History and the Word: Form and Tonality in Schoenberg's Phantasy for Violin with Piano Accompaniment1
For Schoenberg, there is meaning to the word, as there is meaning to the gesture. The sense of frustration, or perhaps inadequacy, that seems to pervade analyses and commentaries of his music
The virtually universal sense of unfulfillment with which musicians are left after reading any technical discussion of Schoenberg's music... is not due to the failure of the music, George Perle to the contrary, but rather to the manifest inadequacy of our theoretical equipment to cope with its richness and complexity.2
comes not ultimately from the "complexity" of the work, but from the fact that there is no space left over for the meaningless. No word, no pitch, and no action occurs without an elaborate historical, internally musical, and logical cross-referencing. There is, in his life and art, a semantic saturation that we can only in part reconstruct.
That Schoenberg was aware, even intimately, of the Brahms Fantasien (op. 116), the great Bach Chromatic Fantasy, the Mozart Fantasy in D Minor, the Beethoven Fantasy in C Minor (op. 77) and Choral Fantasy (op. 80), among other works, is undoubtable. What his own perception of the relation of his music and reference to this 'literature, the history of interpretation of a word, is less clear.
The tendencies which Schoenberg seemed to embrace, rather, the paths he chose to take late in life, have been discussed at some length. Classicism, harmonic conservatism, a return to basic formal principles- all these have been attributed to him, yet each is incomplete. These ideas describe only single aspects of his development, for the whole is described by the work itself.
Thirty-five years after writing the op. 11 Piano Pieces (not to mention Erwartung), for Schoenberg to finally and explicitly name a work "phantasy" is peculiar. It is as if he wanted all that time to show that mastery of forms is equivalent to being able to return to them with a clear conscience, and then depart from them once more. The phrasing, structure, even the harmonic language of the violin Phantasy remind one more of these earlier, free period pieces, than of the later works like the Fourth String Quartet and the two concertos. Yet there is a difference: here Schoenberg's reference is not that of a young composer boldly stepping out into new realms, new sounds: but rather that of a composer who has gone so far forward himself that he is in fact an integral part of the history he wished to refer to. "Violin with piano accompaniment" perhaps this would not have made much sense in 1909.
Two "facts" limit conjecture on the Phantasy: 1) that it was the last instrumental music Schoenberg composed and 2) that, according to Rufer's catalog, the violin part was written first, independent of the piano accompaniment.
With regard to the first, any consideration of Schoenberg's work must take into account the composer's own search for musical meaning and enlightenment. not only through the localized consistencies of a single work, but almost equally through the more globalized intentions of the entire opus. These intentions are clearly both historical and ontological. To know Schoenberg solely through the parodistic styles of the Serenade, even though virtually all characteristics of his conception and virtuosity are present there, is perhaps not to know him at all. As is evidenced by Schoenberg's penchant for being his own best historian, no one can know (or make use of) his development better than he himself. He both constructed and obeyed it, responding to the apparent, entertaining qualities of the Serenade with the abstract austerity of the Wind Quintet; to the complex angularity of the third quartet with the lyricism of the fourth; to the astounding complexities and mammoth scope of Erwartung, Five Orchestra Pieces, and the Book of Hanging Gardens with the Six Little Piano Pieces (op. 19); and perhaps, to a lifetime of the most severely critical and embracive musical thought with the uninhibited grace and pure musical-melodic joy of the Phantasy.
With regard to the second "fact" , that of the alleged manner of the piece's composition, both Leonard Stein and. Josef Rufer seem to be sure that the violin part was composed first, but with the piano accompaniment kept in mind throughout, with regard to both its hexachordal and textural characteristics.
Since Schoenberg wanted to write a phantasy for violin with piano accompaniment and not a duo, he first composed the entire violin part alone.... During the course of the violin part, the row forms which are being used, as well as those which are planned for the piano part, are noted in red, green or black pencil.3
Rufer does not explain how it is known that the violin part was composed first. Leonard Stein, in his liner notes to the ARS NOVA,, ARS ANTIQUA recording Inner Chambers, states it a little differently:
"Schoenberg emphasized this distinction between the two instruments by writing the manuscript twice: the first time, for solo violin only; the second time with the piano part added. Certain calculations concerning the disposition of the twelve-tone row forms and their transpositions show that Schoenberg was planning the inclusion of the piano part...." 4
- but again, it is unclear how the order of composition is known with such certainty, since presumably only one manuscript exists.
