Ricercare in C Major II

I find it fascinating how different pieces come together in different ways and with different levels of difficulty associated with them. It was probably four years from the time I came up with the Sonata Zero fugue theme and the time I had the exposition worked out. To be fair to myself, I wasn't really working on it all that time, as I was following a non-musical career detour during that period, but part of the problem was that I was going for a four-voice exposition... on the guitar. When - after a long layoff from even thinking about it at all - I returned to it and decided to make it only three voices, the piece came together within a couple of weeks. In trying to recall particular problems I had with it, I'm drawing a blank: It pretty much wrote itself within that relatively short time span.

The Extempore was even easier. That piece took only a few days. Working on it was, in fact, a very pleasant experience. Well, this Ricercare is different: It seams like every single measure is frought with problems that stem from the ridiculous number of possibilities I have combined with my attempting to have the finished piece turn out to be organically related to both the first movement Extempore and the final movement Fugue. You would think with all of the material I have developed through the other two pieces, that this ought to be a snap, but it most decidedly isn't. Once I get a phrase off of the ground, I get a few measures done relatively quickly, but it's the decision-making processes for the different areas of the piece that are giving me headaches. I think I want to do x, but that doesn't work, so I try y: When that doesn't pan out either, I'm left scratching my pounding head. Then, seemingly out of nowhere, the idea for z comes to me, and I have it (Today's z occurred to me as I was showering before my gig this evening... er... Probably more information than you actually needed).




As you can see, the first page hasn't changed at all. Actually arriving at this strange "exposition" took several days though, during which time I tried many different constructs that... sounded terrible. Part of the problem is that I'm having to think "upside down" both thematically and modally, and this inverseness leads to the "natural" solution being very different than what I arrived at for the fugue. The reason is, of course, that the tonal/modal system does not simply invert as a symmetrical mirror image: Parts of it do (The perfect consonances), but parts of it do not (The imperfect consonances), and what makes a "harmonic progression" in the rectus has a very strange modal effect with the quasi-inversus form of the subject I'm using here. This modal effect is also beyond my ability to predict in terms of usable effectiveness, so I have to do a lot of experimentation before the correct solutions become apparent to me. Inverted forms of the episodes turned out to be so bizarre that I abandoned them entirely: Not only did they not end up where I wanted them, but they had no real cadential effect at all.




Here is the second area of the piece on the dominant level of G major. I also tried a boatload of various ideas here that didn't work out. What I really wanted was the thematic statement that begins above in measure twenty to begin in measure sixteen. But it just wasn't cutting it with the cadence to the widely-spaced G major chord there. It was a couple of days before I got "The Shower Revelation": This turns out to be the perfect point to "look back" to the Extempore because G major is only a whole step below A minor, and the expository passage from the Extempore was never stated in the major mode. Voila.

Since this is a Ricercare and not a fugue, there is a lot more freedom in terms of handling both key areas and thematic material. Personally, I prefer to let a form (or process) influence the range of choices I allow myself: A fugue is only in closely-related keys (Tonic, dominant, and relative usually, with the possibility of the relative of the dominant and the Subdominant key-pair as well: This reduces to the tonic key-pair and key-pairs one sharp and one flat in either direction of the tonic. Some of Bach's "Contrapuncti" never modulate at all). A ricercare can venture further afield (Beethoven's Op. 133 Grosse Fugue is actually a ricercare, as it's all over the tonal map: He even put "sometimes researched, sometimes free" in the title, which would have been unnecessary had he given it the correct categorization. But hey: He was Beethoven; the man did whatever he wanted).

A ricercare can also treat thematic material more freely than a proper fugue. Limiting yourself to one subject - or a series of subjects as in a multiple fugue - and treating it exclusively and exhaustively is not required. In fact, one of the distinguishing features of a ricercare is that it has a lot of various subjects that the composer picks up and sets down at his discression.

So, I re-used the exposition from the Extempore at the modulation to G, and it is heard here in the major mode for the first time in the Sonata. As you can see, I changed the fourth measure to reflect the tail figure of the subject/answer pairs of the piece, which also allowed me to avoid the riot of sixteenths that permiates the Extempore. Not only that, but the two upper voices now cadence into a unison - which since it is an open string is perfectly playable - that launches the phrase I initially wanted back in measure sixteen very effectively.

