually follows the E major chord. In that case the b hard-diminished chord could be understood as an

altered II in a minor. The key signature and the clear 8 bar phrase structure justify in my opinion the

choice for the first interpretation.Brahms, Symphony n. 4, IV, mm. 1-8

Another situation concerning the hard-diminished seventh chord deserves attention:

If here the hard-diminished chord is understood as a V in e, then we would have a deceptive cadence V-

VI. Nevertheless this situation is at least ambiguous, and especially depending on the context, the hard-diminished chord could work more like a modified seventh degree in C (thus VII-I) than a modified

dominant in e resolving deceptively.



65

Symbol vs. Letter

The acceptance of all these array of chordal formations as self-standing chords requires

to pay special attention to the labels used to denote such chords. One of the main

characteristics of the system here proposed is that the label shows the quality of the

chord. As we already discussed, in diatonic contexts it can be achieved by the use of arelatively small array of symbols added to the right of the roman numerals (-,¼, +...). In

the case of altered chords no symbols are used (let alone standarized) in conventionalliterature, but nicknames and/or figured bass symbols. If we want to dispense with that

we could invent new symbols. We would need at least one symbol for each new chord:

double-diminished and hard-diminished. Other solution would be to use letters as to

denote the quality (dd for double-diminished and hd for hard-diminished). Compound

names will consequently require compound labels (for instance hd/d for hard-

diminished/diminished). This approach has the benefit of not introducing a plethora of

new symbols.

Altered chords in use

Although theoretically the possibilities introduced by the concept of artificial leading-

tones as explained by Louis & Thuille are quite copious, in practice some of the

possibilities are virtually discarded (especially those resulting in chords containing

major sevenths).

Among the altered chords appearing more or less often in literature we find the

following types:

- Diminished/doubly diminished/diminished87

- Minor/doubly diminished/diminished

- Double-diminished- Double-diminished/minor

- Hard-diminished/diminished

- Minor/diminished

- Major/diminished

- Hard-diminished

- Augmented dominant

Some examples of the use of these chords in the repertoire can be found in theAppendix (page 86).

The following figure shows the most frequent contexts in which those chords may

appear:

87 In the names used in Willemze the successive description of the qualities of the chord (triad and

seveventh, or third, fifth and seventh) are separated by a dash (-). Since the translation of Doppelt

vermindert (Ger.) /dubbelverminderd (Dutch) makes use of the dash (double-diminished) we suggest touse a slash (/) to separate the qualities in the English descriptions of the chords, e. g. Double-

diminished/minor.



66

Ex. 3.49 Altered chords in their most frequent contexts



67

Enharmonic possibilities

The acknowledgment of all these chordal possibilities will result in the widening of

enharmonic possibilities. Apart form the well known enharmonic properties of the

diminished seventh chord, and the double-diminished seventh chord/dominant seventh

relationship, the complete set of altered chords will present new enharmoniccorrespondences.

Ex. 3.50 Enharmonic possibilities involving altered chords

Altered chords also produce an ambiguous situation concerning enharmonic

relationships. Example 3.51 a) and b) show two enharmonic notations of a diminished

chord resolving to a cadential , that we could label as #IV¼7 or #II¼

5

6. Considering the

voice leading it seems that the second possibility in which the d# resolves to the e» is the

most suitable, however it should be noted that this note of resolution is actually a

suspension and the real note of resolution is really the following d (5th of the dominant).

In that case voice-leading considerations will plead for understanding the succession as

e -d and thus the correct label would be #IV¼7. Besides in minor (ex. f) the chord can

only be understood as #IV¼7.

In case that the diminished chord is not resolving to the cadential but to other position

of the tonic triad (usually the first inversion) the situation changes. In that case in major,

(ex. e) the e is the real resolution of the d#, that cannot in any case (in major) be

understood as an e . Thus it should be labelled #II¼7. It would be tempting to generalize

that if the diminished chord progress to a cadential it should be labelled as #IV and if

it progresses to a different position of the tonic triad it should be labelled as a #II,

nevertheless it should be noted that if the same diminished chord progress to a minor

tonic (ex. g) it must be also labelled as #IV.



