Harmony

In music, harmony is the combining of notes simultaneously, to produce chords, and successively, to produce chord progressions. The term is used descriptively to denote notes and chords so combined, and also prescriptively to denote a system of structural principles governing their combination. In the latter sense, harmony has its own body of theoretical literature.

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Definitions

1. Combination or adaptation of parts, elements, or related things, so as to form a consistent and orderly whole; agreement, accord, congruity.

pre-established harmony, in the philosophy of Leibnitz, a harmony between mind and matter, e.g. between the body and soul, established before their creation, whereby their actions correspond though no communication exists between them.

c1532 G. DU WES Introd. Fr. in Palsgr. 1058 Others have sayd that it [the operation of God] is a maner of armonie. 1597 HOOKER Eccl. Pol. V. xxxviii. §1 The soule it selfe by nature is, or hath in it, harmonie. 1605 BACON Adv. Learn. I. iv. §6 (1873) 32 The harmony of a science, supporting each part the other, is..the true and brief confutation..of all the smaller sort of objections. 1745 De Foe's Eng. Tradesman ii. (1841) I. 18 Here is a harmony of business, and everything exact. 1814 SOUTHEY Roderick XXI. 382 To heavenliest harmony Reduce the seeming chaos. 1847 LEWES Hist. Philos. (1867) II. 273 His [Leibnitz's] favourite hypothesis of a Pre-established Harmony (borrowed from Spinoza). 1860 TYNDALL Glac. II. xxiv. 353 Where other forces mingle with that of crystallization, this harmony of action is destroyed.

b. Phr. in harmony: in agreement or accordance, consistent, congruous. So out of harmony.

1816 KEATINGE Trav. (1817) I. 42 He may always be sure of finding nature in harmony with herself. 1849 MACAULAY Hist. Eng. II. 149 This mode of attack..was in perfect harmony with every part of his infamous life. 1853 MAURICE Proph. & Kings i. 11 The vox populi was the vox Dei even when the two voices seemed most utterly out of harmony.

2. Agreement of feeling or sentiment; peaceableness, concord.

1588 GREENE Pandosto (1843) 25 Coveting no other companion but sorrowe, nor no other harmonie but repentance. 1667 MILTON P.L. VIII. 605 Harmonie to behold in wedded pair More grateful then harmonious sound to the eare. 1780 COWPER Progr. Err. 140 Love, joy, and peace make harmony more meet. 1844 H. H. WILSON Brit. India III. 408 The harmony which had thus been re-established with the Court of Baroda.

Historical definition

In Greek music, from which derive both the concept and the appellation, ‘harmony’ signified the combining or juxtaposing of disparate or contrasted elements – a higher and a lower note. The combining of notes simultaneously was not a part of musical practice in classical antiquity: harmonia was merely a means of codifying the relationship between those notes that constituted the framework of the tonal system. In the course of history it was indeed not the meaning of the term ‘harmony’ that changed but the material to which it applied and the explanations given for its manifestation in music.

According to the conception of classical writers, taken over by medieval theorists, harmony was a combining of intervals in an octave scale – a scale understood not as a series but as a structure. Consonances based on simple, ‘harmonic’ numerical proportions – the octave (2:1), the 5th (3:2) and the 4th (4:3 – form the framework of a scale (e′–b–a–e), and in addition to the octave structure resulting from the interlocking of consonances, the consonance itself also qualifies as harmony, as a combining agent (see Consonance, §1).

In the Middle Ages the concept of harmony referred to two notes, and in the Renaissance to three notes, sounded simultaneously. An anonymous writer of the 13th century (CoussemakerS, i, 297) defined the concordantia (the simultaneity employed in polyphonic music, and not merely used to test out the relationship between notes) as ‘the harmony [harmonia] of two or more sounds produced simultaneously [in eodem tempore prolatorum]’. Gaffurius and Zarlino spoke of three-note harmonies, though Gaffurius (A1496, bk 3, chap.10) considered only combinations of octaves, 5ths and 4ths. Zarlino (A1558, bk 3, chap.31) was the first also to include in his concept of harmony triads consisting of 5ths and 3rds; this he was able to do because, besides perfect consonances, he defined imperfect ones – the 3rds – by means of simple, ‘harmonic’ numerical proportions (5:4 and 6:5) rather than by the complicated Pythagorean proportions.

In addition to the simultaneous sounding of two or three notes in isolation, the concept of harmony takes in the relationships between such sounds. In 1412 Prosdocimus de Beldemandis designated the regulated alternation of perfect and imperfect consonances as ‘harmony’ (CoussemakerS, iii, 197); in the 17th century, among other prerequisites for the composition of contrappunto moderno, Christoph Bernhard (ed. Müller-Blattau, A1926, p.40) described ‘harmonic counterpoint’ as an articulated sequence of ‘well-juxtaposed consonances and dissonances’ and d’Alembert, Rameau's commentator and popularizer, defined harmonie, in contradistinction to accord (three or four notes sounded simultaneously, forming a unit), as a progression of simultaneously sounded notes intelligible to the ear: ‘l’harmonie est proprement une suite d’accords qui en se succédant flattent l’organe’ (1766, pp.1–2).

The word ‘harmony’ has thus been used to describe the juxtaposition of the disparate – of higher and lower notes – both in the vertical (in the structure of chords or intervals) and in the horizontal (in the relationship of intervals or chords to one another). There is a widespread tendency, probably too deep-rooted to be corrected, to take harmony as meaning no more than the vertical aspect of music, disregarding the fact that chordal progression is one of the central categories dealt with in the teaching of harmony. This tendency entails a bias that not only misrepresents the terminology but can also influence the listener's way of hearing, which is not wholly independent of a verbal understanding of what is involved in music.

Moreover, the concept of harmony refers less to actual musical structures than to the structural principles underlying intervals and their combinations or chords and their relationships. (In Riemann's theory of harmonic function, a harmony is the essence of all chords having a like function and thus exists at a much more abstract level than chords with their inversions and notes ‘foreign to the harmony’.) However, harmony considered as a structural principle is just as much an intrinsic part of ancient and medieval music as it is of the tonal system of modern times. The two-note consonance constituted the foundation of the old tonal system, the three-note consonance that of the new. From the 18th century onwards, the scale of any key has been explained as being the result of a reduction of the three principal chords, the tonic, dominant and subdominant: C–E–G + G–B–D + F–A–C = C–D–E–F–G–A–B–C.

Carl Dahlhaus

Basic concepts

The chord

The harmonic theory of recent times, which evolved gradually between the 16th and the 18th centuries, is based on the idea that a chord – three or four notes sounded simultaneously – is to be taken as primary, as an indivisible unit. While in earlier counterpoint two-part writing was regarded as fundamental (with four-part writing as a combination of two-part counterpoints), in the later study of harmony a chord was regarded as a primary element rather than as an end-product and was indeed considered as such regardless of the difference between homophonic and polyphonic style. (J.S. Bach's counterpoint, despite the complexity of its polyphony, was undoubtedly based on harmony and not merely regulated by it: the harmonic aspect arises as a foundation, not as a resultant.)

In the idea of the chord as a given entity, it is necessary to distinguish between two aspects: that of psychology and that of musical logic. Stumpf defined or characterized the psychological entity as a ‘fusion’ of the notes in a consonant triad (and to a lesser extent in the chord of a 7th too). The logical factor, however, is to a large extent independent of the psychological, although the conception of a chord as a logical entity could not have arisen in the first place without the psychological phenomenon of fusion. Whether, in terms of logic, a chord presents an entity, and not merely a combination of intervals, depends on the function it fulfils in the musical context. A chord each of whose notes is resolved in contrapuntal fashion will nevertheless be conceived as a primary element in the mind of the composer if – for example as a supertonic preceding the dominant and the tonic – it is intrinsic to the harmonic continuity.

Chordal inversion

The doctrine of chordal inversion, namely the proposition that the root-position chord C–E–G, and 6-3 chord E–G–C and the 6-4 chord G–C–E are different manifestations of an identical harmony and that the bass note of the 5-3 form (C–E–G) must count as the fundamental note, the basis and centre of reference (centre harmonique) of the other notes, was for a long time thought to be the remarkable and epoch-making discovery of Rameau; on the basis of this theorem he was held to have been the founder of harmonic theory. The concept of the inversion had in fact been anticipated by a number of theorists of the 17th and early 18th centuries – by Lippius (A1612) and his follower Baryphonus (A1615), by Campion (Ac1613), by Werckmeister (A1702) and also by the ingenious dilettante Roger North (c1710). It was not anticipated, as Riemann claimed, by Zarlino. What was decisively new in Rameau was not the theorem as such but its incorporation into a comprehensive theory of musical coherence, in which the conception of the chord as a unit, primary and indivisible, the concept of the root note, the doctrine of the fundamental bass (basse fondamentale) and the establishment of a hierarchy between the fundamental degrees were interdependent elements, complementing and modifying each other.

The roots of chords (which are no longer their bass notes when the chords are inverted) link together to give the basse fondamentale. The latter is a purely imaginary line, in contrast to the basso continuo, which is the sequence of actual bass notes. It is a construction designed to explain why a progression can become a compelling, intelligible coherence rather than a mere patchwork of separate chords. According to Rameau (whose theories were further developed in the 19th century by Simon Sechter) the cohesive principle is the fundamental progression, from root to root; in fact, the 5th is reckoned a primary, stronger fundamental progression, the 3rd a secondary, weaker one. A fundamental progression or apparent progression of a 2nd (according to Rameau the 2nd, which as a simultaneity is a dissonance, is also not self-sufficient as a bass progression) is reduced to progressions of a 5th: in the chord sequence G–C–D minor, C is indeed a tonic degree (I), related to G as the dominant (V), but it is also a fragment of the 7th on the submediant (VI7), related to the supertonic (II) – the root A being introduced for the sake of the 5th progression A–D: see ex.1 (Rameau, B1722, p.204). For similar reasons, the chord F–A–C–D in C major is considered as a triad of F major with added 6th (sixte ajoutée) provided that it is followed by the C major triad (fundamental 4th progression); on the other hand it is considered an inversion of the 7th on D if it moves on to G major (fundamental 5th progression): F–A–C–D is a chord with two applications (double emploi).

Dissonance

The idea that a chord presented not a mere combination of intervals but a unit, primary and indivisible, was associated with a far-reaching change in the concepts regarding the nature and the functions of dissonances. In counterpoint before 1600, a dissonance in strict writing was a relation between two voices; and the dissonant note was regarded as being the one that had to advance to the resolution of the dissonance, hence in ex.2a the lower note, in ex.2b the upper. (The second, or ‘reference’, voice could either remain on the same note or move to a new one at the resolution.) A suspension on the strong beat is produced by a step in the ‘reference’ voice (ex.2a), and a dissonant passing note on a weak beat by a step in the voice making the dissonance (ex.2b).

From the 17th century the formation of dissonances in strict counterpoint was complemented by new approaches that extended the system of contrapuntal writing without violating any of its fundamental characteristics. But at the same time it was confronted with phenomena grounded in basic principles of a different kind, relating to other categories of musical listening. Among the new approaches to dissonance that merely extended the system were some that characterized the modern counterpoint of the 17th century, the seconda pratica: for example, changing note quitted by downward leap (ex.3a), the suspension resolved by downward leap (ex.3b), and the accented passing note (ex.3c), which in Palestrina’s style was permitted only in a rudimentary form as a weakly accented passing note of the duration of a semiminima (ex.3d).

These dissonances, which are to be found in contrapuntal practice, in no way invalidate the strict rules of counterpoint. They require, if their intended effect is to be understood, an awareness of the norms with which they conflict. It is precisely as exceptions to the conventional rule, which they infringe and in so doing confirm as valid, that they gain their expressive or symbolic meaning: the downward-leaping suspension (ex.3b) owes its character of pathos to its very deviation from the normal resolution by step.

The modern counterpoint of the seconda pratica may seem in consequence an accumulation of licences, of artificial infringements of earlier rules. On the other hand, the distinction between dissonant chords and notes foreign to the chord (or to the harmony) involved a fundamentally new conception of dissonances. It was this differentiation that gradually came to permeate compositional practice from the 17th century onwards and theory from the 18th. Not only was the stock of approaches to dissonance changed, but at the same time so was the basic idea of what a dissonance actually was.

In harmonically based writing a dissonance is accounted a dissonant chord – that is, a chord of which the dissonance is an essential component – if two conditions apply: if, in the first place, the dissonant chord can be meaningfully explained as a piling up of 3rds (‘meaningfully’ because at a stretch it is possible to reduce absolutely any chord to piled-up 3rds); and if, in the second place, the resolution of the dissonance is associated with a change of harmony (a change of root, which moreover will be in some part the result of the dissonance’s pull towards resolution). According to the modern conception (Kurth, C1920), the dissonance factor is not so much a note (F as an adjunct to the harmony G–B–D) or an interval (the dissonance G–F as distinct from the consonances G–B and G–D) as rather the chord, which as a whole is permeated by the character of dissonance, originally a property of individual intervals.