The rectus and quasi-inversus pairs in the phrases starting at measure twenty use the 4-3 suspension chain that is found in the fugue, but the third voice completely redefines them. Unfortunately, the unison in measure twenty-five is not playable (Not easily enough for me to bother with it, anyway) - so the top voice is forced to rest - but that the third voices are slightly varied mirrors of each other is still pretty obvious, and the overall effect is quite nice.

At the second episode I use a very wide open voicing, which is made possible by the open strings involved, and this is entered via a deceptive resolution of the previous dominant sonority. In the concluding fugue the deceptive resolution comes at the end of the corresponding episode. The next section will be in B minor, which is two sharps away from the tonic, but since this is a ricercare, no problem. B minor is also the key of the upcoming Scherzo so it will tie in quite well to the overall plan.

I have no idea what I'm going to do now. Other than turn the heat up, that is: The unseasonably warm weather we've had for the past week or so has suddenly vanished, and it's below freezing now.




Yes, fireplaces do have certain advantages over forced-air heat.


I am ramping back up into busy-gigging mode, so I will have the next installment of Convertible Counterpoint ready Saturday or Sunday. Chapter III is quite short, but not Chapter IV, so knocking out two of them may be too big a bite to chew. We'll see.

Though Taneiev says that this book is not the place for the rules of simple counterpoint, he nevertheless spends all sixteen pages of Chapter IV reviewing them, and there are some groovy musical examples there, so it will be nice to get Chapters II and III out of the way. Unfortunately, II and III contain more unfamiliar new concepts, so it may be slow going.

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Convertible Counterpoint VI

As I have been reading ahead I am reminded again just how deep a treatise this is. There really isn't any way I can summarize what Taneiev presents because... uh... I just don't understand it well enough. So, what I am going to do is basically transcribe the treatise and add commentary: If I can explain it to myself, I should be able to explain it to the reader, and perhaps retain enough to make use of it as I want so much to do. In light of this, I have decided to further delimit the scope of this book-blogging session to only two-voice vertical-shifting complex counterpoint. This is my primary area of interest anyway. That will take me to page 154 of the 301 page treatise (Excluding indicies): Almost exactly half way, which is a significant enough chunk of something so dense.

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: TWO-VOICE VERTICAL-SHIFTING COUNTERPOINT

CHAPTER I: INTERVALS

The Notation of Intervals


§ 1. The subject of the study of vertical-shifting 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 pre-destined 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 re-wrote 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 re-read § 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 bold-face 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 vertical-shifting 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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Ricercare in C Major

As I was falling asleep a few nights ago, I was playing permutations of the Sonata Zero fugue subject in my noggin, and I came up with a version that is sort of an inversion, and sort of not. The head and tail figures are from the rectus version, but the middle line is ascending versus the previous decending one. Because of this, the subject must start on fa instead of sol to cadence to the tonic, and the answer starting on do creates a real cadence to sol instead of a half-cadence to the dominant chord, as the original did.

At first, I was trying out various inverted versions of the original exposition, but they didn't really work out: This subject demanded it's own special treatment. When I finally figured out what it "wanted" to do, it ended up working in C major starting out on the lowest F on the guitar. I was still able to incorporate inverted elements of the final fugue's exposition - so it is obviously organically related to that fugue - but the exposition is quite unique, as you can see. Since it is so unusual, I decided that it would be more appropriate to call it a Ricercare, which implies a degree of freedom beyond the stricter fugue.

The result is that I ended up shuffling the movements of Sonata Zero around yet again: It will now be laid out as Extempore, Ricercare, Scherzo, and Fugue, and the keys will be A minor, C major, B minor, and A minor. In terms of length, the movements will be 2:05 for the Extempore, circa 2:00 for the Ricercare, 3:33 for the Scherzo, and 2:05 for the Fugue: The entire sonata will be performable withinin a ten minute window. It will be a small fugue cycle with the Scherzo added for variety. Typical Hucbaldian approach. ;^)




This is all I have at the moment, but you can see how the exposition and first episode are related to the corresponding parts of the fugue, and yet they are quite different at the same time.


