by David W. Maves
© 1999
Table
of Contents
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= Paragraph numbers)
INDEX
Introduction: There is nothing better than good counterpoint. And there is no musician, no mater how talented, who can not benefit from a careful study of this almost limitless subject. To that end I supply the following notes. (The ideas and examples in this study come [with the exception of a few of my own examples] from the original 1909 Russian edition [in Russian] of the book. Counterpoint by Serge Ivanovitch Taneiev.) Any mistakes and inaccuracies are mine. As this is a work in (very slow) progress the reader should feel free to use the materials contained herein in any way. I will be adding to it and changing it frequently for awhile.
(It is assumed that the reader/student has at this point experienced the equivalent of a good college theory course, and at least an introduction to species counterpoint in the strict, sixteenth century style. If this is not the case, we shall wait patiently until you have completed your preliminary studies.)
1.
First: we need to renumber the intervals in order to work with their
mathematical values. Thus a unison = zero, etc. and there will be -
values.
Unison = 0 Unison = 0 Second = 1 Negative 2nd = -1 Third = 2 Negative 3rd = -2 Fourth = 3 Negative 4th = -3 Fifth = 4 Negative 5th = -4 Sixth = 5 Negative 6th = -5 Seventh = 6 Negative 7th = -6 Octave = 7 Negative 8th = -7 Ninth = 8 Negative 9th = -8 Tenth = 9 Negative 10th = -9 Eleventh = 10 Negative 11th = -10 Twelfth = 11 Negative 12th = -11 Thirteenth = 12 Negative 13th = -12 Fourteenth = 13 Negative 14th = -13 2 Octaves = 14 Negative 2 octaves = -14 and so on. and so on. 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 Example: The interval 25÷7 = 3
(octaves) with a remainder of 4 (3 X 7= 21. 25 - 21= 4). The
interval is a compound FIFTH (3 octaves + a fifth). It takes
a little while to get used to the number 4 being = to a
fifth but we shall soon become used to this and find it a
comfortable way to deal with refreshingly complex
counterpoint. A notational example is below. Note
indications of perfect and imperfect consonances.
Thus this above list of intervals and
their compounds up to three octaves. To determine what the
simple interval is, devide by seven, whatever the remainder
is, is the simple interval.
2.
Rules for Suspensions. In
summarizing the rules for suspensions we will revisit many of the
rules of 16th century contrapuntal style&emdash;a good refresher for
those of you a little bit rusty regarding the rules of simple two
voice counterpoint. The following examples outline the
possibilities.
The rhythmic possibilities are shown above. In imperfect time (three-two time) the resolution would be like the second example resolving on the second of three beats.
The above example summarizes the melodic possibilities for suspensions in both
direct (one or more voices shifting to another interval but not crossing over or
under the other voice) and double (invertible) counterpoint. (-) means a suspension is forbidden, - means that a suspension is allowed. You should print out and/or memorize the chart above. Examples follow:
1. The (-) above the number 1 indicates that a second should not be used as a
suspension when it is above the free (as opposed to the tied) note.
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Not Good |
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2. The - below the number 1 indicates that a second may be used as a suspension when it is below the free note.
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1 |
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3. Conversely, the (-) below the number -1 indicates that when the voices are inverted, the following is forbidden.
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-1 |
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4. The - above the number -1 indicates that a second (when the voices are inverted) may be used as a suspension when it appears as follows.
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OK |
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5. The - above the number 6 means that a seventh may be used as a suspension above the free note.
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OK |
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6. The (-) below the number 6 indicates that the following suspension is forbidden.
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6 |
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7. The (-) above the number -6 indicates that the following suspension is forbidden.
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Not Good |
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8. While the - below the number -6 indicates that the following suspension is acceptable.
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-6 |
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9. The - above AND below the + or -3 and the + or -8 indicate that the + or - fourth and ninth may be used as suspensions (as the tied notes in any appearances above or below the free note. This will be clear shortly as we study some examples.
3.
We shall begin with double counterpoint at the twelfth as that index
allows the greatest freedom with regard to movement of intervals.
This will be indicated as 1JV = -11>. Indices (JV)
proceeded by the superscript 1 are those which permit
parallel motion with at least one interval and its derivative; the
-11> indicates that the example covers no intervals greater than a
twelfth. Now study the following chart (bold numbers indicate
consonances).
original derivative fixed
The top line of intervals = ORIGINAL INTERVALS. The second line of numbers = the DERIVATIVE INTERVALS (the interval that double counterpoint at the twelfth yields). The bottom line of numbers = FIXED CONSONANCES. (A consonance in the original version yields a consonance in the derivative version.) This is the easiest interval of inversion for the beginning exercise since all of the original consonances may be treated as freely as the original interval allows (e.g. parallel motion with thirds and tenths) except the sixth (the "5") because it yields a -6 (a seventh) in the derivative interval &emdash; therefore a sixth in the original version must be treated as a dissonance.
4.
This next example will illustrate double counterpoint at the tenth.
