Half-diminished seventh chord

half-diminished seventh chord
Component intervals from root
	minor seventh
	diminished fifth (tritone)
	minor third
	root
Tuning
	5:6:7:9 or 25:30:36:45
Forte no. / Complement
	4-27 / 8-27

Half-diminished seventh chord on C (Audio file "Half-diminished seventh chord on C.mid" not found).

Leading-tone seventh chord resolution in C major: vii^ø⁷-I <phonos file="Leading-tone seventh chord resolution in C major.mid">Play</phonos>.

In music theory, the half-diminished seventh chord—also known as a half-diminished chord or a minor seventh flat five (m7♭5)—is formed by a root note, a minor third, a diminished fifth, and a minor seventh. Its consecutive intervals are minor 3rd, minor 3rd, major 3rd.

It can be created by playing the tonic, flattened third, flattened fifth, and flattened seventh degrees of any major scale (1, ♭3, ♭5 and ♭7)—for example, C half-diminished is (C E♭ G♭ B♭). Alternatively, assuming even temperament, it can be derived simply by sharpening (by a semitone) the root note of the major seventh chord whose tonic lies a semitone immediately below its tonic. (cf. Bar 6 of the twelve-bar blues where the half diminished is sometimes used to provide a semitone lift to the harmonic root without apparent change to the remaining harmonizing components).

In diatonic harmony, the half-diminished chord occurs naturally on the 7th scale degree of any major scale including a seventh.^[3] (for example, B^ø⁷ in C major). By the same virtue, it also occurs on the second degree of any natural minor scale (e.g., D^ø⁷ in C minor). It occurs as a leading-tone seventh chord in major^[3] and can be represented by the integer notation {0, 3, 6, 10}. It has been described as a "considerable instability".^[4]

Chord symbols and terminology

Half-diminished seventh chords are often symbolized as a circle with a diagonal line through it, as in C^ø. It also can be represented as m7♭5, -7♭5, m7(♭5) etc.

The terms and symbols for this chord break expectations that derive from the usual system of chord nomenclature. Normally a symbol like "B^dim" indicates a diminished triad and "B7" indicates a major triad plus a minor seventh. Thus one would expect the term "Bdim7" to indicate a diminished triad plus a minor seventh. Instead, it means a diminished triad plus a diminished seventh. To make this distinction clear, the term "half-diminished" and the ^ø symbol (^ø) were invented. Since the term dim7 (as in B^dim7) meant something else, the accurate but unwieldy term "minor seventh flat five" (as in B^ø⁷))(Audio file "Half diminished_seventh.mid" not found) came to be used.^[5]

Despite the appearance of the word "diminished" in the name of this type of seventh chord, its sound differs considerably from that of a diminished seventh chord. In fact, the only sonic connection between the two chords is the single diminished triad found in the half-diminished seventh chord. As composer-theorist Milton Babbitt has astutely pointed out, the "half-diminished" seventh chord should be called the "one-third" diminished seventh chord. ... Whatever its deficiencies might be in the label department, however, the half-diminished seventh chord is in many respects the star of the seventh chord harmonic cast. Many songs in the classic American popular song repertoire reserve it for their most intensely expressive moments.^[6]

Jazz musicians typically consider the half-diminished chord as built from one of three scales: the seventh (Locrian mode) of the major scale, the sixth mode of the melodic minor scale (the latter scale is nearly identical to the Locrian mode, except that it has a natural 9 rather than a ♭9, giving it a somewhat more consonant quality), or the "half-whole" diminished scale (see octatonic scales.) See: chord-scale system.

The "Tristan chord" is sometimes described as a half-diminished seventh chord; however, the term "Tristan chord" is typically reserved for a very specific harmonic function, especially determined by the order of the notes from bottom to top, and sometimes even the way the chord is spelled (e.g. is it G♭ or F♯?).

Function

The half-diminished chord has three functions in contemporary harmony: predominant function, diminished, and dominant function. The vast majority of its occurrences are on the II chord in the minor mode—where it takes a predominant function, leading naturally to the dominant V chord. Not including the root motion, there is only a one note difference between a half diminished chord and a V7 chord with a flat 9th. Since it is built on the diatonic II chord of the minor scale, most of the time the II-V pattern resolves to a minor tonic (such as in the progression Dm7(♭5) - G7(♭9) - Cm)., but there are instances where there is a major tonic resolution.

