Wednesday, January 11, 2006

Convertible Counterpoint VII

Taneiev is being very methodical, so be patient with his pace. The scholarly treatise is something I used to find difficult to approach simply because of my nature: I'm naturally impatient. I know what information I want, and I want it now. It was several years - and many experiences with treatises - before I began to understand the nature of the beast: They exhaustively treat a subject. Every last aspect of a subject. In the case of Convertible Counterpoint, I now appreciate this attention to detail.

For example, because I am poor with numbers, I'm thankful that Taneiev displayed the mechanical comparison of successive series during Chapter I. If I can see it, I can internalize it. One of the problems I have with numbers is that - in my mind - they neither look like anything, nor do they sound like anything. My imagination works in terms of geometrical and sonic shapes and relationships: Numbers just don't fit in at all because they are a kind of pure abstraction my particular brain vapor locks on. However, I have learned that if I am patient and I take the time to associate a speciffic set of numbers with a more physical abstraction, I can eventually come to terms with them. I guess I'm maturing (i.e. I'm becoming a slow old fart).

During the Introduction and Chapter I Taneiev gave some examples of the concept and the mechanical basis for that concept. He also introduced an interval designation scheme that will allow for mathematical calculation of vertical-shifting countrapuntal combinations. Chapter II will introduce more terminology for writing the formulas necessary to carry out those calculations. As usual, I will transcribe the treatise in itallics and put my commentary in plain text.

There are also diagrams Taneiev uses that cannot be exactly duplicated with a QWERTY keyboard's fonts, but I have come up with textual approximations that will serve the purpose just fine (These examples must have been nightmarish for 60's-era printing technology). There are also superscripts and subscripts in the formulas, and I'm so much of a computer Luddite that I have no idea how to do this (Searching those terms in Blogger Help gave no results, so it's probably not possible to use them anyway). Due to those shortcomings, I'll simply have to ignore them.

Some of the abbreviations Taneiev uses in his formulas do not make any direct logical sense. For example, Jv means "index of vertical shift." The v is fine, but the J doesn't relate. He may have wanted to keep I as the designation for "voice one" to avoid confusion with a capital I, but it is another quirk of his designation scheme that makes things just that much more difficult to come to terms with.

I will again make entire pages for for the musical examples to keep with my sizing scheme's simplicity, and I'll prompt the reader as to when to consult them if that's necessary.




Notation of the vertical shift: v, vv.

Formulas for Original and Derivative Combinations

§ 20. In the preceeding chapter the shifting of the voices forming separate intervals was investigated. The present subject is the shifting of the contrapuntal union of melodies. To show that two voices form correct counterpoint the Roman numerals indicating the voices (melodies) will be used, united by the plus sign: I + II. Each voice in the derivative combination is indicated by the same figure that it had in the original. I + II is the formula for the original combination.

Obs. - The sign +, used as indicated, refers to addition as meaning a combination of voices (two-voice addition, multi-voice, &c.), and is to be taken in this sense only when employed in connection with the roman numerals for the voices.

§ 21. The letter v, for "vertical" (plural vv) refers to the vertical shift of a voice, and is placed to the right and slightly above the roman numeral corresponding to this voice.
[He is referring to a superscript, which I cannot display. - Ed.] The number indicating the the direction and interval of the shift is united to v by the sign of equality. For example, the expression Iv=5 means the upper voice shifts a sixth upward; Iv=-2 + IIv=-7 indicates a shift of the upper voice a third downward, the lower an octave upward.; the sign + means that the voices so shifted form correct counterpoint.

Such an expression, indicating the shifts of the voices united by the sign +, is the formula for the derivative combination. Formulas for derivative combinations may differ, but that of the original can only be I + II.

§ 22. When a voice remains unshifted neither the letter v nor v=0 need be associated with it's Roman numeral.
[Note: Capitalization of "Roman" is inconsistent in the original text. -Ed.] Therefore the expression I + IIv=1 and Iv=0 + IIv=1 are synonymous, both meaning that the upper voice remains in place but that the lower is shifted a second downward.

The Shifts

§ 23. The relationship of melodies in the derivative combination may present one of the following three cases:

(1) The Direct Shift. - In this the melodies retain their relative positions; the upper voice stays above, the lower underneath, though they may approach each other or recede.

