Chapter XVIII.


Up to this point we have considered only the relations of the tones in a piece of music with its tonic, and of its chords with its tonic chord. On these relations depends the connection of the parts of a mass of tone into one coherent whole. But besides this the succession of the tones and chords must be regulated by natural relations. The mass of sound thus becomes more intimately bound up together, and, as a general rule, we must aim at producing such a connection, although, exceptionally, peculiar expression may necessitate the selection of a more violent and less obvious plan of progression. In the development of the scale we saw that the connection of all the notes by means of their relation to the tonic, if originally perceived at all, was at most but very dimly seen, and was apparently replaced by the chain of Fifths; at any rate, the latter alone was sufficiently developed to be recognised in the Pythagorean construction of tonal systems. But by the side of our strongly developed feeling for the tonic in modern harmonic music, the necessity for a linked connection of individual tones and chords is still recognised, although the chain of Fifths, which originally connected the tones of the scale, as

$$ \begin{gathered} f \pm c \pm g \pm d \pm a \pm e \pm b \end{gathered} $$

has been interrupted by the introduction of perfect major Thirds, and now appears as

$$ \begin{gathered} f \pm c \pm g \pm d .... d_1 \pm a_1 \pm e_1 \pm b_1 \end{gathered} $$

The musical connection between two consecutive notes may be effected:

1. By the relation of their compound tones.

This is either:

a.) direct, when the two consecutive tones form a perfectly consonant interval, in which case, as we have previously seen, one of the clearly perceptible partial tones of the first note is identical with one of the second. The pitch of the following compound tone is then clearly determined for the ear. This is the best and surest kind of connection. The closest relationship of this kind exists when the voice jumps a whole Octave; but this is not usual in melodies, except with the bass, as the alteration of pitch is felt to be too sudden for the upper part. Next to this comes the jump of a Fifth or Fourth, both of which are very definite and clear. After these follow the steps of a major Sixth or major Third, both of which can be readily taken, but some uncertainty begins to arise in the case of minor Sixths and minor Thirds. Esthetically it should be remarked, that of all the melodic steps just mentioned, the major Sixth and major Third have, I might almost say, the highest degree of thorough beauty. This possibly depends upon their position at the limit of clearly intelligible intervals. The steps of a Fifth or Fourth are too clear, and hence are, as it were, drily intelligible; the steps of minor Thirds, and especially minor Sixths, begin to sound indeterminate. The major Thirds and major Sixths seem to hold the right balance between darkness and light. The major Sixth and major Third seem also to stand in the same relation to the other intervals harmonically.

b.) indirect, of the second degree only. This occurs in the regular progression of the scale, proceeding by Tones or Semitones. For example:

$$ \underbrace{c...d}_G \quad \underbrace{d...e_1}_G \quad \underbrace{e_1...f}_C $$

The whole major Tone \(c...d\) proceeds from the Fourth to the Fifth of the auxiliary tone \(G\), which Rameau supposed to be subjoined as the fundamental bass of the above melodic progression. The minor Tone \(d...e_1\) proceeds from the Fifth to the major Sixth of the auxiliary tone \(G\), and the Semitone \(e_1..f\) from the major Third to the Fourth of the auxiliary tone \(C\). But in order that these auxiliary tones may readily occur to both singer and hearer, they must be among the principal tones of the key. Thus the step \(a_1...b_1\) in the major scale of \(C\) causes the singers a little trouble, although it is only the interval of a major Tone, and could be easily referred to the auxiliary tone \(E_1\) But the sound of \(e_1\) is not so firm and ready in the mind, as the sound of \(C\) and its Fifth \(G\) and Fourth \(F\). Hence the Hexachord of Guido of Arezzo, which was the normal scale for singers throughout the middle ages, ended at the Sixth.[1] This Hexachord was sung with different pitches of the first note, but always formed the same melody:

Ut Re Mi Fa Sol La
either \(G\) \(A\) \(B\) \(C\) \(D\) \(E\)
or \(C\) \(D\) \(E\) \(F\) \(G\) \(A\)
or \(F\) \(G\) \(A\) \(B\flat\) \(C\) \(D\)

So that the interval \(Mi... Fa\) always marked the Semitone. [2]

For the same reason Rameau preferred, in the minor scale, to refer the steps \(d_1 ... e^1\flat\) and \(e^1\flat ...f\) to \(G\) and \(C\) as auxiliary tones, rather than to \(B\flat\), the Seventh of the descending scale,[3] which had not a sufficiently close relationship to the tonic, and hence was not well enough impressed on the singer’s mind for such a purpose. Taking \(g\) and \(c\), the Octaves of \(G\) and \(C\) as the auxiliary tones, the motion in \(d...e^1\flat\) is from the Fourth below \(g\) to the major Third below it, and in \(e_1\flat ...f\) from the major Sixth below \(c\) to the Fifth below it. On the other hand, it is impossible to reduce the step \(a^1\flat ... b_1 \)[cents 274] in the minor scale to any relationship of the second degree. [See p. 301c, note 14.] It is therefore also decidedly unmelodic and had to be entirely avoided in the old homophonic music, just as the steps of the false Fifths and Fourths, as \(b_1 ...f'\) [cents 610], or \(f'...b_1'\) [cents 590]. Hence the alterations in the ascending and descending minor scales already mentioned.

In modern harmonic music many of these difficulties have disappeared, or become less sensible, because correct harmonisation can exhibit the connections which are absent in the melodic progression of an unaccompanied voice. Hence also it is much easier to take a part at sight in a harmony, written in pianoforte score, which shews its relations, than to sing it from an unconnected part. The former shews how the tone to be sung is connected with the whole harmony, the latter gives only its connection with the adjacent tones. [4]

2. Tones may be connected by their approximation in pitch.

This relation has been considered previously with reference to the leading note. The same holds good for the intercalated tones in chromatic passages. For example, if in \(C\) major, we replace \(C...D\) by \(C...C\sharp ...D\), this \(C\sharp\) has no relation either of the first or second degree with the tonic \(C\), and also no harmonic or modulational significance. It is nothing but a step intercalated between two tones, which has no relation to the scale, and only serves to render its discontinuous progression more like the gliding motion of natural speech, or weeping or howling. The Greeks carried this subdivision still further than we do at present, by splitting up a Semitone into two parts in their enharmonic system (p. 265a). Notwithstanding the strangeness of the tone to be Struck, chromatic progression in Semitones can be executed with sufficient certainty to allow it to be used in modulational transitions for the purpose of suddenly reaching very distant keys.

