Chapter XV.

The Consonant Chords of The Tonal Modes

Polyphony was the form in which music for several voices first attained a certain degree of artistic perfection. The peculiar characteristic of this style of music was that several voices were singing each its own independent melody at the same time, which might be a repetition of the melodies already sung by the other voices, or else quite a different one. Under these circumstances each voice had to obey the general law of tonality common to the construction of all melodies, and, moreover, every tone of a polyphonic passage had to be referred to the same tonic. Hence each voice had to commence separately on the tonic or some tone closely related to it, and to close in the same way. In practice each part of a polyphonic piece was made to begin with the tonic or its Octave. This fulfilled the law of tonality, but necessitated the closing of a polyphonic piece with a unison.

The reason why higher Octaves might accompany the tonic at the close, lies, as we saw in the last chapter, in the fact that higher Octaves are merely repetitions of portions of the fundamental tone. Hence by adding its Octave to the tonic at the close, we merely reinforce part of its compound tone no new compound or simple tone is added, and the union of all the tones contains only the constituents of the tonic itself.

The same is true for all the other partial tones which are contained in the tonic. The next step in the development of the final chord was to add the Twelfth of the tonic. Now the chord \(c...c'±g'\) contains no element which is not also a constituent of the compound tone \(c\) when sounded alone, and consequently, being a mere representative of the single musical tone \(c\), it is suitable for the termination of a piece of music having the tonic \(c\).

Nay even the chord \(c'±g'...c''\) might be so used, for when it is struck we hear weakly indeed, but still sensibly, the combinational tone \(c\), so that the whole mass of tone again contains nothing more than the constituents of the tone \(c\). It must be owned, however, that this combination would answer to a rather unusual quality of tone, with a proportionably weak prime partial.

On the other hand, it was not possible to use the chords \(c...c'...f'\) or \(c'...f'±c''\) to end a piece having the tonic \(c\), although these chords are consonances as well as the preceding, because \(f\) is not an element of the compound tone \(c\), and hence the closing chord would contain something which was not the tonic at all. It is here probably that we have to look for the reason why some medieval theoreticians wished to reckon the Fourth among the dissonances. But perfect consonance was not sufficient to make an interval available for the final chord. There was a second condition which the theoreticians did not clearly understand. The tones of the final chord had to be constituents of the compound tone of the tonic. This was the only case in which those tones could be employed.

The Sixth of the tonic is as ill suited as the Fourth for use in the final chord. But the major Third can be used, because it occurs as the fifth partial tone of the tonic. Since the qualities of tone which are fit for music generally allow the fifth and sixth partial tone to be audible, but make the higher partials either entirely inaudible or at least very faint, and since, moreover, the seventh partial is dissonant with the fifth, sixth, and eighth, and is not used in the scale, the series of tones available for the closing chord terminates with the Third. Thus we actually find down to the beginning of the eighteenth century, that the final chord has either no Third, or only a major Third, even in tonal modes which contain only the minor and not the major Third of the tonic. To attain fulness, it was preferred to do violence to the scale by using the major Third in the closing chord. The minor Third of the tonic can never stand for a constituent of its compound tone. Hence it was originally as much forbidden as the Fourth and Sixth of the tonic. Before a minor chord could be used to close a piece of music the feeling for harmony had to be cultivated in a new direction.

The ear is the more satisfied with a closing major chord, the more closely the order of the tones used imitates the arrangement of the partial tones in a compound. Since in modern music the upper voice is most conspicuous, and hence has the principal melody, this voice must usually finish with the tonic. Bearing this in mind, we can use any of the following chords for the close (combinational tones are added as crotchets): —

In the chords 1 and 2 all the notes coincide with partials of \(C\), and they therefore most closely resemble the compound tone \(C\) itself. And then closer positions of the chord can be substituted, provided they resemble the first by having \(C\) for the fundamental tone as in 3, 4, 5. They still retain sufficient resemblance to the compound tone of the low \(C\) to be used in its place. Moreover, the combinational tones, written as crotchets in 3, 4, and 5, assist in the effect of making the deeper partials of the compound \(C\), at least faintly, audible. But the first two positions always give the most satisfactory close. The tendency towards a deep final tone in harmonic music is very characteristic, and I believe that the above is its proper explanation. There is nothing of the kind in the construction of homophonic melodies. It is peculiar to the bass of part music.

Precisely in the same way that the tonic, when used as the bass of its major chord at the close, gives it a resemblance to its own compound tone, and is hence felt as the essential tone of the chord, all major chords sound best when the lowest tone of their closest triad position (No. 4. p. 219) is made the bass. The other major chords in the scale are those in its Fourth and Fifth, and hence for the scale of \(C\) major, are \(F+A_1-C\) and \(G+B_1-D\). Hence if we make the harmony of a piece of music to consist of these major chords only, each having its fundamental tone in the bass, the effect is almost that of a compound tonic in different qualities of tone passing into its two nearest related compound tones, the Fourth and Fifth. This makes the harmonisation transparent and definite, but it would be too uniform for long pieces. Modern popular tunes, songs and dances, are however, as is well known, constructed in this manner. The people, and generally persons of small musical cultivation, can be pleased only by extremely simple and intelligible musical relations. Now the relations of the tones are generally much easier to feel with distinctness in harmonised than in homophonic music. In the latter the feeling of relationship of tone depends solely on the sameness of pitch of two partials in two consecutive musical tones. But when we hear the second compound tone we can at most remember the first, and hence we are driven to complete the comparison by an act of memory. The consonance, on the other hand, gives the relation by an immediate act of sensation; we are no longer driven to have recourse to memory; we hear beats, or there is a roughness in the combined sound, when the proper relations are not preserved. Again, when two chords having a common note occur in succession, our recognition of their relationship does not depend upon weak upper partials, but upon the comparison of two independent notes, having the same force as the other notes of the corresponding chord.

When, for example, I ascend from \(C\) to its Sixth \(A_1\), I recognise their mutual relationship in an unaccompanied melody, by the fact that \(e_1'\), the fifth partial of \(C\), which is already very weak, is identical with the third partial of \(A_1\). But if I accompany the \(A_1\) with the chord \(F+A_1-c\) , I hear the former \(c\) sound on powerfully in the chord, and know by immediate sensation that \(A_1\) and \(C\) are consonant, and that both of them are constituents of the compound tone \(F\).

When I pass melodically from \(C\) to \(B_1\) or \(D\), I am obliged to imagine a kind of mute \(G\) between them, in order to recognise their relationship, which is of the second degree. But if I audibly sustain the note \(G\) while the others are sounded , their common relationship becomes really sensible to my ear.

Habituation to the tonal relations so evidently displayed in harmonic music, has had an indisputable influence on modern musical taste. Unaccompanied songs no longer please us; they seem poor and incomplete. But if merely the twanging of a guitar adds the fundamental chords of the key, and indicates the harmonic relations of the tones, we are satisfied. Again, we cannot fail to see that the clearer perception of tonal relationship in harmonic music has greatly increased the practicable variety in the relations of tones, by allowing those which are less marked to be freely used, and has also rendered possible the construction of long musical pieces which require powerful links to connect their parts into one whole.

The closest and simplest relation of the tones is reached in the major mode, when all the tones of a melody are treated as constituents of the compound tone of the tonic, or of the Fifth above or the Fifth below it. By this means all the relations of tones are reduced to the simplest and closest relation existing in any musical system — that of the Fifth.

The relation of the chord of the dominant \(G\) to that of the tonic \(C\), is somewhat different from that of the chord of the subdominant \(F\) to the tonic chord. When we pass from \(C+E_1-G\) to \(G+B_1-d\) we use a compound tone, \(G\), which is already contained in the first chord, and is consequently properly prepared, while \(e\) at the same time such a step leads us to those degrees of the scale which are most distant from the tonic, and have only an indirect relationship with it. Hence this passage forms a distinct progression in the harmony, which is at once well assured and properly based. It is quite different with the passage from \(C+E_1-G\) to \(F+A_1-c\) . The compound tone \(F\) is not prepared in the first chord, and it has therefore to be discovered and struck. Hence the justification of this passage as correct and closely related, is not complete until the step is actually made and it is felt that the chord of \(F\) contains no tones which are not directly related to the tonic \(C\). In the passage from the chord of \(C\) to that of \(F\), therefore, we miss that distinct and well-assured progression which marked the passage from the chord of \(C\) to that of \(G\). But as a compensation, the progression from the chord of \(G\) to that of \(F\) has a softer and calmer kind of beauty, due, perhaps, to its keeping within tones directly related to the tonic \(C\). Popular music, however, favours the other passage from the tonic to the Fifth above (hence called the dominant of the key), and many of the simpler popular songs and dances consist merely of an interchange of tonic and dominant chords. Hence also the common harmonicon (accordion, German concertina), which is arranged for them, gives the tonic chord on opening the bellows, and the dominant chord on closing them. The Fifth below the tonic is called the subdominant of the key. Its chord is seldom introduced at all into usual popular melodies, except, perhaps, once near the close, to restore the equilibrium of the harmony, which had chiefly inclined towards the dominant.