(ex. 1). This shift may be illustrated by the diagram:

Orig. Deriv.
I - I
[I'm having to approximate these, but you get the idea. - Ed.]

The symbol for the direct shift is: =
[Here Taneiev has a graphic that looks like a very elongated equal sign, so the equal sign is what I will use to represent a direct shift. - Ed.]

(2) The Inverse Shift. - Here the melodies change their relative positions; the upper voice goes below, the lower above.


Orig. Deriv.

Symbol: x
[Here Taneiev's symbol is an elongated x, so an x will have to do. - Ed.]

This shift is what is commonly known as "double counterpoint,",
[Two commas in the original. - Ed.] but it is only a special case of the vertical shift.

The final consonance in this example is the unison (0); in the derivative combination it can occur in either the direct shift or the inverse shift.

(3) The Mixed Shift. - This is partly direct, partly inverse:

Relations of Original to Derivative Intervals:

§ 24. The combination of melodies I + II forms a series of intervals: a, b, c, d, .... n. If one of the voices shifts ±s, that is, takes for the derivative formula Iv=±s (§ 21), then a new series of intervals is obtained: a + (±s), b + (±s), c + (±s), .... n. (§ 10). For example, the original combination of Ex. 1 represents the series of intervals: 4, 7, 6, 5, 4, 2, 3, 4.

Its first derivative is I + IIv=3. Adding 3 to each original interval gives the intervals of the derivative combination.
[7, 10, 9, 8, 7, 5, 6, 7. - Ed.]

In the successive series (§ 11) each of these derivative intervals lies from its own original at a fourth to the right (§ 13). In the same way a series of intervals could be obtained for the second derivative, Iv=-2 + II, by adding -2 to each interval of the original.

In Ex. 2 the series of intervals for the original combination is: 7, 6, 5, 2, 3, 7, 9.

Derivative formula: I + IIv=-9. Adding -9 to each interval of the original combination gives a series of negative intervals, each of which lies from it's own original in the successive series at a tenth to the left (§ 13), and showing that the shift is inverse.

§ 25. From the definition of the shifts it follows that at the direct shift the derivative intervals take the same signs as those of the original, and at the inverse shift the opposite signs.

Index of Vertical-Shifting Counterpoint (Jv)

§ 26. To obtain the result of the simultaneous shifting of two voices it is necessary to add to each interval the algebraic sum of the quantities indicating the shifting of either or both voices, i.e. the algebraic sum of their vv (§ 10). This rule is of general application, since the idea of the algebraic sums includes those cases where one of the voices has v=0, that is, remains stationary.

§ 27. The algebraic sum vv of two voices contrapuntally united is termed the index of vertical-shifting counterpoint, and is indicated by Jv (plural JJv), J standing for "index" and v for "vertical shift." In distinction to the sign v, referring to the individual voice (§ 21), the sign Jv can refer only to the combination of two voices.

§ 28. To indicate that a given shift applies at a certain index, the formula of the derivative combination will be put into parentheses and after it Jv; e.g. (Iv=-2 + IIv=-7) Jv=-9. When the formula of the original combination is presented in a similar manner it will mean that the voices admit of a shift at the index indicated; e.g. (I + II) Jv= 2, (I + II) Jv=-2, &c. If one and the same combination admits of shifts at two or more indicies their respective figures are placed after the sign of equality and are separated by commas. For instance, Ex. 1 admits of two shifts: at Jv= 3 and at Jv=-2; this is indicated: (I + II) Jv= 3, -2. Such a Jv, referring to a single original but stating the conditions of two or more indicies, is termed a compound index. A compound index may be double, triple &c., according to how many indicies are united in it (this has no reference to double or triple counterpoint). A compound index is always printed in the singular: Jv, but if several indicies are to be considered that refer to different original combinations the sign is printed in the plural: JJv. If, for example, it is required to list the indicies that correspond to perfect consonances, the expression will read: JJv=±3, ±4 &c., equivalent to Jv= 3, Jv=-3, Jv= 4, Jv=-4 &c.

§ 29. Shifts of voices I + II at a given index may be replaced by other shifts that give as a result the same series of derivative intervals if the algebraic sum of these shifts remains without change (§ 26). A derivative combination will therefore be reproduced on other degrees.

