Invertible Counterpoint

by David W. Maves

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
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.

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.

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.

 I (-) 1 Not Good II

2. The - below the number 1 indicates that a second may be used as a suspension when it is below the free note.

I

OK

1

II

#### -

3. Conversely, the (-) below the number -1 indicates that when the voices are inverted, the following is forbidden.

 II Not Good -1 I (-)

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.

 II - -1 OK I

5. The - above the number 6 means that a seventh may be used as a suspension above the free note.

 I - 6 OK II

6. The (-) below the number 6 indicates that the following suspension is forbidden.

 I Not Good 6 II (-)

7. The (-) above the number -6 indicates that the following suspension is forbidden.

 II (-) -6 Not Good I

8. While the - below the number -6 indicates that the following suspension is acceptable.

 II OK -6 I -

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).

 p. imp. p. imp. p. imp. p. o 1 2 3 4 5 6 7 8 9 10 11 original p. imp. p. imp. p. imp. p. -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 o derivative 0 2 4 7 9 11 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.

 p. imp. p. imp. p. imp. o 1 2 3 4 5 6 7 8 9 original imp. p. imp. p. imp. p. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 derivative 0 2 4 7 9 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:

1JV = -11>

(IV=-11+ II)

 - -x - - 3 6 8 10 0 2 4 7 9 11 1 3 5 8 - - -x -

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:

2JV = -9>

 p. imp. p. imp. p. imp. o 1 2 3 4 5 6 7 8 9 original imp. p. imp. p. imp. p. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 derivative 0 2 4 7 9 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.

2JV = -9>

(I+ IIV=-9)

 - - 6 8 0 2 4 5 7 9 1 3 - -
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.

1JV = -7>

(I+ IIV=-7)

 p - - -x 6 3 4 0 2 5 7 1 p - 3 4 -x -

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.

INTRODUCTION

INDEX

D

Derivative Interval.

F

I

Intervals, Numbering of, (description/examples).

N

Numbering intervals (description/examples)

O

Ornamented resolution.

P

"p"

R

Rules for Suspensions.

S

Suspensions, Rules for.

X

"x"

V

Variable consonance.
The text is a detailed explanation of invertible counterpoint, including the renumbering of intervals for mathematical ease, rules for suspensions, and examples of double counterpoint at various intervals. It contains several examples and charts to illustrate the concepts.
no_question

the text does not contain any explicit mathematical questions or provide answers to such questions. it is primarily instructional, explaining concepts and providing examples.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Invertible Counterpoint

by David W. Maves

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
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.

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.

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.

 I (-) 1 Not Good II

2. The - below the number 1 indicates that a second may be used as a suspension when it is below the free note.

I

OK

1

II

#### -

3. Conversely, the (-) below the number -1 indicates that when the voices are inverted, the following is forbidden.

 II Not Good -1 I (-)

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.

 II - -1 OK I

5. The - above the number 6 means that a seventh may be used as a suspension above the free note.

 I - 6 OK II

6. The (-) below the number 6 indicates that the following suspension is forbidden.

 I Not Good 6 II (-)

7. The (-) above the number -6 indicates that the following suspension is forbidden.

 II (-) -6 Not Good I

8. While the - below the number -6 indicates that the following suspension is acceptable.

 II OK -6 I -

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).

 p. imp. p. imp. p. imp. p. o 1 2 3 4 5 6 7 8 9 10 11 original p. imp. p. imp. p. imp. p. -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 o derivative 0 2 4 7 9 11 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.

 p. imp. p. imp. p. imp. o 1 2 3 4 5 6 7 8 9 original imp. p. imp. p. imp. p. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 derivative 0 2 4 7 9 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:

1JV = -11>

(IV=-11+ II)

 - -x - - 3 6 8 10 0 2 4 7 9 11 1 3 5 8 - - -x -

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:

2JV = -9>

 p. imp. p. imp. p. imp. o 1 2 3 4 5 6 7 8 9 original imp. p. imp. p. imp. p. -9 -8 -7 -6 -5 -4 -3 -2 -1 0 derivative 0 2 4 7 9 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.

2JV = -9>

(I+ IIV=-9)

 - - 6 8 0 2 4 5 7 9 1 3 - -
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.

1JV = -7>

(I+ IIV=-7)

 p - - -x 6 3 4 0 2 5 7 1 p - 3 4 -x -

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.

INTRODUCTION

INDEX

D

Derivative Interval.

F

I

Intervals, Numbering of, (description/examples).

N

Numbering intervals (description/examples)

O

Ornamented resolution.

P

"p"

R

Rules for Suspensions.

S

Suspensions, Rules for.

X

"x"

V

Variable consonance.
user
The text is a detailed explanation of invertible counterpoint, including the renumbering of intervals for mathematical ease, rules for suspensions, and examples of double counterpoint at various intervals. It contains several examples and charts to illustrate the concepts.

TYPE: NO_QUESTION

The text does not contain any explicit mathematical questions or provide answers to such questions. It is primarily instructional, explaining concepts and providing examples.