Notes foreign to the chord (or to the harmony) are distinguished from fundamentally dissonant chords in that, in the first place, they appear as dissonant adjuncts to the chords (Kirnberger spoke of ‘incidental’ dissonances) and, in the second place, their resolution is not dependent on a change of harmony (ex.4). In 16th-century counterpoint there was no essential difference between a suspended 4th (ex.4a) and a suspended 7th (ex.4b). In the 18th and 19th centuries, however, the suspended 4th was regarded as a note foreign to the chord: it is neither intelligible as a result of piling up 3rds (unless one accepts the sort of far-fetched explanation propounded by Sechter in 1853), nor does the resolution of the dissonance correlate with a change of harmony. However, the note C in ex.4b is not free from ambivalence: if, either in the imagination or in musical reality, the root notes D and G are inserted underneath, a fundamental discord (II7) results; but if the C is merely conceived as a suspension to a 6-3 chord on D the dissonance is a note foreign to the chord.

Constructional technique and note relationships

In a composition whose structure is determined by rules governing the progress of the basse fondamentale and the treatment of dissonances, two factors can be distinguished, just as in counterpoint at an earlier date. These factors are in themselves abstract, but their combined effect forms the basis from which actual composition proceeds. One of these is constructional technique, the other the regulation of relationships between the notes.

In earlier, more precise musical terminology, it was the regulated note relationships, as distinct from constructional technique or counterpoint, that were covered by the term ‘harmony’ (in keeping with the ancient and medieval meaning of the term). From the 18th century onwards both aspects of composition became subsumed in the concept of harmony – that is, in addition to the note relationships, the chord progressions built on a basse fondamentale, for which the expression ‘harmony’ was used in an attempt to distinguish it from ‘counterpoint’. This constitutes a linguistic confusion and produces a blurring of the distinction between constructional technique and harmony, in the narrower sense of the word, that has marred many methods of teaching harmony.

The rules of early counterpoint refer not to precise, diastematically defined intervals such as major or minor 3rds but to classes of intervals – 3rds in general. And a distinction must be made between the rules of counterpoint, which are exclusively concerned with abstract musical construction, without regard to the difference between minor, major and augmented 6ths, or between perfect 4ths and tritones, and the directions for a harmonic (in the stricter sense of the word) arrangement of the composition – directions formulated as rules governing the use of mi and fa: a diminished 5th appearing in the place of a perfect 5th did not count as a dissonance, for whose legitimate application there were constructional rules, but as a ‘non-harmonic note relationship’ (relatio non harmonica) which was supposed to be avoided (though could not always be avoided in fact; many of the controversial problems of accidentals are insoluble). Tinctoris spoke of a falsa concordantia in contradistinction to a dissonantia (CoussemakerS, iv, 124b) in order to indicate the discrepancy between the contrapuntal (concordantia) and the harmonic (falsa) import of the interval. The essence of the ‘true concords’ (verae concordantiae), the relationes harmonicae, is represented by the hexachord that excludes both the chromatic intervals and the tritone.

Constructional technique and note relationships, which normally work together in perfect agreement, can sometimes get out of proportion with one another: in technical terms, the mannerism of Gesualdo depends on the device of clothing what is in abstract thoroughly regular, even conventional, counterpoint in an excess of chromaticism. A similar divergence between simplicity of constructional technique and complexity of note relationships can occasionally be observed in late Baroque tonal structure, in sequences of 5ths in the bass: for instance, the C major chord progression I–IV–VII–III–VI–II–V–I can be realized chromatically as C major–F major–B major–E minor–A major–D major–G♯ major–C♯ minor; and it is to the simplicity of the basse fondamentale, made up of steps of a 5th, that the tonally extremely complicated chord progression (a headlong modulation from C major via E minor and D major to C♯ minor) owes its compelling, intelligible effect.

Since the 17th century, keys have been constituted by note relationships as well as by constructional means. Taken in itself, a major or minor scale is not sufficient to define a key; the fact that a chord progression remains within the bounds of a major scale in no way precludes the possibility of its remaining tonally indeterminate. Conversely, as has already been demonstrated, a sequence of 5th steps (which in its tonally self-contained form is often spoken of in German parlance as a ‘Sechter cadence’) can lead into remote and alien areas of tonality rather than circumscribe a particular key. It is only by correlating a given scale with a fundamental bass relying primarily on steps of a 5th that a key can be unmistakably defined.

The cadence I–IV–V–I, or tonic–subdominant–dominant–tonic, relies first on the wholeness of the scale, second on the clear effect of the 5th progressions in the fundamental bass and third on the effect of ‘characteristic dissonances’ (Riemann) in establishing continuity: these would include the 7th on V and also the 6th on IV (related to I, the 6-5 chord is a subdominant with sixte ajoutée, and with respect to V it is an inversion of II7, in which the 6th emerges as the root and the 5th as a dissonance requiring resolution). Individual features, since they alone do not define the cadence, may be altered or even omitted without the sense of the whole being lost; the scale may be chromatically altered, the subdominant replaced by a double dominant or a Neapolitan 6th, and the characteristic dissonances may be dropped.

Rival theories maintain either that the key is founded on the scale or that the scale is founded on the key. What is known as the theory of Stufen, or degrees, ascribes intrinsic importance to the scale. It asserts that seven chord degrees coalesce into a key by virtue of the fact that they form a unique scale. It sees the sequence of 5ths (I–IV–VII–III–VI–II–V–I), which passes through all the degrees of the scale, as a paradigm of the comprehensive realization of a key. (The emphasis on the gamut as the foundation of a key would have impeded the explanation of chromatic alterations as tonal phenomena if the theory of Stufen had not been linked by Sechter with a theory of fundamental steps; the alteration of the supertonic degree to a chord of the Neapolitan 6th, which does not fall within the scale, is accordingly justified by the fact that the altered degree, like the unaltered one, is usually associated with a step of a 5th leading to the dominant, thereby integrating the degree into the tonality.)

In contrast to the theory of Stufen, Riemann’s theory of function starts from the tonic–subdominant–dominant–tonic cadence in order to establish the key, and deduces the scale by analysing the three principal chords (C–E–G, F–A–C, G–B–D = C–D–E–F–G–A–B–C). The chords and their relationships to each other are taken as given; the scale results from them. Furthermore, as the derived phenomenon, the secondary product, the scale is susceptible to virtually unlimited alteration without the key becoming unrecognizable; by interpolations and chromatic inflections in the chords, hence by modifications whose consequence is an extension of the scale, the cadence is in no way restricted in its function of defining the key, but rather it is aided.

Tonality and key

The term ‘tonality’ has now become widespread in addition to the older term ‘key’. It was first used by Choron in 1810 to describe the arrangement of the dominant and subdominant above and below the tonic. In 1844 it was defined by Fétis as the essence of the ‘rapports nécessaires, successifs ou simultanés, des sons de la gamme’. Its currency during the long and complex history of the concept is due to a variety of causes. In the first place the expression ‘tonality’ designates the intrinsic governing principle of key (Fétis: ‘le principe régulateur des rapports’) as distinct from its outward aspect, the individual key. In the second place, the concept of key is usually associated with the idea of a given diatonic scale; but tonality also covers chords with notes foreign to the scale (and even with roots foreign to the scale) provided that they are integrated into the tonal context and do not bring about any impression of a change of key. And in the third place, tonality can be taken to mean a complex of several related keys, a broad key-area.

Thus on the one hand the term refers to the principle that governs a key from within, instead of to the key’s audible exterior. On the other hand it refers to broader relationships, the consequence of carrying the intrinsic principle a stage further and transcending the bounds of the key as it is defined in material terms. Thus Réti (1958), in order to emphasize the specific relation of the notes to a centre, a basic note or chord, in place of the general ‘rapports des sons’, distinguished between the concept of tonality and that of ‘tonicality’ (see Tonality).

Carl Dahlhaus

Historical development

To the end of the Baroque

That modern tonal harmony began about 1600 has been a commonplace since Fétis, who extolled certain spectacular dissonances in Monteverdi’s madrigal Cruda Amarilli as the beginning of the modern era in music. The standard account of this as the replacement of a contrapuntal ‘horizontal’ style by a harmonic ‘vertical’ way of writing, however, is unsatisfactory. It was not that counterpoint was supplanted by harmony (Bach’s tonal counterpoint is surely no less polyphonic than Palestrina’s modal writing) but that an older type both of counterpoint and of vertical technique was succeeded by a newer type. And harmony comprises not only the (‘vertical’) structure of chords but also their (‘horizontal’) movement. Like music as a whole, harmony is a process. (For discussion of the principles of consonance underlying medieval counterpoint see Discant; see also Organum and Counterpoint.)

To be understood from a historical point of view, tonal harmony must be seen in the context of compositional technique in the 15th and 16th centuries, a technique founded first on the opposition between imperfect and perfect consonances, second on the principle of the semitone as a means of connecting consonance with consonance (the leading note), and third on the treatment of dissonances as relationships between two voices. The progression from an imperfect to a perfect consonance – from a 6th to an octave, or from a 3rd to a 5th or unison – was experienced as a tendency comparable to that ‘instinctual life of sounds’ of which Schoenberg spoke in tonal harmony. And intervallic progressions that contain movement by a semitone in one of the voices were reckoned specially intelligible or compelling (ex.5).

As a means of effecting semitone steps, chromatic alteration, or Musica ficta, often (though not always) led to progressions that in retrospect look like anticipations or prefigurations of tonal harmony. The consonance with the leading note (ex.5a) can be heard as a dominant. But the Phrygian cadence (ex.5b) was understood as a self-contained progression and not as a fragment (subdominant–dominant) of a D minor cadence; and the fact that in spontaneous chromatic alteration the Dorian (and also the Aeolian) form of the progression 6th-to-octave (ex.5a) could be exchanged for the Phrygian (ex.5b) is just as alien to tonal feeling (which tends towards harmonic unambivalence) as the alteration of the 3rd before the 5th (ex.5c–d).

That dissonances were treated as relationships between two voices means that they were neither integrated into chords nor contrasted with chords as notes foreign to the harmony. The 7th, C, in ex.6 is neither a component of a chord (II7) which is in itself dissonant nor a dissonant suspension preceding a 6–3 chord on the seventh degree of the scale; it is a dissonance in relation to D and a consonance in relation to F.

Tonal composition using chords, as it gradually evolved during the 17th and 18th centuries, can be distinguished from modal composition using intervals, first (as already mentioned) by its conception of the chord as a primary, indivisible unit, second by its referral of every chord to a single tonal centre and third by its segregation of intervallic dissonances into the categories of dissonant chords and notes foreign to the harmony. In modal composition using intervals the penultimate chord of the Dorian cadence (ex.7a) was conceived as a secondary combination of two intervallic progressions (ex.7b–c); and the progression could be specified as cadential by altering the 3rd, G, to G♯. In tonal harmony the chord of the 6th (ex.7a) presents itself as an indivisible unit and as a fragment of a dominant 7th chord on A, whose root has to be supplied by the imagination if the tonal coherence (dominant–tonic) is to become discernible. The 3rd, G, thus becomes a dissonant 7th which is resolved downwards on to the 3rd of the tonic chord. The change in quality between the penultimate chord and the final chord, perceived tonally, no longer lies in the antithesis of imperfect and perfect consonances but in the contrast between the chord of the 7th and the triad. This change is experienced not merely as a juxtaposition but as a logical sequence in which the second chord forms the goal of the first.

The integration of all the chords, and not merely some of them, in a tonal context related to a single centre was a new principle in the 17th century. While in the modal use of intervals in the 15th and 16th centuries the cadence points (properly clausulas) represented one of the means of defining a modal centre, the intervallic progressions that occurred elsewhere in the context than at cadence points remained on the whole modally neutral.

The relationship of the practice of figured bass to the development of tonal harmony was an ambivalent one. On the one hand, figured bass encouraged the conception of the chord as the primary unit by designating vertical structures; the simultaneity was thought of as a tactile gesture rather than as the result of interwoven melodic parts. On the other hand, the experiencing of chords as based on roots, and the perceiving of the relationships between roots that build harmonic cohesion, were obstructed by the practice of figured bass: the emphasis on the actual basso continuo discouraged the awareness of the imaginary fundamental bass that was essential to the harmonic logic. Those theorists who in the 17th century formulated the principle of chordal inversion and root (Lippius in 1612 and Baryphonus in 1615) did not indeed concern themselves with figured bass but rather with the technique of Lassus; and Rameau’s decisive establishment of the concept of inversion is more a symptom of the end of the figured-bass era than one of its typical manifestations.

The relationship of contrappunto moderno (‘licentious’ counterpoint) to tonal harmony was similarly complex in the 17th century. The dissonant structures that deviate from the norms of classical counterpoint can certainly often be interpreted from the point of view of tonal harmony, though by no means always. The 7th on the subdominant (ex.8a) soon became established as a dissonant chord that did not have to be prepared in order to be understood. (In Monteverdi, IV7 is an even more frequent 7th than V7.) But the irregularity at the downward-leaping suspension (ex.8b) does not give it independent status as a harmonic phenomenon; it is rather a case of a dissonant figure which is understood in terms of intervallic writing and whose pathos derives from the fact that it forms a striking exception to the conventional resolution.