Italian melodies are especially rich in such intercalated tones. Investigations of the laws under which they occur will be found in two essays of Sig. A. Basevi.[5] The rule is without exception that tones foreign to the scale can be introduced only when they differ by a Semitone [6] from the note of the scale on to which they resolve, while any tones belonging to the scale itself can be freely introduced although out of harmony with the accompaniment, and even requiring the step of a whole Tone for their resolution.

In the same way steps of a whole Tone may be made, provided the notes lie in the scale, when they serve merely to connect two other tones which belong to chords. These are the so-called passing or changing notes. Thus if while the triad of \(C\) major is sustained, a voice sings the passage \(c...d...e_1 ...f...g\), the two notes \(d\) and \(f\) do not suit the chord, and have no relation to the harmony, but are simply justified by the melodic progression of the single voice. It is usual to place these passing notes on the unaccented parts of the bar, and to give them a short duration. Thus in the above example \(c, \; e_1, \; g\) would fall on accented parts of the bar. Then \(d\) is the passing note between \(c\) and \(e_1\) and \(f\) between \(e_1\) and \(g\). It is essential for their intelligibility that they should make steps of Semitones or whole Tones. They thus produce a simple melodic progression, which flows on freely, without giving any prominence to the dissonances produced.

Even in the essentially dissonant chords the rule is, that dissonant tones which intrude isolatedly on the mass of the other tones must proceed in a melodic progression, which can be easily understood and easily performed. And since the feeling for the natural relations of such an isolated tone is almost overpowered by the simultaneous sound of the other tones which force themselves much more strongly on the attention, both singer and hearer are thrown upon the gradual diatonic progression as the only means of clearly fixing the melodic relations of a dissonant note of this description. Hence it is generally necessary that a dissonant note should enter and leave the chord by degrees of the scale.

Chords must be considered essentially dissonant, in which the dissonant notes do not enter as passing notes over a sustained chord, but are either accompanied by an especial chord, differing from the preceding and following chords, or else are rendered so prominent by their duration or accentuation, that they cannot possibly escape the attention of the hearer. It has been already remarked that these chords are not used for their own sakes, but principally as a means of increasing the feeling of onward progression in the composition. Hence it follows for the motion of the dissonant note, that when it enters and leaves the chord, it will either ascend on each occasion or descend on each. If we allowed it to reverse its motion in the second half, and thus return to its original position, there would seem to have been no motive for the dissonance. It would in that case have been better to leave the note at rest in its consonant position. A motion which returns to its origin and creates a dissonance by the way, had better be avoided; it has no object.

Secondly it may be laid down as a rule, that the motion of the dissonant note should not be such as to make the chord consonant without any change in the other notes. For a dissonance which disappears of itself provided we wait for the next step, gives no impetus to the progress of the harmony. It sounds poor and unjustified. This is the principal reason why chords of the Seventh which have to be resolved by the motion of the Seventh, can only permit the Seventh to descend. For if the Seventh ascended in the scale, it would pass into the octave of the lowest tone, and the dissonance of the chord would disappear. When Bach, Mozart, and others use such progressions for chords of the dominant Seventh, the Seventh has the effect of a passing note, and must be so treated. In that case it has no effect on the progression of the harmony.

The pitch of a single dissonant note in a chord of many parts is determined with greatest certainty, when it has been previously heard as a consonance in the preceding chord, and is merely sustained while the new chord is introduced. Thus if we take the following succession of chords:

$$ \begin{gathered} G ... d ... g + b_1 \\ c + e_1 - g + b_1 \end{gathered} $$

the \(b_1\) in the first chord is determined by its consonance with \(G\). It simply remains while the tones \(c\) and are introduced in place of \(G\) and \(d\), and thus becomes a dissonance in the chord of the Seventh \(c + e_1 - g + b_1\) In this case the dissonance is said to be prepared. This was the only way in which dissonances might be introduced down to the end of the sixteenth century. Prepared dissonances produce a peculiarly powerful effect: a part of the preceding chord lingers on, and has to be forced from its position by the following chord. In this way, an effort to advance against opposing obstacles which only slowly yield, is very effectively expressed. And for the same reason the newly introduced chord (\(c + e_1 - g\) in the last example) must enter on a strongly accented part of the bar; as it would otherwise not sufficiently express exertion. The resolution of the prepared dissonance, on the contrary, naturally falls on an unaccented part of the bar. Nothing sounds worse than dissonances played or sung in a dragging or uncertain manner. In that case they appear to be simply out of tune. They are, as a rule, only justified by expressing energy and vigorous progress.

Such prepared dissonances, termed suspensions, may occur in many other chords besides those of the Seventh. For example:

$$ \begin{gathered} \text{Preparation:} \: G ...c + e_1 \end{gathered} $$ $$ \begin{gathered} \text{Suspension:} \: G ...c ... d \end{gathered} $$ $$ \begin{gathered} \text{Resolution:} \: G + B_1 - d \end{gathered} $$

The tone \(c\) is the prepared dissonance; in the second chord, which must fall on an accented part of the bar, \(d\) the Fifth of \(G\) is introduced and generates the dissonance \(c...d\), and then \(c\) must give way, and according to our second rule, must go further from \(d\), which results in the resolution \(G + B_1 - d\). The chords might also be played in the inverse order, and then \(d\) would be a prepared dissonance which was forced away by \(c\). But this is not so good, because descending motion is better suited than ascending motion to an extruded note. Heightened pitch always gives us involuntarily the impression of greater effort, because we have continually to exert our voice in order to reach high tones. The dissonant note on descending seems to yield suitably to superior force, but on ascending it as it were rises by its own exertion. But circumstances may render the latter course suitable, and its occurrence is not unfrequent.

