Chapter XVII.


When voices move forward melodically in part music, the general rule is that they must form consonances with each other. For it is only as long as they are consonant, that there is an uninterrupted fusion of the corresponding auditory sensations. As soon as they are dissonant the individual parts mutually disturb each other, and each is a hindrance to the free motion of the other. To this esthetic reason must be added the purely physical consideration, that consonances cause an agreeable kind of gentle and uniform excitement to the ear which is distinguished by its greater variety from that produced by a single compound tone, whereas the sensation caused by intermittent dissonances is distressing and exhausting.

However, the rule that the various parts should make consonances with each other, is not without exception. The esthetic reason for this rule is not opposed to an occasional and temporary dissonance among the various parts, provided the motion of the parts is so contrived as to make the directions of the different voices perfectly easy to follow by the ear. Hence, in addition to the general laws of scale and key, to which the direction of every part is subject, there are particular rules for the progression of voices through discords. Again, dissonances cannot be entirely excluded because consonances are physically more agreeable. That which is physically agreeable is an important adjunct and support to esthetic beauty, but it is certainly not identical with it. On the contrary, in all arts we frequently employ its opposite, that which is physically disagreeable, partly to bring the beauty of the first into relief, by contrast, and partly to gain a more powerful means for the expression of passion. Dissonances are used for similar purposes in music. They are partly means of contrast, to give prominence to the impression made by consonances, and partly means of expression, not merely for peculiar and isolated emotional disturbances, but generally to heighten the impression of musical progress and impetuosity, because when the ear has been distressed by dissonances it longs to return to the calm current of pure consonances. It is for this last reason that dissonances are prominently employed immediately before the conclusion of a piece, where they were regularly introduced even by the old masters of medieval polyphony. But to effect this object in using them, the motion of the parts must be so conducted that the hearer can feel throughout that the parts are pressing forward through the dissonance to a following consonance, and, although this may be delayed or frustrated, the anticipation of its approach is the only motive which justifies the existence of the dissonances.

Since any relation of pitch which cannot be expressed in small numbers is dissonant, and it is only the number of the consonances which is limited, the number of possible dissonances would be infinite were it not that the individual parts composing a discord in music must necessarily obey the laws of melodic motion, that is, must lie within the scale. Consonances have an independent right to exist. Our modern scales have been formed upon them. But dissonances are allowable only as transitions between consonances. They have no independent right of existence, and the parts composing them are consequently obliged to move within the degree of the scales, by the same laws that were established in favour of the consonances.

On proceeding to a detailed consideration of the separate dissonant intervals, it should be remembered that in theoretical music the normal position of discords is taken to be that which arranges their tones as a series of Thirds. This, for example, is the rule for the chord of the dominant Seventh, which consists of the root, its Third, Fifth, and Seventh. The Fifth forms a Third with the Third, and the Seventh forms a Third with the Fifth. Hence we can consider a Fifth to be composed of two, and a Seventh of three Thirds. By inverting Thirds we obtain Sixths, by inverting Fifths we obtain Fourths, and by inverting Sevenths we obtain Seconds. In this way all the intervals in the scale are reproduced.

Using the present modification of Hauptmann’s notation, it is easily seen how different intervals of the same name must differ from each other in magnitude. We have only to remember that \(c^1\) is a comma higher than \(c\), and \(c_1\) two commas lower than \(c^1\) and one comma lower than \(c\), and that the comma is about the fifth part of a Semitone.

To obtain a general view of both the magnitude and roughness of the dissonant intervals, I have constructed fig. 61 (p.333a), in which the curve of roughness is copied from fig. 60 A (p.193b). The base line X Y signifies the interval of an Octave, upon which the individual consonant and dissonant intervals are set off from X, according to their magnitude on this scale.[1] On the lower side of the base are marked the twelve equal Semitones of the equally-tempered scale [each distant from the other by 100 cents], and on the upper side the consonant and dissonant intervals which occur in justly-intoned scales. The magnitude of the interval is always to be measured on the base line from X to the corresponding vertical line. The vertical lines corresponding to the consonances have been produced to the upper margins of the diagram, and those for the dissonances have been made shorter. The length of the verticals intercepted between the base and the curve of roughness shews the comparative degree of roughness probably possessed by the interval when played in a violin quality of tone.

[Tabular Expression of the Diagram, Fig. 61 Opposite.
Intervals No. Helmholtz's Notation, as in Diagram Ellis's Notation of Intervals reckoned from c Ratio Cents Roughness
Unison 1 c : c c : c 1 : 1 0 0
Minor Seconds ∥2 c : c♯ c : c♯ 1 : ¹²√2 100 76
3 b₁ : c' c : d¹♭ 15 : 16 112 70
Major Seconds 4 d : e₁ c : d₁ 9 : 10 182 38
∥5 c : d c : d 1 : ¹²√4 200 25
6 c : d c : d 8 : 9 204 32
7 b₁ : d¹♭ c : e²♭♭ 225 : 256 224 30
Minor Thirds 8 a¹♭ : b₁ c : d₂♭ 64 : 75 274 24
9 d : f c : e♭ 27 : 32 294 26
∥10 c : d♯ c : d♯ 1 : ¹²√8 300 24
*11 c : e¹♭ c : e¹♭ 5 : 6 316 20
Major Thirds *12 c : e₁ c : e₁ 4 : 5 386 8
∥13 c : e c : e 1 : ¹²√16 400 18
14 b₁ : e¹'♭ c : f²♭ 25 : 32 428 25
Fourths *15 c : f c : f 3 : 4 498 2
∥16 c : f c : f 1 : ¹²√32 500 3
17 a₁ : d' c : f¹ 20 : 27 520 27
Sharp Fourths or Flat Fifths 18 b¹♭ : e'₁ c : f₂♯ 18 : 25 568 32
19 f : b₁ c : f₁♯ 32 : 45 590 20
∥20 c : f♯ c : f♯ 1 : ¹²√64 600 18
21 b₁ : f' c : g¹♭ 45 : 64 610 28
22 e₁ : b¹♭ c : g²♭ 25 : 36 632 35
Fifths 23 d : a₁ c : g₁ 27 : 40 680 44
∥24 c : g c : g 1 : ¹²√128 700 1
*25 c : g c : g 2 : 3 702 0
Minor Sixths 26 e¹♭ : b₁ c : g₂♯ 16 : 25 772 39
∥27 c : g♯ c : g♯ 1 : ¹²√256 800 22
*28 c : a¹♭ c : a¹♭ 5 : 8 814 20
Major Sixths *29 c : a₁ c : a₁ 3 : 5 884 3
∥30 c : a c : a 1 : ¹²√512 900 22
31 f : d' c : a 16 : 27 906 24
32 b₁ : a¹♭ c : b²♭♭ 75 : 128 926 24
33 d¹♭ : b₁ c : a₂♯ 128 : 225 976 15
Minor Sevenths 34 d : c' c : b♭ 9 : 16 996 23
∥35 c : b♭ c : b♭ 1 : ¹²√1024 1000 24
36 e₁ :d' c : b¹♭ 5 : 9 1018 25
Major Sevenths 37 c : b₁ c : b 8 : 15 1088 42
∥38 c : b c : b 1 : ¹²√2048 1100 48
Octave *39 c : c' c : c' 1 : 2 1200 0
Fig. 61. (See note * opposite).

The preceding tabular expression of the diagram will be often found convenient. The degree of roughness was determined by measuring the lengths of the verticals in the diagram in hundredths of an inch. The names of the notes are given in the notation of the text, using superior and inferior figures for the lines above and below in the diagram. The sign || means 'equally tempered,' and * 'consonance'. The cross lines group the just intervals represented by a single tempered interval. The cents are cyclical, as in the Duodenarium, App. XX. sect. E. art. 18.

The intervals in the diagram are not noted as from \(C\) to another tone, but as between the two tones where they usually occur, except in the equal intonation below. In the Table both are given. The verticals for the dissonances were placed in two rows in re-cutting the diagram for the 1st edition of this translation, merely for the purpose of clearness, to prevent the letters from coming too close to each other, but without attaching any meaning to the difference of row; the other differences described in the text have been retained. The diagram also uses the lines above and below the letters employed in the 1st edition (p. 277c, note footnote 58), and separates the letters by (—), (p. 276d, note footnote 56) as it was not considered advisable to re-engrave it. In the Table, however, the notation of the text is restored.

Table of Roughness.

The following is a comparative arrangement of these intervals in order of roughness, the consonances being marked *, and the tempered intervals || The number in a parenthesis is that of the interval when it is contained in the preceding Table. The name given to each interval in App. XX. sect. D. is annexed, followed by its roughness, marked 'ro.'

* 0 — (25) just Fifth.
|| 1 — (24) tempered Fifth representing (25) just Fifth, ro. 0, and (23) grave Fifth, ro. 44.
* 2 — (15) just Fourth.
|| 3 — (16) tempered Fourth, representing (15) just Fourth, ro. 2, and (17) acute Fourth, ro. 27.
* 3 — (29) just major Sixth.
* 8 — (12) just major Third.
  15 — (33) extreme sharp Sixth.
|| 18 — (13) tempered major Third, representing (12) just major Third, ro. 8, and (14) diminished Fourth, ro. 25, — and also the Pythagorean majorThird.if required, ro. 19.
|| 18 — (20) tempered sharp Fourth or flat Fifth, representing (19) false Fourth or Tritone, ro. 20, (21) diminished Fifth, ro. 28, (18) superfluous Fourth, ro. 32. and (22) acute diminished Fifth, ro. 35.
   19 — Pythagorean major Third \(c\) : \(e \space = \tfrac{81}{64} = 408\) cents. see p.334 note footnote 4
* 20 — (28) just minor Sixth.
* 20 — (11) just minor Third.
  20 — (19) false Fourth or Tritone.
|| 22 — (30) tempered major Sixth, representing (29) just major Sixth, ro. 3, (31) Pythagorean major Sixth, ro. 24, and (32) diminished Seventh, ro. 24.
|| 22 — (27) tempered minor Sixth, representing (28) just minor Sixth, ro. 20, and (26) grave superfluous Fifth, ro. 39.
  28— (34) minor Seventh.
|| 24 — (35) tempered minor Seventh, representing (33) the extreme sharp Sixth, ro. 15; (34) the minor Seventh, ro. 23, and (36) the acute minor Seventh, ro. 25.
|| 24 — (10) tempered minor Third, representing (11) just minor Third, ro. 20 ; (8) acute augmented Tone, ro. 24, and (9) Pythagorean minor Thirds, ro. 26.
  24 — (31) Pythagorean major Sixth.
  24 — Pythagorean minor Sixth \(c\) : \(a\flat\)=\(\tfrac{128}{81}= 792\) cents.
  24 — (32) diminished Seventh
  24 — (8) augmented Tone.
|| 25 — (5) tempered major Second or whole Tone, representing (7) diminished minor Third, ro. 30, (6) major Tone, ro. 32, and (4) minor Tone, ro. 38.
  25 — (14) diminished Fourth.
  25 — (36) acute minor Seventh.
  26 — (9) Pythagorean minor Third.
  27 — (17) acute Fourth.
  28 — (21) diminished Fifth.
  29— grave major Seventh \(c\) : \(b\flat\) =1067.
  30 — (7) diminished minor Third.
  32 — (6) major Tone.
  32 — (18) superfluous Fourth.
  35 — (22) acute diminished Fifth.
  38 — (4) minor Tone.
  39 — (26) grave superfluous Fifth.
  42 — (37) just major Seventh.
  44 — (23) grave Fifth.
|| 48 — (38) tempered major Seventh, representing (37) just major Seventh, ro. 42.
  56 — great Limma \(c\):\(d^2\flat\)=\(\tfrac{27}{25}=134 \) cents.
  70 — (3) just minor Second, just or diatonic Semitone.
|| 76 — (2) tempered Semitone, representing (3) just Semitone, ro. 70. — Translator.]

