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Introduction to Counterpoint

Minimum Subjects Covered

This syllabus summarizes the content of the Level "III-2" class taught at the New England Conservatory's Preparatory School for several years. (For the last few years, Level III-2 has been a "Harmony" class, q.v., Syllabus elsewhere on this site). A more condensed "student edition" of the Counterpoint Syllabus was typically distributed to the students in photocopied form containing the basic rules.

Required Text: The Study of Counterpoint by Alfred Mann, based on excerpts from Gradus ad Parnassum by Johann Joseph Fux

Suggested Text: Counterpoint In Composition by Felix Salzer and Carl Schacter

Counterpoint is the art of combining two or more melodic lines. It begins with rules and guidelines for the creation of melodies, and then shows how to combine melodies with each other according to a powerful syntax of consonant and dissonant harmonic intervals.

The course uses Fux's text not only for study of the subject itself, but also to provide historical perspective on the pedagogical importance of Fux's work as well. Many master composers including Haydn, Mozart, Beethoven, Chopin and Brahms used Fux's work for their own studies.

But in fact, the rules and guidelines used in the following syllabus are not entirely based on Fux, but rather are more of a digest or compendium of many different pedagogical approaches, e.g. Counterpoint In Composition by Salzer and Schacter, among others.

The class first works through the five species of Fux ("strict" counterpoint in the vocal style of Palestrina and Lassus) first using two voices, then three. At the middle of the term, a written exam on "strict" counterpoint is given.

More freedom in the use of dissonance is introduced, and then figured bass and the subject of harmony are discussed and related to the contrapuntal studies. Students now attempt exercises in the more harmonically sophisticated and potentially more "instrumental" style of Bach.

Canonic techniques are explored and selected works from the baroque and classical period are analyzed, and the students compose a final project for performance by instruments in the class.

The topic of the correct order in which to teach harmony and counterpoint has been the subject of considerable debate. It is the author's belief that counterpoint is usually postponed too long, and should at least be introduced, but not fully elaborated, when melodic and harmonic intervals are first discussed. This introduction would normally be simultaneous with or possibly even before the introduction of the concept of triads. Then harmony can be elaborated, and then counterpoint as taught in a class like this one would make sense. But at first, the order of subjects should follow the order of complexity: 1. melodic lines (one note at a time), 2. intervals (two notes at a time) , and 3. chords (three or more notes at a time).

This class attempts more than the simple "introduction" just described, and so would fit later into the curriculum; but obviously, total mastery of this subject could not be expected within one academic year even at the college level, therefore this class approaches the subject with the premise that there is a smaller, more basic set of rules and guidelines that can be mastered in the allotted time by students of this younger age group (typically middle school or high school). It may form the basis for not only technically "correct" exercises, but for ones that may have real artistic merit as well. The list of rules for the "strict" exercises is therefore kept as small as possible, while including all the basic necessities of contrapuntal understanding.

It is important that the student understand the reasons for studying this subject. Though not all-emcompassing, it is one of the most efficient and elegant systems of compositional pedagogy in existence. Simple musical materials, melodies, circumscribed in range, consisting of pitches from the major and minor scales (and possibly the older modes, since they are used on Fux) are combined within a well defined syntax of consonance and dissonance, using simple rhythms, and are performed in class (usually sung by the class around the piano using fixed-do solfege syllables and/or moveable scale-degree numbers). Ideally, every homework assignment by every student is performed with the participation of every student, but the feasibility of this may depend on class size; in any case, in-class performance of the students' work is a high priority.

It is not the mastery of this or that musical "style" per se that is important, but rather the development of the students' musical awareness. The mastery of the basic materials of music in the context of compositional craftsmanship is of great benefit to the development of this awareness.

For example, motion in parallel octaves is prohibited in this class; when a student writes parallel octaves, this indicates that (s)he is not aware of the parallel octaves. This ability to decipher and comprehend the messages in a page of music is a primary goal of our studies. It must be aural and musical, not just intellectual. The knowledge of the rules and prohibitions is important, but only as it increases this awareness, and the student's compositional craftsmanship, for music of any "style."

The "rules and prohibitions" that follow are intended to clarify and supplement the Fux text. Relevant passages from the text are read aloud in class by the students as appropriate, but the only rules the students are expected to follow (on the homework exercises, the midterm test and final project) are those appearing in this syllabus.

On the first day of class, an overview of counterpoint is given and the course is described as in the preceding paragraphs. Cantus Firmus ("CF," the given melody, usually in whole notes) Counterpoint ("CP," the added melody or melodies) and Species ("type" defined by the ratio of speeds between the to the CF and CP) are defined and the basic principles of good melodic writing are discussed.

The C. F. should be written in whole notes, and be 8-12 measures long.