In the other case, especially frequent for chords of the Seventh, when the dissonant note is not prepared but is struck simultaneously with the chord to which it is dissonant, the significance of the dissonance is different. Since these unprepared Sevenths must usually enter by the descent of the preceding note, they may be always considered as descending from the Octave of the root of their chord, by supposing a consonant major or minor chord having the same root as the chord of the Seventh to be inserted between that and the preceding chord. In this case the entrance of the Seventh merely indicates that this consonant chord begins to break up immediately and that the melodic progression gives a new direction to the harmony. This new direction, leading to the chord of resolution, must be emphasised, and hence the dissonance necessarily falls on the preceding unaccented part of a bar.

The introduction of an isolated dissonant note into a chord of several parts cannot generally be used as the expression of exertion, but this character will attach to the introduction of a chord as against a single note, supposing that this single note is not too powerful. Hence it lies in the nature of things that the first kind of introduction takes place on unaccented and the last on accented parts of a bar.

These rules for the introduction of dissonances may be often neglected for the chords of the Seventh in the reverted system, in which the Fourth and Second of the scale occur, and notes from the subdominant side are mixed up with notes from the dominant side. These chords may also be introduced to enhance the dynamical impression of the advancing harmony, for they have the effect of keeping the extent of the key perpetually before the feeling of the hearer, and this object justifies their existence.

Of several voices which are leaving the chord of the tonic \(C\), it is quite easy for some to pass on to notes of the dominant chord \(g + b_1 - d'\), and for others to proceed to the notes of the subdominant chords \(f + a_1 - c\) or \(f - a^1\flat + c\), as each voice will be able to strike the new note with perfect certainty, on account of the close relationship between the chords. When, however, the dissonant chord has been thus formed and sounded, the dissonant notes will have the feeling for their more distant relations obscured by the strangeness of the other parts of the chord, and must generally proceed according to the rule of resolution of dissonances. Thus the singer who sounds \(f\) in the chord \(g + b_1 - d \;|\; f\), would vainly endeavour to picture to himself the sound of the \(a_1\) which is related to \(f\) with sufficient clearness to leap up or down to it with certainty; but he is easily able to execute the small step of half a Tone, by which \(f\) descends to \(e_1\) in the chord \(c + e_1 - g\). But the note \(g\) itself, on the other hand, having its own compound tone approximatively indicated by the chord of the Seventh, has no difficulty in passing by a leap to its related notes, as \(c\) for example, or \(b_1\) to \(g\).

In the chords \(b_1 - d \;|\; f + a_1\) and \(b_1 - d \; | \; f - a^1\flat \), in which neither dominant nor subdominant prevails, it would not be advisable to let any note proceed by a leap.

And it would also not be advisable to pass by a leap into the chords of the reverted system from any other chord but the tonic, because that chord alone is related to both dominant and subdominant chords at the same time.

It is not possible to pass to chords of the Seventh in the direct system, from another chord related to both extremities of the chord of the Seventh, and hence in this case the dissonance must be introduced in accordance with the strict rules.

Musicians are divided in opinion as to the proper treatment of the subdominant chord with an added Sixth, \(f + a_1 - c ... d\) in \(C\) major. The rule of Rameau is probably correct (p. 347d), making \(d\) the dissonant note, to be resolved by rising to \(e_1\). This is also decidedly the most harmonious kind of resolution. Modern theorists, on the other hand, regard this chord as a chord of the Seventh on \(d\), and take \(c\) as the dissonant note to be resolved by descent; whereas when \(c\) remains, \(d\) is quite free and may therefore even descend.

Chordal Sequences.

Just as the older homophonic music required the notes of a melody to be linked together, modern music endeavours to link together the series of chords occurring in a tissue of harmony, and it thus obtains much greater freedom in the melodic succession of individual notes, because the natural relationship of the notes is much more decisively and emphatically marked in harmonic music than in homophonic melody. This desire for linking the chords together was but slightly developed in the sixteenth century. The great Italian masters of this period allow the chords of the key to succeed each other in leaps which are often surprising, and which we should at present admit only in exceptional cases. But during the seventeenth century the feeling for this peculiarity of harmony also was developed, so that we find Rameau laying down distinct rules on the subject in the beginning of the eighteenth century. In reference to his conception of fundamental bass, Rameau worded his rule thus: 'The fundamental bass may, as a general rule, proceed only in perfect Fifths or Thirds, upwards or downwards.' According to our view the fundamental bass of a chord is that compound tone which is either exclusively or principally represented by the notes of the chord. In this sense Rameau’s rule coincides with that for the melodic progression of a single note to its nearest related notes. The compound tone of a chord, like the voice of a melody, may only proceed to its nearest related notes. It is much more difficult to assign a meaning to progression by relationship in the second degree for chords than for separate notes, and similarly for progression in small diatonic degrees without relationship. Hence Rameau’s rule for the progression of the fundamental bass is on the whole stricter than the rules for the melodic progression of a single voice.

Thus if we take the chord \(c + e_1 - g\), which belongs to the compound tone of \(C\), we may pass by Fifths to \(g + b_1 - d\), the compound tone of \(G\), or to \(f+a_1- c\), the compound tone of \(F\). Both of these chords are directly related to the first \(c + e_1 - g\), because each has one note in common with it, \(g\) and \(c\) respectively. But we can also allow the compound tone to proceed in Thirds, and then we obtain minor chords, that is, provided we keep to the same scale. The transition from the compound tone of \(C\) to that of \(E_1\) is expressed by the sequence of chords \(c + e_1 - g\) and \(e_1 - g + b_1\) which are related by having two notes, \(e_1, g\), in common. The sequence \(c + e_1 - g\) and \(a_1 - c + e_1\) from the compound tone of \(C\) to that of \(A_1\) is of the same kind. The latter is even more natural than the former, because the chord \(a_1 - c + e_1\) represents imperfectly the compound tone of \(A_1\) into which that of \(C\) intrudes, so that the compound tone of \(C\), which was clearly given in the preceding chord, persists with two of its tones, \(c, e_1\) in the second chord, a relation which did not exist in the former case.