The old and the new ways of conceiving music were inextricably interlinked in the minds of many contemporaries. Bernhard, who sought to codify ‘licentious’ counterpoint about 1660 (ed. in Müller-Blattau, A1926, pp.84–5), described in his explanation of the passage in ex.9a the dissonant 4th (c–f″) as the result of an ‘ellipsis’ – an omission of the preparatory consonance d–f″ on the first beat. Thus, in keeping with the 16th-century way of listening, f ″ is a dissonant suspension leading to a resolution on the consonance c–e″. However, Bernhard made his own reduction (ex.9b), intended to demonstrate the real sense behind the unreal outward manifestation. This shows how the traditional interpretation has become coloured, or even overshadowed, by an interpretation based on tonal harmony. Since the underlying progression II–V–I, consisting of two steps of a 5th, is stronger and more intelligible than II–I–V–I, the bass note c should be heard as a passing note (and is thus omitted in the reduction). Thus f″, as the 3rd on the supertonic, is a consonance and e″ a dissonant passing note; and the fact that it forms a consonance with the equally dissonant transitional c in the bass is secondary.

By about 1600 chromaticism had reached a culmination that it is difficult to distinguish from excess. During the 17th century this became both simplified and tonally integrated, the simplification being a necessary part of the integration. Gesualdo’s technique, which in historical terms represents an end and not a beginning, was virtually without consequence for the development of tonal harmony. It relied on the use of extreme chromaticism to render a contrapuntal sequence ‘strange’ while completely obeying the rules of traditional intervallic writing by, for instance, displacing a ‘reference note’ chromatically by a semitone in the course of resolving a dissonance over it.

Chromaticism can qualify as being tonally integrated when the directional pulls of the leading notes, which arise through chromatic alteration, are in agreement with the fundamental progression. The sense of the plagal cadence in a major key (IV–I) is underlined by chromatic alteration of the subdominant chord to a minor triad, just as the sense of the perfect cadence in a minor key (V–I) is underlined by converting the dominant chord to a major triad. And the indeterminate direction of the 5th on the chord of the dominant can be resolved by raising or lowering (G–B–D♯–F or G–B–Dâ™­–F) or by both at once (what Kurth termed Disalteration: G–B–Dâ™­–D♯–F; see ex.11 below). Chromatic alteration gives the note a directional tendency.

Carl Dahlhaus

The Classical era

Among the most striking features that distinguish harmony after about 1730 from that of the Baroque era are the slowing down of harmonic rhythm, the change in function of the bass and the presence of a formally constructive harmonic technique leaning on the principle of correspondences. The fact that the rate of harmonic rhythm (measured as the average distance between changes of harmony) became slower was associated with the stylistic ideal of noble simplicité as opposed to Baroque ostentation; at the same time, it was necessary if the tonal outline of larger-scale form was to be accessible to a public comprising more ordinary music lovers than connoisseurs. In instrumental music above all, where the music’s unity had to be conveyed without the assistance of a text, it was necessary to provide a view of the tonal layout of any movement that was intended to be a closed form, so that it should not appear to be a mere jumble of ideas.

The music of the Baroque era was undoubtedly tonal, but it evolved a type of tonal harmony different from the Classical or Romantic. In contrast to Rameau’s doctrine, the real basso continuo could by no means always reasonably be reduced to a basse fondamentale; many of the harmonic procedures were determined by genuine melodic movement in the bass rather than by an imaginary fundamental progression. (In the 17th-century bass formulae the melodic formula itself appears as the primary factor, and the harmonic elaboration, which is variable, as the secondary factor.) On the other hand, after the decline of figured bass, which represented not only an aspect of performing practice but also a form of musical thought in its own right, it was the tonal functions that established harmonic continuity serving in the background of the music as an abstract regulative force. It was only in the pre-Classical and Classical eras that harmony was moulded into the system that Riemann described. The bass, expressed with emphasis, took the guise of an audible signal of the intended tonal functions, and not, or only occasionally, that of a part on whose individual progress the harmonic sequence depends.

The harmony of the Viennese Classical composers, if it is to be properly understood rather than merely identified by chord names, must be analysed in relation to metre, syntax (i.e. the laws by which musical phrases combine to make larger units) and form. The metrical relationship between anacrusis and termination, or between weak and strong beats, and also the syntactical relations between statement and answer, or between antecedent and consequent, are all relationships of tonal harmony: syntax is founded on, or partly determined by, harmony, and conversely harmony derives its meaning from the syntactical functions it fulfils. Tonal functions do not exist in their own right. They arise as a result of chords of differing tonal strengths – some prominent, some fleeting. And the strength with which a chord fulfils the function of, for example, the dominant depends on its position within the surrounding metrical and syntactic schemes. These factors determine whether it is simply a passing chord or a half-close marking the end of an antecedent phase. The fact that musical forms spanning hundreds of bars are sustained in their effect as a unity primarily through a comprehensible layout of keys was well known to 19th-century theorists such as A.B. Marx. But harmonic theory has still not really accepted the idea that Classical harmony, whose theory it purports to be, cannot be adequately understood other than in relation to musical form.

The meaning of any sequence of chords must depend on where, formally, it occurs. The widespread theory that in Classical music all harmonic relationships can be seen as expansions or modifications of the cadence is thoroughly mistaken. It is necessary to distinguish between closing sections, whose harmony constitutes a cadence, and opening and middle sections. The astonishing harmony at the beginning of Beethoven’s Waldstein Sonata op.53, for example, would be out of place at the end of a movement: its effect as a beginning is compelling and forward-driving. And harmonic sequences characteristic of development sections cannot convincingly be traced to the cadence; nor could they be used as beginnings or endings.

Moreover, the force of a harmonic model is not independent of the formal level on which it appears. The cadence I–IV–V–I, taken as a sequence of chords, cannot be reversed to form I–V–IV–I without some loss of effect; yet as an arrangement of keys, a harmonic outline for an entire movement, the reversed form is commoner than the original. The fact that the dominant key must be arrived at and established in spite of the conflicting pull of the dominant chord back towards the tonic – a goal generally achieved via the dominant of the dominant – imbues the harmonic process with a tension that it would not have if it were merely a reiteration of the same cadential model on different levels of formal organization.

Carl Dahlhaus

The Romantic era

The development of harmony in the 19th century reflected in its ideas the thinking of the age as a whole: the idea of continuous progress, the postulate of originality and the conception of an organism as a self-contained network of functions. In the 17th and 18th centuries the proportion of chromaticism and unusual dissonant figures that seemed admissible or adequate for any one composition depended largely, together with the emotional content of a text, on the genre to which the work belonged. The notion that harmony at any one point was in a single ‘general state of evolution’ is a 19th-century idea that has been applied retrospectively to earlier times. Theatrical style was in reality sustained by criteria different from those applicable to ecclesiastical or chamber style, and the harmony of a recitative or of a fantasia was hardly comparable with that of an aria or a sonata movement. In contrast to this, it is possible (as shown by Kurth, C1920) to describe the history of harmony in the 19th century as a totally interconnected development, propelled by the conviction that every striking dissonance and every unusual chromatic nuance was another step forward in musical progress, towards freedom, provided that the discovery could somehow be successfully integrated into a musical structure. What was of decisive importance about the ‘Tristan chord’ was not the simultaneity as such, which as II7 in D♯ minor would have been a mere trifle, but Wagner’s clever discovery that it could be interpreted as an inversion of a chord of the 7th on the dominant of the dominant (B), with a lowered 5th (Fâ™®) and a suspended 6th (G♯) leading to the 7th (ex.10a).

The idea of originality, which imposed itself as the dominant aesthetic principle in the late 18th century, combined the demand that in ‘authentic’ music the composer should express the emotions of his inner self with the postulate of novelty. Alongside melodic ideas, what the 19th century valued most as ‘inspirations’ were chords that were surprising and yet at the same time intelligible. Such chords were felt to be expressive – the word ‘expression’ being used in a strong sense to refer to the representation of out-of-the-ordinary inner experience by the use of unusual means – and were expected to take their place in the historical evolution of music, an evolution that was seen as a chain of inventions and discoveries. Thus, for instance, the chord of the dominant 7th with raised 5th (see ex.11a), otherwise viewed as the transferring of the augmented triad to the chord of the 7th, provoked as its antithesis the construction of a chord of the dominant 7th with lowered 5th (ex.11b); later the two chords were combined to form a 7th with doubly altered (disalteriert) 5th that could be complemented with a 9th (ex.11c); and finally, by a transposition (forbidden in harmonic doctrine) of the 9th to a lower octave (ex.11d), the tonal phenomenon of the chromaticized dominant chord was transformed into a metatonal phenomenon, the whole-tone scale (ex.11e)

The interpretation of a musical structure as an organism was one of the arguments used to justify the principle of aesthetic autonomy, that is the claim of music to be listened to for its own sake. The orientation to the organism model means that the harmonic as well as the motivic structure of a work represents a self-contained network of functions in which, ideally, there is not a single superfluous note. One particular problem arose for the composer: if he accepted the principle of autonomy, he committed himself to finding new ways of refining the chords on the various degrees of the scale, the types of chromaticism and the uses of dissonance, such as was demanded by the idea of originality and progress, and at the same time to making all these elements appear integrated increasingly closely into the tonal context. The multiplicity of chords on the degrees of the scale that Schoenberg praised in Brahms was intended to consolidate rather than to loosen the tonal articulation. The more comprehensive a supply of chords a key has, the more emphatically must its gravitation round a tonal centre be experienced. Thus the opening of Brahms’s G minor Rhapsody op.79 no.2 for piano contains in its first four bars (ex.12) the harmonies D minor–Eâ™­–C–F–C–D9–G. The underlying key of G minor is implied but never explicitly stated; and yet it is the common denominator among the fragmentary key centres of D minor (I–IIâ™­ Neapolitan), F major (V–I) and G major (IV–V–I), in that the listener, because of the thematic character and the formal position of the opening bars, feels impelled to look for tonal unity among so many harmonic steps.

Similarly, Wagner’s chromaticism was not designed to achieve merely momentary effects of harmonic colour. It also served to link chords more closely together. For example, the progression in the opening bars of Götterdämmerung may at first seem wayward, but even though it cannot be heard as a basic tonal progression the ear can (if it is not prejudiced) recognize it as a strong harmonic movement (ex.10b).

Carl Dahlhaus

Early 20th century

Since the 19th century there has been an alternative to chromatic harmony as a means of extending tonality, namely modal harmony. It arose as part of a general interest in the past, in folk music and in oriental music and served to introduce ‘foreign’ elements into tonal harmony by drawing from other historical and cultural areas. It was not so much a system of harmony in itself as a way of deviating from the normal functions of tonal harmony to achieve particular effects. It was unlike the modality of the 16th century in that it was the relationships between chords, rather than melodic considerations, that determined the key centre. In the 19th century the modes came to be thought of as variants of major and minor, and this is implied by phrases such as ‘Mixolydian 7th’ and ‘Dorian 6th’. The Mixolydian 7th, with the chords of D minor and F major, for example, in the key of G major, is not ‘modal in character’ in the medieval and Renaissance modal system (where the 3rd and 4th were just as much determinants of the modal centre as was the 7th); only against the background of major and minor did it become significant. Modal harmony, for all its apparent dependence on the past, was thus a 19th-century innovation.

The most significant aspects of 20th-century harmony (if indeed harmony is still the appropriate term) include, first, its decline in importance as a factor in composition; second, the ‘emancipation’ of the dissonance, leading directly to atonality; and third, the construction of individual systems.

Whereas in the 19th century harmony appeared to be the central factor by whose evolution the progress of music as a whole was measured, in the new music of the 20th century rhythm, counterpoint and timbre came to the fore, since the structural function of harmony was either too indistinct or too difficult to perceive for it to be capable of establishing musical continuity for long stretches. Harmony, which in a good deal of 20th-century music is regulated solely by negative rules (instructions about what to avoid), became both more intractable and less significant.

The emancipation of the dissonance was something that Schoenberg resolved upon in the years 1906–7 – not in any spirit of iconoclasm, in fact rather reluctantly but with a sense of inner inevitability. What is meant by it is that a dissonance no longer needed to be resolved since it could be understood in its own right; it no longer needed to rely on a consonance as its goal and its justification. The obverse of this was that the dissonance became isolated; its pull towards resolution and forward movement may indeed have been a restraining force, but it had also been a force for coherence, for the relationship of parts. Schoenberg felt the inconsequentiality of the emancipated, self-sufficient dissonance as a deficiency. To counter this isolation he adopted in particular two procedures: the principle of complementary harmony, and the conception of a chord as a motif. By complementary harmony is meant the procedure of relating chords to one another through the number of notes by which they differ. Thus in ex.13 (no.5 of Das Buch der hängenden Gärten op.15) the two four-note chords in bar 1 of the accompaniment have only one note in common, and those in bar 2 no note at all (with the exception of the second bass note, sounded a beat later). The conception of a chord as a motif means that it can present in vertical form a configuration of notes or a structure that can also be presented in horizontal form without losing its identity. The fact that that configuration exists as a common denominator between the different presentations establishes a musical coherence that brings the chord – the emancipated dissonance – out of its isolation. Inherent in the techniques that provide a solution to the problem of emancipation is the change to dodecaphony: complementarity tends towards the 12-note principle; equivalence of horizontal and vertical is a basic feature of serial technique.

In free atonality and 12-note writing the borderline between consonance and dissonance was considerably higher than in tonal music. The emancipation of the dissonance by no means implies, however, that the degrees of dissonance between different intervals had lost their significance. On the contrary, the combining of sounds continued to be governed by what Hindemith called the ‘harmonic fluctuation’ (i.e. the graph of harmonic intensity from chord to chord in a progression).