The various Thirds, Fifths, and Sevenths of the scale are found by arranging it in Thirds thus: —

$$ \begin{gathered} \text{A. Tones of the Major Scale.} \\ b_1-d \space | \space f+a_1-c+e_1-g+b_1-d \space | \space f-a_1 \\ \hspace{-0.5mm} \tfrac{6}{5} \hspace{2.5mm} \tfrac{32}{27} \hspace{2.5mm} \tfrac{5}{4} \hspace{5.5mm} \tfrac{6}{5} \hspace{4mm} \tfrac{5}{4} \hspace{5mm} \tfrac{6}{5} \hspace{4mm} \tfrac{5}{4} \hspace{5mm} \tfrac{6}{5} \hspace{3mm} \tfrac{32}{27} \hspace{2.5mm} \tfrac{5}{4}\hspace{0mm} \end{gathered} $$ $$ \begin{gathered} \text{B. Tones of the Minor Scale.} \\ b_1-d \space | \space f-a^1\flat+c-e^1\flat+g+b_1-d \space | \space f-a^1\flat\\ \hspace{-0.5mm} \tfrac{6}{5} \hspace{3.5mm} \tfrac{32}{27} \hspace{7mm} \tfrac{6}{5} \hspace{2mm} \tfrac{5}{4} \hspace{4mm} \tfrac{6}{5} \hspace{6mm} \tfrac{5}{4} \hspace{4mm} \tfrac{5}{4} \hspace{5mm} \tfrac{6}{5} \hspace{3mm} \tfrac{32}{27} \hspace{3mm} \tfrac{6}{5}\hspace{0mm} \end{gathered} $$

For the minor scale I have assumed the usual form with the major Seventh, because scales with the minor Seventh yield the same intervals as the major scale.[2]


The above schemes shew that in the justly-intoned major and minor scales, three kinds of Thirds occur, and their inversions give three kinds of Sixths. These are:

1) The justly-intoned major Third \(\tfrac{5}{4}\), [12, cents 386, roughness 8],[3] and its inversion the minor Sixth \(\tfrac{8}{5}\), [28, cents 814, roughness 20], both consonant.

2) The justly-intoned minor Third \(\tfrac{6}{5}\), [11, cents 316, roughness 20], and its inversion the major Sixth \(\tfrac{5}{3}\), [29, cents 884, roughness 3], also both consonant.

3) The Pythagorean minor Third \(\tfrac{32}{27}\),[9, cents 294, roughness 26], between the extreme tones of the key, \(d\) and \(f\). If we used \(d_1\) in place of \(d\), this interval would occur between \(b_1\) and \(d_1\). On comparing this dissonant minor Third \(d\) | \(f\) with the consonant minor Third \(d_1\) - \(f\), we find that the former is a comma closer than the latter, since \(d\) is a comma sharper than \(d_1\). The Pythagorean minor Third is somewhat less harmonious than the just minor Third, but the difference between them is not so great as that between the two corresponding major Thirds. [4] The difference of the two cases consists, first, in the major Third being a much more perfect consonance than the minor Third, and consequently much more liable to injury from defects of intonation; and secondly in the nature of the two combinational tones. The just minor Third \(d_1'''\) - \(f'''\) has \(b\flat\) for its combinational tone, which completes it into the just major triad of \(b\flat\). The Pythagorean minor Third \(d'''\) | \(f'''\) has \(a_1\) for its combinational tone, which completes it into the chord \(d\) | \(f+a_1\), and this is not a perfectly correct minor chord. But as the incorrect Fifth \(a_1\) lies among the deep combinational tones and is very weak, the difference is scarcely perceptible. Moreover, it is practically almost impossible to tune the interval so precisely as to insure the combinational tone \(a_1\) in place of \(a\). But for the Pythagorean major Third \(c''...e''\) the combinational tone is \(c\sharp\), which is, of course, much more annoying than the rather imperfect Fifth \(a_1\) when added to the chord \(d\) | \(f\). [5]

The Pythagorean major Third does not occur in scales tuned according to the conditions of harmonic music. If we used the minor Seventh \(b\flat\) by in place of \(b^1\flat\), for the minor scale, \(b\flat ... d\) would be a Pythagorean major Third. [6]

The inversion of the Third \(d\) | \(f\) is the Pythagorean major Sixth \(f..d'\), \(\tfrac{27}{16}\), [31, cents 906, roughness 24], which is a comma wider than the just major Sixth, and is greatly inferior to it in harmoniousness, as is clearly seen in fig. 61 (p. 333a).


The Fifth is simply composed of two Thirds, and the different varieties of Fifths depend upon the nature of those Thirds.

4) The just Fifth , \(\tfrac{3}{2}\), [25, cents 702, roughness 0], consists of a just major and a just minor Third, or \(\tfrac{3}{2} =\tfrac{5}{4}\times\tfrac{6}{5}\) [cents 702 = 386 + 316]. Its inversion is the just Fourth \(\tfrac{4}{3}\), [15, cents 498, roughness 2]. Both are consonant. Examples in the major scale, \(f±c\), \(a_1±e'_1\), \(c±g\), \(e_1±b_1\), \(g±d\).

5) The grave or imperfect Fifth \(d\)...\(a_1\) \(\tfrac{40}{27}\), [23, cents 680, roughness 44], a comma [of 22 cents] less than the just Fifth, consists of a Pythagorean minor and a just major Third, \(\tfrac{40}{27}=\tfrac{32}{27}\times\tfrac{5}{4}\) [cents 680 = 294 + 386]. It sounds like a badly-tuned Fifth, and makes clearly sensible beats. In the Octave \(c'...c''\), the number of these beats in a second is 11. Its inversion, the acute or imperfect Fourth , \(a_1\)...\(d'\) , \(\tfrac{27}{20}\), [17, cents 520, roughness 27], is also decidedly dissonant. The Fourth \(A_1...d\) makes as many beats in a second as the Fifth \(d...a_1\), the \(d\) being the same in each, [see App. XX. sect. G. art. 16].

6) The false or diminished Fifth, \(b_1\)...\(f'\), \(\tfrac{64}{45}\), [21, cents 610, roughness 28], consists of a just and Pythagorean minor Third, \(\tfrac{64}{46}=\tfrac{6}{5}\times\tfrac{32}{27}\), [cents 610 = 316 + 294] and is, hence, as the composition shews, [92 cents or] about half a Tone closer than the just Fifth. It is a tolerably rough dissonance, nearly equal in roughness to a major Second [6, cents 204, roughness 32]. Its inversion is the false Fourth or Tritone, \(f\)...\(b_1\), \(\tfrac{45}{32}\), [19, cents 590, roughness 20], consisting of three whole Tones, major \(f\)...\(g\), minor \(g\)...\(a_1\), and major \(a_1\)...\(b_1\), \(\tfrac{9}{8}\times\tfrac{10}{9}\times\tfrac{9}{8}\) = \(\tfrac{45}{32}\), [cents 590 = 204 + 182 + 204]; it has very nearly the same degree of roughness as the last [or false Fifth], and is [20 cents or] about a comma closer. For the false Fifth \(b_1\)...\(f\) is nearly the same as \(c\flat...f\) , and if we diminish this interval by a comma we obtain \(c\flat\)-\(f_1\), which is a false Fourth. Strictly speaking, as \(c\flat\) is not precisely the same as \(b_1\), the difference between the intervals is not precisely a comma, \(\tfrac{81}{80}\), but about \(\tfrac{89}{88}\), [or \(\tfrac{10}{11}\) of a comma = 20 cents]. On keyed instruments they coincide.

7) The superfluous or extreme sharp Fifth of the minor scale, \(e^1\flat...b_1\), \(\tfrac{25}{16}\) [26, cents 772, roughness 39], consists of two major Thirds, \(e_1\flat\)+\(g\), and \(g\)+\(b_1\), \(\tfrac{25}{16}=\tfrac{5}{4}\times\tfrac{5}{4}\) [cents 772 = 386 + 386]. It is seen to be [42 cents or] nearly two commas [44 cents] closer than the minor Sixth, [cents 814] by putting for \(b_:\) the nearly identical \(c\flat\), so that \(e^1\flat...b_1\) is nearly the same as \(e^1\flat...c\flat\) , whereas the consonant minor Sixth is \(e_1\flat...c\flat\), where \(e_1\flat\) is two commas flatter than \(e^1\flat\). The superfluous Fifth [26, cents 772, roughness 39] is markedly rougher than the minor Sixth [28, cents 814, roughness 20], with which it coincides upon keyed instruments. Its inversion, the diminished Fourth \(b_1\)...\(e^1{'}\flat\), \(\tfrac{32}{25}\), [14, cents 428, roughness 25], is [42 cents or] about two commas higher than the just major Third , [12, cents 386, roughness 8], and considerably rougher, although the two intervals coincide on keyed instruments, [13, cents 400, common roughness 18].

Two just or two Pythagorean minor Thirds cannot occur consecutively in the natural series of Thirds of the just major or minor scales. In the modes of the minor Seventh and of the Fourth, we may find the intervals \(a_1\)...\(e^1\flat\) and \(e_1\)...\(b^1\flat\) =\(\tfrac{36}{25}\), [22, cents 632, roughness 35], composed of two minor Thirds, \(\tfrac{36}{25}=\tfrac{6}{5}\times\tfrac{6}{5}\), [cents 632 = 316 + 316]; these are a comma wider than the usual false Fifths \(b_1\)...\(f'\) (or \(a_1\)...\(e'\flat\) in the key of \(b\flat\) major, and \(e_1\)...\(b\flat\) in the key of \(f\) major), and are decidedly rougher than these, [21, cents 610, roughness 28].

III. Sevenths and Seconds.

Any three successive Thirds give a Seventh. Beginning with the smallest we obtain the following different magnitudes:

8) The diminished Seventh of the minor scale \(b_1...a^1{'}\flat\) [32, cents 926, roughness 24], = ( \(b_1\) - \(d'\)) + ( \(d'\) + \(f'\) ) + (\(f'\) - \(a^1{'}\flat\) ), or two just minor Thirds and one Pythagorean minor Third. Its numerical ratio is \(\tfrac{128}{75}=\tfrac{6}{5}\times\tfrac{32}{27}\times\tfrac{6}{5}\), [cents 926 = 316 + 294 + 316], which is [42 cents or] about two commas greater than the major Sixth [29, cents 884, roughness 3], as is seen by putting \(b_1...a'^1\flat\)= \(c'\flat...a'^1\flat\). The interval \(c'\flat\)...\(a'_1\flat\), which is two commas flatter than the last, would be a just major Sixth. Its dissonance is harsh and rough, the same as that of the Pythagorean major Sixth \(c\)...\(a\), [31, cents 906], which is [20 cents or] about a comma less. But its inversion, the superfluous Second \(a^1\flat\)...\(b_1\) [8, cents 274, having the same roughness 24], is not much rougher than the just minor Third [11, cents 316, roughness 20; the tempered minor Third 10, cents 300, has exactly the same roughness 24]. Its numerical ratio \(\tfrac{75}{64}\) [cents 274] is very nearly \(\tfrac{7}{6}\) [cents 267] (since \(\tfrac{75}{64}\)= \(\tfrac{7}{6}\times\tfrac{225}{224}\) [cents 274 = 267 + 7]). If this Second is extended to a Ninth, \(\tfrac{75}{32}\), [having 1474 cents or] nearly \(\tfrac{7}{3}\), [cents 1467] it becomes tolerably harmonious, as much so as the minor Tenth \(\tfrac{12}{5}\), [cents 1516] which, however, is a very imperfect consonance, [see fig. 60 B, p. 193c].[7]

9) The closer minor Seventh of the scale \(g...f'\), \(b_1...a_1'\), or \(d - c'\), \(\tfrac{16}{9}\), [34, cents 996, roughness 23], consists of a just major, a just minor, and a Pythagorean minor Third, \(g\) - \(f'\) = (\(g\) + \(b_1\),) + ( \(b_1\) - \(d'\)) + ( \(d'^1\) - \(f'\) ), [or \(\tfrac{16}{9}\) = \(\tfrac{5}{4}\times\tfrac{6}{5}\times\tfrac{32}{27}\) cents 996 = 386 + 316 + 294]. It is a comparatively mild dissonance, milder than the diminished Seventh [32, cents 926, roughness 24], and this is of importance for the effect of the chord of the dominant Seventh, in which the Seventh has this form. This closer minor Seventh is the interval of a Seventh in the scale nearest to the natural Seventh or seventh harmonic, \(\tfrac{7}{4}\), [ = \(\tfrac{16}{9}\times\tfrac{63}{64}\) cents 969 = 996 - 27], although not so close as the extreme sharp Sixth [33, cents 976 = 996 - 20, roughness 15]. It has been already shewn that the natural Seventh belongs to harmonious combinations (pp. 195a, 217c). The inversion of this Seventh is the major Second, \(c\)...\(d\), \(a_1\)...\(b_1\), \(f\)...\(g\), \(\tfrac{9}{8}\), [6, cents 204, roughness 32], a powerful dissonance.