But if we prefer to leave the key of \(C\) major, we can pass to perfect compound tones in Thirds, as from \(c + e_1 - g\) to \(e_1 + g_2\sharp - b_1\) or \(a_1 + c_2\sharp - e_1 \), as is very usual in modulations.

Rameau will not allow a simple diatonic progression of the fundamental bass of consonant triads, except where major and minor chords alternate, as from \(g + b_1 - d\) to \(a_1 - c + e_1\) that is from the compound tone of \(G\) to that of \(A_1\), but calls this a 'licence.' In reality this progression is readily explicable from our point of view, by considering \(a_1 - c + e_1\) a compound tone of \(C\) with an intrusive \(a_1\). The transition is then one of the usual close relationship, from the compound tone of \(G\) to that of \(C\), and the \(a_1\), appears as a mere appendage to the latter. Every minor chord represents two compound tones in an imperfect manner. Rameau first formulated this ambiguity (double emploi) for the minor chord with added Seventh, which, in the form \(d_1 - f + a_1 - c\) , may represent the compound tone of \(D_1\) and in the form \(f + a_1 - c ... d\) that of \(F\), or in Rameau’s language its fundamental bass might be \(D_1\) or \(F\).[7] In this chord of the Seventh the ambiguity is more marked because it contains the compound tone of \(F\) more completely; but the ambiguity belongs in a less marked degree to the simple chord also.

With the false cadence in the major key

\(g + b_1 - d\)   to   \(a_1 - c + e_1\)

must be associated the corresponding cadence in the minor key,

\(g + b_1 - d\)   to   \(a^1\flat - c + e^1\flat\)

where the chord \(a^1\flat - c + e^1\flat\) replaces the normal resolution \(c - e^1\flat g\). But here there is only a single note of the compound tone of \(C\) remaining, and the false cadence therefore becomes much more striking. It will be rendered milder by adding the Seventh \(f\) to the \(G\) chord, because \(f\) is related to \(a^1\flat\).

When two chords having only a relationship of the second degree, are placed in juxtaposition, we usually feel the transition to be very abrupt. But if the chord which connects them is one of the principal chords of the key, and has consequently been frequently heard, the effect is not so striking. Thus in the final cadence it is not unusual to see the succession \(f + a_1 - c\) and \(g + b_1 - d\) , the two chords being related through the tonic chord \(c + e_1 - g\), thus:

$$ \begin{alignedat}{2} \overbrace{f+a_1-c}& &\mkern-9mu\overbrace{g+b_1-d} \\ &\mkern-9mu\underbrace{c-e_1-g}& \end{alignedat} $$

Generally we must remember that all these rules of progression are subject to many exceptions, partly because expression may require exceptional abruptness of transition, and partly because the hearer’s recollection of previous chords may sufficiently strengthen a naturally weak relationship. It is clearly an entirely false position which teachers of harmony have assumed, in declaring this or that to be 'forbidden' in music. In point of fact nothing musical is absolutely forbidden, and all rules for the progression of parts are actually violated in the most effective pieces of the greatest composers. It would have been much better to proceed from the principle that certain transitions, which are disallowed, produce striking and unusual effects upon the hearer, and consequently are unsuitable except for the expression of what is unusual. Generally speaking, the object of the rules laid down by theorists is to keep up a well-connected flow of melody and harmony, and make its course readily intelligible. If that is what we aim at, we had best observe their restrictions. But it cannot be denied that a too anxious avoidance of what is unusual places us in danger of becoming trivial and dull, while, on the other hand, inconsiderate and frequent infringement of rules makes compositions eccentric and unconnected.

When disconnected triads would come together it is frequently advantageous to transform them into chords of the Seventh, and thus create a bond between them. In place of the preceding sequence of two triads

\(f + a_1 - c\)   to   \(g + b_1 - d\)

we can use a sequence of chords of the Seventh which represent the same compound tones

\(f + a_1 - c ... d\)   to   \(g + b_1 - d \; | \; f\)

In this case two of the four notes remain unchanged; in the chord of \(F\), the \(d\) belongs to the compound tone of the dominant, and in the chord of \(G\) the \(f\) to that of the subdominant.

Hence chords of the Seventh come to play an important part in modern music for the purpose of effecting well-connected and yet rapid transitions from chord to chord, and urging them forward by the action of dissonances. In this way particularly, transitions to the compound tone of the subdominant are easily effected.

Thus, for example, beginning with the triad \(g + b_1 - d\) we can not merely pass to the chord of \(C\), or \(c + e_1 - g\), but, letting \(g\) remain as a Seventh, to the chord of the Seventh \(a_1 - c + e_1 - g\), which unites the two chords \(c + e_1 - g\) and \(a_1 - c + e_1\), and then immediately pass to \(d_1 - f + a_1\) which is related to the latter chord, so that two steps bring us to the other extremity of the system of \(C\) major. This transition also gives the best progression for the Seventh (\(g\) in the example), because it has been prepared in the previous chord, and is resolved by descent (to \(f\)) in the succeeding chord. If we tried the same transition backwards, we should have to obtain the Seventh \(g\) by progression from \(a_1\) in the chord of \(d_1 - f + a_1\), and then be compelled to introduce the \(c\) of the chord of the Seventh abruptly, because we should have a prohibited succession of Fifths \(d_1 \pm a_1\) and \(c ± g\) if we tried to descend from \(d_1\) We must rather obtain \(c\) by a leap from \(f\), because \(a\), in the first triad must furnish both the \(a_1\) and \(g\) of the chord of the Seventh. Thus the transition to the dominant is by no means easy, fluent, and natural; it is much more embarrassed than the passage to the subdominant. Consequently the regular and usual progression of the chord of the Seventh is for its Seventh to descend to the triad whose Fifth is the root of the chord of the Seventh. Supposing we denote the root of the chord of the Seventh by I, its Third by III, &c., a falling Seventh will lead us to either of these chords:

$$ \begin{alignedat}{2} &\textrm{I -- III -- V -- VII} \quad \textrm{and} \quad && \textrm{I -- III -- V -- VII} \\ &\large{\textrm{| \;\;\; \textbackslash \;\;/ \quad /}} && \large{\textrm{| \;\;\; | \quad\; \textbackslash \; / }} \\ &\textrm{I \; -- \; IV \; -- \; VI} &&\textrm{I -- III \; -- \; VI} \end{alignedat} $$