Diminished chord function is rarer, but it still exists. Half-diminished chords can function in the same way as fully diminished chords do, such as in the chord progression Cmaj7 - C♯dim7 - Dm7, or Em7 - E♭dim7 - Dm7, where the diminished chord serves as a chromatic passing chord preceding a chord with a diatonic root. A typical example of this is when ♯IVm7(♭5) progresses to IVm7, such as in the Cole Porter song "Night and Day", where you have the progression F♯m7(♭5) - Fm7 - Em7 - E♭dim7 - Dm7 - G7 - Cmaj7. If analyzed in its predominant function, it wouldn't sufficiently explain how it functions preceding the Fm7 chord.

In dominant function, the VII half diminished chord, like its fully diminished counterpart, can take the place of the dominant V chord at a point of cadential motion. This generally occurs in a major key, since the flattening of the sixth degree in the natural minor scale renders a dominant diminished seventh chord fully diminished if played within the scale. Indeed, the VII half diminished chord in a major key is identical to a dominant ninth chord (a dominant seventh with an added ninth) but with its root omitted.

The dominant function of the half-diminished seventh chord may also occur in a secondary dominant context, i.e., as part of a progression where the chord performs the dominant function with respect to the overall key's dominant chord. In this scenario, the half-diminished seventh chord is built on the tritone of the overall key and is equivalent to a secondary dominant seventh chord with added ninth and omitted root. If written with respect to the overall key, this chord is styled "♯iv^ø7," but in terms of its function in the progression, the styling "vii^ø7/V" is more descriptive. Examples of the ♯iv^ø7-to-V or vii^ø7/V-to-V transition include the half-line "know when to run" in the Kenny Rogers song "The Gambler",^{[citation needed]} and measures 11-12 of Scott Joplin's "The Entertainer".^[7]

Leading-tone with minor seventh

Leading-tone seventh chord (b^ø7) in C major <phonos file="Leading-tone seventh chord in C major.mid">Play</phonos>.

The leading-tone diminished triad with minor seventh chord is represented with the Roman numeral notation vii^ø7. In the key of C, this is B^ø⁷), the root of which is the leading-tone to the tonic.^[3]

Supertonic with minor seventh in minor

Half-diminished seventh chord on supertonic in C minor <phonos file="Half-diminished seventh chord on supertonic in C minor.mid">Play</phonos>.

One variant of the supertonic seventh chord is the supertonic half-diminished seventh (ii^ø7) in minor. It may be considered a minor seventh chord with a flattened fifth (e.g., ACE♭G rather than ACEG) and is used in the ii-V-I in minor.^[8] For example, over the first three bars of the Ciaccona or Chaconne movement of J.S. Bach's Violin Partita No. 2 in D minor, the tonic in the first measure progresses to the ii^ø7 chord (here in third inversion: DGB♭E) for the first beat of the second measure, then to the dominant (here a V7 in first inversion: C#GAE) for the second and third beats of the second measure, and then back to the tonic for the first beat of the third measure.

Sharpened supertonic with minor seventh

Sharpened supertonic seventh chord (d♯^ø7) in C major <phonos file="Half-diminished seventh chord on sharpened supertonic in C.mid">Play</phonos>.

One variant of the supertonic seventh chord is the supertonic half-diminished seventh (♯ii^ø7) with the raised supertonic, which equals the lowered third through enharmonic equivalence (in C: D♯=E♭).

D♯F♯A C♯ = F♯A C♯E♭
d♯^ø7      = f♯mADD^o7

Sharpened subdominant with minor seventh

Half-diminished seventh chord on sharpened subdominant (f♯^ø7), vii^ø7/V, in C <phonos file="Half-diminished seventh chord on sharpened subdominant in C.mid">Play</phonos>.

The sharpened subdominant diminished triad with minor seventh chord is represented with the Roman numeral notation ♯iv^ø7; the root of this chord is the raised subdominant (sharpened fourth). That root also serves as the leading tone to the dominant when used in the vii^ø7/V function described above; such a function is the diminished, secondary-dominant equivalent of a backdoor progression. For example, in the key of C major, the chord playing this role is F♯^ø7.

Virtual Augmented 6 chord

The half-diminished seventh chord is the involution of the German augmented sixth chord.^[9] Its interval of minor seventh (between root and seventh degree, i.e.: { C B♭ } in { C E♭ G♭ B♭ } ) can be written as an augmented sixth { C E♭ G♭ A♯ }.^[10] Rearranging and transposing, this gives { A♭ C♭ D F♯ }, a virtual minor version of the French augmented sixth chord.^[11] Like the typical +6, this enharmonic interpretation gives on a resolution irregular for the half-diminished seventh but regular for the augmented sixth chord, where the 2 voices at the enharmonic major second converge to unison or diverge to octave.^[12]