Write like Mozart – lecture notes

Major Diatonic Chords

Diatonic chords are chords from the same key (e.g. C major). Chords from more than one key are called chromatic.

Start with the C major scale:

The numbers are the degrees of the scale, with 1 being the tonic and 7 being the leading tone, so called because it leads back to the tonic. The leading tone plays a major role in classical music.

Triads

If we add two successive thirds on top of each note in the scale, we get the set of triads in the major scale.

The roman numerals label each triad, and are derived from the scale degree on which each triad is built. There are three types of triad in a major scale:

1. Uppercase numerals indicate a major triad (I, IV and V), which is composed of a major third followed by a minor third above that. Thus C to E is a major third, and E to G is a minor third, so the I triad is a major triad.

2. Lowercase numerals indicate a minor triad (ii, iii and vi), which is composed of a minor third (e.g. D to F in the ii triad) followed by a major third (F to A).

3. Finally there is one diminished triad (viio ) which is composed of two minor thirds (B to D and D to F).

Seventh chords

By adding another third on top of a triad, we get the seventh chords:

The types of seventh chords are:

1. The chords I^7 and IV^7 are major-major seventh (MM7) chords because they are composed of a major triad with a major seventh interval (C to B in I^7) on top.

2. At ii^7, iii^7 and vi^7 we have minor seventh (m7) chords, which are composed of a minor triad and a minor seventh interval (D to C in ii^7) on top.

3. The V^7 chord is unique in the scale as it consists of a major triad with a minor seventh on top, known as a major-minor seventh (Mm7).

4. The viio7 is a half-diminished chord, consisting of a diminished triad with a minor seventh on top (dim7).

Minor Diatonic Chords

Minor Triads

We can follow the same procedure for the minor scale. In C minor, we have the triads:

We now have the following triad types:

1. The i and iv triads are minor (minor third + major third).

2. The III, V and VI are major triads. The III and VI are naturally major triads (that is, just by using notes from the scale they automatically come out as major triads). The V triad, however, would be minor if we had used the Bb from the scale. The note is changed to a B natural as a minor triad in the dominant position does not fit in well with the rules of harmonic progression. (More on this later.)

3. There are two diminished triads at iio and viio (minor third on top of minor third). Again, the Bb in the vii triad is raised to a B natural to make it fit in with harmonic progression. If we had used the Bb here, then this triad would be major (VII), but this is used mainly when we are progressing out of minor.

Minor Sevenths

The C minor seventh chords are as shown:

The chord types here are:

1. i^7 and iv^7 are minor 7ths.

2. iio7 and viio7 are diminished 7ths.

3. III^7 and VI^7 are major-major 7ths

4. V^7 is a major-minor 7th.

In classical music, some of these 7th chords (iio7, iv^7 , V^7 and viio7) appear frequently. The i^7 and III^7 chords are very rare. VI^7 occurs usually as part of a progression of 7th chords.

Note that the only note that is modified in the minor chords is the Bb to a B natural, which is the leading tone in the major scale. The leading tone plays such an important role that these chords have been modified to include it.

The triads and 7th chords shown above represent the overwhelming majority of chords that are used in classical music.

Other chord types

Other types of music, such as jazz and pop, often contain other types of chords than those shown above. For example, the first chord in the excerpt below is often found in jazz:

The notes in the base clef are F, C and A, and together with the Eb in the treble, form a dominant 7th chord (that is, a V^7 chord with a major-minor structure). A jazz musician would take the top two notes (the G and D) as part of the overall chord, which then becomes an F13 chord.

In classical music, the bottom four notes form the dominant F 7th chord, but the top two notes are classified as non-chord tones or NCTs; they do not form part of the chord. As such, these two notes must be resolved into part of the dominant 7th chord, so the G moves up to the A and the D moves down to the C, with the result that all six notes in the second chord are part of the dominant 7th chord.

In diatonic classical music, harmony is essentially restricted to the chord types described here.

Voicing

To start, we'll deal only with homophony, which is basically a succession of chords. In other words, all voices or parts share the same rhythm, although the intervals between the various parts can change. This is distinct from polyphony, where each part can have a different rhythm.

Since we're dealing exclusively with a succession of chords, we need to consider a couple of points:

1. Where do we put each element of a given chord? That is, in which octave should we place the root, third, fifth and (in a 7th chord), the seventh? How far apart (that is, in which octave) should these tones be?

2. How do we move from one chord to the next? This is known as voice leading, and we'll consider this later.

Vocal Ranges

Since much early music was for voices, we'll build our chords around the ranges of the four main voices: soprano, alto, tenor and bass. The ranges of these four voices are as shown:

In what follows, we'll work in four-part homophony, with one voice from each range.

Open and Close Structure

We'll now look at the spacing of the tones within each chord. There are two structures of chord known as open and close structure. An open chord is one in which the top three voices span more than octave, and a close chord is one where the top three voices span less than an octave. Here are some examples:

In the first example, there is an octave between the bass and tenor, a fifth between the tenor and alto, and a sixth between the alto and soprano. The total range of the top three voices is a tenth, which is greater than an octave. (Note the fact that you can't get the total range by adding a fifth and a sixth to get an eleventh!)