When a section of a piece of music terminates with the passage of the dominant into the tonic chord, musicians call the close a complete cadence. We thus return from the tones most distantly related to the tonic, to the tonic itself, and, as befits a close, make a distinct passage from the remotest parts of the scale to the centre of the system itself. If, on the other hand, we close by passing from the subdominant to the tonic chord, the result is called an imperfect or plagal cadence. The tones of the subdominant triad are all directly related to the tonic, so that we are already close upon the tonic before we pass over to it. Hence the imperfect cadence corresponds to a much quieter return of the music to the tonic chord, and the progression is much less distinct than before.

In the complete cadence the chord of the tonic follows that of the dominant, but to preserve the equilibrium of the system in relation to the subdominant, its chord is made to precede that of the dominant as in 1 or 2.

This succession really forms the complete close, by bringing all the tones of the whole scale together again, and thus in conclusion collecting and fixing every part of the key.

The major mode, as we have seen, permits the requisitions of tonality to be most easily and completely united with harmonic completeness. Every tone of its scale can be employed as a constituent of the musical tone of the tonic, the dominant, or the subdominant, because these fundamental tones of the mode are also fundamental tones of major chords. This is not equally the case in the other ancient tonal modes.

$$ 1. \mathrm{Major \space Mode \space = \space \begin{cases} \rlap{$\underbrace{f+a_1-c}_{major}$}f+a_1-\rlap{$\overbrace{c+e_1-g}^{major}$}c+e_1-\underbrace{g+b_1-d}_{major} \end{cases}} $$ $$ 2. \mathrm{Mode \space of \space the \space fourth \space = \space \begin{cases} \rlap{$\underbrace{f+a_1-c}_{major}$}f+a_1-\rlap{$\overbrace{c+e_1-g}^{major}$}c+e_1-\underbrace{g-b^1\flat+d}_{minor} \end{cases}} $$ $$ 3.\mathrm{Mode \space of \space the \space Minor \space Seventh \space = \space \begin{cases} \rlap{$\underbrace{f+a_1-c}_{major}$}f+a_1-\rlap{$\overbrace{c-e^{\hphantom{1}}\flat+g}^{minor}$}c-e^1\flat+\underbrace{g-b^1\flat+d}_{minor} \end{cases}} $$ $$ 4.\mathrm{Mode \space of \space the \space Minor \space Third \space = \space \begin{cases} \rlap{$\underbrace{f-a_1\flat+c}_{minor}$}f-a_1\flat+\rlap{$\overbrace{c-e^{\hphantom{1}}\flat+g}^{minor}$}c-e^1\flat+\underbrace{g-b^1\flat+d}_{minor} \end{cases}} $$ $$ 5.\mathrm{Mode \space of \space the \space Minor \space Third \space = \space \begin{cases} \rlap{$\underbrace{b\flat-d_1\flat+f}_{minor}$}b\flat-d_1\flat+\rlap{$\overbrace{f-a^{\hphantom{1}}\flat+c}^{minor}$}f-a^1\flat+\underbrace{c-e^1\flat+g}_{minor} \end{cases}} $$ [1]

In the minor chords, the Third does not belong to the compound tone of its fundamental note, and hence cannot appear as a constituent of its quality; so that the relation of all the parts of a minor chord to the fundamental note is not so immediate as that for the major chord, and this is a source of difficulty in the final chord. For this reason we find almost all popular dance and song music written in the major mode[2]; indeed, the minor mode forms a rare exception. The people must have the clearest and simplest intelligibility in their music, and this can only be furnished by the major mode. But there was nothing like this predominance of the major key in homophonic music. For the same reason the harmonic accompaniment of chorales in major keys was developed with tolerable completeness as early as the sixteenth century, so that many of them correspond with the cultivated musical taste of the present day; but the harmonic treatment of the minor and the other ecclesiastical modes was still in a very unsettled condition, and strikes modern ears as very strange.

In a major chord \(c+e_1-g\), we may regard both \(g\) and \(e_1\) as constituents of the compound tone of \(c\), but neither \(c\) nor \(g\) as constituents of the compound tone of \(c_1\), and neither \(c\) nor \(e_1\) as constituents of the compound tone of \(g\).[3] Hence the major chord \(c+e_1 -g\) is completely unambiguous, and can be compared only with the compound tone of \(c\), and consequently \(c\) is the predominant tone in the chord, its root, or, in Rameau’s language, its fundamental bass; and neither of the other two tones in the chord has the slightest claim to be so considered.

In the minor chord \(c-e^1\flat+g\), the \(g\) is a constituent of the compound tones of both \(c\) and \(e^1\flat\). Neither \(e^1\flat\) nor \(c\) occurs in either of the other two compound tones \(c\), \(g\). Hence it is clear that g at least is a dependent tone. But, on the other hand, this minor chord can be regarded either as a compound tone of \(c\) with an added \(e^1\flat\) or as a compound tone of \(e^1\flat\) with an added \(c\). Both views are entertained at different times, but the first usually prevails. If we regard the chord as the compound tone of \(c\), we find g for its third partial, while the foreign tone \(e^1\flat\) only occupies the place of the weak fifth partial \(e_1\) But if we regarded the chord as a compound tone of \(e^1\flat\), although the weak fifth partial \(g\) would be properly represented, the stronger third partial, which ought to be \(b^1\flat\), is replaced by the foreign tone \(c\). Hence in modem music we usually find the minor chord \(c-e^1\flat+g\) treated as if its root or fundamental bass were \(c\), so that the chord appears as a somewhat altered and obscured compound tone of \(c\). But the chord also occurs in the position \(e^1\flat+g...c\) (or better still as \(e^1\flat+g...c^1\)) even in the key of \(B_1\flat\) major, as a substitute for the chord of the subdominant \(e^1\flat\). Rameau then calls it the chord of the great Sixth [in English 'added Sixth'], and, more correctly than most modern theoreticians, regards \(e^1\flat\) as its fundamental bass.[4]

When it is important to guide the ear in selecting one or other of these two meanings of the minor chord, the root intended may be emphasised by giving it a low position or by throwing several voices upon it. The low position of the root allows such other tones as could be fitted into its compound tone, to be considered directly as its partials, whereas the low compound tone itself cannot be considered as the partial of another much higher tone. In the first half of last century, when the minor chord was first used as a close, composers endeavoured to give prominence to the tonic by increasing the loudness of the tonic note in comparison with its minor Third. Thus in Handel’s oratorios, when he concludes with a minor chord, most of the conspicuous vocal and instrumental parts are concentrated on the tonic, while the minor Third is either touched by one voice alone, or merely by the accompanying pianoforte or organ. The cases are much rarer where in minor keys he gives only two voices to the tonic in the closing chord, and one to its Fifth and another to its Third, which is his rule in major modes.

When the minor chord appears in its second subordinate signification, as \(e^1\flat+g...c\) with the root \(e^1\flat\), this fact is shewn partly by the position of \(e^1\flat\), in the bass, and partly by its close relationship to the tonic \(b^1\flat\). Modern music even makes this interpretation of the chord still clearer by adding \(b^1\flat\) as the Fifth of \(e^1\flat\), so that the chord becomes dissonant in the form \(e^1\flat-g+b^1\flat...c^1\).[5]

The disinclination of older composers to close with a minor chord, may be explained partly by the obscuration of its consonance from false combinational tones, and partly because, as already mentioned, it does not give a mere quality of the tonic tone, but mixes foreign constituents with it. But in addition to the minor Third, which does not fit into the compound tone of the tonic, the combinational tones of a minor chord are equally foreign to it. As long as the feeling of tonality required a definite single compound tone for the connecting centre of the key, it was impossible to form a satisfactory close except by a reproduction of the pure compound tone of the tonic with no foreign admixture. It was not till a further development of musical feeling had given the chords of the mode an independent significance, that the minor chord, notwithstanding its possession of constituents foreign to the compound tone of the tonic, could be justified in its use as a close.

Hauptmann[6] gives a different reason for avoiding the minor chord at the close. He asserts that before the chord of the dominant Seventh came into use, there was no voice-part suitable for falling into the minor Third of the tonic. Thus if the final cadence consisted of the chords \(G+B_1-D\),\(C-E^1\flat+G\), the \(D\) of the first chord was the only one which could pass melodiously in but this would have appeared like the passage of the leading note \(D\) in the key of \(E^1\flat\) major into its tonic \(e^1\flat\), and hence have called up the feeling of \(E^1\flat\) major in lieu of \(C\) minor. We may admit that this relation of the leading note would have drawn the hearer’s special attention to the two tones in question, and to a certain extent disturbed his recognition of the key, but yet it is clear that even without the help of this chord of the dominant Seventh, there were several ways for the voices to pass through dissonances into the minor Third of the closing chord, if composers had felt any wish to do so. Thus in the plagal cadence

$$ \begin{gathered} \mathrm{c-e^1\flat+g\quad...c'}\\ \mathrm{F...f\quad-a^1\flat+c'}\\ \mathrm{C-e^1\flat+g\quad...c'} \end{gathered} $$

which is so often used on other occasions, the Fourth \(f\) could be made to descend to the minor Third \(e^1\flat\) without any inconvenience. Indeed, we find that when the chord of the dominant Seventh had actually come into use, and the Seventh F of the chord \(G+B_1-D\space|\space F\) ought by every right to have descended into the minor Third \(E^1\flat\) of the closing chord, musical pieces of the fifteenth century[7] avoid this progression, and make this Seventh \(F\) either ascend to the Fifth \(G\), or descend to the major Third \(E_1\) of the final chord, instead of to \(E^1\flat\), its minor Third. This custom prevailed down to Bach’s time.