It must begin and end on the "tonic" (scale-degree one) preferably in the same register.

It must end with scale-degree two moving to scale-degree one.

There must be a unique high or low point, usually, but not always, slightly after the midpoint of the exercise.

After a large leap in either direction (defined here as a leap larger than a third), there must follow stepwise motion in the opposite direction; for lines other than the CF the motion in the opposite direction need not be stepwise, but stepwise motion is preferable.

No melodic leaps larger than a minor 6^th are allowed with the exception of a leap of a perfect 8ve.

No melodic augmented or diminished intervals are allowed. After Species I, in voices other than the C.F., diminished intervals are permitted as melodic leaps, but motion in the opposite direction immediately afterwards is required and should almost always be stepwise.

Melodic motion should be mostly stepwise, but there should be some leaps and changes of direction. The goal is balance between these types of motion, and the melodies should be singable.

No three consecutive notes in the same direction may outline an augmented 4^th if that sequence of notes is isolated by changes of direction both before and after it.

The total range of a single melody must not exceed a 10^th.

Easily sung triadic arpeggiations are acceptable, but no more than three consecutive leaps in the same direction are permitted in this context.

No consecutively repeated notes are permitted in the C.F., because, we want it to represent "pure melody" -- almost an abstraction of melodic shape. Later in the year when students are writing in a more instrumental style, this rule (and many others) will be loosened.

Sequential repetition is discouraged.

At least two Cantus Firmi are composed for homework, at least one in major and one in minor. At the next class they are sung and critiqued.

When the students are ready for composition in two parts, the distinction between melodic and harmonic intervals is emphasized, and consonance and dissonance are defined in historical and acoustical perspective. It is assumed that students have a familiarity with naming, hearing and writing intervals. If remedial work on intervals is necessary, it should be done at this point and continued throughout the course. The following classification is given for use in the class:

The P8^ve and P5^th are the "perfect" consonances -- 3^rds and 6^ths are "imperfect." The P4^th is a special case, dissonant in one situation and consonant in another. The harmonic series and Pythagoras' studies of the mathematics of what we now call "just intonation" can be invoked to explain the various uses of a P4^th (it only appears early in the series between two "upper partials," not above the fundamental) and it can be pointed out that in the "Ars Antiqua" it was considered a consonance.

As a necessary digression, it should be explained to the students that acoustics alone are not adequate to explain cons./diss. distinctions. For example, equal temperament allows four semitones to sound as a consonant M3rd or a dissonant d4th -- the same sound can be either consonant and dissonant depending on context. Definitions of consonance and dissonance have varied throughout history, and the above is only one possible classification system, but it is a useful and widely used one. It held full sway in western music for about six hundred years and still has a bearing on much of today's music.

The word "counterpoint" comes from the latin "Punctus Contra Punctum" meaning "note against note."

Begin with a perfect consonance and with the tonic in the lowest voice. If the C. F. is not the lowest voice, only an 8ve is possible at the beginning.

The counterpoint line ("C.P.") must end scale-degree 7 to scale-degree 8 as the C.F. ends 2 to 1. In minor keys, scale degree 7 must be raised one semitone by the use of an accidental, so that the distance from 7 to 8 is a semitone (i.e. use the "leading tone" at the cadence). The "Melodic Minor" form of the scale should be used to avoid the melodic augmented 2^nd between scale-degrees 6 and 7 in minor.

Consecutively repeated notes are allowed in the C.P. in Species I and IV, but not II and III. The reason is that this would in effect constitute switching species (two consecutive half notes on the same pitch is really either a whole note or a suspension, contrapuntally speaking, and two quarter notes on the same pitch are similarly equivalent to a half note).

Contrary, parallel, similar and oblique motions are defined.

Contrary: the voices move in opposite directions

Parallel: the voices move in the same direction the same distance measured in scale steps (not necessarily in semitones, although that might be the method of measurement especially in chromatic music)

Similar: the voices move in the same direction different distances

Oblique: one voice moves while the other doesn't (either by sustaining or repeating the same pitch)

Contrary motion is preferable, but variety of these motions is also necessary. By the way, Fux uses the term "direct" motion which is like a combination of similar and parallel motions.

No parallel P8^ves or parallel P5^ths are allowed.

No similar motion into a perfect consonance is allowed.

No more than three consecutive 3^rds or 6^ths in parallel motion are allowed.

At any given point in the exercise, in two voices the space between the two voices should not exceed two 8^ves; in three or more voices this still applies between the lowest voice and the next voice above it, but otherwise no more than one 8^ve between any pair of adjacent voices.

No voice crossing is allowed.

No overlapping of voices is allowed; this is defined as a voice moving higher than where the voice above it just was, or lower than where the voice below it just was.