12-note harmony moves between two extremes: at one extreme, the principle of ‘combinatoriality’ (Babbitt), the bringing together of fragments from the different forms of a row that make up the material of the 12-note system, so that any undue predominance of individual notes (which might suggest tonality) is avoided; at the other, overt or latent association with tonal chord structures and progressions, such as is found in Berg’s Violin Concerto and the beginning of the Adagio of Schoenberg’s Third String Quartet op.30 (ex.14). The idea of stating all or part of a row in vertical form presents problems: in simultaneous presentation the notes of a row are interchangeable, because the vertical order of the row (upwards or downwards) has to follow not only the succession of pitch classes of that row, which are fixed, but also their octave register, which is not.

Harmonic tonality, which broke down about 1910, had dominated the scene for three centuries. It had been a universal system of reference, marking out the boundaries within which a composition had to move in order to correspond with the European concept of what music was. In contrast, none of the systems projected in the early 20th century, apart from the 12-note technique as such, extended beyond specific validity for any individual composer. Skryabin based his later works on a central sound that determined both vertical and horizontal structures – the ‘mystic chord’, which has been interpreted first as a piling-up of 4ths (C–F♯–Bâ™­–E–A–D), second as a chord of the 9th with lowered 5th (C–E–Gâ™­–Bâ™­–D) with unresolved suspension of the 6th (A), and third as a section of the natural harmonic series (upper partials 8–11 and 13–14 imprecisely pitched). The first interpretation is prompted by Skryabin’s way of using the chord in his late works, the second has regard for the chord’s historical provenance, and the third adopts the premise on which scientific rationalizations of harmonic phenomena were based during the 18th and 19th centuries.

Stravinsky frequently used the technique of overlaying triads, for example the chords of C major and Eâ™­ major in the third movement of the Symphony of Psalms (ex.15a). By this means (a manifestation of his wider use of the Octatonic collection) he created bitonal effects and an ambiguity between major and minor (C major and C minor) and gained the possibility of further transformations by interchanging major and minor 3rds, so that, for example, C♯–E–G♯ may appear in place of C–E–G. ex.15b, from the same movement, shows D major and F major alternately superimposed on C♯ major/minor.

Bartók, continuing certain ideas explored by Liszt, developed a harmonic system based primarily on the principle of symmetrical octave division (C–f♯–C′, C–e–g♯–C′, C–eâ™­–f♯–a–C′): a region of harmony that had lain at the edge of traditional tonality or beyond it was established as the centre by Bartók and subjected to systematic organization. The tonal organization that Hindemith developed from the natural harmonic series (using methods that were sometimes idiosyncratic) was intended as a comprehensive system equally valid for Machaut and Bach as for Schoenberg. Seen in perspective it was a projection of Hindemith’s own stylistic peculiarities into a natural scheme that did not have the universality that Hindemith claimed for it. Nonetheless, it was in principle one that sought to measure dissonance level in complex chords and make possible the controlled gradation of dissonance in chord progressions. It used a type of abstracted fundamental bass comparable, though not identical, with that of Rameau.

Carl Dahlhaus

Late 20th century

By the mid-20th century many composers and commentators no longer regarded harmony as a discrete musical category in its own right, independent of more general questions of pitch organization. Certainly it seemed difficult, above all in 12-note and serial music, to conceive of harmony in diachronic terms, as regulating the succession of simultaneities over time. But still there was a recognition among a number of postwar serial composers that Schoenbergian 12-note technique, in which pitch classes had been ordered at least predominantly in the horizontal dimension, had left the vertical dimension underdetermined. Two possible ways of compensating for this perceived arbitrariness were widely explored: on the one hand, techniques that sought to assert more direct control over the construction of chords or pitch-class simultaneities; and, on the other, attempts to create greater harmonic definition through the distribution of pitches in register.

Stravinsky’s sensitivity to questions of harmony and intervallic polarity by no means lessened with his turn to serialism in the 1950s. In a number of later works, including the Variations (Aldous Huxley in memoriam) for orchestra, he employed the technique of hexachordal rotation pioneered by Krenek in his Lamentatio Jeremiae prophetae, which involves splitting the 12-note row into two hexachords each of which is then successively rotated, the rotations being transposed each time back onto the same initial pitch (see Twelve-note composition, §7). By forming successions of chords from the homophonic superimposition of these rotated forms, Stravinsky created a highly personal form of serially generated harmony, often incorporating the split octaves and simultaneous major and minor 3rds which had been prominent features of his middle period. These rotation techniques have been adapted by a number of younger composers, including Wuorinen and Knussen.

Boulez, likewise convinced of the inadequacy and arbitrariness of complementary harmony, developed from 1951 onwards the technique of multiplication. This involves partitioning the series into unequal segments (blocs sonores), each of which is then taken and in turn transposed onto each of the component pitch classes of another segment, the product of the ‘multiplication’ consisting of all the pitches of each of these transpositions combined (see Boulez, pierre, §3). In practice the distinctiveness of the harmonic results obtained from multiplication depends on the intervallic constitution, and above all the density, of the harmonic objects in question. Where the sonorities being multiplied share a concentration of the same interval class, that concentration will be reinforced in the sonority that results. Elsewhere, however, multiplication results in dense aggregates approaching full chromatic saturation, in which such individual intervallic characteristics are neutralized.

The products of multiplication are groups of pitch classes whose articulation in both time and in register remains unspecified. Hence Boulez, along with other composers of his generation such as Stockhausen and Berio, also explored the potential of pitch register for creating a sense of harmonic definition within a chromatically saturated texture. The device, observable in Webern’s late works, of fixing each pitch class in a single registral position over a substantial number of bars, features in a number of serial works of the 1950s and beyond (such as Stockhausen’s Kontra-Punkte), while the third cycle of Boulez’s Le marteau sans maître (see L. Koblyakov, Pierre Boulez: a World of Harmony, Chur, 1990) systematically exploits the transposition of a ‘vertical row’, in which the 12 notes are ordered in register rather than time. Within these fixed-register ‘harmonic fields’ the hard and fast distinction between horizontal and vertical textures is transcended. Such a field can be articulated as pure simultaneity (12-note chord), pure succession (12-note row) or as part-simultaneity and part-succession. While a row or similar ordering might regulate the horizontal dimension (the temporal succession of pitches), the vertical dimension (the registral distribution of pitches) can be structured according to quite different criteria: such pitch fields, for instance, might display the kind of inversional symmetry around a central pitch or interval found earlier in the century in works of Bartók (Music for Strings, Percussion and Celesta, Piano Concerto no.2) and Webern (Symphony op.21, Cantata no.2, op.31), where its function had often resided in the harmonic control of canon or other forms of imitative polyphony.

Often associated with this kind of symmetrical harmony is the attempt to give 12-note chords or pitch fields a distinctive harmonic identity by restricting the number of interval types occurring between registrally adjacent pitches. Such limited interval (or interval class) construction had been adumbrated by Schoenberg (B1911, p.454), who had not only observed that the chord of 4ths employed to notable effect in the Kammersymphonie no.1 could be extended to produce a 12-note chord, but had created further 12-note chords whose adjacent intervals consisted solely of major and minor 3rds (ibid., 456–7). Textures dominated by such ‘characteristic intervals’ interested both Pousseur and Stockhausen in the mid-1950s, while independently LutosÅ‚awski developed his own rich vocabulary of 12-note chords based on limited interval-class construction. Many of these chords restrict themselves to just two adjacent interval types (such as perfect 4ths and minor 3rds, or tritones and semitones). Other composers meanwhile have gravitated to all-interval chords, in which the 11 pitch intervals within the octave each occur only once between pairs of registrally adjacent pitches. In the music of Elliott Carter, these prove ideally suited to controlling the vertical superimposition of separately evolving musical layers, enabling each to be defined by its own tessitura and limited repertory of intervals.

The global effect created by such fixed-register distributions is that of a static articulation of space rather than a dynamic movement through time. This view of harmony as an essentially synchronic phenomenon was not restricted to composers of a serialist persuasion. A similar sense of stasis is provided by harmonic fields based on the omnipresence not of the total chromatic but of a more limited pitch-class collection. What Slonimsky termed ‘pandiatonicism’, the free, non-functional employment of diatonic modes as neutral pitch ‘collections’ rather than as scales with a hierarchy of degrees, had been a prominent feature of mid-century neo-classicism, but it equally came to characterize the minimalist works of Reich, Glass and later Adams. The harmonic consequence of Reich’s phasing processes, in which unison statements of diatonic (but generally non-triadic) melodic fragments move gradually out of synchronization, is often this kind of static modal field. Likewise the later works of Pärt, while often homophonic in texture, are characterized by unpredictable harmonic encounters generally within a closed modal or diatonic collection.

The reintegration of consonance was, from around 1970, perhaps the most noticeable harmonic development in the work of composers of almost any stylistic persuasion. The harmonic taboos characteristic of serial music (such as the avoidance of octaves and triadic formations) started to be abandoned in the 1960s even by such composers as Berio and Pousseur who still saw their work as belonging broadly within the serial tradition. Many composers reintroduced the materials of functional tonality in the context of non-functional syntaxes: the use of triads linked by conjunct motion between voices (a combination of common tones and stepwise progession by tone or semitone) characterizes works as diverse in sound and aesthetic as John Adams’s Phrygian Gates and Scelsi’s Anahit. In the work of neo-Romantic composers such as Rihm or Holloway, the rehabilitation of consonance involved explicit reference to Romantic harmony, even at times outright quotation, but often in a way that fostered a sense of historical distance from the model, resulting in a sense of rupture and dislocation rather than an overarching harmonic unity. In his symphonic works of the 1970s onwards Peter Maxwell Davies has attempted to reinvent a directional harmonic syntax capable of sustaining large-scale tonal structures: such attempts, however, encounter the problem of adequately affirming points of harmonic arrival in the absence of communally recognized criteria of tonal stability.

Electronic transformation has proved an especially fruitful way of exploring the continuum between acoustic consonance and dissonance. In each of the 13 cycles of Stockhausen’s Mantra the sounds of the two pianos are ring-modulated by sine tones whose frequency corresponds to the ‘fundamental’ of that cycle. The degree of dissonance of the modulated complex consequently depends on the dissonance of each piano note in relation to the fundamental note. The simulation of techniques of ring-modulation in the instrumental domain became just one of the techniques associated with Spectral music, especially in the 1980s. This movement evolved under the influence of the work of Messiaen and Stockhausen, as well as the new possibilities opened up by the computer analysis of timbre. Murail, a leading spectralist, compared the many dense harmonic complexes found in Messiaen’s later work to inharmonic resonances of bell sounds, whose spectra can be replicated by means of frequency modulation. Spectral music takes such complex inharmonic spectra as a unifying model for both harmonic and temporal structuring. At the end of a century in which harmonic theory had been sparing in its appeals to science or ‘nature’, spectral music seemed to revive, in principle at least, the acoustic rationalizations that had been central to harmonic thinking from its very beginnings.

Julian Anderson, Charles Wilson

Theoretical study

The difference between the theoretical and the practical study of harmony consists not so much in a divergence between the reasons for rules and their application but rather in dissimilarities that arise out of the differing historical origins of the two disciplines. The theoretical study of harmony owes its inception to a remodelling of musica theoretica – that is, of mathematical speculations on the foundations and structures of the tonal system. The practical study of harmony proceeded from the doctrine of figured bass, which was expanded at the beginning of the 18th century and elevated into a theory of free composition, as opposed to counterpoint, the theory of strict composition (see §6 below). Part of the legacy of musica theoretica is the claim of harmonic theory to be scientific, a claim that has constantly shifted its ground but has never been abandoned. Harmonic theory affects to be a ‘theory’ in the strongest sense of the word instead of being merely a collection of rules for musical craftsmanship.

On the philosophical assumption that numbers and numerical proportions, conceived of in the Platonic sense as ideal numbers, represent founding principles and not mere measurements, the numerical demonstration of intervals and intervallic complexes was taken to be the mathematical basis of musical phenomena up to the 17th century (and in peripheral traditions up to the present day). Zarlino (A1558, bk 3, chap.31), who took ‘harmonic proportion’ (15:12:10) as the basis of the major triad and ‘arithmetical proportion’ (6:5:4) as that of the minor triad (from measurements of string lengths), propounded the musical priority of the major triad by virtue of the mathematical and philosophical prestige of the concept of ‘harmonic’. The same result, the exalting of the major triad, was arrived at two centuries later by Rameau but in a different way. First, his measurements were made by reckoning vibrations instead of string lengths, so the proportion 4:5:6 shifted from the minor triad to the major and the proportion 10:12:15 shifted from the major to the minor. Second, following contemporary natural science rather than humanism, he gave precedence to the simple (hence the proportion 4:5:6) rather than the ancient (the term ‘harmonic’, which was saturated in ideas). Third, he sought to discover the scientific foundations of musical actualities not in Platonically interpreted mathematics but in physics, the most advanced discipline of his times: the Platonic idea of number had lost its force and had sunk from a scientific status to a poetic or sectarian one. The basis of musical phenomena was henceforth discovered in the physically determinable nature of the note, that is, in the natural harmonic series. There was one fundamental fact which from the 18th century on was invoked by authors of textbooks on harmony in order to claim scientific legitimacy: this was that the major triad is contained within, or prognosticated by, the natural harmonic series (in the upper partials 1–6, C–c–g–c–e′-g′, notably as partials 4–6).