68

Ex. 3.51

Further implications of altered chords

The discussed theory on altered chords may also be helpful to understand some progressions that at first sight seem no to have a functional explanation, such as third

relationships.LetÕs consider the following progression:

Ex. 3.52

No connection can be established between the two chords neither in terms of direct

diatonic relationship nor through secondary functions in any direction. Nevertheless it is possible to find a functional explanation by respelling the first chord88 so it is

understood as an incomplete altered seventh chord.

By doing this the third between c and e becomes a second, d - e , that can be seen as

the inversion of the seventh between the root and the seventh of a seventh chord. The

seventh chord thus formed would be incomplete, lacking its fifth. Assuming that we

interpret the chord in relation to the following one, the most likely implied fifth would

be b , forming thus a major/diminished seventh chord on the fifth degree in relationship

to the a minor chord (ex. 3.53). Another possibility would be a missing b , resulting in

a hard-diminished/diminished chord.Ex. 3.53

A similar approach can be used to functionally understand the following progression:

88 The respelling is in any case necessary due to voice leading considerations (c-c

). The implications of

respelling the second chord in this example will be later considered, in the discussion on non-tertian

chords.



70

4. Ninth chords

To complete the description of chordal formations in common practice we must have a

look to ninth chords.

Originally ninth chords appear as a result of a 9_8 suspension. In that sense we can just

apply the suggested method for suspensions in which the ornamental role of the ninth isshowed in the labelling ([9] 8). However sometimes the ninth is not resolved until next

chord. In that case it might be considered a structural (dissonant) component of thechord like in the famous beginning of Franck violin sonata.

Ex. 4.1 Franck, Violin sonata, I, mm. 1-8

Independent ninth chords will be usually associated to the dominant, giving place totwo possible chords, the one with major ninth and the one with minor ninth. We find a

problem to state the quality of these chords in the label: if we use V9 as the label for the

major ninth, we should make explicit the minor ninth in the other label. Gauldin

proposes the label V-9, however our choice for the symbol minus (-) to represent minor

as chord quality makes us discard this option. Figured bass solution as proposed by

Aldwell & Schachter (V 97) has the already discussed disadvantage of varying according

the different tonalities. In absence of a better solution we suggest here a compromise:

accepting V 9

, assuming the meaning of the flat in general terms as lowered pitch

independently of the corresponding accidental in a precise key. Non-dominant ninths occur infrequently except in sequences (ex. 4.2 shows one of

those unusual non-dominant ninths, in which the ninth is not even prepared).

Occassionally the suspended ninth does not resolves before the change of the chords,

resulting in self-standing ninth chords. Although there is not a standarized method for

labelling such chords it seems that the easier way to do it is to relate them to the labels

of the seventh chords plus a 9, adding if necessary the indication of a minor ninth ( 9).

Ex. 4.2 Schumann, Scheherezade from ÒAlbum fŸr die JugendÓ, op. 68, mm. 9-12



71

Ex. 4.3 Ninth chords within a sequence in Mozart, Serenade in c minor, K. 388



72

5. The limits of the system

a) Non-tertian formations

Occasionally we may find chords that seem to behave in a very similar way to the

altered chords, nevertheless they can not be explained as a tertian structure. This is thecase in the following example:

Ex. 5.1

What we hear is a half-diminished seventh on the (natural) fourth degree. But if wename the notes according to their voice-leading within the key it is not possible to

arrange the notes in superposed thirds. The problematic pitch is the b/c : if we adhere to

the tertian structure it should be a c , but this pitch is resolving to c so it must be

understood as a b . Besides, the downward alteration of the tonic would lack any

functional meaning. On the other hand, when interpreted as b , all the voices present a

regular voice-leading and the fundamental progression is a standard descending fourth

(plagal cadence). The term ÒQuasi half-diminished seventh chordÓ has been proposed

for such chords that sound as half-diminished seventh but are spelled in a different way.

These kind of chords may occur in complex chromatic contexts but paradoxically they

are in themselves not chromatic chords: all of its pitches are diatonic.

Ex. 5.2 ÒQuasi half-diminished seventh chordÓ in Franck, ÒPrŽlude, Choral et FugueÓ,

Choral, mm. 44-48

In fact the close relationship of this chord with altered chords becomes evident when wecompare the Òquasi half-diminished seventhÓ (only possible in minor) with the hard-

diminished/diminished chord on the seventh degree (only possible in major). Thesounding pitches are the same in both chords but the spelling is different due to the

different resolution to either a major or a minor tonic.

