Convertible Counterpoint VI
Taneiev uses music notation to illustrate many of his points, but unless the illustrations are key to the conversion technique, I'm not going to transcribe them. I have managed to get these key illustrations onto a single page, which I'll post at the beginning of his exposition on the Successive Series, and I'll prompt the reader to go back for the relevant examples as the need arises.
I love his introductory quotation:
Nissuna humana investigatione si po dimandare vera scientia, s' essa non passa per le mattematiche dimostrationi.
Leonardo da Vinci
Libro di pittura, Parte prima, §1
Which I translate to mean, "There is not one human investigation that does not demand scientific verification, and there is none that surpasses mathematical demonstration." Feel free to correct me if I got it wrong.
PART ONE: VERTICAL SHIFTING COUNTERPOINT
DIVISION A: TWOVOICE VERTICALSHIFTING COUNTERPOINT
CHAPTER I: INTERVALS
The Notation of Intervals
§ 1. The subject of the study of verticalshifting counterpoint consists of an investigation of those combinations from which derivatives are obtained by means of shifting the voices upward or downward. Such alterations in the relative positions of the voices are effected by changing the intervals that are formed by these voices in combination. For the analysis of these changes the best method is that of mathematics, by which the quantitative differences in the sizes of intervals are expressed in figures; mathematical operations are derived therefrom. For this purpose it will be necessary to employ a more accurate method of indicating intervals than that in general use. This new method, used in the present work, consists in taking the interval between two adjacent scale degrees, i.e. a second, as the unit. The interval is then indicated by a figure of these units it contains. The unison is indicated by 0, since in it this quantity is equal to zero. Therefore each interval is represented by a figure that is one less than its usual numerical designation: a third by 2, a fourth by 3, &c.
Here, Taneiev lays the groundwork for the method he will present. The numbers you will encounter will take some getting used to, as they are one less than the standard system, but only in this way can his calculations be performed. The numbers are the representation of the amount of units the interval contains, this irreducible unit being the second, or 1. Through the octave, the numerical designations match up to the standard system as follows:
Unison= 0
Second= 1
Third= 2
Fourth= 3
Fifth= 4
Sixth= 5
Seventh = 6
Octave= 7
Addition and Subtraction: Negative Intervals
§ 2. By indicating intervals according to this method processes of addition and subtraction became possible. An interval may be added to another, either up or down. In the former case the upper voice is shifted upward, in the latter the lower voice downward. In both cases one voice moves away from the other. For example a fifth added to a fourth  3 + 4= 7  gives an octave, an interval equal to the sum of the terms: 3 + 4= 7.
Taneiev is explaining positive shifts, where the voices move away from each other. It does not matter if the upper voice moves up, or the lower voice moves down, the shift is positive when the voices move away from each other. In this example, the voices begin a fourth apart and one or the other shifts away by a fifth, giving the resultant interval of 7 in the derivative(An octave).
Addition is also possible both up and down at the same time; here the result is the sum of three terms: 4 + 3 + 2= 9.
Here the voices still move apart, but both of them shift: The initial interval is 4 (A fifth), the upper voice shifts +3 away from the lower (A fourth) while at the same time the lower voice shifts away by +2 (A third), giving the resultant total of 9 for the derivative (A tenth). It is not the up or down movement of the voices that determines positive or negative shifts, but rather the fact that the interval between the voices increases: A lower voice decending is still a positive shift, not a negative one, because the voices end up farther apart in the derivative.
Other combinations of the same terms yeild the same result; the order are taken does not affect the total:
4 + 3 + 2= 9; 4 + 2 + 3= 9; 5 + 2 + 2= 9; 3 + 2 + 4= 9; &c.
This is a treatise. Taneiev is just being thourough. The point is, if the derivative is predestined to work at a given positive shift, the individual voices can move in any combination of intervals that adds up to the resultant that has been planned. Far from being a minor technical detail, the appropriate shifts will allow the modal transformations to be maximally effective, and for disallowed intervals such as augmented fourths and diminished fifths to be avoided.