Of these two transitions, the first, which leads to a chord of which IV is the root, is the liveliest, because it introduces a chord with two new tones. The other, which leads to a triad having VI for its root, introduces only one new tone. Hence, the first is regarded as the principal method of resolving chords of the Seventh.

For example:

$$ \begin{alignedat}{5} & g + b_1 - d \;|\; f \quad \quad \quad && e_1 - g + b_1 - d \\ &\large{\textrm{| \;\;\;\; \textbackslash \;\;/ \;\; /}} && \large{\textrm{| \;\;\;\; \textbackslash \;\;/ \quad /}}\\ & g \;\; ... \;\; c \; + e_1 && e_1 \; ... \;\; a_1 - \; c \\ & c + e_1 - g + b_1 \quad \quad \quad && b_1 - d \;|\; f + a_1 \\ &\large{\textrm{| \;\;\;\; \textbackslash \;\;/ \;\; /}} && \large{\textrm{| \;\;\;\;\; \textbackslash \;/ \quad /}}\\ & c \;\; ... \;\; f \; + a_1 && b_1 \; ... \;\; e_1 - \; g \end{alignedat} $$

The descent of the tone VII introduces the tone VI. In the first case this is the Third of the new triad, and in the second its root.[8] But it may be its Fifth:

$$ \begin{gathered} \textrm{I \;\;--\;\; III \;\;--\;\; V \;\;--\;\; VII} \\ \large{\textrm{\textbackslash \;\;\;\;\;\; \textbackslash \;\;\;\;/ \quad \quad /}} \\ \textrm{II \; -- \; IV \; -- \; VI} \end{gathered} $$

This, however, could only occur naturally in the two chords:

$$ \begin{alignedat}{5} &b_1 \; - \; d \; | \; f \; + \; a_1 \quad \text{and} \quad &&b_1 \; - \; d \; | \; f \; - \; a^1\flat \\ & \;\; \large{\textrm{\textbackslash \;\;\;\;\; \textbackslash \;/ \;\;\;\; /}} && \; \large{\textrm{\textbackslash \;\;\;\;\; \textbackslash \;/ \;\;\;\; /}} \\ & \;\;\;\; c + e_1 - g && \;\;\; c - e^1\flat + g \end{alignedat} $$

because the two chords of the Seventh represent the compound tone of \(G\) and the tonic chord establishes the bond of the union between its two sections. In other cases our scheme gives so-called false cadences:

$$ \begin{alignedat}{5} &g \; + \; b_1 - d \;\; | \;\; f \quad \text{and} \quad &&g \; + \; b_1 - d \;\; | \;\; f \\ & \;\; \large{\textrm{\textbackslash \;\;\;\;\; \textbackslash \;/ \;\;\;\; /}} && \; \large{\textrm{\textbackslash \;\;\;\;\; \textbackslash \;\;/ \;\;\;\; /}} \\ & \;\;\;\; a_1 - c + e_1 && \;\;\; a^1\flat + c - e^1\flat \end{alignedat} $$

which are justified (the first as most natural) by the fact that either \(c + e_1\) or \(c - e^1\flat \) belongs to the chord of the normal resolution. Rameau therefore justly observes that this kind of resolution is only permissible when the IV of the second chord is the normal Fourth of the I on the chord of the Seventh.

This exhausts the resolutions by the descent of the Seventh. Those in which it remains unchanged take place according to the schemes:

$$ \begin{alignedat}{2} &\textrm{I \;--\; III \;--\; V \;--\; VII} \quad \textrm{and} \quad && \textrm{I\; --\; III\; -- \;V \;--\; VII} \\ & \;\; \large{\textrm{\textbackslash \;\;\;\; \textbackslash \;\;/ \quad \;\;|}} && \; \large{\textrm{\textbackslash \;\; / \quad\; | \;\;\;\;\;\; | }} \\ & \;\;\; \textrm{II -- \; IV \; -- \; VII} && \;\; \textrm{II \; -- \; -- \; V \; -- \; VII} \end{alignedat} $$
[9]                                                  [10]

In the first the Seventh becomes the root, in the second the Third of the new chord. If it were the Fifth, the new chord would coincide with part of the chord of the Seventh:

$$ \begin{gathered} \textrm{I \;-- \;\; III\;-- \; V --\; VII}\\ | \quad\quad | \quad\quad | \quad\quad | \\ \textrm{VII -- III -- V -- VII} \end{gathered} $$

In these connections the resolution is towards the dominant side. The transition is most decisive in the first, where the Seventh becomes the root. These resolutions are on the whole less usual, because we pass more easily and frequently from chords on the dominant side into chords of the Seventh of the direct system. In the chords of the reverted system these transitions occur more frequently, because their Sevenths may enter by ascent, and hence we avoid the sequences of Fifths, which greatly embarrass the transitions from a triad to a chord of the Seventh on its dominant side.

As to the transitions from one chord of the Seventh to another, or to a dissonant triad of the direct system which may be regarded as a mutilated chord of the Seventh, all these matters are sufficiently developed in the ordinary manuals of Thorough Bass, and offer no difficulties that would justify us in dwelling upon them here.