In two voices, a unison is only allowed at the beginning and end of the exercise. In three voices, a unison is allowed if at least one, and preferably both, of the voices involved approach the unison in stepwise motion.

In any voice, care must be taken not to use too many varieties of scale-degrees 6 and 7 in close proximity in the minor mode. "Melodic" minor is the closest approximation of the proper use of these tones, (although "natural" minor may of course be freely used anywhere except at the cadence).

Specific Rules for Each Species:
("Species" can be defined here as "ratio of speeds.")

Consonant harmonic intervals only are allowed (no dissonant harmonic intervals).

In Species II, we are allowed to use dissonance for the first time, but its usage is restricted to a "Passing Tone." This is defined as a dissonance filling in stepwise between two notes that are a third apart (i.e. no leaps into or out of dissonances are allowed for now). Later, chromatic passing tones filling in between two notes a second apart may be used as well, but not for now.

"Thesis" and "Arsis" (from Fux) are defined simply as "Strong" and "Weak" and the following chart indicates the possibilities for harmonic intervals in each measure but the last (which must be "Species I"):

The standard cadences (given in Fux fig. 25, pg. 42) should be used in all Species II exercises. This means that the final measure is "Species I" and this is true in Species III, IV and V as well. Standard cadences for all Species are given in Fux in the appropriate chapters, and for now cadences should be limited to these.

The first C.P. note must be a perfect consonance as before but may be on the "weak" beat of the first measure with a rest on the "strong" beat.

The following chart illustrates the definition of parallel perfect intervals if the same type in Species II:

The Xs represent perfect octaves or perfect fifths.

The melodic leap of a diminished fifth (but not an augmented fourth) is now permissible, but must be followed by motion in the opposite direction, almost always stepwise.

In Species III a new legal usage of dissonance is introduced, namely the "Neighbor Note" (N.N.) or "Auxiliary" note. It is defined as a note moving a step away in either direction to a dissonance from a consonance, and returning to the original note.

These are the metric possibilities for harmonic intervals: (the numbers refer to beats in common time)

The rule against parallel perfect intervals is the same as for Species II, except that we treat beats 3 and 4 of the first measure in question and beats 1 and 2 of the second measure as we would the "strong" and "weak" beats of consecutive measures in Species II. In this class, we do not prohibit octave to octave or fifth to fifth on consecutive first beats in Species III.

The "Cambiata" is described by Fux (p. 51, fig.50) and defined in this class as a five note idiomatic melodic shape: down a step, down a third, up a step, up a step, beginning on the first beat of one measure and ending on the first beat of the next, and in which any harmonic dissonances are allowed if the first and last notes of the five-note pattern are consonant, (even if the dissonances are approached and/or left by leap). In addition to the Cambiata, this course introduces a second allowable idiomatic melodic shape (not covered in Fux) which can also justify otherwise illegal dissonances: a "Double Neighbor" (or "Double Auxiliary") of a specific type: another five note melodic shape beginning and ending on a consonance on consecutive first beats, describing the shape up a step, down a third, up a step, up a step. The Cambiata ends a step below where it began, and the Double Neighbor ends a step above where it began. These idiomatic patterns may in fact be regarded as ornamentations of the slower-moving motion down or up a step. They may be used at will, but the student's judgment must be exercised in not over-using either device.

The standard cadences for Species III are given by Fux on pp 52 and 53 (figs. 53 and 54). The first beat may be a rest as in Species II (but in this case a quarter note rest). As before, these are the only ways to begin and end the exercise allowed (for now).

The four note pattern down a third, up a step, up a step (within one measure) and its inversion, retrograde, and retrograde inversion are all useful prolongations of a single pitch and may be used freely (but again not excessively). This leap of a third must be into a consonance.

These patterns, along with the two "idiomatic" patterns mentioned above, are all ornamentations of a more "skeletal" (Species I) progression and in particular, the Cambiata may be understood as deriving from three distinct layers of ornamentation: Species I moving down a step is ornamented first by Species II down a third and up a step, and then the "arsis" Species II note is itself ornamented by its own "double neighbor."

A diminished seventh melodic interval is now added to the list of allowable leaps, again followed by motion in the opposite direction, almost always stepwise as before.

In Species IV, the beats are defined as "strong" and "weak" as in Species II. What Fux calls a "ligature" we call a "tie." Tying from the weak beat of one measure to the strong beat of the next may result in simply a consonance, or a type dissonance known as a "Suspension." This is the first instance of a legal dissonance occurring on the first beat of a measure.

The Suspension may be understood as having three parts:

• Preparation
• Suspension
• Resolution

These will be illustrated in the chart below. The basic rhythm of Species IV is on-against-one but now syncopated by the delay of one of the voices by half a measure. There must always be a common tone between the weak beat of one measure and the strong beat of the next in the CP line if possible, and for now they must be tied (later in the year suspensions or other tied notes may be re-struck). When ties are not possible (to be explained below) Species II should be temporarily used "ad hoc."