In Chapter XIII. (p. 249a) we characterised modern harmonic music, as contrasted with medieval polyphony, by its development of a feeling for the independent significance of chords. In Palestrina, Gabrieli, and still more in Monteverde and the first composers of operas, we find the various degrees of harmoniousness in chords carefully used for the purposes of expression. But these masters are almost entirely without any feeling for the mutual relation of consecutive chords. These chords often follow one another by entirely unconnected leaps, and the only bond of union is the scale, to which all their notes belong.

The transformation which took place from the sixteenth to the beginning of the eighteenth century, may, I think, be characterised by the development of a feeling for the independent relationship of chords one to the other, and by the establishment of a central core, the tonic chord, round which were grouped the whole of the consonant chords that could be formed out of the notes of the scale. For these chords there was a repetition of the same effort which was formerly shewn in the construction of the scale, where interrelations of the tones were first grounded on a chain of intervals, and afterwards on a reference of each note to a central compound tone, the tonic.

Two chords which have one or more tones in common will here be termed directly related.

Chords which are directly related to the same chord will be here said to be related to each other in the second degree.

Thus \(c+e_1-g\) and \(g+b_1-d\) are directly related, and so are \(c+e_1-g\) and \(a_1-c+e_1\); but \(g+b_1-d\) and \(a_1-c+e_1\) are related only in the second degree.

When two chords have two tones in common they are more closely related than when they have only one tone in common. Thus \(c+e_1-g\) and \(a_1-c+e_1\) are more closely related than \(c+e_1-g\) and \(g+b_1-d\)

The tonic chord of any tonal mode can of course only be one which more or less perfectly represents the compound tone of the tonic, that is, that major or minor chord of which the tonic is the root. The tonic note, as the connecting core of all the tones in a regularly constructed melody, must be heard on the first accented part of a bar, and also at the close, so that the melody starts from it and returns to it; the same is true for the tonic chord in a succession of chords. In both of these positions in the scale we require to hear the tonic note, accompanied not by any arbitrary chord, but only by the tonic chord, having the tonic note itself as its root. This was not the case even as late as the sixteenth century, as is seen by the example on p. 247c taken from Palestrina.

When the tonic chord is major, the domination of all the tones by the tonic note is readily reconciled with the domination of all the chords by the tonic chord, for as the piece begins and ends with the tonic chord, it also begins and ends at the same time with the pure unmixed compound tone of the tonic note. But when the tonic chord is minor, all these conditions cannot be so perfectly satisfied. We are obliged to sacrifice somewhat of the strictness of the tonality in order to admit the minor Third into the tonic chord at the beginning and end. At the commencement of the eighteenth century we find Sebastian Bach using minor chords at the end of his preludes, because these were merely introductory pieces, but not at the end of fugues and chorales, and at other complete closes. In Handel and even in the ecclesiastical pieces of Mozart, the close in a minor chord is used alternately with the close in a chord without any Third, or with the major Third. And the last composer cannot be accused of external imitation of old habits, for we find that in these usages they always observe the expression of the piece. When at the close of a composition in the minor mode, a major chord is introduced, it has the effect of a sudden and unexpected brightening up of the sadness of the minor key, producing a cheering, enlightening, and reconciling effect after the sorrow, grief, or restlessness of the minor. Thus a close in the major suits the prayer for the peace of the departed in the words, 'et lux perpetua luceat eis,' or the conclusion of the Confutatis maledictis, which runs thus: —

$$ \begin{gathered} \mathrm{Oro \space supplex \space et \space acclinis,}\\ \mathrm{Cor \space contritum \space quasi \space cinis;}\\ \mathrm{Gere \space curam \space mei\space finis.} \end{gathered} $$

But such a closing major chord is certainly somewhat startling for our present musical feeling, even though its introduction may, at one time, add wondrous beauty and solemnity, or, at another, dart like a beam of hope into the gloom of deepest despair. If the restlessness remains to the last, as in the Dies irae of Mozart’s Requiem, the minor chord, in which an unresolved disturbance exists, forms a fitting close. Mozart was wont to terminate ecclesiastical pieces of a less decided character with a chord that had no Third. There are many similar examples in Handel. Hence although both masters stood on the very same platform as modern musical feeling, and themselves gave, as it were, the finishing touch to the construction of the modern tonal system, they were not altogether strangers to the feeling which had prevented older musicians from using the minor Third of the tonic in the final chord. They followed no strict rule, however, but acted according to the expression and character of the piece and the sense of the words with which they had to close.

Those tonal modes which furnish the greatest number of consonant chords related to one another or to the common chord, are best adapted for artistically connected harmonies. Since all consonant chords, when reduced to their closest position and simplest form, are triads consisting of a major and a minor Third, all the consonant chords of any key can be found by simply arranging them in order of Thirds, as in the following tables. The braces above and below connect the chords together. The ordinary round braces, which are placed above, point out minor chords; the square braces below indicate major chords. The tonic chord is printed in capitals.

$$ \begin{gathered} \text{1) Major Mode}\hspace{45mm}\\ \mathrm{\overbrace{d_1-f+a_1}\overbrace{-C+E_1}\overbrace{-G+b_1}-d}\\ \hspace{-5mm}\hphantom{d_1---}|\underline{\hphantom{f+-"}}||\underline{\hphantom{C+-"}}||\underline{\hphantom{+b_1--}}|\\ \end{gathered} $$ $$ \begin{gathered} \text{2) Mode of The Fourth}\hspace{45mm}\\ \mathrm{b\flat+\overbrace{d_1-f+a_1}\overbrace{-C+E_1}\overbrace{-G-b^1\flat+d}}\\ \hspace{-15mm}|\underline{\hphantom{\flat+d_1"-}}||\underline{\hphantom{+a_1-}}||\underline{\hphantom{+E_1--"}}| \end{gathered} $$ $$ \begin{gathered} \text{3) Mode of The Minor Seventh}\hspace{45mm}\\ \mathrm{b\flat+\overbrace{d_1-f+a_1}-\overbrace{C-E^1\flat+G}\overbrace{-b^1\flat+d}}\\ \hspace{-7mm}|\underline{\hphantom{b\flat+d_1-}}||\underline{\hphantom{+a_1-c}}|\hspace{7mm}|\underline{\hphantom{l^1\flat+G-}}| \end{gathered} $$ $$ \begin{gathered} \text{4) Mode of The Minor Third}\hspace{45mm}\\ \mathrm{b\flat+d_1-\overbrace{f-a^1\flat+}\overbrace{C-E^1\flat+G}\overbrace{-b^1\flat+d}}\\ \hspace{-8mm}|\underline{\hphantom{\flat+d_1-''''}}|\hspace{5mm}|\underline{\hphantom{\flat+b-j}}||\underline{\hphantom{\flat+G-b}}| \end{gathered} $$ $$ \begin{gathered} \text{5) Mode of The Minor Sixth}\hspace{45mm}\\ \mathrm{\overbrace{b\flat-d^1\flat+f}\overbrace{-a^1\flat+}\overbrace{C-E^1\flat+G}-b^1\flat}\\ \hspace{5mm}|\underline{\hphantom{\flat+f-l}}||\underline{\hphantom{\flat+C-}}||\underline{\hphantom{\flat+G-b^1}}| \end{gathered} $$

In this arrangement I have introduced the different intonations of the Second and Seventh of the key, which we found in the construction of the scales for homophonic music.[8] But we observe that the chords directly related to the tonic chord contain every tone in the scale, excepting in the mode of the minor Sixth. The Second and Seventh of the tonic occur first in the chord of \(G\), which is directly related to the tonic chord, and next in chords containing \(F\), which are, however, not directly connected with the tonic chord. The supplementary tones of the scale which are related to the dominant thus acquire in harmonic music an important preponderance over those related to the subdominant. We must necessarily prefer direct to indirect relations for determining scalar degrees. Hence by confining ourselves to the chords which are directly related to the tonic chord, we obtain the following arrangement of the tonal modes:[9]

$$ \begin{gathered} \text{1) Major Mode}\hspace{45mm}\\ \mathrm{f+\overbrace{a_1-C+E}\overbrace{_1-G+b_1}-d}\\ \hspace{0mm}|\underline{\hphantom{f+a_1-}}||\underline{\hphantom{+E_1+-}}||\underline{\hphantom{+b_1-dl}}| \end{gathered} $$ $$ \begin{gathered} \text{2) Mode of The Fourth}\hspace{45mm}\\ \mathrm{f+\overbrace{a_1-C+E_1}-\overbrace{G-b^1\flat+d}}\\ \hspace{-16mm}|\underline{\hphantom{f+a_1-}}||\underline{\hphantom{+E_1+-}}| \end{gathered} $$ $$ \begin{gathered} \text{3) Mode of The Minor Seventh}\hspace{45mm}\\ \mathrm{f+a_1-\overbrace{C-E^1\flat+G}\overbrace{-b^1\flat+d}}\\ \hspace{-8mm}|\underline{\hphantom{a^1\flat+ccc}}|\hspace{6mm}|\underline{\hphantom{^1\flat+lGb^1}}| \end{gathered} $$ $$ \begin{gathered} \text{4) Mode of The Minor Third}\hspace{45mm}\\ \mathrm{\overbrace{f-a^1\flat+}\overbrace{C-E^1\flat+}\overbrace{G-b^1\flat+d}}\\ \hspace{0mm}|\underline{\hphantom{^1\flat+C-l}}||\underline{\hphantom{^1\flat+G-b}}| \end{gathered} $$ $$ \begin{gathered} \text{5) Mode of The Minor Sixth}\hspace{45mm}\\ \mathrm{d^1\flat+\overbrace{f-a^1\flat+}\overbrace{C-E^1\flat+G}-b^1\flat}\\ \hspace{0mm}|\underline{\hphantom{^1\flat+f-}}||\underline{\hphantom{^1\flat-+C}}||\underline{\hphantom{^1\flat+G-b^1}}| \end{gathered} $$

A glance at this table shews that the major mode and mode of the minor Third (minor mode) possess the most complete and connected series of chords, so that these two are decidedly superior to the rest for harmonic purposes. This is also the reason which led to the preference given to them in modern music.

