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In tonal music theory, a function (often called harmonic function, tonal function or diatonic function, or also chord area) is the relationship of a chord to a tonal center. The concept of harmonic function has never been clearly defined, but it appears to say something of the tonal significance of chords (or of their role) in tonal music. It rests on the recognition of essential hierarchies between the degrees of the tonal scale[clarification needed] and the harmonies that they support, and of possible equivalences between some of these hierarchic values.[not in citation given]
Two main theories of tonal functions exist today, both dealing with the relation of the chords to their tonic:
Both theories find part of their inspiration in the theories of Jean-Philippe Rameau, starting with his Traité d'harmonie of 1722. Even if the concept of harmonic function was not so named before 1893, it could be shown to exist, explicitly or implicitly, in many theories of harmony before that date. Early usages of the term in music (not necessarily in the sense implied here, or only vaguely so) include those by Fétis (Traité complet de la théorie et de la pratique de l'harmonie, 1844), Durutte (Esthétique musicale, 1855), Loquin (Notions élémentaires d'harmonie moderne, 1862), etc.
The concept of harmonic function originates in theories about just intonation. It was realized that three perfect major triads, distant from each other by a perfect fifth, produced the seven degrees of the major scale in one of the possible forms of just intonation: for instance, the triads F-A-C, C-E-G and G-B-D produce the seven notes of the major scale. These three triads were soon considered the most important chords of the major tonality, with the tonic in the center, the dominant above and the subdominant under.
This symmetric construction may have been one of the reasons why the fourth degree of the scale, and the chord built on it, were named "subdominant", i.e. the "dominant under [the tonic]". It also is one of the origins of the dualist theories which described not only the scale in just intonation as a symmetric construction, but also the minor tonality as an inversion of the major one. Dualist theories are documented from the 16th century onwards.
The term functional harmony derives from Hugo Riemann and, more particularly, from his Harmony Simplified. Riemann's direct inspiration was Moriz Hauptmann dialectic description of tonality. Riemann described three abstract functions, the tonic, the dominant (its fifth) and the subdominant (its fourth). He considered in addition that the minor scale was the inversion of the major one, so that the dominant was the fifth above the tonic in major, but below the tonic in minor; the subdominant, similarly, was the fifth below the tonic (or the fourth above) in major, and the reverse in minor.
Despite the complexity of his theory, Riemann's ideas had huge impact, especially where German influence was strong. A good example in this regard are the textbooks by Hermann Grabner. More recent German theorists have abandoned the most complex aspect of Riemann's theory, the dualist conception of major and minor, and consider that the dominant is the fifth degree above the tonic, the subdominant the fourth degree, both in minor and in major.
The three tonal functions are denoted by the letters T, D and S, for Tonic, Dominant and Subdominant respectively; the letters are uppercase for functions in major (T D S), lowercase for functions in minor (t d s). Each of these functions can be fulfilled by three chords: the main chord corresponding to the function, the chord a third lower and that a third higher. The relation between triads a third apart resides in the fact that they differ from each other by one note only, the two other notes being common to the two triads. In addition, within the diatonic scale, triads a third apart necessarily are of opposite mode. Two cases must be considered (see also Neo-Riemannian theory):
It will be noted that several chords could fulfill different functions depending on the context. In the simplified theory where the functions in major and minor are on the same degrees of the scale, the possible functions of chords on the seven degrees of the scale could be summarized as follows:
|Function||T or t||Sp or sL||Tl or tP
Dp or dL
|S or s||D or d||Sl or sP
Tp or tL
|Dl or dP|
The Viennese theory on the other hand, the "Theory of the degrees" (Stufentheorie), represented by Simon Sechter, Heinrich Schenker and Arnold Schoenberg among others, considers that each degree has its own function and refers to the tonal center through the cycle of fifths; it stresses harmonic progressions above chord quality.
Each degree of a diatonic scale, as well as each of many chromatically-altered notes, has a different diatonic function as does each chord built upon those notes. A pitch or pitch class and its enharmonic equivalents have different meanings. For example, a C? cannot substitute for a D?, even though in equal temperament they are identical pitches, because the D? can serve as the minor third of a B? minor chord while a C? cannot, and the C? can serve as the fifth degree of an F? major scale, while a D? cannot.
In music theory, as it is commonly taught in the US, there are seven different functions. In Germany, from the theories of Hugo Riemann, there are only three, and functions other than the tonic, subdominant and dominant are called their "parallels" (US: "relatives"). See Functional harmony. For instance, in the key of C major, an A minor (chord, scale, or, sometimes, the note A itself) is the Tonic parallel, or Tp. (German musicians use only uppercase note letters and Roman numeral abbreviations, while in the US, upper- and lowercase are usually used to designate major or augmented, and minor or diminished, respectively.) In the US, it would be referred to as the "relative minor".
