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Stephen Emmons
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    Education Standards

    A Survey of Music Theory for the College Classroom: Diatonic Harmony

    A Survey of Music Theory for the College Classroom: Diatonic Harmony

    Overview

    A Survey of Music Theory for the College Classroom is a concise, practical, and readable text and workbook for use in the freshman and sophomore music theory curriculum.

    Introduction

    A Survey of Music Theory for the College Classroom is a concise, practical, and readable text and workbook for use in the freshman and sophomore music theory curriculum. The text is divided into four parts which are linked by a table of contents, allowing for a seamless transition between them.

    A Survey of Music Theory for the College Classroom: Fundamentals

    A Survey of Music Theory for the College Classroom: Diatonic Harmony

    A Survey of Music Theory for the College Classroom: Chromatic Harmony I

    A Survey of Music Theory for the College Classroom: Chromatic Harmony II and 20th Century Music

    Most chapters include PDFs of exercises and excerpts for analysis. I encourage the instructor to supplement the course materials through the analysis of lengthy excerpts or complete pieces of music, especially during the sophomore year. I also highly recommend the use of Rising Software’s Musition (theory) and Auralia (aural skills) software. The Musition and Auralia software package is robust, customizable, and can be used to reinforce most concepts covered in freshman and sophomore theory. I have also written a comprehensive set of exams as well as about 1100 objectives questions for use in Blackboard. Please email me at semmons@angelo.edu from a valid faculty email address if you would like to access these tests or Blackboard pools.

    In addition to the extensive excerpts for analysis, the reader will note the prevalence given to part-writing, especially in the books Diatonic Harmony and Chromaticism I. In my experience, part-writing is the laboratory in which the student can learn many of the fundamentals of music theory. Additionally, I have found that students enjoy and even prize the time spent at the boards actively learning through part-writing and the completion of similar in class exercises.

    The examples from the literature in the text and workbook were taken from the IMSLP that are public domain in the United States. I wrote all the part-writing and short examples in the text, all the exercises for the workbook, and a few pieces for analysis that demonstrate various techniques.

    I dedicate this book to the late Dr. Edward Pearsall and the late Dr. Mary Jeanne van Appledorn. I studied Schenkerian analysis and 20th century theory with Ed during my doctorate at Texas Tech. Through these classes and as a member of my dissertation committee, he had a profound influence on my understanding of music and on my writing. Dr. Van was one of a kind. I studied composition and theory with her during my doctorate, and she was a master teacher of composition, the use of scales, and the music of Debussy. I have happy memories of sitting with her at the piano banging through my music or analyzing various 20th century works. She taught with great care and with humor. I shall always be in her debt.

    Stephen D. Emmons, Ph.D.

    Table of Contents

    Harmonic Progression

    Harmonic Progression - Blackboard pools

    • Harmonic Progression

    Overview

    In tonal harmony, chords are used in a hierarchy. The goal of a tonal progression and of a tonal piece of music is the tonic chord. The most common and most useful way of highlighting the tonic chord in both major and minor keys is to precede it with a V or V7 chord. The V(7) – I progression is the most important chord progression in tonal music, and it is possible to create a piece wherein most chords are V(7) and I chords.

    This example is the first eight measures of a short piece by Beethoven, the Ecossaise in G. Observe how the only two chords used are the tonic (I) and dominant seventh (V7) chords. This example does contain a few nonchord tones, a concept that will be discussed later.

    Beethoven excerpt

    While most tonal works of the Common Practice Period (ca. 1600-1900) feature a sizable percentage of tonic and dominant chords, many other chords are used. Tonal harmony is based on the “circle-of-fifths progression” in which the roots of the chords are separated by a descending P5 or an ascending P4. The resulting harmonic sequence is very important to the history of tonal music and is seen time and time again in the music of the 17th through 19thcenturies, as well as in tonal music in the 20th and 21st centuries.

    bass root movement circle of fifths

    The circle-of-fifths progression can be expressed in the following charts of harmonic progression. Tonic chords (I or i) can go to any other chord. The other chords usually proceed in accordance with the circle of fifths, but there are exceptions. The mediant chord occasionally skips over the submediant chord by going directly to the subdominant chord. The dominant chord sometimes goes to the submediant triad rather than the tonic chord in the deceptive progression and deceptive cadence. Additionally, the subdominant chord often progresses to the tonic chord in the plagal progression and plagal cadence.

    The subdominant IV and iv chords serve as substitutes for the supertonic chord in both major and minor keys. Similarly, the viio chord serves as a substitute for the dominant chord in both major and minor keys. Note that chord progression is essentially the same in both major and minor keys, with the addition of the major subtonic (VII) chord in minor. Also note the chord types most used in minor keys.

    Major Keys

    chart of harmonic progression for major keys

                                                        

    Minor Keys

    chart of harmonic progression for minor keys

    The following bullets points provide additional clarity:

    • The I (i) chord can go to any other chord.
    • V and viio chords usually progress to I, but V chords will with some regularity move to vi (VI) in the deceptive progression. Similarly, the V chord will occasionally move to IV (iv) or ii (iio) chords.
    • The IV (iv) chord usually proceeds to V or viio chords but can also move to the ii chord. At times the IV (iv) will go to vi (VI) chords. The IV (iv) will also progress to the I (i) chord in the plagal progression and plagal cadence.
    • The ii (iio) chord functions in the same ways as the IV (iv) chord, but the progression of the ii(o) to the viio is not as common as it is with the IV (iv) to viio. Unlike the IV (iv), the ii (iio) does NOT move to the I (i) chord.
    • The vi (VI) chord advances to the IV (iv) or ii (iio) chords.
    • The iii (III) chord usually progresses to the vi (VI) chord, but with some frequency will advance to the IV (iv) chord. The iii (III) almost never progresses to the ii (iio), however.
    • The subtonic VII in minor keys almost invariably moves to the III chord.

    Most chords used in the Bach chorales are tonic, dominant, leading tone, supertonic, and subdominant harmonies, with other chords such as the mediant and submediant chords being used less frequently. Bach’s usage of chords is a good general guideline for you to follow when you write SATB progressions modeled on Bach’s chorale harmonizations.

    The way a composer uses chords and chord progressions is part of that composer’s style. You should pay attention to and compare the harmonic progressions in the music of composers who were active during the Common Practice Period. When part-writing within the tonal idiom, it is important to follow conventional progression in your writing. Most of your chord choices should follow the table of harmonic progression from left to right. A supertonic chord, for example, normally progresses to dominant harmony. You can occasionally repeat a harmony, for example, moving from the supertonic to the subdominant, from a leading tone to a dominant chord, or from a root position tonic chord to a first inversion tonic chord. At least some movement from right to left in the table is necessary toward the beginning of a phrase to set up the progression. For example, the tonic chord may simply move back to the dominant to set up the motion back to tonic, or it can skip all the way to the mediant triad to set up a more involved progression. Last, you can skip over a class of chords in the motion toward tonic, such as when the mediant triad skips over the submediant by moving directly to the subdominant chord.

    Return to Table of Contents

    Texture, Voice Leading, and Counterpoint

    Texture, Voice Leading, and Counterpoint - Blackboard pools

    • Texture, Voice Leading and Counterpoint

    Texture

    Prior to our study of two-part counterpoint and part-writing, we will briefly survey the basic textures found in music. Texture is one of the basic elements of music, along with rhythm, melody, harmony, dynamics, articulation, register, and timbre. When you describe the texture of a piece of music, you are describing the density of the music, or how much is going on in the music at any given moment. In some styles and cultures, texture can change dramatically within a single musical work.

    The four textures in music include the following:

    • Monophony – an unaccompanied melodic line – a melody sounding by itself [monophonic].
    • Heterophony – simultaneous performance of two or more melodic variants of the same melodic line [heterophonic].
    • Homophony – multiple voices moving homorhythmically (at the same time), or a melody accompanied by chords [homophonic].
    • Polyphony – two or more simultaneous but somewhat independent lines that have distinct melodies and rhythms [polyphonic].