Be that as it may, the idea is not hard to accept. The piano part has virtually no melodic figuration, even of a secondary sort. The closest things to a leading melodic figure are the cadential melody at ms. 24 and the accompaniment ostinato idea at mm. 42-44. For a composer of Schoenberg's contrapuntal intensity, the rest of the piano part is quite unusual in its strict homophonic subordination. It has been frequently noted that the conception of the "accompaniment" is extended even into the row forms, for Schoenberg often supplies the piano part with the complimentary hexachord to that of the violin.
The form of the piece is organically conceived with whatever extramusical intentions Schoenberg might have had for the piece (although Schoenberg's musical conception is so broad that it almost seems as if nothing in his realm of existence falls under that rubric, which, after all, tells us much more about the speaker than the object under consideration). The sense of pun, both in the small and the grand design, is never absent from his work. The naming of the piece, the description of the piano as "accompaniment" , the reflections on and allusions to Brahms, Mozart, Schubert, and even Chopin and Beethoven, and the meaning of the particular compositional gesture at a point in his life and in the history of music when his own tendencies towards classicism were an issue, are all factors to be considered if the music, that is the score itself, is to be appreciated to any degree.
In his liner notes to the ARS NOVA... recording, Stein calls it a "multi-sectional structure, with many strong contrasts in tempo, character and form...." and that " the recurrence of the opening main section serves to unify the entire piece, much in the manner of a Rondo or Sonata- as suggested by the following outline: (Ms. #'s supplied by LP)
1 Main Section (A): Grave-dramatic in character (mm. 1-31)
2 First Episode (B): Meno Mosso (ms. 32), Lento (ms. 40), Grazioso (ms. 52) - mostly lyrical.
3 Second Episode (C): Scherzando (ms. 85)- form and character of a scherzo.
4 Main Section (condensed) (Ins. (133) 135) and coda (A2) (ms. 154?)5
It is a tribute to Schoenberg's mastery of the ambiguity of form, such as that which he admires in the assymetry of Brahms' phrasing, that no such outline of the Phantasy can really contribute much to its understanding. Any such sectionalization only describes its own axioms for perceptual distinction- Schoenberg always knew this and sought to integrate all the varied parameters of his craft in such a way as to at once clarify form and at the same time confound the superficial perception of it.
Form in music serves to bring about comprehensibility through memorability. Evenness, regularity, symmetry, subdivision repetition, unity, relationship in rhythm and harmony and even logic- none of these elements produces or even contributes to beauty. But all of them contribute to an organization which makes the presentation of the musical idea intelligible. The language in which musical ideas are expressed in tones parallels the language which expresses feelings or thoughts in words, in that its vocabulary must be proportionate to the intellect which it addresses, and in that the aforementioned elements of its organization function like the rhyme, the rhythm, the metre, and the subdivision into strophes, sentences, paragraphs, chapters, etc. in poetry and prose.6
and in even a more direct statement,
Let me say at once that I am more inclined- unconsciously, for sure, and often even consciously- to blur motives, a tendency that will certainly meet with the approval of those who feel in music 'life on several levels' and who therefore prefer to hear a kind of 'counterpoint' between motive and phrase: a complimentary opposition. 7
Even a quick glance at the Phantasy reveals that at once Schoenberg is paying homage to those forms which he respects in the works of earlier composers, and at the same time paying them the even greater tribute of participating in their development by taking them further. For example, mm. 52-63, the Grazioso section, appears to be an almost deliberate paraphrase of traditional ternary form. It is twelve measures long, in 9/8, with simple dance-like rhythms (especially the opening and closing four measures) and seems to be the only place in the Phantasy where Schoenberg is willing to relax the feelings of ambiguity. The twelve measures are in clear ABA form, the last four measures nearly identical in rhythm and intervallic content to the first four, although many of the intervals are inverted in the use of the row form permutations:
Example 1: MM. 52-53, 60-61 (Vln. part); "Outside" Measures of Ternary Form Compared
However, this local bow to a certain type of clarity is not indicative of the formal and structural intentions of the piece:
Ternary, rondo, and other rounded forms appear in dramatic music only occasionally, as episodes, mostly at lyrical resting points where the action stops or at least slows down- in places where a composer can proceed along formal concepts and can repeat and develop without being forced to mirror moods or events not included in the character of the material.8
The 'theme' of 'the Phantasy (which also happens to be the row), is stated in mm. 1-2: and is restated in part, whole, and in allusion at significant points throughout the piece. At ms. 32, it serves to introduce the section marked Meno Mosso (ms. 34) but there is clearly some formal development prior to that. The first major cadence occurs at ms. 24, and it seems to me that it is here that the piece first reveals its affinity not only for the highly sectionalized Violin Concerto, but even more for the so called first-period pieces like op. 11 and op. 17 in which formal structure is primarily effected through motivic distinction. In fact, although the Phantasy represents, and has been duly recognized as a masterwork in twelve-tone technique, David Lewin's excellent article to the contrary, I feel that by this time Schoenberg had such great facility with the various manipulations of hexachordal permutations that he was able to use them simply as one more determinant in the formal gestalts: they are not distinguishing as themselves. Rhythm seems to predominate throughout all of his works, and this piece is no exception. As he does in the Suite (op. 25), Schoenberg feels the need to demonstrate to history that row manipulation is a heuristic device to escape the lack of meaning, through overdevelopment, that tonality and atonality had assumed. Form is not determined by pitch. Motive and morphology, rhythm, temporal density, dynamics, and timbre offer the composer more than enough, and perhaps the Phantasy is the last time Schoenberg states his case that twelve-tone technique is indeed an "emancipation", and not a restriction.
Example 2: MM. 1-2; Violin; Row/Theme
A clear formal break occurs at ms. 25. After the fermata in the solo piano line, the Piu Mosso enters in clear rhythmic and melodic contrast to the more songlike first twenty-four measures. Lewin calls it a "second theme" and although as in Ives, it is more like the simplest development of a theme that never occurs but underlies many different structures, it is clearly the prototype for most of the contrasting material, if only in its heightened rhythmic activity, close intervals, and melodic density. In a sense then, the aborted but verbatim return of the first theme at ms. 32, along with the dramatic ritard preceding it, is a kind of conclusion for all before it, and announces the beginning of a development. But the Meno Mosso section beginning at ms. 34 relaxes both the rhythmic and melodic density, and one is reminded of another great Fantasy, the Mozart D minor for piano, K. 397, where the same sort of dramatic textural contrast occurs so often. C Mozart has to be considered above all as a dramatic composer" 9 The section that follows, mm. (32) 34-52, is one of the most extraordinary in the piece. Like the Grazioso, it is basically ternary. MM. 34-38(b)10 present a relatively song-like idea, with a rather typical (Brahmsian) broken chord accompaniment, followed by what seems to be an even more deliberately Brahmsian passage (mm. 40-44(6)), with the piano ostinato dreamily underscoring the slowest violin melody in the work:
Example 3: MM. 44- ; Piano Ostinato Figure
Here, the term fantasy is interpreted not with respect to overall form, but in regard to mood. It is not surprising that the Brahms Fantasien (op. 116), with all their diverse formal characteristics but clear "modal" consistency were among Brahms' last works. Measures (45) 47-51 return to the idea of mm. 34-38b, but the piano clearly signals the upcoming major structural division at ms. 52 with its arpeggiated chords in mm. 48-51.
The Grazioso (52-64) has already been discussed as to its internal formal structure, but now we can see it in its formal relation to what precedes it. In a kind of free rondo, Schoenberg has emphasized local three part form, with the thematic return at 32, and the sectionality of mm. 34-51. In more traditional terms, the piece up to ms. 52 can be seen as two episodes, variations, etc. MM, 52-64 can be seen as a kind of structural pivot and point of least motion around which the piece centers. That this section is, in itself, highly complex does not detract from the strong sense of structural completeness one gets from it and its relationships to the other sections. It is followed by a short, free, chorale-like interlude (mm. 64-72) which in register and rhythmic activity (at least in the violin) is closely related to the "outside" measures of the Meno Mossa (34-51) or that section which immediately preceded the Grazioso.