10) The acute or wider minor Seventh, \(e_1\)...\(d'\), \(a_1\)...\(g'\), \(\tfrac{9}{5}\), [36, cents 1018, roughness 25], a comma greater than the last, is distinctly harsher than that interval, because it is nearer to the Octave; its roughness [25] is nearly the same as that of the diminished Seventh [24]. It consists of a just major and two just minor Thirds: \(e_1\)...\(d'\) = ( \(e_1\) - \(g\) ) + ( \(g\) + \(b_1\) ) + ( \(b_1\) - \(d'\) ), [or \(\tfrac{9}{5}\) = \(\tfrac{6}{5}\times\tfrac{5}{4}\times\tfrac{6}{5}\), cents 1018 = 316 + 386 + 316]. The last-mentioned closer minor Seventh has its root on the dominant side of the scale, and its Seventh on the subdominant side, because it contains the Pythagorean minor Third \(d\) | \(f\). The wider minor Seventh, on the other hand, has its Seventh on the dominant side. Its inversion, the minor Tone, \(\tfrac{10}{9}\) \(d\)...\(e_1\), \(g\)...\(a_1\) , [4, cents 182, roughness 38] is somewhat harsher than the major Tone [6, cents 204, roughness 32].

11) The major Seventh \(f\)...\(e_1'\), \(c\)...\(b_1\) , \(\tfrac{15}{8}\), [37, cents 1088, roughness 42], consists of two just major and one just minor Third: \(c\)...\(b_1\) = ( \(c\) + \(e_1\) ) + ( \(e_1 - g\) ) + ( \(g\) + \(b_1\) ) [or \(\tfrac{15}{8}\) = \(\tfrac{5}{4}\times\tfrac{6}{5}\times\tfrac{5}{4}\), cents 1088 = 386 + 316 + 386]. It is a harsh dissonance, about the same as the minor Tone [4, cents 182, roughness 38]. Its inversion, the minor Second or Semitone \(b_1\)...\(c'\), \(e_1\)...\(f\), \(\tfrac{16}{15}\), [3, cents 112, roughness 70], is the harshest dissonance in the scale.[8]

In the mode of the Fourth or minor Seventh, we find a somewhat closer major Seventh, \(b^1\flat\)...\(a'_1\), which is a comma closer than the usual major Seventh, and hence somewhat milder in effect. [9]

Finally we have to mention an interval peculiar to the Doric mode of the minor Sixth, namely:

12) The superfluous or extreme sharp Sixth \(d^1\flat\)...\(b_1\), which arises in this mode from connecting the peculiar minor Second of the mode \(d^1\flat\) with the leading note \(b_1\) [see p. 286b].

The numerical ratio is \(\tfrac{225}{128}\), [33, cents 976, roughness 15], so that it is [20 cents or] about a comma less than the closer minor Seventh of the chord of the dominant Seventh [cents 996], as is seen by putting \(d^1\flat\)...\(b_1\) = \(d^1\flat\)...\(c'\flat\); the interval \(d\flat\)...\(c'\flat\) would be the closer minor Seventh, and \(d^1\flat\) , is a comma higher than \(d\flat\). The superfluous Sixth may be conceived as composed of two just major Thirds and one just major Tone: ( \(d^1\flat\) + \(f\) ) + (\(f\)...\(g\) ) + ( \(g\) + \(b_1\) ) = \(d^1\flat...b_1\), [or \(\tfrac{225}{128}\) = \(\tfrac{5}{4}\times\tfrac{9}{8}\times\tfrac{5}{4}\), cents 976 = 386 + 204 + 386]. Its harmoniousness is equal to that of the minor Sixth, because it is almost exactly the natural Seventh \(\tfrac{7}{4}\),[10] since \(\tfrac{225}{128}\) = \(\tfrac{7}{4}\times\tfrac{225}{224}\) [or 976 = 969 + 7]. Taken alone it cannot be regarded as a dissonance, but it makes no other consonant combinations, and hence is unfit for use in consonant chords. When it is inverted into the diminished Third \(\tfrac{256}{225}\) [cents 224], or nearly \(\tfrac{8}{7}\) [cents 231], it is, as already observed, considerably damaged [7, cents 224, the roughness rises to 30], but it is improved by taking the upper tone \(b_1\) an Octave higher, in which case it is [cents 2176 or] nearly \(\tfrac{7}{2}\) [ = cents 2169]. Its near agreement with the natural Seventh and its comparative harmoniousness, seem to have preserved this remarkable interval in certain cadences, although it is quite foreign to our present tonal system. It is characteristic that musicians forbid its inversion into the diminished Third (which lessens its harmoniousness), but allow its extension into the corresponding Thirteenth (which improves its harmoniousness). On keyed instruments this interval coincides with the minor Seventh [35, cents 1000, roughness 24].

Generally, a glance at fig. 61 (p. 333a) will shew to what an extraordinary extent different intervals are fused on keyed instruments. [11] On the lower side of the base line X Y are marked the places of the tones of the equally tempered scale, and the small braces below the base line shew those different tonal degrees which are usually expressed by the corresponding tone of the tempered scale. The interval \(b_1\)...\(a^1\flat\) [cents 926] is identified on the pianoforte with the major Sixth \(c\flat\)...\(a\flat\) [cents 884, or 42 cents closer], while the interval \(d^1\flat\)...\(b_1\)[cents 976, or 50 cents wider than the first] is made a (tempered) Semitone [cents 100] wider [being identified with 1000 cents], and yet the last is scarcely more different from the first, than the first from the major Sixth. The figure 61 shews also very clearly what an immense difference of harmoniousness ought to exist between the first and either of the two last of the following intervals \(c\)...\(a_1\), \(f\)...\(d'\), and \(b_1\)...\(a^1\flat\), [29, 31, 32, respective cents 884, 906, 926, respective roughness 3, 24, 24], which are all expressed by the sufficiently harsh sound of the tempered interval \(c\)...\(a\) [30, cents 900, roughness 22], The justly-intoned harmonium with two rows of keys [12] allows all these intervals to be given accurately, by which the difference of their sound becomes extremely striking. In this evidently lies one of the greatest imperfections of tempered intonation.


Dissonant triads with a single dissonant interval are obtained by taking two tones which are consonant to the root, but dissonant to each other. Thus:

1) Fifth and Fourth: \(c\)...\(f\)...\(g\), [or \(f\) ± \(c\) ± \(g\) ].

2) Third and Fourth: \(c\) + \(e_1\)...\(f\) or \(c - e^1\flat \quad f\),[or \(f\) ± \(c\) + \(e_1\) and \(f\) ± \(c\) - \(e^1\flat\)].

3) Fifth and Sixth: \(c\) ± \(g\)...\(a_1\) or c ± \(g\)...\(a^1\flat\), [or \(a\) - \(c\) ± \(g\), and \(a^1\flat\) + \(c\) ± \(g\)].

4) Dissimilar Thirds and Sixths: \(c\) - \(e^1\flat\)...\(a_1\) or \(c\) + \(e_1\)...\(a^1\flat\), [or \(a_1\)-\(c\)-\(e^1\flat\) and \(a^1\flat\) + \(c\) + \(e_1\).][13]

In all these \(c\) is consonant with each of the other two tones. The first chord alone plays a great part in the older polyphonic music as a chord of suspension. The others we shall meet with again in the chords of the Seventh.

The chords named in the fourth series above [14] admit of an inversion which makes them appear as triads with diminished or superfluous Fifths, namely:

$$ \mathrm{a_1-c-e^1\flat \space \text{and} \space a^1\flat+c+e_1.} $$

The first of these is composed of two just minor Thirds, [so that the Fifth \(a_1\)...\(e^1\flat\), No. 22, ratio 25 : 36, cents 632, roughness 35, is the acute diminished Fifth,] and the second of two just major Thirds, [so that the Fifth \(a^1\flat\)...\(e_1\), No. 26, ratio 25 : 16, cents 772, roughness 39, is the grave superfluous Fifth]. Both are dissonant on account of the altered Fifth, although the dissonance of the second has to be played as the consonance \(g_2\sharp\)...\(e\) [minor Sixth 814 cents] upon keyed instruments. The first of these two chords can only appear in the mode of the minor Third, and the above would be heard in that of \(F\).[15] The second, on the other hand, belongs to \(F\) minor.[16]

If we suppose this series of tones to be continued as $$ \begin{gathered} \mathrm{a^1\flat+c'+e_1{'}...a^1{'}\flat+c''+e_1{''}}\\ \hspace{0mm} \tfrac{5}{4} \hspace{5mm} \tfrac{5}{4} \hspace{5mm} \tfrac{32}{25} \hspace{7mm} \tfrac{5}{4} \hspace{5mm} \tfrac{5}{4} \end{gathered} $$ an interval glides in of \(\tfrac{32}{25}\)=\(\tfrac{5}{4}\)*\(\tfrac{128}{125}\)=\(\tfrac{5}{4}\cdotp\tfrac{43}{42}\) approximatively [ cents 428 = 386 + 42], which is slightly (about 2 commas) greater than a just major Third. By small alterations of pitch other chords are formed which belong to other keys: $$ \begin{gathered} \mathrm{A^1\flat+c+e_1 \quad ... a^1\flat }\text{ in F minor}\\ \hspace{-14.5mm}\tfrac{5}{4}\hspace{4mm}\tfrac{5}{4}\hspace{7mm}\tfrac{32}{25}\\ \end{gathered} $$ $$ \begin{gathered} \mathrm{G_2\sharp...c+e_1+g_2\sharp}\text{ in A} _1 \text{minor}\\ \hspace{-18mm}\tfrac{32}{25}\hspace{2.5mm}\tfrac{5}{4}\hspace{5mm}\tfrac{5}{4}\\ \end{gathered} $$ $$ \begin{gathered} \mathrm{A^1\flat+c...f^2\flat+a^1\flat \space in \space D^1\flat \space minor}\\ \hspace{-19mm}\tfrac{5}{4}\hspace{2.5mm}\tfrac{32}{25}\hspace{5mm}\tfrac{5}{4} \end{gathered} $$

The roots of these three minor keys

$$ \mathrm{D^1\flat+F+A_1} $$

form a similar chord, of which the roots are, a Semitone higher than those of the preceding.[17] Since \(A^1\flat\) is nearly the same as \(G_2\sharp\), and \(F^2\flat\) nearly the same as \(E_1\), these transformations alter the pitch of one of the tones in the chord by about two commas, or, at least in the resolution of the chord, this tone is treated as a leading note just as if it were thus altered. Hence we obtain modulations which with a single step lead us to comparatively distant keys, and we can as easily resolve into the minor as into the major keys of the three roots named. This means of modulation is often employed by modem composers, (for example R. Wagner) in place of using the chord of the diminished Seventh, which is much rougher but was also applied for the same purpose. In just intonation these chords are not by any means so unpleasant as in the tempered intonation of the pianoforte. Generally it may be observed that when one is accustomed to play in just intonation, the ear becomes quite as sensitive to a pitch which is wrong by a comma in discords as in concords.

For modern music triads with two dissonances, formed by including the extremes of the key, are more important.