And in this way we obtain a final settlement of the proper intonation of the supplementary tones of the scale, at least for the first four modes. Hauptmann, with whom I agree, considers the tone \(D\) alone to be the essential constituent of both the major and minor modes of \(C\). This \(D\) forms an imperfect (Pythagorean) minor Third with \(F\) , so that the chord \(D \space|\space F+A_1\) must be considered as dissonant.[10] This chord thus intoned is in reality most decidedly dissonant to the ear. On the other hand, Hauptmann admits a major mode which reaches over to the subdominant, and uses \(D_1\) in place of \(D\). I consider this conceptation to be a very happy expression of the real state of things. When the consonant chord \(D_1-F+A_1\) occurs in any composition it is impossible to return immediately, without any transitional tone, to the tonic chord \(C+E_1-G\). The result would be felt as an harmonic leap without adequate notice. Hence it is a correct expression of the state of affairs to look upon the use of this chord as the beginning of a modulation beyond the boundaries of the key of \(C\) major, that is, beyond the limits of direct relationship to its tonic chord. In the minor mode this would correspond to a modulation into the chord of \(D^1\flat+F-A^1\flat\). Of course in the modern tempered intonation the consonant chord \(D_1-F+A_1\) is not distinguished from the dissonant \(D \space|\space F+A_1\), and hence the feeling of musicians has not been sufficiently cultivated to make them appreciate this difference on which Hauptmann insists.[11]

As regards the other supplementary tone \(b^1\flat\) which may occur in the chords \(e^1\flat+g-b^1\flat\) and \(g-b^1\flat+d'\), I have already shewn in the last chapter that even in homophonic music it is almost always replaced by \(b_1\). Harmonic considerations likewise favour the use of \(b_1\), independently of melodic progression. It has been already shewn that when the two tones of the scale which are but distantly related to the tonic, make their appearance as constituents of the dominant, they enter into close relation to the tonic. Now this can only be the case with the compound tones of the major chord \(g+b_1-d\), and not with those of the minor chord \(g-b^1\flat+d\). Considered independently, the tones \(b^1\flat\) and d are quite as closely related to \(c\) as the tones \(b_1\) and \(d\). But by regarding the two latter as constituents of the compound tone \(g\), we connect them with \(c\) by the same closeness of relationship that \(g\) is itself connected with \(c\). Hence, in all modern music, wherever \(b^1\flat\) might occur as a constituent of the dominant chord of the key of \(c\) minor, or of some dissonant chord replacing the dominant chord, it is usual to change it into \(b_1\) and otherwise to use either \(b^1\flat\) or \(b_1\) according to the melodic progression, but more frequently the latter, as I have already remarked when treating of the construction of minor scales. It is this systematic use of the major Seventh \(b_1\) in place of the minor Seventh \(b^1\flat\) of the key which now distinguishes the modern minor mode from the ancient Hypodoric,[12] or the mode of the minor Third. Here again some part of the consistency of the scale is sacrificed in order to bind the harmony closer together.

The chain of consonant chords in the mode of the minor Third is certainly impaired when that mode is transformed into our minor by the introduction of \(b_1\) In place of the chain

$$ \begin{gathered} \mathrm{\overbrace{f-a^1\flat+C}\overbrace{-E^1\flat+G}\overbrace{-b^1\flat+d}}\\ \hspace{0mm}|\underline{\hphantom{^1\flat+C-}}||\underline{\hphantom{\flat+G-b^1}}| \end{gathered} $$

our minor furnishes only

$$ \begin{gathered} \mathrm{\overbrace{f-a^1\flat+C}\overbrace{-E^1\flat+G}+b_1-d}\\ \hspace{7mm}|\underline{\hphantom{^1\flat+C-}}|\hspace{8mm}|\underline{\hphantom{g+b_1-l}}| \end{gathered} $$

which has one triad less. But the composer is at liberty to alternate the two tones \(b^1\flat\)and \(b_1\).

The introduction of the leading note \(b_1\), into the key of c minor generated a new difficulty in the complete closing cadence of this key. When the chord \(g+b_1-d\) is followed by the chord \(c-e^1\flat+g\), the first being a perfectly harmonious major chord, and the latter an obscurely harmonious minor chord, the defect in the harmoniousness of the latter is made much more evident by the contrast. But it is precisely in the final chord that perfect consonance is essential to satisfy the feeling of the hearer. Hence this close could not become satisfactory until the chord of the dominant Seventh had been invented, which changed the dominant consonance into a dissonance.

The preceding explanation shews that when we try to institute a close connection among all the chords peculiar to a mode similar to the close connection among the tones of the scale (that is, when we require all the consonant triads in the harmonic tissue to be related to one of their number, the tonic, triad, in a manner analogous to that in which the notes of the scale are related to one of their number, the tonic tone), there are only two tonal modes, the major and minor, which properly satisfy such conditions of related tones and related chords.

The major mode fulfils the two conditions of chordal relationship and tonal relationship in the most perfect manner. It has four triads which are immediately related to the tonic chord

$$ \begin{gathered} \mathrm{f+\overbrace{a_1-C+E}\overbrace{_1-G+b_1}-d}\\ \hspace{0mm}|\underline{\hphantom{+a_1jl-}}||\underline{\hphantom{+E_1--}}||\underline{\hphantom{+b_1dll-}}| \end{gathered} $$

Its harmonisation can be so conducted (indeed, in popular pieces which must be readily intelligible, it is so conducted), that all tones appear as constituents of the three major chords of the system, those of the tonic, dominant, and subdominant. These major chords, when their roots lie low, appear to the ear as reinforcements of the compound tones of the tonic, dominant, and subdominant, which tones are themselves connected by the closest possible relationship of Fifths. Hence in this mode everything can be reduced to the closest musical relationship in existence. And since the tonic chord in this case represents the compound tone of the tonic immediately and completely, the two conditions — predominance of the tonic tone and of the tonic chord — go hand in hand, without the possibility of any contradiction, or the necessity of making any changes in the scale.

The major mode has, therefore, the character of possessing the most complete melodic and harmonic consistency, combined with the greatest simplicity and clearness in all its relations. Moreover, its predominant chords being major, are distinguished by full unobscured harmoniousness, when such positions are selected for them as do not introduce inappropriate combinational tones.

The major scale is purely diatonic, and possesses the ascending leading note of the major Seventh, whence it results that the tone most distantly related to the tonic is brought into closest melodic connection with it.

The three predominant major chords furnish tones sufficient to produce two minor chords, which are closely related to them, and can be employed to diversify the succession of major chords.

The minor mode is in many respects inferior to the major. The chain of chords for its modern form is —

$$ \begin{gathered} \mathrm{\overbrace{f-a^1\flat+C}\overbrace{-E^1\flat+G}+b_1-d}\\ \hspace{0mm}|\underline{\hphantom{^1\flat+C1-E^1}}|\hspace{7mm}|\underline{\hphantom{+Bb-d}}| \end{gathered} $$

Minor chords do not represent the compound tone of their root as well as the major chords; their Third, indeed, does not form any part of this compound tone. The dominant chord alone[13] is major, and it contains the two supplementary tones of the scale. Hence when these appear as constituents of the dominant triad, and therefore of the compound tone of the dominant, they are connected with the tonic by the close relationship of Fifths. On the other hand, the tonic and subdominant triads do not simply represent the compound tones of the tonic and subdominant notes, but are accompanied by Thirds which cannot be reduced to the close relationship of Fifths. The tones of the minor scale can therefore not be harmonised in such a way as to link them with the tonic note by so close a relationship as in the major mode.

The conditions of tonality cannot be so simply reconciled with the predominance of the tonic chord as in the major mode. When a piece concludes with a minor chord, we hear, in addition to the compound tone of the tonic note, a second compound tone which is not a constituent of the first. This accounts for the long hesitation of musical composers respecting the admissibility of a minor chord in the close.

The predominant minor chords have not the clearness and unobscured harmoniousness of the major chords, because they are accompanied by combinational tones which do not fit into the chord.

The minor scale contains an interval \(a^1\flat...b_1\) which exceeds a whole Tone in the diatonic scale,[14] and answers to the numerical ratio 75 : 64 [= 274 cents]. To make the minor scale melodic it must have a different form in descending from what it has in ascending, as mentioned in the last chapter.

The minor mode, therefore, has no such simple, clear, intelligible consistency as the major mode; it has arisen, as it were, from a compromise between the different conditions of the laws of tonality and the interlinking of harmonies. Hence it is also much more variable, much more inclined to modulations into other modes.

