Musicology fancies itself a science, and decks itself out in the trappings of mathematical models. Unfortunately, however, there are no definitive theories for many of the basic problems of musicology. One of the most glaring of these deficiencies is the lack of a universally applicable methodology for harmonic analysis.

Many harmony textbooks offer students analytical tools that are useful only for the accompanying homework exercises, but woefully inadequate for the encounter with real music by real composers. No matter that each treatise or textbook prides itself on its unbiased inclusiveness. Any system cannot help being parochial if not ethnocentric, limited by the prejudices of its author, its period, or its style. Every system inevitably enunciates analytical principles that are unsuitable for the harmonic phenomena in some other kind of music. In this sense, all systems of harmonic analysis are to some extent reductive. A few examples:

Rameau stated, in his Traite de L'harmonie (1722, pp 29ff) that chords must be formed by superposition of thirds. A harmonic analysis based on this would be unable to deal with the "mystic chord" in Skryabin's Promethee, le Poeme du Feu, Op. 60 (1910), or the whole-tone harmonies in Debussy's Voiles in Preludes, Book 1, No. 2 (1910).
     Harmonic analysis based on Mattheson's Grosse General-Bass-Schule (1731) would be helpless when faced with the towering chords in Charles Ives's "Concord" Sonata (1915) or the non-transposable modes in Messiaen's La Transfiguration de Notre Seigneur Jesu-Christ (1969).
     Fetis propounded his theory, in his Traite Complet (1844, p 249) that chords are derived from the notes of the major and minor scales. This theory would not even be able to explain the harmonies in a work that is almost exactly its contemporary: Tristan und Isolde (1859).
     Schenker, in his Harmonielehre (1906) posited for every piece an Urlinie (or Ursatz) oriented toward (or away from) a dominant and a tonic. It is true that Schenker confined the validity of his method to the period from Bach to Brahms. But even within these limits, it would be useless to apply his theory to Liszt's Bagatelle sans Tonalite (1885).
     Allen Forte, in The Structure of Atonal Music (1973, pp 179-81), presented his tabulation of all the possible "unordered pitch-class sets" together with the procedures for extracting derivative sets. His point was, that this tabulation includes all the possible chords in the twelve-pitch tempered system . A symptom of the perversity and futility of this pseudo-scientific analytical model, is the absence of the major triad from its tables. In this, Forte even surpassed the older pseudo-Pythagorean theory that stated that the minor triad has no independent existence but is derived from the major triad (Rameau, Traite de L'harmonie, 1722, p 36; Hindemith, The Craft of Musical Composition, 1937/1942, pp 74ff). In Forte's pitch-class tables only the minor triad is a "prime form," and the major triad must be derived from it by inversion. Thus Forte's tables would be of no use, at least from this point of view, in analyzing the harmony of Mozart and Beethoven.

A universally applicable analytical model, free of every kind of bias, is probably impossible. Musicology -- indeed any of the humanities - is not an exact science like physics. The nature of the case necessitates a non-dogmatic approach.

A non-dogmatic analysis of any work of art should not be conducted according to predetermined procedures, nor should its goal be conclusions based on preconceived criteria. The first consideration should be the mustering of all the possible elements in such a work. Then the analysis should concentrate on those elements which have been made momentous and significant in the specific work. (Sometimes it is no less important to note which elements are absent.) A simple, obvious example of such an element in poetry, would be the way the words look on the printed page. In a poem by Alexander Pope, this element is not crucial; one must seek elsewhere for the import of the poem. On the other hand, in a poem by e. e. cummings, this element is of cardinal importance. (It even spills over into the lower-case name of the poet.)

The selection of the elements that are significant, although ultimately based on the internal evidence, should be supported by context: the career and oeuvre of the composer, the historical and cultural milieu, and the criteria that were applied in the work's original environment. This involves circular reasoning, but that need not cause dismay. Musicology is not physics, nor is it mathematics, despite the best efforts of many musicologists armed with computers. Circular reasoning enriches the context, and so affords a deeper insight. A few examples:

The first example is taken from a treatise called Harmony in its Systemic and Phenomenological Aspects by Yizhak Sadai (translated by J. Davis and M. Shlesinger,1980, pp xxviii-xxix) In the Introduction, the author proposes "to demonstrate once again the importance of a phenomenological attitude toward harmonic analysis" by discussing a piece from Schumann's Album fuer die Jugend, Opus 68, No. 41. He calls attention to the third chord in m13, weighing the significance of the C, and debating whether or not the eighth-note B-flat could be deleted. He calls the C a "nonchord tone" as if there were no such thing as a suspension or an appoggiatura, as if this commonplace "nonchord tone" were more problematic than the many other ornaments and dissonances in the piece.

The principal blunder, however, is the failure to take into account the extra-musical context: the title of the piece, and the alphabet game concealed in the music. This piece is named Nordisches Lied (Gruss an G.). The "G." refers to the composer Niels W. Gade; whose settings of Danish folk songs Schumann admired. The whole piece is based on the four notes that spell out his name: G-A-D-E. Of course, this is not the only such alphabet game that Schumann played. There are also the "ABEGG" Variations, Opus 1, and the letters Carnaval, Opus 9, and the Sechs Fugen ueber den Namen B-A-C-H, Opus 60.