I have the first chapter of the next Convertible Counterpoint post almost ready - which covers the entire first chapter of the book - and I will probably post that sometime after midnight tonight when I get home from my gig.

I'm really thinking I'll need a vacation soon to avoid burnout.



I could make that work.

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Convertible Counterpoint V

A lot of my musical philosophy comes directly from Taneiev. It's been twenty years since my first time through this book, so I had forgotten just how much I owe him in that area. When he speaks of the super-chromatic harmony of the ultra-romantics and how it contributed to the decline of large-scale works, I really don't think there can be any arguing the point. This is even more true for the serialists. He was borderline psychic in his forward-looking thought on this matter.

The prescriptive he offers (A fantastic example of constructive criticism!) - that of using imitation and convertible counterpoint to re-achieve unifying principles that would again allow for coherent large-scale forms - seems to be the perfect answer for those of us who wish to substitute the chromatic idiom for the diatonic one, and to allow again for "any harmony to be followed by any other harmony", but on a chromatic basis versus the sixteenth century diatonic one. In this we can see that though this book covers complex counterpoint in the strict style, once the techniques are internalized, there is no barrier to substituting chromatic lines for diatonic ones: The underlying natural laws that govern contrapuntal motion are immutable.

Years ago I did some counterpoint exercises - experiments actually - in which I made my first stabs at this kind of chromatic contrapuntal texture: I called them "colorpoint" exercises, because I used five voices and contrapuntal rules to connect all of the "beautiful dissonant" sonorities that I love; added ninths, major sevenths with added augmented elevenths, &c. Though I didn't have the technique to really expand on these early efforts - which amounted to little more than individual phrases in isolation - the results were nevertheless quite striking and effective. My goal is to work back to that idea but to add to it the techniques in this book.

Starting in the second half of the introduction, Taneiev begins to use musical examples. In order to figure out how to effectively present these, I had to take a time-out to decide upon some formatting issues that have to do with limitations inherant in my notation software as well as the image sizing limitations of my Smugmug account. What I came up with is the idea that all examples will be presented on full pages at 75% magnification: This will allow for the examples to be all the same size, and they will comfortably fill up the column width of my template. However, there will be a lot of very "white" pages. The advantage is that I will have to make exactly zero further formatting decisions, which will make things easier and speed up the process considerably.

I am also going to put all of the examples into a single file (I may make different files for different chapters though, just so you can find the examples a bit easier), which will allow me to post single PDF and MIDI files on my Fileshare page for those who would like to listen to the examples and follow along using the score.

In the first examples presented here in the introduction, Taneiev uses a single two-voice combination that produces eleven derivatives, for a total of twelve versions of the same two melodies. The original combination has the voices starting out a perfect fifth apart (+4), and the vertical shifts allow for them to start out an octave apart (+3 from the original), a third apart (-2 from the original), and with the voices inverted at a sixth apart (-13 from the original, which is the octave inversion of the -2 version).

After that, he presents horizontal shifts at two beats of delay and four beats of delay from the original, double-shifts of both vertical and horizontal conversions, and finally, he has the melodies doubled in imperfect consonances. Note that in the original and all of the derivatives there is no parallel motion at all: Oblique and contrary motion is all that is allowed. My first time through this book, I limited myself to internalizing the shifts that could be reduced to simple rule restrictions such as this, and skipped over the algebraic formulas. I got a lot out of it nevertheless. This time, I want to fill in the gaps and really come to an understanding of the underlying mathematical technology. For someone who is practically retarded when it comes to anything having to do with numbers, this will be no insignificant challenge, but I'm highly motivated, so we'll see what we shall see.


INTRODUCTION (Continued)

by Serge Taneiev


Complex counterpoint is divided into categories according to the methods by which derivative combinations are obtained. The principle methods are; 10 the shifting of voices; 2) duplication in imperfect consonances, and 3) transmutation; hence the three aspects of complex counterpoint: 1) Shifted, 2) duplicated, 3) metamorphosed.

