Chapter XII.


We have hitherto examined the effect of sounding together only two tones which form a determinate interval. It is now easy to discover what will happen when more than two tones are combined. The simultaneous production of more than two separate compound tones is called a chord. We will first examine the harmoniousness of chords in the same sense as we examined the harmoniousness of any two tones sounded together. That is, we shall in this section deal exclusively with the isolated effect of the chord in question, quite independently of any musical connection, mode, key, modulation, and so on. The first problem is to determine under what conditions chords are consonant, in which case they are termed concords. It is quite clear that the first condition of a concord is that each tone of it should form a consonance with each of the other tones; for if any two tones formed a dissonance, beats would arise destroying the tunefulness of the chord. Concords of three tones are readily found by taking two consonant intervals to any one fundamental tone as \(c\), and then seeing whether the new third interval between the two new tones, which is thus produced, is also consonant. If this is the case each one of the three tones forms a consonant interval with each one of the other two, and the chord is consonant, or is a concord.[1]

Let us confine ourselves in the first place to intervals which are less than an Octave. The consonant intervals within these limits, we have found to be : 1) the Fifth \(c \; g\), \(\tfrac32\) ; 2) the Fourth \(c \; f\), \(\tfrac43\) ; 3) the major Sixth \(c \; a\), \(\tfrac53\) ; 4) the major Third \(c \; e\), \(\tfrac54\) ; 5) the minor Third \(c \; e\flat\), \(\tfrac65\) ; 6) the minor Sixth \(c \; a\flat\), \(\tfrac85\) ; to which we may add 7) the subminor or natural Seventh \(c \; ^7b\flat\), \(\tfrac74\), which approaches to the minor Sixth in harmoniousness. The following table gives a general view of the chords contained within an Octave. The chord is supposed to consist of the fundamental tone \(C\), some one tone in the first horizontal line, and some one tone of the first vertical column. Where the line and column corresponding to these two selected tones intersect, is the name of the interval which these two latter tones form with each other. This name is printed in italics when the interval is consonant, and in Roman letters when dissonant, so that the eye sees at a glance what concords are thus produced. [Under the name, the equivalent interval in cents has been inserted by the Translator.]

\(G \tfrac32\)
\(F \tfrac43\)
\(A \tfrac53\)
\(E \tfrac54\)
\(E\flat \tfrac65\)
\(A\flat \tfrac85\)
\(G \tfrac32\)
\(F \tfrac43\)
\(A \tfrac53\)
\(E \tfrac54\)
\(E\flat \tfrac65\)
\(A\flat \tfrac85\)
\(^7B\flat \tfrac74\)

From this it follows that the only consonant triads or chords of three notes, that can possibly exist within the compass of an Octave are the following: —

1) \(C \; E \; G\)                           2) \(C \; E\flat \; G\)
3) \(C \; F \; A\)                            4) \(C \; F \; A\flat\)
5) \(C \; E\flat \; A\flat\)                        6) \(C \; E \; A\).

The two first of these triads are considered in musical theory as the fundamental triads from which all others are deduced. They may each be regarded as composed of two Thirds, one major and the other minor, superimposed in different orders. The chord \(C \; E \; G\), in which the major Third is below, and the minor above, is a major triad. It is distinguished from all other major triads by having its tones in the closest position, that is, forming the smallest intervals with each other. It is hence considered as the fundamental chord or basis of all other major chords. The triad \(C \; E\flat \; G\), which has the minor Third below, and the major above, is the fundamental chord of all minor triads.

The next two chords, \(C \; F \; A\) and \(C \; F \; A\flat\), are termed, from their composition, chords of the Sixth and Fourth, written \(6 \atop 4\) [\(C\) to \(F\) being a Fourth, and \(C\) to \(A\) a major, but \(C\) to \(A\flat\) a minor Sixth]. If we take \(G\), instead of \(C\) for the fundamental or bass tone, these chords of the Fourth and Sixth become \(G, \; C \; E\) and \(G, \; C \; E\flat\). Hence we may conceive them as having been formed from the fundamental major and minor triads \(C \; E \; G\) and \(C \; E\flat \; G\), by transposing the Fifth \(G\) an Octave lower, when it becomes \(G_{\prime}\).

The two last chords, \(C \; E\flat \; A\flat\) and \(C \; E \; A\), are termed chords of the Sixth and Third, or simply chords of the Sixth, written \(6 \atop 3\) [\(C\) to \(E\) being a major Third, and \(C\) to \(E\flat\) a minor Third; and \(C\) to \(A\) a major Sixth, and \(C\) to \(A\flat\) a minor Sixth], If we take \(E\) as the bass note of the first, and \(E\flat\) as that of the second, they become \(E \; G \; c\), \(E\flat \; G \; c\), respectively. Hence they may be considered as the transpositions or inversions of a fundamental major and a fundamental minor chord, \(C \; E \; G\), \(C \; E\flat \; G\), in which the bass note \(C\) is transposed an Octave higher and becomes \(c\).

Collecting these inversions, the six consonant triads will assume the following form [the numbers shewing their correspondence with the forms on p. 212d]: —

1) \(C \; E \; G\)                          2) \(C \; E\flat \; G\)
   3) \(\phantom{C \;} E \; G \; c\)                        4) \(\phantom{C \;} E\flat \; G \; c\)
          5) \(\phantom{C \; E \;} G \; c \; e\)                     6) \(\phantom{C \; Eb \;} G \; c \; e\flat\).

We must observe that although the natural or subminor Seventh \(^7B\flat\) forms a good consonance with the bass note \(C\), a consonance which is indeed rather superior than inferior to the minor Sixth \(C \; A\flat \), yet it never forms part of any triad, because it would make worse consonances with all the other intervals consonant to \(C\) than it does with \(C\) itself. The best triads which it can produce are \(C \; E \; ^7B\flat \; = 4 : 5 : 7\), and \(C \; G \; ^7B\flat \; = 4 : 6 : 7\). In the first of these occurs the interval \(E \; ^7B\flat \; = 5 : 7 \) (between a Fourth and Fifth,) in the latter the subminor Third \(G \; ^7B\flat \; = 6 : 7.\) [3] On the other hand the minor Sixth makes a perfect Fourth with the minor Third, so that this minor Sixth remains the worst interval in the chords of the Sixth and Third, and of the Sixth and Fourth, for which reason these triads can still be considered as consonant. This is the reason why the natural or subminor Seventh is never used as a consonance in harmony, whereas the minor Sixth can be employed, although, considered independently, it is not more harmonious than the subminor Seventh.

The triad \(C \; E \; A\flat\) to which we shall return [Chap. XVII. Dissonant Triads, No. 4], is very instructive for the theory of music. It must be considered as a dissonance, because it contains the diminished Fourth \(E \; A\flat \), having the interval ratio \(\tfrac{32}{25}\). Now this diminished Fourth \(E \; A\flat\) is so nearly the same as a major Third \(E \; G\sharp\), that on our keyed instruments, the organ and pianoforte, the two intervals are not distinguished. We have in fact $$ E \; A\flat \; = \; \tfrac{32}{25} \; = \; \tfrac54 \cdot \tfrac{128}{125} $$ or, approximatively

\((E \; A\flat) \; = \; (E \; G\sharp)\cdot\tfrac{43}{42}\)[4]

On the pianoforte it would seem as if this triad, which for practical purposes may be written either \(C \; E \; A\flat\) or \(C \; E \; G\sharp\), must be consonant, since each one of its tones forms with each of the others an interval which is considered as consonant on the piano, and yet this chord is one of the harshest dissonances, as all musicians are agreed, and as any one can convince himself immediately. On a justly intoned instrument [as the Harmonical] the interval \(E \; A\flat\) is immediately recognised as dissonant. This chord is well adapted for shewing that the original meaning of the intervals asserts itself even with the imperfect tuning of the piano, and determines the judgment of the ear.[5]

The harmonious effect of the various inversions of triads already found depends in the first place upon the greater or less perfection of the consonance of the several intervals they contain. We have found that the Fourth is less agreeable than the Fifth, and that minor are less agreeable than major Thirds and Sixths. Now the triad

\(C \; E \; G \;\) has a Fifth, a major Third, and a minor Third
\(E \; G \; c \;\) has a Fourth, a minor Third, and a minor Sixth
\(G \; c \; e\; \) has a Fourth, a major Third, and a major Sixth
\(C \; E\flat \; G \;\) has a Fifth, a minor Third, and a major Third
\(E\flat \; G \; c \;\) has a Fourth, a major Third, and a major Sixth
\(G \; c \; e\flat \;\) has a Fourth, a minor Third, and a minor Sixth

For just intervals the Thirds and Sixths decidedly disturb the general tunefulness more than the Fourths, and hence the major chords of the Sixth and Fourth are more harmonious than those in the fundamental position, and these again than the chords of the Sixth and Third. On the other hand the minor chords of the Sixth and Third are more agreeable than those in the fundamental position, and these again are better than the minor chords of the Sixth and Fourth. This conclusion will be found perfectly correct for the middle parts of the scale, provided the intervals are all justly intoned. The chords must be struck separately, and not connected by any modulation. As soon as modulational connections are allowed, as for example in a concluding cadence, the tonic feeling, which finds repose in the tonic chord, disturbs the power of observation, which is here the point of importance. In the lower parts of the scale either major or minor Thirds are more disagreeable than Sixths.

Judging merely from the intervals we should expect that the minor triad \(C \; E\flat \; G\) would sound as well as the major \(C \; E \; G\), as each has a Fifth, a major 11 and a minor Third. This is, however, far from being the case. The minor triad is very decidedly less harmonious than the major triad, in consequence of the combinational tones, which must consequently be here taken into consideration. In treating of the relative harmoniousness of the consonant intervals we have seen that combinational tones may produce beats when two intervals are compounded, even when each interval separately produced no beats at all, or at least none distinctly audible (pp. 200b-204b).