Ex. 5.3 Hard-diminished/diminished chord on the seventh degree resolving to I6

To make explicit on the labelling the particular configuration of the ÒQuasi half-diminished seventh chordÓ we propose to indicate them in quotation marks: ÒIV¯7Ó



73

Another example of these kind of chords can be found in the beginning of Ò FrŸhling Ó,

the first of R. StraussÕ Vier Letzte lieder . There we find a c minor triad followed by an

a minor triad. Understanding the c minor as the tonic, we find the same problem as we

have found in the Òquasi half-diminished chordÓ, that is in the a minor chord we have a

downward alteration of the tonic note. Following the same reasoning as before that

pitch cannot be part of the key of c minor, so it must actually be a b (indeed we see it

notated as a b in the melody when it directly resolves to c in bar 9). This results in a

Òquasi minor triadÓ in the sixth degree: Ò VI-Ó.

Ex. 5.4

This chord is closely related to the Òquasi half-diminishedÓ discussed before. The

difference is that it lacks the root tone, 4 (maybe not surprisingly this note appears as an

incomplete neighbor in the melody in bar 7). The relationship between ÒIV¯7

Ó and Ò VI-

Ó is similar to the relationship between V7 and VII¼: chords whose root is a third apart

and that belong to a same category, dominant in this case and subdominant in the case

of ÒIV¯7

Ó and Ò VI-Ó.

b) Overcoming functions I. Sequential patterns

In general terms we have so far defended the idea that in common-practice harmonyevery chord has a function within the predominant tonality, and this function will

condition the harmonic progression. However, musical examples show contexts werethe functions of the chords are subordinated to other aspects of musical technique or

expression.Maybe the clearest example of this kind of overcome functionality is found in passages

involving falling fifth sequences. In those situations the direction of the progression is

not implied anymore by the function of the single chords, but is suggested by the pattern

that shapes the model for the sequence: the listener expectations are not so much

conditioned by tonal function but for the continuation of the model. In this context, the

diminished triad looses its dominant function (as VII¼), but in fact it simply behaves as

a single link in the sequential progression. Although the progression I-IV-VII¼-III --VI--

II--V-I in itself denotes a falling fifth sequence, since its sequential behavior prevails

over chord functions it is desirable to make this explicit in the labelling.



74

Ex. 5.5 Falling fifth sequence in Mozart, Piano sonata k. 545, I, mm. 20-23

Chord functions are even more compromised in sequences of stepwise sixth chords,90

where the linear succession of parallel chords predominates and even precludes from

hearing the functional implications of single chords. The degree of negation of function

can vary from case to case depending on several factors such us the speed of theharmonic rhythm or the length of the passage.

In example 5.6 the fast speed makes it out of the question to understand the chords in

the right hand as self-standing chords, but rather an extension of the tonic chord, that is

the first and last chord of the section of chords.

Ex. 5.6 Beethoven, Sonata op. 2, n. 3, IV, mm. 1-5

Example 5.7 shows a similar passage, in which the succession of chords outlines an

extension of a single harmonic function, dominant in this case. The chords that framethe succession have dominant function (the first chord as a suspension, I .Taking into

consideration that the bass stays throughout the progression, the three upper voices in I

form the chord that will move down sequentially). Even when the speed of the chord

changes is here slower and allows us to hear (and label) each chord, the linear motion

still predominates on such a long progression. Moreover in this case the overall

dominant function is reinforced by the dominant pedal.

90 Sometimes this progression is called Faux-bourdon, making reference to a technique of either

improvised singing or shorthand notation particularly associated with sacred music of the 15th

century. Its

basic form consisted in adding two voices to a melody, one a fourth and other a sixth below, resulting insixth chords. In Faux-bourdon the intervallic succession is not restricted to stepwise movement, but

follows the shape of the original melody.



75

Ex. 5.7 Beethoven, Sonata op. 7, n. 4, I

Another typical place where we may find this kind of progressions is in descending basses from tonic to dominant. This gesture, known as Lamento figure has appeared in

different manners through time, but its basic form is based in parallel chords.

