### Convertible Counterpoint VIII

This reminds me of when I was learning to program in BASIC years ago. I had to learn a lot of the language before I got to the "critical mass" point where ideas for how to use it actually started to occur to me. If I recall correctly, it was past the mid-semester point before I wrote my first program, which basically (Ha, ha; very punny) just mapped pitches onto various user-definable growth and decay series. Later I added dynamics and modal filters to it, and by the end of the semester, I had created a very cool Synclavier piece out of it which had canonic entrances of the "wedges" - as I called them - over a timbre-frame sound sculpture that functioned as a pedal point. Which reminds me: I still have a DAT tape of that I ought to record into my computer and convert to MP3. I guess the sins of my youth ought to be archived, anyway.

This is going to be a similar "...journey, quest... thing" - Peregrine Took: Much frustration will have to be endured coming to an understanding of the concepts before the method for applying them becomes clear.

In this installment, I have begun to omit some of the simple examples where the description is clear enough, figuring that if I'm getting it any reasonably intelligent individual ought to as well (It seems my intelligence is currently being brought into question by a broad range of members of the new music community anyway. LOL!). Besides, I'm bogged down enough with text and commentary.

CHAPTER III

THE GROUPING OF INDICIES

CHAPTER III

THE GROUPING OF INDICIES

*[Example omitted. - Ed.]*

List of indicies

§ 46. The indicies of which the conditions are presently to be examined correspond to intervals of the successive series taken within the limits of three octaves. In the following list they accordingly fall into three groups, with seven indicies in each group. Beginning with

Inverse Shifts: -14 to -4

-14> -13> -12> -11> -10> -9> -8>, -7> -6>\< -5>\< -4>\<| -3< -2< -1<, 0 1 2 3 4 5 6.

Direct Shifts: -6 to 6

§ 47. The positive indicies end with

List of indicies

§ 46. The indicies of which the conditions are presently to be examined correspond to intervals of the successive series taken within the limits of three octaves. In the following list they accordingly fall into three groups, with seven indicies in each group. Beginning with

**Jv= 0**the positive indicies proceed to the right up to nearly an octave; the negative indicies down to the left to two octaves. Shifts of negative indicies are indicated by their proper signs (§ 41).**JJv=**Inverse Shifts: -14 to -4

-14> -13> -12> -11> -10> -9> -8>, -7> -6>\< -5>\< -4>\<| -3< -2< -1<, 0 1 2 3 4 5 6.

Direct Shifts: -6 to 6

§ 47. The positive indicies end with

**Jv= 6**for the reason that to continue to**Jv= 7**would merely shift a voice of the original combination an octave higher:*(by an octave)*

It is obvious that anything conforming to the conditions of simple counterpoint (i.e. a combination at

It is obvious that anything conforming to the conditions of simple counterpoint (i.e. a combination at

**Jv= 0**) would also be correct for**Jv= 7**, so this index does not require special rules. Similarly, to separate a derivative combination at**Jv= 1***gives a derivative at*Example omitted. - Ed.]

**Jv= 8**:

A combination at

§ 48. Proceeding to the negative indicies: those of the values -1<, -2< and -3< will always refer to the direct shift, because those of the same values for the inverse shift would result in limiting the movements of the voices to too narrow a range (§ 38). For the same reason it is advisable to regard -4, -5, and -6 as indicies applying to the direct shift, though these values also admit of the inverse shift. In the former case they are indicated -4<, -5<, -6<; in the latter -4>, -5>, -6> (§ 41).

A combination at

**Jv= 2**will serve equally well for**Jv= 9**, one at**Jv= 3**for**Jv= 10**; the same relation holds between**Jv= 4**and**Jv= 11**&c. It is therefore unnecessary to formulate rules for positive indicies equal to compound intervals; those equal to the corresponding simple intervals can be used instead.§ 48. Proceeding to the negative indicies: those of the values -1<, -2< and -3< will always refer to the direct shift, because those of the same values for the inverse shift would result in limiting the movements of the voices to too narrow a range (§ 38). For the same reason it is advisable to regard -4, -5, and -6 as indicies applying to the direct shift, though these values also admit of the inverse shift. In the former case they are indicated -4<, -5<, -6<; in the latter -4>, -5>, -6> (§ 41).

Here Taneiev is being tiresome in his exactitude - in my humble opinion - though I understand why he's doing it in the context of a scholarly treatise. From a practical viewpoint, negative shifts up to the octave can be considered as being direct; those an octave and beyond can be considered inverse. It is the octave that I consider to be the "pivot point" for day-to-day writing of combinations admitting of shifts.

§ 49. The next three indicies -4>, -5>, -6> begin the series of shifts in double counterpoint; the fifth, sixth and seventh. The indicies beyond them to the left, beginning with

§ 50. One index remains:

§ 49. The next three indicies -4>, -5>, -6> begin the series of shifts in double counterpoint; the fifth, sixth and seventh. The indicies beyond them to the left, beginning with

**Jv= -7**and continuing to**Jv= -13**inclusive, are regarded as inverse shifts (double counterpoint at the octave, ninth, tenth &c.). Only exceptionally will they be treated as direct, in which case they will take the sign <.§ 50. One index remains:

**Jv=-14**, but neither this nor the ones beyond (**Jv= -15, Jv= -16**&c.) require special study. Any combination written at**Jv= -7**can be shifted at**Jv= -14**, since here the derivative is only separated by an octave as compared to the derivative at**Jv= -7**; similarly with a shift at**Jv= -21**, where the voices are separated two octaves.One nice thing about Taneiev's numbering system is that, once you get used to it, it is far more logical: All octaves are multiples of seven instead of proceeding unison, octave, fifteenth, twenty-second &c. And, dividing by seven leaves the remainder for the simple interval.