Hence we must determine the combinational tones of the major and minor triads. We shall confine ourselves to the combinational tones of the first order produced by the primes and the first upper partial tones. In the following examples the primes are marked as minims, the combinational tones resulting from these primes are represented by crotchets, those from the primes and first upper partials by quavers and semiquavers. A downward sloping line, when placed before a note, shews that it represents a tone slightly deeper than that of the note in the scale which it precedes.

1.) Major Triads with their Combinational Tones:[6]

2.) Minor Triads with their Combinational Tones:[7]

In the major triads the combinational tones of the first order, and even the deeper combinational tones of the second order (written as crotchets and quavers) are merely doubles of the tones of the triad in deeper Octaves. The higher combinational tones of the second order (written as semiquavers) are extremely weak, because, other conditions being the same, the intensity of combinational tones decreases as the interval between the generating tones increases, with which again the high position of these combinational tones is connected. I have always been able to hear the deeper combinational tones of the second order, written as quavers, when the tones have been played on an harmonium, and the ear was assisted by the proper resonators: [8] but I have not been able to hear those written with semiquavers. They have been added merely to make the theory complete. Perhaps they might be occasionally heard from very loud musical tones having powerful upper partials. But they may be certainly neglected in all ordinary cases.

For the minor triads, on the other hand, the combinational tones of the first order, which are easily audible, begin to disturb the harmonious effect. They are not near enough indeed to beat, but they do not belong to the harmony. For the fundamental triad, and that of the Sixth and Third [the two first chords], these combinational tones, written as crotchets, form the major triad \(A\flat \; C \; E\flat\), and for the triad of the Sixth and Fourth [the third chord], we find entirely new tones, \(A\flat, \; B\flat\), which have no relations with the original triad.[9] The combinational tones of the second order, however (written as quavers), are sometimes partly above and generally partly below the prime tones of the triad, but so near to them, that beats must arise; whereas in the corresponding major triads the tones of this order fit perfectly into the original chord. Thus for the fundamental minor triad in the example, \(c' \; e'\flat \; g'\), the deeper combinational tones of the second order give the dissonances \(a\flat \; b\flat \; c'\), and similarly for the triad of the Sixth and Third, \(e'\flat \; g' \; c''\). And for the triad of the Sixth and Fourth \(g' \; c'' \; e''\flat\) we find the dissonances \(b\flat \; c'\) and \(g' \; a'\flat\) . This disturbing action of the combinational tones on the harmoniousness of minor triads is certainly too slight to give them the character of dissonances, but they produce a sensible increase of roughness, in comparison with the effect of major chords, for all cases where just intonation is employed, that is, where the mathematical ratios of the intervals are preserved. In the ordinary tempered intonation of our keyed instruments, the roughness due to the combinational tones is proportionably less marked, because of the much greater roughness due to the imperfection of the consonances. Practically I attribute more importance to the influence of the more powerful deep combinational tones of the first order, which, without increasing the roughness of the chord, introduce tones entirely foreign to it, such as those of the \(A\flat\) and \(E\flat\) major triads in the case of the \(C\) minor triads. The foreign element thus introduced into the minor chord is not sufficiently distinct to destroy the harmony, but it is enough to give a mysterious, obscure effect to the musical character and meaning of these chords, an effect for which the hearer is unable to account, because the weak combinational tones on which it depends are concealed by other and louder tones, and are audible only to a practised ear.[10] Hence minor chords are especially adapted to express mysterious obscurity or harshness. [11] F. T. Vischer, in his Esthetics (vol. iii. § 772), has carefully examined this character of the minor mode, and shewn how it suits many degrees of joyful and painful excitement, and that all shades of feeling which it expresses agree in being to some extent 'veiled' and obscure.

Every minor Third and every Sixth when associated with its principal com binational tone, becomes at once a major chord. \(C\) is the combinational tone of the minor Third \(e' \; g' \) ;\(c\) of the major Sixth \(g \; e'\), and \(g\) of the minor Sixth \(e' \; c''\).[12] Since, then, these dyads naturally produce consonant triads, if any new tone is added which does not suit the triads thus formed, the contradiction is necessarily sensible.

Modern harmonists are unwilling to acknowledge that the minor triad is less consonant than the major. They have probably made all their experiments with tempered instruments, on which, indeed, this distinction may perhaps be allowed to be a little doubtful. But on justly intoned instruments [13] and with a moderately piercing quality of tone, the difference is very striking and cannot be denied. The old musicians, too, who composed exclusively for the voice, and were consequently not driven to enfeeble consonances by temperament, shew a most decided feeling for that difference. To this feeling I attribute the chief reason for their avoidance of a minor chord at the close. The medieval composers down to Sebastian Bach used for their closing chords either exclusively major chords, or doubtful chords without the Third; and even Handel and Mozart occasionally conclude a minor piece of music with a major chord. Of course other considerations, besides the degree of consonance, have great weight in determining the final chord, such as the desire to mark the prevailing tonic or key-note with distinctness, for which purpose the major chord is decidedly superior. More upon this in Chapter XV.

After having examined the consonant triads which lie within the compass of an Octave, we proceed to those with wider intervals. We have found in general that consonant intervals remain consonant when one of their tones is transposed an Octave or two higher or lower at pleasure, although such transposition has some effect on its degree of harmoniousness. It follows, then, that in all the consonant chords which we have hitherto found, any one of the tones may be transposed some Octaves higher or lower at pleasure. If the three intervals of the triad were consonant before, they will remain so after transposition. We have already seen how the chords of the Sixth and Third, and of the Sixth and Fourth, were thus obtained from the fundamental form. It follows further that when larger intervals are admitted, no consonant triads can exist which are not generated by the transposition of the major and minor triads. Of course if such other chords could exist, we should be able by transposition of their tones to bring them within the compass of an Octave, and we should thus obtain a new consonant triad within this compass, whereas our method of discovering consonant triads enabled us to determine every one that could lie within that compass. It is certainly true that slightly dissonant chords which lie within the compass of an Octave are sometimes rendered smoother by transposing one of their tones. Thus the chord \(1 \; : \; \tfrac76 \; : \; \tfrac74 \) , or \(C, \; ^7E\flat , \; ^7B\flat ,\) [14] is slightly dissonant in consequence of the interval \(1 \; : \; \tfrac76\); the interval \(1 \; : \; \tfrac74\), or subminor Seventh, does not sound worse than the minor Sixth; the interval \(\tfrac76 \; : \; \tfrac74\) is a perfect Fifth. Now transposing the tone \(^7E\flat\), an Octave higher to \(^7e\flat\), and thus transforming the chord into \(1 \; : \; \tfrac74 \; : \; \tfrac73\) we obtain \(1 \; : \; \tfrac73\), in place of \(1 \; : \; \tfrac76\), and this is much smoother, indeed it is better than the minor Tenth of our minor scale \(1 \; : \; \tfrac{12}{5}\), [15] and a chord thus composed, which I have had carefully tuned on the harmonium, although its unusual intervals produced a strange effect, is not rougher in sound than the worst minor chord, that of the Sixth and Fourth. This chord, \(C, \; ^7B\flat, \; ^7e\flat\), is also much injured by the unsuitable combinational tones \(G\), and \(F\).[16] Of course it would not be worth while to introduce such strange tones as \(^7B\flat, \; ^7e\flat\), into the scale for the sake of a chord which in itself is not superior to the worst of our present consonant chords, and for which the tones could not be transposed without greatly deteriorating its effect.[17]

The transposition of some tones in a consonant triad, for the purpose of widening their intervals, affects their harmoniousness in the first place by changing the intervals. Major Tenths, as we found in Chapter X. p. 195b, sound better than major Thirds, but minor Tenths worse than minor Thirds, the major and minor Thirteenth worse than the minor Sixth (p. 196a). The following rule embraces all the cases: — Those intervals in which the smaller of the two numbers expressing the ratios of the pitch numbers is EVEN, are IMPROVED by having one of their tones transposed by an Octave, because the numbers expressing the ratio are thus diminished.

The Fifth \(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 2 \; : \; 3 \;\;\) becomes the Twelfth \(\;\;\;\;\;\;\;\;\;\;\;\;\) \(2 \; : \; 6 = 1 \; : \; 3\)
The major Third \(\;\;\;\;\;\; 4 \; : \; 5 \;\;\) becomes the major Tenth \(\;\;\;\;\; 4 \; : \; 10 = 2 \; : \; 5\)
The subminor Third \(\; 6 \; : \; 7\) becomes the subminor Tenth \(\; 6 \; : \; 14 = 3 \; : \; 7\).

Those intervals in which the smaller of the two numbers expressing the ratio of the vibrational numbers is ODD, are MADE WORSE by having one of their tones transposed by an Octave, as the Fourth 3 : 4 [which becomes the Eleventh 3 : 8], the minor Third 5 : 6 [which becomes the minor Tenth 5 : 12], and the Sixths [major] 3 : 5, and [minor] 5 : 8 [which become the Thirteenths, major 3 : 10 and minor 5 : 16].

Besides this the principal combinational tones are of essential importance. An example of the first combinational tones of the consonant intervals within the compass of an Octave is given below, the primary tones being represented by minims and the combinational tones by crotchets, as before.[18]

The upwards sloping line prefixed to \(f''\) denotes a degree of sharpening of about a quarter of a Tone [53 cents]; and the downwards sloping line prefixed to \(b'\flat\) flattens it [by 27 cents] to the subminor Seventh of \(c\). Below the notes are added the names of the intervals, the numbers of the ratios, and the differences of those numbers, giving the pitch numbers of the several combinational tones.

We find in the first place that the combinational tones of the Octave, Fifth, Twelfth, Fourth, and major Third are merely transpositions of one of the primary tones by one or more Octaves, and therefore introduce no foreign tone. Hence these five intervals can be used in all kinds of consonant triads, without disturbing the effect by the combinational tones which they introduce. In this respect the major Third is really superior to the major Sixth and the Tenth in the construction of chords, although its independent harmoniousness is inferior to that of either.