Ex. 5.8 Different versions of Lamento chord progression



76

Example 4.8 a) shows the basic form of the Lamento figure. The melodic bass

conditions the whole progression. The use of the subtonic as an harmonic note results in

a minor triad on the V degree, that completely looses its dominant character. Example

b) shows a typical elaboration in which the higher voice sustained over the bar line

produces a series of 7 6 suspensions. The addition of chromatic steps in the bass line

(example c) results in apparent chords that are still subjected to the linear progression.

As we saw falling fifth sequence, Òfaux-bourdonÓ and lamento progressions have incommon that harmonic functions are overcome by another factors. For that reason the

labels also loose their functional implications and they are just used to identify the notes

of the chords.91

The explicit indication of these progressions is thus needed to put in

context the special behavior of the chords in such concrete situations. 92

91 We may question why to use roman numerals at all then. In my opinion they are helpful to identify the

chords in contexts that using only figured bass may lead to confusion, such us in the following examples:

Both progressions are based on diatonic sequences alternating with chords. However the sequential

patterns are different.92

In these situation De la Motte also proposes a different labelling. In his case he changes the letters

indicating functions (inherited from Riemann) for Roman numerals, to denote the lack of functional

relationship.De la Motte, Harmonielehre, p. 113



77

c) Modal contexts

Unquestionably the establishment of a tonal center is one of the main pillars on which

we base our harmonic analysis. The substantial use of the so-called modal degrees93

(Piston), i.e. the mediant and submediant as substitutes for the functionally strongertonal degrees (I, IV and V) weakens the feeling of tonal center.

Ex. 5.9 Chopin, Nocturne op 15 n. 3, bars 79-108

In ex. 4.9 the ambiguity of tonal center is already given by the fragment previous to the

a tempo in which the c# on the bass can be seen as tonic for C# or dominant for F#. The

enharmonic reinterpretation and chromatic voice leading of the continuation maybe pleads for the second interpretation (F#:V7 = F: #IV dd). The expected dominant on c is

however substituted by an a chord in first inversion. This chord functions as a weaker

dominant in F (III6). Moreover, this weak dominant is resolved deceptively, to the VI-

degree. The tonal centre, F, is perceived but not firmly established until we hear a V-I

progression.

In fact the extensive use of modal degrees can lead us to perceive them as tonal degrees

in another key.Ex. 5.10 Ambiguity of modal degrees in Piston, Harmony, p. 33

93

This designation can be understood in two ways: Modal in the sense that they are rooted in 3 and 6, thatare the degrees that define the difference between the major mode and the minor mode. On the other hand

the chords on the mediant and the submediant can suggest the use of a church mode.



78

Other aspect of modality implies alterating pitches of a key so that the pattern of whole

tones and half tones is changed, and match up the pattern of a church mode. If we

compare the patterns of the major and minor scales with those of the modes we find that

we can suggest the following modes:

1. from natural minor- using the natural scale also in ascending direction (subtonic-tonic):

aeolian

- using 6: dorian

- using 2: phrygian

2. from major

- using 7: mixolydian

- using #4: lydian

A very relevant feature of modal influence in tonal contexts is the use of the subtonic. Ifwe look at the church modes we can see how in four of them (Dorian, Phrygian,

Mixolydian and Aeolian) there is a whole tone between 7 and 1 (subtonic), while only

in two of them we find a leading tone (Lydian and Ionian). Considering that the Ionian

corresponds to the major mode, and that the Lydian in practice includes a flat in the

fourth step converting the mode in transposed Ionian, the subtonic will give a

characteristic modal flavor to a progression. J. T. Stell94 suggests that the step from

modality towards tonality produced by the use of musica ficta in cadences, that changed

the subtonics into leading tones, could be compared with the use of subtonic in tonal

contexts, that turn progressions into modal contexts. The modal meaning of 7 is only

firmly established when the subtonic proceeds to the tonic. In fact in common practice

music the 7 will usually have tonal meaning, belonging to a secondary dominant to IV

or II in major or to IV or VI in minor.

94 Stell, J. T. The flat-7

th scale degree in tonal music, Ph. D. diss., Princeton University, 2006.