*[This is more clear in the graphic that the book has than in my approximation, in which I used commas. If you scroll up, you'll see the dividing points. - Ed.]*

The same applies to a shift at

The only difference between these cases concerns their limiting intervals, which always are equal to the absolute value of the index (§ 38). Thus, at

Columns of Indicies

§ 51. The indicies listed in § 46 presented three series of figures, seven in each.

The same applies to a shift at

**Jv= -8**, also possible at**Jv= -15**and**Jv= -22**&c. Therefore by double counterpoint at the octave will be understood not only**Jv= -7**, but also**Jv= -14**and**Jv= -21**; by double counterpoint at the tenth,**JJv= -9, -16, -25**&c. (cf. table, § 6).The only difference between these cases concerns their limiting intervals, which always are equal to the absolute value of the index (§ 38). Thus, at

**Jv= -14**the voices may be separated by two octaves, but at**Jv= -7**by not more than one.Columns of Indicies

§ 51. The indicies listed in § 46 presented three series of figures, seven in each.

*Putting these series in numerical order, one underneath another, gives the seven columns below. Four of these, shaded, contain indicies corresponding to intervals of the second group (2int.)*(= intervals that appear in four forms: major, minor, augmented and diminished)

*. The other columns, unshaded, contain indicies corresponding to the first group (1int.)*(= intervals that appear in three forms: perfect, augmented, and diminished).

I'm going to have to substitute labels for the shading he uses: 0, 3, and 4 are

*1int.*groups and 1, 2, 5, and 6 are

*2int.*groups. Also, since this isn't an ASCII format, I'm going to put them in a column versus a row.

_0_1int.

-14>

-7>

0

_1_2int.

-13>

-6>\<

1

_2_2int.

-12>

-5>\<

2

_3_1int.

-11>

-4>\<

3

_4_1int.

-10>

-3<

4

_5_2int.

-9>

-2<

5

_6_2int.

-8>

-1<

6

_0_1int.

-14>

-7>

0

_1_2int.

-13>

-6>\<

1

_2_2int.

-12>

-5>\<

2

_3_1int.

-11>

-4>\<

3

_4_1int.

-10>

-3<

4

_5_2int.

-9>

-2<

5

_6_2int.

-8>

-1<

6

I had forgotten just how many printing errors there are in this book. The direct shifts of -2< and -1< had the arrows in the inverse shift direction, which I have corrected.

*[At the top. - Ed.]*

These columns, except the first, are numbered; each referrence number corresponds to the figure of its lowest

These columns, except the first, are numbered; each referrence number corresponds to the figure of its lowest

**Jv**. The first column to the left*is not counted, it will be referred to as the zero column. It contains*

§ 52. In these seven columns are placed the indicies corresponding to the intervals of the successive series, taken in a range of three octaves; the positive indicies within the limits of one octave, the negative of two. If more indicies are needed to indicate an octave extension on either side, each column will contain four indicies instead of three; for an extension of four octaves will have five indicies, &c. Therefore the value of any index determines its place in one of the seven columns. But, as already stated, for practical purposes three indicies in each column are enough.

§ 53. It is not difficult to remember the indicies contained in each column. The lowest indicies are all positive, each of them corresponds to the number of the given column. The middle and upper indicies are all negative and may easily be learned if attention is paid to the following relations:

(1) The sum of the absolute values of the lowest and middle indicies in each column is seven.

(2) The sum of the absolute values of the upper and middle indicies is 14.

(3) The difference between the absolute values of the upper and middle indicies is 7.

**Jv= 0**, i.e. the index of simple and not one of vertical-shifting counterpoint (§ 44). The upper index in this column,**Jv= -14**, represents the same double counterpoint at the octave as does the middle index,**Jv= -7**, hence it is not necessary to discuss it further as a distinct index. The only essential index in this column is**Jv= -7**. Therefore the zero column will often be referred to in an incomplete form, restricted to the middle index.§ 52. In these seven columns are placed the indicies corresponding to the intervals of the successive series, taken in a range of three octaves; the positive indicies within the limits of one octave, the negative of two. If more indicies are needed to indicate an octave extension on either side, each column will contain four indicies instead of three; for an extension of four octaves will have five indicies, &c. Therefore the value of any index determines its place in one of the seven columns. But, as already stated, for practical purposes three indicies in each column are enough.

§ 53. It is not difficult to remember the indicies contained in each column. The lowest indicies are all positive, each of them corresponds to the number of the given column. The middle and upper indicies are all negative and may easily be learned if attention is paid to the following relations:

(1) The sum of the absolute values of the lowest and middle indicies in each column is seven.

(2) The sum of the absolute values of the upper and middle indicies is 14.

(3) The difference between the absolute values of the upper and middle indicies is 7.

Obviously, the indicies within a column are octaves apart, which Taneiev goes into next. Further, they can be grouped by the commonality of the absolute values; i.e. the column with 1 at the bottom goes with the column with -1< in the middle &c., which he will describe later. The reason for these columns and the later grouping of related columns is to amalgamate the indicies that have the same rule restrictions, which is a couple of chapters down the road. By that time, there should be some "critical mass" action going on, which is a very rewarding point to arrive at. Though progress through this seems tortuously slow, one of the benefits is that with the constant repetition of the terminology in various contexts, it has more of a chance to sink in and stick.

One of the things I am going to do when I get to the end of this section on two-voice vertical-shifting counterpoint is going to be to round up all of the links to the individual posts and put them in order in a sidebar section. If you are joining this program in progress, you'll have to look up the previous posts, though they have not fallen off of the first page as of this posting.

Until next time...

Yeah, I kinda broke that yesterday. Don't think your elastic will work as well as the medical tubing and ball bearings did.

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