The double Octave introduces the Fifth as a combinational tone. Hence if the fundamental tone of a chord is doubled by means of the double Octave, the chord is not injured. But injury would ensue if the Third or Fifth of the chord were doubled in the double Octave.

Then we have a series of intervals which are made into complete major triads by means of their combinational tones, and hence produce no disturbance in major chords, but are injurious to minor chords. These are the Eleventh, minor Third, major Tenth, major Sixth, and minor Sixth.

But the minor Tenth, and the major and minor Thirteenth cannot form part of a chord without injuring its consonance by their combinational tones.

We proceed to apply these considerations to the construction of triads.

1. Major Triads.

Major triads can be so arranged that the combinational tones remain parts of the chord. This gives the most perfectly harmonious positions of these chords. To find them, remember that no minor Tenths and no [major or minor] Thirteenths are admissible, so that the minor Thirds and [both major and minor] Sixths must be in their closest position. By taking as the uppermost tone first the Third, then 51 the Fifth, and lastly the fundamental tone, we find the following positions of these chords, within a compass of two Octaves, in which the combinational tones (here written as crotchets as usual) do not disturb the harmony.

The most Perfect Positions of Major Triads.[19]

When the Third lies uppermost, the Fifth must not be more than a major Sixth below, as otherwise a [major] Thirteenth would be generated. But the fundamental tone can be transposed. Hence when the Third is uppermost the only two positions which are undisturbed are Nos. 1 and 2. When the Fifth lies uppermost, the Third must be immediately under it, or otherwise a minor Tenth would be produced; but the fundamental tone may be transposed. Finally, when the fundamental tone is uppermost, the major Third can lie only in the position of a minor Sixth below it, but the Fifth may be placed at pleasure. Hence it follows that the only possible positions of the major chord which will be entirely free from disturbance by combinational tones, are the six here presented, among which we find the three close positions Nos. 2, 4, 6 already mentioned [p. 215a], and three new ones Nos. 1, 3, 5. Of these new positions two (Nos. 1, 3) have the fundamental tone in the bass, just as in the primary form, and are considered as open positions of that form, while the third (No. 5) has the Fifth in the bass, just as in the chord of the Sixth and Fourth [of which it is also considered as an open position]. The chord of the Sixth [and Third] (No. 6), on the other hand, admits of no opener position [if it is to remain perfectly free from combinational disturbance].

The order of these chords in respect to harmoniousness of the intervals is, perhaps, the same as that presented above. The three intervals of No. 1 (the Fifth, major Tenth, and major Sixth) are the best, and those of No. 6 (the Fourth, minor Third, and minor Sixth) are relatively the most unfavourable of the intervals that occur in these chords.

The remaining positions of the major triads present individual unsuitable combinational tones, and on justly intoned instruments are unmistakably rougher than those previously considered, but this does not make them dissonant, it merely puts them in the same category as minor chords. We obtain all of them which lie within the compass of two Octaves, by making the transpositions forbidden in the last cases. They are as follows, in the same order as before, No. 7 being made from No. 1, and so on: —

The less Perfect Positions of Major Triads.[20]

Musicians will immediately perceive that these positions of the major triad are much less in use. The combinational tone \(^7b'\flat\) gives the positions 7 to 10 something of the character of the chord of the dominant Seventh in the key of \(F\) major, \(c \; e \; g \; b\flat\). The two last, 11 and 12, are much the least pleasing; indeed they are decidedly rougher than the better positions of the minor chord.

2. Minor Triads.

No minor chord can be obtained perfectly free from false combinational tones, because its Third can never be so placed relatively to the fundamental tone, as not to produce a combinational tone unsuitable to the minor chord. If only one such tone is admitted, the Third and Fifth of the minor chord must lie close together and form a major Third, because in any other position they would produce a second unsuitable combinational tone. The fundamental tone and the Fifth must never be so placed as to form an Eleventh, because in that case the resulting combinational tone would make them into a major triad. These conditions can be fulfilled by only three positions of the minor chord, as follows: —

The most Perfect Positions of Minor Triads[21]

The remaining positions which do not sound so well are: —

The less Perfect Positions of Minor Triads.[22]

The positions Nos. 4 to 10 each produce two unsuitable combinational tones, one of which necessarily results from the fundamental tone and its [minor] Third; the other results in No. 4 from the Eleventh \(G \; c'\), and in the rest from the transposed major Third \(E\flat \; G\). The two last positions, Nos. 11 and 12, are the worst of all, because they give rise to three unsuitable combinational tones [two of which beat with original tones].

The influence of the combinational tones may be recognised by comparing the different positions. Thus the position No. 3, with a minor Tenth \(c' \; e''\flat\) and major Third \(e''\flat \; g''\), sounds unmistakably better than the position No. 7, with major Tenth \(e'\flat \; g''\) and major Sixth \(e'\flat \; c''\), although the two latter intervals when struck separately sound better than the two first. The inferior effect of chord No. 7 is consequently solely due to the second unsuitable combinational tone, \(b'\flat \).

This influence of bad combinational tones is also apparent from a comparison with the major chords. On comparing the minor chords Nos. 1 to 3, each of which has only one bad combinational tone, with the major chords Nos. 11 and 12, each of which has two such tones, those minor chords will be found really pleasanter and smoother than the major. Hence in these two classes of chords it is not the major and minor Third, nor the musical mode, which decides the degree of harmoniousness, it is wholly and solely the combinational tones.

Four Part Chords or Tetrads.

It is easily seen that all consonant tetrads must be either major or minor triads to which the Octave of one of the tones has been added.[23] For every consonant tetrad must admit of being changed into a consonant triad by removing one of its tones. Now this can be done in four ways, so that, for example, the tetrad \(C \; E \; G \; c\) gives the four following triads: —

\(C E G,\)      \(C E c,\)      \(E G c,\)      \(C G c.\)     

Any such triad, if it is not merely a dyad, or interval of two tones, with the Octave of one added, must be either a major or a minor triad, because there are no other consonant triads. But the only way of adding a fourth tone to a major or minor triad, on condition that the result should be consonant, is to add the Octave of one of its tones. For every such triad contains two tones, say \(C\) and \(G\), which form either a direct or inverted Fifth. Now the only tones which can be combined with \(C\) and \(G\) so as to form a consonance are \(E\) and \(E\flat \); there are no others at all. But \(E\) and \(E\flat \) cannot be both present in the same consonant chord, and hence every consonant chord of four or more parts, which contains \(C\) and \(G\), must either contain \(E\) and some of the Octaves of \(C, \; E, \; G,\) or else \(E\flat \) and some of the Octaves of \(C, \; E\flat , \; G\).

Every consonant chord of three or more parts will therefore be either a major or a minor chord, and may be formed from the fundamental position of the major and minor triad, by transposing or adding the Octaves above or below some or all of its three tones.

To obtain the perfectly harmonious positions of major tetrads, we have again to be careful that no minor Tenths and no [major or minor] Thirteenths occur. Hence the Fifth may not stand more than a minor Third above, or a Sixth below the Third of the chord; and the fundamental tone must not be more than a Sixth above the Third. When these rules are carried out, the avoidance of the minor Thirteenths is effected by not taking the double Octave of the Third and Fifth. These rules may be briefly enunciated as follows: Those major chords are most harmonious in which the fundamental tone or the Fifth does not lie more than a Sixth above the Third, or the Fifth does not lie more than a Sixth above or below it. The fundamental tone, on the other hand, may be as far below the Third as we please.

The corresponding positions of the major tetrads are found by combining any two of the more perfect positions of the major triads which have two tones in common, as follows, where the lower figures refer to the positions of the major triads already given.

The most Perfect Positions of Major Tetrads within the Compass of Two Octaves.[24]

We see that chords of the Sixth and Third must he quite close, as No. 7 ; [25] and that chords of the Sixth and Fourth [26] must not have a compass of more than an Eleventh, but may occur in all the three positions (Nos. 5, 6, 11) in which it can be constructed within this compass. Chords which have the fundamental tone in the bass can be handled most freely.

It will not be necessary to enumerate the less perfect positions of major tetrads. They cannot have more than two unsuitable combinational tones, as in the 12th position of the major triads, p. 220c. The major triads of \(C\) can only have the false combinational tones marked \(^7b\flat \) and \(^{11}f\), [that is, with pitch numbers bearing to that of \(C\) the ratios 7 : 1, or 11 : 1].

Minor tetrads, like the corresponding triads, must at least have one false combinational tone. There is only one single position of the minor tetrad which has only one such tone. It is No. 1 in the following example, and is compounded of the positions Nos. 1 and 2 of the minor triads on p. 221b. But there may be as many as 4 false combinational tones, as, for example, on combining positions Nos. 10 and 11 of the minor triads, p. 221c.

Here follows a list of the minor tetrads which have not more than two false combinational tones, and which lie within the compass of two Octaves. The false combinational tones only are noted in crotchets, and those which suit the chord are omitted.

Best Positions of Minor Tetrads.[27]

The chord of the Sixth and Fourth [marked \(6 \atop 4\)] occurs only in its closest position, No. 5 ; but that of the Sixth and Third [marked \(6 \atop 3\)] is found in three positions (Nos. 3, 6, and 9), namely, in all positions where the compass of the chord does not exceed a Tenth ; the fundamental chord occurs three times with the Octave of the fundamental note added (Nos. 1, 2, 4), and twice with the Octave of the Fifth added (Nos. 7, 8).

In musical theory, as hitherto expounded, very little has been said of the influence of the transposition of chords on harmonious effect. It is usual to give as a rule that close intervals must not be used in the bass, and that the intervals should be tolerably evenly distributed between the extreme tones. And even these rules do not appear as consequences of the theoretical views and laws usually given, according to which a consonant interval remains consonant in whatever part of the scale it is taken, and however it may be inverted or combined with others. They rather appear as practical exceptions from general rules. It was left to the musician himself to obtain some insight into the various effects of the various positions of chords by mere use and experience. No rule could be given to guide him.

