1. Oswald Jonas, Introduction to the Theory of Heinrich Schenker, translated and edited by John Rothgeb (New York: Longman, 1982), 127-8.

2. Dirk Haagmans, Tonal Function: Harmony, Scales, and Intervals, Book 1 (New York: J. Fischer & Bro., 1916). John Comfort Fillmore's translation of Riemann's 1882 lecture, "Die Natur der Harmonik" (in New Lessons in Harmony [Philadelphia: Presser, 1887]) should be noted as the first published appearance of Riemannian ideas in North America. Also noteworthy is the persistent, sharp criticism of Riemann's work by Bernhard Ziehn, a German expatriate living in Chicago whose writings were followed in the German-American community as well as in Germany itself.

3. Journal of Music Theory 26.1 (1982): 32-60.

4. Journal of Music Theory 42.2 (1998): 167-80.

5. Riemann's writing desk--at which he worked, 4 am to 10 pm, every day of the year except Christmas--was called "Papa's Altar" by his kids. Michael Arntz, Hugo Riemann (1849-1919): Leben, Werk und Wirkung (Cologne: Concerto Verlag, 1999), 43.

6. Hermann Helmholtz, On the Sensations of Tone, trans. and ed. Alexander Ellis (New York: Dover, 1954), 235.

7. To be sure, the central place of the major third in Riemann's thought, an interval he understands as primarily a harmonic one, is the starting point for any reconstruction of Riemann's theory-building method. But yet there are (and could have been then as well) other entry points into dualistic systems besides the triad.

8. Rehding's extract leaves out the positive attributes of the major mode, quite understandable given the direction of his argument at that point. His redaction of the second and third sentences is: "The major mode�is quite unsuited" etc. Also, in connection with Rehding's description of the passage as "added," it is appropriate to note that the passage was in the first German edition (1863) of Helmholtz's treatise (p. 463) and was not an addition to subsequent editions.

9. I sketch this situation in Chapter 7 of Harmonic Function in Chromatic Music (Chicago: University of Chicago Press, 1994).

10. On the other hand, dualism did seem to engender interest in symmetrical musical structures even among opponents of Riemann's dualism. See David W. Bernstein, "Symmetry and Symmetrical Inversion in Turn-of-the-Century Theory and Practice," in Music Theory and the Exploration of the Past, ed. Christopher Hatch and David W. Bernstein (Chicago: University of Chicago Press, 1993), pp. 377-407.

11. Helmholtz's translator, Alexander Ellis, notes that meantone temperament may have played a significant role here. The minor third of the tonic triad was quite sharp and hence "much rougher" than just, whereas the major third was just or nearly so. Helmholtz, 301, note �.

12. See Helmholtz, pp. 301-2, 365, and vii.

13. A Riemannian rebuttal of Helmholtz's arguments would, I am quite sure, involve evidence of composers treating the two modes in some equal fashion, and exhibit A would be the Well-Tempered Clavier. Here, the place of Bach and of his great minor-mode works (e.g., the B-minor Mass) in the reception history of German music would be critical.

Before leaving the subject of Riemann and his predecessors, I should correct an apparent misreading of Moritz Hauptmann's minor-triad structures and explanations, which Robert Wason caught and pointed out to me. While not a full-gospel dualist, Hauptmann consistently thought about musical structures in dualistic, oppositional ways. The inversional relationship between major and minor triads was thus of great interest to him. In adapting his dialectical system (succinctly described by Rehding on p. 24) to the minor triad, his first impulse was to invert the representation, so that the starting point, I, was reckoned from the upper note of the fifth, like so:

Hauptmann, p. 17, 
figure no. 1
II -- I
F ab C
  III --I

This analysis of the minor triad, shortened to II-III-I, is used throughout the rest of his book and seems to be entirely satisfactory. However, through some unexplained calculus, Hauptmann claims that the above structure "is the same as" the following analysis:

Hauptmann, p. 17, 
figure no. 2
= Rehding, Ex. 1.4a
I -- II
F ab C
  I --III

I will avoid a detour into Hauptmann's opposition of "determine" and "is determined," which seems to be the way of balancing the equation of first and second figures, and instead point out that the two figures claim very different things. The second, in fact, is consonant with a "co-generation" theory of minor, in which the chord appears to have two "roots": in this case, F and ab. Curiously, it is this figure that appears first in Rehding's discussion of the "dialectial" minor triad--as an inverted major.

The following figure then appears in Rehding's book as an illustration of Hauptmann's "alternative explanation of the minor triad, bottom-up but no longer dialectical" (25):

Rehding, Ex. 1.4b
I -- II
F ab C
I-- III  

This representation is found nowhere in Hauptmann, and is even at odds with Hauptmann's basic claims of interval intelligibility (that is, there is no intelligible relationship in the minor third F-ab.) Something has gone wrong here, and I suspect a kind of typographical error involving Rehding's examples, since the text commentary suits the "correct" versions of the examples much better. Since more readers of Rehding's book may be consulting this review than combing library stacks for a dusty copy of Hauptmann, I submit that discussing the typo here is better than merely calling attention to it.

14. See Arntz, 341-44, for a list.

15. Ludwig van Beethoven, Sonaten für Klavier, kritisch revidirt mit Fingersatz und genauer Bezeichnung der Phrasirung herausgegeben von Dr. Hugo Riemann (Berlin: Simrock, 1885), II: 4. Translation by the author. Schenker's response to this kind of work was his essay "Abolish the Phrasing Slur" (The Masterwork in Music, vol 1 [1925], translated by William Drabkin [Cambridge: Cambridge University Press, 1994]), in which he voices his indignation that "�editors can have the audacity to incorporate their interpretations into a work of art of which they have not the slightest understanding" (20).

16. See, for example, Hugo Riemann, Ludwig van Beethovens sämtliche Klavier-Solosonaten, 2nd edition (Berlin: Max Hesse), II: 201, for rebarring the opening of op. 27, no. 1.

17. Heinrich Schenker, Counterpoint, trans. John Rothgeb and J�rgen Thym, ed. John Rothgeb (New York: Schirmer, 1987), I: xvii.

18. Quotation from Riemann, Musikalische Syntaxis (Leipzig: Breitkopf & H�rtel, 1877), 120. It is the peroration of the work.

19. Arntz, 120.

20. To be sure, a terrible mutant strain of this attitude came back with the swastika, and reading Riemann in the backwards-reflecting light of this tragic development is quite chilling.

21. Consult, for example, the thread initiated by John Clevenger entitled "The nature of tonality in the Western European tradition" at http://societymusictheory.org/pipermail/smt-talk/2002-December/.

22. Artnz's book is given a thorough review by Rehding in Music Theory Spectrum 24.2 (2002): 283-93. Some material in the review also appears in Rehding's book. Renate Imig, Systeme der Funktionsbezeichnung seit Hugo Riemann (Düsseldorf: Verlag der Gesellschaft zur Förderung der systematischen Musikwissenschaft, 1971).

23. Hugo Riemann's Theory of Harmony and History of Music Theory, Part III (Lincoln: University of Nebraska Press, 1977).

24. David Kopp, Chromatic Transformations in Nineteenth-Century Music (Cambridge: Cambridge University Press: 2002).

End of footnotes