As opposed to a physical foundation for musical theory, Hauptmann (B1853) presented a foundation that was dialectical and idealistic. He construed such phenomena as the triad or the cadence as instances of the Hegelian model of thesis, antithesis and synthesis. According to Hauptmann, in the cadence I–IV–I–V–I the tonic is first ‘set up as a direct entity’ (as if it were being stated unquestioningly rather than being argued); it is then ‘divided in itself’ (as the dominant of IV and sub-dominant of V), finally to be ‘reinstalled as a result’ and thus confirmed (retrospectively, IV and V appear no longer as tonics to which I relates but as subdominant and dominant of I: I’s ‘existence as dominant’ turns to ‘possession of dominant’). This dialectic, which he evolved in detail, Hauptmann thought of as an active dialectical process, an intellectual process in the subject matter itself, not a mere mode of description. He saw dialectical construction as a valid theoretical representation of a principle that was active in music and that established it as ‘logic in sound’ and to that extent as a science.

Riemann made various attempts to forge a link between the physical explanation and the dialectical. He gradually came to minimize the importance of the former and finally almost to deny it altogether. In accordance with the philosophical tendencies of his times, he transformed Hauptmann’s Hegelianism into a kind of Kantianism in his later writings, above all in the ‘Ideen zu einer “Lehre von den Tonvorstellungen”’ of 1914–15. But the prevailing means by which the theory of harmony was given a scientific basis in much of the 20th century was the explanation of harmonic phenomena in historical terms. It was no longer Nature (physically definable) but History that constituted the final court of appeal. After mathematics, physics, Hegelian dialectics and psychology (Kurth, B1913, C1920), it was the role of history to guarantee the scientific character of music theory. The next step, in recent years, has been a psychological approach dealing with musical cognition and perception (see §5 below). The mathematical, physical and idealistic dialectical theorems have indeed not been forgotten; but, in spite of important efforts to restore them (Handschin’s ‘Pythagoreanism’, 1948), they have become peripheral.

Since Rameau there has been a constant endeavour to explain the historical phenomenon of tonal harmony during the 17th, 18th and 19th centuries by reference to the physical properties of the single note, and thus to establish harmonic theory as a strict science. This has involved music theory in problems that would have been less difficult to solve were it not for this fixation on the natural harmonic series. Above all, the minor triad presented this approach through physics with a challenge that it expanded to meet only with difficulty and with some questionable theorizing.

None of the endeavours to discover an acoustical explanation for the minor triad analogous to the rationalization of the major triad (by means of the natural harmonic series) met with success. Significantly, the one physicist among music theorists, Hermann von Helmholtz, forwent any reduction to physical terms. The repeated attempts that were made, persistently and futilely, from Rameau’s postulation (B1737) of a sympathetically vibrating f and Aâ™­ beneath c″ through to Riemann’s idea of a series of lower partials (as a symmetrical inversion of the series of upper partials) all started with certain assumptions. They took for granted on the one hand that the major triad derived from the physical properties of the single note, and on the other that the minor triad enjoyed a status equal to that of the major; from these it followed logically that there must be a physical explanation for the minor triad. The notion that the minor was a shading of the major, an artificial variant of the natural triad (Helmholtz), was not really taken as a satisfactory explanation. It was regarded as no more than a description, and an admission of defeat in the face of the impossibility of explanation.

A third hypothesis, or group of hypotheses, proceeded from the idea that the 5th and the major 3rd were the only ‘directly intelligible’ intervals, that is, intervals essential for note relationships. The major triad C–E–G would thus be a combination of C–G and C–E with C as the common point of reference (centre harmonique) of the intervals, and the minor triad C–Eâ™­–G would be a combination of C–G and Eâ™­–G with G as the common point of reference (Hauptmann, B1853) or with C and Eâ™­ as a double root. The assertion that the minor triad must be read from the top downwards contradicts the musical fact that it attempts to rationalize. The division of the root, if it is to be empirically valid, must be perceived as an ambivalence and not as a simultaneous appearance of two roots: in other words, as the possibility of allowing either C or Eâ™­ to become alternately the centre harmonique (Helmholtz, E1863). A variant of Hauptmann’s theory of the minor triad appears in Oettingen’s thesis (C1866) that the unity of the chords resides in the fact that the notes of a major triad (c′–e′–g′) have a ‘tonic’ root (C) and those of a minor triad (c–eâ™­–g) have a ‘phonic’ overtone (g″) in common. As a rationalization of the unity of sound the principle of ‘tonicity’ is complemented by that of ‘phonicity’.

According to Kurth (B1913), who substituted psychological for physical explanations, the major and minor triads are to be regarded not as stable, self-contained structures but as states of tension: the 3rd, E, in the (dominant) major triad on C pushes towards F (as the root of the major triad on F), while the 3rd, Eâ™­, of the (subdominant) minor triad on C pulls towards D (as the 5th of the G major triad). Rationalization is sought no longer in the nature of things, definable in physical terms, but in the nature of people, interpretable in psychological terms.

The understanding of a theory does not merely mean the grasping of the principles from which it derives. It also particularly means the apprehension of the questions that prompted it in the first place. The difficulty of explaining the minor chord through physical premises – the conviction that acoustics must form the foundations of music theory – represents one of the challenges that harmonic ‘dualism’, most clearly exemplified in Riemann’s theory of harmonic functions, set out to meet. A second objective that was in Riemann’s mind was to formulate a theory of the secondary degrees II, III and VI which would establish their differences from the primary degrees I, IV and V more precisely than had the theory of degrees. The term ‘mediant’ is not an explanation but merely a descriptive label without any theoretical content; and the idea that primary degrees are connected to the tonic by direct relationships of a 5th whereas secondary degrees are indirectly so connected may do justice to some applications of the secondary degrees (for example, their role in the sequence of 5ths, I–IV–VII–III–VI–II–V–I, the prime harmonic model in the early 18th century) but cannot do justice to all their functions.

Riemann proceeded from the observation that the secondary degrees can sometimes appear as ‘representatives’ of primary degrees. Thus, in major keys II can fulfil the function of IV, III that of V or I, and VI that of I or IV. (VII, except in sequential passages where the bass moves by 5ths, is an incomplete form of the dominant 7th.) Riemann failed to take into account, or else his system led him to suppress, the affinity or intrinsic proximity of II to the dominant of the dominant. Impressed by the musical experience of functional similarities between the degrees, Riemann attempted to rationalize them through his theory of ‘apparent consonances’ or ‘understood dissonances’. Here, IV can be represented by II (so that II appears as a relative minor to the subdominant – Subdominantparallele, abbreviated as ‘Sp’) since II has its basis in IV, having arisen from it by a substitution of the 5th on the subdominant (in C major, C) by the 6th (D). The note D, regarded by the theory of fundamental progressions as the root of the chord on the supertonic, is according to Riemann only apparently a consonance. For its true importance for the harmonic context to be recognized it must be perceived as a dissonant adjunct to the subdominant. Thus in Riemann’s system the terms ‘consonance’ and ‘dissonance’ refer less to phenomena of actual perception than to categories in musical logic. Indeed it might be asked whether the speculative theory of ‘apparent consonances’, which flatly contradicts the usual empirical perception of the supertonic degree, is necessary at all in order to explain the functional similarity between degrees II and IV: that two structures fulfil the same or an analogous function in no way presupposes that the one must be materially derivable from the other.

The theories of function and of fundamental progressions, which are generally presented as alternatives, can in large part be understood as contrary but complementary. In the first place the distinction made by the theory of fundamental progressions between primary and secondary steps is certainly not stated as such by proponents of functional theory, but neither is it ignored or actively denied. And modern proponents of the fundamental progression, like Kurth, have not found the explanations attempted by functional theory superfluous; rather they find them too speculative on the one hand and too narrow on the other to do justice to all the functional differentiations between the degrees of the scale. In the second place it is obvious that the theory of fundamental progressions is primarily orientated to early 18th-century harmony (namely to the harmonic model of the sequence of 5ths), while the theory of functions, in common with Riemann’s doctrine of metre and rhythm, is developed from the music of Beethoven. It is thus to some degree not a case of competitive theories dealing with the same matter in hand, but of theses concerning different stages of a historical development. Third, the fundamental progressions described or reconstructed by that theory and the direct and indirect relationship of chords to the tonic conceptualized in the theory of functions are factors in composition that are perfectly capable of existing side by side. The description of the degree of relatedness between individual chords and the tonal centre, and the delineation of the path followed by the basse fondamentale within the chord sequence, are not mutually exclusive but complementary. Similarly, the different stages of historical development mentioned above represent not a total change of principles but merely a shift of emphasis between progression from chord to chord, and the relationship of chord to tonal centre.

Carl Dahlhaus

Theory since 1950

Since the mid-20th century theoretical approaches to harmony have developed in several new directions, particularly in the anglophone world. Of primary importance is the inflence of heinrich Schenker, whose theory of harmony is ultimately absorbed into a thorough-going account of tonal structure (see Analysis and Tonality). Schenker assigns to a vertical formation the status of a harmony only if it is heard to represent a Stufe (scale-step, from Sechter) that is prolonged through a (finite) span of time. Formations that fail this criterion are interpreted as an amalgam of conjunct melodic lines, whose interaction is constrained by principles of counterpoint rather than harmony. Furthermore, a vertical formation that achieves the status of a harmony when considered within the context of its own time-span of prolongation dissolves into linear motion when considered beyond those boundaries. With a single exception, every formation thus ultimately fails the Stufe criterion and reverts to prior linear and contrapuntal processes. The exception is the Ursatz-generating tonic triad, whose time-span of prolongation is co-extensive with the entire piece. It alone is generated by harmonic rather than contrapuntal principles. Thus although harmony, represented by the tonic triad, is the foundation of the Schenkerian view of tonality, the status of individual harmonic structures is nonetheless attenuated. Those aspects of music that traditionally count as harmonic – chord and chord progression – are subsumed by, and inseparable from, the broader concept of tonality. In this sense, ‘harmony’ reclaims the earlier sense of the Greek harmonia (see §1 above).

A similar absorption of harmony is evident in recent analytic approaches to atonal repertories, but for entirely different reasons. Once the diatonic gamut gives way to the set of 12 equal-tempered pitch classes, the distinction between the diatonic step and leap that underwrites the dichotomy of tonal melody and harmony is altered. Harmonic formations are composed of the same materials and relationships as pitch categories (scales, melodies, motifs, contrapuntal conjunctions) from which they are traditionally held distinct. Each of these categories is considered, rather, as a specific mode of projecting (or formatting) a more abstract category, the pitch-class set, the properties, potentials and interrelations of which exist independently of its realization in time and in register. Consequently, although atonal theory, in the tradition of Forte, Perle and Morris, furnishes a descriptor for all harmonic formations, the study of harmony now warrants different treatment. It is for this reason that there is no distinction in subject matter between books entitled Harmonic Materials of Modern Music (Hanson, D1960), The Book of Modes (Vieru) or The Structure of Atonal Music (Forte, D1973), although each of these works pursues its subject in a different way.

Where ‘harmony’ (as a category designating the description and interpretation of simultaneously attacked or perceptually fused pitches) most retains its categorical integrity is in the analysis of the chromatic repertories of the later 19th century, specifically those connecting consonant triads and dominant-7th and half-diminished-7th chords via semitonal part-writing. While some such chromatic events elaborate or replace diatonic ones and are readily interpreted in terms of linear generation along Schenkerian lines, others resist a determinate diatonic interpretation, and are a hybrid of tonality and atonality. They maintain the leap/step and hence harmony/melody distinction of diatonic tonality, but reinterpret it in reference to the tempered 12-note gamut that also underlies atonal music: semitones are melodic, all other intervals harmonic. In this larger, symmetric gamut, tonics are mercurial and Stufen lack identity, so the distinction between prolonged and prolonging events is occluded. Vertical formations are thus safeguarded against absorption into a linear-generated framework at higher structural levels. At the same time, the leap/step and harmony/melody demarcation protects harmony from subsumption under the context-free pitch-class-set umbrella of atonal theory: because harmony and melody are made up of different intervals, each retains its categorical autonomy.

In 12-note triadic and 7th-chord chromaticism, then, harmony retains the primary cognitive status that it bears in the approaches of Rameau and Riemann to classical tonality. Yet the ontology and behaviour of harmonic objects in a symmetric, tempered chromatic universe are different from those in a (conceptually) just or Pythagorean diatonic universe, and these differences warrant distinct interpretative strategies. The tonal indeterminacy of chromatic harmony leads to two classes of interpretation: individual harmonies are orientated either towards multiple tonics simultaneously or towards no tonics at all. The first approach (multiple tonics) traces its origin to early 19th-century notions of Mehrdeutigkeit (Vogler, Weber), leads through Schoenberg’s notion of hovering tonality, and is represented by theories of double tonality (Bailey, C1977–8) and functional extravagance (Smith, C1986). The second approach (no tonics) traces its origin to Fétis’s observation (B1844) that diatonic sequences suspend a listener’s sense of tonality; leads through writings of Capellen (C1902) and Kurth (C1920); is represented by more recent work (e.g. Proctor, C1978) on transposition and symmetric division, and has achieved a particularly systematic synthesis in the neo-Riemannian transformational theory arising from the writings of David Lewin.

