An eclectic assortment of essays treating not only such familiar music-theoretical topics as the triad and temperament, but also more exotic subjects as Männlich und Weiblich and Ironie der Kunst, Moritz Hauptmann’s Opuscula: Vermischte Aufätze was published posthumously in 1874, having been prepared by his son, Ernst G. Hauptmann. As the younger Hauptmann explains in his introduction to the volume, the collection was intended as such by the theorist, though he desired to modify some of the essays, writing to prominent baritone Franz Hauser that, “—I wish that some of the essays may still be further worked out and expanded, if they have demonstrated the thought, essays that are indeed long enough to me, but for the reader must be somewhat worked out and fortified.”[1] Hauptmann would not, of course, live to see to such modifications, but it is interesting to compare his explanations intended for a more general audience with those found in the Theory of Harmony and Metre, the locus classicus of his theorizing. A few short essays dealing with core music-theoretical subject matter are to be discussed in the following, with special attention to relevant passages in Harmony and Metre.

Der Dreiklang und seine Intervalle is a modest introduction to the construction of the major and minor triads and their scales. The first half of the essay introduces a number of key ideas for Hauptmann. He begins by introducing the ratios of the intervals of the triad as they relate to a vibrating string: 1/2 for the octave, 2/3 for the fifths, 4/5 for the third. Never content with allowing the familiar ratios stand on their own, he goes on to explain how their inner meaning arises and, “to what extent through them a distinctive sense is able to express itself musically …”[2] This marks the introduction of the dialectical moments as they are embodied in the major triad, in which octave unity, fifth duality and third two-times duality, or unity renewed anew, are given sensuous form beyond mere sound.

Such is the introduction of the dialectic, though here unfreighted by any logical musing such as that which appears in Harmony and Metre; in that work, the structure of the dialectic is responsible for music’s making any sense at all, and represents the very condition for comprehensibility and, indeed, value:

That which is musically inadmissible is not so because it is against a rule determined by musicians, but because it is against a natural law given to musicians fro mankind, because it is logically untrue and of inward contradiction. A musical fault is a logical fault, a fault for the general sense of mankind, and not for a musical sense in particular.[3]

Concerning the characteristics of what animates this logic, Hauptmann will add that:

This notion is and remains everywhere the same, in every formation and transformation. It is the notion, that something, which at first subsists for intuition in immediate totality (Octave), parts from itself into its own opposite (Fifth), and that then this opposite is in its turn abolished, to let the whole be produced again as one with its opposite (Third), as a whole correlated in itself.”[4]

In the treatise, the characterization of the dialectical process given in the latter quotation infuses his discussion of the very ratios themselves, giving the numbers a certain musical texturing or nuance very particular to Hauptmann himself. In this explantion, the half of the ratio 1/2 stands along side itself, pointing towards unity. Similarly in the case of the fifth, 1/3 of the string stands alongside twice itself, representing the principle of a self opposed to itself.[5]

Hauptmann’s account of the minor triad in Der Dreiklang, although substantial in itself and comprising the majority of the essay, like other explanations, lacks certain aspects of those found in the treatise. Common to both is the premise that a minor triad can convey tonic meaning within in a key, and thus in it must consist some capacity for unity. But while in both writings he espies a negative unity found in the reciprocal ratios for the major triad, the treatise makes a special point of having and being had. Indeed, they are his first thoughts on the distinction between major and minor: “If the first [major] may be expressed by saying, that a note has a Fifth and Third, then the opposite meaning will lie in a note being a Fifth and Third. Having is an active state, being a passive one.”[6]

Hauptmann then gives an account of the minor triad as containing a doubly-determined (and thus negatively doubly determining) fifth, yielding the structure a-C-e | II-III-I.[7] An alternative structure is given to complement the initial one: F-ab-C | I-II [F-C] + I-III [ab-C].[8] He states that since the C of the F minor triad is the same unity C as C major only negatively determined, the two pictures amount to the same thing, with the C being acknowledged in its determining faculty and its determinedness. Hauptmann is clearly trying to have it both ways—by acknowledging oppositional structure while maintaining the same fundamental note for a key, in this case F, thus permitting the same tonic meaning. The passage dealing with the minor triad concludes with a few qualitative remarks, in which Hauptmann distinguishes between major’s rising up and minor’s weight and dependence, “the hanging boughs of the weeping willow as contrasted with the aspiring arbor vitæ”[9]

In Der Dreiklang the explanation of the minor triad is largely the same, though, as noted, lacking any discussion of active vs. passive qualities.[10] Here Hauptmann presents a major triad emanating from C and an F minor triad hanging down from C, with the octave C indicated below, thus making a 6/4 chord on C with the bass doubled. These pictures are adduced as visual aids to the discussion of the reciprocal ratios. Then he directs our attention to the same 6/4 triad on its own and acknowledges that while we know that F-ab is not the relationship that we perceive, we nevertheless do hear the triad as a 6/4 triad with a root on F.