A suspension is defined as follows: a dissonance on the strong beat prepared (directly preceded) by common tone (tied) and followed by stepwise motion downwards to a consonance. Although upward-resolving suspensions ("retardations") do occur in music literature, they are prohibited for now for the sake of simplicity. Their occurrence in literature should be mentioned, however.

The following diagram illustrates the relationship between the three parts of a Suspension:

The reason that downward resolutions are more common may be explained as follows: ascending melodic motion is associated with increasing tension of the vocal cords, and downward melodic motion is associated with a release or relaxation of this tension. The relaxation of this type of tension can sometimes be psychologically and musically associated with resolution of dissonance to consonance, and this phenomenon is often useful for a performer in "shaping a phrase."

The possibilities for metric placement of harmonic intervals in Species IV are illustrated here:

Note the two prohibited suspensions; if in tying over the barline one of these suspensions is encountered, the student should use Species II momentarily and find a suitable consonance on the subsequent weak beat ("arsis") to prepare the next suspension.

The most common error in Species IV is tying out of a dissonance; this is prohibited until later in the year when harmony and figured bass are discussed, at which time dissonances in seventh chords (root position or inverted) are allowed as preparations for suspensions.

The CP must end with scale-degree 7 to 8 as before; if possible, the leading tone at the cadence should be the resolution of a suspension.

For the first time in the course, eighth notes may be used in the context of ornamentation. The following are allowable "ornamental resolutions" in Species IV:

• suspension (quarter note) down a third (quarter note) up a step to resolution (half note)
• suspension (quarter note) up a step (quarter note) down a third to resolution (half note)
• suspension (quarter note) down a step (eighth note) down a step (eighth note) up a step to resolution (half note)
• suspension (quarter note) up a step (eighth note) down a step (eighth note) down a step to resolution (half note)
• suspension (quarter note) leaping up or down to a consonance (quarter note) and back up or down to resolution (half note)

Later in "Free Counterpoint," this type of ornamentation may be more elaborate and imaginative.

Species V is a combination of the previous Species. No more than two consecutive measures of the same rhythm are allowed. Common time is assumed (for now).

No note values shorter than a quarter note are allowed except in ornamental resolutions of suspensions. The preparation for a suspension must be a half note, but the suspension itself may be either a half note or a quarter note (i.e. it may resolve on the second quarter note).

The rhythm quarter-quarter-half is prohibited unless there is a tie out of the half note into the next measure. Quarter-half-quarter is prohibited (too syncopated). Half-quarter-quarter is allowed, provided all dissonances follow the rules of the "species-of-the-moment."

The Species V rules for parallel perfect intervals are those of the species-of-the-moment.

Species I is only used in the final measure.

The Species of the penultimate measure determines the "standard cadence.".

Two of the voices are written in whole notes, and the third may move in whole notes or faster according to the rules of one of the five species. The rules for each Species are the same as they were in two voices, but "consonances" are now defined as follows:

The conventions of "figured bass" are being used here and so, with the exception of the special cases of "10" and "13" above, compound intervals are represented as simple ones in the figures e.g., a "3^rd" in the figures can appear as a 10^th in the music, a "5^th" as a 12^th, etc.

Complete root position triads with diminished or augmented fifths are dissonant. The diminished triad in "1st inversion," however, is defined as an imperfect consonance, and in fact must occur as the penultimate chord if the CF (which will be expressing scale-degree two) is in the bass; no other cadence is possible in this situation. Tritones should usually resolve augmented 4^th to 6^th and diminished 5^th to 3^rd, but sometimes the augmented 4^th is followed by a perfect 4^th.

Six-three and eight-six sonorities may not occur as the first and/or last chord; eight-five, eight-eight and eight-one may only occur as the first and/or last chord. Thirteen-six and ten-three are not as satisfactory harmonically, and should only be used as a "last resort" depending on the musical situation.

Complete five-three chords are always preferable, but sometimes this is superseded by voice-leading considerations. Good three part writing may be seen as a balance between GOOD LINES and FULL HARMONY. If one must be sacrificed or compromised for the other, the line usually takes precedence over the harmony; hence the common occurrence of incomplete and inverted triads as consonances.

The top two voices should not be spaced more than an octave apart at any given moment in the exercise, but the bottom two voices may be as far apart as two octaves.

As before, no voice crossings or overlappings are allowed (for now).

