CHAPTER XV. 
The Double-Fugue.

161. The Double-fugue is one in which two Subjects appear, and are manipulated, simultaneously.

It is not to be confounded with the forms cited in par. 52, par. 106, par. 155, and par. 157, in which two (or more) Themes are treated successively, each by itself for the time being; but takes its name from the fact that its thematic basis is a Double-theme — thematic duet.

It is not necessary that the two Subjects should always be together, for, as will be seen, either (or each) may be announced alone, or even have a separate Exposition; but, in order to achieve the idea and purpose of the Double-fugue, the Themes must appear in conjunction, distinctly, and during a reasonably extended section of the whole.

162. The Double-fugue has its inception in that variety of the single Fugue in which, as shown in par. 132, the first Counterpoint is retained, more or less persistently, throughout a part, or the whole, of the Fugue; for such companionship between the Subject and its Counterpoint approaches the effect of an intentional double thematic basis. Review par. 132, and context, thoroughly. But there still remains a characteristic difference between such examples of retained Counterpoints and the genuine Double-fugue.

163. An important intermediate grade, sufficiently independent to merit separate treatment, is

The Fugue with Thematic Counterpoint.

An ordinary contrapuntal associate of the Theme, such as is found in the majority of simple Fugues, could not be called “thematic,” even if “retained” and introduced occasionally in subsequent announcements, for the reason that it provides no especial thematic feature, does not enrich the thematic resources, and takes no active part in the development of the design.

In order to be thematic, the Counterpoint must be to a certain extent characteristic, must contain rhythmic and melodic features which distinguish it somewhat from the Theme; must recur often enough and emphatically enough to become a recognizable (though not obtrusive) line in the physiognomy of the Fugue; must influence the development, and enter actively (though not vitally, or perhaps even essentially) into the thematic texture as a whole. All this the “Counterpoint” may be, without becoming a genuine Secondary Theme, or subject to any special conditions with regard to time, place, frequency, or accuracy of recurrence.

164. During the Exposition the Thematic Counterpoint appears, at each succeeding announcement of the Theme, in the part which last contained the latter; i.e., the Counterpoint follows the Theme, each time, in the same part (whether interrupted by episodes or not). After the Exposition, however, the location of the Counterpoint, like that of the Theme, is optional; and it may be occasionally omitted, or more or less modified.

EXERCISE 47.

One or two examples of the Fugue (3- or 4-voice) with Thematic Counterpoint.

The Genuine Double-Fugue.

165. In the authentic Double-fugue, the two Subjects are strictly coördinate. The secondary Theme (sometimes so called merely by way of distinction), or Counter-theme, is not a “thematic counterpoint,” but an essential part of the thematic material, subject to precisely the same specific limitations as the principal Theme.

Rule 1. Each Subject must be a perfectly good melody, in and by itself. Hence, each must be tested alone, in order to verify its individual validity as thematic sentence.

Rule 2. Each Subject must be well individualized; must differ from its fellow (chiefly as regards its rhythmic formation) sufficiently to contrast perceptibly in character, though not obtrusively. They should not begin exactly together, nor need they end upon the same beat.

Rule 3. The two Subjects must constitute peculiarly faultless counterpoint, — not by any means to the exclusion of characteristic dissonances, but without any awkward irregularities. They should not cross each other; and, as a rule, should not run much beyond an octave or tenth apart, in their original associated form, when designed for a Fugue with 4 or more parts.

Compare these rules with the primary definition of polyphony on page 1, of which they are simply a singularly severe and just exposition.

Rule 4. After the Double-subject has been completed, it must be tested in inverted form, i.e., the parts so exchanged that the upper becomes the lower. Also, though less urgently, in their joint Contrary motion.

This is necessary, not only because it is impossible to avoid Inversion, in the development of the Double-subject, but because the art of the Double-fugue consists in a great measure in these, and many similar, changes in the relations of the two Themes to each other. Review pars. 55 to [par.] 58, especially par. 57; and refer, briefly, to Artificial Double-counterpoint, par. 169, etc.

