# Functional theory

The functional theory is a branch of music theory and belongs to the harmony . It describes the relationships and tensions between the chords in major-minor tonal music. Hugo Riemann developed it in 1893. It was worked out and expanded primarily by Wilhelm Maler and Diether de la Motte .

## Use

Function theory is used for music analysis . Harmonic processes can be determined and described on this basis. It presupposes the listener's expectation of certain sequences of harmonies, for example cadences and sequences . The structure of longer pieces of music can also be viewed with their help.

Functional theory can be applied primarily to the harmony of baroque , classical and most romantic music, but only to smaller parts of modern music. Many harmonious relationships within jazz and pop music can also be grasped with the function theory. In jazz theory, however, the analysis according to the degree theory and the chord scale theory predominates. In popular music literature, individual terms from functional analysis and stage theory are often used synonymously.

Both systems are legitimate and mostly easily comprehensible models for describing harmonic relationships. Which method is preferred depends on the context. However, the choice between step or function theory is also highly location-dependent. Outside of Germany, for the most part, only step theory is used, while the much-criticized functional theory is still often used today, mainly exclusively in Germany.

## The functions

### Main functions

 The main functions as chords of the C major scale The main functions as chords of the harmonic A minor scale The parallels as chords of the C major scale The parallels as chords of the pure A minor scale

In functional theory, the triad on the first level of a key that manifests itself in a certain period of time is considered the tonic (in level theory: 1st level) of this section. It is joined by two other main functions , namely the next pure fifths : the dominant (upper fifth, 5th degree) and the subdominant (fourth or lower fifth , 4th degree). The functions themselves are designated by letters in function theory, with major functions being given capital letters and minor functions being given lower case letters.

### Secondary functions

In addition, there are the secondary functions that are in a third from the main functions. A letter is added to the main functions for symbolic representation of the secondary functions.

The auxiliary functions comprise several groups:

• the parallels in small spaces to the main
function Main function in major: parallel sound (in minor) downwards –Tp–, or
main function in minor: parallel sound (in major) upwards –tP–
• the counter-sounds in the major third
main function in major: counter-sound (in minor) upwards –Tg–, or
main function in minor: counter-sound (in major) downwards -tG–
• the mediants in large or small interdigits upwards or downwards, which cannot be formed from the scale's own tones and which are achieved by reducing or reducing the main function or the secondary function.

Examples of parallels: Tp in C major is A minor. tP in A minor is C major.
Examples of counter-sounds: Tg in C major is E minor. tG in A minor is F major.
A mediant would be E major in C major: TG (the secondary function is lost),
another E major in C minor: tp (the main and secondary functions are missing).

There is also diminution and thinning for every main function. They are always indicated by upper or lower case (the omitted subdominant in a major system is denoted by s instead of S, for example).

The tonic is strengthened by cadences . The simplest cadences are DT ( authentic ending ) and ST ( plagal ending ). The full cadence TSDT is generally accepted as the basic model for cadences.

 Simple full cadence in C major sound sample The same cadenza in A major sound sample

If there is a transition from one key to another through diatonic, chromatic or enharmonic modulation in a work passage , the new key is considered unconfirmed in the analysis until an authentic or, more rarely, plagal cadence follows. If a chord can be understood not only as a function of the previous key, but also as a main function of the new key, one speaks of main intermediate functions. These include in particular the intermediate dominant, which is only possible in the case of diatonic modulations.

## Additions in the form of numbers

All symbols can be provided with additions in the form of numbers and letters. Superscript numerals after the function name indicate additional tones. Numbers written under the function symbol indicate the bass note of the chord in an interval relationship to the root note of the respective function. Triads in their basic position are written without an addition.

The dominant of B is F. A dominant seventh chord on F is a four-note chord with the additional note E flat.

The dominant of G is D. A dominant seventh chord on D is a four-note chord with the additional note C.

• Seventh chords exist in the basic position (the triad is followed by a seventh) and in three inversions: 1st inversion = fifth position, 2nd inversion = third quarter position, 3rd inversion = second position. The names indicate in which intervals to the lowest note of the chord are the two notes that make up the second dissonance. In the case of the designation according to the theory of functions, however, the superscript 7 is retained, and the bass note is referred to in relation to the chord root instead. A subscript 3 is added for the 1st inversion, because it contains the third of the chord in the basic position in the bass, for the 2nd inversion a subscript 5 (fifth in the bass), and for the 3rd inversion a subscript 7 (seventh in the Bass). In the latter case, the seventh is only noted under the function symbol, since it already shows that it is a seventh chord.

• Function theory knows the possibility of adding an additional sixth to a chord ( sixte ajoutée ). This is indicated by superscripting 5 and 6. This chord is usually valued as a subdominant in function theory. According to the degree theory, it is a seventh chord of the second degree in the fifth position.

• The following example appears to contain the functional progression T - T 5 - D - T, but the second chord is interpreted as a fourth-fourth lead to the following, real dominant, because it is resolved like this:

(This lead is also referred to as D T-D .)