On the other hand, we have to say a few words on certain rules respecting the progression of the individual parts in polyphonic compositions. Originally, as we have already remarked, all these parts were of equal importance, and had usually to repeat the same melodic figures in succession. The harmony was a secondary consideration, the melodic progression of the individual voices was the principal matter. Hence it was necessary to take care that each voice should stand out clear and distinct from all the others. The relation between the importance of harmony and melody has certainly altered essentially in modern music; the former has attained a much higher independent significance. But, after all, perfection of harmony must arise from the simultaneous performance of several voices, each of which has its own beautiful and clear melodic progression, and each of which therefore moves in a direction that the hearer has no difficulty in understanding.

On this rests the prohibition of consecutive Fifths and Octaves. The meaning of this prohibition has given rise to much disputation. The meaning of prohibiting consecutive Octaves has been made clear by musical practice. In polyphonic music two voices which lie one or two Octaves apart, are forbidden to move forward in such a way that after their next step they should be also one or two Octaves apart. But precisely in the same way, two voices in a polyphonic piece are forbidden to go on in unison for several notes, while for complete musical compositions it is not forbidden that two voices, or even all the voices, should proceed in Unisons or Octaves, for the purpose of strengthening the melodic progression. It is clear that the reason of this rule must lie in the limiting the wealth of the progression of parts by Unisons and Octaves. This is allowable when it is intentionally introduced for a whole melodic phrase, but it is not suited for a few notes in the course of a piece, where it can only give the impression of reducing the richness of the harmony by an unskilful accident. The accompaniment of a lower part by a voice singing an Octave higher, merely strengthens part of the compound tone of the lower voice, and hence where variety in the progression of parts is important, does not essentially differ from a Unison.

Now in this respect the nearest to an Octave are the Twelfth, and its lower octave, the Fifth. Hence, then, consecutive Twelfths and consecutive Fifths partake of the same imperfection as consecutive Octaves. But the case is somewhat worse. It is possible to accompany a whole melody in Octaves when desirable, without committing any error, but this cannot be done for Fifths and Twelfths without changing the key. It is impossible to proceed by a single diatonic step from the tonic as root with an accompaniment of Fifths, without departing from the key. In \(C\) major, we ascend from the Fifth \(c \pm g\) to the Fifth \(d + a\), but a does not belong to the scale, which requires the deeper \(a_1\); we descend to \(b_1 \pm f_1\sharp\), and there is no \(f_1\sharp\), in the scale at all. The other upward steps from \(d\) exclusive to \(a_1\)can of course be accompanied by perfect Fifths in the scale, as \(e_1 \pm b_1\), \(f \pm c'\), \(g \pm d'\), \(a_1 \pm e_1'\). It is therefore impossible to use the Twelfth consistently for increasing the richness of the tone. But again, when the intervals of a Twelfth or Fifth are continued for a few steps in melodic progression, they have simply the effect of strengthening the root. For the Twelfth this arises from its directly corresponding to one of the upper partial tones of the root. For the Fifth \(c \pm g\), the \(c\) and \(g\) are the two first upper partials of the combinational tone \(C\), which necessarily accompanies the Fifth. Hence an accompaniment in Fifths above, when it occurs isolatedly in the midst of a polyphonic piece, is not only open to the charge of monotony, but cannot be consistently carried out. It should therefore be always avoided.

But that consecutive Fifths merely infringe the laws of artistic composition, and are not disagreeable to the natural ear, is evident from the simple fact that all the tones of our voice, and those of most instruments, are accompanied by Twelfths, and that our whole tonal system reposes upon that fact. When the Fifths are introduced as merely mechanical constituents of the compound tone, they are therefore fully justified. So in the mixture stops of organs. In these stops the pipes which give the prime tones of the compound, are always accompanied by others which give its harmonics, as the Octaves, Twelfths variously repeated, and even the higher major Thirds. By this means the performer is able to compose a tone of a much more penetrating, piercing quality, than it would be possible to produce by the simple organ pipes with their relatively weak upper partial tones. It is only by such means that an organ is able to dominate over the singing of a large congregation. Almost all musicians have blamed an accompaniment of Fifths, or even Thirds, but fortunately have not been able to effect anything against the practice of organ-builders. In fact the mixture stops of an organ merely reproduce the masses of tone which would have been created by bowed instruments, trombones, and trumpets, if they had executed the same music. It would be quite different if we collected independent parts, from each of which we should have to expect an independent melodic progression in the tones of the scale. Such independent parts cannot possibly move with the precision of a machine; they would soon betray their independence by slight mistakes, and we should be led to subject them to the laws of the scale, which, as we have seen, render a consistent accompaniment in Fifths impossible.

The prohibition of Fifths and Octaves extends also, but with less strictness, to the next adjacent consonant intervals, when two of them are so placed as to form a connected group of upper partials in a compound tone. Thus transitions like

\(d \; ... \; g + b_1\)        to        \(c \; ... \; f + a_1\)

are ruled by musical theorists to be inferior to transitions like

\(b_1 - d' \; ... \; g'\)        to        \(a_1 - c' \; ... \; f'\)

For \(d, g, b_1,\) are the third, fourth, and fifth partial tones of the compound \(G_\prime\), but \(b_1, d', g'\) could only be regarded as its fifth, sixth, and eighth. Hence the first position of the chord expresses a single compound tone much more decidedly than the second, which is often allowed to be continued through long passages, when of course the nature of the Thirds and Fourths varies.

The prohibition of consecutive Fifths was perhaps historically a reaction against the first imperfect attempts at polyphonic music, which were confined to an accompaniment in Fourths or Fifths, and then, like all reactions, it was carried too far, in a barren mechanical period, till absolute purity from consecutive Fifths, became one of the principal characteristics of good musical composition. Modern harmonists agree in allowing that other beauties in the progression of parts are not to be rejected because they introduce consecutive Fifths, although it is advisable to avoid them, when there is no need to make such a sacrifice.