This assertion that the minor system is much less consistent than the major, will be combated by many modern musicians, just as they have contested the assertion already made by me, and by other physicists before me, that minor triads are generally inferior in harmoniousness to major triads. There are many eager assurances of the contrary in recent books on the theory of harmony.[15] But the history of music, the extremely slow and careful development of the minor system in the sixteenth and seventeenth centuries, the guarded use of the minor close by Handel, the partial avoidance of a minor close even by Mozart, — all these seem to leave no doubt that the artistic feeling of the great composers agreed with our conclusions.[16] To this must be added the varied use of the major and minor Seventh, and the major and minor Sixth of the scale, the modulations rapidly introduced and rapidly changing, and finally, but very decisively, popular custom. Popular melodies can contain none but clear transparent relations. Look through collections of songs now preferred by those classes among the Western nations which have often an opportunity of hearing harmonic music, as students, soldiers, artisans. There are scarcely one or two per cent, in minor keys, and those are mostly old popular songs which have descended from the times of homophonic music. It is also characteristic that, as I have been assured by an experienced teacher of singing, pupils of only moderate musical talent have much more difficulty in hitting the minor than the major Third.

But I am by no means of opinion that this character depreciates the minor system. The major mode is well suited for all frames of mind which are completely formed and clearly understood, for strong resolve, and for soft and gentle or even for sorrowing feelings, when the sorrow has passed into the condition of dreamy and yielding regret. But it is quite unsuited for indistinct, obscure, unformed frames of mind, or for the expression of the dismal, the dreary, the enigmatic, the mysterious, the rude, and whatever offends against artistic beauty; — and it is precisely for these that we require the minor mode, with its veiled harmoniousness, its changeable scale, its ready modulation, and less intelligible basis of construction. The major mode would be an unsuitable form for such purposes, and hence the minor mode has its own proper artistic justification as a separate system.

The harmonic peculiarities of the modern keys are best seen by comparing them with the harmonisation of the other ancient tonal modes.

Major Mode.

Among the melodic tonal modes the Lydian of the Greeks (the ecclesiastical Ionic [p. 274, note No. 1]), in agreement with our major, is the only one which has an ascending leading note in the form of a major Seventh. The four others had originally and naturally only minor Sevenths, which even in the later periods of the middle ages began to give place to major Sevenths, in order that the Seventh of the scale, which was in itself so loosely connected with the tonic, might be more closely united to it by becoming the leading note to the tonic at the close.

Mode of the Fourth.

The mode of the Fourth (the Greek Ionic, and ecclesiastical Mixolydian) is principally distinguished from the major mode by its minor Seventh. By merely changing this into the major we obliterate the difference between them. Taking \(C\) as the tonic the chain of chords in the unaltered mode are as on p. 298, No. 2,

$$ \begin{gathered} \mathrm{f+\overbrace{a_1-C+E_1}-\overbrace{G-b^1\flat+d}}\\ \hspace{-16mm}|\underline{\hphantom{f+a_1-}}||\underline{\hphantom{+E_1+-}}| \end{gathered} $$

If we attempt to form a complete cadence in this mode, as in the following examples 1 and 2, they will sound dull from want of the leading note, even when the dominant chord is extended to a chord of the Seventh \(g-b^1\flat+d|f\), as in 2.


The second example, in which the leading note \(b^1\flat\) lies uppermost, is even duller than the first example, in which that note \(b^1\flat\) is more concealed. The \(b^1\flat\) in these examples has a very uncertain sound. It is not closely enough related to the tonic, it is not part of the compound tone of the dominant note g, it is not sufficiently close in pitch to serve as a leading note to the tonic, and it has no tendency to push on to the tonic. Hence when the older composers wished to distinguish pieces written in the mode of the Fourth from those in the major mode, by their close, they employed the imperfect or plagal cadence, as in example 3. And as such a cadence wants the decisive progression required for a close, the sluggishness previously caused by the absence of a leading tone ceases to be striking.[18]

In the course of a piece written in this tonal mode, the leading note \(b_1\) may of course be used in ascending passages, provided the minor Seventh \(b^1\flat\) is employed often enough in descending passages. But the effect of the mode is destroyed when an essential tone of the scale is changed at the close. Hence pieces in the mode of the Fourth sound like pieces in a major mode which have a decided inclination to modulate into the major mode of the subdominant.[19] For reasons already given, transition to the subdominant appears to be less active than transition to the dominant. This tonal mode has also no decided progression at the close, whereas major chords, of which the tonic is one, predominate in it owing to their greater harmoniousness. The mode of the Fourth is consequently as soft and harmonious as the major mode, but it wants the powerful forward impetus of major movement. This agrees with the character assigned to it by Winterfeld.[20] He describes the ecclesiastical Ionic (major) mode, as a scale which 'strictly self-contained and founded on the clear and bright major triad — a naturally harmonious and satisfactory fusion of different tones, — also bears the stamp of bright and cheerful satisfaction'. On the other hand, the ecclesiastical Mixolydian (mode of the Fourth) is a scale 'in which every part by sound and movement hastens to the source of its fundamental tone' (that is, to the major mode of its subdominant), 'and this gives it a yearning character in addition to the former cheerful satisfaction, not unlike to the Christian yearning for spiritual regeneration and redemption, and return of primitive innocency, though softened by the bliss of love and faith'.

Mode of the minor Seventh.

The mode of the minor Seventh (Greek Phrygian [p. 274, note, No. 4] ecclesiastical Doric) has a minor chord on \(c\) as the tonic, and originally another on \(g\) as the dominant, while it has a major chord on its subdominant \(f\), and this last chord distinguishes the mode from the mode of the minor Third (Eolic [p. 294, note, No. 3]); thus

$$ \begin{gathered} \mathrm{f+a_1-\overbrace{C-E^1\flat+G}\overbrace{-b^1\flat+d}}\\ \hspace{-8mm}|\underline{\hphantom{a^1\flat+ccc}}|\hspace{6mm}|\underline{\hphantom{^1\flat+lGb^1}}| \end{gathered} $$

Both of these modes of the minor Seventh and minor Third may, without destroying their character, change the minor Seventh \(b^1\flat\) into a leading note \(b_1\) and our minor mode is a fusion of both. The ascending minor scale belongs to the mode of the minor Seventh, in which the leading note is used, and the descending to the mode of the minor Third. But when the mode of the minor Seventh admits the leading note, its chain of chords reduces to the three essential chords of the scale

$$ \begin{gathered} \mathrm{f+a_1-\overbrace{C-E^1\flat+G}+b_1-d}\\ |\underline{\hphantom{----j}}|\hspace{13mm}|\underline{\hphantom{----j}}| \end{gathered} $$

This tonal mode has all the character of a minor, but the transition to the chord of the subdominant has a brighter effect than in the normal minor, where the subdominant chord is also minor. On forming the complete cadence both dominant and subdominant chords are major, while the tonic remains minor. This has of course an unpleasant effect in the close, because it makes the final Chord obscurer than either of the other two principal chords. Hence it is necessary to introduce strong dissonances into these two chords, to restore the balance. But if we follow the old composers and make the final chord major, we give the closing cadence of this mode an unmistakably major character. As in ecclesiastical modes it is always allowed to change \(A_1\) into \(A^1\flat\) which would change the subdominant chord of the mode of the minor Seventh into a minor chord,[21] we can protect the mode of the minor Seventh from confusion with the major mode in its final cadence, but then again it will entirely coincide with the old minor cadence.

Sebastian Bach introduces the major Sixth of the tonic, which is peculiar to this tonal mode, into other chords for the closing cadence, and thus avoids the major triad on the subdominant. He very usually employs the major Sixth as the Fifth of the chord of the Seventh on the Second of the scale,[22] as in the following examples. No. 1 is the conclusion of the chorale: Was mein Gott will, das gescheh' allzeit, in the St. Matthew Passion-Music. No. 2 is the conclusion of the hymn Veni redemptor gentium, at the end of the cantata: Schwingt freudig Euch empor zu den erhabenen Sternen. In both the tonic is \(b_1\), the major Sixth \(g_2\sharp\).[23]

There are many similar examples. He evidently evades a regular close.

Minor-Major Mode.