Similar motion into a perfect consonance is now sometimes allowed, under the following conditions:

• The remaining voice must move in contrary or oblique motion
• If the similar motion occurs in the top two voices, the top voice must move stepwise

Three part parallel motion may only occur as in "Faux-Bourdon" style, i.e. parallel six-three ("first inversion") chords, of which there should be no more than three in a row, as before. The student should also watch for parallel fifths in this situation.

At the cadence, the CF must of course end 2-1. If the CF is not in the bass, the other upper voice must end 7-8 with 5-1 in the bass. If the CF is in the bass, the other two parts must end 4-3 (or occasionally 4-5) and 7-8, in either of the possible inversions. As mentioned earlier, this means that the penultimate chord must be a diminished triad in 1st inversion in this situation (because the penultimate CF note must be scale-degree two, and a six-four chord above this note would be dissonant).

The same rules as before concerning parallel perfect intervals now apply with respect to any pair of voices within the three part texture.

The first new element of "freedom" is in the rhythmic flow of the music: the "cantus firmus" is now understood simply as the slower-moving voice at any given moment, and need not always be represented by the same voice throughout the exercise. At one moment the top voice may be the CF, at the next, the bottom may be.

But the most far-reaching new element of "freedom" is in the treatment of dissonance, and several new usages are now introduced and allowed henceforth. These are sometimes referred to as "ancillary tones." They are numerous enough to suggest classification into the categories "unaccented" and "accented." Up until now, there had been only two allowable types of unaccented (PT and NN) and one accented (suspension) dissonance. These are listed first on the following chart and separated from the remaining usages by a dotted line.

Pedal point should also be included in this discussion, and a good starting point for this might be Fux, fig. 142 on p. 99 (in the context of Species IV in three voices).

Note that all the dissonances in the chart resolve stepwise into consonances, with the exception of the escape tone and the anticipation, which are connected stepwise with an immediately preceding consonance, and all serve to elaborate an underlying stepwise motion.

For the remainder of the term "Species" is taken to mean "ratio of speeds" and any of the voices may at any given moment play the role of CF or CP. As mentioned above, the CF is defined simply as the slowest moving voice. Henceforth the proviso that all the exercises written for class should be "singable" is relaxed somewhat, so that "instrumental" textures may now be experimented with; sixteenth notes (and even occasional thirty-second notes) are now admissible (within reason) provided that all harmonic intervals are either consonances or one of the freer allowable dissonances. All exercises must be playable if not singable, however!

Repeated notes are now freely allowed, within reason, as are re-struck (non-tied) suspensions.

Similar motion into a perfect fifth is now allowed, but much more caution is needed with similar motion into a perfect octave, which is normally reserved for cadences, although exceptions could be considered on a "case-by-case" basis.

More than three consecutive thirds or sixths in parallel motion are now acceptable (again within reason).

Occasional voice crossing and overlapping is acceptable now, the harmonic interval then being measured from the lowest pitch, whichever voice is presenting it.

Now we are ready to introduce the study of Figured Bass and the subject of harmony in general. The study of Harmony is the study of chords and their interrelationships, as opposed to harmonic intervals as consonances and dissonances only, as in strict counterpoint. This would include a discussion of the theory of roots and inversions, and might (should ideally, if time allows) also include a brief description/discussion of the modern day chord-symbol shorthand for harmony as compared to the older figured bass system. It should also include Roman Numeral Analysis, but it should be noted in class that use of roman numerals in this way is a fairly recent invention (mid nineteenth century) and so was not a part of the thinking of Bach, et al. Obviously, we can only really "scratch the surface" of these topics.

We will review harmonic vocabulary: major, minor, diminished, and augmented triads, as well as dominant seventh chords (from Theory Level II) and we will learn the other types of seventh chord (augmented major 7^th, major 7^th, minor/major 7^th, minor 7^th, half-diminished 7^th, and diminished 7^th) both as abstract "sonorities" and in the context of major and minor scales. We will see how the seventh chords evolved historically from the contrapuntal use of dissonance. This review of Harmony will have to be somewhat "efficient" (it is hoped without being cursory...), since the primary focus of the class is how it all relates to Counterpoint. We will practice using our contrapuntal skills to prolong and elaborate a single harmony, and to see how harmonies can elaborate each other in ways that can be understood contrapuntally.