next section, mm. 72-84, clearly set off by "cadential"
material in measure
Example 4: Ms. 72; Violin; Reoccurence of the "Theme"
- but its rhythmic and
morphological distinctiveness from the preceding material marks it as a clearly recognizeable
motivic return. The greatly reduced rhythmic activity, along with violin melodic ideas clearly
reminiscent of mm. 34-7, 47-51, and 64-70, make
this a kind of two part mini-recapitulation of the formal and motivic
Example 5: Ms. 84; Vln. and Piano; Cadential Figure of the " Piu Mosso"
If all that precedes ms. 85 could be considered a structural unit in itself (A), then ms. 85-154 can certainly be considered a kind of middle section (B). Although these large formal descriptions tend to overlook the tremendous inner conplexity that exists, it seems clear that Schoenberg is working, as he does in the Serenade and in Pierrot, with forms whose association with musical convention is only coincidental to him. He viewed these forms as expressions of a larger set of meanings, tonality being simply one semantic representation, the twelve-tone system being another- tonality's logical and inevitable replacement. Tonality did not cause the predominance of the ternary idea, but was rather a parallel result of a deeper generative idea, that of distinction:
For in a key, opposites are at work, binding together. Practically the whole thing consists exclusively of opposites, and this gives the strong effect of cohesion. To find means of replacing this is the task of the theory of twelve-tone composition.11
Thus, the "simple" rounded forms he found himself so comfortable with later on are not a renunciation of the revolutionalry formal achievement of works like op. 17, but a result of maturation and deep acceptance of law, and his own profound instinctive affinity with the " cosmos:
Hauer looks for laws, Good. But he looks for them where he will not find them. I say that we are obviously as nature around us is, as the cosmos is. So that is also how our music is. But then our music must also be as we are (if two magnitudes both equal a third...). But then from our nature alone I can deduce how our music is (bolder men than I would say 'how the cosmos is!'). Here, however, it is always possible for me to keep humanity as near or as far off as my perceptual needs demand....12
Thus the rather startling fact that the Phantasy, a work whose formal intricacies, rapid character shifts, and high density of information suggest an absence of a simple large scale structure, can be divided so "neatly" into a three part form (1-84, 85-153, 154166). This is a result of Schoenberg's intuitive sense of the simplest and most direct, while at the same time creating high levels of micro-structural activity. One can not help but compare this "style" with that of the later Brahms, for example his Capriccio (D minor) and Intermezzo (A minor), both from op. 116, where a tremendous amount of localized activity is contrasted to a "simple" encompassing three-part form.
The Scherzando (85-92), Poco Tranquillo (93-116), Scherzando (117-134), and Meno Mosso (135-153) together comprise the second major section of the work. Although there is a fundamentally different rhythmic character present here, more dance-like and certainly less complex, most of the material in this section is derived readily from that of the first. The three against four feel of mm. 34-38b; the melodic ideas of mm. 125-33, of ms. 47-51, 34-38b, 76-81; and the repetitive ideas of 135-139 are a kind of condensation (rather than development) of the same idea which has been prevalent throughout (e. g. yin. ms. 5; yln. ms. 12a; yin. ms. 27; piano mm. 52-54, 56-7, etc.). The effect of mm. 135-53 is to coalesce two disparite elements, the repeated pitch and the rather complicated melodic line, into one unified concept. Almost all of the ideas in this section are based on this union (even the premature return of the theme at ms. 143). To reach for stasis of some kind at the end of a highly developmental section is not unusual, but it is certainly not the kind of creative convention taught in composition classes. It is a subtle, highly intuitive gesture which bespeaks a compositional mind of the greatest maturity, and one to whom the actual traditional forms are not of primary interest, but rather the nuance of their realization. It is once again a matter of opposites: rather than climaxing developmental agitation by bringing it to 4 frenzied conclusion, Schoenberg understands, through his meticulous knowledge of the "masters" that the most astonishing perceptual effects are brought about by distinction: stasis in context is as active as any motion. Gestures of this type are of course well-known, but one of the most breathtaking (and relevant here) is the section of the Eroica (1st movement, near the end of the development, mm. 250-280, but especially 270-280) where, after some of the most complex melodic and harmonic invention, Beethoven surprisingly relaxes the harmonic and orchestral motion.
Comparisons with the Eroica are not so haphazard as they might appear, for the section mm. 143-153 (of the Phantasy) brings to mind an even more famous event in the former, the " premature" entry of the theme, near the end of the development, in the horn (in the " wrong key" ). The "reprise" of the Phantasy takes place ten measures later (at ms. 153). Depending on one's terminology, we might call this either a coda or a recapitulation. Lewin, in his analysis of the work almost entirely in terms of its hexachordal "regions" states:
I have divided the piece into three sections.... It Will be noted that the change of area at measure 32 corresponds to an obvious major formal articulation of the piece, and that the AO area between ms. 143 and ms. 161 1/2 contains a formal "reprise" (ms. 154).13
Example 6: MM. 143-145; Piano; Use of "Tonic" and "Dominant" Functions
- (in section III., I will comment briefly on the "functions" of certain of these harmonic sonorities).