A) Shifting Counterpoint

A derivative is obtained by shifting the voices. he following classification exhausts all possible shifts:


1) Vertical shifting - upward or downward - hence vertical shifting counterpoint:



In the third example (above), the upper voice is shifted underneath and the lower voice above, a special case of the vertical shift known as "double counterpoint."


2) Horizontal shifting, in which the time-intervals between entries of the voices are changed, hence horizontal-shifting counterpoint:




3) Vetical and horizontal shifting together, hence double-shifting counterpoint:




In this work double-shifting counterpoint is included in the divisions devoted to horizontal-shifting and is explained in parallel with it, as the methods of writing both are similar. [I was afraid of that. I'm still hoping to be able to skip the double shifts if I'm having a rough time with the math. - Ed.]


The subdivisions of shifting counterpoint are therefore:

Shifting Counterpoint (a derivative from the shifting of voices):

1) Vertical-shifting (Upward or downward);
2) Horizontal-shifting (changing relationship between entries);
3) Double-shifting (the combination of the two preceeding).


B) Duplicated Counterpoint

A derivative combination is obtained by duplicating one or more voices in imperfect consonances. Therefore the number of voices in the derivative is increased: at the duplication of one voice to three; at the duplication of two voices to four.




An original three-voice combination yeilds derivatives of four, five, or six voices, according to how many voices in the original are duplicated. Examples will be found in chapter XV.

The connection of this counterpoint with the vertical-shifting counterpoint is obvious: each duplication is nothing but the vertical transference of a voice at an interval equal to an imperfect consonance. Therefore the study of counterpoint admitting of duplications is included in the divisions dealing with vertical-shifting counterpoint.

In the various phases enumerated of complex counterpoint, forming the contents of the present work, the detailed treatment of double counterpoint is of the utmost value in music theory. It is of practical importance both in connection with counterpoint admitting of duplication and with certain cases of multi-voice vertical-shifting counterpoint, for instance in triple and quadruple counterpoint, especially at the octave. In theoretical literature little reference is found to any aspects of vertical-shifting counterpoint other than double, and still less to horizontal-shifting counterpoint, the study of which, as a special department of shifting counterpoint, is here presented for the first time. [Emphasis mine. - Ed.]


C) Metamorphosed Counterpoint

The derivative is obtained by a process of transmutation. By metamorphosis is meant such a change of the original combination as would correspond to it's reflection in a mirror - this is known as "mirror counterpoint." To avoid ambiguous terminology "metamorphosis" will be used only in this sense, and will not refer to the shifting voices in double counterpoint. Since metamorphosed counterpoint does not enter into the plan of the present work it will not be considered further. [I'm betting this is going to be found in the Doctrine of Canon, where mirror canons are certain to be covered. - Ed.]

The statement has been made that the transference of voices is characteristic of shifting counterpoint. The changes that this counterpoint makes in a melody amount only to its transference vertically to other degrees or horizontally to other measures or parts of measures. Every other change, such as metamorphosis. augmentation, diminution, made simultaneously with shifting, places the given combination beyond the scope of convertible counterpoint, and the shifting ceases to be a vital characteristic.

Another feature of complex counterpoint remains to be mentioned: the existence of rules in relation to its various subdivisions and to simple counterpoint. The study of the latter, in either the strict or free style, is that of a system of rules to which every union of voices must conform. In complex counterpoint the same rules apply to both original and derivative combinations. The significance of the rules of complex counterpoint is that if they are ignored in the original combination the derivative will show progressions that violate the rules of simple counterpoint. The mutual relations of the aspects of complex counterpoint and their relations to simple may be illustrated symbolically by the accompanying diagram, where the large circle represents the domain of simple counterpoint, and the small intersection circles, with the portions of their areas coinciding, the various aspects of complex counterpoint.


Here Taneiev has a diagram of four circles: A large circle with three smaller circles contained within it. The interior circles have areas all their own, as well as smaller areas that only two circles share, and a small central area (That looks like the rotor of a Wankel engine, if you are a gearhead like me) which all three circles encompass. If you are a Christian, you've seen this diagram many times as a representation of the Holy Trinity. If I had my old OS 9 Paint application I could duplicate it, but unfortunately OS X has nothing I can find that will do the trick. Yes, I need a scanner.