§ 3. The reverse process, subtraction, causes voices to approach, i.e. the higher voice is shifted downward or the lower voice upward, or both. For example, subtracting a third from an octave leaves a sixth: 7  2= 5.
If the subtracted equals the value of the first interval the result is 0, i.e. a unison: 4  4= 0.
If the subtracted interval is greater than the first interval, the result is a negative quantity: 4  5= 1
§ 4. A negative quantity therefore refers to an interval of which the lowest tone belongs to the upper voice and the highest tone to the lower voice. These intervals are termed negative. The same mathematical processes may be applied to them as to positive intervals.
It is worth remembering here that strict counterpoint is a vocal idiom, and so unisoni and voice crossings are not uncommon. Not only that, but when dealing with invertible versions of complex counterpoint, all of the resulting intervals will be nominated in negative terms (But they can, of course, be exchanged to positives at that point as well, since a negative 1 is actually the same as a positive 1 &c).
§ 5. It is possible to regard the addition and subtraction of intervals in the algebraic sense; i.e. to consider both processes as addition, in which the amounts concerned may be either positive or negative quantities. Results so obtained are algebraic. The sum of two or more positive numbers is only a special case.
§ 6. The order in which the terms are taken does not affect the total. Therefore when the two voices shift simultaneously it will be found more convenient to add their algebraic values all at once, not to add each item in turn to the given interval. Suppose that the given interval is a fourth and that one of the voices shifts 9 and the other +1. The sum of the quantities is 8. Adding 8 to the value of the interval 3 gives 3 + (8)= 6, i.e. a negative sixth.
OK. There is an obvious error here in the translation, but it is a simple one. The actual result for a sixth is 5. The situation described is not as weird as it sounds (To me, anyway: I always wonder if the original Russian is clearer than this translation). Basically, the situation describes a combination that begins a fourth apart (3). One of the voices moves 9 (A tenth), and the other moves +1 (A second). Either the upper voice or the lower voice can make either move, so there are two possibilities. Keeping in mind that a positive movement of the lower voice makes it decend from its original position (The positive movement, remember, indicates the original interval increases in the derivative), and a negative movement of the lower voice makes it ascend from its original position (And vice versa for the upper voice), it works out like this for the two instances encompassed by the description.
1) Starting from a fourth (3), the upper voice moves 9 (It decends a tenth), and the lower voice moves +1 (It decends a second): The resulting interval is therefore a sixth (5). Since the voices cross, the result is negative (5).
2) Starting from a fourth (3), the lower voice moves 9 (It ascends a tenth), and the upper voice moves +1 (It ascends a second): The resulting interval is therefore a sixth (5). Since the voices cross, the result is negative (5).
In both cases the shift is described by the equation 3 + 1  9= 5, or as Taneiev suggests, 3 + (8)= 5.
If other algebraic shifts are substituted of which the algebraic sums are the same the result remains unchanged:
3  3  5= 5, 3 + 2  10= 5 &c.
Again, the derivative can appear as the result of any combination of shifts that gives the required final interval. And again, this property can be used to obtain desirable modal transmutations and to avoid forbidden leaps and intervals of augmented fourths and diminished fifths.
Compound Intervals
§ 7. If an interval contained within the octave limits is increased by one or more octaves an interval is obtained that is termed compound, in relation to the first. To separate the voices forming an interval by an octave, add seven to it's absolute value: 2 + 7= 9, 2 + (7)= 9, &c.
To separate the voices two octaves, add 14 to the absolute value of the interval; for three octaves add 21, &c., in multiples of 7.
§ 8. The following table is a list of simple and compound intervals within the limits of four octaves:
Unison 0, 7, 14, 21
Second 1, 8. 15, 22
Third 2, 9, 16, 23
Fourth 3, 10, 17, 24
Fifth 4, 11, 18, 25
Sixth 5, 12, 19, 26
Seventh 6, 13, 20, 27
§ 9. To find what interval within the octave limits corresponds to a given compound interval, divide the latter by 7. The remainder will be the desired interval and the quotient will indicate by how many octaves the voices are separated. Suppose the given interval is 30. Dividing this by 7 gives 4 as a quotient, with 2 as a remainder. The desired interval is therefore a third, and the voices in the given interval are separated by four octaves in addition to the third.
