Monday, January 02, 2006

Convertible Counterpoint V

A lot of my musical philosophy comes directly from Taneiev. It's been twenty years since my first time through this book, so I had forgotten just how much I owe him in that area. When he speaks of the super-chromatic harmony of the ultra-romantics and how it contributed to the decline of large-scale works, I really don't think there can be any arguing the point. This is even more true for the serialists. He was borderline psychic in his forward-looking thought on this matter.

The prescriptive he offers (A fantastic example of constructive criticism!) - that of using imitation and convertible counterpoint to re-achieve unifying principles that would again allow for coherent large-scale forms - seems to be the perfect answer for those of us who wish to substitute the chromatic idiom for the diatonic one, and to allow again for "any harmony to be followed by any other harmony", but on a chromatic basis versus the sixteenth century diatonic one. In this we can see that though this book covers complex counterpoint in the strict style, once the techniques are internalized, there is no barrier to substituting chromatic lines for diatonic ones: The underlying natural laws that govern contrapuntal motion are immutable.

Years ago I did some counterpoint exercises - experiments actually - in which I made my first stabs at this kind of chromatic contrapuntal texture: I called them "colorpoint" exercises, because I used five voices and contrapuntal rules to connect all of the "beautiful dissonant" sonorities that I love; added ninths, major sevenths with added augmented elevenths, &c. Though I didn't have the technique to really expand on these early efforts - which amounted to little more than individual phrases in isolation - the results were nevertheless quite striking and effective. My goal is to work back to that idea but to add to it the techniques in this book.

Starting in the second half of the introduction, Taneiev begins to use musical examples. In order to figure out how to effectively present these, I had to take a time-out to decide upon some formatting issues that have to do with limitations inherant in my notation software as well as the image sizing limitations of my Smugmug account. What I came up with is the idea that all examples will be presented on full pages at 75% magnification: This will allow for the examples to be all the same size, and they will comfortably fill up the column width of my template. However, there will be a lot of very "white" pages. The advantage is that I will have to make exactly zero further formatting decisions, which will make things easier and speed up the process considerably.

I am also going to put all of the examples into a single file (I may make different files for different chapters though, just so you can find the examples a bit easier), which will allow me to post single PDF and MIDI files on my Fileshare page for those who would like to listen to the examples and follow along using the score.

In the first examples presented here in the introduction, Taneiev uses a single two-voice combination that produces eleven derivatives, for a total of twelve versions of the same two melodies. The original combination has the voices starting out a perfect fifth apart (+4), and the vertical shifts allow for them to start out an octave apart (+3 from the original), a third apart (-2 from the original), and with the voices inverted at a sixth apart (-13 from the original, which is the octave inversion of the -2 version).

After that, he presents horizontal shifts at two beats of delay and four beats of delay from the original, double-shifts of both vertical and horizontal conversions, and finally, he has the melodies doubled in imperfect consonances. Note that in the original and all of the derivatives there is no parallel motion at all: Oblique and contrary motion is all that is allowed. My first time through this book, I limited myself to internalizing the shifts that could be reduced to simple rule restrictions such as this, and skipped over the algebraic formulas. I got a lot out of it nevertheless. This time, I want to fill in the gaps and really come to an understanding of the underlying mathematical technology. For someone who is practically retarded when it comes to anything having to do with numbers, this will be no insignificant challenge, but I'm highly motivated, so we'll see what we shall see.


by Serge Taneiev

Complex counterpoint is divided into categories according to the methods by which derivative combinations are obtained. The principle methods are; 10 the shifting of voices; 2) duplication in imperfect consonances, and 3) transmutation; hence the three aspects of complex counterpoint: 1) Shifted, 2) duplicated, 3) metamorphosed.

A) Shifting Counterpoint

A derivative is obtained by shifting the voices. he following classification exhausts all possible shifts:

1) Vertical shifting - upward or downward - hence vertical shifting counterpoint:

In the third example (above), the upper voice is shifted underneath and the lower voice above, a special case of the vertical shift known as "double counterpoint."

2) Horizontal shifting, in which the time-intervals between entries of the voices are changed, hence horizontal-shifting counterpoint:

3) Vetical and horizontal shifting together, hence double-shifting counterpoint:

In this work double-shifting counterpoint is included in the divisions devoted to horizontal-shifting and is explained in parallel with it, as the methods of writing both are similar.
[I was afraid of that. I'm still hoping to be able to skip the double shifts if I'm having a rough time with the math. - Ed.]