From this diagram it is clear that the combinations used in complex counterpoint must also belong to the domain of the simple, but not vice versa; portions of what is permitted in simple counterpoint are found outside of the circles that represent the various aspects of complex. The intersections of the circles show that certain phases of complex counterpoint may be combined, as was illustrated in the examples given.

This work is divided into two parts; the first part deals with vertical-shifting counterpoint and counterpoint admitting of duplications; the second with horizontal-shifting and double-shifting counterpoint. Each part consists of two divisions, one devoted to two-voice counterpoint [Which I'll cover. - Ed.], the other to three voice. [Which I'll skip. - Ed.] The investigations are limited to two or three voices. More than these are found only as the result of duplications; they are given at the end of the first and second divisions of Part One, where duplications are found using a larger number of voices, up to six inclusive.

Whew!



"Fabulous, darling!"

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Convertible Counterpoint IV

Happy '06 everybody. I managed to survive the hectic gigging of the last two days of aught-five without incident, but I was asleep by 11:00PM last night, so I didn't "ring in" the new year. No problemo, as I don't drink alcohol anymore anyway, and hanging out with drunken revelers just doesn't hold the same appeal for me that it used to. Thank goodness I was just the opening act at last night's New Year's Eve party, becasuse by 9:30 PM I was toooooast, and was having quite a bit of trouble actually, you know, performing: Lots of little brain-fade errors going on, but a drunk crowd is a happy crowd, so it really wasn't much of an issue. When I saw the main act arrive, I actually just quit mid-song and packed up: I was pooped. I ordered dinner (Free!), ate, came home, and crashed like a high-rise imploding. I seldom ever sleep any more than five hours at a stretch (Afternoon naps rock!), but I slept an entire eight hours last night, and awoke feeling like a million quatloos. I'm betting most in Alpine are nursing hangovers right now, but not moi. Once I quit drinking entirely, it was no more than a month before I felt better than I had since my teenage years, and just the thought of beer now makes me wince. You never say never about these kinds of things, but due to "career goals" I'm planning to wait for New Year's Eve of '07 before I treat myself to some Krug Champagne and Chimay Ale (Thoughts of which decidedly do not make me wince). Anyhoo...


I'm going to reprint Taneiev's entire introduction in two parts, complete with musical examples (Actually transcribing these into Encore and hearing them ought to be a huge plus: I'm going to do that with all the examples in the book), because getting a firm grasp on the elemental concepts is absolutely required. It's 11:15 AM CDT here right now, so it will be interesting to see just how long this takes. I'm going to launch my Studio TVR TV display, stuff it in the corner of my monitor, and spend the rest of the day here at the keyboard. I refuse to even touch a guitar today (Cool! "The Bourne Supremacy" just started!).


INTRODUCTION

by Serge Taneiev


The art of counterpoint has passed through two eras: That of the strict style, which attained its highest development in the sixteenth century (Palestrina and Orlando Lasso), and the period of the free style, of which the crowning achievements are found in the works of Bach and Handel. The differences between the contrapuntal writing of these two eras are to be found both in the nature of the melodies themselves and in the character of the harmonies formed by these melodies in combination.

Strict counterpoint, developed on the basis of the so-called ecclesiastical modes, was a pre-eminently vocal style that had not been exposed to the kind of influence that instrumental music later exerted, antedating such influence, it attained to complete self-fulfilment. Strich counterpoint excludes everything that presents difficulty to voices singing without instrumental accompaniment. Melodies in the strict style show evidences of their origin in the chants of the Catholic Church - they exhibit many characteristics of these early canticles. They are strictly diatonic, are written in the ecclesiastical modes, and in them are no progressions of intervals that are difficult of intonation, such as sevenths, ninths, augmented or diminished intervals, &c. [I am actually looking forward to learning how to write more closely to this style, as all of my work thus far has been in free style harmonic counterpoint, and in instrumental idioms. - Ed.]