Not much need be said about mm. 154-166. The gestures are clear and effective. With a nearly literal repetition of the theme (a major second below, for the original pitches have been stated ten measures previous), reference is quickly made to many of, the predominant motives of the piece (ms. 158, vin., triplet and triplet-like figures in the piano, and at ms. 161 (second half) a tune closely resembling that of the "second theme" (ms. 25)). The work ends, appropriately enough, as Stein puts it, with the "final liquidation in double stops, tremolos and repeated notes (bringing) the composition to a brilliant ending" .14 (from Stein's liner notes).
The row for the Phantasy is chosen, I believe, with some very definite harmonic considerations in mind. The order of the second hexachord really depends on what one accepts as the prime of the row, or whether there is one. Lewin seems to believe that this is not the prime, for the second hexachord is merely a transposed inversion of the first. However, since hexachordal manipulation is used so readily throughout, the actual "ur" form of the row is of little interest, and probably undeterminable. Stein seems to support the form I have given, noting as well that the "piano part often complements the given violin part by supplying the tones of the opposite hexachord, as in the opening phrase". Thus, the order of the first twelve pitches stated in the violin part is not of restrictive importance.
Example 7: Two Hexachords of the "Row"
If we reorder the hexachord (either one) we see some interesting harmonic characteristics:
A/C#/F and G/B flat /B
of slightly less importance:
G/B/C# and A/B flat /F.
It is significant that the hexachord contains no major or minor (" tonic" ) triads, and that the two triads present (augmented and flat-fifth) are those which least imply a tonal center out of all possible triadic combinations, since, by Schoenberg's own reasoning, their respective fifths are drawn from the eleventh and thirteenth partials. Schoenberg's concern with this aspect of consonance (and dissonance) can be seen easily in his lifelong devotion to harmonic principles and theory, whether we consider the Harrnonielehre and Structural Functions..., his music, or his short essays. One fine and interesting example of the latter is the famous "Problems of Harmony" from Style and Idea where he relates all harmonies not major or minor to the use of partials higher than the sixth.15
One of the most convenient clues to the functions of these triads in the Phantasy (aug., b5th, and the chord consisting of a root with a minor and major third above it, which I will call the maj/min) comes from Schoenberg himself:
...In my Harmonielehre I have shown how every diminished seventh chord and every augmented triad belong to all major and minor keys, and what is more, in many a different sense. This is probably the place to, point out that J. S. Bach in many 'introductions', for example, and especially such pieces or parts labeled 'Fantasia' prefers a disposition of the harmonic structure which neither in its entirety nor even in its detail can be easily referred to a key. It is not uninteresting that in just such instances these old masters use the name 'Fantasia' and unconsciously tell us that fantasy, in contradistinction to logic, which everyone should be able to follow, favours a lack of restraint and a freedom in the 'manner of expression, permissible in our day only perhaps in dreams; in dreams of future fulfillment; in dreams of a possibility of expression which has no regard for the perceptive faculties of a contemporary audience; where one may speak with kindred spirits in the language of intuition and know that one is understood if one uses the speech *of the imagination- of fantasy.16
Even thirteen years prior to the composition of the Phantasy, such a self-analysis is essential to our understanding of the work. In constructing the particular row and compositional texture (piano accompaniment) Schoenberg makes the grandest of references to the tradition and solves the problem of how to continue writing in it. Thus, the chordal accompaniment, which actually sounds as if it has 'tonal' functions, and indeed frequently does, is never traditionally tonal, for no single fundamental can be truly established with these chords. In the piece, the augmented triad acts as a sort of tonic, or point of reference (though often combined vertically with the maj/min), and the flat fifth acts as its polar opposite, or dominant. A careful analysis of the piano part will reveal an overwhelming preponderance of these three sonorities, in rather undisguised fashion. Almost all three part simple sonorities are of these types, as are most exposed chords.