There is also another point in the prohibition of Fifths to which Hauptmann has drawn attention. We are not tempted to use consecutive Fifths when we pass from one consonant triad to another which is nearly related to it, because other progressions lie nearer at hand. Thus we pass from the triad of \(C\) major to the four related triads in the following manner, the fundamental bass proceeding by Thirds or Fifths:

$$ \begin{alignedat}{20} & c + e_1 - g \quad\quad\quad && c + e_1 - g \quad\quad\quad && c + e_1 - g \quad\quad\quad \text{and} && \; c + e_1 - g \\ \text{to}& \; c + e_1 \; ... \; a_1, \quad\quad\text{to} &&\; c \; ... \; f + a_1, \quad\quad \text{to} &&\; B_1 \; ... \; e_1 - g, \quad\quad\text{to} &&\; B_1 \; ... \; d \; ... \; g \end{alignedat} $$

But when the fundamental bass proceeds in Seconds, and hence does not pass to a directly related chord, the nearest position of the new chord is certainly one which produces consecutive Fifths. For example:

$$ \begin{alignedat}{20} & g + b_1 - d' \quad\quad \text{or} && g + b_1 - d' \\ \text{to} \; & a - c' + e_1', \quad\quad \text{to} \; && f + a_1 - c \end{alignedat} $$

In such cases, then, we must have recourse to other transitions by larger intervals, as:

$$ \begin{alignedat}{20} & g + b_1 - d' \quad\quad \text{or} && g + b_1 - d' \\ \text{to} \; & e_1 \; ... \; a_1 - c', \quad\quad \text{to} \; && a_1 - c \; ... \; f \end{alignedat} $$

which avoid consecutive Fifths.

Hence when the chords are closely connected by near relationship and small distance in the scale, consecutive Fifths do not present themselves. When they occur, therefore, they are always signs of abrupt chordal transition, and it is then better to assimilate the progression of parts to that which spontaneously arises in the case of related chords.

This consideration respecting consecutive Fifths, which was emphasised by Hauptmann, appears to give the law greater importance. That it is not the only motive for the prohibition of consecutive Fifths appears from the fact, that the forbidden sequence

\(g + b_1 - d'\)        to        \(f + a_1 - c'\)

is allowed, when the chords are in the position

\(b - d' \; ... \; g'\)        to        \(a_1 - c' \; ... \; f'\) ,

although the step in the fundamental bass is the same.

The prohibition of so-called hidden Fifths and Octaves has been added on to the prohibition of consecutive Fifths and Octaves, at least for the two extreme voices of a composition in several parts. This prohibition forbids the lowest and uppermost voice in a piece to proceed by direct motion [that is, both parts ascending or both parts descending] into the consonance of an Octave or Fifth (including Twelfth). They should rather come into such a consonance by contrary motion (one descending and the other ascending). In duets this would also hold for the unison. The meaning of this law must certainly be, that whenever the extreme voices unite to form the partial tones of a compound, they ought to have reached a state of relative rest. It must be conceded that the equilibrium will be more perfect when the extreme parts of the whole mass of tone approach their junction from opposite sides, than when the centre of gravity, so to speak, of the sonorous mass is displaced by the parallel motion of the extreme voices, and these voices catch one another up with different velocities. But where the motion proceeds in the same direction, and no relative rest is intended, the hidden Fifths are also not avoided, as in the usual formulae:

in which the \(g \pm d\) is reached by passages involving hidden Fifths.

Another rule in the progression of parts, prohibiting false relations, must have had its origin in the requirements of the singer. But what the singer finds a difficulty in hitting, must naturally also appear an unusual and forced skip to the hearer. By false relations is meant the case when two tones in consecutive chords, which belong to different voices, form false Octaves or false Fifths. For example, if one voice in the first chord sings \(b_1\) and another voice in the next chord sings \(b\flat\), or the first has \(c\) and the second \(c_2\sharp\) there are false Octave relations. False Fifth relations are forbidden for the extreme voices only. Thus in the first chord the bass has \(b_1\) in the second the soprano has \(f\), or conversely, where \(b_1...f\) is a false Fifth. The meaning of this rule is, probably, that the singer would find it difficult to hit the new tone which is not in the scale, if he had just heard the next nearest tone of the scale given by another singer. Similarly, when he has to take the false Fifth of a tone which is prominent in present harmony as lowest or highest. There is therefore a certain sense in the prohibition, but numerous exceptions have arisen, as the ear of modem musicians, singers and hearers, has become accustomed to bolder combinations and livelier progressions. All these rules were essentially intended for the old ecclesiastical music, where a quiet, gentle, well-contrived, and well-adjusted stream of sound was aimed at, without any intentional effort or disturbance of the smoothest equilibrium. Where music has to express effort and excitement, these rules become meaningless. Hidden Fifths and Octaves and even false relations of Fifths are found in abundance in the chorales of Sebastian Bach, who is otherwise so strict in his harmonies, but it must be admitted that the motion of his voices is much more powerfully expressed than in the old Italian ecclesiastical music.