79

Ex. 5.11 Use of the modal 7 at the end of GriegÕs Piano Concerto

In GriegÕs example the use of 7 is ambiguous. In one hand the minor dominant

following the fff tonic and the scales on m. 437 suggest A mixolydian. However if we

consider the second half of m. 435 we could understand the e minor chord as a

secondary subdominant to the second degree (A: (IV-)!II

-) leading to a plagal cadence

(foreshadowing that of mm. 438-9), thus maintaining the progression within tonalterms.

The frequent use of modal alterations again weakens the feeling of a tonal center

because it produces a conflict of perception between context and the tonal center

deduced from position finding .95 The modal alteration varies the whole tone/ half tone

pattern of the original scale, matching it with the pattern of another diatonic scale (for

instance, the use of the flat seventh in C major ÐC mixolydianÐ produces a whole tone/

half tone pattern of F major). However the modal tonic may be emphasized in thecontext (repetition, ending or beginning of a section or movement) so that it prevails

over the information provided by position finding. In ex. 4.12 the key signature (D /b )

is confirmed in mm. 40-41 (b : V-I-). The following measures are completely diatonic

what could imply D as key. However the absence of any D chord and the emphasis on

the A chord suggest A as the tonal centre (A mixolydian), reinforced by the modal

progression V- -I in mm. 48-50.

95 This term coined by R. Browne refers to the ability to infer the tonic in relation to concrete intervalic

relationships. Browne explains that Òwhen one hears a tritone, or a minor second, oneÕs tonal

ÒknowledgeÓ offers a greater sense of the Òpossible places one may be inÓ than when one hears arelatively common interval (like a fourth or a major second) which could hold any number of places in a

diatonic field.Ó (ÒTonal implications of the diatonic setÓ, In theory only 5, n. 6-7)



80

Ex. 5.12 Tchaikovsky, Symphony n. 4 , II, mm. 40-55

The use of modal inflections opens also more chordal possiblities within the group of

altered chords. One clear example is found in R. StraussÕ Don Juan, in which thecombination of 7 (modal alteration) and 4 and 2 (leading tones to tonic triad notes)

produces a diminished chord on V.

Ex. 5.13 R. Strauss, Don Juan , op. 20 , 8 bars before the end

In summary modal contexts weaken the harmonic functions, making in many cases that

our choice for a tonal centre is ambiguous, what conditions our understanding and

labelling of a progression.

d) Overcoming functions II

We have reached a point in which thanks to all the harmonic procedures described

(secondary dominants and subdominants, mixture, altered chords, modal

progressions...) we are able to find a functional connection (sometimes strong,

sometimes weak) between chords in almost any context. However the functionalexplanations we can give of a concrete progression are occasionally so extravagant that

seem to work only as a theoretical speculation that in fact do not reveal the fundamental

meaning of a progression. The next examples are representatives of that kind of

progressions.



81

Ex. 5.14 Dvo! ‡k, Symphony n. 9, II, mm. 1-4

In ex. 5.14 the tritone relationship between the E major and B major chords that open

the movement can be functionally interpreted in a (major or minor) as V and II6

respectively. Nevertheless the key that is finally established (and also given by the key

signature) is D major. Besides the initial progression is not II to V but the opposite,what makes unlikely to hear the E major chord as a dominant. On the other hand if we

hear the E chord as (local) tonic (the position of the chord, with the octave on top, and

the fact that is the beginning of the movement and the last movement ends also in e-

minor- could plead for that) then the B chord lacks any functional explanation (it

should be interpreted as a kind of VII min-aug: d-f-a# ! what makes it a theoretical

nonsense). The D chord on bar two, that could be seen as a secondary dominant for VI

in A, will eventually be the tonic in D and the next chords can be understood as

borrowed chords from d minor.

If we just observe the upper part in the first 5 chords it seems that Dvo! ‡k is playing

with the possible harmonizations of two notes without regard for tonal function. On the

contrary it seems that he is trying to avoid the more conventional (functional)

possibilities. In that sense our functional interpretations, although feasible, are in a way

out of the question.

The next two examples are taken from LisztÕs Sonata in b minor.