    Voice Leading and Counterpoint

    Voice leading refers to the ways in which harmonies are created by the motions of two or more separate but complementary lines in a musical texture. Understanding voice leading greatly aids the understanding of western music. We will begin our study of voice leading by learning to compose simple melodies paired with tonal bass lines.  

    Counterpoint is the art of combining two or more musical lines into a coherent whole and is perhaps the most significant item that distinguishes the Western art music tradition. The most important lines in tonal music are the melody and the bass. The inner voices complete and support the harmonies suggested by the melody and the bass line. These practices are in part based on the ancient practice of writing species counterpoint, which developed several hundred years ago. In species counterpoint a second voice is added to a cantus firmus (Latin for “fixed melody”) either above or below the original melody. 

    Vocal Ranges for Part-Writing

    The following ranges are to be followed when writing two-part counterpoint for this chapter, as well as for part-writing in an SATB texture.

    voice ranges for part writing

    First Species Counterpoint

    Adding a melody to an existing tonal bass line or adding a bass line to an existing tonal melody is a fundamental skill in music. Learning to write basic counterpoint has been a vital part of theory and composition instruction for several hundred years in the Western music tradition. We will freely adapt the rules of strict voice leading practices in traditional first species counterpoint and apply them to writing tonal melodies and bass lines. 

    The Four Types of Motion

    There are four types of motion that can occur between two voices – parallel motion, contrary motion, similar motion, and oblique motion. All these types of motion can be used when writing counterpoint, but contrary motion is especially valuable in maintaining the independence of the voices and is a hallmark of contrapuntal writing.

    • Parallel motion happens when two voices move in the same direction while keeping the same interval between them. Parallel thirds and sixths are commonly encountered in tonal part-writing. Parallel fourths are also occasionally encountered in this style. Parallel fifths and octaves are to be avoided.

    parallel motion

    • Contrary motion results when two voices move in opposite directions. Contrary motion between the soprano (melody) and bass lines should be sought whenever practical. 

    contrary motion

    • Similar motion occurs when two voices move in the same direction, but by different intervals.

    similar motion

    • Oblique motion ensues when one voice is stationary, while the other voice moves toward or away from it. 

    oblique motion

    Basic Rules and Traditions

    When adding a soprano melody to a tonal bass line or when writing in an SATB texture, you should pay special attention to certain traditions that have evolved over the past few hundred years.

    • Final Note – The final notes of the melody and bass should occur on a strong beat.
    • Chord Member – Each melody note should fit in the chord that is harmonizing it. 
    • Steps – The melody should feature mostly conjunct (stepwise) motion, have a single high note (focal point), include some variety, and should end with a re – do (\(\hat{2}\) – \(\hat{1}\)) or ti – do (\(\hat{7}\) – \(\hat{1}\)).
    • Leaps – Melodies can feature some leaps, but certain leaps are to be avoided.
    • Intervals to avoid are the +2, +4, m7, M7, and intervals larger than a P8.
    • Leaps of a fourth or larger are to be followed by a step in the opposite direction.
    • No more than two leaps should be used in a row, and typically these will outline a triad.
    • Leading Tone – The leading tone should progress to the tonic unless it is part of a descending \(\hat{1}\) – \(\hat{7}\) – \(\hat{6}\) line.
    • Repeated Notes – A single repeat of a melody note is allowed. For example, it is fine to have two C’s on successive beats.
    • Independence – The bass and soprano lines must be independent of each other. They should not move in parallel motion (thirds and sixths) for more than a few notes. Do not allow the lines to move in parallel octaves, parallel fifths, hidden fifths/octaves, or otherwise violate the rules of voice leading. 
      • Identify the intervals between the soprano and bass and write that number above the staff. This provides a quick check against voice leading errors and excessive use of parallel thirds and sixths. 
    • Consonances – Emphasis should be placed upon thirds, sixths, fifths, and octaves. The interval of a second should be avoided, and the interval of a fourth against the bass will be introduced in second inversion part-writing.
    • Minor Mode – The melodic minor scale should be used when in the minor mode.
    • Distance and Crossing – The voices should be within about 2 ½ octaves and should not cross.
    • Final Notes – Assuming a V-I progression in the bass at the end of the example, the final two soprano notes should be scale degrees \(\hat{2}\) – \(\hat{1}\) (re – do) or possibly scale degrees \(\hat{7}\) – \(\hat{1}\)  (ti  do). This is also true in the V-vi (V-VI in minor keys) deceptive progression and deceptive cadence.

    Adding a Melody to a Tonal Bass

    The first step to harmonizing a given tonal bass line is to analyze the bass line with Roman numerals. The second step is to determine what notes can go in the melody. This can be done by spelling the various triads that can be used to harmonize the bass line. Because of the special rules that surround part-writing seventh chords, we will avoid those for now. We will also only use bass lines comprised of the roots of root position chords. In the following example, all the possible triad spellings that include the notes in the bass line are listed, but we will only use the ones in root position at this stage.

    adding a melody to a tonal bass

    The last step is to create a melodic line that conforms to the conventions expressed in this text. You can write above the staff the numbers for the simple intervals that occur between the bass and soprano as a check for good voice leading. Any time you have a 5 or an 8, you should check for the errors such as parallel fifths and octaves, direct fifths and octaves, hidden fifths and octaves, and unequal fifths, all of which are illustrated in the Root Position Part-Writing chapter.

    adding a melody to a tonal bass 2

    You should always play your two-part compositions and SATB part-writing examples at the piano because doing so will allow you to detect errors and will also help develop your piano, aural and theoretical skills.

    Adding a Bass to a Tonal Melody

    Just as a melody can be added to an existing bass line, a bass line can be added to an existing melody. The bass is the harmonic generator or foundation in Western music, and the notes in the bass voice will most often be the root or third of a chord and sometimes the fifth or the seventh. For our initial examples, the bass line will consist of the roots of root position chords. The charts of harmonic progression should always be of foremost importance when harmonizing a melody. Random successions of chords are not progressions!

    The first step is to analyze a melody to determine what chords can harmonize it. Each melodic note can be the root, third or fifth of a triad. Because of the special rules that surround part-writing seventh chords, we will continue to avoid those.

    adding a bass to a tonal melody

    The second step is to carefully choose a progression that conforms to the charts of harmonic progression set forth earlier. Bass lines in root position will typically feature more leaps than the average melody. Later, we will use first and second inversion chords to smooth out the bass line and to add variety. Always remember that the goal for our progression is the motion from V to I. In the next chapter we will learn how to complete the inner voices of an SATB texture. 

    adding a bass to a tonal melody 2

    Return to Table of Contents

    Root Position Part-Writing

    Root Position Part-Writing - Blackboard pools

    • Root Position Part-Writing

    Overview

    Part-writing is a laboratory in which we can learn how individual voices move, how chords function, and how to write basic counterpoint. There are several considerations, including chord voicing, doubling of chord tones, voice leading, and harmonic progression. We will begin by writing SATB textures in root position only. The rules of root position writing will form the foundation for the more advanced, open-ended part-writing that we will study in the future.

    Now that you know the basic principles of counterpoint, the next step is to add the alto and tenor lines to the soprano/bass textures you have created. Adding these inner voices should be the last step when writing in SATB texture, so always begin by producing good two-voice contrapuntal textures. The inner voices almost “write themselves” if the soprano/bass counterpoint has been competently written.

    Voicing a Triad – Chord Spacing

    The structure of a chord in SATB part-writing has an influence on the way it sounds. You should experiment with chord voicings at the piano keyboard to develop an awareness of the differences in sound.