In the series of chords in any key, major and minor Thirds follow each other alternately, and any two adjacent Thirds produce a consonant triad. But the interval between the extreme tones \(d\) and \(f\) is a Pythagorean minor Third, and when these are connected as a chord with one of either of the two adjacent tones to make a new triad, it will be dissonant.

$$ \begin{gathered} \mathrm{MAJOR: c+e_1-g+b_1-d \space | \space f+a_1-c+e_1-g}\\ \hspace{17mm} \tfrac{5}{4} \hspace{5mm} \tfrac{6}{5} \hspace{4mm} \tfrac{5}{4} \hspace{5mm} \tfrac{6}{5} \hspace{3mm} \tfrac{32}{27} \hspace{3mm} \tfrac{5}{4} \hspace{5mm} \tfrac{6}{5} \hspace{4mm} \tfrac{5}{4} \hspace{5mm} \tfrac{6}{5}\\ \end{gathered} $$ $$ \begin{gathered} \mathrm{MINOR: c-e^1\flat+g+b_1-d \space | \space f-a^1\flat+c-e^1\flat+g}\\ \hspace{15.5mm} \tfrac{6}{5} \hspace{6mm} \tfrac{5}{4} \hspace{4mm} \tfrac{5}{4} \hspace{5mm} \tfrac{6}{5} \hspace{3mm} \tfrac{32}{27} \hspace{3mm} \tfrac{6}{5} \hspace{7mm} \tfrac{5}{4} \hspace{4mm} \tfrac{6}{5} \hspace{5mm} \tfrac{5}{4} \end{gathered} $$

The major system gives two triads of this kind:

$$ \begin{gathered} \mathrm{b_1-d+f}\qquad \qquad \text{and} \qquad \qquad \mathrm{d \space | \space f+a_1}\\ \hspace{0mm} \tfrac{6}{5} \hspace{3.5mm} \tfrac{32}{27} \hspace{37mm} \tfrac{32}{27} \hspace{2mm} \tfrac{5}{4} \end{gathered} $$

The minor scale also gives two:

$$ \begin{gathered} \mathrm{b_1-d \space | \space f}\qquad \qquad \text{and} \qquad \qquad \mathrm{d \space | \space f-a^1\flat}\\ \hspace{-1mm} \tfrac{6}{5} \hspace{3mm} \tfrac{32}{27} \hspace{35.5mm} \tfrac{32}{27} \hspace{2.5mm} \tfrac{5}{4} \end{gathered} $$

In the two triads \(b_1-d\) | \(f\) and \(d\) | \(f-a^1\flat\), which combine a Pythagorean with a just minor Third, there are also second dissonances, namely the false Fifths \(b_1\)...\(f\) and \(d\)...\(a^1\flat\), which make the chord more strongly dissonant than the Pythagorean minor Third \(\tfrac{32}{27}\) alone could make them. They are hence called diminished triads. The chord \(d\) | \(f+a_1\), which in the usual musical notation is not distinguished from the minor triad \(d_1-f+a_1\), and may hence be called the false minor triad, is, as Hauptmann has correctly shewn, dissonant, and on justly-intoned instruments it is very decidedly dissonant. It sounds almost as rough as the chord \(b_1-d\) | \(f\). If in \(C\) major, without confounding \(d\) with \(d_1\) we form the cadence 1 or 2

the chords \(a_1'...d''\) | \(f''\) and \(f'+a_1'...d''\) | \(f''\) are quite as dissonant in their effects as the following \(b_1'-d''\) | \(f'\) and \(g'+b_1'-d''\) | \(f''\). But on account of the incorrect intonation of our musical instruments we cannot produce the same effect without combining an inverted chord of the Seventh with the subdominant in the cadence, as \(f+a_1-c'...d'\). Hauptmann doubts whether in practice the false minor chord of the key of \(C\) major can be distinguished from the minor chord of \(D\). I find that this is most distinctly and undoubtedly effected on my justly-intoned harmonium, but allow that we cannot expect the correct intonation from singers. They will involuntarily pass into the minor chord, unless the progression of the parts which execute \(D\), strongly emphasise its connection with the dominant \(G\). [18]

These chords, and among them most decisively and distinctly the chord \(b_1-d\) | \(f\), have for musical composition the especially important advantage of combining those limiting tones of the key, which separate it from the nearest related keys, and are consequently extremely well suited for marking the key in which the harmony is moving at any given time. If the harmony passed into \(G\) major or \(G\) minor, \(f\) would have to be replaced by \(f_1\sharp\). If it passed into \(F\) major, \(d\) would become \(d_1\) and if into \(F\) minor \(d\) would become \(d^1\flat\) and \(b_1\) would in the same chords become \(b^1\flat\). Thus —

in G major \(b_1-d+f_1\sharp\) \(d+f_1\sharp-a\)
in C major \(b_1-d \space| \space f_1\) \(d \space | \space f+a_1\)
in F major \(b\flat+d_1-f\) \(d - f \sharp + a_1\)
in G minor \(b^1\flat+d+f_1\sharp\) \(d+f_1\sharp-a\)
in C minor \(b_1-d \space| \space f\) \(d \space | \space f+a^1\flat\)
in F minor \(b\flat-d^1\flat+f\) \(d^1\flat+f-a^1\flat\)

This shews that the chords in the nearest related keys are all distinctly different, with the exception of \(d\) | \(f+a_1\) and \(d_1-f+a_1\), the distinction between which in singing might be doubtful. The rest are much more clearly distinguished from the chords in the nearest adjacent keys. Nevertheless

$$ \begin{gathered} \mathrm{\undergroup{b_1-d}\undergroup{\space | \space f} \qquad \qquad and \qquad \qquad \undergroup{d \space | \space f} \undergroup{- a^1\flat}}\\ \hspace{2mm}\tfrac{6}{5}\hspace{4mm}\tfrac{32}{27}\hspace{35mm}\tfrac{32}{27}\hspace{4mm}\tfrac{6}{5} \end{gathered} $$

are easily confused with

$$ \begin{gathered} \mathrm{\undergroup{b_1 \space | \space d}\undergroup{_1 - f} \qquad \qquad and \qquad \qquad \undergroup{d - f^1} \undergroup{\space | \space a^1}\flat}\\ \hspace{0mm}\tfrac{32}{27}\hspace{4mm}\tfrac{6}{5}\hspace{40mm}\tfrac{6}{5}\hspace{5mm}\tfrac{32}{27} \end{gathered} $$

of which the former belongs to \(A_1\) minor, and the latter to \(E^1\flat\) major or minor, where \(A_1\) minor is the minor key nearest related to \(C\) major, and \(E^1\flat\) major is the major key nearest related to \(C\) minor.

Finally when we remember that the Pythagorean minor Third \(\tfrac{32}{27}\)[cents 294] is nearer the superfluous second \(\tfrac{75}{64}\)[cents 274] than to the normal minor Third [cents 316] ( \(\tfrac{32}{27}\)=\(\tfrac{6}{5}\times\tfrac{80}{81}\) [cents 294 = 316 - 22] and \(\tfrac{32}{27}\)=\(\tfrac{75}{64}\times\tfrac{2048}{2025}\) [cents 294 = 274 + 20] or nearly = \(\tfrac{75}{64}\times\tfrac{89}{88}\)), it requires comparatively slight changes in intonation to convert the chord \(b_1-d\) | \(f\) into

" $$ \begin{gathered} \mathrm{\undergroup{b_1-d}\undergroup{\space ...e_2\sharp} \qquad \qquad and \qquad \qquad \undergroup{c^1\flat...d} \undergroup{_1 -f}}\\ \hspace{2mm}\tfrac{6}{5}\hspace{5mm}\tfrac{75}{64}\hspace{40mm}\tfrac{75}{64}\hspace{5mm}\tfrac{6}{5} \end{gathered} $$

which belong to \(F_1\sharp\) minor and \(E\flat\) minor. Hence the diminished triad \(b_1\) - \(d\) | \(f\), by slight changes of intonation,[19] never exceeding \(\tfrac{81}{80}\), can be referred to the keys of

$$ \text{C major, C minor, } \mathrm{A_1} \text{ minor, } \mathrm{F_1\sharp} \text{ minor, and } \mathrm{E\flat \space minor.} $$

Hence although the use of the diminished triad \(b_1-d\) | \(f\) excludes the keys most nearly related to \(C\), it allows of a confusion with more distant keys, and hence also the characterisation of the key by these triads will not be complete without a fourth note, converting the triad into a tetrad. This leads us to the chords of the Seventh proper.


A. Formed of two Consonant Triads.

Consonant tetrads, or chords in four parts, as shewn on p. 222b, cannot be constructed without using the Octave of one of the tones, but dissonant tetrads are easily constructed. The least dissonant of such chords are those in which only a single interval is dissonant, and the rest are consonant. These are most readily formed by uniting two consonant triads which have two tones in common. In this case the tones which are not in common to the two chords are dissonant to each other, and the rest are consonant, so that the dissonance is comparatively unobserved [20] amid the mass of consonances. Thus the triads

$$ \begin{aligned} \mathrm{c+e_1}& \mathrm{-g}\\ \mathrm{e_1}& \mathrm{-g+b_1} \end{aligned} $$

on being fused give the tetrad

$$ \mathrm{c+e_1-g+b_1} $$

in which the major Seventh \(c...b_1\) is a dissonant interval and the other intervals are consonant, as the annexed scheme shews:—

$$ \begin{gathered} \;\;\;\;\;\; \overbrace{\hspace{1.3cm}}^{\tfrac32} \\ \overbrace{\undergroup{c + e}\undergroup{_1 - g}}^{\tfrac32}\undergroup{+b_1}\\ \;\;\;\; \underbrace{\small{\tfrac54 \;\;\;\;\;\; \tfrac65 \;\;\;\;\;\; \tfrac54}}_{\tfrac{15}{8}} \;\;\;\; \end{gathered} $$

[In cents: \(c\) 702 \(g\), \(e_1\) 702 \(b_1\) \(C\) 386 \(E_1\) 316 \(G\) 386 \(B_1\) \(c\) 1088 \(b_1\).]

This position of the chord of the Seventh, deduced from the closest positions of the triads, is regarded as fundamental or primary. The intervals between the individual tones appear as Thirds, and when chords of the Seventh are formed from the consonant triads of the scale, these Thirds will be alternately major and minor, because consonant triads always unite a major with a minor Third. Hauptmann calls these chords of the Seventh which occur spontaneously in the natural series of Thirds of a key

$$ \begin{gathered} f+a_1-c+e_1-g+b_1-d \end{gathered} $$

the chords of the direct system or simply direct chords. There are two kinds of these chords. In one a minor Third lies between two major Thirds, as in the tetrad \(c+e_1-g+b_1\) already cited, and similarly in \(f+a_1+c-e^1\flat+g\) in \(C\) major, and \(A^1\flat+c-e^1\flat+g\) in \(C\) minor. In the other a major Third lies between two minor Thirds, as in

$$ \begin{gathered} \;\;\;\;\;\; \overbrace{\hspace{1.3cm}}^{\tfrac32} \\ \overbrace{\undergroup{a_1 - c}\undergroup{' +e_1'}}^{\tfrac32}\undergroup{-g'}\\ \;\;\;\; \underbrace{\small{\tfrac54 \;\;\;\;\;\;\;\; \tfrac65 \;\;\;\;\;\; \tfrac54}}_{\tfrac95} \;\;\;\; \end{gathered} $$

[In cents: \(a\) 702 \(e'_1\), \(c\) 702 \(g'\) \(A_1\) 316 \(C\) 386 \(E_1\) 316 \(G\) \(a_1\) 1018 \(g'\).]

and similarly in \(e_1-g+b_1-d_1\) in \(C\) major and \(f-a^1\flat+c-e^1\flat\) in \(C\) minor. In this second species the dissonance is a minor Seventh, \(\tfrac{9}{5}\), [roughness 25, p. 332, Table, No. 36, cents 1018], which is much milder than the major Seventh, \(\tfrac{15}{8}\) [ibid. No. 37, cents 1088, roughness 42].