82

Ex. 5.15 Liszt, Sonata in b minor

The key of f# major is firmly established prior to this passage. After a dominant chord

on c# (F#:V7) we hear a C major chord. The most common functional interpretation of

such a progression would be to reinterpret the dominant chord as a double diminished

chord resolving to a dominant in F, or even to the tonic in C (C:VIIdd -I). Two facts

contradict these interpretations. First of all the voice leading the c# in the bass

reinterpreted as a d should resolve to the c natural, however the c chord in the next baris in first inversion, with the e in the bass. Secondly the melody over the c chord uses an

f# in a descending line, what contradicts the assumption of F as governing key. In fact

according to the melody it will be theoretically possible to understand the c chord as a

subdominant in G major (IV of II). The immediate return to the dominant chord on c#

also difficults to establish the connection of the c chord to a tonic key.96 The question

again is to what extent the theoretical abstraction that we may infer reflects the way we

actually hear the music. Do we actually hear such a distant relationship as the

subdominant to the Neapolitan? In places like this the description reflects the content

but not the meaning of the progression. Like in Dvo! ‡kÕs example maybe a melodic

aspect is the main issue, in this case the chromatic descending line from c# to c

(different register), then continued by b, b ... The very end of the piece also defies

functional explanation.

96 Eventually the progression is repeated and the subdominant role of the c chord is confirmed by the

following g chord, and this succession is repeated sequentially in F and E .



83

Ex. 5.16 Liszt, Sonata in b minor, final measures.

After two plagal cadences in B major we hear a descending scale (motif from the

beginning of the piece) that combines elements from b minor (first descending

tetrachord) continuing in B major (d#) and finally using the 2 (c natural). This 2 is

harmonized first with an a minor chord and after by a F major chord, the triad a tritone

apart from the B major that ends the piece. The status of the a chord and F chord in

relation to B major can be again explained in relation to II (C: VI- and IV

respectively)97 but once more the theoretical explanation seems not to reveal the real

content of the music. One more we could say that melodic considerations (upper voice

e-f-f#) condition the harmonic progression to such extent that functions are not the mostimportant factor.

It is remarkable how these three fragments discussed here, that are using just consonanttriads put into question a functional approach in a greater extent that some passages

containing dissonant seventh chords.

In conclusion the labels we use are valid to give descriptions of the music, but these

descriptions are not always revealing of the true sense of the music.

The following example show a different context in which we may question to what

extent the insight given by the labelling is revealing of the content of a piece of music.

97 Actually the whole progression could be understood in E/e: IV- II-V, what is not a satisfying

explanation, because the last chord is certainly perceived as a tonic.



84

5. 17 Schoenberg, ÒSchenk mir deinen goldenen KammÓ, op.2 n. 2, mm. 1-8

The highly chromatic writing produces fast changes in the tonal centre, so that the

connection of chords makes sense only at a very local level. Tonal centres seem not tohave strong structural or dramatic relevance anymore, and seem more (although not

exclusively) the result of chromatic voice leading, that incidentally we can understand at

local level. Our functional explanations seem possible whenever two basic features are

kept in the music writing: voice leading according to resolution of dissonances andleading tones and chords built in thirds. The descending chromatic lines in the higher

parts seem to be the framework that conditions the harmonic progressions but in fact the

model for this progression could be a falling fifth sequence in which the upper voices

progress at different speeds resulting in those complex harmonic progressions. It would

be difficult to say at what moment voice leading overrules the harmonic meaning of a

progression but in examples like this one it seems that again the information given bythe harmonic analysis tells only one part of the history, that is not necessarily the most

relevant.



86

Appendix. Examples of altered chords from the repertoire

R. Strauss, Till Eulenspiegels lustige Streiche

C. Franck, Symphonic variations, mm. 171-173

F. Chopin, Nocturne in F major op. 15 n. 1, mm. 15-19

R. Wagner, Tristan und Isolde, Prelude, mm. 32-37



87

R. Wagner, Tristan und Isolde, Liebestod

W. A. Mozart, Fantasia in c minor, K. 475, mm. 12-15



88

Bibliography

- Aldwell E., Schachter, C., Harmony and voice leading , 4th ed. Boston: Schirmer,

2011.

- Benwards, B., Saker, M., Music in theory and practice, 8th

ed., New York:McGraw-Hill, 2009.