Now is a good time to study the proper uses of a six-four ("2nd inversion") triad. In this course, six-fours may only be used as follows:

• Cadential six-four (the fourth is an appoggiatura or suspension) over scale-degree five ("of the moment") resolving to a five-three over the same bass note; it need not appear only at cadences, but any time the composer wishes to prolong or elaborate the dominant ("V") harmony.
• Neighbor Note or Auxiliary six-four resolving to five-three over scale-degree one; it may or may not be preceded by the five-three into which it resolves
• Passing Tone six-four (the 4^th is the "CF" around which two other voices in the texture move in an exchange of tones, with harmonic intervals either 10-8-6 or 6-8-10; the result is motion from root position to 1^st inversion or vice-versa, and thus really a prolongation of a single triad
• Six-four as a result of two or more simultaneous arpeggiations of a single triad, at the moment when the "fifth" (i.e. the note that would be the fifth in root position) of the chord is in the lowest voice (the fourth may be approached entirely by leap in this situation)
• Six-four resulting when a triad is prolonged in the upper voices while the primary melodic activity is in the lowest voice, at the moment when the "fifth" of the chord is prominently expressed in the lowest voice's melody

The reason that six-four chords are so restricted is that they are dissonant chords.

In a four-part (SATB) figured bass realization, one tone of the triad will have to be doubled at the octave or unison (the triad only contains three "pitch classes" even in a four-part texture). The choice of what note to double may be made, first of all, according to two rules:

• Never double a dissonance
• Never double the leading tone (i.e. scale degree seven in a "dominant function" situation)

For further guidance in the choice of doublings, we may refer to the following chart (but these guidelines are not as "hard-and-fast" as the two rules above).

Please note that under secondary five-three chords, the root should not be doubled on a diminished triad, since it is usually the leading tone (in a major key, the only diatonic diminished triad is on "vii"), and even if the chord does not happen to be "vii" such doubling emphasizes the tritone in the chord too strongly.

The boxes on the chart represent the most common but not always the only choice(s) of doubling for the indicated chords. Roman numerals are used here because they are pedagogically useful even though as mentioned they were not used for musical analysis by the composers, e.g. Palestrina, Lassus, Bach, et al., whose techniques we might use as our models. Note also that the roman numerals are all capitalized (even though the author prefers to use the lower case for chords containing a minor third) because they are all meant to apply to both major and minor modes.

Seventh chords normally contain four different notes and therefore no doublings in SATB texture, but occasionally (again for reasons of voice-leading) may appear with doubled root, one third and one seventh or even (usually on secondary degrees) with one root, doubled third and one seventh; in both cases the fifth is the omission of choice (because of its strong presence in the harmonic series of the root anyway). The note that would be the "seventh" if the chord in root position (no matter what the inversion of the chord actually is) must resolve stepwise down (for now) into the next chord.

The standard figured bass notations for triads and seventh chords in root position and all inversions should be memorized by the student:

By tradition, these may be abbreviated by omitting the numerals in parentheses, so that if no numbers appear, "five-three" is understood, etc. These seven sonorities should be memorized. It should be understood that Figured (i.e. "Numbered") Bass was simply a practical shorthand for harmonic notation used in the Baroque period, much like modern-day chord symbols, and is useful for the study of harmony.

The exact quality (e.g. major, minor, diminished, etc.) of the intervals represented by the figures is determined by

• the key signature, if any
• any accidentals appearing in the figures/music

In connection with accidentals, the author recommends that the following guideline is the least confusing for classroom use: any accidentals appearing in the music (or desired in the "realization" of the bass) should exactly match those appearing in the figures, even though this was not always the case in the historical practice of the system. For example, some composers, including Bach, used "five flat" in the figures to represent a diminished fifth interval above the bass, even in a sharp key, where the accidental needed in the music to create this interval would have been a natural, not a flat. To avoid confusion, again simply for pedagogical purposes, in this course it is understood that the accidentals must match, so that we might write "five natural" in one key, and "five flat" in another to create the same interval.

An accidental without a number always refers to "3." On occasion we will see slashes through the numbers representing sharps, but the normal sharp may also be used. This can also mean "raised" as opposed to literally sharped, as already mentioned. Standard editorial practice of the period was to place the accidental after the number to which it refers, but accidentals before the numbers are also seen. If there is a chromatic alteration to any figure, that figure may not be omitted in any of the standard abbreviations.

At first, we will try connecting root position triads, then inverted triads and seventh chords, and finally adding the middle two voices when the soprano and figured bass are given.

For connecting root position triads over a given bass line, the following three rules should be observed:

• Retain any common tones between two adjacent chords in the same voice(s) in both chords
• Each voice should move as stepwise as possible when not retaining a common tone
• No parallel octaves or fifths as before

When working with these three rules note that there are really only three types of root motion between root position triads: up or down a step, third, or fourth. fifths, sixths and sevenths are simply inversions of these three, e.g. root motion "down a fifth" equals (for voice-leading purposes) "up a fourth" etc. For each type of root motion, there will always be a certain number of common tones between the two chords in question, as illustrated in the following table:

The prohibition against parallel octaves and fifths can now be understood this way: duplication of one voice of the texture by another constitutes a missed opportunity for fuller harmony. Duplicating what another voice is already doing is the least imaginative way to proceed, contrapuntally speaking. It makes four voices sound like three with one voice doubled or "thickened" for emphasis or to add color. More generally, "N" voices becomes "N-1" voices with a doubling. However, intentional doubling for emphasis or color (especially in the context of instrumentation or orchestration) is something else entirely, and is not prohibited or even germane to "counterpoint" per se. Octave doubling of a single voice is in fact very common as a surface feature of many musical textures, while the underlying counterpoint is more skeletal.