In the second example, the interval between tenor and alto is a minor third, and between alto and soprano it is a fourth. The total range from alto to soprano is a minor sixth, which is less than an octave so this chord is in close position. Note that the fact that there is more than an octave between bass and tenor is irrelevant to deciding between open and close.

In close structure, the three upper voices must be three chord notes that are closest to each other. In the second example, the tenor plays E, so the alto must be the first note that is part of the chord that is above the E (which is G), and the soprano must be the first note above the G that is part of the chord (the upper C).

We could also have given the tenor middle C, in which case the alto would be E and the soprano G.

In the third example, the interval between tenor and soprano is an eleventh (a sixth from tenor to alto and a minor sixth from alto to soprano) so this is also an open chord.

It is often a good idea to stick to either open or close structure for the duration of a phrase, but this isn't a strict rule; if a switch between structures gives you what you want, feel free to do it.

Two Things to Avoid

1. Avoid gaps greater than an octave between adjacent voices in the top three voices. In the first example shown below there is a gap of an eleventh between the tenor G and the alto C.

The reason for this is that a large gap detracts from the unity of the chord.

However, a gap of more than an octave between bass and tenor is OK. Usually a gap up to a tenth is acceptable, although larger intervals are also possible. The gap shown in the second example above is a twelfth.

2. Voice crossing occurs when a note in an upper register is lower than a note in a lower register. In the third example above, the alto A is lower than the tenor middle C. This is bad since it's desirable to be able to follow each individual voice within the chord, and if the voices cross, it's hard to tell which note belongs to which voice.

Doubling

If we use a triad in four-part harmony, we have only 3 notes for 4 voices. To provide all 4 voices, we double one of the notes. Working with chords in root position (where the lowest note is the root of the key), we should double the root to provide the fourth voice. This is shown in the first example below:

The bass and tenor both play a G, with the D (the fifth) in the alto and the B (third) in the soprano.

If we are using a full 7th chord, then we simply write one note for each voice. In the second example, the bass is the root G, the tenor plays F# (seventh), the alto D (fifth) and the soprano B (third).

However, it is quite common for a note to be dropped from a seventh chord; in such a case we need to double one of the remaining notes. The dropped note should be the fifth, since the root is always required for a chord in root position, the seventh is required to make the chord a 7th chord, and the third is required to determine if the chord is major or minor.

Basic Progressions

Harmonic Progressions

As classical music contains a limited number of chords, we need to look at what progressions of these chords are used. That is, we need to look at harmonic progressions. The usual progressions are shown in the following chart:

First, look at the I chord on the right. The dotted arrow means that from the I chord we can go to any other chord. Almost every progression starts with the I chord.

Conversely, the I chord is also the goal or endpoint of a piece. It is usually preceded by the V chord, and the V chord is usually in root position.

Thus a simple, but common, progression is to begin with I, then move to V, then back to I, then back to V and so on, eventually ending on I.

The Circle of Fifths

A fundamental progression is the circle of fifths. In the diagram, we start at I, and then move up or down a fifth to get the next chord, then up or down another fifth to get the next chord and so on. If we descend through a series of fifths, we traverse the diagram from left to right, so we begin at I, then go to IV, then viio, then iii, vi, ii, V and back to I.

Ascending, we start at I, then move through V, ii, vi, iii, viio, IV and back to I.

To find the next fifth from a given chord (if, like me, you have trouble visualizing it), for descending, subtract 4 (not 5!) from the starting chord. If this gives you a number less than or equal to zero, add 7. Thus from I, we have 1 – 4 = -3. Then add 7 to get -3+7=4. Ascending, first add 4. If this gives you a number greater than 7, subtract 7. Thus starting from I, we have 1+4=5, so the next chord is V. Then 5+4=9, which is greater than 7 so subtract 7 to get 9-7=2, and so on.

The first two chords in the diagram are shown in brackets since we usually encounter them in these positions when we're doing a circle of fifths progression.

Substitutions

There are two places where we may choose from two chords, as shown by the two boxes in the diagram. The I chord is also called the tonic chord, with the IV being the sub-dominant and V the dominant. However, we must distinguish between a label for a given chord and the function of the chord. The dominant function can be performed by either the V or the viio, and the sub-dominant function by either the IV or the ii chord.

Some common progressions

Usually, it is only chords from the group I, vi, IV and V that are found in classical music.

The I-V-I progression is quite common, as is I-IV-V-I.

A reduced circle of fifths goes from I to vi, then ii, V and I.

Voice Leading

Harmonic progression is the sequence of chords, such as I-V-I. However, each roman-numeral-labelled chord can be produced in a variety of ways (e.g. open and close structure) so, once we have decided on the progression, we need to choose the specific notes that make up each chord. This is known as voice leading.

The strategies used vary from one era of classical music to another. We'll concentrate on the strategies used in the late 18th to early 19th centuries.

Principles of voice leading

- In the upper three voices (SAT):

- if the two adjacent chords have any common tones (e.g. I and V always share one note), keep this common tone

- otherwise, move the voice by step, if possible

- if neither of these is possible, move by the smallest possible interval

- Restrict our attention to root position chords, so the base always provides the root of each chord.

- Use contrary motion (moving two voices in opposite directions) if possible. It's not always possible to do this, however.