    • Close Structure: the upper three voices are close together, with less than an octave between soprano and tenor
    • Open Structure: the upper three voices are more widely spaced, with an octave or more between soprano and tenor

    open and close position

    General Rules for Part-Writing

    There are a few rules for part-writing to which you should adhere. Doing so will eliminate the possibility of a great many errors in your work.

    • Leading Tones – Leading tones in outer voices need to resolve up by a half step. Internal leading tones in the alto and tenor can leap down by a third to the fifth of the tonic chord. It is also possible to triple the root and omit the fifth by resolving the leading tone up to the tonic on the last chord of an example.

    leading tone resolution 1

    leading tone resolution 2

    leading tone resolution 3

    • Sevenths – Sevenths are active tones and should be resolved down by step.

    resolve sevenths down

    • Common Tones – Keep common tones when possible.

    keep common tones

    • Leaps – In SATB textures, the upper 3 voices will generally move within a third with occasional leaps of a fourth within the phrase. At a new phrase, voices can move by more than a third. Bass leaps can form triads. All leaps in all voices that do not outline a triad should be followed by a step in the opposite direction. No leaps of a seventh are allowed.
    • Bass – The bass is the harmonic generator upon which everything is built. You can use leaps in the bass up to an octave within phrases and more than an octave between phrases.
    • Augmented Intervals – Avoid augmented 2nds (+2) and 4ths (+4). Diminished fifths (o5) can be used if the line immediately changes direction by step. In minor keys, don’t approach the leading tone (raised scale degree \(\hat{7}\)) from scale degree \(\hat{6}\) (natural or harmonic minor) because an +2 will result.

    augmented intervals

    • Contrary Motion – Use contrary motion between soprano and bass whenever possible.
    • High Tenors – High tenors (A3 and above) are good if they do not interfere with the alto line.

    General Rules for Doubling

    Doubling refers to the practice of placing more than one voice on a given chord member. The guidelines are relatively simple and should be memorized.

    • Leading Tones and Sevenths – Do not double any altered chord tones, leading tones, or sevenths.
    • Root Position – In root position chords double the root. Occasionally doubling the third or more rarely the fifth is permitted. The final chord in a progression can feature a tripled root and a single third.
    • First Inversion – In first inversion, the following scale degrees are always good for doubling
      • Scale degrees \(\hat{1}\)\(\hat{2}\)\(\hat{4}\), and \(\hat{5}\) are important in the maintenance of tonality.
      • Scale degrees \(\hat{3}\) and \(\hat{6}\) are modal scale degrees, affecting the tonality of major and minor.
      • For first inversion triads, double scale degrees \(\hat{1}\)\(\hat{2}\)\(\hat{4}\), \(\hat{5}\), then \(\hat{6}\), then \(\hat{3}\), and never \(\hat{7}\) (the leading tone).
    • Second inversion – In second inversion, double the bass which is also the fifth of the chord.
    • Subtonic – It is fine to double the root in a subtonic VII chord.
    • Doubling Exceptions – There are a few exceptions to these rules.
      • Deceptive resolution of V to vi – In major keys, it is acceptable to resolve an inner voice leading tone down to the tonic in a V – vi deceptive progression. In minor keys, it is necessary to double the third in the VI chord to avoid writing an +2. Our preference is to double the third in the vi chord in major keys as well.

    deceptive progression in major

    deceptive progression in minor

    Common Part-Writing Errors

    Many errors can be prevented by writing good soprano/bass counterpoint and by following the general rules of part-writing and doubling. There are several common errors that you should watch for in your work, especially during your first attempts at SATB writing.

    • Voice Crossing – No part should be allowed to cross above the soprano or below the bass. Bach would allow the alto and tenor lines to cross briefly. We will not allow this, however.

    crossed voices

    • Spacing Errors – Keep adjacent upper parts (soprano, alto, and tenor) within an octave of one another. The bass may be an octave or more from the tenor, however. Keeping a high tenor usually prevents this problem from occurring in the upper voices.

    spacing error

    • Parallel Fifths and Octaves – Do not write parallel fifths and parallel octaves. Parallel thirds, fourths, and sixths may be used. Parallel fifths and octaves occur when two parts that are a P5 or P8 apart move to new pitches that are separated by the same interval. This includes both simple and compound fifths and octaves.

    parallel fifths and octaves

    • Consecutive Fifths and Octaves – Do not write consecutive P5s and P8s by contrary motion because the effect is like that of a parallel fifth or octave.

    consecutive fifths

    consecutive octaves

    • Direct Fifths and Octaves – Direct fifths or octaves – also called hidden fifth and octaves – occur when the soprano and bass voices move in the same direction into a P5 or P8. The soprano part must leap, as shown in the following examples.

    direct fifth

    direct octave

    • Unequal Fifths – Unequal fifths refer to a P5 followed by a °5 or a °5 followed by a P5. These intervals are fine unless they involve a °5 – P5 between the bass and any upper voice.

     

    unequal fifths

    Examples of Root Position Part-Writing

    The following examples are constructed on the bass and melody lines encountered earlier. Here is the melody that was added to the existing bass line in the Texture, Voice Leading, and Counterpoint chapter.

    root position part writing example 1

    After completing the soprano melody, the next step is to fill in the alto and tenor lines. The goal is to have smooth lines that fill in the harmonies, so keep in mind the general rules for part-writing and doubling listed earlier. Note the doubled third in the submediant chord in the V - vi progression. Also notice that the tenor begins on a relatively high note. Beginning with a high tenor allows it more freedom to move, keeps it out of the way of the bass line, and helps to eliminate spacing errors between the tenor and the alto voices.

    Observe that in this example the inner voice leading tone in the tenor is resolved down by leap to the fifth in the final chord. While such a low tenor might be problematic earlier in the example, it is not a problem at a cadence or on the final chord.

    root position part writing 2

    The following is the bass line that was added to the existing melody at the end of the Texture, Voice Leading, and Counterpoint chapter.

    root position part writing 3

    Compare the resolution of the inner voice leading tone in the tenor on the last two beats to the previous example. The decision here was to triple the root on the ultimate chord rather than move the tenor down to the fifth of the tonic chord.

    root position part writing 4

    In both examples, common tones were kept whenever possible. Keeping common tones helps to ensure smooth writing as well as the elimination of errors in voice leading.

    Return to Table of Contents

    First Inversion Triads

    First Inversion Triads - Blackboard pools

    • First Inversion Triads

    Overview

    The use of first inversion triads provides enhanced note choices for bass lines and modifies the sound of chord progressions. First inversion triads occur when melodies appear in the bass (melodic bass), when bass lines are arpeggiated, in nonfunctional passages, and when they are substituted for root position chords.

    Melodic Bass and First Inversion Chords

    The following example shows how first inversion chords are created by a melodic bass line. The harmony throughout the example is the tonic chord in d minor, but first inversion chords are created when the melody switches to the left hand.

    melodic bass and first inversion triads

    Bass Arpeggiation and First Inversion Chords

    The accompaniments in pieces for piano can include figurations in which the chords are sounded in arpeggiation, and this technique was particularly popular during the Classical period in works by Mozart, Haydn, and others. The following example is taken from the third movement of Mozart’s Piano Sonata in B♭, K. 281. The first note of each measure in the left hand effectively serves as the bass note for each measure because it is on the metrically accented downbeat. These notes are also repeated on successive beats, further contributing to their importance.

    Mozart K. 281 example

    Alberti Bass

    Another technique involving bass arpeggiation is the Alberti bass, named after the composer Domenico Alberti (1710-1740) who used the technique extensively. The Alberti bass is also commonly found in the works of Mozart, Haydn, and early Beethoven. The pattern of arpeggiation for the pitches involved is lowest, highest, middle, and highest. This example is taken from the opening of the first movement of Mozart’s Piano Sonata in C major, K. 545. As with the first example, the metrically accented note in each beat serves as the bass note for the entire beat. This example also illustrates how the technique of Alberti bass can be extended to second inversion triads and to seventh chords.

