A _diminished seventh_ has three minor and three major seconds. C--B[double-flat].

A _minor seventh_ has two minor and four major seconds.

C--B[flat].

A _major seventh_ has one minor and five major seconds. C--B.

192. An _octave_ is an interval comprising seven seconds.

A _diminished octave_ has three minor and four major seconds.

C--C[flat].

_A perfect octave_ has two minor and five major seconds. C--C.

An _augmented octave_ has one minor and six major seconds.

C--C[sharp].

193. A _ninth_ is usually treated as a second, a _tenth_ as a third, etc. The interval of two octaves is often referred to as a _fifteenth_.

194. If the major diatonic scale be written and the interval between each tone and the key-tone noted, it will be observed that the intervals are all either major or perfect. See Fig. 64.

[Ill.u.s.tration: Fig. 64.]

In this connection also it will be noted that the interval next smaller than _major_ is always _minor_, while that next smaller than _perfect_ or _minor_ is always _diminished_: but that the interval next larger than both _major_ and _perfect_ is _augmented_.

195. An interval is said to be _inverted_ when the tone originally the upper becomes the lower. Thus C--E, a major third, inverted becomes E--C, a minor sixth.

CHAPTER XVIII

CHORDS, CADENCES, ETC.

196. A _chord_ is a combination of several tones sounding together and bearing an harmonic relation to each other. The simplest chord is the _triad_, which consists of a fundamental tone called the _root_, with the third and fifth above it. C--E--G is a triad, as are also D--F--A, F--A--C, and G--B--D.

197. Triads are cla.s.sified as _major_, _minor_, _diminished_, or _augmented_.

A _major triad_ has a major third and a perfect fifth, _i.e._, it is a major third with a minor third on top of it. Ex.

C--E--G.

A _minor triad_ has a minor third and a perfect fifth, _i.e._, it is a minor third with a major third on top of it. Ex.

C--E[flat]--G.

A _diminished triad_ has a minor third and a diminished fifth, _i.e._, it is a minor third with another minor third on top of it. Ex. C--E[flat]--G[flat].

An _augmented triad_ has a major third and an augmented fifth, _i.e._, it is a major third with another major third on top of it. Ex. C--E--G[sharp].

198. A triad may be built on any scale-tone, but those on I, IV, and V, are used so much oftener than the others that they are often called the _common chords_. In referring to triads the Roman numerals are used to show on what scale-tone the triad is based, the size of the numeral (with other signs) indicating the kind of triad found on each tone of the scale. Thus _e.g._, the large I shows that the triad on the first tone (in major) is a _major triad_, the small II shows that the triad on the second tone is minor, etc. The following figure will make this clear.

[Ill.u.s.tration: Fig. 65.]

The triads in the minor scale are as follows:

[Ill.u.s.tration]

199. A triad is said to be _in fundamental position_ when its root is the lowest tone. It is said to be in the _first inversion_ when the _third_ is the lowest tone, and in the _second inversion_ when the fifth is the lowest tone. Thus _e.g._, in Fig. 66 the same chord (C--E--G) is arranged in three different positions, at (a) in fundamental position, at (b) in the first inversion, and at (c) in the second inversion.

[Ill.u.s.tration: Fig. 66.]

200. When the root is not the ba.s.s note, figures are sometimes used to show what chord is to be played or written. Thus, _e.g._, the figure 6 over a ba.s.s note means that the note given is the _third_ of a chord, the root being found by going up a sixth from the ba.s.s note: _i.e._, the chord is to be sounded in its first inversion. In the same way the figures 6/4 indicate that the note given is the _fifth_ of the chord, the root and fifth being found by going up a sixth and a fourth from the note given; _i.e._, the chord is to be sounded in its second inversion.

The use of these and other similar figures and signs is called _figured ba.s.s_ (or _thorough ba.s.s_) _notation_. An example of a _figured ba.s.s_ is given in Fig. 67.

[Ill.u.s.tration: Fig. 67.]

_Thorough ba.s.s notation_ was formerly used extensively in writing accompaniments to vocal works, the accompanist having to interpret the notes and signs given, and then to make up an interesting accompaniment as he went along. Much of Handel's music was written in this way, but in modern editions of these works the chords have been printed in full and the signs omitted.

201. A _seventh chord_ consists of a fundamental tone with its third, fifth, and seventh. The fifth is sometimes omitted. A _ninth chord_ consists of a fundamental with its third, fifth, seventh, and ninth.

202. A _cadence_ is the close of a musical phrase: in melody it refers to the last two tones; in harmony to the last two chords.

The word _cadence_ is derived from _cadere_, a Latin word meaning to _fall_, the reference being to the falling of the voice (_i.e._, the dropping to the normal pitch) at the close of a sentence.

203. The most frequent cadence in harmony is that involving the chord on I preceded by the chord on V. Because of its directness the cadence V--I is called the _authentic cadence_.

204. The most satisfactory form (to the ear) of the authentic cadence is that in which the highest voice (the soprano) of the final chord is the _root_ of that chord. When the final chord appears in this position the cadence is called _perfect_[36] _authentic_, and when the third or fifth of the chord appear in the soprano, the cadence is called _imperfect authentic_. Fig. 68 shows the chord G--B--D cadencing to C--E--G in three different ways. The first one (a) is called a _perfect authentic cadence_, but the last two (c) and (d) are _imperfect authentic_.

[Footnote 36: Many theorists (including Durand in his monumental "Treatise on Harmony") consider the V--I cadence to be the only one which may legitimately be called _perfect_, but the majority of writers seem to take the view that either authentic or plagal cadence may be either perfect or imperfect, depending upon the soprano tone, as noted above.]

[Ill.u.s.tration: Fig. 68.]

205. A _plagal cadence_ is one in which the tonic chord is preceded by the sub-dominant chord (IV--I). The _plagal cadence_ (sometimes called the _church cadence_, or _amen cadence_), like the authentic, is described as being _perfect_ when the soprano of the tonic chord is the root of that chord, and _imperfect_ when the soprano of the final chord is the third or fifth of that chord. Fig. 69 shows the chord F--A--C cadencing to C--E--G in three ways. The first one (a) is called a _perfect plagal cadence_, the last two are _imperfect plagal_.

[Ill.u.s.tration: Fig. 69.]

206. A _half-cadence_ occurs when the dominant chord is used as the final chord of a phrase, and is immediately preceded by the tonic chord.

This form is used to give variety in the course of a composition, but is not available at the end because it does not give a definite close in the tonic key. Fig. 70 shows the use of the _half-cadence_ at the close of such a phrase.

[Ill.u.s.tration: Fig. 70. BACH]

207. A _deceptive cadence_ is the progression of the dominant chord to some other chord than the tonic, the word _deceptive_ implying that the ear expects to hear V resolve to I and is deceived when it does not do so. The most common form of _deceptive cadence_ is that in which V (or V^7) resolves to VI. It is used to give variety, but as in the case of the half-cadence, is not available at the end of a composition. Fig. 71 gives an example.

[Ill.u.s.tration: Fig. 71. WM. MATHER]

208. A _sequence_ is a succession of similar harmonic progressions, these resulting from a typical or symmetrical movement of the ba.s.s part.

See Fig. 72.

[Ill.u.s.tration: Fig. 72.]