- Resolve the leading tone to the tonic if it occurs in an outer voice (soprano or bass). If a leading tone occurs in an inner voice (alto or tenor), it may be resolved to the tonic or sometimes to the dominant (fifth).

Types of motion

There are four types of relative motion within two voices:

- Parallel motion occurs when both voices move in the same direction.

- Oblique motion occurs when one voice stays the same and the other moves, either up or down.

- Contrary motion is when the voices move in opposite directions.

- Static is when both voices stay the same. This can occur when the two chords share 2 notes (e.g. the I and vi).

Voice Leading with the Common Tone Approach

Here are a few examples of voice leading using the harmonic progressions I-V-I and I-IV-I.

I-V-I

The first two bars show the I chord in closed position. We then move to the V chord, retaining closed position. Applying the rules in the last lecture, we first move the bass to the root of the V chord, which is G. Then we look for the common note in the SAT parts. The note common to I and V is G, so we retain the G in the soprano part. To fill in the alto and tenor parts, we attempt to move them by a single step, which is possible in both cases. The alto moves from E to D, and the tenor from C to B.

We've also included some contrary motion, since the bass moves up while the alto and tenor move down.

To move back to I, we just invert the procedure.

The second example shows the same harmonic progression but this time in open mode (note that the SAT parts in the I chord span more than an octave).

This time we move the bass down to low G and retain the G in the tenor. The soprano can move by a single step from C to B, and the alto from E to D. Again, we reverse the procedure to return to the I chord.

I-IV-I

This example is similar to the first one, except here we use the progression I-IV-I. In the closed position, we first move the bass to the root of the IV chord (F). The I and IV chords share one note (C), so we retain the C in the tenor. The soprano moves by step from G to A, and the alto from E to F.

In the open case, we again move the bass to F. This time we retain the C in the soprano and move the alto by step from E to F and the tenor from G to A.

Voice leading without common tones (I-IV-I and I-V-I)

I-IV-I

It's also possible to use voice leading without common tones between adjacent chords. First, we'll look at the I-IV-I progression:

In bar 1, we start with a I chord in closed position, then we want to move to a IV chord, but without any common tones. Since we're still treating the bass as the root position, we move the bass from C to F. The tenor C (which we used as the common tone in the earlier example) now moves to its closest tone in the IV chord, which is the A. This also introduces some contrary motion relative to the bass.

If we want the alto and soprano to move down also, we pick the smallest intervals in each case so that the E moves down to C and the G down to F. To get back to I, we reverse the process (although this isn't always the effect we might want, but we do need to revert back to some form of the tonic I chord).

Notice that what matters when deciding whether or not we are using a common tone is whether the same voice repeats the tone, not whether the same tone occurs in two adjacent chords. In the first example, middle C occurs in the tenor in I and in the alto in IV, but although it is the same note, it occurs in two different voices so this is still voice leading without common tones.

The second example shows a combination of progressions without and with a common tone. We start with a close structure I chord and use the same voicing to get to the IV chord as before. Now suppose the soprano goes down to the E (as part of the melody). In this case we can use a common tone in the alto and retain the middle C. The tenor moves from A to G to fill in the fifth.

I-V-I

We can now try the same technique with a I-V-I progression. Our first attempt is shown on the left here:

We make the transition from I to V without common tones, and using contrary motion between SAT (moves up) and bass (moves down). We might be tempted to just reverse the process to get back to the I chord, as shown. However, this is a bad thing to do since the soprano note in the V chord is a leading tone (B), and a leading tone must always resolve to the tonic when it's in an outer voice (soprano or bass). We therefore must move the soprano to a C as shown in the second example. The alto part uses a common tone (G) and the tenor moves by step from D to E.

If the leading tone occurs in an inner voice (alto or tenor), the leading tone can resolve to either the fifth or the third as well as the tonic. For example:

On the left, the leading tone (B) in the V chord occurs in the tenor and we resolve it to the tonic (middle C) when we return to I. On the right, the first two chords are the same, but this time we resolve the B to a G. The soprano and alto each move down to the first available tone in the I, so we've done the transition without using common tones.

This problem with the leading tone always occurs in the V chord, so this is an important rule to remember.

Resolution of the V7 chord

The V^7 chord is probably the most common seventh chord in classical music.

Tendency tones

The leading tone is the seventh scale degree (e.g. B in C major) and is so-called because it leads to the tonic (and must always resolve to the tonic in an outer voice). However, the fourth scale degree (e.g. F in C major) also tends to the third degree (E in C major). The fourth degree is not called a leading tone, but rather a tendency tone because it 'tends' to the third. (The leading tone can also be called a tendency tone.)

In C major, one voicing of the V^7 chord is as shown by the first chord in bar 1:

The root (G) is in the bass, with the third (B) in tenor, fifth (D) in the soprano and the seventh (F) in the alto. In C major, B is the leading tone and F is a tendency tone, so if we follow the rules for resolving these notes we get the I chord shown, assuming we fill in the fifth (G) in this chord in the soprano.

However, if this progression is ending a piece, it might be more natural for the soprano to resolve to the tonic, so we might fix this by using the I chord in bar 2. This version, although ok and sometimes found in compositions, is a little sparse since it contains 3 roots and a single third, with no fifth. The presence of the third makes this acceptable, since the third determines the nature of the chord (major or minor).