Modern composers, when they wish to insert a tonal mode which lies between Major and Minor, to be used for a few phrases or cadences, have generally preferred giving the minor chord of the mode to the subdominant and not to the tonic. Hauptmann calls this the minor-major mode (Moll-Dur tonart)[24] Its chain of chords is —

$$ \begin{gathered} \mathrm{\overbrace{f-a^1\flat+C}+\overbrace{E_1-G+b_1}-d}\\ \hspace{17mm}|\underline{\hphantom{----}}||\underline{\hphantom{----.}}| \end{gathered} $$

This gives a leading note in the dominant chord, and a complete final cadence in the major chord of the tonic, while the minor relation of the subdominant chord remains undisturbed. This minor-major mode is at all events much more suitable for harmonisation than the old mode of the minor Seventh. But it is unsuitable for homophonic singing, unless in the ascending scale \(a^1\flat\) is changed into \(a_1\), because the voice would otherwise have to make the complicated step \(a^1\flat ...b_1\)[ = 274 cents, see p. 301. The old modes were derived from homophonic singing, for which the mode of the minor Seventh is perfectly well fitted, as we know from its being still used as our ascending minor scale.[25]

Mode of the Minor Sixth

While the mode of the minor Seventh oscillates indeterminately between major and minor without admitting of any consistent treatment, the mode of the minor Sixth (Greek Doric[p. 274, note No. 7], ecclesiastical Phrygian), with its minor Second, has a much more peculiar character, which distinguishes it altogether from all other modes. This minor Second stands in the same melodic connection to the tonic as a leading note would do, but it requires a descending progression. Hence for descending passages this mode possesses the same melodic advantages as the major mode does for ascending passages. The minor Second has the more distant relationship with the tonic, due entirely to the subdominant. The mode cannot form a dominant chord without exceeding its limits. If we keep \(c\) as the tonic, the chain of chords is

$$ \begin{gathered} \text{5) Mode of The Minor Sixth}\hspace{45mm}\\ \mathrm{d^1\flat+\overbrace{f-a^1\flat+}\overbrace{C-E^1\flat+G}-b^1\flat}\\ \hspace{0mm}|\underline{\hphantom{^1\flat+f-}}||\underline{\hphantom{^1\flat-+C}}||\underline{\hphantom{^1\flat+G-b^1}}| \end{gathered} $$ [26]

In this case the chords \(b\flat-d^1\flat+f\) and \(d^1\flat+f-a^1\flat\) are not directly related to the tonic. The tone \(d^1\flat\) cannot enter into any consonant chord which is directly related to the tonic. But since \(d^1\flat\) is the characteristic minor Second of the mode, such chords cannot well be avoided, not even in the cadence. Although, then, there is a close relationship between the consecutive links of the chain of chords, some of its indispensable terms are only distantly related to the tonic. Moreover, in the course of a piece in this mode, it will always be necessary to form the dominant chord \(g+b_1-d\) [27] although it contains two tones foreign to the original mode, as otherwise we could not prevent the prevalence of the impression that \(f\) is the tonic and \(f-a^1\flat+c\) the tonic chord. It follows, therefore, that the mode of the minor Sixth must be still less consistent in its harmonisation and still more loosely connected than the minor mode, although it admits of very consistent melodic treatment. It contains three essential minor chords, namely the tonic \(c-e^1\flat+g\), the subdominant \(f-a^1\flat+c\), and the chord which contains the two tones slightly related to the tonic \(b\flat-d^1\flat+f\). It is exactly the reverse of the major mode, for whereas that mode proceeds towards the dominant, this mode proceeds towards the subdominant.

$$ \begin{alignedat}{14} &\text{Major:} &&f + a_1 &-&C + E_1 &-&G + b_1 -d\\ & &&\;| &&\;| &&\;|\\ &\text{Mode of minor Sixth:} \quad b\flat - d^1\flat &+&f - a^1\flat &+&C -E^1\flat &+& G \end{alignedat} $$

For harmonisation the difference of the two cases is, first, that the related tones introduced into the scale by the subdominant \(f\), namely \(b\flat\) and \(d^1\flat\), are not partials of the compound tone of the subdominant, whereas tones \(b_1\) and \(d\), which are introduced by the dominant, are some of the partials of the tonic; and, secondly, that the tonic chord always lies on the dominant side of the tonic tone. Hence in the harmonic connection, the tones \(b\flat\) and \(d^1\flat\) cannot be so closely united with either the tonic tone or the tonic chord, as is the case with the supplementary tones introduced by the dominant. This gives a kind of exaggerated minor character to the mode of the minor Sixth, when harmonised. Its tones and chords are certainly connected, but much less clearly and intelligibly than those of the minor system. The chords which can be brought together in this key, without obscuring reference to \(c\) as the tonic, are \(b\flat\) minor and \(d^1\flat\) major on the one hand and \(g\) major on the other, chords which in the major system could not be brought together without extraordinary modulational appliances.[28] The esthetical character of the mode of the minor Sixth corresponds with this fact. It is well suited for the expression of dark mystery, or of deepest depression, and an utter lapse into melancholy, in which it is impossible to collect one’s thoughts. On the other hand, as its descending leading note gives it a certain amount of energy in descent, it is able to express earnest and majestic solemnity, to which the concurrence of those major chords which are so strangely connected gives a kind of peculiar magnificence and wondrous richness.

Notwithstanding that the mode of the minor Sixth has been rejected from modern musical theory, much more distinct traces of its existence have been left in musical practice than of any other ancient mode; for the mode of the Fourth has been fused into the major, and the mode of the minor Seventh into the minor. Certainly a mode like that we have described is not suitable for frequent use; it is not closely enough connected for long pieces, but its peculiar power of expression cannot be replaced by that of any other mode. Its occurrence is generally marked by its peculiar final cadence which starts from the minor Second in the root. In Handel the natural cadence of this system is used with great effect. Thus in the Messiah, the magnificent fugue And with his stripes we are healed, which has the signature of \(F\) minor, but by its frequent use of the harmony of the dominant Seventh on \(G\), shews that \(C\) is the real tonic, introduces the pure [ecclesiastical Phrygian] Doric cadence as follows[29]:

Similarly in Samson[30], the chorus, Hear, Jacob's God, which, written in the Doric mode of \(E\), finely characterises the earnest prayer of the anxious Israelites as contrasted with the noisy sacrificial songs of the Philistines in \(G\) major, which immediately follow. The cadence here also is purely Doric.[31]

The chorus of Israelites which introduces the third part: In Thunder come, O God, from heaven! and is chiefly in \(A\) minor, has likely an intermediate Doric section.

Sebastian Bach also, in the chorales which he has harmonised, has left them in the mode of the minor Sixth, to which they melodically belonged, whenever the text requires a deeply sorrowful expression, as in the De Profundis or the Aus tiefer Noth schrei’ ich zu dir, and again in Paul Gerhardt’s song, Wenn ich einmal soll scheiden, so scheide nicht von mir. But he has harmonised the same melody arranged for other texts, as Befiehl Du deine Wege, and O Haupt von Blut und Wunden, &c., as major or minor, in which case the melody ends on the Third or Fifth of the key, instead of on the Doric tonic.

Fortlage[32] had already observed that Mozart had applied the Doric mode in Pamina’s air in the second act of Il Flauto Magico [No. 19], One of the finest examples for the contrast between this and the major mode occurs in the same composer’s Don Giovanni, in the Sestette of the second act [No. 21], where Ottavio and Donna Anna enter. Ottavio sings the comforting words —

$$ \begin{gathered} \text{Tergi il ciglio, o vita mia,}\\ \text{E dii calma al tuo dolore} \end{gathered} $$

in \(D\) major, which, however, is peculiarly coloured by a preponderating, although not uninterrupted, inclination to the subdominant, as in the mode of the Fourth. Then Anna, who is plunged in grief, begins in perfectly similar melodical phrases, and with a similar accompaniment, and after a short modulation through \(D\) minor, establishes herself in the mode of the minor Sixth for \(C\), with the words —

$$ \begin{gathered} \text{Sol la morte, o mio tesoro,}\\ \text{Il mio pianto pud finir.} \end{gathered} $$

The contrast between gentle emotion and crushing grief is here represented with a most wonderfully beautiful effect, principally by the change of mode. The dying Commandant also, in the introduction to Don Giovanni, ends with a Doric cadence.[33] Similarly the Agnus Dei of Mozart's Requiem — although, of course, we are not quite certain how much of this was written by himself.

Among Beethoven’s compositions we may notice the first movement of the Sonata, No. 90, in \(E\) minor, for the pianoforte, as an example of peculiar depression caused by repeated Doric cadences, whence the second (major) movement acquires a still softer expression.

Modern composers form a cadence which belongs to the mode of the minor Sixth, by means of the minor Second and the major Seventh, the so-called chord of the extreme sharp Sixth,[34], \(f^1+a...d_1\sharp\), where both \(f^1\) and \(d_1\sharp\) have to move half a tone to reach the tonic \(e\) [p. 286b]. This chord cannot be deduced from the major and minor modes, and hence appears very enigmatical and inexplicable to many modern theoreticians. But it is easily explained as a remnant of the old mode of the minor Sixth, in which the major Seventh \(d_1\sharp\), which belongs to the dominant chord \(b+d_1\sharp-f\sharp\), is combined with the tones \(f^1+a\), which are taken from the subdominant side.[35]

These examples may suffice to shew that there are still remnants of the mode of the minor Sixth in modern music. It would be easy to adduce more examples if they were looked for. The harmonic connection of the chords in this mode is not sufficiently firm and intelligible for the construction of long pieces. But in short pieces, chorales, or intermediate sections, and melodic phrases in larger musical works, it is so effective in its expression, that it should not be forgotten, especially as Handel, Bach, and Mozart have used it in such conspicuous places in their works.[36]

Similar relations exist for the mode of the Fourth and of the minor Seventh, although these are less specifically different from the major and minor modes respectively. They are, however, capable of giving a peculiar expression to certain musical periods, although difficulties would arise in consistently carrying out these peculiarities through long pieces of music. The harmonic phrases which belong to these two last-named modes can of course also be executed within the usual major and minor systems. But perhaps it would facilitate the theoretical comprehension of certain modulations, if the conception of these modes and of their system of harmonisation were definitely laid down.

The only point, then, as historical development and physiological theory alike testify, for which modern music is superior to the ancient, is harmonisation. The development of modern music has been evoked by its theoretical principle, that the tonic chord should predominate among the series of chords by the same laws of relationship as the tonic note predominates among the notes of the scale. This principle did not become practically effective till the commencement of last century, when it was felt necessary to preserve the minor chord in the final cadence.