§ 10. The propositions following are based on what has been established. Considering each voice separately, the vertical shift in one direction is a positive operation, in the reverse direction a negative operation. The voice for which the upward shift is regarded as a positive operation will be termed upper, first, and indicated by the roman numeral I; that for which the positive operation is the downward shift will be termed lower, second, and indicated by the roman numeral II.
The positive and negative shifts may be represented by the following diagram:
+
I

___

II
+
I have changed this diagram  In fact, I rewrote it in the margin my first time through the book  so that the voice representations are above and below. Taneiev (Or the translator) had them side by side, which I think is less clear, even though their positions can in fact be exchanged, which is coming up next.
When the voices are aranged in the order I/II (I over II) the intervals formed by their union are positive; in the order II/I (II over I) they are negative.
If two voices forming an interval a shift by intervals of which the algebraic sum is ±s, then from a is obtained a + (±s) (§ 6). The same result is obtained if one voice shifts at ±s and the other remains stationary, s being the algebraic designation by which the voice is shifted up or down.
Just reread § 6 and my explanation again at this point. Taneiev is here beginning to introduce the algebraic terminology and associated symbols which will be used in the upcoming formulas: ±s is just to be read as "any given positive or negative shift."
Successive Series of Intervals; Division into Two Groups: 1int. and 2int.
This is the page with all of the pertinent examples of how the successive series is used as the mechanical tecnique to calculate conversions.
§ 11. Intervals may be put in a successive series, such as that from the unison (0) positive intervals are ar on one side, negative on the other. In the following series the consonances are in boldface figures, with p. or imp. added, for perfect or imperfect. [See the top two staves in the example page above.  Ed.]
Positive and negative intervals are divided into two groups; (1) intervals that appear in three forms: perfect, augmented, and diminished; and (2) intervals that appear in four forms: major, minor, augmented, and diminished. The first group consists of 0, 3, 4, and 7 and the corresponding intervals beyond the octave. The second group consists of 1, 2, 5, 6 and their compounds. Intervals of the first group are indicated 1int. and those of the second group 2int.
The first froup includes the perfect consonances, the second group the imperfect.
Obs. These groups of intervals have other characteristics. For example, each 1int., counted upward from the first degree of the major scale, is identical in size to the same interval counted downward; both intervals are perfect.
On the contrary, the quality of each 2int. is changed, under the same conditions; those counted upward, those downward, minor.
Also notice that the first four notes of the harmonic series include all the perfect intervals of the first group; the fifth note forms one of the intervals of the second group from each of the preceding notes. [This was where I began to understand that the laws governing contrapuntal motion were inextricably linked to the harmonic series, as well as are the laws governing harmonic root progression patterns.  Ed.]
§ 12. The distance between two given intervals in the successive series is determined by the interval at which one voice is shifted, the other remaining stationary, a process required in order that from a given interval another may be derived. [Several seeminly redundant examples.  Ed.]
§ 13. If a positive interval (termed here a) is added to a given interval, the interval obtained will lie in the successive series to the right of the given interval at the distance indicated by a. If a negative interval, a, is added, the interval obtained will be [Typographical error: "will be" is repeated: Omitted.  Ed.] found to the left of the given interval at the distance a. For example, adding a positive sixth to a third gives an octave (2 + 5= 7), lying a sixth to the right of the third. Conversely, adding a negative sixth to a third gives a negative fourth (2  5= 3) lying a sixth to the left of the third.
Here Taneiev is referring to the chart presented earlier where the positive and negative intervals are listed to the right and left of a unison. This is the mechanical aspect of conversion: An interval lies either to the right or left of the beginning interval on the chart depending on whether the shift is positive or negative, and by the distance corresponding to the absolute value of the shift. By having this resultant it can be determined whether the shift is of the 1int. or a 2int. category, which is required to determine the relevant rule restrictions. And please note that since this is the strict style, the perfect fourth is always classified as a dissonance. Taneiev mentions some exceptions to this later, but they are so rare as to constitute no threat to his conservative classification.