In bar 3, we have 3 roots and a fifth, which produces a bare, empty sound and is in general not acceptable (although it is sometimes found in earlier music).

Bar 4 shows a resolution to a full I chord, with all 3 notes of the triad present. This produces the fullest sound.

Bar 5 shows the two tendency tones in the soprano and alto parts. The resolution into the I chord follows the tendency of these two notes, and uses the common tone in the tenor.

In the first four bars, the two tendency tones occur with the B below the F, while in bar 5, the F is below the B. The interval B-to-F is a diminished fifth, since it covers 5 letter names over the interval (BCDEF) and is a semitone below a full fifth (B to F#). The interval F-to-B is an augmented fourth, since it covers four letter names (FGAB) and is a semitone greater than a normal fourth (F to Bb). However, both intervals are identical in terms of the number of tones separating the two notes in the interval, as there are 3 full tones (known as a tritone) in each case.

The rule for resolving these two intervals is that if the interval's spelling makes it a diminished fifth, the voices move towards each other, while if the interval's spelling makes it an augmented fourth, the voices move away from each other.

Voice leading from IV-V

We'll now consider the IV-V progression, thus allowing us to complete a I-IV-V progression.

With chords a fifth apart (I-IV and I-V) there is always a common tone, so we have the option of using voicing with a common tone. Going from IV-V the two chords are a second apart, and have no common tones.

In bar 1, we start off with a IV chord in close position:

You might think that the easiest progression to the V is just to raise all the tones in the IV up to the corresponding tones in the V. While this is ok in some forms of music (e.g. pop music), it is not acceptable in classical music.

Parallel fifths and octaves

The bass-tenor interval in the IV chord is a fifth, and the bass-alto interval is an octave. This pattern is repeated in the V chord. These progressions are known as parallel fifths and parallel octaves, and should be avoided when writing homophonic and homorhythmic music.

One correct way of voice leading from IV to V is to use the non-common tone technique, as shown in bar 2. Raise the bass (we're still considering root position only) and then lower the other three notes to the nearest V chord tone. Thus the tenor C goes down to B, the alto F goes to D and the soprano from A to G.

Although there are still intervals of a fifth and an octave, these intervals occur between different pairs of voices than they did in the IV chord. The fifth was between bass and tenor in the IV but is now between bass and alto, and the octave was between bass and alto in the IV but moves to bass-soprano in the V. Fifths and octaves are parallel only if they occur between the same two voices in two successive chords.

Open structure

Bar 3 shows an example with an open chord. We apply the rules for non-common tone voicing. The bass moves up to G, and we try to move the other tones by a single step. This is possible for the tenor (A to G) and soprano (C to B) but the alto has no V chord tone within a single step, so we move it by the smallest possible step, from F to D.

Voice leading with chords that have a mediant relationship

We'll look now at voice leading with chords that are a third apart. The third degree in a scale is midway between the tonic and dominant, so it is called the mediant. Because triads are based on successive thirds, shifting a triad by a third leaves two notes in common.

In bar 1 in the diagram, we start with the I triad in C major. If we shift this triad down a third, we get the A minor triad (the vi triad). Note that it has two tones (C and E) in common with the I triad. The third triad in bar 1 is the E minor triad (the iii triad), which is the I triad shifted up by a third. It too has 2 notes (E and G) in common with the I triad.

Because of these two common tones, we can use the common-tone approach to voice lead from the I to either the vi or iii chord. In bar 2, we start with the I chord in close position and move to the vi chord (A minor). After moving the bass from C to A, we retain the two common tones (middle C and E) and shift the soprano by a single step from G to A.

Bar 3 starts from an open I chord. Again, we retain the two common tones (C and E in the soprano and alto) and shift the tenor G by one step to A.

Texture types

There are four main textures:

- monophony, in which the same melody is played in unison by all voices (although some voices might be whole octaves above or below others). That is, all voices use the same rhythm and the same pitch.

- homophony, in which the same rhythm is used by all voices, but pitches vary. This is the texture we've used in all our voice leading examples so far.

- heterophony, in which the same pitch is used by all voices, but the rhythms of separate voices differ. Heterophony is never found in classical music (although it is found in some contemporary music, and in music from other cultures).

- polyphony, in which different voices use different rhythms and different pitches. Fugues use polyphony extensively, with Bach's fugues being particularly good examples.

These four types can be summarized in this table:

Same

rhythm

Different

rhythm

Same

pitch

monophony

heterophony

Different

pitch

homophony

polyphony

In classical music, homophony occurs in two main types:

- homorhythmic homophony, in which all voices play exactly the same rhythm (as we've done in voice leading examples);

- melody and accompaniment, in which all voices use the same underlying rhythm, with the accompaniment providing harmonic support for the melody rather than containing a separate melodic line.

Voice leading example – 1

We'll look at a complete example of harmonic progression and voice leading, using the poem How Do I Love Thee? by Elizabeth Barrett Browning set to a simple melody, to which we'll add the harmony. Here's our starting point:

The first 5 bars have been filled in, so we'll look at those first. The key is Bb (two flats could also be G minor, but due to the presence of Bb chords, we can tell the key is Bb). The major triads in Bb are:

- I: Bb – D – F.

- IV: Eb – G – Bb.

- V: F – A – C.

