Convertible Counterpoint II
Convertible counterpoint does not require that the voices invert, just that the voices convert to a different intervallic distance. For example, if a two voice combination is written with the proper set of restrictions, it can make proper counterpoint starting an octave apart or starting at a perfect fifth apart. In Convertible Counterpoint, Taniev exhaustively lists all possibilities for conversion. In order to do this, he defines intervals of conversion in mathematical terms that take a little getting used to. The interval of a unison is not one, but zero; a second is not 2, but one, &c. By employing this terminology, Taneiev is able to reduce the process down to a mechanically precise technique wherein there is no needless and time-wasting experimentation: The convertible combination is properly written from the beginning with the possible conversions being known at the outset.
Familiarity with the rules of convertibility will also allow the composer to exhaustively list all possible conversions of pre-existing contrapuntal combinations: Simply analyze the intervallic progression, and the non-usable conversions will eliminate themselves and the usable ones will be revealed.
Obviously, this technological approach is never taught in any university or conservatory counterpoint classes (Unless Taneiev's book is being used as a text, and I've never heard of this being done), and that is a shame because the technology is obviously supremely superior.
The applications for this technology go far beyond fugue writing, as they can be applied to sonata process themes, song melodies and accompaniment, or anything else you can imagine applying them to. Taneiev himself said that in miniatures there are rich possibilities for applications of these techniques (I used convertible counterpoint in a sonatina where the second theme and it's counterpoint work an octave, a tenth, and a twelfth apart, which simply requires no parallel movement at all, and so is quite easy to do: In that piece the theme appears in E minor, G major, and C major, but always the theme is on the same tonal level; only the bass line's level and the respective key-related incidentals are changed. The theme also, therefore, works harmonized in thirds with the bass counterpoint: This is a very cool effect!).
by Dr. G. Ackley Brower
The works of Serge Ivanovitch Taneiev (1856-1915), both musical and literary, seem to be little known outside of his native country, though recent years have witnessed an increased interest of one who as a composer, theorist, concert pianist, critic, and teacher, became one of the outstanding figures in the musical life of Russia and who is now gaining the wider recognition that he deserves. Taneiev's compositions must speak for themselves; the present purpose is neither to attempt a critical estimate of them nor to give a biographical sketch of their composer, but to introduce to teachers and students of composition his work on advanced counterpoint.
It is difficult to discuss this book in terms of restraint. Since the fourteenth century, or earlier, many books on music theory have appeared. Amid a mass of indifferent writings and others of considerable value but not outstanding there are a few that have made history; one of the greatest was the Dodechachordon of Glanarious, another was the Gradus ad Parnassum of Josph (sic) Fux. Not without good reason has Lazare Saminsky referred to Convertible Counterpoint as "having the same meaning for for musical science that Newton's Principia has for cosmology." It may be said without reservation that the student has at hand the greatest work on counterpoint ever written. It is a book that will reveal possibilities for the art of composition that have hitherto been but vaguely realized or actually unknown. Though applying, according to the title, only to the strict style of the Polyphonic Period, it's principles, as the author himself says, may be extended to the free style of later times and to the modernism of today and of the future. To study and master its contents will give the student a command over the resources of composition that can be obtained in no other way. The variety of subjects in it, the thoroughness and clearness with which they are presented, their logical arrangement, the examples illustrating the text, and above all the outstanding originality of the author's thought - all this makes this a work compared to which other books on counterpoint seem elementary.
From this it may be inferred that Convertible Counterpoint is not a beginner's text, yet it's study may be undertaken sooner than might be expected. The author says (in § 175) that the exercises he suggests suggests should start as soon as three voice counterpoint can be satisfactorally managed [I had to wait until I could manage three voices comfortably on the guitar - Ed.], and that thereafter simpls convertible counterpoint should be studied concurrently. To this I would add that there seems to be no reason why the exercises in two voices should not be successfully attempted as soon as two-voice mixed counterpoint (i.e. both voices in the fifth species) has been studied. [The reason Taneiev suggests that one wait until three voices are mastered is because to make the conversions effective, and to apply them in an actual composition, a third free voice must often be composed. - Ed.] The student should then be well able to cope with the simpler of the fascinating problems set by Taneiev in the earlier chapters of Parts One and Two, though the more difficult ones will require require skill in the manipulation of from three to six voices. [Exactly: I can handle five voices off of the guitar now, so I should be able to cope with writing purely theoretical examples with all of the conversion possibilities presented. I can't imagine a student who can write in only two voices being acute enough to grasp some of these concepts and apply them unless only the simplest possibilities are attempted. In my opinion, it's good to be exposed to these concepts early, but to study through this book thoroughly will require a fairly advanced facility with counterpoint in at least three voices. - Ed.]
The first thing that is likely to surprise the reader who may think that this is "just another book on counterpoint" is the proposition advanced by the author in his Preface - that the study of counterpoint is here put on a mathematical basis - algebra in fact. [I score only in the forty-second percentile in numerical ability, and got D's consistently in math classes all of my life (Except for geometry, where I got an A), so hold on to your butts: This could get messy. - Ed.] Yet this is quite in accordance with present-day tendencies, and the fact that Taneiev thought about it as far back as 1870 shows that he is a pioneer in a field of research that now includes several prominent names in the musical world. But the student may be assured that he is not expected to know more than the fundamentals of algebra; of this more will be said presently. [I had to look up the definition of algebra! - Ed.] Taneiev's method opens up an enormous extent of untried resources, heretofore inaccessible because of the lack of an adequate approach - and only mathematics can provide it. Let no one get the idea that such an approach will stifle the imagination and yeild the unwelcome result of writing music that sounds mathematical. The effect of Taneiev's method is quite the opposite; it releases the imagination, pointing the way to endless possibilities that otherwise would never have been thought of. Here a few statistics may be enlightening, as showing how inadequately a vast subject has hitherto been treated.