    Mozart K. 545 excerpt

    Nonfunctional Uses of First Inversion Chords

    Passages of nonfunctional first inversion passing chords can be interpolated into a circle of fifths progression to connect chords with stronger harmonic function. The parallel sixth chords used in this manner are not functional because they do not follow the typical circle of fifths progression. This example from the third movement of Beethoven’s Piano Sonata Op. 2 No. 1 beautifully illustrates this concept. The IV chord in measure 59 is connected to the V chord in measure 63 by a stream of nonfunctional first inversion chords.

    Beethoven Op. 2 No. 2 excerpt

    Function of First Inversion Chords in Substitution

    Triads in first inversion are most frequently used as substitutes for root position chords. The inversion does not change that chord’s function within the chart of harmonic progression. A V6 chord, for example, continues to function as a normal V chord whose goal is tonic harmony. Likewise, the ii6 chord continues to function as a subdominant (predominant) chord whose goal is the dominant. The next example shows how first inversion triads can be substituted in an SATB texture. The substituted chords allow a more interesting bass line, the freedom to double notes other than the bass, and generally provide aural relief from the stronger bass motion and sound of root position chords in progression.

    substitution of first inversion triads

    How to Analyze the Harmonies in Music

    At this stage of our studies, we begin to examine music in greater detail and with a deeper focus on excerpts from a broader range of music literature. Musical parameters such as harmony, melody, form, scales, pitch sets, pitch structures, rhythms, timbres, dynamics, texture, and programmatic elements can all be the focus of various analytical techniques developed over the centuries. This text focuses on the analysis of harmonies, melodies, and form, particularly regarding the tonal music of the past four centuries. 

    The basic techniques of harmonic analysis can be applied to a diversity of musical textures within the tonal system. For example, some music is homorhythmic, with the supporting lines changing at about the same pace as the rhythm of the melody, while other homophonic pieces feature blocked chords or arpeggiated accompaniments such as the Alberti Bass. Still other pieces are primarily polyphonic in texture and feature two or more relatively independent lines. 

    The first step in analyzing harmonies is to determine the harmonic rhythm, which is the speed or rate at which the chords change. Harmonic rhythm is influenced by musical genre, period, composer, meter, tempo, and style. Harmonic rhythm can be very fast, with chords changing every quarter note or even every eighth note, while in other works the harmonic rhythm can be much slower, with chords changing every measure or even less frequently. A good place to start when examining harmonic rhythm is to consider the meter. Chords often change on strong beats within a measure. For example, a passage in common time may feature chord changes on beats one and three.

    The second step is to take a pitch class inventory starting with the lowest notes in the texture. The lowest notes – often but not always written in the bass clef – typically serve as the harmonic generator for a passage. After taking the pitch class inventory, find the root of the chord and stack the notes into thirds. Nonchord tones will often be present and are customarily found in the middle to upper voices within a texture because that is where melodies usually occur. We will gain a better understanding of nonchord tones later in this text.

    The last step is to use Roman numerals, bass position symbols, and lead-sheet symbols to label the chords, allowing for a clear comprehension of the progression.

    First Inversion Part-Writing

    Using first inversion chords in SATB part-writing is easier than part-writing using only root position chords because of the increased choices in doubling. The guidelines for doubling were given earlier and are restated here. While some scale degrees such as the tonic, supertonic, subdominant, and dominant are emphasized more than others, essentially any tone except the leading tone can be doubled when part-writing chords in first inversion. 

    In first inversion part-writing, the following scale degrees are always good for doubling

    • Scale degrees \(\hat{1}\)\(\hat{2}\)\(\hat{4}\), and \(\hat{5}\), are important in the maintenance of tonality.
    • Scale degrees \(\hat{3}\) and \(\hat{6}\) are modal scale degrees, affecting the tonality of major and minor.
    • For first inversion triads—double scale degrees \(\hat{1}\)\(\hat{2}\)\(\hat{4}\)\(\hat{5}\), then \(\hat{6}\), then \(\hat{3}\), never \(\hat{7}\) (leading tone).

    The following are examples with a fixed bass and soprano but with a few different tones chosen for doubling. Playing these examples at the piano will show the subtle differences in sound these changes in doubling make.

    first inversion doublings

    Diminished triads (viio6 and iio6) are usually used in first inversion. When the viio triad is used in root position or in inversion, the third or fifth of the chord must be doubled because the leading tone (root of the chord) cannot be doubled. This may necessitate larger than normal leaps in one voice to avoid parallel octaves if the viiprogresses to the tonic chord with doubled root.

    part writing diminished triads in first inversion

    The vi6 (VI6) should not be substituted for the root position vi (VI) in a V – vi (VI) deceptive progression because the aural impression is very weak.

    avoid first inversions in deceptive cadences

    The IV6 (iv6) can be substituted for the vi (VI) chord in a deceptive progression because the ear hears the motion from the dominant note to the submediant note in the bass line. While this is not as common as the V – vi (VI) progression, it is seen with some frequency in the middle of phrases.

    substitution in deceptive progression

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    Second Inversion Triads

    Second Inversion Triads - Blackboard pools

    • Second Inversion Triads

    Overview

    The use of second inversion triads augments the chord and bass choices available to composers. Second inversion triads arise when melodies appear in the bass, when accompaniments in pieces for piano are sounded in arpeggiation, when bass lines are arpeggiated over several beats or measures, and as passing, pedal, and cadential chords. 

    Bass lines can be arpeggiated over several beats or in multiple measures, as shown in the next example. In a case such as this, the most appropriate way to analyze the passage is as a tonic chord throughout.

    six four arpeggiated ass

    Functional Six-Four Chords

    Second inversion triads that function within the circle of fifths progression include three types – the passing six-four, pedal six-four, and cadential six-four. The first two types – the passing six-four and pedal six-four – are harmonically weak and have a subsidiary and decorative function related to the chords around them. The cadential six-four is a more abundant and much stronger chord that functions as dominant harmony. 

    Passing Six-Four

    The passing six-four chord occurs when a second inversion triad is used to fill in the gap of a third in a bass line. The passing six-four chord is harmonically weak and serves merely to connect the other two chords. It normally occurs on a weak beat or weak part of a beat. Contrary motion is typical between the bass and soprano, and the bass must be doubled in the passing six-four chord.

    passing six four

    The opening measures of the first song in Beethoven’s song cycle An Die Ferne GeliebteOp. 98 No. 1 illustrate the passing six-four chord.

    Beethoven excerpt of passing six four

    Pedal Six-Four

    The pedal six-four chord happens when a chord – typically the I or V – is decorated with a six-four chord. The aural effect is like the pedal point (pedal tone) because of the repeated or sustained bass note. Like the passing six-four, the chord usually falls on a weak beat, and the bass must be doubled.

    peel six four examples

    The following example taken from the closing measures of Schumann’s Wild Rider, Op. 68 No. 8 clearly illustrates a typical use of the pedal six-four chord.

    Schumann Wild Rider pedal six four

    Cadential Six-Four

    The cadential six-four chord is an exceptionally useful companion to the dominant V chord in cadences. Cadences are resting places at the ends of phrases that are described in greater detail later in this text. Cadential six-four chords are very common in western art music and on rare occasions even substitute for the dominant chord in a half-cadence. They have a strong dominant function and serve to heighten the listener’s interest in the expected V(7) chord. They are almost always found on strong beats and like the other functional six-four chords always feature the doubling of the bass note in a four-part texture.

    Carefully observe the descent of scale degrees \(\hat{3}\)\(\hat{2}\), and \(\hat{1}\) in the following examples. This descent is often in the soprano, especially when driving to a perfect authentic cadence (PAC), which is discussed in the next chapter. It can also appear in an inner voice if the desired cadence is an imperfect authentic cadence (IAC).

    cadential six four examples

    The next example is taken from the beginning of the first movement of Mozart’s Piano Sonata in A K. 331. A cadential six-four chord is featured at the ends of both the first and second phrases, which end with a half cadence (HC) and perfect authentic cadence (PAC) respectively.