The physiological phenomenon which this esthetical principle brought into action, is the compound character of musical tones which are of themselves chords composed of partials, and consequently, conversely, the possibility under certain circumstances of replacing compound tones by chords. Hence in every chord the principal tone is that of which the whole chord may be considered to express its compound form. Practically this principle was acknowledged from the time that pieces of music were allowed to end in chords of several parts. Then it was immediately felt that the concluding tone of the bass might be accompanied by a higher Octave, Fifth, and, finally, major Third, but not by a Fourth, or minor Sixth, and for a long time also the minor Third was rejected; and we know that the first three intervals (the Octave, Fifth, and major Third) occur among the partials of the compound tone which lies in the bass, and that the others do not.

The various values of the tones of a chord were first theoretically recognised by Rameau in his theory of the fundamental bass, although Rameau was not acquainted with the cause here assigned for these different values. That compound tone which represents a chord according to our view, constitutes its Fundamental Bass, Radical Tone or Root, as distinguished from its bass, that is, the tone which belongs to the lowest part. The major triad has the same root whatever be its inversion or position. In the chords \(c+e_1-g\), or \(g...c+e_1\) the root is still \(c\). The minor chord \(d-f^1+a\) has also as a rule only \(d\) as its root in all its inversions, but in the chord of the great [or added] Sixth \(f^1+a...d^1\) we may also consider \(f^1\) as the root, and it is in this sense that it occurs in the cadence of \(c^1\) major. Rameau’s successors have partly given up this last distinction; but it is one in which Rameau’s fine artistic feeling fully corresponded with the facts in nature. The minor chord really admits of this double interpretation, as we have already shewn (p. 294d).

The essential difference between the old and new tonal modes is this: the old have their minor chords on the dominant, the new on the subdominant side.

The reasons for the following construction have been already investigated.[37]

In the The chord of the
Subdominant is Tonic is Dominant is
Old Mode of the minor Third Minor Minor Minor
Mode of the minor Seventh Major Minor Minor
Mode of the Fourth Major Major Minor
  Major Mode Major Major Major
New Minor-Major Mode Minor Major Major
Minor Mode Minor Minor Major
[1][Of course when the modes are thus reduced to harmonic combinations, the tones of the old modes, as given in footnote to p. 268c, are all altered, and become those in footnote to p. 274c. See also App. XX. sect. E. arts. 9 and 10.— Translator.]
[2][This remark does not apply to old English music.— Translator.]
[3] [Taking only six partials, we have for —

Compound Tones Simple Partial Tones
1 2 3 4 5 6
\(C\) \(C\) \(c\) \(g\) \(c'\) \(e_1'\) \(g'\)
\(E_1\) \(E_1\) \(e_1\)