The subject has been treated here at such length in order to shew that a right view of the cause of consonance and dissonance leads to rules for relations which previous theories of harmony could not contain. The propositions we have enunciated agree, however, with the practice of the best composers, of those, I mean, who studied vocal music principally, before the great development of instrumental music necessitated the general introduction of tempered intonation, as any one may easily convince himself by examining those compositions which aimed at producing an impression of perfect harmoniousness. Mozart is certainly the composer who had the surest instinct for the delicacies of his art. Among his vocal compositions the Ave verum corpus is particularly celebrated for its wonderfully pure and smooth harmonies. On examining this little piece as one of the most suitable examples for our purpose, we find in its first clause, which has an extremely soft and sweet effect, none but major chords, and chords of the dominant Seventh. All these major chords belong to those which we have noted as being the more perfectly harmonious. Position 2 occurs most frequently, and then 8, 10, 1, and 9 [of p. 223c]. It is not till we come to the final modulation of this first clause that we meet with two minor chords, and a major chord in an unfavourable position. It is very striking, by way of comparison, to find that the second clause of the same piece, which is more veiled, longing, and mystical, and laboriously modulates through bolder transitions and harsher dissonances, has many more minor chords, which, as well as the major chords scattered among them, are for the most part brought into unfavourable positions, until the final chord again restores perfect harmony.

Precisely similar observations may be made on those choral pieces of Palestrina, and of his contemporaries and successors, which have a simple harmonic construction without any involved polyphony. In transforming the Roman Church music, which was Palestrina’s task, the principal weight was laid on harmonious effect in contrast to the harsh and unintelligible polyphony of the older Netherland[28] system, and Palestrina and his school have really solved the problem in the most perfect manner. Here also we find an almost uninterrupted flow of consonant chords, with some dominant Sevenths, or dissonant passing notes, charily interspersed. Here also the consonant chords wholly, or almost wholly, consist of those major and minor chords which we have noted as being in the more perfect positions. Only in the final cadence of a few clauses, on the contrary, in the midst of more powerful and more frequent dissonances, we find a predominance of the unfavourable positions of the major and minor chords. Thus that expression which modern music endeavours to attain by various discords and an abundant introduction of dominant Sevenths, was obtained in the school of Palestrina by the much more delicate shading of various inversions and positions of consonant chords. This explains the harmoniousness of these compositions, which are nevertheless full of deep and tender expression, and sound like the songs of angels with hearts affected but undarkened by human grief in their heavenly joy. Of course such pieces of music require fine ears both in singer and hearer, to let the delicate gradation of expression receive its due, now that modern music has accustomed us to modes of expression so much more violent and drastic.

The great majority of major tetrads in Palestrina’s Stabat mater are in the positions 1, 10, 8, 5, 3, 2, 4, 9 [of p. 223c], and of minor tetrads in the positions 9, 2, 4, 3, 5, 1 [of p. 224a]. For the major chords one might almost think that some theoretical rule led him to avoid the bad intervals of the minor Tenth and the [major or minor] Thirteenth. But this rule would have been entirely useless for minor chords. Since the existence of combinational tones was not then known, we can only conclude that his fine ear led him to this practice, and that the judgment of that ear exactly agreed with the rules deduced from our theory.

These authorities may serve to lead musicians to allow the correctness of my arrangement of consonant chords in the order of their harmoniousness. But any one can convince himself of their correctness on any justly intoned instrument [as the Harmonical]. The present system of tempered intonation certainly obliterates somewhat of the more delicate distinctions, without, however, entirely destroying them.


Having thus concluded that part of our investigations which rests upon purely scientific principles, it will be advisable to look back upon the road we have travelled in order to review our gains, and examine the relation of our results to the views of older theoreticians. We started from the acoustical phenomena of upper partial tones, combinational tones and beats. These phenomena were long well known both to musicians and acousticians, and the laws of their occurrence were, at least in then-essential features, correctly recognised and enunciated. We had only to pursue these phenomena into further detail than had hitherto been done. We succeeded in finding methods for observing upper partial tones, which rendered comparatively easy an observation previously very difficult to make. And with the help of this method we endeavoured to shew that, with few exceptions, the tones of all musical instruments were compounded of partial tones, and that, in especial, those qualities of tone which are more particularly favourable for musical purposes, possess at least a series of the lower partial tones in tolerable force, while the simple tones, like those of stopped organ pipes, have a very unsatisfactory musical effect, although even these tones when loudly sounded are accompanied in the ear itself by some weak harmonic upper partials. On the other hand we found that, for the better musical qualities of tone, the higher partial tones, from the Seventh onwards, must be weak, as otherwise the quality, and every combination of tones would be too piercing. In reference to the beats, we had to discover what became of them when they grew quicker and quicker. We found that they then fell into that roughness which is the peculiar character of dissonance. The transition can be effected very gradually, and observed in all its stages, and hence it is apparent to the simplest natural observation that the essence of dissonance consists merely [29] in very rapid beats. The nerves of hearing feel these rapid beats as rough and unpleasant, because every intermittent excitement of any nervous apparatus affects us more powerfully than one that lasts unaltered. With this there is possibly associated a psychological cause. The individual pulses of tone in a dissonant combination give us certainly the same impression of separate pulses as slow beats, although we are unable to recognise them separately and count them; hence they form a tangled mass of tone, which cannot be analysed into its constituents. The cause of the unpleasantness of dissonance we attribute to this roughness and entanglement. The meaning of this distinction may be thus briefly stated: Consonance is a continuous, dissonance an intermittent sensation of tone. Two consonant tones flow on quietly side by side in an undisturbed stream; dissonant tones cut one another up into separate pulses of tone. This description of the distinction at which we have arrived agrees precisely with Euclid’s old definition, 'Consonance is the blending of a higher with a lower tone. Dissonance is incapacity to mix, when two tones cannot blend, but appear rough to the ear.'

After this principle had been once established there was nothing further to do but to inquire under what circumstances, and with what degree of strength, beats would arise in the various combinations of tones through either the partial or the combinational tones. This investigation had hitherto been completely worked out by Scheibler for the combinational tones of two simple tones only. The law of beats being known, it became easy to extend it to compound tones. Every theoretical conclusion on this field can be immediately checked by a proper observation, when the analysis of a mass of tone is facilitated by the use of resonators. All these beats of partial and combinational tones, of which so much has been said in the last chapter, are not inventions of empty theoretical speculation, but rather facts of observation, and can be really heard without difficulty by any practised observer who performs his experiments correctly. The knowledge of the acoustic law facilitates our discovery of the phenomena in question. But all the assertions on which we depend for establishing a theory of consonance and dissonance, such as was given in the last chapters, are founded wholly and solely on a careful analysis of the sensations of hearing, an analysis which a practised ear could have executed without any theoretical assistance, although of course the task was immensely facilitated by the guidance of theory and the assistance of appropriate instruments of observation.

For these reasons the reader is particularly requested to observe that my hypothesis concerning the sympathetic vibration of Corti’s organs inside the ear has no immediate connection whatever with the explanation of consonance and dissonance. That explanation depends solely upon observed facts, on the beats of partial tones and the beats of combinational tones. Yet I thought it right not to suppress my hypothesis (which must of course be regarded solely as an hypothesis), because it gathers all the various acoustical phenomena with which we are concerned into one sheaf, and gives a clear, intelligible, and evident explanation of the whole phenomena and their connection.

The last chapters have shewn, that a correct and careful analysis of a mass of sound under the guidance of the principles cited, leads to precisely the same distinctions between consonant and dissonant intervals and chords, as have been established under the old theory of harmony. We have even shewn that these investigations give more particular information concerning individual intervals and chords than was possible with the general rules of the former theory, and that the correctness of these rules is corroborated both by observation on justly intoned instruments and the practice of the best composers.

Hence I do not hesitate to assert that the preceding investigations, founded upon a more exact analysis of the sensations of tone, and upon purely scientific, as distinct from esthetic principles, exhibit the true and sufficient cause of consonance and dissonance in music. One circumstance may, perhaps, cause the musician to pause in accepting this assertion. We have found that from the most perfect consonance to the most decided dissonance there is a continuous series of degrees, of combinations of sound, which continually increase in roughness, so that there cannot be any sharpy line drawn between consonance and dissonance, and the distinction would therefore seem to be merely arbitrary. Musicians, on the contrary, have been in the habit of drawing a sharp line between consonances and dissonances, allowing of no intermediate links, and Hauptmann advances this as a principal reason against any attempt at deducing the theory of consonance from the relations of rational numbers.[30]

As a matter of fact we have already remarked that the chords of the natural or subminor Seventh 4 : 7 [\(c'\) to \(^7b'\flat\) on the Harmonical], and of the subminor Tenth 3 : 7 [\(g\) to \(^7b'\flat \)] in many qualities of tone sound at least as well as the minor Sixth 5 : 8 [\(e'\) to \(c''\)], and that the subminor Tenth really sounds better than the ordinary minor Tenth 5 : 12 [\(e'\) to \(g''\)]. But we have already noticed a circumstance of great importance for musical practice which gives the minor Sixth an advantage over the intervals formed with the number 7. The inversion of the minor Sixth gives a better interval, the major Third [\(e' \; c''\) inverted gives \(c' \; e'\)], and its importance as a consonance in modern music is especially due to this very relation to the major Third; it is essentially necessary, and justified, just because it is the inversion of the major Third. On the other hand the inversion or transposition of an interval formed with the number 7 leads to intervals worse than itself. Hence, as it is necessary, for the purposes of harmony, to have the power of transposing the parts at pleasure, we have a sufficient reason for drawing the line between the minor Sixth on the one hand, and the intervals characterised by 7 on the other. It is not, however, till we come to construct scales, which we shall have to consider in the next chapter, that we find decisive reasons for making this the boundary. The scales of modern music cannot possibly accept tones determined by the number 7.[31] But in musical harmony we can only deal with chords formed of notes in the scale. Intervals characterised by 5, as the Thirds and Sixths, occur in the scale, as well as others characterised by 9, as the major Second 8 : 9, but there are none characterised by 7, which should form the transition between them. Here, then, there is a real gap in the series of chords arranged according to the degree of their harmonious effect, and this gap serves to determine the boundary between consonance and dissonance.