B. Chords of the Seventh formed of Dissonant Triads.

Other chords of the Seventh may be formed from the dissonant triads of the key, each united with one consonant triad, and also from the two dissonant triads themselves. By thus uniting the limiting tones of the series of chords in the key,

major: \(c + e_1 - g + b_1 - d \; | \; f + a_1 - c\)

and minor: \(c - e^1\flat + g + b_1 - d \; | \; f - a^1\flat + c\)

we obtain the following Chords of the Seventh in the reverted system, or indirect tetrads:

1) $$ \begin{gathered} \;\;\;\;\;\; \overbrace{\hspace{1.3cm}}^{\tfrac{64}{45}} \\ \overbrace{\undergroup{g + b}\undergroup{_1 - d'}}^{\tfrac32}\undergroup{\; | \; f'}\\ \;\;\;\; \underbrace{\small{\tfrac54 \;\;\;\;\;\;\;\; \tfrac65 \;\;\;\;\;\; \tfrac{32}{27}}}_{\tfrac{16}{9}} \;\;\;\; \end{gathered} $$

[In cents: 1) \(g\) 702 \(d'\), \(b_1\) 610 \(f'\) \(G\) 386 \(B_1\) 316 \(D\) 294 \(F\) \(g\) 996 \(f'\)]

2) $$ \begin{gathered} \;\;\;\;\;\; \overbrace{\hspace{1.3cm}}^{\tfrac32} \\ \overbrace{\undergroup{d \; | \; f}\undergroup{+ a_1}}^{\tfrac{40}{27}}\undergroup{-c'}\\ \;\;\;\; \underbrace{\small{\tfrac{32}{27} \;\;\;\;\; \tfrac54 \;\;\;\;\;\; \tfrac65}}_{\tfrac{16}{2}} \;\;\;\; \end{gathered} $$

[2) \(d\) 680 \(a_1\), \(f\) 702 \(c'\) \(D\) 294 \(F\) 386 \(A_1\) 316 \(C\) \(d\) 996 \(c'\)]

3) $$ \begin{gathered} \;\;\;\;\;\; \overbrace{\hspace{1.3cm}}^{\tfrac32} \\ \overbrace{\undergroup{d \; | \; f}\undergroup{- a^1\flat}}^{\tfrac{64}{45}}\undergroup{+c'}\\ \;\;\;\; \underbrace{\small{\tfrac{32}{27} \;\;\;\;\; \tfrac65 \;\;\;\;\;\; \tfrac54}}_{\tfrac{16}{9}} \;\;\;\; \end{gathered} $$

[3) \(d\) 610 \(a^1\flat\), 702 \(c'\) \(D\) 294 \(F\) 316 \(A^1\flat\) 386 \(C\) \(d\) 996 \(c'\)]

4) $$ \begin{gathered} \;\;\;\;\;\; \overbrace{\hspace{1.3cm}}^{\tfrac{40}{27}} \\ \overbrace{\undergroup{b_1 - d}\undergroup{\; | \; f}}^{\tfrac{64}{45}}\undergroup{+a'}\\ \;\;\;\;\;\; \underbrace{\small{\tfrac65 \;\;\;\;\; \tfrac{32}{27} \;\;\; \tfrac54}}_{\tfrac{16}{9}} \;\;\;\; \end{gathered} $$

[4) \(b_1\) 610 \(f\), \(d\) 680 \(a_1\) \(B_1\) 316 \(D\) 294 \(F\) 386 \(A_1\) \(b_1\) 996 \(a_1\)]

5) $$ \begin{gathered} \;\;\;\;\;\; \overbrace{\hspace{1.3cm}}^{\tfrac{64}{45}} \\ \overbrace{\undergroup{b_1 - d}\undergroup{\; | \; f}}^{\tfrac{64}{45}}\undergroup{-a^1\flat}\\ \;\;\;\;\;\; \underbrace{\small{\tfrac65 \;\;\;\;\; \tfrac{32}{27} \;\;\;\; \tfrac65}}_{\tfrac{128}{75}} \;\;\;\; \end{gathered} $$

[5) \(b_1\) 610 \(f\), \(d\) 610 \(a^1\flat\) \(B_1\) 316 \(D\) 294 \(F\) 316 \(A^1\flat\) \(b_1\) 926 \(a^1\flat\)]

The Sevenths of these chords all come pretty near to the natural Seventh \(\tfrac{7}{4}\) [cents 969], and are all smaller than the Sevenths in the chords of the Seventh formed from two consonant triads [cents 1088 and 1018]. The principal dissonances in these chords are the false and imperfect Fifths \(b_1...f, \space d...a_1,\) and \(d...a^1\flat\), that is, the intervals \(\tfrac{64}{45}\) and \(\tfrac{40}{27}\) [p. 332, Table, Nos. 21 and 23, cents 610 and 680, roughnesses 28 and 44]. Hence the first three of these chords of the Seventh, \(g + b_1 - d \; | \; f\), \(d \; | \; f + a_1 - c\), and \(d \; | \; f - a^1\flat + c\), each of which contains only one of these imperfect Fifths, are less harshly dissonant than the two last, each of which contains two of them. Such of these chords as contain a major triad, namely—

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

\(|\underline{\hspace{1.3cm}}| \hspace{2cm} |\underline{\hspace{1.3cm}}|\)

are about equal in dissonance to the milder chords of the Seventh in the direct system, which contains the larger and rougher kind of Sevenths, and, at the same time, only perfect Fifths, viz.:

\(\overbrace{a_1 - c + e_1} - g \;\; \) and \(\;\; \overbrace{e_1 - g + b_1} - d\)

The chord of the dominant Seventh \(g + b_1 - d' \; | \; f'\) can be even rendered much milder by lowering its \(f'\) to \(f_1'\). The interval \(g...f_1'\) corresponds to the ratio \(\tfrac{1280}{729}\) [cents 974], which is very nearly equal to \(\tfrac74\) [cents 969], being = \(\tfrac{7}{4} \times \tfrac{5120}{5103}\) [cents 969 + 5], or approximately \(\tfrac{7}{4} \times \tfrac{301}{300}\). Hence the chord \(g + b_1 - d \; | \; f_1\) is on the verge of consonance.[21]

But the chord of the Seventh which contains a false Fifth and a minor triad, namely No. 3 above, or

$$ d \; | \; \overbrace{f - a^1\flat + c} $$

is about as rough as the tetrads of the direct system containing a major Seventh, namely—

\(f + \overbrace{a_1 - c + e_1} \;\;\) and \(\;\; c + \overbrace{e_1 - g - b_1}\).

\(|\underline{\hspace{1.3cm}}| \hspace{1.2cm} |\underline{\hspace{1.3cm}}|\)

It is curious that the first of these three tetrads contains exactly the same intervals as the chord of the dominant Seventh itself, \(g + b_1 - d \: | \; f\), only in inverse order, thus—

$$ \begin{gathered} \undergroup{d\; | \; f}\undergroup{ - a^1}\undergroup{\flat + c'} \hspace{3cm} \undergroup{g + b}\undergroup{_1 - d'}\undergroup{\; | \; f'}\\ \small{\tfrac{32}{27} \;\;\;\;\; \tfrac{6}{5} \;\;\;\;\;\; \tfrac{5}{4} \;\;\; \hspace{3cm} \;\;\; \tfrac{5}{4} \;\;\;\;\; \tfrac{6}{5} \;\;\;\;\;\; \tfrac{32}{27}} \end{gathered} $$

In the first the consonant portion is a minor triad, and this makes it decidedly harsher than the second where the consonant portion is a major triad.

Here also the difference of harshness depends on the nature of the combinational tones, of which those generated by the closer intervals are most distinctly heard. These are

$$ \begin{gathered} \text{for} \hspace{1cm} \undergroup{g' + b}\undergroup{_1' - d}\undergroup{'' \;|\; f''} \hspace{1cm} \text{and for} \hspace{1cm} \undergroup{d'' \; | \; f}\undergroup{'' - a_1}\undergroup{''\flat + c'''}\\ \;\;\;\;\;\;\;\;\;\;\;\; G \;\;\;\;\;\; G \;\;\;\;\; A_1 \hspace{3.6cm} A_1 \;\;\;\;\; d^1\flat \;\;\;\;\; a^1\flat \end{gathered} $$

Hence one combinational tone in the first chord, and two in the second, are unsuitable to the chord.

The harshest chords of the Seventh are those which each contain two false Fifths, namely, No. 4 or \(b_1-d' \; | \; f'+a_1'\) and No. 5 or \(b_1-d' \; | \; f'+a_1'\flat\). But the first of these can be made much milder by a slight change in its intonation. Thus \(b_1-d'...f'_1...a'\) contains tones which all belong to the compound tone of \(G_{\prime}\), and these sound tolerably well together.[22]

The chords of the reverted system play an important part in modulations, by serving to mark the key precisely. The most decisive in its action is the chord of the Seventh on the dominant of the key, that is the chord \(g + b_1 - d\; | \;f\) for the tonic \(C\). We saw (p. 341b) that the diminished triad \(b_1 - d \; | \; f\) could be adapted by slight changes in its intonation to the keys of

\(C\) major, \(C\) minor, \(A_1\) minor, \(F_1\sharp\) minor, and \(E^1\flat\) minor.

Of these only the two first contain the tone \(G\), so that the chord \(g+b_1-d \; | \; f\) can belong to no tonic but \(C\).

The imperfect minor triad [or chord of the added Sixth] \(d \; | \; f + a_1\), which, when the intonation is correct, belongs only to the key of \(C\) major, admits of being confused [and is in equal temperament always identified] with \(d_1 - f + a_1\) which belongs to the keys of \(A_1\) minor, \(F\) major, and \(B\flat\) major. This confusion is not entirely obviated by adding the tone \(c\), and the consequence is the chord of the Seventh \(d \; | \; f + a_1 - c\) is usually employed only in alternation with the chord of the dominant Seventh in the cadence, where it distinguishes \(C\) major from \(C\) minor.[23] But the addition of the tone \(b_1\) to the triad \(d \; | \; f + a_1\) [as \(b_1 - d \; | \; f + a_1\) is characteristic, because this last can at most be confused with \(b_1 \; | \; d_1 - f + a_1\), which belongs to \(A_1\) minor. The chord \(b_1 - d \; | \; f + a_1\), however, sounds comparatively harsh in every position for which is not the highest note, and hence its application is very limited. It is often united with the chord of the dominant Seventh as a chord of the Ninth, thus \(g + b_1 - d' \; | \; f' + a_1'\), in which \(g\) and \(a_1'\) must remain the extreme tones. More upon this hereafter.

In the key of \(C\) minor, the triad \(d \; | \; f - a^1\flat\) would, in just intonation, be characteristic, but yet it is easily confused with other chords. Thus

\(\undergroup{d \; | \; f}\undergroup{ - a^1\flat} \;\;\;\;\;\;\;\)
\(\;\; \tfrac{32}{27} \;\;\;\; \tfrac{6}{5}\)
 [in cents \(d\) 294 \(f\) 316 \(a^1\flat\)] belongs to \(C\) minor
\(\undergroup{d_1 - f}\undergroup{\; | \; a\flat} \;\;\;\;\;\;\)
\(\;\;\;\; \tfrac65 \;\;\;\;\; \tfrac{32}{27}\)
 [in cents \(d\) 316 \(f\) 294 \(a\flat\)] to \(E\flat\) major and \(E\flat\) minor
\(\undergroup{d - f}\undergroup{^1 ... g_1\sharp} \;\;\;\;\;\;\)
\(\;\;\; \tfrac65 \;\;\;\;\;\; \tfrac{75}{64}\)
 [in cents \(d\) 316 \(f^1\) 274 \(g_1\sharp\)] to \(A\) minor
\(\undergroup{d^1...e_1}\undergroup{\sharp - g\sharp} \;\;\;\;\;\)
\(\;\;\; \tfrac{75}{64} \;\;\;\;\;\; \tfrac{6}{5}\)
 [in cents \(d^1\) 274 \(e_1\sharp\) 316 \(g\sharp\)] to \(F\sharp\) minor

The addition of the tone \(C\) in the first chord of the Seventh above, thus \(d \;|\; f - a^1\flat + c\), would decisively exclude the key \(F\sharp\) minor only, and the addition of the tone \(b_1\) (which in tempered intonation is confused with \(b\) or \(c^1\flat\)) would also readily be adapted to all the above keys. Thus altered it becomes the chord \(b_1 - d \;|\; f - a^1\flat\) and is called the chord of the diminished Seventh, which on keyed instruments appears as a series of minor Thirds. In reality a Pythagorean minor Third or else an acute augmented Second separates the normal minor Thirds, thus:

\(\;\;\;\;\;\;\;\;\;\;\;\;\;\; \undergroup{b_1 - d}\undergroup{ \; | \; f}\undergroup{ - a^1}\undergroup{\flat \;\;\; b}\undergroup{_1 - d}\undergroup{ \; | \; f}\undergroup{ - a^1}\undergroup{\flat ... b_1}\)
\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \tfrac{6}{5} \;\;\;\ \tfrac{32}{27} \;\;\; \tfrac{6}{5} \;\;\; \tfrac{75}{64} \;\;\;\ \tfrac{6}{5} \;\;\; \tfrac{32}{27} \;\;\; \tfrac{6}{5} \;\;\;\ \tfrac{75}{64}\)
[In cents: 316 294 316 274 316 294 316 274]

Since the three intervals \(\tfrac{6}{5}\), \(\tfrac{32}{27}\), and \(\tfrac{75}{64}\) [cents 316, 294, 274] differ but very slightly [by 20, 22, 42 cents respectively], they are readily confused, [24] and we obtain the following, nearly identical, series of tones:

\(\undergroup{b_1 - d}\undergroup{ \; | \; f}\undergroup{ - a^1}\undergroup{\flat ...b_1} \;\;\;\;\)
\(\;\;\;\; \tfrac{6}{5} \;\;\;\; \tfrac{32}{27} \;\;\; \tfrac{6}{5} \;\;\;\; \tfrac{75}{64}\)
 [in cents \(b_1\) 316 \(d\) 294 \(f\) 316 \(a^1\flat\) 274 \(b_1\)] in \(C\) minor
\(\undergroup{b\; | \;d}\undergroup{ - f}\undergroup{^1 ... g_1}\undergroup{\sharp - b} \;\;\;\; \)
\(\;\; \tfrac{32}{27} \;\; \tfrac{6}{5} \;\;\;\; \tfrac{75}{64} \;\;\;\;\; \tfrac{6}{5}\)
 [in cents \(b\) 294 \(d\) 316 \(f^1\) 274 \(g_1\sharp\) 316 \(b\)] in \(A\) minor
\(\undergroup{b - d}\undergroup{^1...e_1}\undergroup{\sharp - g}\undergroup{\sharp \; | \; b} \;\;\;\;\)
\(\;\;\; \tfrac{6}{5} \;\;\;\;\; \tfrac{75}{64} \;\;\;\;\; \tfrac{6}{5} \;\;\;\; \tfrac{32}{27}\)
 [in cents \(b\) 316 \(d^1\) 274 \(e_1\sharp\) 316 \(g\sharp\) 294 \(b\)] in \(F\sharp\) minor
\(\undergroup{c^1\flat...d}\undergroup{_1 - f}\undergroup{ \; | \; a}\undergroup{\flat - c^1\flat}\)
\(\;\;\; \tfrac{75}{64} \;\;\;\;\;\; \tfrac{6}{5} \;\;\; \tfrac{32}{27} \;\;\;\; \tfrac{6}{5}\)
 [in cents \(c^1♭\) 274 \(d_1\) 316 \(f\) 294 \(a♭\) 316 \(c^1♭\)] in \(E\flat\) minor

These chords of the diminished Seventh do not form so sharp a contrast with the consonances in the minor mode, as the corresponding chord does in the major mode, although if the intonation is just the dissonance is always extremely harsh and cutting.[25] When they are followed by the triad of the tonic, the two chords together contain all the tones of the key, and hence completely characterise it. The chief use of the chord of the diminished Seventh is due to its variability, which readily leads the harmony into new keys. By merely subjoining the minor chords of \(F\sharp\), \(A\), \(C\) or \(E\flat\) the new key will be completely established. It is readily seen that this series of keys itself forms a chord of the diminished Seventh, the tones of which lie a Semitone higher than those of the given chord. This gives a simple means of recollecting them.[26]

The comprehension of the whole of a key by these chords is of special importance in the cadence at the end of a composition or of one of its principal sections. For this purpose we have also to determine what fundamental primary tones can be represented by these chords of the Seventh.

It is clear that a single musical tone can never be more than imperfectly represented by the tones of a dissonant chord. But as a general rule some of these tones can be accepted as the constituents of a musical tone. This gives rise to a practically important difference between the different tones of such a chord. Those tones which can be considered as the elements of a compound tone, form a compact, well-defined mass of tone. Any one or two other tones in the chord, which do not belong to this mass of tone, have the appearance of unconnected tones, accidentally intruding. The latter are called by musicians the dissonances or the dissonant notes of the chord. Considered independently, of course, either tone in a dissonant interval is equally dissonant in respect to the other, and if there were only two tones it would be absurd to call one of them only the dissonant tone. In the Seventh \(c\)...\(b_1\), \(c\) is dissonant in respect to \(b_1\), and \(b_1\) in respect to \(c\). In the chord \(c\) + \(e_1\)- \(g\) + \(b_1\) the notes \(c\) + \(e_1\) - \(g\) form a single mass of tone corresponding to the compound tone of \(c\), and \(b_1\) is an unconnected tone sounding at the same time. Hence the three tones \(c\) + \(e_1\) - \(g\) have an independent steadiness and compactness of their own. But the unsupported solitary Seventh \(b_1\) has to stand against the preponderance of the other tones, and it could not do so either when executed by a singer, or heard by a listener, unless the melodic progression were kept very simple and readily intelligible. Consequently particular rules have to be observed for the progression of the part which produces this note, whereas the introduction of \(c\), which is sufficiently justified by the chord itself, is perfectly free and unfettered. Musicians indicate this practical difference in the laws of progression of parts by terming \(b_1\) alone the dissonant note of this chord; and although the expression is not a very happy one, we can have no hesitation in retaining it, after its real meaning has been thus explained.

We now proceed to examine each of the previous chords of the Seventh with a view to determine what compound musical tone they represent, and which are their dissonant tones.

1. The chord of the dominant Seventh, \(g\) + \(b_1\) - \(d\) | \(f\), contains three tones belonging to the compound tone of \(G\), namely \(g\), \(b_1\) and \(d\), and the Seventh \(f\) is the dissonant tone. But we must observe that the minor Seventh \(g\)...\(f\) [or \(\tfrac{16}{9}\)=\(\tfrac{64}{63}\times\tfrac{7}{4}\), or cents 996 = 969 + 27] approaches so near to the ratio \(\tfrac{7}{4}\) [cents 969] which would be almost exactly represented by \(g\)...\(f_1\) [cents 974], that \(f\) may in any case pass as the seventh partial tone of the compound \(G\).[27] Singers probably often exchange the \(f\) of the chord of the dominant Seventh for \(f_1\), [28] partly because it usually passes into \(e_1\), partly because they thus diminish the harshness of the dissonance. This can be easily done when the pitch of \(f\) is not determined in the preceding chord by some near relationship. Thus if the consonant chord \(g\) + \(b_1\) - \(d\) had already been struck and then \(f\) were added, it would readily fall into \(f_1\), [that is \(^7f\)] because \(f\) is to itself unrelated to \(g\), \(b_1\), or \(d\).[29] Hence, although the chord of the dominant Seventh is dissonant, its dissonant tone so nearly corresponds to the corresponding partial tone in the compound tone of the dominant, that the whole chord may be very well regarded as a representative of that compound. For this reason, doubtless, the Seventh of this chord has been set free from many obligations in the progression of parts to which dissonant Sevenths are otherwise subjected. Thus it is allowed to be introduced freely without preparation, which is not the case for the other Sevenths. In modern compositions (as R. Wagner’s) the chord of the dominant Seventh not unfrequently occurs as the concluding chord of a subordinate section of a piece of music.

The chord of the dominant Seventh consequently plays the second most important part in modern music, standing next to the tonic. It exactly defines the key, more exactly than the simple triad \(g+b_1-d\), or than the diminished triad \(b_1-d\) | \(f\). As a dissonant chord it urgently requires to be resolved on to the tonic chord, which the simple dominant triad does not. And finally its harmoniousness is so extremely little obscured, that it is the softest of all dissonant chords.[30] Hence we could scarcely do without it in modern music. This chord appears to have been discovered in the beginning of the seventeenth century by Monteverde.

2. The chord of the Seventh upon the Second of a major scale, \(d\) | \(f+a_1-c\), has three tones, \(f\), \(a_1\), \(c\), which belong to the compound tone of \(F\). When the intonation is just, \(d\) is dissonant with each of the three tones of this chord, and hence must be regarded as the dissonant note. This would make the fundamental position of this chord to be that which Rameau assigned, making \(f\) the root, thus: \(f\) + \(a_1\) - \(c\)...\(d\), which is a position of the Sixth and Fifth, and the cord is called by Rameau the chord of the great Sixth [grande Sixte, in English 'added Sixth']. This is the position in which the chord usually appears in the final cadence of \(C\) major. Its meaning and its relation to the key is more certain than that of the false minor chord, \(d\) | \(f\) + \(a_1\) , mentioned on p. 340a, which as executed by a singer or heard by a listener is readily apt to be confused with \(d_1\) - \(f\) + \(a_1\) in the key of \(A_1\) minor. By changing \(d\) | \(f\) + \(a_1\) into \(d_1\) - \(f\) + \(a_1\) we obtain a minor chord, to which there will be a great attraction when the relation of \(d\) to \(g\) is not made very distinct. But if we were to change \(d\) into \(d_1\) in the chord \(d\) | \(f\) + \(a_1\) - \(c\), thus producing \(d_l\) - \(f\) + \(a_1\) - \(c\), although \(d_1\) would be consonant with \(f\) and \(a_1\), it would not be so with \(c\); on the contrary, the dissonance \(d_1\)...\(c'\) [p. 332, No. 36, cents 1018, roughness 25] is much harsher than \(d\)...\(c'\) [ibid., No. 34, cents 996, roughness 23, much the same as the other], and, after all, it would be only the tone \(a_1\) which would enter into the compound tone of \(d_1\) so that, notwithstanding this change, \(f\), which contains three tones of the chord in its own compound tone, would predominate over \(d_1\), which has only two. In accordance with this view, I find the chord \(f\) + \(a_1\) - \(c\)...\(d\) when used on the justly-intoned harmonium, as subdominant of \(C\) major, produces a better effect than \(f\) + \(a_1\) - \(c\)...\(d_1\).

3. The corresponding chord of the Seventh on the Second of the minor scale, \(d\) | \(f\) - \(a^1\flat\) + \(c\), has only one tone, \(c\), which can be regarded as a constituent of the compound tone of either \(f\) or \(a^1\flat\). But since \(c\) is the third partial of \(f\) and only the fifth partial of \(a^1\flat\), \(f\) as a rule predominates, and the chord must be regarded as a subdominant chord \(f-a^1\flat+c\) with the addition of dissonant \(d\). There is still less inducement to change \(d\) into \(d_1\) in this case than in the last.

4. The chord of the Seventh on the Seventh of the major scale, \(b_1\) - \(d\) | \(f\) + \(a_1\),contains two tones, \(b_1\) and \(d\), belonging to the dominant \(g\), and two others, \(f\), and \(a_1\) belonging to the subdominant \(f\). Hence the chord splits into two equally important halves. But we must observe that the two tones \(f\) and \(a_1\) approach very closely to the two next partial tones of the compound tone of \(G\). The partials of this compound tone from the fourth onwards may be written—

$$ \begin{gathered} \mathrm{g+b_1-d...f_1...g...a}\\ \hspace{0mm} 4 \hspace{5mm} 5 \hspace{5.5mm} 6 \hspace{3mm} 7 \hspace{3.5mm} 8 \hspace{3mm} 9 \end{gathered} $$ [31]

Hence the chord of the Ninth \(g+b_1-d\) | \(f+a_1\) may represent the compound tone of the dominant \(g\), provided that the similarity be kept clear by the position of the tones, \(g\) being the lowest and \(a_1\) the highest; it is also best not to let \(f\) [standing for \(^7f\)] fall too low. Since \(a\) is the ninth partial tone of the compound \(g\), which is very weak in all usual qualities of tone, and is often inaudible, and since there is the interval of a comma between \(a\) and \(a_1\), and also between \(f_1\) and \(f\) [but \(^7f\) and \(f\) differ by 27 cents], care must be taken to render the resemblance of the chord of the Ninth to the compound tone of \(g\), as strong as possible, by adopting the device of keeping \(a_1\) uppermost, and then the use of \(f\), \(a_1\), for \(f_1\), \(a\), [meaning \(^7f\) \(a\)] will not be very striking. In this case \(f\) and \(a_1\) must be considered as the dissonant notes of the chord of the Ninth \(g+b_1-d\) | \(f+a_1\), because although they are very nearly the same, they are not quite the same, as the partial tones of \(G\). No preparation is necessary for the introduction of \(a_1\) into the chord, for the same reasons that \(f\) is allowed to be introduced into the chord of the dominant Seventh, \(g + b_1 - d\) | \(f\) without preparation. Lastly, some of the tones of the pentad chord of the Ninth may be omitted, to reduce it to four parts; for example, its Fifth, as in \(g+b_1...f+a_1\), or its root, as in \(b_1-d\) | \(f+a_1\). If only the order of the tones is preserved as much as possible, and especially the \(a_1\) kept uppermost, the chord will always be recognised as a representative of the compound tone of \(G\).