The basis of multi-voice counterpoint of the strict style is of course the two-voice texture. Two-voice counterpoint is subject to the rules governing the progression of intervals, these being employed in a way that for the normal hearing is the most simple and natural. A knowledge of the rules of simple counterpoint in the strict style is essential in order to understand the present work, though this is not the place to explain them. In strict writing the rules of two-voice counterpoint apply also to more intricate polyphony. With a few exceptions it is observable that in a multi-voice combination each voice together with every other forms correct two-voice counterpoint.; that multi-voice counterpoint is an association of several two-voice combinations, as a result of which is obtained a series of varied consonant and dissonant harmonies, foreign to contemporary harmony and often sounding strange to us. Although isolated harmonies may be classified under the heads of certain chords, the term "harmony," in the sense in which it is used in the music of today, is not applicable to the old contrapuntal style. Harmony in the strict sense is not subordinated to the requirements of our modern tonal system, in which a series of chords is grouped around a central tonic chord: a system that in the course of a composition allows the tonic to be shifted (modulation), and groups all secondary tonalities around the principle key, besides which the tonality of one division influences those of others, from the beginning of the piece to it's conclusion.

In the harmony of the strict style there is no such dependence of some parts upon others, or of what may be called harmonic action at a distance. Only in the perfect cadence, where as a result of the ascent of the leading tone to the tonic the gravitation of dominant to tonic harmony is temporarily brought about, can be seen the embryo of our present tonal system. Aside from such cadences the strict style does not present a series of harmonies that are unified in this sense; key-continuity may be entirely absent, and any chord may follow any other, on a strictly diatonic basis.

In music of the Polyphonic Period - essentially vocal - coherence was provided first of all by a text. But besides the text - an external factor not belonging to the domain of music - the works of the period possessed another - purely musical - resource, by which took on coherence and unity; a resource all the more valuable inasmuch as harmony did not as yet possess the unifying power that it subsequently acquired. This was imitation: the recurrence of a melody in one voice immediately after its presentation in another. The result of this use of a single melody that appeared in different voices was to distribute the thematic material equally among all of them, giving to the whole a high degree of coherence. An imitating melody often entered before the preceeding melody had closed, and then did not end until still another imitation had begun, a process that served to knit still closer the contrapuntal texture.

For two or more centuries the working-out of imitative forms in the strict style received much attention from composers. There arose many different phases of this device; imitations on a given voice and without it, canonic imitation, imitation in contrary motion, augmentation, diminution - forms that in the course of time culminated in the highest contrapuntal form of all - the fugue. From the introducing of one melody in all voices it was natural to take a further step and apply the same process to two melodies at once; hence double imitation, double canon, double fugue. At this transference to different voices of two melodies simultaneously the question must have come up as to the possibility of changing their relationship at the successive recurrences, and thereby from an original combination to obtain another, the derivative. Thus the origin of complex counterpoint, i.e. the obtaining of derivative combinations, also came in the era of the strict style.

In multi-voice music melodic and harmonic elements are subject to the influences of the time and to the nationality and individuality of composers. But the forms of imitation, canon and complex counterpoint - either as actualities or as possibilities - are universally valid; they are independent of such conditions, capable of entering into the plan of any harmonic system and adaptable to any melodic idiom. The idea is prevalent that the old contrapuntalists of the Flemish Schools exhausted the resources of imitation, especially as regards the canonic forms, but in reality they worked out completely only a few of them; the rest received only incidental treatment or were not touched upon at all. The outstanding merit of the Flemish composers was that they invented these forms and from them developed a flexible and efficient system of technical proceedure.