The fundamental insight of neo-Riemannian theory is that the relationships of the harmonic structures of 12-note chromaticism are direct, unmediated by the tonal centres inherent to both functional and Stufen theories. The connection to Riemann is not through his influential theory of harmonic functions but rather to his development, in Skizze einer neue Methode der Harmonielehre (1880), of the system of Schritte (triadic transpositions) and Wechsel (exchanges of major for minor triads) first introduced by von Oettingen in 1866. Initially conceived in the context of just intonation, the Schritte and Wechsel are translated by neo-Riemannian theory into equal temperament, with particular attention to the characteristic 19th-century transformations that maximize common tones and semitonal motion. The Table of Tonal Relations developed by Leonhard Euler (E1739) and appropriated by Oettingen and Riemann furnishes an elegant geometric model for triadic progressions consisting of such transformations. When conceived in equal temperament, the geometry of the table is circularized in multiple dimensions, yielding a torus. Analogous transformations and geometrical representations are available for progressions involving various species of 7th chords.

The torus, as a model of harmonic relations, also turns up in empirical studies of musical cognition and perception, a branch of scholarship with a methodological and conceptual legacy quite independent from that of music theory. Recent work in these areas merges the traditional concerns of Tonpsychologie (Helmholtz, Stumpf) with the epistemology of contemporary cognitive science, which emphasizes learning and memory and hence affords a role for style and culture as well as for phenomena characteristically associated with ‘nature’. Cognitive and perceptual work includes psychoacoustic studies of harmonic similarity; strategies for modelling assignments of tonal centres to harmonic progressions and for assessing their closural strengths; and probes of the psychological reality of concepts of historical significance to music theorists. Harmony figures heavily as a primary category in such studies, which are generally more concerned than analytically orientated approaches with well-defined atomic musical relationships, and less with their absorption into an integrated vision of the artistic masterwork. (See Psychology of music, §II, 1.)

Richard Cohn

Practice

Writings on harmony are practical rather than speculative if they satisfy purposes other than the theorist’s passion for explanation. The distinction is porous, if not flimsy: there is a clear pragmatic value when ‘laws’ of harmony are pressed into service as aesthetic filters in support of political or nationalist agendas (as in post-Revolutionary France or Stalinist Russia), for example. In standard usage, however, ‘harmonic practice’ refers more narrowly to writings that place harmonic knowledge at the service of the education of performers and composers. It has an unbroken history whose origin roughly coincides with that of harmonic tonality, reinforced, during the 19th and 20th centuries, by a robust institutional framework of conservatories, universities and preparatory academies that continue to place harmony at the core of the elementary training of performers, composers and musical amateurs.

All works of practical musicianship project, perhaps only implicitly, a theory of harmonia (in its original sense – see §1 above). If the focus is limited to the subset of writings that take ‘chord’ and ‘chord progression’ as primary lexical and syntactic categories, the boundary that divides theory from practice is still difficult to locate. Explicitly speculative tomes such as Lippius’s Synopsis (A1612) offer compositional instruction in part-writing, while classroom primers develop concepts that eventually migrate into the mainstream of speculative writings (e.g. the seminal treatments, in the Lehrbücher of E.F. Richter (B1853) and Mayrberger (B1878), of passing chords, which were to become conceptually crucial to Schenker’s theories of Schichten and Ursatz).

Practical harmony has served largely as preparatory training for three activities – improvisation, composition and analysis – that are distinct in principle but frequently overlap. Most practical works on harmony in the 17th century were aimed primarily at training continuo players in the improvisational art of thoroughbass realization, while the greatest thoroughbass treatise of the 18th century, by Heinichen, was entitled Der General-Bass in der Composition. On the other hand, the most influential of all practical writings in the years between Rameau and Riemann, Gottfried Weber’s Versuch einer geordneten Theorie der Tonsetzkunst (B1817–21), has a primarily analytical purpose, despite its title (‘Attempt at a systematic theory of composition’): to give the reader a means ‘to conceive and communicate a rational idea of the good or ill effect of this or that combination of tones and of the beauty or deformity of this or that musical passage or piece’.

The origins of practical training in harmony may be traced to Italian and German counterpoint treatises of the early 16th century that enumerate the combinations usable in four-voice composition. A century later, the concept of triadic invertibility stimulated an emerging awareness of such combinations as primary categories. Although this awareness is adumbrated in German composition manuals (e.g. Baryphonus, A1615) that focus on principles of four-part writing, it is in contemporary improvisation primers that chords are first fully hypostasized. The new chordal autonomy is more evident in guitar treatises from the Iberian peninsula (notably Amat, A1639) than in the better-known thoroughbass manuals of central and southern Europe, which only began to integrate principles of traidic invertibility around the mid-17th century.

In 18th-century Germany, thoroughbass realization was increasingly cultivated not only by continuo players but also as part of composition training. C.P.E. Bach reported that his father’s composition pupils ‘had to begin their studies by learning pure four-part thorough bass. From this he went to chorales; first he added the basses to them himself, and they had to invent the alto and tenor. Then he taught them to devise the basses themselves’. The more advanced of these tasks required understanding not only of part-writing but also of harmonic syntax, a topic that had first been addressed in thoroughbass manuals of the late 17th century dealing with the realization of unfigured basses (e.g. Penna, A1672). As the role of the basso continuo waned in the later 18th century, improvisation ceased to be a primary goal of harmonic training. Although isolated passages on ‘preluding’ (e.g. Friedrich Wieck, B1853) attest that a thorough understanding of harmony was prerequisite to the improviser’s art in the 19th century, there is no continuing tradition of improvisation primers again until the 1950s, when jazz musicians began to codify their methods.

Rameau’s theory of fundamental bass, which has constituted a nearly universal foundation of harmonic training since 1800, was initially disseminated in his own writings for composers and accompanists. It was only after Rameau’s death in 1764 that his ideas became the basis of a standard pedagogical practice. During the second half of the 18th century several other core concepts of harmonic teaching began to appear in a form resembling current usage. In the 1750s, Marpurg developed a taxonomy of non-harmonic tones. In his Two Essays, published in 1766, John Trydell indicated fundamentals as scale degrees in relation to a tonic, representing them by Arabic numerals; this system subsequently reappeared, with Roman numerals, in Vogler’s writings. Perhaps most importantly, the composition treatises of Kirnberger (B1771–9) and H.C. Koch (B1782–93) began to consider the role of harmony as a formal articulator on the scale of phrases and movements. Each of these concerns (non-harmonic tones, scale-degree root representation, and form) was put to analytical as well as compositional purposes in the following century. (See also Analysis.)

Harmonic analysis, in the form of fundamental basses laid under existing compositions, is already present, albeit sparsely, in the writings of Rameau and Kirnberger. Vandermonde (1778) advocated fundamental bass analysis as part of preparatory training in composition, and by the turn of the 19th century analysis had become a central concern of writings on music. In the empirical compendia of Momigny, Weber and Reicha, fragments from 18th-century works provide exemplars of all manner of musical phenomena, and stimulate nuanced formulations of harmonic principles in place of the coarse normative pronouncements that often dominated composition pedagogy of the era. Harmonic analysis is recommended to students as a fruitful activity for its own sake. Reicha suggests the following procedure: "First, all the inessential notes of the piece are eliminated, and the essential notes alone are entered; then the fundamentals of the chord are entered on a separate staff; the piece in its original form is compared with that giving just the essential notes; then the succession of fundamentals is examined so as to see how the chord successions form progressions."

Reicha’s procedure has endured since, although the fundamentals entered on a staff were soon replaced by Roman numerals, which were disseminated in Weber’s influential treatise and put to more scholasticist ends in the didactic harmony writings of Jelensperger (E1830) in France, Sechter (B1853–4) in Austria and Richter (B1853) in Germany, Richter’s Lehrbuch der Harmonie staying in print for a full century, through 36 editions, and being translated into ten European languages. These three textbooks are among the earliest or most influential representatives of a stream, eventually a raging torrent, of prescriptive, ahistorical harmony textbooks created to fill the demand created by the institutionalization of advanced musical training during the first half of the century in northern and western Europe and the British Isles, and by 1860 in North America and Russia. Such primers typically offered lessons in figured-bass realization and Roman-numeral analysis of synthetic examples, without reference to any particular repertory.

Roman-numeral analysis was challenged at the end of the 19th century by Riemann, who proposed the theory of dual principles, together with the theory of tonal functions, in a series of pedagogical writings. In 1917 Riemann wrote that ‘the Roman numeral method is being more and more marginalized as outmoded’, and that dualism and functional theory ‘ever more certainly takes its place’. His vision of universal monopoly, however, failed to materialize. After Riemann’s death, functional theory was divorced from dualism, which has found few advocates among modern writers (an exception is Levarie, E1954). The theory of harmonic functions has dominated harmonic pedagogy in Germany, Scandinavia and eastern Europe, but even there it has not totally supplanted the Roman numeral method. Riemann’s influence in southern Europe and in the anglophone world has been minimal.

The Harmonielehren of Schenker (B1906) and of Louis and Thuille (C1907) marked the revival of a thoroughly empirical attitude toward practical harmony for the first time since Weber. These are not exclusively, or even primarily, works of practical harmony, despite their titles: each presents an original conceptual synthesis, bolstered by presentation and discussion of abundant examples from the German tradition since 1700. The repertorial orientation represented by these writings soon took hold in didactic works, particularly in the United States. Of central importance are the harmony textbooks of Mitchell (E1939) and Piston (B1941), both of which are explicitly empirical and analytical rather than normative and pre-compositional. Almost every page of Piston’s Harmony includes at least one illustrative score fragment from the musical canon, while Mitchell’s Elementary Harmony, the first harmony text influenced by Schenker’s theory of tonality, leads the reader through a set of rudimentary linear-graphic analyses of movements from Beethoven’s sonatas.

Since the mid-20th century, Schenker’s influence on tonal theory in North America has had an increasing impact on practical harmony. Sessions (E1951) analyses a progression from Brahms’s Violin Concerto, which uses a variety of diatonic harmonies, as an ‘elaboration of a tonic triad’, and suggests that the prolongational attitude behind this claim is broadly applicable. Forte’s Tonal Harmony (E1962) advances a view of harmonic progression as dynamic motion towards a cadential goal, and generates local harmonic formations via linear motion or temporal displacement. The first page of the ambitious and widely used textbook of Aldwell and Schachter (E1978) reproduces the opening of a Mozart piano sonata; its second page introduces a two-voice reduction that serves as a platform from which to explore and express some rudimentary observations. And in Gauldin (E1997), linear reduction of examples from the literature sits alongside Roman numeral analysis as an activity that develops the student’s sense for the inner workings of tonality.

In retrospect, the most significant articulation point in the history of practical harmony was its separation from contemporary compositional practice in the early 19th century, a development that roughly coincided with the rise of harmonic analysis and institutional musical training, the waning of improvisational primers and the consolidation of the western musical canon. The first stage of this isolation was marked by a synthetic, scholastic and arepertorial approach to elementary compositional training; the second stage, a century later, sought to overcome the limitations of the first by reorientating harmonic knowledge towards the development of an analytical practice directed at classical repertories, and whose explanatory power wanes as chromatic writing is liberated from diatonic tonality. Several developments in the late 20th-century academy – notably a suspicion of historicizing teleologies and the re-evaluation of the distinction between classical and vernacular – stimulated a recognition of diatonic tonality as a living tradition. Perhaps the most important trend in practical harmony at the beginning of the 21st century is the reintroduction of contemporary music, in the form of folk music, jazz, show-tunes, rock and so on into manuals of practical harmony, in both Europe and North America, in the service of compositional and improvisational as well as analytical training.

