I wish I'd have thought of that.
There is now a Serge Taneiev section in the Sidebar to the right which has links to all of the Convertible Counterpoint posts in order: As the next one on the list gets to the bottom of the blog's main page, I will continue to add to it. I decided to create a Serge Taneiev archive due to the fact that I intend to blog through my translation of his Doctrine of Canon next: That way one section will serve for both treatises.
We left off in Chapter III where Taniev was about to explain the octave relationships of the indicies.
CHAPTER III (CONT.)
THE GROUPING OF INDICIES
Octave relationship of JJv
§ 54. The lowest Jv of each column corresponds to a positive interval within the limits of an octave; the middle Jv to a negative interval, also within an octave, and the upper Jv to a negative interval beyond the octave limit. If -7 is added to the lower Jv the middle one is obtained; if -7 is added to the middle Jv (or -14 to the lower) the upper Jv of the same column is obtained. Therefore if in a combination of I + II one of those voices remains in place while the other shifts in conformity to each of the three JJv of a given column, shifts of the melody will result, to the same degrees though in other octaves. All three indicies of the same column are thus in an octave relationship. This is clearly shown in the following table, using the indicies of the third column: [Example omitted. - Ed.]
§ 55. In each column the lower Jv has the direct shift, the upper the inverse shift. Of the middle numbers, three: -6>\<, -5>\<, and -4>\< have both the direct and the inverse shift, and three; -3<, -2< and -1< have the direct shift only (§ 46).
As I hinted at earlier, practical considerations make -7 the most natural pivot point for the division between inverse and direct shifts, but as you gain familiarity with the system problems associated with the mixed shifts at other indicies will be easily avoided by using appropriate melodic ranges and octave displacements as required.
§ 56. If a derivative is written at a middle Jv and one of the two voices is separated by an octave, a derivative is obtained at an outer Jv of the same shift (column) of the middle Jv.
Take for example the table in § 54. [Example omitted. - Ed.] Writing a combination at Jv= -4< (i.e. with the direct shift) and separating a voice of the derivative an octave gives another derivative at Jv= 3, i.e. at the lowest Jv of the same column, also a direct shift. Taking a derivative at Jv= -4>, i.e. at the Jv giving the inverse shift, and separating a voice an octave, gives the derivative at Jv= -11, the highest Jv of the same column and also an inverse shift. Similarly, a derivative at Jv= -5<, separated an octave, yeilds another at a Jv in its own column: Jv= 2; one at Jv= -5> in the same way a derivative at Jv= -12. The same relation holds between Jv= -6> and Jv= -13, &c.
In general, every combination written at a middle Jv yeilds a valid derivative at each outer Jv of the same column and with the shift indicated. But the contrary is not necessarily true; writing at an outer Jv may be unsuitable for shifts at the middle, as will be seen later (§ 72).
§ 57. The statement was made in § 51 that the unshaded columns contained JJv corresponding to to intervals of the first group (1int.) (= intervals that appear in three forms: perfect, augmented, and diminished), and those shaded, JJV corresponding to intervals of the second group (2int.) (= intervals that appear in four forms: major, minor, augmented, and diminished). On this fact is based a division of indicies into two groups that has great importance for the whole study of vertical-shifting counterpoint. In the first group are JJV corresponding to the 1ints. (i.e. those in columns 3, 4 and the zero column). In the second group are JJv corresponding to the 2ints. (i.e. columns 1, 2, 5 and 6). When it is necessary to refer the characteristics of a given Jv to either of these groups, use will be made of the indications 1Jv and 2Jv.
Grouping of the columns in pairs
§ 58. In the following table six columns (1-6) are grouped in pairs; the zero column is isolated. In each pair of columns the sum of the lowest indicies equals 7, and each lowest index is of the same value as the middle index in the other column of the same pair, but with the opposite sign; the lowest Jv is positive, the middle Jv negative. The zero column is unrelated and cannot pair with any of the others.