For bass lines including inversions and seventh chords, follow the same three rules but note that the wider variety of doublings allowed in the doubling chart given earlier may make choices concerning the first two rules more complicated. Perhaps the best generalization would be "move from one chord to the next as smoothly as possible while avoiding parallel octaves and fifths."

As suggested, when the soprano is given along with the figured bass, strict observance of the first two rules may not be appropriate, since the soprano may contain motivic features including leaps which it would not contain as simply the highest voice of a simple bass realization, as might the alto and tenor voices the student is asked to add. Here the student must balance "Good Lines" with "Full Harmony" as before, while still observing the third rule scrupulously.

We will attempt several four-part realizations of not only figured bass, with or without the soprano given, but also beginning with the soprano only, and adding bass tenor and alto.

As mentioned previously, the subject of "instrumental texture" may now illuminate the distinction between "parallel octaves" and "octave doubling of a single voice" as an instrumentation or orchestration technique. In piano music this distinction can be extremely slippery and confusing. Indeed, composers take advantage of this confusion by moving in and out of a great variety of such textures within even a single measure of a composition. It should be emphasized, however, that although "octave doubling" may be ubiquitous, actual "parallel octaves" are rare in pre-20^th century western practice and are, as we should now begin to realize, much more aesthetically objectionable than "octave doubling."

For the purposes of this class, chromaticism (other than the leading tone in minor and the "three forms of minor" already studied) will almost always be associated with the use of chromatically altered ancillary tones (normally chromatically raised when approaching from below, but diatonic approaching from above) and "Secondary Dominants" (a.k.a. "Applied Dominants"). Other instances of chromaticism e.g. Neapolitan sixth, Augmented sixth chords, neighbor-note diminished seventh resolutions (Raised IIdim7 and raised VIdim7), and Modal Interchange may be briefly mentioned and explained, but students' familiarity with these other usages is not necessarily expected.

The four most common "dominant function" chords are:

• V (major triad)
• V⁷ (dominant seventh chord)
• vii (diminished triad, usually in first inversion)
• vii⁷ (diminished seventh or half-diminished seventh chord)

These may be freely inverted except: six-fours are restricted as previously, and the vii (diminished triad, item three above) should almost always be in first inversion.

Now each of these "dominant function" chords may be used as a "secondary" or "applied" dominant, i.e., each may be transposed in such a way that it resolves to any diatonic (for now) major or minor triad. A diminished or augmented triad may not be approached by a secondary dominant function chord.

The standard roman numeral notation for these chords is to write the dominant function chord and the chord to which it is (expected) to resolve in a form appearing like a "fraction," for example:

We should be familiar with this in connection with the dominant triad and seventh chord from Level II, but have not seen the vii and vii⁷ as dominant function or secondary dominant function chords.

As before, the note that would make the "seventh" interval in root position must resolve down stepwise no matter what the inversion of the chord.

The "dominant" function of these chords is really an extension or elaboration of the "leading tone" function of scale degree seven (which, along with scale degree two, all the chords contain)

The distinction between "tonicization" and "modulation" can now be understood as follows: if a passage of chromatic music ends in the same key in which it began, any secondary dominants "tonicize" the harmonies to which they resolve; if the passage ends (i.e. with a cadence) in a different key than the one in which it began, there has been a "modulation" to the key in which there was a cadence.

Standard cadences will be reviewed (authentic, plagal, deceptive and half). More elaborate, Bach-like cadential formulae such as:

A melodic line may actually be carrying the information of two or more lines. Large melodic leaps, previously prohibited in strict counterpoint, are now admissible provided all the underlying voice-leading connections are completed convincingly. The discontinuity of such a leap is simply the result of switching from one underlying "voice" to another, in the manner of juggling. Harmonies of four or more parts may be arpeggiated, sometimes somewhat obscuring which voice is the "CF." We will attempt several such exercises, perhaps beginning with a given four-part chord progression and composing two-part counterpoint expressing the progression, or even beginning with simply a sequence of roman numerals provided by the instructor. Remember that all dissonances and "tendency tones" e.g. the leading tone, scale-degree four or the dissonant tones in seventh chords, should be resolved properly.