Examples of Double-subjects:

166. Double-fugues are divided into three classes or Species, according to the degree of importance assumed by Theme B.

The First Species of Double-Fugue.

167. In this Species, the two Subjects are announced together at the beginning of the Fugue.

a. They may appear in any two of the parts to be employed, but are almost certain to be in neighboring parts (either the upper, inner, or lower pair). During the Exposition, for instance in 4 parts, the rules of par. 123 (which review) apply to each Theme independently of the other; i.e., each one takes its way through the four voices in the usual order. For instance (assuming them to start in the inner pair of voices):

If announced in parallel parts, a collision is possible, but always easily to be avoided. Thus:

These principles can also be readily applied to the Fugue for 3 or 5 voices.

b. After the Exposition these regulations are relaxed, as usual; the two Themes may appear in any pair of parts, in optional frequency; and, possibly, one or the other of the Themes may occasionally appear alone, though this should occur very rarely, as it militates against the specific conditions of this species of the Double-fugue. Review par. 128. Either Subject may, as usual, be slightly modified or abbreviated, as occasion or necessity arises, but not to the extent permitted with a mere “Thematic Counterpoint.”

The formal designs, and all other structural details, correspond exactly to those given in Chapters XII, XIII, and XIV.

Analyze, very minutely, the following examples of the First Species:

EXERCISE 48.

The Second Species of Double—Fugue.

168. In the Second Species, Theme B enters as first contrapuntal associate, in connection with the second announcement of Theme A (i.e., with the first “Response”). It therefore corresponds, in this respect, to the Fugue with Thematic Counterpoint; but all the conditions detailed in par. 165 must be strictly observed.

The order of voices during the Exposition is simpler than in the First Species, because Theme B naturally (and almost invariably) follows Theme A in the same voice,— precisely as shown in Exs. 164, [ex.] 165, excepting that the contrapuntal associate is a genuine Counter-theme, and not merely a thematic counterpoint. But, as Theme A thus appears once alone, the Exposition is generally extended by one extra announcement, in order that Theme B may appear in every part; i.e., in the 4-voice Fugue the Exposition usually embraces five announcements of Theme A, the 5th time in the voice which began the Fugue.

See Bach, Well-temp. Clav., Vol. I, Fugue 12; Theme A is announced in Tenor, then in Alto (meas. 4), Bass (meas. 7 — this is irregular — par. 124), Soprano (meas. 13), and again in Tenor (meas. 19); Theme B follows, each time, in the same voice; the Exposition closes in meas. 22. Exactly the same in Vol. II, Fugue 16, in every particular, excepting that the order of parts is regular; the Exposition closes in meas. 24. — In Vol. II of the Org. Comp., Fugue 9, the Exposition is not thus extended.

The distinction between the 1st and 2nd Species is limited, thus, to the Exposition alone. The subsequent development is identical. Analyze, very carefully, the following; observe, here again, the degree and quality of freedom of detail exercised, and endeavor to recognize the factors that must be regarded as vital and inviolable:

EXERCISE 49.

A number of Double-fugues, 2nd Species; 3-, 4-, or 5-voice; different character and tempo; instrumental or vocal. Some of the Double-subjects of Exercise 48A may be utilized.

Artificial Double-Counterpoint.

169. The term “artificial” is adopted (in its most serious sense) in distinction to “natural,” with reference to those varieties of Double-counterpoint in which the harmonious result is not a matter of course, but must be assured by calculation. Review par. 55.

As stated in pars. 55 and [par.] 56, when the Double-counterpoint is obtained by inversion in the 8ve, no change of interval-relations takes place, and an equally harmonious agreement results, naturally. But no such natural guarantee exists when any other interval of inversion is used. Therefore, Double-counterpoint in the 8ve, without any other modification, is the only Natural kind, while Double-counterpoint in any other interval, or with any other modification of either part, belongs to the Artificial kind, and is to be determined only by calculation or experiment.

170. Double-counterpoint in the Twelfth. Double-counterpoint is obtained by transferring one part towards and past the other; and the species is defined by the interval of inversion involved. For example, if the lower part remains, and the upper part is shifted downward a 12th (an 8ve and a 5th, or perhaps two 8ves and a 5th) the result is Double-counterpoint in the 12th. For illustration:

(a) The Original Counterpoint.