Arising in the era of the strict style, these forms survived without material change until the end of the seventeenth and the beginning of the eighteenth century, when under the powerful influence of instramental music they were enriched by acquisitions up to that time had constituted mere technical virtuosity. ["mere technical virtuosity": I like that. - Ed.]; also by harmonic, figurative and other elements that in the preceeding era had been absent. This free counterpoint of the time of Bach and Handel, essentially the same as our own, was sharply distinguished from the counterpoint of former times, and it's subsequent development naturally contained its own elements. The new counterpoint was not based on the ecclesiastical modes but upon the present major-minor tonal system. [Well, The Bourne Supremacy is over. Time out while I channel-surf for another show. Ah. Rounders is next. Believe I'll take a lunch break. - Ed.] Not only in instrumental but also in vocal melody progressions are found that are difficult of intonation for voices and which are unconditionally forbidden in the strict style, such as leaps of sevenths and of augmented and diminished intervals, figuration based on dissonant chords, chromatics and other resources unknown to the older order.

The harmony of the free style is no less sharply distinguished from that of the preceeding era. The free style enables entire groups of harmonies to be consolidated into one organic whole and then by means of modulation to dissect this whole inot factors that are totally interdependent. This characteristic, absent in the former harmony, provided the conditions for the development of the free forms of instrumental music that appeared at the end of the eighteenth and during the first half of the nineteenth century. This new tonal system made possible the writing of works of large dimensions that possessed all the qualities of effective structural style that did not have to be reinforced by texts or by immitative forms per se, but contained within themselves the necessity for the later. By degrees this system widened and deepened and its spreading circle embraced newer and newer resources and laws governing the relations between remote harmonies. Such were the broad horizons opened up for harmony; the creative activity of Beethoven then appears, and he, by a further expansion od the modulatory plans as they stood at the end of the eighteenth century, showed how much variety of key-relationships a composition could exhibit, both in its larger and smaller aspects.

Superseding the ecclesiastical modes, this tonal system was in turn affected by a new one that tended to endanger key-sense by the substitution of a chromatic for a diatonic basis; this lead to a transformation of musical form. Applying the principle that by the use of chromatic progression any chord may follow any other, and pushing it too far, is likely to compromise key-relatinship and to exclude those factors by which the smaller units of form are grouped and amalgamated into one organic whole. Neither did the harmony of the strict style, in which any chord could follow any other, though on a diatonic basis, exhibit the characteristics of tonality and form as now understood. The new harmony, as it now stands and which Fetis called "omnitonal," is inimical to the logic of tonality and form; the chief difference between the old and the new is that the diatonic basis is replaced by the chromatic. Omnitonal harmony, though adding to the resources of composition, at the same time lacks the virility characteristic of the diatonic method. To remain for a time in one key, as opposed to more or less rapid modulation, the contrasts afforded by passing gradually to a new key, with a return to the principle key - all this, by contributing to the clearness of long movements and enabling the listener to comprehend their forms, has little by little disappeared from music since the time of Beethoven and far more rapidly since the beginning of the twentieth century. [The original reads "...has little by little disappeared from contemporary music," but as this was written in 1906 it can hardly be considered a violation of the author's thought to bring the statement up to date. - Tr.] The result has been the production of small works and a general decline in the art of composition. Unity of construction appears with less and less frequency. Works are written not as consistent organisms but as formless masses of mechanically associated parts, any of which might be replaced by others. [Amen: Insights like these are why I mark the end of the tradition at 1915, when Taneiev died. - Ed.]

As for the music of today, the harmony that has gradually lost it's virility would be greatly benefited by the strength that the contrapuntal forms can infuse. [Bingo. - Ed.] Beethoven, who in his later works reverted to the the technical methods of the old contrapuntalists, sets the best example for composers of the future. The music of today is essentially contrapuntal. Not only in large orchestral works, where the abundance of independent parts often results in obscurity, or in opera, where leitmotifs are worked out contrapuntally, but even in pieces of insignificant dimensions, can counterpoint be employed to the greatest advantage. The study of free counterpoint is therefore indespensible for the technical training of composers, but because of its melodic and harmonic intracacy it cannot be studied first. The foundation must be laid by counterpoint of the strict style, more accessible because of it's simplicity. The preliminary steps as regards shifting counterpoint is the subject of the present work.

The term "complex" is used for that kind of counterpoint in which an original combination of melodies yeilds one or more derivatives. The term does not refer to the complexity that results from the union of many voices, nor to the complexity of their melodic or rhythmic features. The essential mark of complex counterpoint is the possibility of obtaining from an original combination of melodies a new one, the derivative..