The decision does not depend, then, on the nature of the intervals themselves but on the construction of the whole tonal system. This is corroborated by the fact that the boundary between consonant and dissonant intervals has not been always the same. It has been already mentioned that the Greeks always represented Thirds as dissonant, and although the original Pythagorean Third 64 : 81, determined by a series of Fifths, was not a consonance, yet even when the natural major Third 4 : 5 was afterwards included in the so-called syntono-diatonic mode of Didymus and Ptolemaeus, it was not recognised as a consonance. It has already been mentioned that in the middle ages, first the Thirds and then the Sixths were acknowledged as imperfect consonances, that the Thirds were long omitted from the final chord, and that it was not till later that the major, and quite recently the minor Third was admitted in this position. It is quite a mistake to suppose, with modern musical theorists, that this was merely whimsical and unnatural, or that the older composers allowed themselves to be fettered by blind faith in Greek authority. The last was certainly partly true for writers on musical theory down to the sixteenth century. But we must distinguish carefully between composers and theoreticians. Neither the Greeks, nor the great musical composers of the sixteenth and seventeenth centuries, were people to be blinded by a theory which their ears could upset. The reason for these deviations is to be looked for rather in the difference between the tonal systems in early and recent times, with which we shall become acquainted in the next part. It will there be seen that our modern system gained the form under which we know it through the influence of a general use of harmonic chords. It was only in this system that a complete regard was paid to all the requisitions of interwoven harmonies. Owing to its strict consistency, we were not only able to allow many licences in the use of the more imperfect consonances and of dissonances, which older systems had to avoid, but we were often required to insert the Thirds in final chords, as a mode of distinguishing with certainty between the major and minor mode, in cases where this distinction was formerly evaded.

But if the boundary between consonance and dissonance has really changed with a change of tonal system, it is manifest that the reason for assigning this boundary does not depend on the intervals and their individual musical effect, but on the whole construction of the tonal system.

The enigma which, about 2500 years ago, Pythagoras proposed to science, which investigates the reasons of things, 'Why is consonance determined by the ratios of small whole numbers?' has been solved by the discovery that the ear resolves all complex sounds into pendular oscillations, according to the laws of sympathetic vibration, and that it regards as harmonious only such excitements of the nerves as continue without disturbance. The resolution into partial tones, mathematically expressed, is effected by Fourier’s law, which shews how any periodically variable magnitude, whatever be its nature, can be expressed by a sum of the simplest periodic magnitudes.[32] The length of the periods of the simply periodic terms of this sum must be exactly such, that either one or two or three or four, and so on, of their periods are equal to the period of the given magnitude. This, reduced to tones, means that the pitch numbers of the partial tones must be exactly once, twice, three times, four times, and so on, respectively, as great as that of the prime tone. These are the whole numbers which determine the ratios of the consonances. For, as we have seen, the condition of consonance is that two of the lower partial tones of the notes combined shall be of exactly the same pitch; when they are not, disturbance arises from beats. Ultimately, then, the reason of the rational numerical relations of Pythagoras is to be found in the theorem of Fourier, and in one sense this theorem may be considered as the prime source of the theory of harmony.[33]

The relation of whole numbers to consonance became in ancient times, in the middle ages, and especially among Oriental nations, the foundation of extravagant and fanciful speculation. ‘ Everything is Number and Harmony,’ was the characteristic principle of the Pythagorean doctrine. The same numerical ratios 1T which exist between the seven tones of the diatonic scale, were thought to be found again in the distances of the celestial bodies from the central fire. Hence the harmony of the spheres, which was heard by Pythagoras alone among mortal men, as his disciples asserted. The numerical speculations of the Chinese in primitive times reach as far. In the book of Tso-kiu-ming, a friend of Confucius (b.c. 500), the five tones of the old Chinese scale were compared with the five elements of their natural philosophy — water, fire, wood, metal, and earth. The whole numbers 1, 2, 3 and 4 were described as the source of all perfection. At a later time the 12 Semitones of the Octave were connected with the 12 months in the year, and so on. Similar references of musical tones to the elements, the temperaments, and the constellations are found abundantly scattered among the musical writings of the Arabs. The harmony of the spheres plays a great part throughout the middle ages. According to Athanasius Kircher, not only the macrocosm, but the microcosm is musical. Even Keppler, a man of the deepest scientific spirit, could not keep himself free from imaginations of this kind. Nay, even in the most recent times, theorising friends of music may be found who will rather feast on arithmetical mysticism than endeavour to hear upper partial tones.

The celebrated mathematician Leonard Euler[34] tried, in a more serious and more scientific manner, to found the relations of consonances to whole numbers upon psychological considerations, and his theory may certainly be regarded as the one which found most favour with scientific investigators during the last century, although it perhaps did not entirely satisfy them. Euler [35] begins by explaining that we are pleased with everything in which we can detect a certain amount of perfection. Now the perfection of anything is determined by the co-operation of all its parts towards the attainment of its end. Hence it follows that wherever perfection is to be found there must be order; for order consists in the arrangement of all parts by a certain law from which we can discover why each part lies where it is, rather than in any other place. Now in any perfect object such a law of arrangement is determined by the end to be attained which governs all the parts. For this reason order pleases us more than disorder. Now order can be perceived in two ways: either we know the law whence the arrangement is deduced, and compare the deductions from this law with the arrangements observed; or, we observe these arrangements and endeavour to determine the law from them. The latter is the case in music. A combination of tones will please us when we can discover the law of their arrangement. Hence it may well happen that one hearer finds it and that another does not, and that their judgments consequently differ.

The more easily we perceive the order which characterises the objects contemplated, the more simple and more perfect will they appear, and the more easily and joyfully shall we acknowledge them. But an order which costs trouble to discover, though it will indeed also please us, will associate with that pleasure a certain degree of weariness and sadness (tristitia).

Now in tones there are two things in which order is displayed, pitch and duration. Pitch is ordered by intervals, duration by rhythm. Force of tone might also be ordered, had we a measure for it. Now in rhythm two or three or four equally long notes of one part may correspond with one or two or three of another, in which the regularity of the arrangement is easily observed, especially when frequently repeated, and gives considerable pleasure. Similarly in intervals we should derive more pleasure from observing that two, three, or four vibrations of one tone coincided with one, two, or three of another, than we could possibly experience if the ratios of the time of vibration were incommensurable with one another, or at least could not be expressed except by very high numbers. Hence it follows that the combination of two tones pleases us the more, the smaller the two numbers by which the ratios of their periods of vibration can be expressed. Euler also remarked that we could better endure more complicated ratios of the periods of vibration, and consequently less perfect consonances, for higher than for deeper tones, because for the former the groups of vibrations which were arranged to occur in equal times, were repeated more frequently than in the latter, and we were consequently better able to recognise the regularity of even a more involved arrangement.

Hereupon Euler develops an arithmetical rule for calculating the degree of harmoniousness of an interval or a chord from the ratios of the periods of the vibrations which characterise the intervals. The Unison belongs to the first degree, the Octave to the second, the Twelfth and Double Octave to the third, the Fifth to the fourth, the Fourth to the fifth, the major Tenth and Eleventh to the sixth, the major Sixth and major Third to the seventh, the minor Sixth and minor Third to the eighth, the subminor Seventh 4 : 7 to the ninth, and so on. To the ninth degree belongs also the major triad, both in its closest position and in the position of the Sixth and Fourth. The major chord of the Sixth and Third belongs, however, to the tenth degree. The minor triad, both in its closest and in its position of the Sixth and Third, also belongs to the ninth degree, but its position of the Sixth and Fourth to the tenth degree. In this arrangement the consequences of Euler’s system agree tolerably well with our own results, except that in determining the relation of the major to the minor triad, the influence of combinational tones was not taken into account, but only the kinds of interval. Hence both triads in their close position appear to be equally harmonious, although again both the major chord of the Sixth and Third, and the minor chord of the Sixth and Fourth, are inferior with him as with us.[36]

Euler has not confined these speculations to single consonances and chords, but has extended them to their results, to the construction of scales, and to modulations, and brought out many surprising specialities correctly. But without taking into account that Euler’s system gives no explanation of the reason why a consonance when slightly out of tune sounds almost as well as one justly tuned, and much better than one greatly out of tune, although the numerical ratio for the former is generally much more complicated, it is very evident that the principal difficulty in Euler’s theory is that it says nothing at all of the mode in which the mind contrives to perceive the numerical ratios of two combined tones. We must not forget that a man left to himself is scarcely aware that a tone depends upon vibrations. Moreover, immediate and conscious perception by the senses has no means of discovering that the numbers of vibrations performed in the same time are different, greater for high than for low tones, and that determinate intervals have determinate ratios of these numbers. There are certainly many perceptions of the senses in which a person is not precisely able to account for the way in which he has attained to his knowledge, as when from the resonance of a space he judges of its size and form, or when he reads the character of a man in his features. But in such cases a person has generally had a large experience in such relations, which helps him to form a judgment in analogous circumstances, without having the previous circumstances on which his judgment depends clearly present to his mind. But it is quite different with pitch numbers. A man that has never made physical experiments has never in the whole course of his life had the slightest opportunity of knowing anything about pitch numbers or their ratios. And almost every one who delights in music remains in this state of ignorance from birth to death.