One-against-one dissonance is henceforth allowed, provided the dissonance resolves stepwise into a consonance. This stepwise motion may happen in more than one of the voices involved in creating the dissonance, but must happen in at least one of the voices. In simultaneous scalewise motion in opposite directions, more than one dissonance in a row is even possible. One seventh chord may resolve into another provided that the "sevenths" of the chords all resolve properly; this very often happens when the "root motion" is down a fifth or up a fourth, wherein the seventh of one chord resolves down to the third of the following chord, and the third prepares the following seventh.

The "CF" need not henceforth remain stationary during the resolution of a suspension or appoggiatura, but rather may move to another note consonant with the resolution.

Sequences are small-scale transposed repetitions of a motive or phrase, and for now they are to be mostly diatonic, although they may include some tonicizations and may also be used to modulate to (for now) closely related keys. It could be pointed out that in many compositions by Bach, sequential repetition down a step is associated with "root motion" of descending fifths; down a fifth and up a fourth (the inversion of a fifth), or vice-versa, equals down a step. The initial motive or phrase to be transposed and repeated is called the "Model" and the transposed repetitions are called the "Sequences." Too many sequences become monotonous; two are often sufficient and three are often too many. Students should develop their own aesthetic judgment in this respect.

The word "inversion" in music has three distinct meanings and associated terms:

• Melodic Inversion
• Harmonic Inversion
• Mirror Inversion

Melodic Inversion means transformation of the melodic shape into its mirror image by reversing the direction of each of its successive melodic intervals. This will be discussed more fully below in the context of canonic devices.

By "Invertible Counterpoint" ("Double" or "Triple" counterpoint, etc.) we mean Harmonic Inversion, wherein a contrapuntal passage is repeated in such a way that the music previously appearing in the upper voice appears in the lower, or vice-versa. This means that one of the voices is transposed relative to the other, far enough to be below when it had previously been above the other; the interval of transposition for our purposes will be one or more octaves, although "double counterpoint at the twelfth" etc. may be mentioned in passing. Harmonic intervals have the following

The only difficulties in this type of inversion are:

• A P5^th must be treated as a dissonance, because it will invert to one, and a P4^th becomes a perfect consonance, with all the associated rules
• The melodies shouldn't wander too far in register because excessive voice-crossing should be avoided in both the original and inverted versions.

We will briefly investigate "Triple Counterpoint" in which three voices may be stacked six different ways:

We will analyze the Bach 3-part Invention (Sinfonia) in D major as an example of this technique.

"Quadruple Counterpoint" (with twenty four possibilities) might also be mentioned.

Some interesting mirror inversion properties of some well known scales:

Canon is the technique in which a melody functions as a counterpoint to itself by means of imitation (delayed repetition in one or more other voices). Students may recognize "Frere Jacques" or "Row, Row, Row Your Boat" as examples of this technique, since the Round is a type of Canon.

It must be shown to the students that a composer cannot simply compose a melody and expect it to automatically work as a canon. Most melodies will serve as vivid examples. Instead, the following procedure must be used:

Each segment of music must be composed specifically to work as a counterpoint to the previously composed segment. The length of this segment is simply chosen by the composer, and is the temporal distance between the initial statement of the melody and its subsequent statements(s). In more than two voices, this distance is, for the purposes of this class, constant; of course in music literature there are beautiful exceptions to this, and students are welcome to try composing this much more difficult type of canon for three or more voices, (e.g., William Byrd, "Non Nobis Domine") but mastery if this is not expected.

The distance in pitch between the initial melody and its repetition(s) may also vary, and although we will be exposed to this, our focus should be on "Canon at the Octave."

The voice initially stating the melody is referred to as the Dux (leader) and the imitating voice is referred to as the Comes (follower). Any voice may function as "Dux."

The music the Dux initially presents may be literally presented at the unison (i.e. beginning at exactly the same pitch), one or more octaves, or some other interval in the Comes, but may also appear

• in Augmentation (slower by some factor, usually two in simple time signatures)
• in Diminution (faster by some factor)
• transformed by Melodic Inversion
• transformed by Retrograde repetition ("Crab" or "Canzicrans" canon)
• transformed by a combination of retrograde repetition and inversion

A canon in which these transformations are not used is called a "Literal" canon. The others are referred to as Canon by Augmentation or Diminution, Canon by Inversion (now meaning melodic inversion as distinct from Invertible Canon which refers to harmonic inversion as described previously) and Retrograde Canon.

The penultimate and final measures may deviate from strict canonic imitation for the sake of a strong cadence, which may be "freely" composed.

We are now ready to start our final composition projects, which are to be composed for and performed by instruments and/or voices in the class. The instructor may assign something very specific, but there is flexibility for the students to devise their own projects; the result should be a substantial piece (say, a minute in length at the minimum).

It is hoped that by the end of this class, the students will be more aware as listeners, performers, composers, and musical thinkers.

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