Aside from Hauptmann’s actual acknowledgement of the minor third F-ab (not a feature of his earlier discussion in the treatise), the inclusion of the 6/4 chord is curious, and one wonders why a simple 5/3 triad on F wouldn’t have sufficed, as it had in the treatise, despite its lack of actual musical notation. Ostensibly to illustrate a mirror inversion of the major triad on C and showing the C as determining the consonances, invoking the 6/4 triad alone, as he does, might have been calculated to highlight the perceived ambiguity of the structure of minor as he understands it, namely as downwardly determined yet having tonic meaning projected from below. In a way he forces our hand to chose F as the actual root of the chord; we have to chose F as the root, so unstable is the 6/4 structure on its own, the status of which he has already called into question earlier in the essay, though without any further qualification.[11]

Hauptmann does not give a formal discussion of the formation of the major key and its triad of triads in Der Dreiklang; evidently it is assumed, along with his representation of keys (or keys, at least) as rows of pitches arranged by thirds, which, at any rate, he references, but does not employ here.[12] Hauptmann does, however supplement his discussion of the minor triad by giving an account of how it is able to serve as tonic within a key system. The minor triad, by Hauptmann’s reasoning, must presuppose the dominant triad, emanating from the tonic fifth, thus engendering both a positive and negative unity. The account of the minor key in Harmony and Metre is only slightly expanded; here he calls attention to the fact that if three minor triads were to be the three primary triads of a key, this key would exist only by analogy to the major key and would have no claim to its own unique status.[13]

Der Dreiklang concludes with a lengthy discussion of the minor scale and how it is constructed. Unlike Weber, who associates minor with the harmonic minor scale, Hauptmann essentially distinguishes between the minor key as a triad of triads, whose pitches do indeed comprise a harmonic minor scale (tonic, major dominant, minor subdominant), and the two forms of the melodic minor scale. Passage through the tones of a scale requires a higher-level mediation for Hauptmann. In other words, successive tones must be discernible as root, fifth or major third of some separate tone.[14] Thus in the C major scale, the first three degrees all take G as unifying or mediating element, degrees 4-6 C, and 6-1 e. Notice how this also gives each tone of the tonic triad a role to play in the mediation of the major scale.

Such an account is not provided for but assumed on the part of the reader in Der Dreiklang, which proceeds directly to the construction of the minor scale. The problem with the minor scale(s), as it was expounded in Harmony and Metre, is the the great impasse between degrees 6 and 7 as they appear as members of primary triads, i.e., Ab and B-natural in C minor. Here no mediation is possible, as no note can project both Ab and B-natural as octave, fifth of major third. Hence the motivation for invoking the melodic minor scale, where the addition of A-natural allows D (the fifth of the dominant) to mediate scale degrees 5-7, while G governs 7-1. In its descent, the minor scale partakes of Bb, thereby smoothing the gap between itself and Ab, resulting in a configuration in which F mediates degrees 1-6 (in descent), C degrees 6-3, and G degrees 3-1.[15] Here a triad is not formed by the pitches of mediation, which doesn’t seem to solicit a comment from Hauptmann.

There is a crucial distinction to be made between the descending Bb and the ascending A-natural and their lower-case counterparts with respect to intonation and tonal meaning. Lower-case letters would indicate properly tuned (and thus fully functional within in a key) thirds above a root. The upper-case letters refer to borrowings from the subdominant side, in the case of B-flat, and from the dominant side, in the case of A. Although Hauptmann says quite clearly in Der Dreiklang that, “It is a completely symmetrical construction, itself constructed out of necessity,” his explanation in Harmony in Metre actually captures the symmetry much more closely. In Der Dreiklang, Hauptmann focuses on the changing role of the tonic note C in the C minor key and its attendant shift in meaning from octave of itself to fifth of subdominant F, which ushers in Bb as subdominant of the subdominant. With A natural, Hauptmann focuses on A as the fifth of the dominant fifth D. While it is clear that pitches are being plucked from opposite directions in the row of tones, making the process, to that extent, indeed symmetrical, in Harmony and Metre, Hauptmann focuses instead on the role of the pitch F in C minor, putting the situation in a way that highlights the symmetry to the greatest extent possible:

Now if in the ascending minor scale progress was impeded from the sixth degree to the seventh, then in the descending scale there will also be no connexion found between the seventh and sixth degrees. As there the minor Sixth could not form the passage to the major Seventh, so here the major Seventh cannot lead into the minor Sixth. The Octave, however, finds a note to conduct it to the minor Sixth, again outside the system, but this time upon the subdominant side. While in ascending the Fifth of the dominant had to become Root, in descending the root of the subdominant must become Fifth; the former change provided the intermediate step to the major Seveth, the latter change provides the step to the minor Sixth degree.[16]

Dreiklang mit der pythagoräischen Terz is Hauptmann’s account of the role of just intonation as concerns tonal meaning, and much of the essay actually recapitulates what has been said concerning the borrowing of tones outside the minor scale in order to traverse scale-degrees 6-8. What is interesting in this essay is the series of examples that demonstrate, in counterpoint, how pitches derived from other keys combine with themselves to form the Pythagorean major third of ratio 81:64 (as opposed to the just 5:4), as he also points out. In the first three examples on page 58, Hauptmann combines two lines in the key of C minor in parallel sixths, lines which are mediated by separate pitches, C and D. Since one line borrows from the dominant side, the combination of F/A (see the asterisk in the third example) will sound improperly to the key of C minor, this interval being of the ratio 81:64. Next Hauptmann fills out the two-voice texture into four-voice texture so as to put triads and seventh chords in play. In the fourth example of page 58, Hauptmann instructs us that we will hear the F dominant-seventh chord as such as long as it is played on the keyboard, this is not at all the case as regards tonal meaning. A parallel example on pages 58-9 provides an instance of how the minor seventh of the C minor scale is incorporated into a contrapuntal and chordal context.

No such demonstration exists in Harmony and Metre, but it is possible to speculate as to what else in the treatise this pertains to. One candidate lies in Hauptmann’s discussion of “diminished triads of the key-system in transit.”[17] Although Hauptmann wouldn’t have put it this way, this discussion essentially treats the issue of tonicizing the dominant via an applied dominant partaking of scale-degree #4 of the home key. Hauptmann’s “limits” of the key are the outer-most pitches in the row of tones representing the key, thus F and D in the key of C major. But when the dominant is tonicized, the row of pitches will click one slot over to the right, leaving F behind and taking on f# as major third of the fifth of the dominant. In the key of C major, chords acting in the capacity of tonicizing the dominant, moreoever, will often combine f# (from the dominant) with C, D or both. To this, the pitch a will also be found, only this lower-case a is native to C major, making an improper fifth with D, which is native to C major but not G major. Thus the key of G major will not be allowed to fully realize itself, and we resist the temptation to read a full change of key.

By analogy, then, perhaps by invoking the improperly tuned intervals, Hauptmann is implementing a stop-gap against calling certain sonorities actual triads, since their inner meaning, as it were, would prevent us for doing so, even if their sound on the keyboard would not. For instance, the Bb-D passing tones in the upper voice on beat 2 of measure 1 of the third example on page 60 are surely not what we would call a Bb major triad in a hierarchical sense; they are passing tones still on their way to members of the F minor triad. Under this interpretation, the same would hold for the asterisked sonorities on page 58 as well as on the two examples that conclude the essay.

Hauptmann’s short essay Zum Quintenverbot actually surpasses his discussion of the issue in Harmony and Metre; there, his discussion consisted of the familiar notion of 5/3 triads’ needing to be followed by chords in other inversions, followed by Hauptmann heaping scorn on the phenomenon in prose slightly more lively than his usual, tortuous style in the treatise.[18] In the later essay, Hauptmann begins by distinguishing between parallel fifths that result from both voices moving by step, and by motions that are combinations of steps and leaps. In the case of fifths resulting from simultaneous leaps, Hauptmann says that this violates the more natural progression—that is, a more parsimonious progression would eliminate fifths and result in a more pleasing sound.