Hence it would certainly be necessary to shew how the ratios of pitch numbers can be perceived by the senses. It has been my endeavour to do this, and hence the results of my investigation may be said, in one sense, to fill up the gap which Euler’s left. But the physiological processes which make the difference sensible between consonance and dissonance, or, in Euler’s language, orderly and disorderly relations of tone, ultimately bring to light an essential difference between our method of explanation and Euler’s. According to the latter, the human mind perceives commensurable ratios of pitch numbers as such; according to our method, it perceives only the physical effect of these ratios, namely, the continuous or intermittent sensation of the auditory nerves.[37] The physicist knows, indeed, that the reason why the sensation of a consonance is continuous is that the ratios of its pitch numbers are commensurable, but when a man who is unacquainted with physics, hears a piece of music, nothing of the sort occurs to him,[38] nor does the physicist find a chord in any respect more harmonious because he is better acquainted with, the cause of its harmoniousness.[39] It is quite different with the order of rhythm. That exactly two crotchets, or three in a triplet, or four quavers go to one minim is perceived by any attentive listener without the least instruction. But while the orderly relation (or commensurable ratio) of the vibrations of two combined tones, on the other hand, undoubtedly affects the ear in a certain way which distinguishes it from any disorderly relation (incommensurable ratio), this difference of consonance and dissonance depends on physical, not psychological grounds.

The considerations advanced by Rameau[40] and d’Alembert[41] on the one side, and Tartini[42] on the other, concerning the cause of consonance agree better with our theory. The last founded his theory on the existence of combinational tones, the two first on that of upper partial tones. As we see, they had found the proper points of attack, but the acoustical knowledge of last century did not allow of their drawing sufficient consequences from them. According to d’Alembert, Tartini’s book was so darkly and obscurely written that he, as well as other well-instructed people, were unable to form a judgment upon it. D’Alembert’s book, on the other hand, is an extremely clear and masterly performance, such as was to be expected from a sharp and exact thinker, who was at the same time one of the greatest physicists and mathematicians of his time. Rameau and d’Alembert lay down two facts as the foundation of their system. The first is that every resonant body audibly produces at the same time as the prime (générateur) its Twelfth and next higher Third, as upper partials (harmoniques). The second is that the resemblance between any tone and its Octave is generally apparent. The first fact is used to shew that the major chord is the most natural of all chords, and the second to establish the possibility of lowering the Fifth and the Third by one or two Octaves without altering the nature of the chord, and hence to obtain the major triad in all its different inversions and positions. The minor triad is then found by the condition that all three tones should have the same upper partial or harmonic, namely, the Fifth of the chord (in fact \(C, \; E\flat \) and \(G\) have all the same upper partial \(g'\)). Hence although the minor chord is not so perfect and natural as the major, it is nevertheless prescribed by nature.

In the middle of the eighteenth century, when much suffering arose from an artificial social condition, it may have been enough to shew that a thing was natural, in order at the same time to prove that it must also be beautiful and desirable. Of course no one who considers the great perfection and suitability of all organic arrangements in the human body, would, even at the present day, deny that when the existence of such natural relations have been proved as Rameau discovered between the tones of the major triad, they ought to be most carefully considered, at least as starting-points for further research. And Rameau had indeed quite correctly conjectured, as we can now perceive, that this fact was the proper basis of a theory of harmony. But that is by no means everything. For in nature we find not only beauty but ugliness, not only help but hurt. Hence the mere proof that anything is natural does not suffice to justify it esthetically. Moreover if Rameau had listened to the effects of striking rods, bells, and membranes, or blowing over hollow chambers, he might have heard many a perfectly dissonant chord. And yet such chords cannot but be considered equally natural. That all musical instruments exhibit harmonic upper partials depends upon the selection of qualities of tone which man has made to satisfy the requirements of his ear.

Again the resemblance of the Octave to its fundamental tone, which was one of Rameau’s initial facts, is a musical phenomenon quite as much in need of explanation as consonance itself.

No one knew better than d’Alembert himself the gaps in this system. Hence in the preface to his book he especially guards himself against the expression: 'Demonstration of the Principle of Harmony,' which Rameau had used. He declares that so far as he himself is concerned, he meant only to give a well-connected and consistent account of all the laws of the theory of harmony, by deriving them from a single fundamental fact, the existence of upper partial tones or harmonics, which he assumes as given, without further inquiry respecting its source. He consequently limits himself to proving the naturalness of the major and minor triads. In his book there is no mention of beats, and hence of the real source of distinction between consonance and dissonance. Of the laws of beats very little indeed was known at that time, and combinational tones had only been just brought under the notice of French savants, by Tartini (1751) and Romieu (1753). They had been discovered a few years previously in Germany by Sorge (1745), but the fact was probably little known. Hence the materials were wanting for building up a more perfect theory.

Nevertheless this attempt of Rameau and d’Alembert is historically of great importance, in so far as the theory of consonance was thus for the first time shifted from metaphysical to physical ground. It is astonishing what these two thinkers effected with the scanty materials at their command, and what a clear, precise, comprehensive system the old vague and lumbering theory of music became under their hands. The important progress which Rameau made in the specially musical portion of the theory of harmony will be seen hereafter.

If, then, I have been myself able to present something more complete, I owe it merely to the circumstance that I had at command a large mass of preliminary physical results, which had accumulated in the century that has since elapsed.

[1][If two tones each consonant with a third are dissonant with each other, I call the result a 'con-dissonant triad'. See App. XX. sect. E. art. 5. — Translator.]
[2][The reader ought to hear the whole set of triads that could be formed from the table, at least all exclusive of those formed by the last line. The ordinary tuning of the harmonium, organ, and piano does not permit this. But they can all (inclusive of those formed by the last line) be played on the Harmonical. — Translator.]
[3][Add the consonance \(G^7 B\flat D = 6 : 7 : 9\). — Translator.]
[4][\(E \; A\flat\) has 428 cents, and \(E \; G\sharp\) has 386 cents, difference 42 cents, the great diēsis. See App. XX. sect. D. — Translator.]
[5][Inserting the values of the intervals in cents, the two chords, \(A^1\flat \; 386 \; C \; 386 \; E_1\) and \(C \; 386 \; E_1 \; 386 \; G_2\sharp\) are seen to be identical, but when the first is inverted to \(C \; 386 \; E_1 \; 428 \; A^1\flat\) it becomes different from the other. Both, however, remain harshly dissonant. On tempered instruments of course they become identical \(C \; 400 \; E \; 400 \; G\sharp , C \; 400 \; E \; 400 \; A\flat \) and are very harsh. The definition of consonant triads does not apply to tempered chords, in none of which are any of the intervals purely consonant. — Translator.]
[6][As all the differentials must be harmonics of \(C \; 66\), if we represent this note by 1, the harmonics and hence differentials will all be contained in the series
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
\(C\) \(c\) \(g\) \({c'}\) \({e'}\) \({g'}\) \({^7b'\flat} \) \({c''}\) \({d''}\) \({e''}\) \({^{11}f''}\) \({g''}\) \({^{13}a''}\) \({^7b''\flat}\) \({b''}\) \({c'''}\)
First Chord. — The notes will then be 4, 5, 6, represented by minims, and their Octaves 8, 10, 12, which are not given in notes.
1) Crotchets, \(5 - 4 = 6 - 5 = 1, 6 - 4 = 2. \)
3) Quavers, \(12 - 10 = 2 , 8 - 5 = 3. \)
4) Semiquavers, \(12 - 5 = 7 , 12 - 4 = 8. \)

Second Chord. — Notes 5, 6, 8; Octaves 10, 12, 16.
1) Crotchets, \(6 - 5 = 1, 8 - 6 = 2, 8 - 5 = 3. \)
2) Quavers, \(10 - 8 = 12 - 10 = 2, 12 - 8 = 4. \)
3) Semiquavers, \(12 - 5 = 7, 16 - 6 = 10\), (but this is also an audible partial,) \(16 - 5 = 11\), which being more than half an equal Semitone (51 cents) above equally tempered \(f''\) is represented on the staff as a flattened \(f''\sharp\).

Third Chord. — Notes, 6, 8, 10; Octaves 12, 16, 20.
1) Crotchets, \(10 - 8 = 8 - 6 = 2, 10 - 6 = 4. \)
2) Quavers, \(12 - 10 = 2, 12 - 8 = 4. \)
3) Semiquavers, \(20 - 6 = 14. \)

How far these higher notes marked by semiquavers are effective, except possibly when they beat with each other, or with some partials of the original notes, remains to be proved. — Translator.]
[7][In minor chords the case is different. On referring to the list of harmonics in the last note, it will be seen that the only minor chord is 10, 12, 15 or \(e'' \; g'' \; b''\) , and this is the chord upon the major Third above the third Octave of the fundamental. Hence in the example where the chord taken is \(c' \; e'\flat \; g'\) and its inversions, the harmonics must be formed on \(A_{\prime\prime}\flat\) which is the same interval below \(c'\). The list of harmonics in these examples is therefore
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
\(A_{\prime\prime}\flat\) \(A_{\prime}\flat\) \(E\flat\) \(A\flat\) \(c\) \(e\flat\) \(^7g\flat\) \(a\flat\) \(b\flat\) \(c'\) \(^{11}d'\flat\) \(e'\flat\) \(^{13}f'\) \(^7g'\flat\) \(g'\)
16 18 20 21 22 24 25 26 27 28 30 32 33 39 40
\(a'\flat\) \(b'\flat\) \(c''\) \(^7d''\flat\) \(^{11}d''\flat\) \(e''\flat\) \(e''\) \(^{13}f''\) \(f''\) \(^7g''\flat\) \(g''\) \(a''\flat\) \(^{11}a''\flat\) \(^{13}c'''\) \(c''\)
The omitted harmonics are not used in this investigation, though differentials of higher orders occur up to the 48th harmonic.