     

    Mozart K. 331 cadential six four

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    Cadences and Small-Scale Form

    Cadences and Small-Scale Form - Blackboard pools

    • Cadences and Small-Scale Form

    Foundations of Form

    Form refers to the organization or overall plan of all the musical elements in a piece, including rhythm, melody, harmony, dynamics, texture, articulation, register, and timbre and how they work together. The study of form is vital to understanding and performing western art music at any level.

    The three foundational techniques that create musical form include the following:

    • Repetition – Because music exists in time, repetition creates a sense of musical unity. Musical ideas gain significance to the hearer as they are repeated.  
    • Contrast – Contrast creates conflict, heightens interest, and enhances the effect of repetition. Most musical pieces include at least a little contrast.
    • Variation – Variation consists of taking a musical idea and making changes to it without completely obscuring its lineage. This combines the concepts of repetition and contrast and is the basis for the theme and variations form.

    The Building Blocks of Form

    The single note is the smallest division in music. Notes combine to make cells, which are referred to by some as figures, figurations, or submotives. Cells can derive from motives and can include arpeggios, scales, and other short collections of notes. Cells are often featured in transitional (or “traveling”) music used to connect larger sections and themes. Motives are short musical ideas that can be primarily melodic, harmonic, rhythmic or a combination of the three. The term motive is used flexibly by musicians, but generally refers to something less than a phrase in length. Motives are integral to the creation of phrases and larger forms in the Western music tradition.

    Motives are melodically or rhythmically distinct and are usually delineated in some way within the music, either through rests, articulations, note length, repetition or in some other way. Motives can be a few notes in length or may be long enough to consist of several smaller cells that stretch over one or more measures. The motives in music with slow tempos usually contain few notes, while the motives in music with faster tempos usually contain more notes. Motives also contain smaller submotives which can themselvses be developed.

    To have importance, motives must be restated (repeated) and developed. Motives can be developed by being sequentially repeated, transposed, subjected to intervallic expansion and contraction, retrograded, modified in rhythm and pitch, imitated, subjected to rhythmic augmentation and diminution, inverted, ornamented, and so forth. If the original rhythmic pattern is preserved, a motive can undergo a great deal of pitch alteration without losing its identity.

    Beethoven’s Symphony No. 5 in c minor provides good ground on which to examine the concept of motive.

    Cadences

    At the heart of musical form is the cadence. Cadences are arrival and resting places that are like punctuation marks such as commas and periods in written language. While cadences are a combination of the musical elements in a piece, this study emphasizes the harmonic component of that mixture. The proper performance of individual phrases and their groupings (periods, phrase groups, etc.) constitute one of the pillars of basic musicianship.

    The typical cadences found in western music are illustrated below. These examples include only the penultimate and ultimate chords found at the ends of phrases and are to be approached as the goals of a longer progression. We will encounter variants of these cadences during future semesters of music theory.

    Perfect Authentic Cadence (PAC)

    Perfect authentic cadences consist of a root position V(7) to I (i). The melodic motion will end on the tonic note in the soprano and is typically approached by either the leading tone or supertonic note. PACs can occur at any cadence point within a piece but are almost always found on the final chord.

    PAC

    Imperfect Authentic Cadence (IAC)

    Imperfect authentic cadences are not as final sounding as PACs. IACs are subdivided into three types.

    Root Position IAC

    As with the PAC, root position imperfect authentic cadences consist of a root position V(7) to I (i).  In a root position IAC, however, the mediant or dominant note of the key is in the soprano in the tonic chord.

    Root IAC

    Inverted IAC

    In an inverted IAC, one or both chords are inverted. 

    Inverted IAC

    Leading Tone IAC

    In a leading tone IAC, the approach to I (i) is by a viio(7) chord. The viio triad is typically found in first inversion, while the viiØ7 and viio7 chords can be used in root position or in any inversion. The leading tone seventh chord is particularly useful because it includes in common three of the four notes of the dominant seventh chord for a given key.

    Leading Tone IAC

    Half Cadence (HC)

    Half cadences end on V(7) chords and are most commonly encountered as the internal cadences in periods or as the cadences in independent phrases or phrase groups. The supertonic scale degree supported by the V chord is by far the most encountered melodic note in this cadence.

    HC

    Plagal Cadence (PC)

    Plagal cadences consist of the IV (iv) to I (i) progression. Plagal cadences are final sounding cadences and are often used in conjunction with the “Amen” at the end of traditional hymns. Plagal cadences are on rare occasions substituted for authentic cadences in periods.

    PC

    Deceptive Cadence (DC)

    The deceptive progression and the special part-writing conventions that accompany it were discussed earlier. Deceptive progressions and deceptive cadences occur when the melodic motion and harmonic motion disagree. The most common deceptive progression is the V – vi (VI), but the V – IV6 (iv6) can also be found in the interior portions of phrases. In a DC, the ear expects the supertonic to the tonic (or alternately the leading-tone to tonic) notes in the melody to be supported by the V – I (i) harmonic progression but is instead surprised by the submediant triad. The usual voice leading of the notes with the V- vi (VI) progression are as follows:

    • Supertonic to Tonic
    • Leading tone to Tonic
    • The other voice skips down a third.

    deceptive cadence

    Phrygian Half Cadence (PHC)

    The Phrygian half cadence occurs only in minor mode and consists of the iv6 to V progression. It is named after the Phrygian mode, a scale that was used in the Medieval period that we will study later. This cadence is most common in the Baroque period and was little used in the Classical and Romantic periods. The part-writing can be a little tricky, so it is best to double the fifth of the iv6 chord and resolve as shown in this example.

    Phrygian HC

    Phrase Groupings

    Phrase

    The phrase is the shortest structural unit in music that ends with a cadence. Phrases are somewhat independent musical ideas that can range from about two measures in length up to several times that, depending on the meter, tempo, and style. Music of the 18th and 19th centuries is often constructed in four measure phrases. Phrases can be marked in form charts using brackets and lowercase letters, as illustrated in the following examples, and these can be combined into larger form charts that show the overall form of a movement or piece.

    Phrases are labeled based on their melodic content. The first phrase in a piece is marked with a lowercase “a.” Phrases whose melodies begin similarly to the first “a” phrase but that end in a different key or end with a different cadence are called a’ (a prime). The first phrase that is different from the initial “a” phrase is labeled with a “b,” and this can logically be extended to b’ (b prime) and so forth. If we encounter another phrase that is similar to a and a’ (or b and b’) we can label it a” or b” (a double prime or b double prime), but for the sake of visual clarity it is our custom to label phrases with a “2” to indicate double prime (for ex., a2, b2).

    Period

    A period is comprised of two phrases in an antecedent-consequent (question-answer) relationship. The second phrase ends with a stronger cadence than does the first phrase, for example HC, PAC or IAC, PAC. Periods are parallel when the melodies of the component phrases begin with similar or identical material and are contrasting when they are not similar.

    paralle and contrasting periods

    An example of a parallel period comes from the first two phrases of Schubert’s Sonata in A Major D. 664. The first phrase, labeled a, concludes with an IAC in m. 4, and the second phrase, labeled a’, terminates with a PAC in m. 8.

    Schubert excerpt

    The following parallel period is from the opening measures of Schubert’s Sonata in E Major D. 157. This illustrates the concept of formal elision, which occurs when phrase endings and beginnings overlap. The HC in m. 5 serves as the jumping off point for phrase a’, which in turn ends with a PAC in m. 9. 

    Schubert example

    A contrasting period is seen at the beginning of the first movement of Haydn’s Sonatina in C, Hob. XVI:35. The first phrase (a) ends in m. 4 on a HC and the second phrase (b) terminates in m. 8 with a PAC. This contrasting period is immediately repeated with ornamentation, making it an example of a repeated contrasting period.