When fifths result from two motions of a diatonic semi-tone, another form of mediation is required. In the set of F/C à G/D fifths introduced at the bottom of page 62, mediation is required in the form of a D minor triad, as shown at the bottom of page 62, or as he does in the immediately following example on the next page. The second example seems to suggest the familiar 5-6 technique familiar from myriad counterpoint and figured bass writings, only the upper voice dips down to form the third of the resulting G chord and does not remain atop the sonority and thus does not confirm the a second fifth above the bass. More likely, Hauptmann has simply converted a IV-V progression into a IV-II6-V progression, as he had done in the previous example, although in a different inversion. Evidently more than voice-leading concerns are required for such a progression (IV-V) to make sense, as Hauptmann makes clear in his discussion of chord succession in Harmony in Metre. Hauptmann distinguishes between motions between chords that share two, one and zero pitch classes. In cases where a C major chord as tonic moves to a D minor chord, Hauptmann says this is as if the progression were mediated by an interstitial A minor chord, a familiar Rameauian tactic.[19] Hauptmann’s discussion of fifths concludes with his consideration of a German augmented-sixth chord that resolves directly into a dominant chord in 5/3 position, a resolution he says does not bother him on the keyboard because he can imagine an alternation progression of individual voices, but which disturbs him to no end when actually sung.[20]

The final essay to be considered here is Zur Auflösung des Dominantseptimenaccordes durch Erweiterung der Septime zur Octav, which, it ends out, deals with the distinction between a dominant seventh chord and an augmented-sixth chord (here only the German variety is treated), but with an unusual constraint: both chords proceed to a 6/4 chord. Hauptmann’s explanation in Zur Auflösung begins by examining the resolution of a dominant seventh chord on G and its resolution to a B minor triad in 6/4 position. Hauptmann states that the dissonance G-F merges into consonance on octave F#s, thereby removing the ambiguity found in the internal inverval b-D, which can be third and fifth of G major or root and third of b^o.[21]

This explanation points to a key passage in Harmony and Metre concerning the resolution of dissonance. To begin with, his definition of dissonance is comprised of two parts: in terms of actual sound, it is a step presented as a simultaneity. In terms of inner meaning, it is the ambiguity that obtains in a certain third note, which stands in a contradictory relationship to the tones of the dissonance, i.e., a G/F simultaneity will engender a contradiction for the note C, which stands as root to G and fifth to F.[22] Regarding suspensions, the note of mediation is contained in the sonority itself and only one addition note has to resolve, as when a-C-G resolves to F, being the note that will reconcile all contradictions. In seventh chords, more voices must progress in order to resolve the conflict between the two dissonant tones.[23] Thus the seventh chord e-G-b-D seeks resolution via the mediation of A; the inner third must therefore clear space for this to be clear, resolving in the intermediate (conceptually—it need not sound accordingly) stage of e-a-D. At this stage, D can resolve down to C or e up to F, both options being capable of resolving the chord around a.

But there is a sense in which the mediating tone can be considered to be already present. For instance, the seventh chord e-G-b-D can resolve around b, which is consonant with e and D but not in the same chordal context. In this case, e moves up to F, bringing it into rapport with b, while G continues to hang around. If the chord resolves around G, D will drop to C, completing the C triad while b endures. In the case of the augmented-sixth, however, the chord resolves around both of the notes in the middle. Turning back to the example in Zur Auflösung, the outer voices resolve out to F#s, while both of the inner voices remain.[24] Such is the relationship, then, of the example at the beginning of Zur Auflösung to the relevant portion of Hauptmann’s larger theory.

Hauptmann goes on so explain how F and E# is not a simple enharmonic transformation and are by no means meant to be conflated. The essay concludes with a glib enjoinder to understand the different between the resolutions of the dominant seventh to the octave and the same resolution through the augmented sixth through a set of examples.[25] Both examples do as they say, but with the preceding interpretation in mind, Hauptmann’s goal seems to be that in the passages exhibiting the dominant seventh, the melodic motion of the voices plays around in the triadic space of B-d-f for the dominant seventh chord and then B-d-F# for the 6/4 chord (showing the clarification of chordal context of the B-d interval), while in the other cases this triadic space is ignored by the motion of the voices, suggesting that the E# partakes of nothing triadic. While perhaps the ultimate motivation for this example is merely to give his idiosyncratic explanation of the resolution a more concrete form, it is unclear how different the examples would need to be in order to arbitrate in favor of E# of F—how, in other words, in assuming nothing on the part of the composer’s spelling, we are to calculate the inner meaning of the progression and, if away from the keyboard, how to tune our performance.

Works Cited:

Hauptmann, Moritz. The Nature of Harmony and Metre. Ed. and trans. W. E. Heathcote. New York: Da Capo Press, 1991.

———. Opuscula: Vermischte Aufätze. Leipzig: Verlag von F. E. C. Leuckart, 1874.