Referring to eighteen standard texts that claim to teach double counterpoint *, I find that while all all of them deal with double counterpoint at the octave (or two octaves), none mentions double counterpoint at the fifth, only three at the sixth, two at the seventh, six at the ninth, seven at the eleventh, all except two at the tenth and twelfth, six at the thirteenth and five at the fourteenth. Two of them deal with double counterpoint only at the octave,. Several speak disparagingly of double counterpoint at intervals other than the octave, tenth, and twelfth. Not one mentions a use of combined themes that is found in Bach but which can be classified as neither simple nor double counterpoint. Put together, these texts provide for only nine ways of writing counterpoint in which the interval-relationship could be changed; Taneiev deals exhaustively with twenty-three, not by giving endless lists of rules and exceptions by by equations in simple algebra that eliminate all trial-and-error methods and that give positive results. All of them are practicable in the strict style, not to mention the free. Furthermore, none of these texts deals with tripple counterpoint at any interval other than the octave, and one of them (Jadassohn) definitely states that such a thing is impossible. Taneiev shows how it is done.
Of the authors cited in the first footnote and who could be expected to know about Taneiev's work only one, George Conus, mentions him. Conus' book A Course in Modal Counterpoint ** refers briefly to double counterpoint at the octave, tenth and twelfth, but he gives credit to Taneiev as having written the only complete treatise on the subject. After all, the sources upon which Taneiev's work is based were available to many who came before him, to his contemporaries, and to all who came after, and the fact that no one took full advantage of them certainly justifies the remark made in § 279 about certain theorists being exposed to "grave suspicions."
Nearly all these texts confuse the issue by treating double counterpoint at the tenth and the twelfth and duplication in imperfect consonances as belonging to the same category, whereas they do not. Again, the changes possible in the time-entrances of two or more melodies, called in this book horizontal shift, are ignored in all texts except Taneiev's, though some authors call attention to this interesting phenomenon in occasional quotations from Bach. But none of them throw the faintest light on how it is done. Finally, the principles of duplicated counterpoint and the horizontal shift combined with other varieties of counterpoint in both two and three voices leave one amazed at the enormous scope of the subject. An inventory of what is still untried in counterpoint would, by application of Taneiev's methods, run almost into astronomical figures.
Now as to algebra: the amount needed is very small - only a knowledge of the meanings of the signs +, -, and [Those right and left arrows code is enclosed in, which vanish in the preview pane! - Ed.], the pribnciples of addition of addition and subtraction [I have enough fingers and toes, I think. - Ed.]; the transposition of terms in equations, and the rules for the removal of parentheses. Three hours spent with any algebra textbook should be enough for the most complicated of Taneiev's problems. [If I had a nickel for every math teacher who said to me, "This isn't really very difficult: Why are you having so much trouble with it?" I'd be a thousandaire. - Ed.]
A few additions of my own are in footnotes or at the ends of chapters. The musical quotations have been verified - a necessary measure as the original edition contains many misprints. On the last page of the original is the word konyetz, which I have omitted, as this book was not "the end" but was followed by a sequel dealing with the canon, doing (for - Ed.) this difficult style of composition what the present work does for convertible counterpoint - (it - Ed.) puts the study on a scientific basis.
I am indebted to several whose interest, advice, and information are in no small measure responsible for the appearance in English of the monumental work of Taneiev. First to be mentioned in (sic) Lazare Saminsky, who, about twenty years ago, told me of Taneiev's book and how radically different and superior it was to other texts. Without his description of the book and his enthusiastic recommendation of it I might never have known about it. Next are the obligations I owe to Alexander Siloti and Nicholas Orloff, both pupils of Taneiev whose reminiscences of their teacher were of the greatest interest. From Serge Rachmaninoff, Leopold Godowsky, Moritz Rosenthal, and Gregor Piatigorski I have received encouragemnent in a project that I entered upon with some doubts as to its interest to a publisher but none as to its value. Dr. Serge Koussevitzky, whose activities in the musical life in America were too well known to need further comment, contributed an Introduction. From books I have got valuable help from the Memoirs of Taneiev ,by Leonid Sabaniev, and from the second volume of the History of Music in Russia, a symposium on Taneiev, published in Moscow.
Tha (sic) manager of Bruce Humphries, Mr. Edmund R. Brown, and the members of the editorial and production staff have solved most successfully the peculiar mechanical problems that the printing of such a complicated text involves - the first of its kind to be done in English.
* The authors consulted were: Bandini, Bridge, Cherubini, Conus, Dubois, Goetschius, Haupt, Jadassohn, Jeppesen, Kitson, Krohn, Marquard, Morris, Prout, Richter, Riemann, Spalding, and Stohr; by no means exhaustive of the literature on the subject but, I think, a fair cross-section.
** In Russian; it is published by the Soviet Government.
I should get some kind of a reward for hunting-and-pecking my way through that.
"What did you have in mind?"
Uh... er... (Huc drops ball: punts)...