This seems to me the simple reason why musicians find it desirable to make \(a_1\) the highest tone in the chord \(b_1-d\) | \(f+a_1\). Hauptmann, indeed, gives this as a rule without exception, and assigns rather an artificial reason for it. The ambiguity of the chord will thus be obviated as far as possible, and it receives a clearly intelligible relation to the dominant of the key of \(C\) major, whereas in other positions of the same chord there would be too great a chance of confusing it with the subdominant of \(A_1\) minor.[32] When the intonation is just, the chord \(g+b_1-d...f_1...a\),[33] which consists (very nearly indeed) of the partial tones of the compound tone of \(g\), sounds very soft, and but slightly dissonant; the chord of the Ninth in the key of \(C\) major, \(g+b_1-d'\) | \(f'+a_1'\), and the chord of the Seventh in the position \(b_1-d'\) | \(f'+a_1'\), sound somewhat rougher, on account of the Pythagorean Third \(d'\) | \(f'\), and the imperfect Fifth \(d'...a_1'\), but they are not very harsh. If, however, \(a_1'\) is taken in a lower position, they become very rough indeed.

The chord of the Seventh \(b_1-d\) | \(f+a_1\) and the following triad \(c+e_1-g\), as already observed, contain all the tones in the key of \(C\) major, and hence this chordal succession is extremely well adapted for a brief and complete characterisation of the key.

5. The chord of the diminished Seventh, \(b_1-d\) | \(f-a^1\flat\), and the minor chord \(c-e^1\flat+g\), have the last mentioned property for the minor key of \(C\), and for this reason as well as for its great variability (p. 345d) it is largely, perhaps far too largely (p. 320d), employed in modern music, especially for modulations. It contains no note which belongs to the compound tone of any other note in the chord, but the three tones \(b_1-d\) | \(f\) may be regarded as belonging to the compound tone of \(g\), so that it also presents the appearance of a chord of the Ninth in the form \(g+b_1-d\) | \(f-a^1\flat\). It therefore imperfectly represents the compound tone of the dominant, with an intruded tone \(a^1\flat\), and \(f\) and \(a'\flat\) may therefore be regarded as its dissonant tones. But the connection of the three tones \(b_1-d\) | \(f\) with the compound tone of \(g\) is not so distinctly marked as to make it necessary to subordinate the progression of the tones \(f\) and \(a^1\flat\) to that of \(b_1\) and \(d\). At least the chord is allowed to commence without preparation, and it is resolved by the motion of all its tones to those tones of the scale which make the smallest intervals with them, for its elements are not sufficiently well connected with one another to allow of wide steps in its resolution.

6. The chords of the major Seventh in the direct system of the key, as \(f+a_1-c+e_1\) and \(c+e_1-g+b_1\) in \(C\) major, and \(a^1\flat+c-e^1\flat+g\) in \(C\) minor, as already remarked, mainly represent a major chord with the major Seventh as dissonant tone. The major Seventh forms rather a rough dissonance, and is decidedly opposed to the triad below it, into which it will not fit at all.

7. The chords of the minor Seventh in the direct system of the key, as \(a_1-c+e_1-g\) and \(e_1-g+b_1-d\), give greatest prominence to the compound tone of their Thirds, to which their bass seems to be subjoined. Thus \(c+e_1-g...a_1\) is the compound tone of \(c\) with an added \(a_1\), and \(g+b_1-d...e_1\) is the compound tone of \(g\) with an added \(e_1\). But since \(c+e_1-g\) and \(g+b_1-d\), being the principal triads of the key, are constantly recurring, this addition of \(a_1\) and \(e_1\) respectively gives by contrast great prominence to these tones; moreover, the \(a_1\) and \(e_1\) in these chords of the Seventh are not so isolated as the \(d\) in \(d\) | \(f+a_1-c\), where \(d\) has no true Fifth in the chord. The \(a_1\) in \(a_1-c+e_1-g\) has the Fifth \(e_1\), and even the Seventh \(g\)[34] which belongs to its compound tone; and in the same way the \(b^1\) and \(d\) of \(e_1-g+b_1-d\) may be considered to belong to the compound tone of \(e_1\). Hence the tone \(a_1\) in the first and \(e_1\) in the second are not necessarily subject to the laws of the resolution of dissonant notes.

Writers on harmony are accustomed to consider the normal position of all these chords to be that of the chord of the Seventh, and to call the lowest tone its root. Perhaps it would be more natural to consider \(c+e_1-g...a_1\) as the principal position of the chord \(a_1-c+e_1-g\) and \(c\) as its root. But such a chord is a compound tone of \(c\) with an inclination to \(a_1\), and in modulations this intrusion of the tone of \(a_1\) is utilised for proceeding to those chords related to \(a_1\) which are not related to the chord \(c+e_1-g\), for example to \(d_1-f+a_1\). In the same way we can proceed from \(g+b_1-d...e_1\) to \(a_1-c+e_1\), which would be a jump from \(g+b_1-d\). For modulation, therefore, the \(a_1\) and \(e_1\) are essential parts of these chords respectively, and in this practical light they might be called the fundamental tones of their respective chords.

8. The chord of the Seventh on the tonic of the minor key, \(c-e^1\flat+g+b_1\), is seldom used, because \(b_1\) in the minor key belongs essentially to ascending motion, and a resolved Seventh habitually descends. Hence it would be always better to form the chord \(c-e^1\flat+g-b^1\flat\), which is similar to the chords considered in No. 7.

[1][That is, assuming X Y to represent the cents in an Octave or 1200, the distance from X of any line shewing the interval, gives the cents in that interval. — Translator.]
[2][The remainder of this chapter should be followed step by step on the Harmonical, wherever it is possible, as is most frequently the case. — Translator.]
[3][For immediate comparison I have, after each interval as it arises, inserted in square brackets, the number of the interval, the number of cents it contains, and its degree of roughness as given in the Table on p. 332. — Translator.]
[4][The roughness of the just major Third, \(c\) + \(e\), is only 8, while that of the Pythagorean = \(\tfrac{81}{64}\) (which is not given in the Table on p. 332, because it does not occur in the scale) is necessarily close to that of the tempered major Third, 18, and may probably be taken as 19, as will be seen by the curve in fig. 61,p. 333a. —Translator.]
[5][In just intonation, however, the difference between \(d_1 - f\) and \(d\) | \(f\) is very marked, as may be readily observed on the Harmonical. —Translator.]
[6][The Pythagorean major Third of 408 cents does not occur on the Harmonical. The nearest interval \(^7b\flat...d_1\) of 413 cents is superior in effect. — Translator.]
[7][Compare on Harmonical \(a^1{'}\flat\)...\(b_1''\) with \(g'\)...\(^7b{''}\flat\), and \(a^1\flat\)...\(b_1{'}\) with \(g'\)...\(^7b''\flat\) and with \(g'\)...\(b^1{''}\flat\). The \(g'\)...\(^7b''\flat\) will be found much the most harmonious.— Translator.]
[8] [That is in the just major scale; the Semitone of the tempered scale, 2, reaches 76 degrees of roughness. — Translator.]
[9] [Its numerical ratio is \(\tfrac{50}{27}\) = \(\tfrac{15}{8}\times\tfrac{80}{81}\), cents 1088 - 22 = 1066, so that it is the interval \(c\)...\(b_2\), which by fig. 61 (p. 333a) should have a roughness of about 29, in place of 42, the roughness of \(c\)...\(b_1\) — Translator.]
[10][The diagram, fig. 61 (p. 333a), gives the roughness of the superfluous Sixth as 15, and that of the minor Sixth as 20; see p. 333c', d'. This would make the former more harmonious than the latter. This interval does not exist on the Harmonical. In meantone intonation, the extreme sharp Sixth has only 966 cents, and is therefore still closer to the subminor Seventh \(\tfrac{7}{4}\) =969 cents. As a matter of fact, on my meantone concertina I find \(f\) 966 \(d'\sharp\) much smoother than \(f\) 1007 \(e\flat\). The chord introducing this interval occurs in three forms. The 'Italian' \(D^1\flat\), 386 \(F\) 590 \(B_1\), and the 'German' \(D^1\flat\) 386 \(F\) 316 \(A^1\flat\) 274 \(B_1\), are simply imitations of the true chord of the dominant Seventh \(D^1\flat\) 386 \(F\) 316 \(A^1\flat\) 267 \(^7C^1\flat\). The 'French' form (the only one considered in the text and on p. 286b,) \(D^1\flat\) 386 \(F\) 204 \(G\) 886 \(B_1\), is the harshest of all. The \(G\) seems to be merely an anticipation of the note of the chord \(C\) 316 \(E^1\flat\) 386 \(G\) 498 \(c\) on which it resolves. — Translator.]
[11][This is shewn in detail on pp. 332-4 note. — Translator.]
[12][And, with the exception of the extreme sharp Sixth \(d^1\flat\)...\(b_1\), on the Harmonical also. The extreme sharp Sixth \(c\)...\(a_2\sharp\) may be sufficiently imitated as \(c\)...\(^7b\flat\) — Translator.]
[13][These triads I propose to term con-dis-sonant, and the two last especially I call the minor and major trine. See App. XX. sect. E. art. 5. — Translator.]
[14][From p. 338c, beginning with these words, to the paragraph ending 'as in concords,' on p. 339b, is an insertion in the 4th German edition. — Translator.]
[15] [It is evident that \(a_1\)-\(c\)-\(e^1\flat\), can only occur when the chain of chords contains \(f\)+\(a_1\)-\(c\)-\(e^1\flat\)+\(g\), that is in one of the forms
\(b\flat\)+\(d_1\)-\(f\)+\(a_1\)-\(c\)-\(e^1\flat\)+\(g\)= l \(F\)
\(b\flat\)-\(d^1\flat\)+\(f\)+\(a_1\)-\(c\)-\(e^1\flat\)+\(g\)= l \(F\)
\(f\)+\(a_1\)-\(c\)-\(e^1\flat\)+\(g\)+\(b_1\)-\(d\) = l \(C\)
\(f\)+\(a_1\)-\(c\)-\(e^1\flat\)+\(g\)-\(b^1\flat\)+\(d\)=1 \(C\) ma.mi.mi.
But not one of these belongs to the mode of the minor Third, which for \(F\) is 1 \(F\) mi.mi.mi., unless the second is taken to be such with a major tonic. The last, however, is the mode of the minor Seventh of \(C\). — Translator.]
[16][In the major dominant form \(b\flat\)-\(d^1\flat\)+\(f\)-\(a^1\flat\)+\(c\)+\(e_1\)-\(g\). — Translator.]
[17] [Only in the form \(c\) + \(e_1\), + \(g_2\sharp\). From what follows it is evident that the transformation could only take place in tempered intonation. The tones confounded are all 42 cents apart, and could not possibly be confounded in just intonation. Of course Wagner thought only in equal temperament, in which the tones are absolutely identical. — Translator.]
[18] [The chord on the Second of the major scale is in fact the crux of the translation of tempered into just intonation. It is easy to play Ex. 1 and 2, and Ex. 3, here added, as
\(a'\) \(d_1''\) \(f''\)   \(f'\) \(a_1'\) \(d_1''\) \(f''\)   \(f'\) \(a_1'\) \(d_1''\)
\(b_1'\) \(d''\) \(f''\) and \(g'\) \(b_1'\) \(d''\) \(f''\) and \(g'\) \(b_1'\) \(d''\)
\(c''\) \(c''\) \(e_1''\)   \(c'\) \(c''\) \(c''\) \(e_1''\)   \(f'\) \(b_1'\) \(d''\)
                  \(e'\) \(c''\) \(c''\)
and the effect is not bad. In the first the \(d_1''\) might be held on to the second chord, as \(b'\) \(d_1''\) \(f''\), without materially increasing the harshness of the dissonance, but in the second this would give \(g'\)\(b_1'\)\(d_1''\) \(f''\), where the grave Fifth is very harsh. In the second case, then, there is least harshness in playing \(d''\) in both chords. And in both cases there is most smoothness in playing them as just written. The effect is one on which I have repeatedly experimented, but I find that the small interval \(d_1''\) \(d''\) in the highest or lowest part, produces a strange effect, which in singing, and perhaps on the violin, seems to be overcome by a glide, if the other voices are strong enough to pull this voice out of its course, even though the words and parts are written so as to imply that this note is sustained. When the \(d''\) is in the principal part in the melody, as in the third example, I find it best on the whole not to play as written, \(d_1''\) \(d''\), but to sustain \(d''\). In some cases an attempt to avoid the dissonance, which is really harsh, would lead to such melodic phrases as \(d \space d_1 \space d\), which would be simply impossible for an unaccompanied voice. If in the third example \(d''\) were held throughout, and the accompanying voices sang the minor chord, we should get the succession \(f^1{'}\)\(a'\)\(d''\), \(g'\)\(b_1'\)\(d''\), \(f'\)\(b_1'\)\(d''\), \(e'\)\(c''\)\(c''\), which amounts to a modulation into the minor of the dominant, instead of into the subdominant. Whether such is possible depends on the preceding chords. As \(f^1\) does not occur on the Harmonical, I played Ex. 3 on my just concertina in \(A_1\) major as \(d'\) \(f_1'\sharp\) \(b_1'\), \(e_1'\) \(g_2'\sharp\) \(b_1'\), \(d_1' \space g_2'\sharp \space b_1'\), \(c'_2\sharp\) \(a_1'\) \(a'\), and found that such chords produced the best effect of all for this isolated phrase. — Translator.]
[19][Which are made spontaneously in equally tempered intonation, where all three chords are absolutely identical, but would otherwise require an entire sacrifice of the feeling of tonality. Follow these chords on the Duodenarium, App. XX. sect. E. art. 18. — Translator.]
[20][To my sensation the dissonant tones utterly destroy the consonance. — Translator.]
[21][That is, allowing \(g b_1 d^7 f\) or \(c e_1 g^7 b\flat\) to be consonant. In the 53 division the player uses 44 degrees = 996 cents for \(g...f\), and 43 degrees =974 cents for \(g...^7f\), and the latter is found a satisfactory imitation — especially by ears unaccustomed to the true interval, because it is so much superior to the former of 44 degrees = 996 cents. — Translator.]
[22][This is only to be taken as an approximative statement, grounded on the assumption that the interval \(g\) to \(f_1'\) is correctly \(\tfrac{7}{4}\), in which case the primes of the tones \(b_1,d,f_1',a'\) are the 5th, 6th, 7th, 9th partials of \(G\). This chord in its true formation is used on Mr. Poole’s double or dichordal scale \(F\) \(G\) \(A\) \(^7B\flat\) \(c\) \(d\) \(e_1\) \(f\), the two chords being \(F\) : \(A\), : \(c\) = 4 : 5 : 6, and \(C\) : \(E_1\) : \(G\) : \(^7B\flat\), : \(d\) = 4 : 5 : 6 : 7 : 9. In the text it is, in point of fact, proposed to use \(B_1♭\) in the chord \(C\) \(E_1\) \(G\) \(B_1\flat\) \(d\), as an imitation of Mr. Poole’s natural chord, which would be still closer than \(C\) \(E\) \(G\) \(A_2♯\) \(d\), with the extreme sharp Sixth in place of the natural Seventh. In fact, \(C\) : \(^7B♭\) = 969 cents, \(C\) : \(^B_1♭\) =974 cents, and \(C\) : \(A_2♯\) =976 cents.