This is a very important point. Taneiev is referring to modal transformation possibilities in the derivative combination: The upper and lower voices can move in any combination of shifts that gives the correct algebraic sum for the index. This gives great modal flexibility and, as I mentioned previously, allows the composer to select the best possibilities and reject the less desirable ones. This is equally important whether the idiom in which the conversions are to be used is modal or tonal.

In this way one and the same index can generate various shifts of voices and therefore can belong to different formulas of the derivative combination.
[Could the translation possibly be less clear? -Ed.]

§ 30. It is possible to get a derivative combination in which one voice remains unshifted. In this case the other voice must take: v= Jv. This follows from the fact that that the index is equal to the algebraic sum vv of both voices (§ 26).

The Formula m + Jv= n; Inferences Therefrom

§ 31. Adding the value of the index to an interval of the original combination gives the corresponding interval of the derivative (§§ 26-7). Indicating the original interval by m and its derivative by n, the relationship is expressed by the equation m + Jv= n.

§ 32. From this equation it follows that -

(1) If m=0 then n= Jv; i.e. from the unison is obtained an interval in the derivative equal to the value of the index.

(2) If m and Jv are equal but have opposite signs, then from m= Jv is obtained n= 0; i.e. from an original combination equal to the index but having the opposite sign is obtained a unison in the derivative combination.

(3) Since at the inverse shift m + (-Jv)=-n (§ 25), then n + (-Jv)=-m. i.e. in double counterpoint, from the derivative intervals are obtained intervals equal to the original. At Jv=-7, for example, from a fourth is obtained a fifth in the derivative (3 - 7=-4), and from a fifth a fourth (4 - 7=-3). At Jv=-9 from a third is obtained an octave (2 - 9=-7), and from an octave a third (7 - 9=-2) &c.

§ 33. Comparing the formula m + Jv= n with the statements in §§ 12 and 30 the conclusion is that in the successive series of intervals (§ 11) a derivative interval lies from an original at an interval equal to Jv. If Jv is positive this distance will be to the right; if negative, to the left (§ 13).

Conditions of the Shifts

§ 34. The index (Jv) may be of positive value, of negative value, or may equal zero. The conditions under which Jv yeilds the direct, the inverse, or the mixed shift are next to be investigated. From the equation m + Jv= n and what was stated in § 25 it follows that -

(1) If m and Jv are both positive or negative the shift is direct.

(2) If one of the quantities of m or Jv is positive and the other negative either the direct or the inverse shift is possible, depending on the value of the intervals in the original combination relative to the value of Jv, namely:

(a) At m with absolute value greater than Jv the shift is direct.

(b) At m with absolute value less than Jv the shift is inverse.

§ 35. From the fact that in the derivative combination a unison (0) may be found in either the direct or the inverse shift, the conditions of the shifts can be stated as follows:

(1) If m > Jv, then with like signs for m and Jv the shift is direct.

(2) If m < Jv, then with like signs for m and Jv the shift is inverse.

§ 36. These principles for the shifts apply without exception to all cases, whether the intervals of the original combination are positive or negative. But since in practice it is advisable not to cross the voices but to use only positive in the original combination, the rules for the shifts applying to the latter may be formulated thus:

This is an important point, because avoiding voice crossings in the derivatives is important to a lot of music and avoids many complications. In fact, for a piano piece (Where unisoni are impossible to play) it may be desirable to avoid 0 as well. Knowing the formula characteristics that will make derivatives either all direct or all inverse are therefore important to note.

(1) At a positive Jv the shift is always direct.

(2) At a negative Jv the shift may be direct, inverse or mixed. Condition of the direct shift: m > Jv (§ 35 [1]); of the inverse shift: m < Jv (§ 35 [2]).
[In the translation the second sign was pointing the wrong direction, which I have corrected. - Ed.] The union in the same original of these and other conditions forms the mixed shift. (§ 23, [3]).

Limiting intervals; Their Signs (< , >)

In the following section, Taneiev uses some musical notation examples that simply relate back to the successive series. I am not going to transcribe these, as they are quite evident from the descriptions. In lieu of this, I will offer simple numerical examples sans the notation.

§ 37. A successive series of original intervals for a positive index (giving always the direct shift § 36, [1]), starts with 0:

Original Intervals:

0, 1, 2, 3, 4 &c.

Jv= 4:

4, 5, 6, 7, 8 &c.