First Chord. — Notes 10, 12, 15; Octaves 20, 24, 30.
1) Crotchets, \(12 - 10 = 2, 15 - 12 = 3, 15 - 10 = 5. \)
2) Quavers, \(20 - 15 = 5, 20 - 12 = 8, 24 - 15 = 9.\)
3) Semiquavers, \(24 - 10 = 14, 30 - 12 = 18, 30 - 10 = 20.\)

Second Chord.— Notes 12, 15, 20; Octaves 24 30 40.
1) Crotchets, \(15 - 12 = 3, 20 - 15 = 5, 20 - 12 = 8.\)
2) Quavers, \(24 - 20 = 4, 24 - 15 = 9, 30 - 20 = 10. \)
3) Semiquavers, \(30 - 12 = 18, 40 - 15 = 25, 40 - 12 = 28. \)

Third Chord. — Notes 15, 20, 24; Octaves 30, 40, 48.
1) Crotchets, \(24 - 20 = 4, 20 - 15 = 5, 24 - 15 = 9.\)
2) Quavers, \(30 - 24 = 6, 30 - 20 = 10, 40 - 24 = 16.\)
3) Semiquavers, \(40 - 15 = 25, 48 - 20 = 28, 48 - 15 = 33. \)

This I have hero represented as \(^{11}a''\flat\) because it is the Twelfth above \(^{11}d'\flat\). but in the text it is called a flattened \(a''\) because it is almost the one-sixth of \(C'' = 528\). In fact on the Harmonical, \(\tfrac53 \times 528 = 880\), and \(A_{\prime\prime} \flat\) would be \(\tfrac45 \cdot C_{\prime}= \tfrac45 \cdot 33 = 26.3\), so that \(33 \times 26.3 = 867.9\) vibrations. The interval 880 : 867.9 has 24 cents, and hence \(a''\) is more than a comma too sharp. The same observation applies as in the last footnote regarding the audible effect of the high notes, when not beating with each other, or with audible partials. — Translator.]
[8][See note 14 on p. 157d. — Translator.]
[9][From the list of harmonics on p. 215c it will be seen that these tones occur as lower harmonics of the tone whence the minorchords are derived. — Translator.]
[10][The Author is of course always speaking of chords in just intonation. When tempered, as on the harmonium, even the major chords are accompanied by unrelated combinational tones, sufficiently close to beat and sufficiently loud for Scheibler to have laid down a rule for counting the beats in order to verify the correctness of the tempered tuning (see p. 203d). But still the different effects of the two chords are very marked. — Translator.]
[11][The English names major and minor were chosen because the first Third in the fundamental position is major in the first case and minor in the second. In German the terms are dur and moll, that is, hard and soft. It is well known that the names dur and moll are not connected with the hard or soft character of the pieces of music written in these modes, but are historically derived from the angular form of \(\natural\) and the rounded form of \(\flat\), which were the B durum and B molls of the medieval musical notation. [The probable origin of the forms \(\flat \; \natural \; h \; \sharp\) is given from observations on the plates in Gaforius’s Theoricum Opus Harmonicae Disciplinae, 1480, the earliest printed book on music, in a footnote, infrà, p. 312d.— -Translator.
[12][For \(e' \; : \; g'\) = 5 : 6, diff. \(6 - 5 = l\) or \(C\); \(g \; : \; e'\) = 3 : 5, diff. \(5 - 3 = 2\) or \(c \; ; \; e' \; : \; c''\) = 5 : 8, diff. \(8 - 5 = 3\) or \(g\). — Translator.]
[13]See Chapter XVI. for remarks upon just and tempered intonation, and for a justly intoned instrument suitable for such experiments. [The Harmonical can also be used. See App. XX. sect. F. for this and other instruments.]
[14][See these intervals examined in p. 195, note 18. — Translator.]
[15][The intervals \(6 \; : \; 7 = g' \; : \; ^7b'\flat\), \(3 \; : \; 7 = g \; : \; ^7b'\flat\) and \(5 \; : \; 12 = e \; : \; g'\) can be tried and compared on the Harmonical. — Translator.]
[16][The ratios are 12 : 21 : 28, and \(21 - 12 = 9\), but \(9 \; : \; 12 = 3 \; : \; 4\), hence if 12 is \(C\), 9 is \(G_{\prime}\). Again \(28 - 12 = 16, \;\; 12 \; : \; 16 = 3 \; : \; 4\) and hence 16 is \(F\). — Translator.]
[17][They are, however, insisted on by Poole, see App. XX. sect. F. No. 6. — Translator.]
[18][Some of the bars and numbers have been changed to make all agree with the footnote to p. 214d. All these notes and their combinational notes can by this means be played on the Harmonical. — Translator.]
[19](Calculation according to list of harmonics, p. 214d, footnote 6.
1) Chord 4, 6, 10. Differentials \(6 - 4 = 2, \; 10 - 6 = 4, \; 10 - 4 = 6\), which is also one of the tones.
2) Chord 6, 8, 10. Differentials \(8 - 6 = 10 - 8 = 2, \; 10 - 6 = 4\).
3) Chord 4, 10, 12. Differentials \(12 - 10 = 2, \; 10 - 4 = 6, \; 12 - 4 = 8\).
4) Chord 8, 10, 12. Differentials \(10 - 8 = 12 - 10 = 2, \; 12 - 8 = 4\).
5) Chord 3, 5, 8. Differentials \(5 - 3 = 2, \; 8 - 5 = 3 \;(\text{which is also one of the tones}), \; 8 - 3 = 5 \;(\text{which is also one of the tones})\).
6) Chord 5, 6, 8. Differentials \(6 - 5 = 1, \; 8 - 6 = 2, \; 8 - 5 = 3\).
These chords should be studied on the Harmonical, and the combinational tones listened for, and afterwards the tones played as substantive notes. — Translator.]
[20][Calculation in continuation of the last note.
7) Chord 3, 4, 10. Differentials \(4 - 3 = 1, \; 10 - 4 = 6, \; 10 - 3 = 7\).
8) Chord 3, 8, 10. Differentials \(10 - 8 = 2, \; 8 - 3 = 5, \; 10 - 3 = 7,\) which gives the interval 7 : 8 with the tone 8.
9) Chord 4, 5, 12. Differentials \(5 - 4 = 1, \; 12 - 4 = 8, \; 12 - 5 = 7,\) the two last differential tones being 7 : 8.
10) Chord 5, 8, 12. Differentials \(8 - 5 = 3, \; 12 - 8 = 4, \; 12 - 5 = 7,\) which gives the interval 7 : 8 with the tone 8.
11) Chord 5, 6, 16. Differentials \(6 - 5 = 1, \; 16 - 6 = 10, \; 16 - 5 = 11,\) which two last form the dissonant trumpet interval 11 : 10 of 165 cents or about three-quarters of an equal tone.
12) Chord 5, 12, 16. Differentials \(16 - 12 = 4, \; 12 - 5 = 7, \; 16 - 5 = 11\), which forms the same dissonant trumpet interval 11 : 10, but this time with one of the tones, and therefore more harshly.
All these 12 chords should be well studied on tho Harmonical, and for the first 10, the differential tones can be played also as substantive notes (remembering that \(^7B\flat\) is on tho \(G\flat\) digital), which will enable the student to acquire a better idea of the roughness. The tones 11 and 13 could not be introduced among the first 4 Octaves on the Harmonical without incurring the important losses of \(f''\) and \(a''\). But if we take the chords an Octave higher we can play \(^{11}f'''\) and \(^{13}a''\).
The chords should also be played in lower and higher positions, not only as Octaves of those given, but from the other major chords on the Harmonical as \(FA_1C_2GB_1D, \; A^1\flat CE^1\flat , \; E^1\flat GB^1\flat\). Particular attention should be paid to the contrasting of the positions 1 and 7, 2 and 8, 3 and 9, 4 and 10, 5 and 11, 6 and 12. Unless the ear acquires the habit of attending to these differences it will not properly form the requisite conceptions of major chords. For future purposes the results should also be contrasted with those obtained by playing the same chords on a tempered instrument, — if possible of the same pitch, A 440. — Translator.]
[21][Calculation according to the list of harmonies on p. 215c, footnote 7.
1) Chord 24, 30, 40. Differentials \(30 - 24 = 6, \; 40 - 30 = 10, \; 40 - 24 = 16\).
2) Chord 20, 24, 30. Differentials \(24 - 20 = 4, \; 30 - 24 = 6, \; 30 - 20 = 10\).
3) Chord 10, 24, 80. Differentials \(30 - 24 = 6, \; 24 - 10 = 14, \; 30 - 10 = 20\).
These can also be studied on the Harmonical, and the differentials to Nos. 1 and 2 can be played as substantive tones. Not so No. 3, but the effect may be felt by playing the chord a major Third higher as \(eg'b'\), being the 10, 24, 30 harmonics of \(C_{\prime\prime}\) and giving the differentials \(G, \; ^7b\flat , \; e'\) which can be played as substantive tones, but being so low will make the effect very rough. — Translator.]
[22][Calculation in continuation of the last note.
4) Chord 12, 15, 40. Differentials \(15 - 12 = 3, \; 40 - 15 = 25, \; 40 - 12 = 28\).
5) Chord 12, 30, 40. Differentials \(40 - 30 = 10, \; 40 - 12 = 18, \; 40 - 12 = 28\).
6) Chord 15, 20, 24. Differentials \(24 - 20 = 4, \; 20 - 15 = 5, \; 24 - 15 = 9\).
7) Chord 12, 20, 30. Differentials \(20 - 12 = 8, \; 30 - 20 = 10, \; 30 - 12 = 18\), where 18 forms the dissonance \(20 \; : \; 18 = 10 \; : \; 9\) with the tone 20.
8) Chord 10, 15, 24. Differentials \(15 - 10 = 5, \; 24 - 15 = 9, \; 24 - 10 = 14\), which forms the dissonant interval 15 : 14 with one of the tones 15.
9) Chord 10, 12, 30. Differentials \(12 - 10 = 2, \; 30 - 12 = 18, \; 30 - 10 = 20\), the two last form together the dissonance \(20 \; : \; 18 = 10 \; : \; 9\).
10) Chord 15, 20, 48 referred to \(A_{\prime\prime\prime}\). Interpret by taking the Octaves below the numbers in p. 215c, note 7. Differentials \(20 - 15 = 5 = 0; \; 48 - 20 = 28 = ^7g'\flat , \; 48 - 15 = 33 = ^{11}a'\flat\), see p. 215d', note 7, towards the end of the observations on the Third Chord.
11) Chord 15, 40, 48 referred to \(A_{\prime\prime\prime}\flat\) as in last chord.
Differentials \(48 - 40 = 8 = A\flat, \; 40 - 15 = 25 = e', \; 48 - 15 = 33 = ^{11}a'\flat\) as in last chord, which see.
12) Chord 15, 24, 40 referred to \(A_{\prime\prime\prime}\flat\). Differentials \(24 - 15 = 9 = B\flat , \; 40 - 24 = 16 = a\flat , \; 40 - 15 = 25 = e' \) where the differentials 16, 25 form the dissonant intervals \(16 \; : \; 15, \; 25 \; : \; 24\) with the two tones 15 and 24 respectively. All these chords can bestudied on the Harmonical, and their differentials can be played as substantive tones in Nos. 6, 7, and 12. No. 8 can be taken a major Third higher as in chord No. 3 of the last note, that is as \(e' \; b' \; g''\) giving the differentials \(e, \; d, \; ^7b\flat \) which can be played. Also No. 9 may be played as \(e' \; g' \; b''\) giving differentials \(c, \; d'', \; e''\). Nos. 4 and 5 do not admit of such treatment because \(e'''\flat \) is not on the instrument. Nos. 10 and 11 cannot be so played because \(^{11}a'\flat \) is not on the instrument. In fact it is the 33rd harmonic of \(A_{\prime\prime\prime}\flat = 13.15\), and this (see footnote p. 215d', remarks on Third Chord,) = \(33 \times 13.15 = 433.95\) vib.; whereas \(a = 440\), and hence is too sharp by the interval \(440 \; : \; 433.95\) or 24 cents, rather more than a comma.
The student should try all the minor chords not only in different positions in Octaves, but with all the other minor chords on the Harmonical, namely, \(FA^1\flat C, \; GB^1\flat D, \; D_1FA_1\) (which contrast with the dissonance \(DFA_1\) for future purposes), \(A_1CE_1, \; E_1GB_1,\) also in different Octaves, till the ear learns to distinguish these 12 different forms.
Finally the 12 forms of the major should be contrasted with the corresponding 12 forms of the minor triad, for the three possible cases \(FA_1C\) and \(FA^1\flat C\); \(CE_1G\) and \(CE^1\flat G\); \(GB_1D\) and \(GB^1\flat D\). To merely read over these pages by eye instead of studying them by ear is useless, and ordinary tempered instruments only impede instead of assisting the investigator. — Translator.]
[23][That is, if we exclude the harmonic Seventh from consideration, as on p. 195d, those who admit it (as Mr. Poole, App. XX. sect. F. No. 6) consider \(CE_1G^7B\flat \) to be a perfectly consonant tetrad. — Translator.]
[24][These major tetrads can all be played on the Harmonical, and should be tried in every position of Octaves and for all the major chords on the instrument, namely \(F A_1 C, \; C E_1 G, \; G B_1 D, \; A^1\flat C E^1\flat , \; E^1\flat G B^1\flat \), till the ear is perfectly familiar with the different forms and the student can tell them at once and designate them by their number in this list on hearing another person play them. — Translator.]
[25][This chord has the Third both lowest and highest and is marked \(6 \atop 3\), but is more commonly marked 6. — Translator.]
[26][This chord has the Third both lowest and highest and is marked \(6 \atop 3\), but is more commonly marked 6. — Translator.]
[27][Calculation of the combinational tones, by the list of harmonics in p. 215c.
1) Chord 20, 24, 30, 40. Differentials \(24 - 20 = 4 = A\flat , \; 40 - 24 = 16 = a'\flat \).
2) Chord 10, 24, 30, 40. Differentials \(24 - 10 = 14 = ^7g'\flat , \; 40 - 24 = 16 = a'\flat \).
3) Chord 12, 15, 20, 30. Differentials \(20 - 12 = 8 = a\flat , \; 30 - 12 = 18 = b'\flat\).
4) Chord 10, 20, 24, 30. Differentials \(24 - 20 = 4 = A\flat , \; 24 - 10 = 14 = ^7g'\flat \).
5) Chord 15, 20, 24, 30. Differentials \(24 - 20 = 4 = A\flat , \; 24 - 15 = 9 = b\flat \).
6) Chord 12, 20, 24, 30. Differentials \(24 - 20 = 4 = A\flat , \; 20 - 12 = 8 = a\flat , \; 30 - 12 = 18 = b'\flat \).
7) Chord 10, 12, 15, 30. Differentials \(12 - 10 = 2 = A_{\prime}\flat , \; 30 - 12 = 18 = b'\flat \).
8) Chord 10, 15, 24, 30. Differentials \(24 - 15 = 9 = b\flat , \; 24 - 10 = 14 = ^7g'\flat \).
9) Chord 12, 15, 20, 24. Differentials \(24 - 20 = 4 = A\flat , \; 20 - 12 = 8 = a\flat , \; 24 - 15 = 9 = b\flat \).
These chords should all be studied on the Harmonical. With the exception of Nos. 2, 4, 7, 8, the differentials can also be played on it as substantive tones. But they can be transposed. Thus No. 2 may be played as \(e \; g' \; b' \; e''\) giving the differentials \(^7b\flat , \; c' \). No. 4 will become \(e' \; e'' \; g'' \; b''\) giving the differential \(^7b'\flat \), which can be played. No. 7 becomes \(e' \; g' \; b' \; b''\) giving the differentials \(C\) and \(d''\). No. 8 becomes \(e' \; b' \; g'' \; b''\) giving the differentials \(d'\) and \(^7b'\flat \). These chords should also be studied in all the minor forms on the Harmonical, not only in different Octaves, but on all the minor chords on that instrument, viz., \(D_1FA_1, \; A_1CE_1, \; E_1GB_1, \; FA^1\flat C, \; CE^1\flat G, \; GB^1\flat D\), till the ear recognises the form, and the student can name the number of the position to another person’s playing. — Translator.]
[28][Including both the modern kingdom of the Netherlands or Holland, and the still more modern kingdom of Belgium. Josquin, 1450-1532, was born in Hainault in the present Belgium. — Translator.]
[29][But see also Prof. Preyer, in App. XX. sect. L. art. 7, infra. — Translator.]
[30]Harmonik und Metrik, p. 4. [At the same time, by accepting equal temperament they accept as consonant a series of tones which really form only one consonant interval (the Octave) and only two others even approximatively consonant (the Fifth and Fourth), while the commonest intervals on which harmony rests, the Thirds, with their inversions the Sixths, are not merely dissonant but, on the sustained tones of the voice for example, grossly dissonant. It is difficult for any ear brought up among these dissonances, to understand the real distinction between consonance and dissonance. Hence the absolute necessity of testing all the above assertions by a justly intoned instrument such as the Harmonical. — Translator.]
[31][Poole’s scale \(f \; g \; a_1 \; ^7b\flat \; c' \; d' \; e'_1 \; f',\) and Bosanquet’s and White’s tempered imitation of \(^7b'\flat \), properly 969 cents, as 974 cents, shew the feeling that exists for using the 7th harmonic, which is the only acoustical justification for the greatly harsher dominant Seventh, — Translator.']
[32]Namely magnitudes which vary as sines and cosines.
[33][The coincidences or non-coincidences of combinational tones, which are independent of Fourier’s law, are also considered of importance by Prof. Preyer. See infra, App. XX. sect. L. art. 7. — Translator.]
[34]Tentamen novae theoriae Musicae, Petro-poli, 1739.
[35]Loc. cit. chap. ii. § 7.
[36]The principle on which Euler calculated the degrees of harmoniousness for intervals and chords, is here annexed, because its consequences are very correct, if combinational tones are disregarded. When \(p\) is a prime number, the degree is \(=p\). All other numbers are products of prime numbers. The number of the degree for a product of two factors \(a\) and \(b\), for which separately the numbers of degree are \(\alpha \) and \(\beta \) respectively \(= \alpha + \beta - 1\). To find the number of the degree of a chord, which can be expressed by \(p \; : \; q \; : \; r \; : \; s\), &c., in smallest whole numbers, Euler finds the least common multiple of \(p, \; q, \; r, \; s\), &c., and the number of its degree is that of the chord. Thus, for example:
The number of the degree of 2 is 2, and of 3 is 3,