    Haydn example

    Repeated Period

    A repeated period happens when a period is immediately repeated, often with embellishments. This is not to be confused with a double period. Repeated periods can be parallel or contrasting. An example of a repeated parallel period occurs immediately after the slow introduction in the first movement of Haydn’s Symphony No. 102 in B.

    repeated period

    Independent Phrase

    A phrase that does not combine with other phrases to form a period relationship or a phrase group is called an independent phrase. Independent phrases are sometimes present in small ternary forms where they form the middle section (Part 2 or Part B).

    independent phrase

    The third phrase of Purcell’s Minuet in a minor provides a clear example of an independent phrase. In this example, the first two phrases form a parallel period, and the overall form is ternary.

    Repeated Phrase

    A repeated phrase transpires when a single phrase is immediately repeated, either verbatim or with embellishments. The phrase labels and the cadences will be the same.

    repeated phrase

    In Chopin’s Nocturne, Op. 9 no. 1 in b minor, we find an example a repeated phrase with ornamentation. The first phrase terminates with an IAC on the down beat of m. 2, and the phrase is immediately repeated but with chromatic and melodic elaboration. Chopin utilizes the same technique a few measures later in mm. 8-12.

    Chopin excerpt

    Phrase Group

    A phrase group consists of a series of phrases – at least two, but often more – in succession that belong together structurally but that do not constitute a period, three phrase period, or double period. The phrases in a phrase group range from being very similar to being highly independent of one another and may end in similar cadences or in a mixture of cadences. Phrase groups are often encountered in developmental sections, in transitional passages, or as the middle section (Part 2 or Part B) of a ternary form.

    phrase group

    This excerpt from Couperin’s Le Petit Rien illustrates the concept. The first phrase of the example finishes with an IAC in m. 20, and the second phrase ends with an HC in m. 24.

    Couperin excerpt

    Three Phrase Period

    This structure occurs when three phrases unite to stand in a period relationship. When the cadences are progressively stronger (HC, IAC, PAC), the period will have one antecedent phrase and two consequent phrases. When two weak cadences are followed by a strong cadence (HC, HC, PAC), the period will have two antecedent phrases and one consequent phrase. The phrase labels will vary according to the similarity of the melodic materials.

    three phrase period

    The beginning of the first movement of Haydn’s Sonata No. 59 in E Major, Hob. XVI:49 provides an example of a three-phrase period with one antecedent phrase (a, which ends in a HC in m. 4 on a viio) and two consequent phrases (a' and b) which conclude with an inverted IAC in m. 8 and a PAC in m. 12.

    Haydn excerpt

    Double Period

    A double period consists of four phrases in two pairs that unite to form a period relationship. The first three cadences will be relatively weak, and the last cadence will be decidedly stronger, for example HC, HC, HC, PAC. Parallel double periods have similar first and third phrases, while contrasting double periods feature dissimilar first and third phrases. Parallel double periods are common, while contrasting double periods are quite rare.

    parallel double period

    An example of a parallel double period comes from the opening measures of the first movement of Beethoven’s Piano Sonata Op. 26. Note that phrases two (b) and four (b’) are also similar, which is occasionally seen in these forms. This example follows the HC, HC, HC, PAC formula.

    Beethoven example

    Sentence

    The sentence is typically an eight-measure long phrase that contains two four-measure components and a single cadence at the end, though other sentence lengths are possible. The first component of a sentence is called the presentation phrase and consists of a basic motive – often two measures in length – that is repeated or varied. The presentation phrase will not end with a clear cadence but does reinforce the tonic. The second component is called the continuation phrase and is less strict in construction. The main purpose of the continuation phrase is to drive toward a clear cadence. Continuation phrases may use sequential repetition, faster surface rhythm (smaller rhythmic values), quicker harmonic rhythm (rate of harmonic change), and melodic fragmentation to help propel the music toward a satisfying cadence.

    Some period forms contain or are overlaid with the melodic structure of the sentence, which allows for varying analyses and interpretations.

    sentence

    The following example from the opening movement of Beethoven’s Piano Sonata in f minor, Op. 2 no. 1 is an example of a sentence that concludes in m. 8 with a half cadence. The basic motive is presented in mm. 1-2, repeated in mm. 3-4, and the continuation phrase propels to the cadence in m. 8.

    Beethoven excerpt

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    Nonchord Tones (Nonharmonic Tones)

    Nonchord Tones (Nonharmonic Tones) - Blackboard pools

    • Nonchord Tones

    Overview

    Nonchord tones (abbreviated NCTs), also known as nonharmonic tones (abbreviated NHTs), are notes that are not a member of a harmonizing chord. The primary purpose of NCTs is to create richer and more varied melodies, but they are also regularly used in interior voices and in bass lines. A common method of identifying NCTs is to place brackets or parentheses around the NCT and then to name it with an abbreviation.

    Some NCTs can be accented or unaccented, while others such as the suspension are always accented. An accented NCT occurs on a strong beat or on the strong part of a beat, while an unaccented NCT happens on a weak beat or on the weak part of a beat. Accented NCTs usually feature a more striking or dissonant sound in relation to the prevailing harmony.

    NCTs can be diatonic (within the key) or chromatic (outside of the key). The majority of NCTs are diatonic, but chromatic NCTs are found with increasing frequency in music of the Romantic period. NCTs can also be shorter than a beat or longer than a beat. Longer NCTs – particularly appoggiaturas – are more frequent in music of the Romantic period.

    Common NCTs in Music

    There are several common NCTs encountered in tonal music, and the names, abbreviations, and approaches/resolutions of these standard NCTs are described below.

    Passing Tone

    Passing tones (pt) are approached by step and left by step in the same direction. Passing tones can be accented or unaccented (most common) and may be diatonic (most common) or chromatic. Passing tones are one of the two NCTs found in second species counterpoint. When adding passing tones to a musical line, look for notes within an individual voice that are a third apart.

    passing tones

    Neighbor Tone

    Neighbor tones (nt) are approached by step and left by step in the opposite direction. Unaccented neighbor tones are the most common. Neighbor tones are one of the two NCTs found in second species counterpoint. When adding neighbor tones to a texture, look for notes that are repeated within a voice.

    neighbor tones

    Neighbor Group

    Neighbor groups (ng) or double neighbor figures consist of a lower and upper neighbor tone, one a step above and the other a step below the chord tone that is embellished. As with neighbor tones, look for notes that are repeated within a voice.

    neighbor group

    Suspension 

    Suspensions (s) are approached by the same tone and left (resolved) by step down. Suspensions are accented and call a lot of attention to the NCT and to the harmonizing chord. They consist of three parts.

    • Preparation – the note to be suspended is prepared by being part of a chord.
    • Suspension – the note is suspended (held) when the chord changes.
    • Resolution – the suspended note resolves down by step to become a member of the chord.

    Suspensions are labeled with numbers that show the simple interval distances between the bass note and the dissonant suspended note and its resolution. The most common suspensions are 9-8 (or 2-1 when the resolution is a unison), 7-6, 4-3, 6-5 (the "consonant" suspension because both notes are consonances), and the 2-3 bass suspension in which the bass features the suspended note and resolution.