\(e_1'\) \(g_2'\sharp\) \(b'\)
\(G_1\) \(G\) \(g\) \(d'\) \(g'\) \(b_1'\) \(d"\)
\(E^1\flat\) \(E^1\flat\) \(e^1\flat\) \(b^1\flat\) \(e^1 \hspace{-2pt}' \flat\) \(g'\) \(b^1 \hspace{-2pt}'\flat\)
[4][The scale of \(B^1\flat\) major has the chords \(e^1\flat+g-b^1\flat+d-f^1+a-c^1\); hence, regarding the chord as made up of the notes of this scale, it would be \(c^1\space|\space e^1\flat+g\), which is not a minor chord at all, like \(c-e^1\flat+g\), because it has a Pythagorean in place of a just minor Third. It was only tempered intonation which confused the two cases. Attention will be hereafter drawn to this important distinction, see p. 299a. — Translator.]
[5][Transposing the \(c^1\) the chord becomes \(c^1\space | \space e^1\flat+g-b^1\flat\) so that we have a major chord with the Sixth of its root added, that is, the subdominant of the key of rendered dissonant by introducing \(c^1\), the Second of the key, or the Sixth above the subdominant \(c^1\flat\). Observe that it is \(c^1\) which is now introduced in the text, in place of \(c\). If \(c\) is retained, thus \(c^1\space | \space e^1\flat+g-b^1\flat\), the chord is one of those chords of the Seventh considered in Chapter XVI. — Translator.]
[6]Harmonik und Metrik, Leipzig, 1853, p. 216.
[7]See an example in Anton Brumel, in Forkel’s Geschichte der Musik, vol. ii. p. 647. Another, with a plagal cadence by Josquin, will be found, ibid. p. 550, where the voices might have easily been led to the minor Third.
[8][These scales differ from those transcribed in pp. 293d and 294d, only in the addition of the secondary forms of intercalary tones, \(d_1\), \(b\flat\),or \(b^1\flat\), which, in fact, imply modulations into adjacent modes, or else give a double and ambiguous character to each mode, as shewn on pp. 277, footnote 59, and by referring to the Duodenarium, App. XX. sect. E. art. 18, it will be seen that there is a real change of duodene, which always must happen when changes of a comma occur. — Translator.]
[9][The first four are the same as in pp. 293d and 294d. The settlement in the text avoids the double modality alluded to in the last note, and fixes the modes in the meanings of App. XX. sect. E. art. 9, as
(1) 1 C
(2) 1 C
(3) 1 C ma.mi.mi.
(4) 1 C mi.mi.mi.
(5) 3 \(A^1\flat\),
In the last scale it is more usual, however, to take \(b\flat\) in the place of \(b^1\flat\), which makes the scale \(b\flat-d^1\flat+f-a^1\flat+C-E^1\flat+G=5F\) mi.mi.mi. But temperament obscures all these differences. — Translator.]
[10][See p. 295, note 5. — Translator.]
[11][This was referred to in p. 294, note 4. Sec App. XX. sect. E. art. 26, example of the use of duodenals. It is a real, though temporary modulation into a new duodene, one Fifth lower. But for \(D \space | \space F+A_1\), we might use \(D-F^1+A\), which is again a modulation into a new duodene, one Fifth higher. This should be traced on the Duodenarium. — Translator.]
[12][Hypodoric, also called Eolic, p.268d, footnote No. 6, but here the harmonic alteration of that mode is meant as in p. 274, footnote No. 3. This confusion is here regular and intentional. — Translator.]
[13][That is, among the characteristic chords. text, contain the tones of one major chords. The two minor chords, as is shewn in the text, contain the tones of one major chord, \(a^1\flat+c-e^1\flat\). — Translator.]
[14][The interval is so strange, when unaccompanied, that if it had to be taken merely as an interval, \(a^1\flat 274b_1\), a singer would probably fail. But the \(a^1\flat\) is taken as the minor Third of \(f\) with ease, and the \(b_1\) is taken as the leading note to \(c'\), with equal ease, so that the perfectly unmelodic and inharmonic interval \(a^1\flat 274b_1\) never comes into consideration at all. To get rid of it, the subdominant is often taken major, producing the chords of 1 C, App. XX. sect. E. art. 10, III., which makes the scale \(c\;204\;d\;ll2\;e^1\;182\;f\;204\;g\;l82\;a_1\;204\;b_1 \;112 \;c'\), and this differs from the major only by having \(e^1\flat\) in place of \(e_1\). In many pianoforte instruction books this is given as the only form of the ascending minor. Mr. Curwen (Standard Course, p. 86) says that this major Sixth 'ascending is very difficult to sing,' and 'has a hard and by no means pleasant effect,' and points out that it leads singers to forget the key, and in such a phrase as \(g a_1 b_1 c' d' e^{1}\text{'}\flat \), the pupils will sing \(e'\), instead of \(e_1'\); and even in singing such a passage as \(g\; a_1\; b_1\; c'\;g\), instead of falling upon the same note with which they began, will take \(e_1'\) the major Third of \(c'\). Hence the difficulty is not avoided but increased by introducing the ambiguity of the major key, into which this is a real modulation from \(g\) onwards. — Translator.]
[15][Can this be due to temperament? The sharp equally tempered major Third of 400 cents is worse of its kind than the flat equally tempered minor Third of 300 cents, which approaches close to 16 : 19 = 298 cents, an interval which many like, and which may be tried as \(c'''\space|\space^{:9}e'''\) on the Harmonical. — Translator.]
[16][These composers played in meantone temperament (App. XX. sect. A. art. 16), in which the minor Third of 310 cents was much rougher than the equally tempered one of 300 cents, having much slower beats. Possibly this difference in the modes of tempering the minor Third, may have led to the difference of opinion mentioned in the text. — Translator.]
[17][The [C] is the duodenal of App. XX. sect. E. art. 26, shewing the exact pitch of all the notes. These examples have been transposed to admit of their being played on the Harmonical. — Translator.]
[18][These can be played on the Harmonical. — Translator.]
[19][This inclination seems to arise from the tempered confusion of \(b^1\flat d\) with \(b\flat d_1\) so that the scale \(c\;d\;d\;e_1\;f\;g\;a_1\;b^1\flat\;c'\) becomes confused with its subdominant \(f\;g\;a_1\;b\flat\;c'\;d'_1\;e'_1\;f'\). — Translator.]
[20]Johannes Gabrieli und sein Zeilalter, vol. i. p. 87.
[21][In the original the scale was \(g+b_1-D-F'+A+C_1\sharp-e\) in order that it might run from D to d; and hence the statement was that it is allowable to change \(B\) into \(B^1\flat\). But in order to keep to the same notes as were used previously, and to allow of the scale being played on the Harmonical, I have transposed it, and hence have had to make the same change here. The result is precisely the same, merely meaning that the Seventh might be taken minor. — Translator.]
[22][In the scale \(f+a_1-c-e^1\flat+g+b_1-d\), \(a_1\) is the major Sixth of the tonic c and d the Second. The chord of the Seventh on the Second of the scale is therefore \(d+f_1\sharp-a\space|\space\), chords are in the duodene of \(B_1\). Then [\(E_1\)] shews that the next two chords are in the duodene of \(E_1\). The difference relates to the chords with A in the first case and \(A_1\) in the second. But the next pair of chords return to the duodene of \(B_1\), which remains till the last bar, when the notes are in the duodene of \(F_1\sharp\). This is rendered necessary by the chord of the Seventh on \(C_1\sharp\) the second of the scale, the Fifth of which is \(G_1\sharp\) and not \(G_2\sharp\), which is the Sixth of the scale of \(B_1\). That is, it is \(C_1\sharp+E_2\sharp\) hence if S. Bach makes this Fifth a agree with the major Sixth of the scale \(a_1\), he is thinking in tempered music. When just intonation is restored, this occasions a restless modulation as shewn by the duodenals which I have introduced over the following examples. — Translator.}
Subdominant \(E_1\) Tonic \(B_1\) Dominant \(F_1\sharp\)
\(D\) \(F_1\sharp\) \(A_2\sharp\) \(A\) \(C_1\sharp\) \(E_2\sharp\) \(E\) \(G_1\sharp\) \(B_2\sharp\)
\(G\) \(B_1\) \(D_2\sharp\) \(D\) \(F_1\sharp\) \(A_2\sharp\) \(A\) \(C_1\sharp\) \(E_2\sharp\)
\(D\) \(E_1\) \(G_2\sharp\) \(G\) \(B_1\) \(D_2\sharp\) \(D\) \(F_1\sharp\) \(A_2\sharp\)
\(F\) \(A_1\) \(C_2\sharp\) \(C\) \(E_1\) \(G_2\sharp\) \(G\) \(B_1\) \(D_2\sharp\)
[23][The notes in the staff notation are the usual tempered scale, but the inserted duodenals convert them into just notes, on the principle of App. XX. sect. E. art. 26. The tonic is taken as \(B_1\) in order to be within the duodene of \(C\), and hence the subdominant is \(E_1\) and the dominant \(F_1\sharp\), giving the three duodenes: In Ex. 1 the [\(B_1\)] indicates that the first two - \(G_1\sharp|B_1\). This is, however, only a temporary modulation, and the piece ends in the duodene of \(B_1\). In Ex. 2 the modulations are only \(B_1.F_1\sharp\) and \(B_1\), that into \(F_1\sharp\) being necessitated by the same chord as before. If those modulations wore not taken, but the duodene of \(B_1\) were persisted in throughout, frightful dissonances (much worse than the old 'wolves') would ensue from the imperfect Fifths \(E_1A\) and \(C_1\sharp G_2\sharp\). — Translator.]
[24][It is 1 C of App. XX. sect. E. art. 9. — Translator.]
[25][After the introduction of the leading note to form a major dominant chord. — Translator
[26][The notes have been transposed in order to keep the same tonic chord \(C-E^1\flat+G\). Observe that both Sevenths \(b\flat\)and \(b^1\flat\) are introduced. If \(b^1\flat\) be omitted, the system of chords is that of 5F mi.mi.mi. On the Harmonical, on account of the absence of \(b\flat\), it is necessary to use the system of chords \(d_1-f+a_1-c+E_1-G+B_1-d\). — Translator.]
[27][The introduction of this chord shews that the composer is writing in the key of \(c\), but has a prevailing tendency to modulate into the subdominant, from which \(b\flat\), \(d^1\flat\) are chosen. When \(b^1\flat\) is used for \(b\flat\), or \(b_1\) for \(b^1\flat\) the modulation into the subdominant does not take place. The major chord \(e^1\flat+g-b^1\flat\) is entirely adventitious. If it is used in ascending, thus, \(c \; 112 \; d^1\flat \; 204 \; e^1\flat \; 182 \; f \; 204 \; g \; 112 \; a^1\flat \; 204 \; b^1\flat \; 182 \; c'\), the result is the scale of 3 \(A^1\flat\) of App. XX. sect. E. art. 9. — Translator.]
[28][This, in fact, lengthens the original chain of chords into \(b\flat-d^1\flat+f-a^1\flat+c-e^1\flat+g+b_1-d\), and leads to the treatment of the mode as merely \(C\) minor, with a tendency to modulate into \(F\) minor. The \(C\) minor is, however, the modern minor \(C\) mi. mi. ma., and the \(F\) minor is \(F\) mi. mi. mi., which is much more gloomy. — Translator.]
[29][The cadence is produced by passing from the minor subdominant \(B\flat-D^1\flat+F\) to the major dominant, \(C+E_l-G\), in the key of \(F\) minor. This is the concluding cadence of the whole fugue, and for this reason apparently, the signature in Novello’s edition is that of \(C\) minor, not of \(F\) minor, and the \(d^1\flat\), is marked as an accidental throughout. That is, Novello takes the key to be \(C\) minor with a constant tendency to modulate into the key of the subdominant, from which it borrows the chord \(B\flat-D^1\flat+F\). But the fugue begins with \(F...f\) in the bass, and the opening subject, in the treble, is \(c"\), \(a^1\text{'}\flat\),\(d^1\text{"}\flat\), \(e_1'\), \(f'\), \(g'\), \(a^1\text{'}\flat\), \(b'\flat\), \(c"\), which is clearly in the scale of \(F\) minor, with the chordal system \(b\flat-d^1\flat+f-a^1\flat+c+e_1-g\), of which it contains every note. In the text the [\(F\).] is the duodenal and refers to the duodene of \(F\), which contains all the tones in the passage. The whole fugue oscillates between the duodenes \(C\) and \(F\). — Translator.]
[30][Mr. H. Keatley Moore informs me that this chorus was taken by Handel from Plorate filiae Israel in Carissimi’s Jephthah. — Translator.]
[31][The duodene is that of \(A_1\). The succession of chords, each reduced to the simplest form, as referred to by the bracketed figures below the notes, is 1. \(e_1-g+b_1\), 2. \(a_1-c+e_1\), 3. \(e_1g+b_1\), 4. \(f+a_1-c\), 5. \(d_1-f+a_1\) 6. \(e_1+g_2\sharp-b\), 7.\(a_1-c+e_1\), 8. \(e_1+g_2\sharp-b_1\). Hence, assuming the scale to have the chordal system \(d_1-f+a_1-c+e_1-g+b_1\), with \(e_1-g+b_1\) as the tonic chord, taken major as \(e_1+g_2\sharp-b_1\), in the close, we have the 'Doric cadence' between chords 5 and 6, which is then lengthened by introducing the remaining tones of the key in 7, the whole closing as in 8. It would be most probably received as in \(A_1\) minor, closing in the dominant. — Translator.]
[32]Examples from instrumental music are mentioned by Ekert in his Habilitationsschrift Die Principien der Modulation und musikalischen Idee. Heidelberg, 1860, p. 12.
[33][No. 1. of the opera. Representing major chords by capitals and minor by small letters, the final chords of the vocal music are \(f\), \(D^1\flat\), \(G^1\flat\), \(f\), \(C\), \(f\), so that all the tones will lie in the scheme \(g^1\flat+b\flat-d^1\flat+f-a^1\flat+c+e_1-g\), or \(c-e^1\flat+g\). The tonic is \(F\). — Translator.]
[34][Callcott (Musical Grammar, 1809, art. 441) calls it 'the chord of the extreme sharp Sixth,' and says that ‘this harmony when accompanied simply by the Third, has been termed the Italian Sixth’. Of course he has no theory for it; the tone is 'accidentally sharpened'. — Translator.]
[35][That is the chords of the scale are taken as \(d-f^1+a-c^1+e-g^1+b+d\sharp-f\sharp\), of which the two notes last are modern additions. See p. 286d, note 72. — Translator.]
[36]Herr A. von Oettingen, in his Harmonic system in dualer Entwickelung (Dorpat and Leipzig, 1866), has carried out, in a most interesting manner, the complete analogy between the mode of the minor Sixth and the major mode, of which it is the direct conversion; and has shewn how this conversion leads to a peculiarly characteristic harmonisation of the mode of the minor Sixth. In this respect I wish emphatically to recommend this book to the attention of musicians. On the other hand, it seems to me that it is necessary to shew by musical practice, that the new principle, which is made the basis of that writer’s theory of the mode of the minor Sixth, considered by him as the theoretically normal minor mode, really suffices for the construction of great musical pieces. The author, namely, considers the minor triad \(c-e^1\flat+g\) as representing the tone \(g''\) which is common to the three compound tones of which it is composed (being a higher Octave of \(g\), of the Fifth of \(c\), and of the major Third of and hence calls it 'the phonic \(g\) tone,' whereas he considers \(c+e_1-g\) in the same way as we do, as the 'tonic \(c\) tone'.
[37][It will be seen that this arrangement does not include the mode of the minor Sixth. It was this tabulation which led me to the richordal theory development in App. XX. sect. E. art. 9, and thence to the general theory of duodenes in that section. - Translator.]
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