The augmented triad first appears in ms. 10, in a texture and tempo which makes its introduction quite clear. From then on, it is never absent. Note also that almost any time a chord is rolled (mm. 12, 14, 26, 46,...) it is the combination of maj/min chord and aug. triad. Sometimes, as in ms. 51 (Ex. 8) the violin supplies the missing note:
Example 8: Ms. 51; Vln. and Piano; Maj/Min and Augmented Chord
Note also the important cadential quality of this measure. In the phrase previously discussed, mm. 143-45, the aug. triad at 145 is clearly a point of rest for a chordal figure beginning on the maj/min at ms. 143, and of course, the final sonority in the piano, at ms. 166, is this same combination of the two. Often, as well, the three sonorities are used in rapid chorale-like succession when a feeling of changing harmony is desired, in perhaps a deliberate evocation of the older and tonal homophonic style (e. g., mm. 68-71; 86-91).
Such an analysis reveals
Schoenberg's intense concern with harmony, in that even
with the powerful tools of the twelve-tone system and his virtuosity in it, he
(All Schoenberg quotations are from Style and Idea).
Lewin, David, "A study of Hexachord Levels in Schoenberg's Violin Fantasy" ; Perspectives of New Music; Fall-Winter 1967; pp. 18-33.
Partch, Harry, Genesis of a Music; Da Capo Press, New York, 1974.
Rufer, Josef, The Works of Arnold Schoenberg, Free Press of Glencoe, 1963.
Schoenberg, Arnold, Style and Idea, Stein, Leonard, ed., St. Martin's Press, 1975.
Stein, Leonard, (liner notes to) ARS NOVA, ARS ANTIQUA recording Inner Chambers (of Phantasy).
1 `I would like to thank Dr. Alexander Ringer, eminent Schoenberg scholar, whose fine seminar at the University of Illinois, Champaign-Urbana motivated this paper; and my colleague in the Music Department at Mills College, Karen Rosenak, whose critical and editorial abilities are only matched by her musicianship and generosity.
3 Rufer, Josef, The Works of Arnold Schoenberg, Free Press of Glencoe, 1963, p. 74.
4 Stein, Leonard, (liner notes to) ARS NOVA, ARS ANTIQUA recording Inner Chambers (of Phantasy).
6 Schoenberg, Arnold, "Brahms the Progressive,' Style and Idea, St. Martin's Press, (Stein, Leonard, ed.), p. 399.
7 Schoenberg, Arnold, "Phrasing," Style and Idea, St. Martin's Press, (Stein, Leonard, ed.), p. 348.
8 Schoenberg, Arnold, "Brahms the Progressive," Style and Idea, St. Martin's Press, (Stein, Leonard, ed.), p. 405.
9 Schoenberg, Arnold, "Brahms the Progressive," Style and Idea, St. Martin's Press, (Stein, Leonard, ed.), p. 411.
10 Note that, in the current edition, there
is no measure 13
(but 12b) or 39 (38b) (3x13). I have no idea why this is so.
Hopefully, it will be clarified in the critical edition, as it occurs in other
scores, and at the present it is far too easy to attribute it solely to
Schoenberg's numerological beliefs-7 in many other works (notably
the Prelude, op. 44) these measures exist! Also, in the Phantasy, mm. 52, 65, etc. exist.
11Schoenberg, Arnold, "Hauer's Theories," Style and Idea, St. Martin's Press, (Stein, Leonard, ed.), p. 209. 36
12 ibid., pp. 209-210
13 Lewin, David, "A Study of Hexachord Levels in Schoenberg's Violin Fantasy," Perspectives of New Music, Fall-Winter, 1967, p. 20.
14 Stein, Leonard, (liner notes to) ARS NOVA, ARS ANTIQUA recording Inner Chambers (of Phantasy).
15 There are of course some problems with Schoenberg's theoretical harmonic ideas. On page 271 he states that "Eb is the 7th overtone of F and the 13th of G" and that "Db is the 13th overtone of F and the 11th of G". Partch (page 48) accurately points out that, among other things, the difference between the aforesaid E flat s is 72 cents, and the D flat is 100 cents, or a semitone. Of course, Schoenberg's thesis that the major scale is derived from the first five partials of I, IV, and V (although there is no "IV" in the overtone series) is fundamentally tenable, but the derivation of the chromatic scale in tempered tuning which is virtually axiomatic to Schoenberg's entire system of harmonic thought, is intonationally inaccurate, to say the least.
16 Schoenberg, Arnold, "Problems of Harmony," Style and Idea, St. Martin's Press, (Stein, Leonard, ed.), pp. 274-5.