Scanning through the first 5 bars, we can identify each chord as one of these three major triads, with the root doubled in each case.

In bar 1, the melody goes from Bb up to the next I chord tone (D). In this case, the harmony doesn't change – we use the I triad in both cases – but the voicing has changed. This is a common technique when the melody moves from one tone to another tone within the same triad: keep the same harmony but change the voicing. In this case, the bass stays the same while the top 3 voices move up to the next chord tone.

In bar 2 we move from I to IV using a common tone in the tenor. The bass moves up to the Eb root, and the top two voices move up one step. We then return from IV to I by reversing the process (recall this is OK for triads a fourth or fifth apart, since we're not using parallel fifths or octaves).

In bar 3, we repeat the I chord but then the melody moves from D down to Bb. Since we want to use the IV harmony here, we move the bass up to Eb, but now we need to find notes for the alto and tenor. If we used a common tone in the tenor, then we would a Bb in both the tenor and soprano so we could fill in the third (G) in the alto. This gives us a chord with one root, one third and two fifths. At this stage, with simple voice leading, it's better to double the root, so we make the alto an Eb and fill in the tenor with a G. This is a non-common tone approach, with each of the middle two voices moving down to the nearest chord tone.

In bar 4, we move from IV to V. We must avoid parallel fifths and octaves, and we can take the opportunity to introduce some contrary motion, so we move the bass up from Eb to F and the top 3 voices down. This results in the bass and tenor having exactly the same note, but this acceptable and counts as two separate roots. The fifth between the soprano and alto in the IV becomes a sixth in the V, and the octave between the bass and alto in the IV becomes a fifth in the V.

The second chord in bar 4 is also a V, but the melody goes up from A to C. This provides an opportunity to move from a relatively compact V chord (not quite closed, since the tenor-soprano interval is greater than an octave) to a broader, more open chord. Again, we have contrary motion, with the bass moving down an octave and the other 3 voices moving up to the next chord tone.

The final bar 5 returns to the tonic by moving the bass up and the other 3 voices down to the nearest chord tone.

Voice leading example – 2

Important: You should do the exercises in the videos before reading these notes so you get some practice in voice leading without seeing all the answers beforehand!

We'll continue with harmonizing How Do I Love Thee? by filling in bars 6 through 12. Here's the completed piece:

Assuming we've been given the melody (in the soprano), the first thing to do is fill in the bass line, so this was done in bars 6 to 8. We're still dealing only with root position chords, so all the bass notes are root notes. There is some flexibility in which octave we use, so the bass notes shown here aren't cast in stone. The bass notes are determined by how we wish the voice leading to go, and may need to be modified as we fill in the inner voices.

Bar 6:

- I to V uses common tone (F) and single step in tenor from B to A.

- V to I. If we tried a common tone approach, the alto would be F and if we then move the tenor by one step we get Bb, but this gives 3 roots and a fifth. A better solution is to use the non-common tone approach as shown.

Bar 7:

- I to vi: This is a mediant progression, so we have 2 common tones. Retaining these in the soprano and alto, we can then move the tenor by one step from F to G. The next chord is just a repeat of this chord. In bar 1, we kept the same harmony I but because the melody moved, we revoiced the chord. In bar 7, the melody stays on the same note, so it makes sense to keep the same voicing in the vi chord.

- vi to ii: This is a progression of a fifth, so we have one common tone. If we retain this G we need an Eb to finish off the chord, which can be obtained by a single step from the alto D.

- ii to V: This is a fourth progression, so again we have one common tone. This is already given since the melody uses the same C from one note to the next. Filling in the inner voices by moving them a single step each gives the chord shown.

Bar 8:

- V to I: Using the common tone approach, we can retain the F in the alto and move the tenor A by a step to Bb. Deciding whether to use common-tone or non-common-tone is often determined by the preceding chord and the current note in the melody. For example, in the current transition, the melody goes from C to D meaning that we need an F and a Bb to fill out the chord. If we didn't use a common F, the tenor would need to jump to F and the alto to Bb, which creates ugly gaps, so the common-tone approach gives the smoothest solution.

- I to IV: Retain the Bb, thus we get a G in the alto.

Bar 9:

- IV to V: Using a bass F as shown, we fill in the inner voices so we get contrary motion with the bass. Need to watch for parallel fifths and octaves, but we don't have any here.

From here to the end of bar 10 is a repeat of bars 6 and 7 so we can just copy the harmony from there.

Bars 11 and 12 are different, so we need to create new harmony. This isn't given in the lecture so the solution here is my own. However, it sounds quite nice and follows the rules. I've used a common-tone for the I to V and a non-common tone along with contrary motion for the V to I at the end.

Inversion notation

Up to now, all our chords have been root-position chords, with the root in the bass. If one of the other notes is in the bass, the chord is said to be inverted. We need some additions to the roman numeral notation to be able to label inversions.

In the diagram, the first chord in bar 1 is the normal I triad. If we invert the chord by putting the third (E) in the bass, the interval between the lowest and highest notes is a sixth, and between the bottom two notes it is a third. To be complete, we could write this as I^6_3 but usually it is abbreviated to just I^6.

The first chord in bar 2 is still a I chord but now the fifth is in the bass. The intervals are now a sixth to the top note and a fourth to the middle note. Since we've already used the notation I^6, we have to include both intervals in the notation to get I^6_4.