This will be indicated as 2JV = -9>. Indices (JV)
proceeded by the superscript 2 are those which permit NO
parallel motion; the -9> indicates that the example covers no
intervals greater than a tenth. Now study the following
chart.
original derivative fixed
You notice that even with two fewer intervals, you have the same number of fixed consonances, but that every imperfect consonance is paired with a perfect consonance, therefore parallel thirds, for example, yield parallel octaves, or parallel sixths would yield parallel fifths &emdash; so each consonance in the original must be treated as a perfect interval.
5.
The "X" indicates a VARIABLE CONSONANCE or VARIABLE DISSONANCE
(meaning that a consonance yields a dissonance or visa versa). These
must be, if used as suspensions, resolved as to a dissonance
requiring either a passing tone or an auxiliary tone, a fixed
consonance, or an ornamented resolution - other freedoms with regard
to this will be explained a later. Study the
following.
Now for our first extended example. We can reduce the numbers needed to summarize the restrictions of double counterpoint at the twelfth as follows:
(IV=-11+ II)
The fixed consonances in the middle row may be used freely with only those restrictions necessary in the original version. The "x-ed" notes (i.e. variable consonances or dissonance's) must resolve to a note that is treated as a dissonance or to a fixed consonance. Below is the original of Jv=-11.
In m. 2, the 7th is resolved to a passing tone. The same in m.3, i.e. the 5x is resolved to a passing tone which assures that the derivative will be correct &emdash; see derivative below.
In m.9, the 7th is resolved to a fixed consonance; in m.11, the 7th is resolved to an ornamented resolution. This assures that the derivative version will be anatomically correct. (See below.)
Below is the derivative of JV=-11.
In m. 2, the 6th is resolved to a passing tone; the same in measure 4. In m.3, the 7th is also resolved to a passing tone and again in measure 7..
In m. 8, the 7th is resolved to a passing
tone. In measure 11, the sixth is resolved ornamentally, as was the
seventh in the original version. So, as you can see this works! Now
if you would pretend for a moment that this derivative is the
original, and that the original original is the derivative, [i.e.
(I + IIV=-11) Jv=-11] you will see that all of the
possibilities outlined for variable intervals in 5
have been illustrated and that each of the possibilities for
suspensions have at least one example. A good practice now would be
to try a few examples at this index (JV=-11) of your own, write out
the derivative and check the results.
Next we try:
original derivative fixed
Again notice that each imperfect interval is paired with a perfect
interval; therefore, as the superscript 2 before the JV
indicates, there can be no parallel motion at this index. A summation
of the possibilities follows.
(I+ IIV=-9)
Again: no parallel motion, and there are fewer allowable
suspensions.
Below is the derivative of JV=-9
Now if you would pretend for a moment that this derivative is the original, and that the original original is the derivative, [i.e. (I V=-9+ II) Jv=-9] you will see that all of the possibilities demonstrated in the table of fixed consonances for this index have been accounted for. Now would be a good time for you to try a few similar examples in this style keeping the table above in front of you while you do an original version. Done correctly, the derivative version will also be anatomically correct as well. NEAT.
6
In our following example we will encounter use of the "p", i.e. why
we need to indicate this in the table of fixed consonances for the
following example. (Since this is a
1JV,
parallel motion is allowed.)
In strict counterpoint
one must not prepare a suspension that resolves to a perfect
consonance (e.g. fifth or octave) with the same interval because the
results come "too close" to sounding like consecutive parallel fifths
and octaves.
So that the "p" (for perfect interval) warns in the following table that any suspension labeled with this letter should be approached by some interval other than the interval to which it resolves.
(I+ IIV=-7)
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The "p" over the fourth in measure 6 (and again in measure 7) warns that the note of resolution may not also be a fourth.
Note above in measure 13 the leap away from the dissonant suspension to a fixed consonance and then the leap back to the note of resolution: a legal and interesting ornamented resolution.
Below is the derivative of JV=-7
Note below in measure 13, the ornamented resolution.
Now if you would pretend for a moment that this derivative is the original, and that the original original is the derivative, [i.e. (I V=-7+ II) Jv=-7] you will see that all of the possibilities demonstrated in the table of fixed consonances for this index have been accounted for.
This will end the two part examples of intvertible counterpoint on this page. I shall begin to create a linked page with examples at other, more difficult indices which most people will probably not have the interest or ability (nor be masochistic enough--most of these examples have only one or two fixed consonances!) to pursue. When those are finished, we shall return to more advanced countrapuntal concepts.
Table
of Contents (
= Paragraph numbers)
1.
Numbering intervals (description/examples)
3.
Fixed Consonances. 1JV = -11>.
4.
Fixed Consonances. 2JV = -9>.
5.
The "X": Variable consonance/Variable dissonance
6.
The "p": When perfect intervals must be indicated in the table of
fixed consonances.
D
F
I
Intervals, Numbering of, (description/examples).
N
Numbering intervals (description/examples)
O
P
R
S
X
V