of \(4 = 2 \times 2\), it is \(2 + 2 - 1 = 3\),

of \(12 = 4 \times 3\), it is \(3 + 3 - 1 = 5\),

of \(60 = 12 \times 5\), it is \(6 + 5 - 1 = 9\).

That of the major triad 4 : 5 : 6 is that of 60, because 60 is the least common multiple of 4, 5, 6, that is, the least number which all of 1 them will divide without a remainder.
[37][With possibly Prof. Preyer’s addition, see App. XX. sect. L. art. 7. — Translator.]
[38][In point of fact, as he always hears tempered tones, he never hears the exact commensurable ratios. Indeed, on account of the impossibility of tuning with perfect exactness, the exact ratios are probably never heard, except from the double siren and wave-siren. — Translator.]
[39][Does a man breathe more easily and aerate his blood better because he knows the constitution of the atmosphere and its relation to his carbonised blood? Does a man feel a weight greater or less, because he knows the laws of gravitation? These are quite similar questions. — Translator.]
[40][Traité de l’harmonie réduite à des principes naturels, 1721. — Translator.]
[41][Traité de l’harmonie réduite à des principes naturels, 1721. — Translator.]
[42][Trattato di Musica secondo la vera scienza dell’ armonia. Padova, 1751. — Translator.]
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