• In the case of a ninth lead, a ninth is added to the basic chord , whose resolution to the octave follows immediately:

## Other signs and symbols

Alterations
High and low alternations are indicated by the symbols <and>. A dominant with an older fifth, for example, is given the symbol D 5 < , in C major: gh-dis
Ellipses
If an expected sound does not occur, it can still be noted in addition to indicate what was expected at this point. The expected sound is put in square brackets (e.g. [T]).
If a sound or a sound sequence relates in its function to a different tonic than the one that was last manifested (e.g. to the future in the case of a modulation), the entire sequence from a meaningful point to the manifestation of the new tonic is put in round brackets .
Intermediate dominants
If a chord is preceded by its dominant (usually not the ladder's own) without any significant modulation (leaving the main key), it is called an intermediate dominant. In the designation it does not have to be related to the actual basic key, but only refers to the following chord and is put in round brackets for identification.
Ligatures
Ligatures are identified by horizontal lines directly following a function symbol and extend over several chords. They indicate that the function described is perceived as unchanged over the duration of the ligature (binding), even in spite of any strange sounds. This is common for. B. in passages . Function symbols can also be placed above ligatures, which then refer to the function to which the ligature started.
Dual functions
Double functions are indicated by two interlocking function symbols. For example, DD denotes the dominant of the dominant ( double dominant ), SS the subdominant of the subdominant ( double subdominant ).
Strikethroughs
Strikethroughs in the chord symbol, which are only used with seventh chords, indicate that the sound is shortened , that is, it lacks the root. This results in the dominant one diminished chord in the stage theory is interpreted as a triad on the 7th degree.
Special chords
Some more common chords, such as the Neapolitan sixth chord , the independent Neapolitan sixth chord, or the diminished seventh chord are given superscript small letters:
s n : The Neapolitan sixth chord (also known as Neapolitan for short ) is derived either from the first inversion of the major chord on the lower-aged second scale level, or alternatively from the minor subdominant with a small, non-scale sixth (i.e. quasi s 6> ). Example for C major: f-a-flat-des.
S N : The Neapolitan sixth chord that has become independent (also briefly independent Neapolitan ) is a Neapolitan sixth chord which, contrary to the name, does not appear in a sixth chord position. Example for C major: D flat-f-flat.
D v : If the diminished seventh chord (in C major: hdf-a-flat) appears dominant, it is interpreted as a shortened dominant seventh chord, whose functional designation would actually be a crossed out D with superscript 7 and 9>. As a simplified notation one often uses D v . If, on the other hand, its appearance is rather subdominant, functional theory regards it as a minor subdominant with a sixth ajoutée and a fourth instead of a fifth, i.e. with a superscript 3, 4 and 6. The hybrid status of this chord is also expressed in the symbol of the interwoven letters D and s.${\ displaystyle {\ cancel {D}} ^ {\ begin {smallmatrix} 9 &> \\ 7 & ~ \ end {smallmatrix}}}$${\ displaystyle s ^ {\ begin {smallmatrix} 6 \\ 4 \\ 3 \ end {smallmatrix}}}$

## Functional harmonic analysis of a Bach chorale

Example of a functional analysis

Although Bach was not familiar with the theory of functions, his chorales can (within limits) be described with it. The following analysis (of course) does not claim to be complete or correct. It is also just an interpretation of the chant, others are quite possible. It is easy to see that because of the many small movements in the individual voices, the composition can only be described vertically , i.e. harmonically, in a very complicated manner, which is due to a strong linear component. The functional theory does not really do justice to this music, since harmonic structures were conceived from the figured bass at that time . Nevertheless: the function-harmonic analysis is common practice, even if it quickly reaches its limits in terms of clarity and completeness.

However, the present analysis is pointless if it is not interpreted. Basically, the translation into function symbols is only a generalizing consideration of the composed special case.

A starting point for the interpretation would be, for example, the description of the harmonic dramaturgy: The first part (up to the repetition mark ) modulates to the dominant, which would be interpreted as a well-known principle of the sonata or later the sonata main clause form . After the tonic was first consolidated at the beginning of the second part (the subdominant plays a decisive role here), the movement moves very far away from it, the two shortened intermediate dominants offer a new sound quality at the same time. After the longest caesura on the subdominant parallel reached, the tonic re-establishes itself, it is also noticeable that the harmonic movement becomes calmer towards the end, and the complete absence of intermediate dominants smooths the final path to the basic sound. Particularly noteworthy here at the end would be the two final turns TSDT, as well as the emphasis (due to strong temporal expansion) of the dominant as the penultimate sound.

Another possible object of consideration would be the treatment of inversions , in particular the voice leading of the bass: sevenths are invariably continued with a second step down, thirds also have a stepped environment, etc.

## Restriction

Not all harmonic relationships and progressions can be grasped with the help of function theory. Functional harmony only takes effect where it is at least a triad harmonics of music that is based on a central major or minor key. Therefore, functional theory is unsuitable as an instrument of analysis to the extent that music does not meet these conditions.