The subdivisions of shifting counterpoint are therefore:

Shifting Counterpoint (a derivative from the shifting of voices):

1) Vertical-shifting (Upward or downward);
2) Horizontal-shifting (changing relationship between entries);
3) Double-shifting (the combination of the two preceeding).

B) Duplicated Counterpoint

A derivative combination is obtained by duplicating one or more voices in imperfect consonances. Therefore the number of voices in the derivative is increased: at the duplication of one voice to three; at the duplication of two voices to four.

An original three-voice combination yeilds derivatives of four, five, or six voices, according to how many voices in the original are duplicated. Examples will be found in chapter XV.

The connection of this counterpoint with the vertical-shifting counterpoint is obvious: each duplication is nothing but the vertical transference of a voice at an interval equal to an imperfect consonance. Therefore the study of counterpoint admitting of duplications is included in the divisions dealing with vertical-shifting counterpoint.

In the various phases enumerated of complex counterpoint, forming the contents of the present work, the detailed treatment of double counterpoint is of the utmost value in music theory. It is of practical importance both in connection with counterpoint admitting of duplication and with certain cases of multi-voice vertical-shifting counterpoint, for instance in triple and quadruple counterpoint, especially at the octave. In theoretical literature little reference is found to any aspects of vertical-shifting counterpoint other than double, and still less to horizontal-shifting counterpoint, the study of which, as a special department of shifting counterpoint, is here presented for the first time.
[Emphasis mine. - Ed.]

C) Metamorphosed Counterpoint

The derivative is obtained by a process of transmutation. By metamorphosis is meant such a change of the original combination as would correspond to it's reflection in a mirror - this is known as "mirror counterpoint." To avoid ambiguous terminology "metamorphosis" will be used only in this sense, and will not refer to the shifting voices in double counterpoint. Since metamorphosed counterpoint does not enter into the plan of the present work it will not be considered further.
[I'm betting this is going to be found in the Doctrine of Canon, where mirror canons are certain to be covered. - Ed.]

The statement has been made that the transference of voices is characteristic of shifting counterpoint. The changes that this counterpoint makes in a melody amount only to its transference vertically to other degrees or horizontally to other measures or parts of measures. Every other change, such as metamorphosis. augmentation, diminution, made simultaneously with shifting, places the given combination beyond the scope of convertible counterpoint, and the shifting ceases to be a vital characteristic.

Another feature of complex counterpoint remains to be mentioned: the existence of rules in relation to its various subdivisions and to simple counterpoint. The study of the latter, in either the strict or free style, is that of a system of rules to which every union of voices must conform. In complex counterpoint the same rules apply to both original and derivative combinations. The significance of the rules of complex counterpoint is that if they are ignored in the original combination the derivative will show progressions that violate the rules of simple counterpoint. The mutual relations of the aspects of complex counterpoint and their relations to simple may be illustrated symbolically by the accompanying diagram, where the large circle represents the domain of simple counterpoint, and the small intersection circles, with the portions of their areas coinciding, the various aspects of complex counterpoint.

Here Taneiev has a diagram of four circles: A large circle with three smaller circles contained within it. The interior circles have areas all their own, as well as smaller areas that only two circles share, and a small central area (That looks like the rotor of a Wankel engine, if you are a gearhead like me) which all three circles encompass. If you are a Christian, you've seen this diagram many times as a representation of the Holy Trinity. If I had my old OS 9 Paint application I could duplicate it, but unfortunately OS X has nothing I can find that will do the trick. Yes, I need a scanner.

From this diagram it is clear that the combinations used in complex counterpoint must also belong to the domain of the simple, but not vice versa; portions of what is permitted in simple counterpoint are found outside of the circles that represent the various aspects of complex. The intersections of the circles show that certain phases of complex counterpoint may be combined, as was illustrated in the examples given.

This work is divided into two parts; the first part deals with vertical-shifting counterpoint and counterpoint admitting of duplications; the second with horizontal-shifting and double-shifting counterpoint. Each part consists of two divisions, one devoted to two-voice counterpoint
[Which I'll cover. - Ed.], the other to three voice. [Which I'll skip. - Ed.] The investigations are limited to two or three voices. More than these are found only as the result of duplications; they are given at the end of the first and second divisions of Part One, where duplications are found using a larger number of voices, up to six inclusive.


"Fabulous, darling!"


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