We've already seen that the 7th chord with the root in the bass is denoted with a superscript 7, as in the V^7 chord shown in bar 2. Since there are 3 notes apart from the bass in a 7th chord, there are 3 different inversions.

The first inversion puts the third in the bass, giving an interval of a sixth to the top note (B to G), a fifth (B to F) to the next note down and then a third (B to D) to the next note. A complete notation would need the numerals 6, 5 and 3 next to the V, but this is more information than we need to specify the chord, so we just call it a V^6_5 chord, labelling the top two intervals.

The next inversion (second chord in bar 3) puts the fifth in the bass and now the intervals from top to bottom are a sixth (D to B), fourth (D to G) and third (D to F). Since we've already used the notation V^6_4 for a normal triad with the fifth in the bass, we use instead the bottom two intervals to get V^4_3.

The final inversion (bar 4) puts the seventh in the bass, and now the intervals from top down are a sixth (F to D), fourth (F to B) and second (F to G). Again, we use the bottom two intervals for the notation V^4_2. Sometimes this notation is abbreviated further to V^2, but we won't use that in this course.

What if the chord is in open position so it spans more than an octave?

In the first chord here, we have notes F-G-D-B which are the notes of a dominant seventh chord, with the seventh (F) in the bass. The facts that the notes span more than an octave, and that the top 3 notes aren't in the same order as the V^4_2 chord in the top example are irrelevant; all that matters is what note is in the bass. Since it is the 7th in the bass, this chord is also labelled V^4_2 .

The same is true of the other two chords. They both contain the notes of the dominant seventh and have the 7th (F) in the bass, so they are both V^4_2 chords.

Progressions with inversions – 1

Triads

Because a triad has 2 possible inversions (third or fifth as root) and a 7th chord has 3 possible inversions (third, fifth or seventh as root), there are many more possibilities for progressions with inversions than with the tonic as root alone. However, some of these inversions occur much more frequently than others, so we'll look at the common ones to begin with.

This week we'll also try to use minor keys for our examples. [Side note: as there are 3 types of minor scale (natural, harmonic and melodic), we need to decide which one we're using. It's not specified in the lecture, but it looks like he's using the harmonic scale in which the 7th degree is augmented by a semitone.]

One problem with inversions is that the simple algorithms we had for voice leading in the root position often don't work well with inversions. We need to be extra careful to avoid things like parallel fifths and octaves and so on.

We'll start with a i-V-i progression with all 3 chords in root position, and then move the V to a V^6 (with the third in the bass, and a sixth between bass and top note). A first attempt is as shown:

To get the V^6, we have just moved the bass up to the third (C#). The problem with this is that we now have a parallel octave in the soprano and bass.

The preferred solution here is to shift the soprano up to get:

This gives us 2 fifths (soprano and tenor), a root (alto) and a third (bass). Two roots or 2 fifths are common in a V^6, but 2 thirds are not found.

Here we've used the rules from last week in that we have a common tone in the alto, the other voices move by a step, and we've included contrary motion.

For a different voicing of the same progression, we try the following:

In bar 3, we start with a different root form of the i chord and again progress to the V^6 by moving the bass to C#. This time we retain the common tone in the tenor and shift up the top two voices to the nearest chord tone. This gives us 2 roots (soprano and tenor) a fifth (alto) and a third (bass). It also includes contrary motion, so is a good solution.

Voice leading back to i is done by retaining the common tone in the soprano, shifting the alto and tenor down and the bass up. We've also avoided parallel fifths and octaves.

Seventh chords

We can also use first inversion with the seventh chord, as shown:

To get the seventh chord we keep the common tone (the root) in the soprano and move the bass to the third. Moving the alto by a step gives us the fifth (E), so to make the chord a seventh, we need to move the tenor to the seventh (G), which is also a single step, so things look OK. To return to i we just reverse the process.

A V seventh chord can be used in any situation where a triad is used; just be careful of voice leading issues.

Second inversions (fifth in the bass)

The V^6_4 is usually used between two i chords, as shown:

When using a second inversion of a triad, you must always double the fifth. In this example, we have a common tone in the soprano and can include the third by a single step from the alto D. Doubling the fifth gives us the E in the tenor.

It is common to progress from a V^6_4 to a i^6 as it provides a smooth transition for the bass (although going back to a i is also fine). We move the tenor to D rather than F since the latter would give parallel octaves.

A V seventh chord in second inversion is shown in the last 2 bars. The progression is the same except the tenor moves from F to G to give the seventh (recall that a V seventh chord in second inversion is a V^4_3 chord since the lower two intervals are used in the labelling). The return to i^6 involves doubling the third, which is fine here as it follows on naturally from the voice leading.

Progressions with inversions – 2

The passing 6-4 chord

The 6-4 chord (second inversion with the fifth in the bass) is used only in particular situations, unlike the 6 chord which can be used wherever it sounds good. In the last video we saw the V^6_4 used as a passing chord, when passing from i to V^6_4 to i^6.

A i^6_4 can also be used as a passing chord between iv^6 and iv, usually going down.

In D minor, the iv chord is G-Bb-D so iv^6 has the Bb in the bass. To progress to the root iv chord we can use the i^6_4 chord, with the A in the bass as shown. This provides a smooth transition in the bass moving down from Bb to A to G.