[1]For the same reason d’Alembert explains the limits of the old Greek heptachord, by means of two connected tetrachords — $$ \begin{gathered} b ... c ... d ... e ... f ... g ... a \end{gathered} $$ in which the step \(a...b\) is avoided. But this explanation would only suit a key in which \(c\) was the tonic, and this was probably not the case for the ancient Greek scale.
[2][Prof. Helmholtz leaves the intonation unmarked. Guido d’Arezzo, the presumed inventor of the Hexachord, is said to have introduced it about 1024 A.D., that is long before meantone temperament existed. Hence we must assume Pythagorean intonation (see p. 313d). Yet in later times the Hexachord was certainly used for training singers in mean tone temperament. It could not have been used for just intonation, because the melody \(c\; d\; e\; f\) is assumed to be identical with \(g\; a\; b\; c\) in the same scale, whereas in just intonation \(c \; 204 \; d \; 182 \; e_1 \; 112 \; f \) and \(g \; 182 \; a_1 \; 204 \; b_1 \; 112 \; c'\) are different. For an excellent account of the Hexachord see Mr. Rockstro's article 'Hexachord,' in Grove's Dictionary. To shew, however, how intonations are mixed up, it may be observed that he illustrates the use of the Hexachord in 'Real Fugue' by an example of Palestrina, who lived in the sixteenth century, and is often credited with just intonation, but who being junior to Salinas and Zarlino must have used meantone temperament. — Translator.]
[3][The Author writes \(B\flat\), and calls it 'the Seventh of the descending scale' of \(C\) minor, which, however, is \(B^1\flat\) and this answers for the first interval \(d...e^1\flat\), owing to \(b^1\flat + d\), and \(e^1\flat \pm b^1\flat\); but it does not answer to the second interval \(e^1\flat ...f\) as \(f...b^1\flat\) is dissonant, and it would not do to use \(b\flat\) although \(b\flat \pm f\) is consonant, because \(e^1\flat ... b\flat\) is dissonant. But \(a^1\flat\) would do, as we see from \(a^1\flat \pm e^1\flat, \; f-a^1\flat\). Hence if the text gives Rameau’s notes, he must have been misled by temperament. — Translator.]
[4][Hence any means of shewing the relation of each tone to the tonic of the moment, as in the Tonic Solfa System, materially facilitates sight-singing, as perhaps the use of the duodenal (App. XX. sect. E. art. 26) when thoroughly understood might also do. — Translator.]
[5] Introduction à un nouveau Système d'Harmonie, traduit par L. Delâtre; Florence, 1855. Studj sull' Armonia, Firenze, 1865.
[6][Of course those who laid down the rule thought only of a tempered Semitone. But in Pythagorean temperament there were two Semitones, the small 90 and the large 114 cents, and the rule was to make the Semitone closest to the note to which it led, thus \(c \;114\; c\sharp \;90\;d, \; d \;114 \; d\flat \; 90 \; d\). And this notation was retained even in meantone temperament, where the relations were reversed, as \(c \;76 \; c\sharp \; 117 \; d, \; d \; 76 \; d\flat \; 117 \; c\); but practically this made no difference except to the singer, as the player had only one Semitone at command. This writing is still continued in equal temperament, although the two Semitones are now equalised as 100 cents, thus \(c \; 100 \; c\sharp \; 100 \; d\), and \(d\; 100 d\flat\; 100 \;c\). But in just intonation we have Semitones of various dimensions, \(c \;114\; c\sharp \; 90 \; d, \; c\; 112 \;d^1\flat \; 92 \; d, \;c \;92, \;c_1\sharp \;112 \;d, \;c \;90 \; d\flat \; 114 \; d,\; c \;70 \; c_2\sharp \; 134 \; d\); which of these is the player to play, or the singer to sing (a question of importance when each part is sustained by many unaccompanied voices)? Practically the player will take the most handy interval, and the singers must arrange in rehearsal, but would possibly take \(c \; 92 \; c_1\sharp \; 112 \; d, \; d \; 92 \; d^1\flat \; 112 \; c\), as these are the intervals used in modulation from \(f\) to \(f_1\sharp\) to the dominant, and \(b_1\) to \(b\flat\) to the subdominant. — Translator.]
[7][Of course Rameau, writing in tempered notation, did not distinguish \(d_1\) and \(d\), so that the actual notes in the two chords \(d_1 - f + a_1 - c\) and \(f + a_1 - c...d\) were to him identical. See pp. 340a, 345a, 348a. — Translator.]
[8][As examples of the second method have been omitted in the text, take $$ \begin{alignedat}{5} & g + b_1 - d \; | \; f \quad \quad \quad && e_1 - g + b_1 - d \\ &\large{\textrm{| \;\;\;\; | \;\;\;\; \textbackslash /}} \quad \quad \quad &&\large{\textrm{| \;\;\;\; | \;\;\;\; \textbackslash \;\; /}} \\ & g + b_1 \; ...\; e_1 \quad \quad &&e_1 - g \; ... \;\;\; c \end{alignedat} $$ -Translator.]
[9][Examples: $$ \begin{alignedat}{10} &g + b_1 - d \; | \; f \quad && c + e_1 - g + b_1 \quad && e_1 - g + b_1 - d \\ & \; \large{\textrm{\textbackslash \;\;\;\;\; \textbackslash \; / \;\; |}} && \; \large{\textrm{\textbackslash \;\;\;\; \textbackslash \; / \;\;\;\;\; |}} && \; \large{\textrm{\; \textbackslash \;\;\;\; \textbackslash \; / \;\;\;\;\; |}} \\ & \;\; a_1 - \; c \; ... \;\; f && \;\; d \;\; | \;\; f \; ... \;\;\;\; b_1 && \;\;\; f + \; a_1 \; ... \;\;\; d \end{alignedat} $$ Here in the first example we obtain the major triad \(f + a_1 - c\); in the second the diminished triad \(b_1 - d \; | \; f\), itself a dissonance; and in the third the imperfect minor \(d \; | \; f + a_1\). - Translator.]
[10][Examples: $$ \begin{alignedat}{10} & g + b_1 - d \; |\; f, \quad &&c + e_1 - g + b_1 \\ & \large{\textrm{\textbackslash \;\; / \;\;\;\;\; | \;\; |}} && \large{\textrm{\; \textbackslash \; / \;\;\;\;\; |}} \\ & \;\; a_1 \;\;\;\;\;\;\;\; d \;\; f, && \;\;d \;\;\; ... \;\;\; g + b_1 \end{alignedat} $$ -Translator.]
[11][Examples: $$ \begin{alignedat}{10} & g + b_1 - d \;\; | \; f \; \quad && c \;+\; e_1 - g + b_1 \\ & \large{\textrm{\;| \;\;\; | \;\;\;\; | \;\;\; |}} && \large{\textrm{\; | \;\;\;\; | \;\;\;\; | \;\;\; |}} \\ & f \; ... \; b_1 - d \; | \; f && b_1 \; ... \; e_1 - g + b_1 \\ \end{alignedat} $$ -Translator.]
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