(b) Inversion in the 12th (affecting Th. A).

An apparently new result is obtained by moving the other Theme down, past its fellow, a 12th. But this, or any other movement of both parts together, equivalent to the 12th, represents simply a transposition of the Inverted form. Thus:

(a) Inversion in 12th (affecting Th. B).

(b) Inversion in the 12th (affecting both Themes).

171. The invention of a Double-subject (or any two-voice-sentence) which may thus submit to Inversion in the 12th — or any other interval excepting the 8ve — is so largely a matter of incessant experiment, tone for tone, that scarcely any valid rules can be given. At the same time there is a certain device for each separate species of Double-counterpoint, which systematizes and simplifies the formation to some extent.

The device for Double-counterpoint in the 12th consists in the frequent use of the interval of a 3rd (or 10th) in the Original form; for this interval, when inverted in the 12th, results again in a 3rd; for example:

This point, and others, are illustrated in the following:

172. Double-counterpoint in the 11th. Frequent use of the interval of a 6th will, as a rule, yield a good Inversion in the 11th (the 6th becoming again a 6th). For example:

This species, Double-counterpoint in the 11th, is the companion (the counterpart) of that in the 12th; and, because of the harmonic similarity between the 6th and the 3rd, is scarcely distinguishable from it. One is, technically speaking, simply the 8ve-inversion of the other. For illustration: invert the parts in Ex. 167a (in the octave), and also invert Ex. 167b in the same manner; upon comparing the two results with each other they will be found to constitute Double-counterpoint in the 11th.

173. a. Double-counterpoint in the 10th. The best device for Double-counterpoint in the 10th consists in doubling one of the Themes in the 3rd, and counterpointing the other Theme against this double line. In order to yield counterpoint in the 10th, the duplication must fall between the principal lines as a so-called inner 3rd; i.e., it must be the lower third of the upper voice, or the upper third of the lower. Thus:

*1) The large notes constitute the Original counterpoint; the inner part (small notes) is a duplication of the upper Theme in the lower or inner 3rd, in order to provide for Double-counterpoint in the 10th by inverting the parts, thus:

b. Although Double-counterpoint always implies Inversion of the parts (as stated in par. 55), it must be understood that the principle is the same whether the Inversion actually takes place or not; and therefore Ex. 172 is as certainly Double-counterpoint, with its duplication of one part in the 3rd, as Ex. 173, or any other of the inverted forms.

c. Such a duplication in the 3rd is less difficult and hazardous than might be suspected, and will quite frequently be found practicable with little or no change, in any ordinary counterpoint. The only necessary condition or limitation is, that the parts shall move in contrary directions (at those points where both are moving). This may be verified in Ex. 172. It is applied, with precisely equal certainty, to the other part (Theme B) of the above example, by Bach himself. Thus:

N.B. Each lower part alone must be valid with the upper.

d. Further, exactly the same result is obtained by duplicating either part in the outer 6th (instead of inner 3rd, as in Exs. 172 and [ex.] 174). Thus:

N. B. Play each duplicated part alone with the other. And write out the Inverted forms of each example, complete, as in Ex. 173.

174. Double-cpt. in the 13th (or 6th). This is the companion-species of that in the 10th, and is obtained by duplicating either Theme in the outer 3rd (comp, par. 173a), or inner 6th (par. 173d). For example:

N. B. Play each upper part alone with the other. Write out the example also with duplication of the lower part in the outer (i.e., lower) 3rd. Write out each example in its Inverted form, complete.

175. It is sometimes possible to apply the device of duplication in the 3rd (or 6th) to both parts at once. Thus:

*1) Slightly changed from the Original (Bach, Well-temp. Clav., Vol. II, Fugue 16, meas. 59–63). Comp. with meas. 9–13; and see, also, meas. 45–49; 51–55; 69–73.

The complication of Double-cpt. “Species” involved in such twofold duplications can be defined only by reference to the Original form (here shown in large notes): it is exceedingly confusing, — and no doubt wholly unnecessary.