    Suspensions can be added whenever a voice steps down a note in the next chord. They can be found anywhere within the phrase but are especially effective at cadences. Suspensions with change of bass appear when the bass moves at the same time as the resolution, creating suspensions such as the 9-6 suspension, which is the most common of the suspensions with change of bass.

    suspensions

    Retardation

    Retardations (r) are approached by the same tone and left (resolved) by step up. Like suspensions, retardations are accented and call extra attention to the NCT and to the harmony. Holding the leading tone over in an authentic cadence creates a common retardation. Retardations are not commonly found in the interior portions of phrases.

    retardation

    Appoggiatura

    Appoggiaturas (app) are incomplete neighbors that are approached by leap and left by step. The leap is usually but not always up, and the resolution by step is usually but not always in the direction opposite to the leap. Appoggiaturas are almost always accented and are most often in the melody. When adding an appoggiatura to a texture, look for places where the melody or another voice steps or leaps up.

    appoggiatura

    Escape Tone

    Escape tones (et) are incomplete neighbors that are approached by step (usually up) and normally resolved by leap in the opposite direction. Escape tones are almost always unaccented, lending a milder sound than the appoggiatura. When adding escape tones to a texture, look for places where a note steps down or leaps down to the next chord tone.

    escape tone

    Anticipation

    Anticipations (ant) are approached by step (usually) or leap (less frequently) and resolve by the same tone (usually) or by leap (less frequently). While they can be used anywhere within a phrase and in any voice, the most typical use is at cadences, where the tonic note is anticipated prior to the tonic chord.

    anticipation

    Pedal Point

    The name for the pedal points (ped) comes from organ music, where pedal points are used in the foot pedals while the organist continues playing the manuals (keyboards) with the hands. Pedal points or pedal tones consist of notes that are held while the chord changes. The pedal note begins as a chord tone in the harmony, then becomes a nonchord tone when harmonies change, and then becomes a chord tone once again when the harmonies resolve.

    Pedal points most frequently sustain the dominant or tonic notes toward the end of compositions or important sections. Pedal points can be held (sustained) notes, repeated notes, or even notes interrupted by rests and then rearticulated. While most pedal points are in the bass of a given texture, inverted pedals in which the pedal point is in the interior voices or in the soprano/melodic voice can also be found.

    In both major and minor keys, the most typical harmonies include the use of the subdominant and dominant chords above the tonic pedal and the tonic chord above the dominant pedal. Note that inversions are not indicated above pedals because the sound of the inverted chord is fundamentally changed by the presence of the pedal.

    pedal points

    Next is an example of an SATB texture illustrating how a texture can be activated by including a variety of NCTs. It is apparent that using a mixture of chords in root position and inversion as well as adding an assortment of NCTs greatly adds to the musical interest of a passage. You should play the following example at the piano both with and without the added NCTs to hear the difference. Note that the F in the alto in measure 5 is the seventh of a V7 chord. While this seventh is approached in passing motion and sounds like a normal passing tone, it functions as the seventh of the chord. We will further examine approaches to the seventh in Dominant Seventh Chords chapter.

    SATB example activated with NCTs

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    Second and Fourth Species Counterpoint

    Second and Fourth Species Counterpoint - Blackboard pools

    • Second & Fourth Species Counterpoint

    Second Species – Passing Tones and Neighbor Tones

    Second species counterpoint exploits consonant skips, passing tones, and neighbor tones. Think of it as a method of “activating” the texture of a simple first species (1:1 or note vs. note) contrapuntal texture. The practices governing adding melodies to existing bass lines and bass lines to existing melodies are still in operation. Additionally, voice leading conventions such as the avoidance of parallel fifths/octaves, direct fifths/octaves and so forth are still to be observed.

    We will confine our second species counterpoint exercises to the adding of a melody to an existing tonal bass line and will adapt the traditional rules of second species counterpoint to this purpose. The general guidelines for writing second species (2:1) counterpoint are as follows:

    • Continue to use the rules and traditions for first species counterpoint but with an eye toward creating opportunities for passing tones and neighbor tones.
    • Continue to follow the guidelines illustrated in the first species counterpoint instructions with respect to the types of motion (contrary, similar, oblique, and parallel) and phrase beginnings and endings.
    • Begin by analyzing the bass line to see what chords can harmonize it. The bass notes can be either the roots or thirds of triads. Do not use second inversion triads or seventh chords.
    • On the offbeats, incorporate consonant (chordal) skips (skips from one chord tone to another), passing tones, and neighbor tones.
    • Passing tones should be used between melody notes that are separated by a third.
    • Neighbor tones should be used between repeated melody notes.
    • Treat the P4 as a dissonance – use it only as a passing or neighbor tone on offbeats.
    • Avoid the common part-writing errors that are illustrated in this text.
    • Write the counterpoint in eighth notes against a cantus firmus in quarter notes.

    This is an example of second species counterpoint in which a melody was added to an existing bass line. The passing tones and neighbor tones are analyzed, as are the intervals from beat to beat. Note the chordal skips that occur in each measure. These add variety to the melody and can also be used to assist in setting up further opportunities for adding passing and neighbor tones.

    example of second species counterpoint

    Fourth Species (Syncopated) – Suspensions

    Fourth species or syncopated counterpoint involves the same two voice textures we have worked with in our earlier counterpoint exercises. Consonant skips may be used when needed to set the stage for suspensions and passing and neighbor tones may be freely used. In fourth species counterpoint we should confine ourselves to writing 4-3, 7-6, 9-8, 6-5 consonant suspensions, and 2-3 bass suspensions.

    When one voice – for our purposes either the soprano or bass – features a stepwise descending line, it is possible to write a chain of suspensions in the other part. For example, a note-against-note series of descending parallel thirds or sixths can be made into a “chain” of 4-3 or 7-6 suspensions, respectively. Chains of suspensions are an important feature of the music of the Baroque composer Arcangelo Corelli.

    The first example contains 9-8, 7-6, and 4-3 suspensions as well as several consonant skips. Also note the chain of 7-6 suspensions found in measure two of the example.

    fourth species example

    The next example demonstrates several 2-3 bass suspensions. As with the more common upper voice suspensions, it is especially useful to suspend notes that resolve to the leading tone of the key.

    bass suspensions example

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    Dominant Seventh Chords

    Dominant Seventh Chords - Blackboard pools

    • Dominant Seventh Chords

    ​​​​​Overview

    In this chapter, we begin our examination of the role and use of seventh chords in music. With a single exception that will be discussed in Nondominant Seventh Chords, the presence of the seventh in a chord does not change the function of that chord within the circle of fifths progression. The addition of the seventh to a triad adds an extra dimension to the sound and creates a layer of tension that is released when the seventh chord resolves.

    In addition to major and minor triads, the dominant seventh chord, which consists of a major triad with a minor seventh (Mm7), is one of the most recognizable and significant chords in the western music tradition during the Baroque, Classical, and Romantic periods. The dominant seventh chord has a strong, forward directed sound that provides extra impetus in a progression.

    Not all major-minor seventh chords are built on the dominant scale degree. For example, the VII7 in natural minor and the IV7 in melodic minor are also dominant seventh sound types but do not share the function of the V7 in relation to the tonic chord. These chords will be discussed in the chapter covering Nondominant Seventh Chords.

    While we will primarily study and apply these chords through SATB part-writing, a survey of music from across the literature for piano, orchestra, and choir reveals that the following guidelines for the resolutions of the various chord tones within a V7 are generally applied.

    Root Position

    Resolution of the root position V7 to I (i) or vi (VI):

    • Resolve the seventh of the chord (scale degree \(\hat{4}\)) down by step to scale degree \(\hat{3}\).
    • Leading tones placed in outer voices must resolve up to the tonic. This results in an incomplete tonic chord with a tripled root and a single third. This option is only available when the V7 is progressing to the final chord in an example.
    • If you wish to write a complete Vto a complete tonic chord, place the leading tone in an inner voice in the V7and resolve it down to the fifth of the tonic chord. Inner voice leading tones can be resolved up to tonic resulting in a tripled root (final chord only) or can be resolved down to the fifth of the I (i) chord which allows for a complete I (i) chord.
    • Scale degree \(\hat{2}\) (the fifth of the V7) resolves down to tonic.
    • Root position V7 chords can be resolved to I6 (i6) chords.
    • In a V7 to vi (VI) deceptive progression, the third of the vi (VI) chord will be doubled.