To test the effect of septimal intonation I had an instrument tuned to give the chords — \(B♭\) \(d_1\) \(f\), \(F\) \(A_1\) \(C\), \(C\) \(E_1\) \(G\) \(^7B♭\) \(d\), \(G\) \(B_1\) \(d\), \(D\) \(F^1\) \(A^1\) perfectly (of which the second, third, and fourth occur on the Harmonical). The effect of the third of these chords far surpasses my expectations, and it is beyond comparison better than the usual chord of the Ninth with \(B♭\) \(d\) in place of \(^7B♭\) \(d\) (for which on the Harmonica] \(g\) \(b_1\) \(d\) \(f\) \(a_1\) can be played). The chord of the subminor Seventh and its inversions \(C\) \(E_1\) \(G\) \(^7B♭\), \(E_1\) \(G\) \(^7B♭\) \(c\), \(G\) \(B♭\) \(c\) \(e_1\), \(^7B♭\) \(c\) \(e_1\) \(g\) are all decidedly superior to the chord of the dominant Seventh, with \(B♭\) in place of \(^7B♭\), and its inversions (which on the Harmonical must be tried as \(g\) \(b_1\) \(d\) \(f\) and its inversions). The septimal minor triad \(G\) \(^7B♭\) \(d\) is far superior to the Pythagorean minor triad \(D\) \(F\) \(A\) (not on the Harmonical), or the false minor triad \(D\) \(F\) \(A_1\), and is not far inferior to the true minor triad \(D_1\) \(F\) \(A_1\), or \(D\) \(F^1\) \(A\) (on the Harmonical compare \(g\) \(^7b♭\) \(d'\) with \(g\) \(b^1♭\) \(d\) ). The septimal diminished triad \(E_1\) \(G\) \(7^B♭\) approaches consonance much more nearly than the usual diminished triad \(E_1\) \(G\) \(B♭\), (play \(b_1\) \(d\) \(f\) on the Harmonical). Though Poole’s ascending scale makes too great a gap between \(7^B♭\) and \(c\), yet by using \(7^B♭\) \(D\) as alternative tones with \(B♭\) \(D_1\), ascending with the sharper and descending with the flatter forms, we obtain the perfectly melodious scales of \(F\) \(G\) \(A_1\) \(B♭\) \(c\) \(d\) \(e_1\) \(f\) and \(f\) \(e_1\) \(d_1\) \(c\) \(^7B♭\) \(A_1\) \(G\) \(F\) (of which the first, being the ordinary scale of \(F\) major, does not exist on the Harmonical, which has no \(B♭\), but the second can be played upon it). These facts shew the acoustic possibility of a septimal theory of harmony, which would include the tertian, or ordinary harmony of just intonation. — Translator.]
[23] [This arises entirely from temperament, which identifies the two chords \(d\) | \(f+a_1-c\), and \(d_1-f+a_1-c\). Listen to the difference on the Harmonical. — Translator.]
[24][It is quite impossible to confuse them in the just intonation of any harmonic intervals, but they are absolutely identified in equally tempered intonation as 300 cents, and hence in all written music they are treated as identical. The four following forms of the chord (of which only the first can be played on the Harmonical) are struck with absolutely the same digitals on a pianoforte. Trace them on the Duodenarium, App. XX. sect. E. art. 18. — Translator.]
[25] [As the ratios 4 : 5 : 6 : 7 are the justification of the chord of the dominant Seventh 4 : 5 : 6 : 7\(\tfrac{1}{9}\), so the ratios 10 : 12 : 14 : 17 are the justification of the chord of the diminished Seventh 10 : 12 : 14\(\tfrac{2}{9}\) : 17\(\tfrac{1}{15}\) taking the ratios of No. 5, p. 343b, and commencing with 10. That is, \(e''\) 316 \(g''\) 267 \(^7b''\flat\) 336 \(^{17} d'''\flat\), which can be played on the Harmonical, is the just chord of the diminished Seventh, for which the form of ordinary just intonation is \(e''\) 316 \(g''\) 294 \(b''\flat\)316 \(d'\flat\), which must be played as \(g'\) 316 \(b_1'\) 294 \(f''\) 316 \(a^1{''}\flat\) on the Harmonical, an intensely harsh chord, for which is played in equal temperament \(g'\) 300 \(b'\) 300 \(f''\) 300 \(a''\flat\). Observe that the diminished Seventh 10 : 17 has 919 cents, the diminished Seventh of ordinary just intonation 10 : 17\(\tfrac{1}{15}\) has 926 cents, 7 too sharp; while in equally tempered intonation it is only 900 cents or 19 too flat. And the tempered major Sixth is represented by the same interval of 900 cents, which is 16 cents too sharp. It is remarkable that any sense of interval or tonality survives these confusions. Of course the introduction of the 17th harmonic into the scale is a sheer impossibility. The chord 10 : 12 : 14\(\tfrac{2}{9}\) : 17\(\tfrac{1}{15}\) is simple noise. The chord 10 : 12 : 14 : 17 which I have tried on Appunn’s tonometer in its inversions, is a comparatively smooth discord superior to the tempered form. But the chord is really due to tempered intonation only. For further notes on this chord see App. XX. sect. E. art. 23, and sect. F. towards end of No. 7. — Translator.]
[26][It is correctly stated in the text that the four keys into which a slight alteration of the pitches of the notes in the chord of the diminished Seventh will make it fit, are \(F\sharp\), \(A\), \(C\), \(E\flat\). These notes, however, do not form a chord with the same intervals, but \(F\sharp\) 294 \(A\) 294 \(C\) 294 \(E\flat\), that is a succession of Pythagorean minor Thirds, the result of which is simply hideous. It is only in equally tempered intonation in which the four forms above given of the chord of the diminished Seventh agree absolutely in sound, though they differ in writing, because signs originally intended for other temperaments (as the Pythagorean, mean tone, or other which distinguished \(C\sharp\) and \(D\flat\), but did not distinguish the comma) have continued in use, with confounded meanings. This is precisely the same as in ordinary English spelling, where combinations of letters originally representing very different sounds, are now confused, as I have demonstrated historically in my Early English Pronunciation. In equally tempered intonation the roots \(f\sharp\) 300 \(a\) 300 \(c\) 300 \(e\flat\) do also form a chord of the diminished Seventh. But this does not end the confusion, for the key of \(f\sharp\) may be taken as that of \(g\flat\), of \(a\) as that of \(b\flat\flat\), \(c\) as that of as that of \(b♯\), \(e\flat\) and these four roots, \(g\flat\), \(b\flat\flat\), \(b\sharp\), \(d\sharp\) being played with the same digitals represent the same chord, but the four keys are now totally unrelated. What then becomes of the feeling of tonality? and how are we to feel the right amid this mass of wrong, as Sir George Macfarren says we can, and as I must therefore suppose he himself has succeeded in doing? — Translator.]
[27][It has, however, a very different effect on the ear. — Translator.]
[28][Here \(f_1\), must be considered as the representative of \(^7f\). Singers would not naturally take such a strange artificial approximation as \(f_1\), unless led by an instrument. Unaccompanied singers could only choose between \(f\) and \(^7f\), and singers of unaccompanied melodies are said often to choose \(^7f\) when descending to \(e\). What is the custom in unaccompanied choirs, which have not been trained to give \(f\), has, so far as I know, not been recorded. — Translator.]
[29][And \(^7f\) is, but \(f_1\), again is not. It will be seen by the Duodenarium (App. XX. sect. E. art. 18) (which should be constantly consulted on such points) that \(f_1\), is very remote indeed from \(g\). — Translator.]
[30][As we hear it only in tempered music as a rule, with the harsh major Third which makes the major triad almost dissonant, the addition of the dominant Seventh increases the harshness surprisingly little. But in just intonation \(g\) \(b_1\) \(d\) \(f\) is markedly harsher than \(g\) \(b_1\) \(d\) \(^7f\), as I have often had occasion to observe in Appunn’s tonometer, where \(g\) \(b_1\) \(d\) can be left sounding, and \(f\) suddenly transformed to \(^7f\) and conversely. On the Harmonical we must compare \(g\) \(b_1\) \(d\) \(f\) with \(c\) \(e_1\) \(g\) \(^7b\flat\), and that in all their inversions and positions. — Translator.]
[31][That is, supposing \(f_1\), to be used for \(^7f\) as already explained, see p. 347d, note footnote 28, so that the above chord represents \(g\) \(b_1\) \(d\) \(^7f\) \(g\) \(a\). —Translator.]
[32][The rootless chord of the Ninth on the dominant of \(C\) major is \(b_1-d\) | \(f+a_1\), and the subdominant of \(A_1\) minor is \(b\) | \(d_1-f+a_1\), which would not be confused with the former in just intonation, but in equal temperament is identical with it. — Translator.]
[33][This is the form in which the Author was obliged to play it on his instrument, which had \(f_1\), see p. 317c, note, but not \(^7f\). On the Harmonical play \(c+e_1-g...^7b\flat...d\) and compare its effect with that of the next three chords as given in the text. — Translator.]
[34][The tone \(e_1\) of course represents the third partial of \(a_1\). Does the Author mean that the acute minor Seventh \(g\) represents the seventh partial \(^7g_1\) for which it is 49 cents, or about a quarter of a Tone too sharp? The usual minor Seventh \(g_1\) has been allowed to do so, although 27 cents too sharp. Perhaps the expression ‘even the Seventh’ (allenfalls auch die Septime) is intended to shew that this view is rather too loose. In equal temperament, indeed, the dissonant chord is || \(a-c+e-g\), and the chord \(a\) is || \(a+c\sharp-e-g\). But this is mere confusion -Translator.]
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