[This table is not in the same form as the book has it because I cannot duplicate the shading and this isn't an ASCII format. - Ed.]
1st Pair:
2int.
-13>
-6>\<*
1
2int.
-8>
-1<
6*
2nd Pair:
2int.
-12>
-5>\<*
2
2int.
-9>
-2<
5*
3rd Pair:
1int.
-11>
-4>\<*
3
1int.
-10>
-3<
4*
Zero Column:
1int.
-14>
-7>
0
* Absolute values are the same within the pairs.
§ 59. The 1JJv in the zero column correspond to perfect consonances; in the third pair of columns some JJv correspond to perfect consonances, others to dissonances of the first group. [Remember, the fourth is a dissonance in this system. - Ed.] The 2JJv in the first pair of columns correspond to dissonances of the [Missing text must be] second group, while the JJv of the second pair to consonances of this group, i.e. imperfect. [Again, there are quite a few errors in the printing of this treatise. - Ed.]
§ 60. Classifying the JJv in this way into two groups, arranging them in columns showing octave relationship, and then combining the columns in pairs that bring indicies in logical order - all this facilitates their study and recall.
Taneiev is about to get into some of the basic furmulas that will be followed up with questions requiring that calculations be made. I'll have to illustrate some of this with the musical examples he presents. These begin to get quite interesting, since they are from the music of Palestrina.
Determination of the value of an Original Interval and of the Index
§ 61. Reference to the formula m + Jv= n (§ 31) shows that a derivative interval (n) is equal to an original interval (m) to which Jv has been added; i.e. the equation constitutes a definition of what the two other quantities m and Jv are equal to. Taking -
(a) m= n - Jv= n + (-Jv)
(b) Jv= n - m= n + (-m)
Though Taniev is about to present examples of his own, it is useful at this point to actually create one out of whole cloth for illustrative purposes in order to internalize these formulas. If the original initial interval (m) of a combination is a third (2), and the index of vertical shift (Jv) is a fourth (3), the resulting initial interval (n) for the derivative is a sixth (5): 2 + 3= 5. In (a) and (b) Taneiev simply exchanges the terms algebraically (Man: If my highschool algebra I and II teacher could only see me now! (I got a C and a D in those courses, I believe)):
(a) 2= 5 - 3= 5 + (-3)
(b) 3= 5 - 2= 5 + (-2)
I can't believe this actually makes more sense to me with numbers involved. The end of the world is truely near at hand.
i.e. (a) the original interval (m) equals the derivative interval (n) plus Jv taken with the opposite sign; (b) Jv equals the derivative interval (n) plus the original interval (m) taken with the opposite sign.
§ 62. This will be illustrated by two examples determining the value of the original interval (m) according to the derivative (n) and Jv (equation (a), § 61).
(1) From what interval is obtained a tenth at Jv= 4?
Since Jv is positive and has the direct shift, the derivative tenth (9) is also positive. Adding it to the index with the opposite sign (equation (a), § 61) gives: 9 - 4= 5. Therefore at Jv= 4 a derivative tenth is obtained from a sixth.
(2) From what interval at Jv= -11 is obtained a derivative ninth?
Since Jv= -11 has the inverse shift (§ 49) a derivative ninth is negative (-8). Adding to this quantity (according to the same equation) the index taken with the opposite sign gives: -8 + 11= 3, i.e. a derivative ninth is obtained from a fourth.
Starting in § 63 Taneiev begins to use musical examples which I will have to transcribe, take screen shots of, and upload to my Smugmug account. Since there has been too long of an intermission between these posts as it is, I'm going to stop here and start on another entry. I don't want to get caught with my pants down on this project like I did with the Beethoven analysis and have to abandon it.
Oh, for crying out loud.
Art Frahm was definitely sick - he did several pinups along these lines - but his warped humor and these physically impossible happenings do make me laugh.
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