The pedal 6-4 chord

Here we hold the bass and tenor notes on the D while moving the top two voices. The term 'pedal' comes out of older organ music, where the bass (played on the pedals) is held while the upper voices add decoration. The pedal 6-4 is used most often with a i-iv-i progression as seen here.

The cadential 6-4 chord

When we have a V-i cadence at the end of a phrase, we can move the V over and precede it with a i^6_4, known as a cadential 6-4 chord. This serves as a decoration of the V.

The first 2 bars show an unadorned cadence from V to i. Bar 3 shows the addition of a cadential i^6_4 chord as an ornament. The tenor and bass are common tones with the V, and the voice leading in the alto and soprano goes by a single step to the V.

A cadential i^6_4 should always be at least as long as the V which it precedes.

In bar 1, the i^6_4 is shorter that its following V, so this should be corrected to either of the next two examples. In bar 4, the cadential i^6_4 is the same length as the V, while in bar 7 it's longer, which is also acceptable.

The i-iv^6-V^7-i progression

A variant on the i-iv-V-i progression uses a iv^6, as shown:

The V^4_2 chord

We can use a V^7 chord with the seventh in the bass, thus making it a V^4_2 chord, as part of a progression from V to i^6. The only note we've changed in the original V chord is the bass, from the root (A) to the seventh (G). Obviously, this sort of progression can be done only with a seventh chord, since we need the seventh as part of the chord.

Recall that the two tendency tones are the leading tone (C# here) and the fourth scale degree (G), and that the leading tone tends to the tonic while the fourth degree tends to the third (F). When a tendency tone is in an outer voice, it is exposed so needs to be voice led properly. For this reason, the V^4_2 almost always resolves to the i^6; the G in the bass must resolve to the F, which puts the i in first inversion.

More voice leading

When using inversions, the rules we developed for root position chords sometimes cause problems. Consider the following:

In going from i to V^6, we've followed the rule of moving by step, but this introduces a parallel octave in the alto & bass. Similarly, when leading back to the i we get a parallel octave again.

Doubled leading tones

Doubling the leading tone should be avoided, since the leading tone tends to resolve to the tonic and if both leading tones are thus resolved, we get a parallel octave.

Augmented seconds

Another example is:

This has a parallel octave in soprano and bass, and a parallel fifth in tenor and bass. It also uses an augmented second in the alto, which is avoided in harmonic minor. [Although no reason is given in the lecture for this, it seems that this is a stylistic choice for music of the classical era.]

Octaves and fifths by contrary motion

Here the tenor and bass start out in unison and move by contrary motion to an octave. Again, this is something to avoid for stylistic reasons.

Direct octaves and fifths

In this example, the soprano and bass start out as a third but both leap into a C, so they leap into the same octave.

In order to get a direct octave, the leaps must occur in the outer voices (so a leap to an octave with one outer and one inner voice, or two inner voices, is OK). Also, a direct octave occurs only if both soprano and bass leap into the octave, rather than one or both moving by step.

Summary of additional voice leading rules

- No parallel fifths or octaves

- No fifths or octaves by contrary motion

- No hidden (direct) fifths or octaves between outer voices

- Avoid the augmented second melodically (harmonically, that is, within the same chord, it is OK).

- Avoid doubling the leading tone.

Using inversions (A Demonstration)

Let's start with a simple i-V-i-iv-V progression in G minor with all chords in the root position (bars 1 to 3; I've corrected the V chord in bar 3 to avoid the parallel octave):

While the melody in the soprano is quite smooth, the restriction to root chords leads to a lot of jumping around in the bass. We can smooth this out using inversions.

In bar 4, we start with the bass an octave lower since we need more room. The first i chord is now in open structure, but still in root position. Putting the i chord in bar 2 into first inversion gives us a Bb in the bass which leads nicely into the C and D that follow. To progress from the i in bar 4 to the i^6 in bar 5 we can use a passing V^6_4, which makes the bass successive notes from the scale. We use a common tone (D) and move the S and T by step, which gives us the required doubled fifth.

To lead to the i^6, retain the common D and move the S and T by step again, giving a doubled root.

Going from i^6 to iv we have two possible common tones (the 2 Gs), so we retain the tenor and move the other voices by step giving a root position iv.

To complete the process we move from iv to V. Since there are no common tones and we want the melody to continue downwards, we move SAT to the nearest chord tones downward.

The overall effect is a smoother progression in bars 4 to 6, even though the harmony is the same as in bars 1 to 3.

We could also change bar 6 to include a cadential i^6_4:

Keyboard voicing

Most of the voicing rules we developed for vocal music apply also to keyboard music. The main difference is that for keyboard we have one note in the left hand (bass clef) and the other three in the right hand (treble clef). As a result, sometimes there is a very large gap between the bass and tenor, but this is fine for keyboard. Because of the limited span of the right hand, most chords are in close position.

Here's an example of a basic progression with a vocal voicing in bars 1 to 3 and a keyboard voicing in bars 4 to 6.

To build the keyboard version, we start with the i in root position and then a passing V^6_4 leading to a i^6. The V^6_4 is produced by single steps from the i, remembering that we must always double the fifth in a 6-4 chord. To go to the i^6, we could try D-Bb-G in SAT, but this produces a parallel octave in the bass and alto, so we use the solution shown even though the leading tone (F#) goes down to the D rather than up to the tonic.