In particular, a large part of the art music of the 20th and 21st centuries, as well as the music of the Renaissance, is often unsatisfactory or almost impossible to grasp with the means of functional theory. The former, since a large part of modern art music is based on non-traditional composition techniques such as polytonality and atonality . The latter, as the music of the Renaissance (which is largely polyphonic vocal music) was both thought and musically implemented much more horizontally than vertically. Of course, harmonies also form in the music of the Renaissance, but due to the special musical architecture, which can cause problems not only due to the horizontal orientation, but also due to the mostly polyphonic, complicated melodies, which the functional theory does not appear to be the method of choice here .

In the same way, functional theory is more likely for certain harmonic progressions, such as sequences of fifths (particularly common in the Baroque) or chords that are based on non-diatonic scale levels (as used particularly often in pop, jazz or film music, for example in tritone substitution or modal interchange) difficult to use. With these harmonic processes, the step theory, which is opposed to the functional theory, is often much better because it is structured in a much more rudimentary and elementary manner. In film music, for example, in addition to chromatic mediants, modal interchange and other harmonic specifics, a change between tonic and a major chord on the high-aged #IV or low-aged bV scale level (one tritone away) is common practice. If one were to assume C major as the key and thus as the tonic, the chord change to the #IV scale level could easily be classified as 'I - #IV' (or confused enharmonically as 'I - bV') with the degree theory; with functional theory, on the other hand, it is almost impossible to grasp, unless one extends the definitions and introduces new terms such as T 'as a tritone tone.

The change from C to G flat can be easily explained by the theory of equating chords: If you set the tonic C equal to the dominant C (i.e. the 5th degree in F major), you can use the G flat chord as an inversion Interpreting the Neapolitan sixth chord in F major: actually a B flat minor chord without a fifth and with an alternate sixth (tones b - d flat - gb). The inverse of this chord then looks like G major. This should be considered primarily as a way of modulating back into the tonic.

If you play the chord progression C - Ges - F, an obvious possibility would be to hear the F as a new tonic (C = dominant, Gb = Neapolitan sixth chord, F = tonic). If you do not play the F chord, no cadential pattern is played and thus no correct goal according to traditional harmony is achieved.

The pre-baroque and late medieval music ( Ars nova ) also works more according to melodic or contrapuntal principles. The harmonic course results from the rules of progression within a voice and the ratio of two voices to one another, not from a superordinate harmonic structure. The resulting sequence of sounds is the origin of our later developing sense of harmony.

## Jazz functional harmony

While European traditional music theory is based on triads, in jazz four notes (three thirds on top of each other) are seen as the basic chords. The numbering of degrees consists of the fact that major chords are numbered in capital letters, while minor chords are written in small letters. For example Ionic: I ii iii IV V vi vii or Doric: i ii III IV v vi VII.
If you build chords on the individual steps of any scale that only consist of tones on this scale, you get a diatonic series. Accordingly, in addition to the horizontal (melodic) aspect, each mode also has a vertical (harmonic) aspect and thus also a ladder-specific harmony. Major and minor are only part of the timbres that are theoretically possible.
The following function table shows the ladder's own harmonics, with function abbreviations for major and minor steps being colored accordingly in addition to upper and lower case.

Vertical view: A "b" or "#" is always put in front if there is a deviation from the Ionic scale, namely in relation to the level and tone gender.

## literature

• Wolf Burbat : The Harmonics of Jazz. 5th edition. Deutscher Taschenbuch Verlag et al., Munich et al. 1998, ISBN 3-423-30140-6 .
• Hermann Grabner : Handbook of functional harmony theory (= Hesse handbooks of music. 15 and 25, ZDB -ID 777229-4 ). 2 volumes (Vol. 1: Textbook. Vol. 2: Exercise book. ). Hesse, Berlin-Halensee et al. 1944.
• Richard Graf, Barrie Nettles: The Chord Scale Theory & Jazz Harmonics. Advance Music, Rottenburg / N. 1997, ISBN 3-89221-055-1 .
• Hanno Hussong: Investigations into practical harmony teaching since 1945. dissertation.de, Berlin 2005, ISBN 3-89825-931-5 (also: Saarbrücken, University, dissertation, 2004).
• Wilhelm Maler : Contribution to the major minor tonal harmony theory. Volume 1: Textbook. 13th edition. Leuckart, Munich et al. 1984, ISBN 3-920587-00-6 .
• Diether de la Motte : Harmony (= dtv 30166). Joint original edition, 13th edition. Deutscher Taschenbuch-Verlag et al., Munich et al. 2004, ISBN 3-423-30166-X .
• Benedikt Stegemann: Theory of Tonality (= pocket books on musicology. 162). Noetzel, Wilhelmshaven 2013, ISBN 978-3-7959-0962-8 .
• Erich Wolf : The music education. Volume 2: Harmony. Chord theory, harmonic functions, modulations, harmonization technique, musical composition, harmony analyzes, exercises. 6th edition. Breitkopf & Härtel, Wiesbaden 1992, ISBN 3-7651-0061-7 .