Richard Cohn

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J.C. Pepusch: A Treatise on Harmony (London, 1730, 2/1731/R)

J.-P. Rameau: Génération harmonique, ou Traité de musique théorique et pratique (Paris, 1737/R)

J. Le Rond d’Alembert: Elémens de musique théorique et pratique suivant les principes de M. Rameau (Paris, 1752/R, enlarged 2/1762)

J.-A. Serre: Essais sur les principes de l’harmonie (Paris, 1753/R)

G. Tartini: Trattato di musica secondo la vera scienza dell’armonia (Padua, 1754/R); Eng. trans. in F.B. Johnson: Tartini’s ‘Trattato di musica secondo la vera scienza dell’armonia (diss., Indiana U., 1988)

J. Riepel: Anfangsgrunde zur musikalischen Setzkunst, ii: Grundregeln zur Tonordnung insgemeun (Frankfurt and Leipzig, 1755)

F.W. Marpurg: Handbuch bey dem Generalbasse und der Composition mit zwey-drey-vier-fünf-sechs-sieben-acht und mehreren Stimmen (Berlin, 1755–81/R, suppl., 1760/R; 2/1762/R [vol.i only])

F.W. Marpurg: Systematische Einleitung in die musicalische Setzkunst, nach den Lehrsätzen des Herrn Rameau (Leipzig, 1757)

J. Trydell: Two Essays on the Theory and Practice of Music (Dublin, 1766)

J. Kirnberger: Die Kunst des reinen Satzes in der Musik, aus sicheren Grundsätzen hergeleitet und mit deutlichen Beyspielen erlaütert, i (1771/R; with new title-page, Berlin and Königsberg, 1774); ii (Berlin and Königsberg, 1776–9/R); both vols. (2/1793; Eng. trans., 1982, as The Art of Strict Musical Composition)

J.B. Mercadier de Belesta: Nouveau système de musique théorique et pratique (Paris, 1776)

G.J. Vogler: Tonwissenshcaft und Tonsetzkunst (Mannheim, 1776/R)

H.C. Koch: Versuch einer Anleitung zur Composition (Rudolstadt, 1782–93/R)

J.-J. Momigny: Cours complet d’harmonie et de composition (Paris, 1803–5)

E.A. Forster: Anleitung zum General-Bass (Vienna, 1805)

A.F.C. Kollmann: A New Theory of Musical Harmony, according to a Complete and Natural System of that Science (London, 1806)

G. Weber: Versuch einer geordneten Theorie der Tonsetzkunst (Mainz, 1817–21, 3/1830–32; Eng. trans., 1851)

A. Reicha: Cours de composition musicale, ou Traité complet et raisonné d’harmonie pratique (Paris, 1818; Eng. trans., 1854)

A.B. Marx: Die Lehre von der musikalischen Komposition, praktisch-theoretisch, i (Leipzig, 1837, rev., 10/1903; Eng. trans., 1852; ii (Leipzig, 1838, rev. 7/1890); iii (Leipzig, 1845, 5/1879); iv (Leipzig, 1847, 5/1888; Eng. trans., 1910)

S.W. Dehn: Theoretisch-praktische Harmonielehre (Berlin, 1840, 2/1860)

F.J. Fétis: Traité complet de la théorie et de la pratique de l’harmonie (Paris and Brussels, 1844, 20/1903)

M. Hauptmann: Die Natur der Harmonik und der Metrik (Leipzig, 1853, 2/1873; Eng. trans., 1888/R)

E.F. Richter: Lehrbuch der Harmonie (Leipzig, 1853, 36/1953; Eng. trans., 1912)

F. Wieck: Clavier und Gesang: Didaktisches und Polemisches (Leipzig, 1853/R; Eng. trans., 1875/R)

S. Sechter: Die Grundsätze der musikalischen Komposition (Leipzig, 1853–4)

F. Hiller: Uebungen zum Studium der Harmonie und des Contrapunkts (Cologne, 1860, 16/1897)

J.C. Lobe: Vereinfachte Harmonielehre (Leipzig, 1869)

M. Mayrberger: Lehrbuch der musikalischen Harmonie (Leipzig, 1878)

O. Hostinsky′: Die Lehre von den musikalischen Klängen: ein Beitrag zur aesthetischen Begründung der Harmonielehre (Prague, 1879)

C. Kistler: Harmonielehre (Munich, 1879, 2/1898)

Y. Kurdyumov: Klassifikatsiya garmonicheskikh soyedineniy (St Petersburg, 1896)

F.-A. Gevaert: Traité d’harmonie théorique et pratique (Paris and Brussels, 1905–7)

H. Schenker: Neue musikalische Theorien und Phantasien, i: Harmonielehre (Stuttgart, 1906/R; Eng. trans., 1954/R)

R. Münnich: ‘Von der Entwicklung der Riemannschen Harmonielehre und ihren Verhältnis zu Oettingen und Stumpf’, Riemann-Festschrift (Leipzig, 1909/R), 60–76

A. Schoenberg: Harmonielehre (Vienna, 1911, 3/1922; Eng. trans., abridged, 1948, complete, 1978)

E. Kurth: Die Voraussetzungen der theoretischen Harmonik und der tonalen Darstellungssysteme (Berne, 1913/R)

H. Schenker: ‘Vom Organischen der Sonatenform’, Das Meisterwerke in der Musik, ii (Munich, 1926; Eng. trans., 1996), 45–54

H. Schenker: Neue musikalische Theorien und Phantasien, iii: Der freie Satz (Vienna, 1935, rev. 2/1956 by O. Jonas; Eng. trans., 1979)

K. Jeppesen: ‘Zur Kritik der klassischen Harmonielehre’, IMSCR IV: Basle 1949, 23–34

A. Dommel-Diény: L’harmonie vivante, i: L’harmonie tonale (Neuchâtel, 1963)

E. Seidel: ‘Die Harmonielehre Hugo Riemanns’, Beiträge zur Musiktheorie des 19. Jahrhunderts, ed. M. Vogel (Regensburg, 1966)

R. Imig: Systeme der Funktionsbezeichnung in den Harmonielehren seit Hugo Riemann (Düsseldorf, 1970)

P. Williams: Figured Bass Accompaniment (Edinburgh, 1970)

C. Rosen: The Classical Style: Haydn, Mozart, Beethoven (New York, 1971, rev.2/1997)

D.W. Beach: ‘The Origins of Harmonic Analysis’, JMT, xviii (1974), 274–307

M. Wagner: Die Harmonielehren der ersten Hälfte des 19. Jahrhunderts (Regensburg, 1974)

P. Rummenhöller: ‘Eine Bezeichnungsweise tonaler Harmonie’, Zeitschrift für Musiktheorie, vi/1 (1975), 28–47

B. Simms: ‘Choron, Fétis, and the Theory of Tonality’, JMT, xix (1975), 112–38

D. de la Motte: Harmonielehre (Kassel, 1976; Eng. trans., 1991)

H.M. Krebs: Third Relation and Dominant in Late 18th- and Early 19th-Century Music (diss., Yale U., 1980)

D.M. Thompson: A History of Harmonic Theory in the United States (Kent, OH, 1980)

M. Federhofer: Akkord und Stimmführung in den musiktheoretischen Systemen von Hugo Riemann, Ernst Kurth und Heinrich Schenker (Vienna, 1981)

A. Keiler: ‘Music as Metalanguage: Rameau's Fundamental Bass’, Music Theory: Special Topics, ed. R. Browne (New York, 1981), 83–100

R. Groth: Die französische Kompositionslehre des 19. Jahrhunderts (Wiesbaden, 1983)

F. Lerdahl and R. Jackendoff: A Generative Theory of Tonal Music (Cambridge, MA, 1983/R)

C. Deliège: Les fondements de la musique tonale: une perspective analytique post-schenkerienne (Paris, 1984)

R.W. Wason: Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg (Ann Arbor, 1985)

F.K. and M.G. Grave: In Praise of Harmony: the Teachings of Abbé Georg Joseph Vogler (Lincoln, NE, 1987)

T. Christensen: ‘Nichelmann Contra C.Ph.E. Bach: Harmonic Theory and Musical Politics at the Court of Frederick the Great’, Carl Philipp Emanuel Bach und die europäische Musikkultur: Hamburg 1988, 189–220

E. Narmour: ‘Melodic Structuring of Harmonic Dissonance: a Method for Analysing Chopin's Contribution to the Development of Harmony’, Chopin Studies (Cambridge, 1988), 77–114

C.M. Gessele: The Institutionalization of Music Theory in France, 1764–1802 (diss., Princeton U., 1989)

J. Lester: Between Modes and Keys: German Theory, 1592–1802 (Stuyvesant, NY, 1989)

J. Lester: Compositional Theory in the Eighteenth-Century (Cambridge, MA, 1992)

T. Christensen: Rameau and Musical Thought in the Enlightenment (Cambridge, 1993)

B. Hyer: ‘Sighing Branches: Prosopopoeia in Rameau's Pigmalion’, MAn, xiii (1994), 7–50

E. Agmon: ‘Functional Harmony Revisited: a Prototype-Theoretic Approach’, Music Theory Spectrum, xvii (1995), 196–214

C. Schachter: ‘The Triad as Place and Action’, Music Theory Spectrum, xvii (1995), 149–69

T. Christensen: ‘Fétis and Emerging Tonal Consciousness’, Music Theory in the Age of Romanticism, ed. I. Bent (Cambridge, 1996), 37–56

C. Schachter: Unfoldings: Essays in Schenkerian Theory and Analysis, ed. J.N. Straus (Oxford, 1999) C: Triadic and post-Tristan chromaticism

G.J. Vogler: Handbuch zur Harmonielehre und für den Generalbass (Prague, 1802)

C.F. Weitzmann: Der übermässige Dreiklang (Berlin, 1853)

C.F. Weitzmann: Der verminderte Septimen-Akkord (Berlin, 1854)

H.J. Vincent: Die Einheit in der Tonwelt (Leipzig, 1862)

A.J. von Oettingen: Harmoniesystem in dualer Entwicklung: Studien zur Theorie der Musik (Dorpat and Leipzig, 1866, enlarged 2/1913 as Das duale Harmoniesysteme)

B. Ziehn: Harmonie- und Modulationslehre (Brunswick, 1888, 2/1910; Eng. trans., 1907, as Manual of Harmony)

G. Capellen: Ist das System Simon Sechters ein geeigneter Ausgangspunkt für die theoretische Wagnerforschung? (Leipzig, 1902)

R. Louis and L. Thuille: Harmonielehre (Stuttgart, 1907, 10/1933 by W. Courvoisier and others); Eng. trans. in R.I. Schwarz: An Annotated English Translation of ‘Harmonielehre’ of Rudolf Louis and Ludwig Thuille (diss., Washington U., 1982)

G. Capellen: Fortschrittliche Harmonie- und Melodielehre (Leipzig, 1908)

R. Mayrhofer: Die organische Harmonielehre (Berlin, 1908)

B. Ziehn: Fünf- und sechsstimmige Harmonien und ihre Anwendung (Milwaukee and Berlin, 1911)

E. Kurth: Romantische Harmonik und ihre Krise in Wagners ‘Tristan’ (Berne, 1920, 2/1923/R); partial Eng. trans. in Ernst Kurth: Selected Writings, ed. L.A. Rothfarb (Cambridge, 1991), 97–147

D.F. Tovey: ‘Tonality’, ML, ix (1928), 341–63

S. Karg-Elert: Polaristische Klang- und Tonalitätslehre (Harmonologik) (Leipzig, 1931)

T.W. Adorno: Versuch über Wagner (Berlin and Frankfurt, 1952, 2/1964; Eng. trans., 1981)

A. Schoenberg: Structural Functions of Harmony (New York, 1954, rev. 2/1969 by L. Stein)

W.J. Mitchell: ‘The Study of Chromaticism’, JMT, vi (1962), 2–31

M. Vogel: Der Tristan-Akkord und die Krise der modernen Harmonie-Lehre (Düsseldorf, 1962)

C. Dahlhaus: Zwischen Romantik und Moderne: vier Studien zur Musikgeschichte des späteren 19. Jahrhunderts (Munich, 1974; Eng. trans., 1980)

R.P. Morgan: ‘Dissonant Prolongation: Theoretical and Compositional Precedents’, JMT, xx (1976), 49–91

R. Bailey: ‘The Structure of the Ring and its Evolution’, 19CM, i (1977–8), 48–61

G. Proctor: Technical Bases of Nineteenth-Century Chromatic Tonality (diss., Princeton U., 1978)

R.R. Subotnik: ‘Tonality, Autonomy and Competence in Post-Classical Music’, Critical Inquiry, vi (1979), 153–63

L. Kramer: ‘The Mirror of Tonality: Transitional Features of Nineteenth-Century Harmony’, 19CM, iv (1980–81), 191–208

D. Lewin: ‘A Formal Theory of Generalized Tonal Functions’, JMT, xxvi (1982), 23–60

E. Lendvai: Workshop of Bartók and Kodály (Budapest, 1983)

D. Lewin: ‘Amfortas's Prayer to Titurel and the Role of D in Parsifal: the Tonal Spaces of the Drama and the Enharmonic Câ™­/B’, 19CM, vii (1983–4), 336–49

R. Taruskin: ‘Chernomor to Kaschei: Harmonic Sorcery or, Stravinsky's “Angle”’, JAMS, xxxviii (1985), 72–142

M. Brown: ‘The Diatonic and the Chromatic in Schenker's Theory of Harmonic Relations’, JMT, xxx (1986), 1–33

C.J. Smith: ‘The Functional Extravagance of Chromatic Chords’, Music Theory Spectrum, viii (1986), 94–139

C. Lewis: ‘Mirrors and Metaphors: Reflections on Schoenberg and Nineteenth-Century Tonality’, 19CM, xi (1987–8), 26–42

L. Rothfarb: Ernst Kurth as Theorist and Analyst (Philadelphia, 1988)

R.W. Wason: ‘Progressive Harmonic Theory in the Mid-Nineteenth-Century’, JMR, viii (1988), 55–60

J. Blume: ‘Analysen als Beispiele musiktheoretischer Probleme: auf der Suche nach der angemessenen Beschreibung chromatischer Harmonik in romantischer Musik’, Musiktheorie, iv (1989), 518–39

B. Shamgar: ‘Romantic Harmony through the Eyes of Contemporary Observers’, JM, vii (1989), 518–39

J. Maegard: ‘Zur harmonischen Analyse der Musik des 19. Jahrhunderts: eine theoretische Erwägung’, Musikkulturgeschichte: Festschrift für Constantin Floros, ed. P. Petersen (Wiesbaden, 1990), 61–86