To be continued...




I'm on the edge of my seat!

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Four Gigs... Two Days

Newyear's weekend is the zenith of silly season. Today? A wedding at four, a dinner gig at seven. Eight-eight miles apart. Tomorrow? A wedding at four, a dinner gig at seven. Eighty-eight miles apart. Anyway...




Happy New Year from Hucbald in Lajitas (Where it was 78 degrees F this afternoon).


Too much of this kind of scheduling and I might take up smoking again.



Oh, that's great. Just... great.

EDIT: 01/15/06 Image re-sized for IE users having trouble with the sidebar.

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Convertible Counterpoint III

In the previous post I explained the phenomenon of vertical shifting conversions, but Taneiev also covers horizontal shifting conversions. This is inextricably linked to the technique of canon, but the melodies are not copies of each other, rather, they are different melodies. So, for instance, with Taneiev's techniques, a two-voice contrapuntal combination could be written wherein the two voices start out simultaneously, but they will also make correct counterpoint if one of the melodies starts out a measure later (Or any other time-delay you might want: He is again exhaustive in his treatment of the subject).

For the present study of this book, I am primarily interested in learning to a competent level of facility the two-voice vertical and horizontal shifting technologies. This means that I will be leaving out significant portions of the book. The book is divided into two parts - Part One and Part Two - and each part has two divisions - Division A, Division B, Division C, and Division D.

In Part One, Division A covers two-voice vertical-shifting counterpoint, so I will work through that, but Division B covers the same subject in three voices, so that will be skipped. In Part Two, Division C covers two-voice horizontal-shifting and double-shifting counterpoint, so I'll go through that, but Division D covers those subjects for three voices, so I'll leave that out too. I may skip over the double-shifting conversions as well, depending on how confident I feel at that point, but the book sort of mingles the two subjects, so that may not be practicable. So, it looks like I will cover about 190 of the book's circa 350 pages.

The reason I only want to deal with two-voice vertical and horizontal shifting combinations is manifold: That is the irreducible essence of the technology (So I want to master that first), the two-voice techniques will more readily be applicable to the guitar (So I can start using them in compositions I'll actually perform right away, especially considering that a third free voice will often be needed to make them effective), I am anxious to get to Dr. Grove's translation of Taniev's Doctorine of Canon - where I'll also restrict myself to the sections dealing with two voices at first - and finally, by grasping these techniques first, I'll be able to cement the three voice techniques easier during a future study through both books. There is also the consideration that this will be a long enough book-blog as it stands.

I really want to get this stuff down. In the worst possible way. Have for years, but now I have the basic contrapuntal chops to do it (I pray).


AUTHOR'S PREFACE

In studying the difficult and involved treatment of counterpoint - especially double counterpoint - as presented in the textbooks of ancient and modern theorists, I encountered various obsticles that seemed to result from faulty classifications, too many useless rules and not enough essential rules. [This is what got me to start thinking about the difference between rule-sets that describe a style and the underlying laws governing contrapuntal motion. - Ed.]

The system expounded in the present work appears to me to be simpler, more accurate and more accessible, as the result of applying the process of elementary algebra to contrapuntal combinations, and by restating certain essential rules in terms of the conventional symbols of mathematics. This enabled me to take into consideration a far greater number of relevant facts, and to bring them under control of a comparitively small number of general principles.

For many years I have used seperate parts of this theory in my classes in counterpoint at the Moscow Conservatory, and I have tried to simplify the treatment at those places where experience has shown that difficulties were encountered by students.

The present book is an exposition, of the utmost comprehensiveness, of convertible counterpoint in the strict style. In using it as a textbook the teacher should select, from amid the detailed development of the subject, what is most necessary for the student. [Exactly my goal in this autodidactic escapade. - Ed.]

I have dedicated this book to the memory of H. A. Laroche, whose articles (especially Thoughts on Musical Education in Russia) have had a profound influence on the trend of my musical activities.

Serge Taneiev

Klin, July 1, 1906


My musical activities ought to include more Country and Western dancing with partners who look like this.

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