§ 38. A successive series of original intervals for a negative Jv, in order to yeild he direct shift, an interval equal to the absolute value of the index is the limiting interval for approaching voices of the original combination; it is indicated by the sign < . At Jv=-2, for example, the voice must not approach closer than a third.

Original Intervals:

2, 3, 4, 4 &c.
2 <


= (Direct Shift Sign)
0, 1, 2, 3, 4, &c.

At Jv=-3, not closer than a fourth, &c.

The way Taneiev uses this sign ( < ), it basically points to the smallest interval that results in a unison at the absolute value of the shift.

§ 39. A successive series of original intervals for a negative Jv, in order to yeild the inverse shift, must start with 0 and end with a positive interval equal to the absolute value of Jv, showing in this case the limiting interval for receeding voices; it is indicated by the sign > , for example, at Jv=-7 (double counterpoint at the octave) the voices must not receed from each other by more than an octav:

Original intervals:

0, 1, 2, 3, 4, 5, 6, 7 >

x (Inverse Shift Sign)
-7, -6, -5, -4, -3, -2, -1, 0

At Jv=-11 (double counterpoint at the twelfth), by not more than a twelfth

Obs. - Limiting intervals for receeding voices are necessary only for indicies giving the inverse shift. For others the series of original intervals can be continued to the right as far as required.

§ 40. If in an original combination at a negative index are taken intervals some of which are less than the absolute value of Jv and some greater, the result is the mixed shift (§ 23, [3]), Cf. Ex. 3, written at Jv=-7. In the second measure, where in the original the voices exceed the limits of an octave, the shift is direct.

Signs for the Shifts. (<, >, < >, >\< )

For the last sign Taneiev has the greater-than sign directly below the less-than sign. This is impossible to reproduce on a QWERTY keyboard, so I have separated them with a backslash.

§ 41. To indicate the shift of a negative Jv the same signs are used as for the limiting intervals; they are placed after the figure for the index. The sign of the limiting interval for approaching voices ( < ) will be used for the direct shift; that for the limiting interval for the receeding voices ( > ) for the inverse shift. Placed in succession ( < > ) the signs will refer to the mixed shift. For example, Jv=-5< that this index gives the direct shift, i.e. that the series of original intervals relevant to it starts with a sixth.; the limiting interval for approaching voices is also shown: 5<.
[To remain a direct shift and avoid voice crossings, the original combination can have no interval smaller than a sixth. - Ed.] The expression Jv=-5> means that the index gives the inverse shift and that the series of original intervals, while starting with a unison, ends with a sixth; the limiting interval for receeding voices is indicated by 5>. [In other words, the arrow points to the smallest interval necessary to maintain the direct shift, and away from the largest interval possible to maintain the inverse shift. - Ed.] Jv=-5< > indicates the mixed shift, i.e. the presence of both of the preceeding cases, more or less in alternation. When one sign is placed above the other ( >\< ) they refer to two negative indicies of identical value but with different shifts. Therefore, JJv=-4 >\< serves for two expressions: Jv=-4< and Jv=-4>.

§ 42. For positive JJv it is not necessary either to indicate or to establish distance limits for the voices.

Jv Equal to Zero

§ 43. The positive and negative indicies have been considered; the index equal to zero (§ 20) remains to be mentioned. This Jv, like all the others, can denote different shifts of the voices; for example (Iv=9 + IIv=-9) Jv= 0; (Iv=-3 + IIv=3) Jv= 0, &c. The result of such shifts is to yeild a series of intervals identical in value to those in the original.
[It is important to not that a second is a second and a third is a third for this system: There is no distinction in the successive series between major and minor or perfect, augmented, and diminished. The value is identical, even though the mode will have been transformed. - Ed.]

§ 44. It is possible to regard every recurrence of a two-voice combination on the same degrees or its removal to other degrees as a shift at Jv= 0.

† The rules of simple counterpoint are the rules of Jv= 0, and therefore simple counterpoint can be understood as a special case of the vertical shift.

I have separated this amazing insight because all rule restrictions pertaining to the various shifts are just added restrictions on the rules of simple counterpoint, as we shall see.

§ 45. The important idea implied by the use of the symbol Jv simplifies the study of vertical-shifting counterpoint; it yeilds numerous possibilities of voice shifting with a comparitively small number of indicies.

I'm definitely going to do the next chapter in two parts.

I don't think it's what's under the bed she should be worrying about.


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