- Berry, W., Structural functions in music, Dover, 1987 (Reprint).- Bister, H., ÒMax Regers harmonik im VerhŠltnis zu Hugo RiemannÓ. In Max Reger

und seiner zeit , ed. Siegfried Kross, Bonn: Rheinische Friedich-Wilhelms-UniversitŠt,

1973.

- Blatter, A., Revisiting music theory, New York: Routledge, 2007.

- Callcott, J., A Musical Grammar , London: Birchall, 1806.

- Chailley, J., TraitŽ historique d'analyse harmonique, Paris: Alphonse Leduc et Cie.,

1977.

- Cook, N., A guide to musical analysis, Oxford University Press, 1987.

- Damschroeder, D., Thinking about harmony; Cambridge: Cambridge UniversityPress, 2008.- Desdren, S., Algemene muziekleer , Groningen: Wolters-Noordhoof, 1972.

- Ellis, M., A chord in time:The evolution of the augmented sixth from Monteverdi to

Mahler , Farnham: Ashgate, 2010.

- Gauldin, R., Harmonic practice in tonal music, New York: Norton, 1997.

- Harrison, D., Harmonic function in chromatic music, Chicago: The University of

Chicago Press, 1994.

- Harrison, D., ÒSupplement to the theory of augmented-sixth chords.Ó Music Theory

Spectrum Vol. 17, No. 2 (Autumn, 1995), pp. 170-195.- van Hage, K, Muziektermen Engels-Nederlands/Nederlands-Engels,

http://wrvh.home.xs4all.nl/kvhage/muziektermen/. Last accesed 17-May-2012.- Hunt, R., A second harmony book , London: Herber Jenkins, 1965.

- Kostka, S., Payne, D., Tonal harmony, 3rd ed., McGraw-Hill, Inc., 1995.

- Leuchtmann, H. et al., Terminorum musicae index sept linguis redactus, 2nd

ed.,

Budapest: Akademiai Kiadš, 1980.

- Levine, M., The Jazz theory book , Petaluma: Sher Music Co., 1995.

- Louis, R., Thuille, L. Harmonielehre,Stuttgart: Ernst Klett Verlag, 1907. Trans. by

Schwartz, R. I., An annotated english translation of ÒHarmonielehreÓ of Rudolf Louis

and Ludwig Thuille, Ph. D. diss. Washington University, 1982.- Marx, A. B., Die Lehre von der musicalischen Komposition, Leipzig: Breitkopf &

HŠrtel, 1847. Posth. Ed. by Riemann, trans. and ed. by. Saroni, H. S as Theory and Practice of Musical Composition, New York: Huntington, Mason & Law, 1851.

- De la Motte, D., Harmonielehre, Dt. Taschenbuch-Verlag, 1999.

- Morris, R. O., Figured harmony at the keyboard, Oxford University Press.

- Mulder, E. W., Harmonie, 2 vols., Utrecht: W. De Haan, 1947.

- Piston, W., Harmony, Norton, 1987

- Riemann, H. Harmony simplified or The theory of the tonal functions of chords,

London: Augener Ltd., 1896.

- Rousseau, J.J., Dictionnaire de musique, Paris, 1768.

- Schenker, H., Harmonielehre, trans. by Mann Borgese, E. as Harmony, Chicago:

The University of Chicago Press, 1954.

- Schouten, H., Eenvoudige muziekleer , Naarden: Strenholt, 1929.



- Smith, Ch. J., ÒThe Functional Extravagance of Chromatic Chords.Ó Music Theory

Spectrum Vol. 8, (Spring, 1986), pp. 94-139.

- Stell, J. T. The flat-7 th scale degree in tonal music, Ph. D. diss., Princeton

University, 2006.

- Tchaikovsky, P. I., Guide to the practical study of harmony, 1900. Reprint Mineola

NY: Dover, 2005. - Weber, G. , Versuch einer geordneten Theorie der Tonsetzkunst , Mainz: Schott,

1817 trans. James. F. Warner (from 3rd ed.) An Attempt at a Systematically ArrangedTheory of Musical Composition, 2 vols. Boston: J.H. Wilkins & R. B. Carter, 1842-46.

GARCÍA, Miguel V. Harmonic Analysis in Practice _ a Critical Review of the Labels Employed to...

Documents

Transcript of GARCÍA, Miguel V. Harmonic Analysis in Practice _ a Critical Review of the Labels Employed to...