      

    root position dominant to tonic and submediant chords

    Incomplete V7 chords are perfectly acceptable within a four-part texture. The usual practice is to omit the fifth and double the root, and though it is possible to omit the third and double the root, this is very rare. Incomplete V7chords will progress to complete I (i) chords, and common tones should be kept. Please note that incomplete V7 chords in SATB writing cannot be correctly resolved in a deceptive progression and should not be used.

    resolution of root position V7

    Approach to the Seventh

    Composers have approached the seventh of a chord in four ways. These approaches can be used in any voice within the texture but are illustrated below in the alto voice and using root position V7 chords. These approaches to the seventh can also be used in the leading tone and nondominant seventh chords that will be discussed later in this text. The most common of these approaches to the seventh are the first two.

    • As a passing tone figure – the seventh of the chord is approached by step from above.
    • As a suspension tone figure – the seventh of the chord is approached by the same tone.
    • As a neighbor tone figure – the seventh of the chord is approached by step from below.
    • As an appoggiatura figure – the seventh of the chord is approached by leap from below.

    approaches to the seventh

    First Inversion

    The first inversion V7 is the most common of the inversions. The resolution is straightforward and is as follows:

    • The seventh of the chord resolves down by step.
    • The leading tone in the bass will resolve up to the tonic.
    • The supertonic resolves to the tonic.
    • The common tone is kept.

    resolution of first inversion V7

    Second Inversion

    The second inversion V7 is sometimes used as passing chord to fill in the gap of a third in the bass line in the manner of a passing V64:

    • The seventh of the chord resolves down by step.
    • The leading tone resolves up to tonic.
    • When resolving the V43 to a I (i), the supertonic resolves to the tonic.
    • When resolving the V43 to a I6 or i6 (i.e., when used as a passing chord), the supertonic will resolve up to the mediant.
    • The common tone is kept.

    resolution of second inversion V7

    Third Inversion

    Third inversion V7 chords almost always progress to a I6 (i6) chord, with the seventh of the chord resolving to the third of the inverted I (i) chord.

    • The seventh of the chord in the bass will resolve down by step to the third of the I6 (i6).
    • The leading tone resolves up to the tonic.
    • The supertonic resolves to the tonic.
    • The common tone is kept.
    • Cadential or passing six-four chords may be used.

    resolution of third inversion V7

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    Leading Tone Seventh Chords

    Leading Tone Seventh Chords - Blackboard pools

    • Leading Tone Seventh Chords

    Overview

    The leading tone seventh chord is sometimes used as a dominant functioning chord in place of the V and V7 in tonal progressions in both major and minor keys. In major keys it is a half-diminished seventh chord, while in minor keys it is a fully diminished seventh chord. Both the viiØ7 and the viio7 typically resolve to the tonic chord, but they can also resolve to a V7.

    Major

    In major mode, use the following guidelines for the resolution of viiØ7 chords in root position and inversion:

    • The seventh of the chord will resolve down by step.
    • The leading tone will resolve up, even when in an inner voice.
    • The third of the I chord may be doubled to avoid parallel fifths, which can happen easily in viiØ7 – I and viiØ43 – I progressions.
    • First inversion viiØ7 chords will progress to I6 chords to avoid parallel fifths.
    • Second inversion viiØ7 chords typically progress to a I6 chord.
    • Third inversion viiØ7uses are infrequent and are not illustrated.

    leading tone sevenths in major

    Minor

    In minor mode the viio7 is used in an equivalent manner as the viiØ7 is used in major mode. The resolution of the viio7 features the following:

    • The seventh of the chord will resolve down by step.
    • The leading tone will resolve up, even when in an inner voice.
    • The third of the I chord may be doubled in the resolution.
    • First inversion viio7 chords should progress to a i6 to avoid unequal fifths between the third (bass) and seventh of the chord.
    • Second inversion viio7 chords usually progress to i6 chords but can go to root position i chords.
    • Third inversion viio7chords can progress to i64 or V7 chords.

    leading tone sevenths in minor

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    Nondominant Seventh Chords

    Nondominant Seventh Chords - Blackboard pools

    • Nondominant Seventh Chords

    Overview

    Nondominant seventh chords are frequently used in tonal music to increase tension and provide interest within a harmonic progression. The part-writing procedures for nondominant seventh chords, regardless of inversion, are as follows:

    • The seventh of the chord resolves down by step.
    • Any common tones are kept.
    • Incomplete chords feature a doubled root and may omit the fifth or less commonly the third.

    The Supertonic Seventh Chord

    The most common of the nondominant seventh chords is the supertonic ii7 (iiØ7). It is a minor seventh chord in major keys and a half-diminished seventh chord in minor keys. When melodic minor is used, the possibility of a minor seventh chord exists in the minor mode, but this is quite rare. The ii7 (iiØ7) chord is freely used in place of the supertonic triad in both major and minor keys and most commonly occurs in first inversion. It can progress directly to the V chord, a viio6, or may first resolve to a cadential six-four chord.

    supertonic seventh chords

    The Subdominant Seventh Chord

    The subdominant seventh chord is a major seventh chord (IVM7) in major keys and a minor seventh chord (iv7) in minor keys. When melodic minor is employed, the major-minor seventh IV7 can infrequently be found. The subdominant seventh chord is typically used prior to the ii(iiØ7), V, viio6, or I64 (i64) chords. When the IVM7 or iv7moves immediately to the V chord, the third of the subdominant seventh chord should be placed above the seventh to avoid parallel fifths.

    subdominant seventh chords

    The Submediant Seventh Chord

    The submediant seventh chord is a minor seventh chord (vi7) in the major mode and a major seventh chord (VIM7) in minor mode. Submediant seventh chords progress to supertonic and subdominant triads and seventh chords. When melodic minor is used, the half-diminished seventh (viØ7) can sometimes be found, and these chords typically progress to viior viio7 chords.

    submediant seventh chords

    The Mediant Seventh Chord

    The mediant seventh chord is a minor seventh (iii7) in major mode and a major seventh (IIIM7) in minor mode. Like the mediant triad, it usually progresses to the submediant or subdominant chords. Notice how bass lines that move up by step should be balanced by contrary motion in the upper parts. When the iii7 (IIIM7) progresses to a vi7 (VIM7) in root position, complete chords will alternate with incomplete chords, as discussed below.

    mediant seventh chords

    The Tonic Seventh Chord

    The tonic seventh chord is a major seventh (IM7) in major mode and a minor seventh (i7) in minor mode. The addition of a seventh to the tonic triad causes a change in the function of the chord. Tonic seventh chords share three tones in common with the mediant triad, and like the mediant triad usually progress to submediant vi (VI) or subdominant IV (iv) chords. Unlike the mediant chord, tonic sevenths sometimes progress to supertonic triads and supertonic seventh chords. The tonic seventh or the chords to which it progresses may be inverted and may also have sevenths.

    tonic seventh chords

    The Subtonic Seventh Chord

    The minor mode has the addition of the subtonic major-minor seventh chord (VII7). This subtonic triad and seventh chord are the least frequently encountered chords in the circle-of-fifths progression in minor. Both progress to the mediant chord, and for this reason they are referred to by some theorists as a V(7)/III.

    subtonic seventh chords

    Seventh Chords and the Circle-of-Fifths Progression

    Seventh chords can be used in the circle-of-fifths progression, for example iii7 – vi7 – ii7 – V7 – I. The part-writing rules for a circle of fifths progression using seventh chords in an SATB texture for both major and minor modes are as follows:

    • Complete root position seventh chords will alternate with incomplete root position seventh chords with the fifth omitted.

    root position sevenths in circle of fifths progression

    • First inversion seventh chords will alternate with third inversion chords, for example iii65 – vi42 – ii65 – V42 – I6. All seventh chords are complete.

    first inversion sevenths in circle of fifths progression

    • Root position seventh chords will alternate with second inversion seventh chords, for example iii7 – vi43 – ii7– V43 – I. All seventh chords are complete.

    second inversion sevenths in circle of fifths progression

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