As d'Indy summarizes:
- There is only one chord, a perfect chord; it alone is consonant because it alone generates a feeling of repose and balance;
- this chord has two different forms, major and minor, depending whether the chord is composed of a minor third over a major third, or a major third over a minor;
- this chord is able to take on three different tonal functions, tonic, dominant, or subdominant.-- D'Indy (1903), 
In the United States, Germany, and other places the diatonic functions are:
|Function||Roman Numeral||English||German||German abbreviation|
|Mediant||iii||Dominant parallel/Tonic counter parallel||Dominantparallele||Dp/Tkp|
|Leading||vii||incomplete Dominant seventh||verkürzter Dominantseptakkord||diagonally slashed D7 (D?7)|
Note that the ii, iii, vi, and vii are lowercase; this is because in relation to the key, they are minor chords. Without accidentals, the vii is a diminished viio.
In the US the minor mode or scale is considered a variant of the major, while in German theory it is often considered, per Riemann, the inversion of the major. In the late 18th and early 19th centuries a large number of symmetrical chords and relations were known as "dualistic" harmony. The root of a major chord in root position is its bass note, but, symmetrically, the 'root' of a minor chord in root position is the fifth (for example CEG and ACE). The plus and degree symbols, + and o, are used to denote that the lower tone of the fifth is the root, as in major, +d, or the higher, as in minor, od. Thus, if the major tonic parallel is the tonic with the fifth raised a whole tone, then the minor tonic parallel is the tonic with the US root/German fifth lowered a whole tone.
|Parallel||Note letter in C||US name||Parallel||Note letter in C||US name|
|Tp||A minor||Submediant||tP||E? major||Mediant|
|Sp||D minor||Supertonic||sP||A? major||Submediant|
|Dp||E minor||Mediant||dP||B? major||Subtonic|
If chords may be formed by raising (major) or lowering (minor) the fifth a whole step, they may also be formed by lowering (major) or raising (minor) the root a half-step to wechsel, the leading tone or leitton. These chords are Leittonwechselklänge (literally: "leading-tone changing sounds"), sometimes called gegenklang or "contrast chord".
Three categories can appear in any one of three chordal guises in either of two modes, eighteen positions in all: T, Tp, Tl, t, tP, tL, S, Sp, Sl, s, sP, sL, D, Dp, Dl, d, dP, dL. Why all this complexity? Perhaps the central reason is that this ingenious, occasionally convoluted system enabled Riemann to achieve a grand and masterful synthesis of both the old and the new in late 19th-century music. Ostensibly remote triads could be interpreted through the traditional terms of the I-IV-V-I, or now T-S-D-T, cadential schema. A sequence of A?-major, B?-major, and C-major chords, for example, could be neatly interpreted as a subdominant (sP) to dominant (dP) to tonic (T) progression in C-major, a reading of these chords not without support in certain late-Romantic cadences. And a chord that often perplexes harmony students, the Neapolitan chord D? major in a C-major context, could be shown to be nothing more than a minor-mode subdominant Leittonwechselklang (sL).-- Carl Dahlhaus
Some may at first be put off by the overt theorizing apparent in German harmony, wishing perhaps that a choice be made once and for all between Riemann's Funktionstheorie and the older Stufentheorie, or possibly believing that so-called linear theories have settled all earlier disputes. Yet this ongoing conflict between antithetical theories, with its attendant uncertainties and complexities, has special merits. In particular, whereas an English-speaking student may falsely believe that he or she is learning harmony "as it really is," the German student encounters what are obviously theoretical constructs and must deal with them accordingly.-- Robert O. Gjerdingen
Another theory regarding harmonic functionality is that "functional succession is explained by the circle of fifths (in which, therefore, scale degree II is closer to the dominant than scale degree IV)." According to Goldman's Harmony in Western Music, "the IV chord is actually, in the simplest mechanisms of diatonic relationships, at the greatest distance from I. In terms of the circle of fifths, it leads away from I, rather than toward it." Thus the progression I-ii-V-I would comply more with tonal logic. However, Goldman, as well as Jean-Jacques Nattiez, points out that "the chord on the fourth degree appears long before the chord on II, and the subsequent final I, in the progression I-IV-viio-iii-vi-ii-V-I."  Goldman also points out that, "historically the use of the IV chord in harmonic design, and especially in cadences, exhibits some curious features. By and large, one can say that the use of IV in final cadences becomes more common in the 19th century than it was in the 18th, but that it may also be understood as a substitute for the ii chord when it precedes V. It may also be quite logically construed as an incomplete ii7 chord (lacking root)."  However, Nattiez calls this, "a narrow escape: only the theory of a ii chord without a root allows Goldman to maintain that the circle of fifths is completely valid from Bach to Wagner." 
Functions during or after modulations, and especially tonicizations, are often notated in relation to the function--in the original key--of the chord being tonicized. For example, in C major, a D major chord is notated as II, but during a tonicization of a G major chord, it would be notated as it is functioning in G major but with the G also notated as it functions as the dominant of C major. The standard notation for this is: V/V (five of five). For example, the twelve bar blues turnaround, I-V-IV-I, considered tonally inadmissible, may be interpreted as a doubled plagal cadence, IV/V-V-IV-I (IV/V-I/V, IV/I-I/I).
From the viewpoint of musical behaviour or structure there are three essential functions:
Other functions serve to support the Tonic and Dominant functions listed above:
The dominant, dominant preparation and the tonic substitution all involve more than one scale degree with only the tonic and subdominant containing only one scale degree. Several scale degrees exercise more than one function.
The tonic includes four separate activities or roles as the: