# Diminished seventh chord

File:Diminished seventh chord on C.png
Diminished seventh chord on C .
File:Diminished7thandMinor9thComparison.png
A comparison of the diminished 7th and dominant 7th[1] (9) chords.
diminished seventh
Component intervals from root
diminished seventh
diminished fifth (tritone)
minor third
root
Tuning
25:30:36:45
125:150:180:216[2]
Forte no. / Complement
4-28 / 8-28

A diminished seventh chord is a four note chord that comprises a diminished triad plus the interval of a diminished seventh (alternatively regarded enharmonically as a major sixth) above the root. Thus it is (1, 3, 5, double flat7), or enharmonically (1, 3, 5, 6), of any minor scale; for example, C diminished-seventh would be (C, E, G, Bdouble flat), or enharmonically (C, E, G, A). It occurs as a leading-tone seventh chord in harmonic minor and can be represented by the integer notation {0, 3, 6, 9}.

Because of this it can also be viewed as four notes all stacked in intervals of a minor third. The diminished seventh contains two diminished fifths, which often resolve inwards.[3]

In most sheet music books, Cdim or C° denotes a diminished seventh chord with root C; but it may also happen, mostly in modern jazz books and some music theory literature, that Cdim or C° or Cm(♭5) denotes a diminished triad, while Cdim7 or C°7 or Cm6(♭5) denotes a diminished seventh chord.

## Uses

File:Diminished seventh chord resolution.png
Diminished seventh chord resolution: both diminished fifths tend to resolve inward, doubling the third of the tonic chord .

The most common form of the diminished seventh chord is that rooted on the leading tone; for example, in the key of C, the chord (B, D, F, A). So its other constituents are the second, fourth, and flatted sixth (flat submediant) scale degrees. These notes occur naturally in the harmonic minor scale. But this chord also appears in major keys, especially after the time of Bach, where it is "borrowed" from the parallel minor. Fétis tuned the chord 10:12:14:17.[4]

Seventh chords may also be rooted on other scale degrees, either as secondary function chords temporarily borrowed from other keys, or as appoggiatura chords: a chord rooted on the raised second scale degree (D-F-A-C in the key of C) acts as an appoggiatura to the tonic (C major) chord, and one rooted on the raised sixth scale degree (A-C-E-G in C major) acts as an appoggiatura to the dominant (G major) chord. Because these chords have no leading tone in relation to the chords to which they resolve, they can not properly have "dominant" function. They are therefore referred to commonly as "non-dominant" diminished seventh chords or "common tone" diminished seventh chords. (See "common tone diminished seventh chord" below)

In jazz, the diminished seventh chord is often based on the lowered third scale degree (the flat mediant) and acts as a passing chord between the mediant triad (or first-inversion tonic triad) and the supertonic triad: in C major, this would be the chord progression E minor - E diminished - D minor. The chord, "plays no role in...jazz."[5] The passing chord is used widely in Brazilian music like Choro, Samba, and Bossa Nova

The diminished seventh chord normally possesses a "dominant" function, and this is most straightforwardly shown when the root of a dominant seventh chord is omitted. The remaining third, fifth and seventh of that chord form a diminished triad (whose new root is the third of the former chord), to which a diminished seventh can be added. Thus in C (major or minor), a dominant seventh chord consisting of G, B, D, F can be replaced by a diminished seventh chord B, D, F, A. (In jazz harmony, a combination of the original chord with its substitute (with G in the bass and A simultaneously in an upper voice) yields the very common "79" chord, which intensifies the dominant function of either a diminished seventh or dominant seventh chord.) Other transformations of this kind facilitate a variety of substitutions and modulations: any of the four notes in a diminished seventh chord are raised by a semi-tone, that raised note is then the flat-seventh of a half-diminished seventh chord. Similarly, if any of the four notes in the diminished seventh chord are lowered by a semi-tone, that lowered note is then the root of a dominant seventh chord.

The diminished seventh chord comprises frequencies that are equally spaced when considered on a logarithmic axis, and thus divides the octave into four logarithmically equal portions, each being a minor third.

File:Alpha chord.png
Two diminished seventh chords in the octatonic scale (one red, one blue) may be rearranged into the alpha chord .

The diminished scale may be conceived of as two interlocking diminished seventh chords, which may be rearranged into the alpha chord.

### Sharpened subdominant with diminished seventh

File:Sharpened subdominant with diminished seventh chord in C.png
Sharpened subdominant with diminished seventh chord in C.

The sharpened subdominant with added diminished seventh chord is another common use of the chord. It can be simply represented with the Roman notation ivo7, but in classical music is more correctly represented as viio7/V, being a very common way for a composer to approach the dominant of any key. In the key of C, this is Fdim7, which may be used for a strikingly fearful effect, as its root is a tritone (augmented fourth) from the tonic. It is also a common chord in jazz and ragtime music. A common traditional jazz or Dixieland progression in E may go:

File:Sharpened subdominant with diminished seventh chord progression in C.png
Sharpened subdominant with diminished seventh chord progression in C .
A - A dim - B7,

which is

IV - ivo7 - V7

Another common usage of ivo7 is often found in Gospel music and jazz progressions such as in the song "I Got Rhythm".

In C:

```| C  C/E | F  F♯dim7 | C/G A7 | Dm7 G7 |
```

### Supertonic diminished seventh chord

File:Supertonic diminished seventh chord in C.png
Supertonic diminished seventh chord in C .

One variant of the supertonic seventh chord is the supertonic diminished seventh[6] with the raised supertonic, which equals the lowered third through enharmonic equivalence (in C: D=E). It may be used as a dominant substitute.[7]

File:Sharp IIdim7 as dominant substitute with tonic chord substitution.png
IIo7 as dominant substitute with III−7 substituted for the tonic (I) chord (D-E) .

#### Common-tone diminished seventh chord

File:Common-tone diminished seventh chord.png
Common-tone diminished seventh chord .

A diminished seventh chord may alternatively resolve to a major or major-minor seventh chord whose root is one of the notes of the diminished seventh chord, the most common being the raised supertonic seventh, which resolves to the tonic in major keys and the raised submediant, which resolves to dominant triad or seventh in major keys, with the altered tones resolving upward by half step.[8]

Diminished seventh chord to dominant cadence (bo-e7-A7-D) .

The diminished chord may also resolve through lowering two of the bottom three voices producing a supertonic seventh chord that may lead to a conventional dominant cadence.[9]

## Diminished seventh root

File:Diminished seventh chord on C supposed root.png
In Rameau's supposition the root of the dominant chord on B, left, is substituted producing a diminished seventh chord on C, right.

Music theorists have struggled over the centuries to explain the meaning and function of diminished seventh chords. Currently, two approaches are generally used. The less complex method treats the leading tone as the root of the chord, and the other chord members as the third, fifth, and seventh of the chord, the same way other seventh chords are analyzed.

File:Diminished seventh chord incomplete ninth in C.png
Diminished seventh chord incomplete ninth in C Minor .

The other method is to analyze the chord as an "incomplete dominant ninth", that is a ninth chord with its root on the dominant, whose root is missing or implied. A vii°7 chord in the minor key (for example, in C minor, B, D, F, A) occurs naturally in the harmonic minor scale and is equivalent to the dominant 7(9) chord (G, B, D, F, A) without its root. This was already proposed by Arnold Schoenberg,[10] and Walter Piston has long been the champion of this analysis.[11] Jazz guitarist Sal Salvador, and other jazz theorists, also advocated this view, rewriting chord charts to reflect this and supplying the "missing" root as part of their bass lines.[12]

The dominant ninth theory has been questioned by Heinrich Schenker. He explained that although there is a kinship between all univalent chords rising out of the fifth degree, the dominant ninth chord is not a real chord formation.[13]

Rameau explained the diminished seventh chord as a dominant seventh chord whose supposed fundamental bass is borrowed from the sixth degree in minor, raised a semitone producing a stack of minor thirds.[14] Thus in C the dominant seventh is G7 (G-B-D-F) and the sixth degree borrowed from minor produces A-B-D-F.[14] He observed in his Treatise on Harmony that three minor thirds and an augmented second make up a chord where the augmented second is such that "the ear is not offended" by it. He may have been talking of the augmented second in quarter-comma meantone, a tuning he favored, which is close to the just septimal minor third of 7/6.

## Inversions

The fundamental tone or root of any diminished seventh chord, being composed of three stacked minor thirds, is ambiguous. For example, Cdim7 in root position: C + E + G + Bdouble flat (each has one and half interval), is just as easily viewed as an Edim7 in its third inversion:

Ddouble flat (enharmonic equivalent of C) + E + G + Bdouble flat.

It can also be viewed as a Gdim7 in its second inversion:

Ddouble flat + Fdouble flat (enharmonic equivalent of E) + G + Bdouble flat.

Delineating this chord in its last possibility, that of Bdouble flatdim7 in its first inversion, is very clumsy and not very useful as it requires the use a triple-flatted note, something that is hardly ever used in a musical score:

Ddouble flat + Fdouble flat + Atriple flat (enharmonic equivalent of G) + Bdouble flat.

However, by enharmonically respelling the Bdouble flat to A, this can also be viewed as a first inversion Adim7 chord:

C + E + G + A (enharmonic equivalent of Bdouble flat).

Other possibilities present themselves by respelling the various roots; for instance:

C + E + F (enharmonic equivalent of G) + A (enharmonic equivalent of Bdouble flat) (second inversion Fdim7).
C + D (enharmonic equivalent of E) + F (enharmonic equivalent of G) + A (enharmonic equivalent of Bdouble flat) (third inversion Ddim7).
B (enharmonic equivalent of C) + D (enharmonic equivalent of E) + F (enharmonic equivalent of G) + A (enharmonic equivalent of Bdouble flat) (root position Bdim7).

All of the chord's inversions have the same sound harmonically. Because of the chord's symmetrical nature (superimposing more minor thirds on top of the dim 7 produces no new notes), there are only three different diminished seventh chords possible.

The diminished seventh chord can appear in first, second, or (least common) third inversion. Each inversion is enharmonic with another diminished seventh chord, and 19th-century composers in particular often make use of this enharmonic to use these chords for modulations. Percy Goetschius calls it the "enharmonic chord."[15]

File:Diminished seventh chord on C enharmonic.png
Diminished seventh chord on C, written four different ways enharmonically (all sounding the same).
File:Diminished seventh modulation.png
Diminished seventh chord's use in modulation.[16]

Using Piston's incomplete-ninth analysis, a single diminished seventh chord, without enharmonic change, is capable of the following analyses: V, V of ii, V of III (in min.), V of iii (in maj.), V of iv, V of V, V of VI (in min.), V of vi (in maj.), V of VII (in min.). Since the chord may be enharmonically written in four different ways without changing the sound, we may multiply the above by four, making a total of forty-eight possible interpretations.[17] More conservatively, each assumed root may be used as a dominant, tonic, or supertonic, giving twelve possibilities.[16]

## Diminished seventh chord table

Chord Root Minor Third Diminished Fifth Diminished Seventh
Cdim7 C E G Bdouble flat (A)
Cdim7 C E G B
Ddim7 D F A C (B)
Ddim7 D F A C
Edim7 E G B D
Fdim7 F A C (B) Edouble flat (D)
Fdim7 F A C E
Gdim7 G B D F (E)
Gdim7 G B D F
Bdim7 B D F A

## References

1. ^ Richard Lawn, Jeffrey L. Hellmer (1996). Jazz: Theory and Practice, p.124. ISBN 0-88284-722-8.
2. ^ Shirlaw, Matthew (1900). The Theory of Harmony, p.86. ISBN 978-1-4510-1534-8. "g-b-d-f."
3. ^ Benward & Saker (2003). Music: In Theory and Practice, Vol. I, p.219. Seventh Edition. ISBN 978-0-07-294262-0.
4. ^ Fétis, François-Joseph and Arlin, Mary I. (1994). Esquisse de l'histoire de l'harmonie, p.139n9. ISBN 978-0-945193-51-7.
5. ^ Tenzer, Michael and Roeder, John (2011). Analytical and Cross-Cultural Studies in World Music, p.157n10. ISBN 978-0-19-538458-1.
6. ^ Kitson, C. H. (2006). Elementary Harmony, p.43. ISBN 1-4067-9372-8.
7. ^ Coker, Jerry (1997). Elements of the Jazz Language for the Developing Improvisor, p.82. ISBN 1-57623-875-X.
8. ^ Benward & Saker (2003). Music: In Theory and Practice, Vol. I, p.220-21. Seventh Edition. ISBN 978-0-07-294262-0.
9. ^ Carrie Adelaide Alchin (1917). Applied Harmony, p.99.
10. ^ Schönberg, Arnold: "Harmonielehre", chapter IX, Wien, Universal Edition, 1911
11. ^ Piston, Walter: "Harmony", pg. 191, Third Edition, W. W. Norton & Company, 1962
12. ^ Sal Salvador: "Chordal Enrichment & Chord Substitution: Jazz Guitar", Mel Bay, ISBN 0871665271
13. ^ Schenker, Heinrich; ed. and annot. Oswald Jonas (1954). Harmony. trans. Elisabeth Mann-Borgese. Chicago: University of Chicago Press. p. 192. OCLC 280916.
14. ^ a b Christensen, Thomas Street (2004). Rameau and Musical Thought in the Enlightenment, p.100. ISBN 978-0-521-61709-3.
15. ^ Goetschius, Percy: "The Material Used in Musical Composition - A System of Harmony", pg. 159, G. Shirmer, Inc., 1913
16. ^ a b Adela Harriet Sophia Bagot Wodehouse (1890). A Dictionary of Music and Musicians: (A.D. 1450-1889), p.448. Macmillan and Co., Ltd.
17. ^ Piston, Walter: "Harmony", pg. 201, Third Edition, W. W. Norton & Company, 1962