Order of Intervals in the Successive Series
§ 14. It is desirable to dwell at some length on certain peculiarities in the order of intervals in the order of intervals of the successive series which will be referred to later on.
(1) The perfect and imperfect consonances alternate, in both directions.
(2) Two consonances are adjacent, 4 and 5 (fifth and sixth), but not two dissonances. (Here the possibility is not considered whereby a consonance can be changed to a dissonance by chromatic alteration.) Thus the fifth and sixth have a dissonance on one side only; all the other consonances have dissonances on both sides.
(3) On both sides of each dissonance is found a consonance, one of which is perfect, the other imperfect.
(4) Of two consonances found at equal distances, right or left, from a dissonance, one will always be perfect, the other imperfect.
§ 15. In the following the consonances only are taken from the successive series: [See the third stave on the example page above.  Ed.]
The calculation of the distance between positive consonances only or negative (consonances) only proceeds as follows:
(1) Two consonances of the same group (i.e. both perfect or both imperfect) are separated from each other by a 1int. (An interval that can appear in three forms: perfect, augmented, or diminished)
(2) Consonances of different groups (i.e. one perfect, the other imperfect) are separated from each other by a 1int. For example, consonance 2 (imp.) is separated from consonance 5 (imp.) by 3 (= 1int.); consonance 7 (p.) is separated from 11 (p.) by 4 (= 1int.). Conversely, consonance 5 (imp.) is separated from 7 (p.) by 2 (= 2int.) (An interval that can appear in four forms: major, minor, augmented, or diminished); consonance 4 (p.) from 9 (imp.) by 5 (= 2int.), &c.
§ 16. Exceptions to the foregoing statements are not found as long as positive consonances only or negative consonances only are compared. But in comparing positive consonances with negative (consonances) the following sole exception is encountered:
Two fifths, negative and positive (thus corresponding to a compound interval) (An interval beyond the octave) are separated from each other by a ninth (or at an interval than a ninth by an octave, or two octaves, &c.). Since the ninth is a 2int, the case represents an exception to what was stated in § 15, (1).
With these exceptions (Actually, just a single exception and it's octave compounds) the statements in the preceding section relative to ther distances between consonances of the same group and those of different groups apply also to all cases where one interval is positive and the other negative.
§ 17. Proceeding to dissonances:
See the fourth stave of the above example page.
First the distance is to be measured between positive dissonances only or negative (dissonances) only.
(1) The second and its compounds (1, 8, 15 &c.) are separated from one another by a 1int.
(2) The other dissonances the fourth, seventh and their compounds (3, 6, 10, 13, &c.)  are also separated by a 1int.
(3) The second and its compounds are separated from the other dissonances by a 2int.
§ 18. Next to be considered are the mixed cases where one dissonance is positive and the other (dissonance) is negative. Here the statements in § 17 regarding dissonances are presented in reverse order:
(1) The second and its compounds are separated from one another by a 2int. (With one positive, the other negative)
(2) The other dissonances (3, 6, 10 &c.) also are separated from one another by a 2int. (With one positive, the other negative)
(3) The second and its compounds are separated from every other dissonance under the same conditions (that one interval is positive, the other negative), by a 1int.
§ 19. If under one successive series of intervals is placed another so that a new interval a comes directly below 0 (unison) in the upper series, then each interval in the lower series will be equal to the algebraic sum of the interval above it + a. Let m equal any interval in the upper series and n the interval in the lower series directly underneath m; then m + a= n. In the following [See the bottom two staves on the example page above.  Ed.] under 0 in the upper voice is placed a= 4. Taking for example 7 in the upper series, m= 7, and adding 4 gives below it n= 3 (7  4= 3). [Which I think is algebraically clearer if you state it as 7 + (4)= 3.  Ed.]
Such comparison of two series of intervals is necessary in working out exercises in verticalshifting counterpoint.
So, after all of this, we are given a simple mechanical comparison method! Sheesh. I'm going to bed.
I wish.
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