The binary form of Schumann's little "Greeting to Gade" is fashioned from several different re-harmonizations of the G-A-D-E motive. Each two-measure phrase begins with the G-A-D-E motive: on the original pitches (m1-2, 5-6, 9-10bass,13-14,17-18) alternating with the motive transposed (m3-4, 7-8, 11-12, 15-16,19-20). Of the original-pitch harmonizations, those in m1-2, 5-6, 17-18 are quite similar to each other. The harmonizations of the transposed motive, m3-4,
7-8,15-16, are also quite similar to each other. But m19-20, obviously saved for the conclusion, is quite different. As is conventional in binary form, the most radical harmonies, those furthest from the tonic, are placed directly after the repeat sign, here in m9-14. This section contains the most chromatic harmonies, and the most dissonances, and the purportedly "problematic" C in m13 is merely one of these.

Another example is J. S. Bach's harmonization of the chorale Es ist genug at the end of Cantata No. 60, O Ewigkeit, du Donnerwort, BWV60. In the chorale melody, each phrase or group of phrases is immediately repeated (phrases 1+2+3=4+5+6, 7=8, 9=10). Bach exploits these repetitions to give new meaning to each phrase of melody by re-harmonizing it. Although these re-harmonizations are radically different, the framework of the tonality is not disturbed. The differences between the progressions occur only within the phrases. At the ends of the phrases -- the melodic resting-points marked with fermata signs which are also the main harmonic resting-points -- the melodic pattern of repetitions is reflected in the harmonic repetitions. For the first and last pairs of phrases, phrases1=4, 9=10, the concluding harmonies are related and analogous. (The final pair. phrases 9=10, are the conventional deceptive cadence followed by an authentic cadence.) For the remaining pairs, phrases 2=5, 3=6, 7=8, the concluding harmonies are exactly the same.

The voice leading, however, is as important as the harmonic progression, and perhaps even more so. This is most conspicuous in the first phrase, to the words Es ist genug (It is enough) whose melody mounts with strident directness to the sharped fourth degree of the scale. Bach emphasizes the exclamatory nature of this cri de coeur, not only by arriving with convulsive suddenness at so distant a chord as V7/III, but also by having all four voices rising or leaping up -- and yet without the slightest suggestion of parallel octaves or fifths. The repetition, phrase 4, to the words Mein Jesus Kommt (My Jesus comes) expresses an entirely different emotion: the confidence of the believer. Accordingly, the inner voices are almost motionless while the bass descends, in contrary motion to the soprano. But no less important is the fact that the motion of the bass in phrase 4 is the opposite of the bass in phrase 1. This inversion is consistently carried through in the voice leading of the entire chorale. Within the limitations imposed by the spellings of the chords, the motion of the bass in each phrase is reversed in the repetition of that phrase.

Yet another, even more complex example is the second movement of Brahms's Symphony No. 4, Opus 98. The melodic line of the theme appears in several different versions, but all the variants have this in common: that they begin by revolving symmetrically around a central note: ascending two or three scale degrees and returning to the initial note, descending similarly and again returning to the initial note, etc. The body of the movement consists of two parallel groups of four theme variants each (mm5ff and mm64ff), the second group being a recapitulation of the first. The first variant is alwaysthe same, and always on the tonic (m5,13,22=m64,72). The second (m30) is different in the recapitulation (m78), but both are harmonized with pedal points. The third and fourth variants are melodically the same in the recapitulation (m36=m84, m41=m88,98) but they are first harmonized on II and V respectively, then in the recapitulation on V and I respectively, as is conventional in binary and sonata-allegro forms. The whole movement is framed by a sort of "motto": a variant of the melodic line that appears at the beginning and at the end (m1=m113) but nowhere else. Taking into account the two symmetrically placed non-thematic sections
(m48-m103) the form-at-large can be schematically expressed as: mottoAAABCD -- AAECDD -- motto.

The "motto" opens the movement ambiguously, in unison without harmony, but concludes it with an apocalyptic resolution of the ambiguity -- and that is the whole point. In the opening "motto" the central note is E, the putative tonic, but all the sharps are altered to naturals. Is this central note E the first degree of the phrygian scale or the third degree of the C-major scale? Since there is no harmony, only unisons, this opening "motto" leaves the listener undecided.

In the first theme variant (m5,13,22=m64,72) the central note is always the third degree (G-sharp) of an E-major scale. In the fourth variant (m41=m88,98) the central note is always the tonic of a major scale (B,E,E). In the second and third variants the situation is more fluid: the scale is sometimes major, sometimes phrygian; and the central note is sometimes the first, sometimes the third degree.

The harmonization of the concluding "motto" (m113) combines the previously contradictory harmonizations of the melodic line into one triumphantly definitive progression. As in the beginning, the central note is E and all the sharps are altered to naturals. At first the bass is E, and the central note E is the first degree of the phrygian scale. Then the bass moves to C, and the same melody is redefined: the central note E is now the third degree of the C major scale. Then the bass returns to E, and the central note E becomes the tonic in the concluding E-major chord; and with this epiphany, the ambiguities of the movement are resolved into a radiant certitude.

In all three of these examples, an important element is the comparison between different harmonizations of the same melodic line. Furthermore, if the analysis takes into consideration such supposedly extraneous elements as the verbal context (in Schumann and Bach) or the ambiguity of the mode (in Brahms) this may alter the entire perspective from which the harmony should be analyzed. Neither a simple series of Roman numerals, nor polysyllabic phenominological ratiocination, nor a diagram that purports to contain a mathematical insight, will even come close to revealing the true significance of the harmony.

The analyst must not stubbornly apply one model, or one set of criteria, to any and every case. Internal evidence and historical and social context should be allowed to direct attention to those aspects, the analysis of which will prove most fruitful.