### Convertible Counterpoint IX

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

CHAPTER III (CONT.)

THE GROUPING OF INDICIES

*[Example omitted. - Ed.]*

Octave relationship of

§ 54. The lowest

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:

§ 55. In each column the lower

§ 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.

*(column)*

§ 56. If a derivative is written at a middle

§ 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*of the middle*

Take for example the table in § 54.[Example omitted. - Ed.]

**Jv**.Take for example the table in § 54.

*Writing a combination at*

In general, every combination written at a middle

§ 57. The statement was made in § 51 that the unshaded columns contained(= intervals that appear in three forms: perfect, augmented, and diminished)

**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.)*, and those shaded,*(= intervals that appear in four forms: major, minor, augmented, and diminished)

**JJV**corresponding to intervals of the second group (2int.)*. 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*

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

[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.]

**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.*[Remember, the fourth is a dissonance in this system. - Ed.]*

§ 59. The

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.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.*The*[Missing text must be] second group, while the

**2JJv**in the first pair of columns correspond to dissonances of 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

§ 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

(a)

(b)

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 (

§ 62. This will be illustrated by two examples determining the value of the original interval (

(1) From what interval is obtained a tenth at

Since

(2) From what interval at

Since

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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