A long and instructive example of Double-counterpoint in the 10th may be found in Bach, “Art of Fugue,” Canon III; meas. 5–39 are inverted in the 10th in meas. 44–78; the lower part one 8ve higher, and the upper a 10th lower, than before.

Bach, 3-voice Inv. No. 8, meas. 17–18; upper, part duplicated in the inner 6th.— Well-temp. Cl., Vol. II, Fugue 22, meas. 6–3 from the end (double duplications, partly in 6ths, and partly in 3rds). — Bach, Org. Comp., Vol. VI, No. 18, meas. 22–23; Theme A in Bass, Theme B doubled in 3rds, slightly embellished, in upper parts.

176. Double-cpt. in the 9th, and its companion, in the 7th. For these extremely difficult and rare species there is no other guide than the rigid application of the two general principles, of (1) stepwise progression and (2) contrary direction. For example:

A double-subject, devised thus strictly in contrary directions, and with exclusive stepwise progressions, will generally prove available for every species of Double-cpt. That is the case with Ex. 178, as the student may (and should) verify.

Other Varieties of Artificial Double-Counterpoint.

177. These are purely experimental, and cannot be obtained by any method of calculation. They consist in multiplying the relations of the two Themes to each other by applying the ordinary modifications (Contrary motion, Shifted rhythm, Augmentation, or Diminution) to either Theme alone, while the other retains its original form. For illustration, Ex. 167 chances to admit of transformation as follows:

And Ex. 170, No. 2, reappears later thus:

*1) See Ex. 178, Note *2).

See further, Bach, 2-voice Invention No. 11; Original cpt. in meas. 1–2, Theme A in upper, Theme B in lower part; inverted in meas. 3–5, Theme A as before, but in lower part, Theme B in upper part in Contrary motion and Shifted rhythm. — Bach, Well-temp. Cl., Vol. II, Fugue 23; comp. meas. 60–63 with meas. 27–30 (Theme B in Shifted rhythm).

EXERCISE 50.

The Third Species of Double-Fugue.

178. In this species, which may be regarded as the most genuine and distinctive of the Double-fugue, the entire Exposition (first section) is devoted to Theme A alone. And, in the broader Song-forms, this Exposition may be extended a section or more, so as to complete the measure of a full First Part.

The following sections or Parts may then be designed in either of two ways:

(1) Theme B may simply join Theme A, so that they appear in conjunction the rest of the way; or (2) Theme B may also be first manipulated alone, for one section or Part, and the conjunction of the two Subjects follow in the final Sections, or Third Part.

In the latter case, where each Subject is manipulated alone before they appear together, care must be taken to avoid contrapuntal associates that resemble the other Theme too closely, especially with regard to their rhythmic formation; otherwise the subsequent conjunction of the Themes will not be striking enough, and the product may be monotonous.

An incipient type of the first class is shown in par. 142, which review.

The first class is illustrated in the following (to be thoroughly analyzed):

Bach, “The Art of Fugue,” Fugue 9; 4-voice; 3 Sections, cadences vague; Theme A alone to meas. 34; A and B together from meas. 35 to the end. Contains Double-cpt. in the 12th. — Same work, Fugue 14; 4-voice; Theme A alone to meas. 21; A and B together from meas. 22 to the end; in Section III (meas. 34) there is a regular Exposition of an auxiliary Motive, derived from the first Counterpoint (meas. 6–7); in Sec. V (meas. 67) another brief auxiliary Figure appears. Contains Double-cpt. in the 10th and 12th.

The second class is illustrated in the following:

Analyze, further, the following miscellaneous examples (irrespective of class):

Irregular (4th) Species of Double-Fugue.

179. An irregular variety of the Double-fugue, which might be designated as the 4th Species, consists of one principal Theme, with two or more contrasting thematic Counterpoints, one or the other (or all) of which may assume the importance of a genuine secondary Theme.

The thematic Counterpoints (or Counter-themes) are likely to appear in successive sections, or Parts, of the usual forms.

EXERCISE 51.