P. McCreless: ‘Schenker and Chromatic Tonicization: a Reappraisal’, Schenker Studies, ed. H. Siegel (Cambridge, 1990), 125–45

D.W. Bernstein: ‘Symmetry and Symmetrical Inversion in Turn-of-the-Century Theory and Practice’, Music Theory and the Exploration of the Past, ed. C. Hatch and D. Bernstein (Chicago, 1993), 377–407

I. Bent, ed.: Music Analysis in the Nineteenth-Century, i: Fugue, Form, and Style (Cambridge, 1994)

D. Harrison: Harmonic Function in Chromatic Music: a Renewed Dualist Theory and an Account of its Precedents (Chicago, 1994)

B. Hyer: ‘Reimag(in)ing Riemann’, JMT, xxxix (1995), 101–38

D. Kopp: A Comprehensive Theory of Chromatic Mediant Relations in Mid-Nineteenth-Century Music (diss., Brandeis U., 1995)

R. Cohn: ‘Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions’, MAn, xv (1996), 9–40

W. Kinderman and H. Krebs, eds.: The Second Practice of Nineteenth-Century Tonality (Lincoln, NE, 1996) [incl. P. McCreless: ‘An Evolutionary Perspective on Nineteenth-Century Semitonal Relations’, 87–113]

M.K. Mooney: The ‘Table of Relations’ and Music Psychology in Hugo Riemann’s Harmonic Theory (diss., Columbia U., 1996)

R. Cohn: ‘Neo-Riemannian Operations, Parismonious Trichords, and their Tonnetz Representations’, JMT, xli (1997), 1–66

R. Cohn: ‘Introduction to Neo-Riemannian Theory: a Survey and a Historical Perspective’, JMT, xlii (1998), 167–80

P.J. Telesco: ‘Enharmonicism and the Omnibus Progression in Classical-Era Music’, Music Theory Spectrum, xx (1998), 242–79

V.F. Yellin: The Omnibus Idea (Warren, MI, 1998) D: Harmony in 20th-century music

D. Alaleona: ‘L'armonia modernissma: le tonalità neutre e l'arte di stupore’, RMI, xxviii (1911), 769–838

R. Lenormand: Etude sur l'harmonie moderne (Paris, 1913; Eng. trans., 1915, enlarged 2/1940–42/R by M. Carner)

A.E. Hull: Modern Harmony: its Explanation and Application (London, 1914/R)

E.L. Bacon: ‘Our Musical Idiom’, The Monist, xxvii (1917), 560–607

A. Gentili: Nuova teorica dell’armonia (Turin, 1925)

H. Erpf: Studien zur Harmonie- und Klangtechniker neueren Musik (Leipzig, 1927, 2/1969)

A. Hába: Neue Harmonielehre (Leipzig, 1927)

H. Cowell: ‘New Terms for New Music’, MM, v/4 (1927–8), 21–7

H. Cowell: New Musical Resources (New York, 1930, rev. 1996 by D. Nicholls)

H.A. Miller: New Harmonic Devices: a Treatise on Modern Harmonic Problems (Boston, 1930)

E. van der Nüll: Moderne Harmonik (Leipzig, 1932)

H.K. Andrews: Modern Harmony (London, 1934)

P. Hindemith: Unterweisung im Tonsatz, i: Theoretischer Teil (Mainz, 1937, 2/1940; Eng. trans., 1942 as The Craft of Musical Composition, 2/1948)

J. Schillinger: The Schillinger System of Musical Composition, ed. L. Dowling and A. Shaw (New York, 1941/R, 3/1946/R)

J. Schillinger: Harmony (New York, 1947)

F. Reuter: Praktische Harmonik des 20. Jahrhunderts (Halle, 1952)

G.A. Russell: The Lydian-Chromatic Concept of Tonal Organization for Improvisation (New York, 1959)

H. Hanson: Harmonic Materials of Modern Music: Resources of the Tempered Scale (New York, 1960)

V. Persichetti: Twentieth-Century Harmony (New York, 1961)

K. Janeček: Základy moderní harmonie [Bases of modern harmony] (Prague, 1965)

V. Dernova: Garmoniya Skryabina (Leningrad, 1968; Eng. trans., 1979)

A. Forte: The Structure of Atonal Music (New Haven, CT, 1973)

G. Perle: Twelve-Tone Tonality (Berkeley, 1977, rev. and enlarged, 2/1996)

J. Samson: Music in Transition: a Study of Tonal Expansion and Atonality, 1900–1920 (London and New York, 1977)

A. Chapman: A Theory of Harmonic Structures for Non-Tonal Music (diss., Yale U., 1978)

S. Strunk: ‘The Harmony of Early Bop: a Layered Approach’, JJS, vi (1979), 4–53

H. Martin: Jazz Harmony (diss., Princeton U., 1980)

J. Rahn: Basic Atonal Theory (New York, 1980/R)

R. Morris: Composition with Pitch Classes (New Haven, CT, 1987)

J. Straus: ‘The Problem of Prolongation in Post-Tonal Music’, JMT, xxxi (1987), 1–21

J. Roeder: ‘Harmonic Implications of Schoenberg's Observations of Atonal Voice Leading’, JMT, xxxiii (1989), 27–62

S. Block: ‘Pitch-Class Transformations in Free Jazz’, Music Theory Spectrum, xii (1990), 181–202

M. Delaere: Funktionelle Atonalität: analytische Strategien für die frei-atonale Musik der Wiener Schule (Wilhelmshaven, 1993)

A. Forte: The American Popular Ballad of the Golden Era, 1924–1950 (Princeton, NJ, 1995)

W. Gieseler: Harmonik in der Musik des 20. Jahrhunderts: Tendenzen, Modelle (Celle, 1996)

J.W. Bernard: ‘Chord, Collection, and Set in Twentieth-Century Theory’, Music Theory in Concept and Practice, ed. J.M. Baker, D.W. Beach and J.W. Bernard (Rochester, NY, 1997), 11–51

Speculative, empirical, pan-historical and pedagogical writings

MGG2 (P. Rummenhöller: ‘Harmonielehre’)

L. Euler: Tentamen novae theoriae musicae, ex certissimis harmoniae principiis dilucide expositae (St Petersburg, 1739; Eng. trans., 1960); repr. in Opera omnia, iii/1, 197–427

J.-J. de Momigny: Encyclopédie méthodique (Paris, 1791–1818)

C.S. Catel: Traité d’harmonie (Paris, 1802, enlarged 2/1848 by A. Leborne)

A. Choron: Sommaire de l’histoire de la musique (Paris, 1810; Eng. trans., 1825)

F.H.J. Castil-Blaze: Dictionnaire de musique moderne (Paris, 1821, 2/1825)

P. de Geslin: Cours d’harmonie (Paris, 1826)

D. Jelensperger: L’harmonie au commencement du dix-neuvième siècle et méthode pour l’étudier (Paris, 1830)

J.A. André: Lehrbuch der Tonsetzkunst (Offenbach, 1832–40)

A. Day: A Treatise on Harmony (London, 1845); ed. G.A. Macfarren (London, 1885)

H. Helmholtz: Die Lehre von den Tonempfindungen (Brunswick, 1863, 6/1913/R; Eng. trans., 1875, as On the Sensations of Tone, 2/1885/R)

F.-J. Fétis: Histoire générale de la musique (Paris, 1869–76/R)

P.I. Chaykovsky: Rukovodstvo k prakticheskomu izucheniyu garmonii [Guide to the practical study of harmony] (Moscow, 1872)

L. Bussler: Praktische Harmonielehre in vierundfünfzig Aufgaben (Berlin, 1875, rev. 10/1929 by H. Leichtentritt)

S. Jadassohn: Lehrbuch der Harmonie (Leipzig, 1883, 23/1923; Eng. trans., 1884–6, 7/1904)

C. Stumpf: Tonpsychologie (Leipzig, 1883–90/R)

N.A. Rimsky-Korsakov: Uchebnik garmonii [Textbook of harmony] (St Petersburg, 1884–5, 2/1886 as Prakticheskiy uchebnik garmonii, 19/1949; Eng. trans., 1930)

S. Jadassohn: Explanatory Remarks and Suggestions for the Working of the Exercises in the Manual of Harmony, with Special Consideration for Self-Instruction (Leipzig, 1886)

E. Prout: Harmony: its Theory and Practice (London, 1889, 20/1903)

S. Jadassohn: Die Kunst zu moduliren und zu präludiren: ein praktischer Beitrag zur Harmonielehre in stufenweise geordnetem Lehrgange dargestellt (Leipzig, 1890, 2/1902)

S. Jadassohn: Aufgaben und Beispiele für die Harmonielehre (Leipzig, 1891, 8/1918)

H. Riemann: Vereinfachte Harmonielehre oder die Lehre von den tonalen Funktionen der Akkorde (London and New York, 1893, 2/1903; Eng. trans., 1896)

H. Riemann: ‘Ideen zu einer “Lehre von den Tonvorstellungen’, JbMP 1914–15, 1–26; Eng. trans., in JMT, xxxvi (1992), 69–117

H. Riemann: ‘Neue Beiträge zur einer Lehre von den Tonvorstellungen’, JbMP 1916, 1–21

M. Shirlaw: The Theory of Harmony (London, 1917, 2/1955/R)

R.O. Morris: Foundations of Practical Harmony and Counterpoint (London, 1925/R)

C. Koechl: Tratié de l’harmonie (Paris, 1927–30)

W. Maler: Beitrag zur Harmonielehre (Leipzig, 1931, rev. 6/1967 by G. Bialas and J. Driessler as Beitrag zur durmolltonalen Harmonielehre)

W. Piston: Principles of Harmonic Analysis (Boston, 1933)

W. Mitchell: Elementary Harmony (New York, 1939, 3/1965)

H. Distler: Funktionelle Harmonielehre (Kassel, 1941)

W. Piston: Harmony (New York, 1941, 2/1948, rev. 5/1987 by M. DeVoto)

P. Hindemith: A Concentrated Course in Traditional Harmony (New York, 1943, 2/1949; Ger. trans., 1949); ii (New York, 1948, 2/1953; Ger. trans., 1949)

A. Katz: Challenge to Musical Tradition: a New Concept of Tonality (New York, 1945/R)

J. Handschin: Der Toncharakter: eine Einführung in die Tonpsychologie (Zürich, 1948)

D.F. Tovey: ‘Musical Form and Matter’, The Mainstream of Music and other Essays, ed. H.J. Foss (Oxford, 1949), 160–82

J. Rohwer: Tonale Instruktionen und Beiträge zur Kompositionslehre (Wolfenbüttel, 1951)

R. Sessions: Harmonic Practice (New York, 1951)

F. Salzer: Structural Hearing: Tonal Coherence in Music (New York, 1952/R)

S. Levarie: The Fundamentals of Harmony (New York, 1954, 2/1962)

J. Mehegan: Jazz Improvisation (New York, 1959–65)

A. Forte: Tonal Harmony in Concept and Practice (New York, 1962, 3/1979)

E. Terhardt: ‘Pitch, Consonance, and Harmony’, JASA, lv (1974), 1061–9

W. Mickelsen: Hugo Riemann’s Theory of Harmony: a Study (Lincoln, NE, 1977)

E. Aldwell and C. Schachter: Harmony and Voice Leading (New York, 1978, 2/1989)

R. Browne: ‘Tonal Implications of the Diatonic Set’, In Theory Only, v/6–7 (1981), 3–21

E.D. Carpenter: ‘Russian Music Theory: a Conspectus’, Russian Theoretical Thought in Music, ed. G.D. McQuere (Ann Arbor, 1983), 1–81

D. Lewin: ‘Music, Theory, Phenomenology, and Modes of Perception’, Music Perception, iv (1986–7), 327–92

D. Lewin: Generalized Musical Intervals and Transformations (New Haven, CT, 1987)

F. Lerdahl: ‘Tonal Pitch Space’, Music Perception, v (1987–8), 315–49

R. Parncutt: Harmony: a Psychoacoustical Approach (Berlin, 1989)

C.L. Krumhansl: Cognitive Foundations of Musical Pitch (Oxford, 1990)

Z. Gardonyi and H. Nordhoff: Harmonik (Wolfenbüttel, 1990)

E. Agmon: ‘Linear Transformations between Cyclically Generated Chords’, Musikometrika, iii (1991), 15–40

J. Clough and J. Douthett: ‘Maximally Even Sets’, JMT, xxxv (1991), 93–173

B. Rosner and E. Narmour: ‘Harmonic Closure: Music Theory and Perception’, Music Perception, ix (1991–2), 383–411

M. Beiche: ‘Tonalität’ (1992), HMT

W. Thomson: ‘The Harmonic Root: a Fragile Marriage of Concept and Percept’, Music Perception, x (1992–3), 385–415

M. Ernst: Harmonielehre in der Schule: Überlegungen zu einem handlungs- und schülerorientierten Musikunterricht in der Sekundarstufe I (Essen, 1993)

R. Gauldin: Harmonic Practice in Tonal Music (New York, 1997)

J. Douthett and P. Steinbach: ‘Parsimonious Graphs: a Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition’, JMT, xlii (1998), 241–63

Richard Cohn, Brian Hyer