To get the iv, we retain the common tone (G) and double the root in the bass and soprano, moving the tenor by a step to get the third. The cadential i^6_4 must have a doubled fifth which we get from the bass and tenor, by moving the latter by a step down. The third is obtained by moving the C down to the Bb. The leading from the cadential i^6_4 to the V must be as smooth as possible, since the cadential chord is a decoration on the V. Thus we retain the two common Ds and shift the top two voices down by a step.

Sometimes the vocal version will be found in keyboard writing if the composer is trying to imitate a vocal texture.

Creating accompaniment

Starting with the keyboard voicing we got in the last video, it is fairly easy to create an accompaniment by adding a rhythm to the bass and treble parts of the chord. Here's a simple example:

Each chord is introduced by the bass in the left hand and then the treble is repeated 3 times in the right.

A slightly more involved accompaniment breaks the treble chord up:

The same up-down pattern is repeated for each chord.

When analyzing a piece, we can apply the same process in reverse, that is, condense phrases into single chords. This is called textural reduction and we'll look at that next.

Textural reduction

If we look at a piece such as Schubert's impromptu op. 90 #4, at first it may appear much more complex than what we've seen so far:

The first 4 bars are from the impromptu itself. First we can do a harmonic analysis. The first bar contains a I chord if we're in Ab major, except that the notes are broken up and appear as a rhythmic pattern over the bar. The treble is just an arpeggiated I chord and the first two notes in the bass are part of this same chord. The only note that doesn't belong is the final G in the bass, which we could take as part of a I^7, but in classical music we never start with a seventh chord, so it's more usual to classify this G as a non-chord tone.

Similarly bars 2 to 4 contain IV, V and I chords, all in root position.

The textural reduction comes in condensing the notes in each harmonic segment into a single chord. This is done in bars 5 to 8, where we simply combine the chord-tone notes in the treble and bass, omitting the non-chord tones. Here the harmony changes once per bar, so each chord in the reduction occupies one bar.

Voice leading

When we texturally reduce the impromptu, we find some violations of our earlier voice leading rules. For example, there are a lot of parallel octaves. However, each chord is spread out over a whole bar, which here makes it acceptable.

In summary, textural reduction allows us to:

- see the harmonic progression more clearly

- see the harmonic rhythm more clearly

- see the voice leading more clearly

However, it should not be used to evaluate the quality of the voice leading.

Reversing Schubert

We can invert the textural reduction process so that we begin with a harmonic progression and then add rhythmic patterns to each chord to produce an accompaniment. If we begin with the same harmonic progression we've been using in our keyboard examples, we might get something like this:

The bass line follows a simple rhythm, and the treble is broken up into arpeggios in a manner similar to Schubert's impromptu. Once we've done this, we may want to fine-tune some of the arpeggios.

Circle of fifths progressions

We'll now look at some more involved harmonic progressions, beginning with the circle of fitfhs, which can be represented by the diagram we saw back in week 1:

Starting at I, we go to IV and then follow the arrows to the right, choosing the ii and V in the two boxes on the right. Thus each chord is down a fifth from the preceding one.

Voice leading in the circle of fifths

We use an alternating common tone/noncommon tone approach. Here's the progression in keyboard voicing, with all chords in root position:

There is a common tone (alto) between the first and second chords, third and fourth, and so on, but no common tones between second and third, fourth and fifth, and so on. The soprano and tenor notes follow the pattern of up a step then down a third.

Descending 5-6 progressions

When using the descending 5-6 (the 6-5 title in the video is an error) progression, the 'circle of fifths' chart isn't much use, since the progression doesn't follow this pattern.

The interval between the lowest and highest notes in a root-position triad is a fifth, but with a triad in first inversion (third at the bottom) this interval is a sixth. The 5-6 progression starts with the I chord and then follows the pattern I → V^6 → vi → iii^6 → IV → I^6 →. The general pattern is:

- Odd numbered chords are in root position, and drop by a third each time. Thus I down a third is 1 – (3 – 1) + 7 (using my little algorithm from earlier), which gives 6, then 6 – 2 = 4, 4 – 2 = 2 (so the next chord after the sequence above would be ii), and so on.

- Even numbered chords are always in first inversion, and are a fifth above the preceding root chord (although the actual notes in the chord will be lower than the preceding chord, it is formed with notes from a chord a fifth above the preceding chord). Looked at another way, each successive even chord drops by a third, so we can work out the next even chord from the previous one using the same algorithm as for the odd chords. Thus from the V^6 we go to 5 – 2 = 3 (so iii^6), then to 3 – 2 = 1 (so I^6), then to 1 – 2 + 7 = 6 (so vi^6) and so on.

The name '5-6' comes from this alternation between root triads (with a fifth interval) and first inversion triads (with a sixth interval).

Music notation

We'll use three-voice chords to illustrate the 5-6 progression.

Starting with the I C major chord, we move to V^6, keeping the common G and moving the other 2 notes by a step. The interval between bass and the lower treble G is a 12th , which is effectively a 5th after removing the octave. The interval between the B and G in the second chord is thus a 6th so we don't have parallel fifths.

To go from V^6 to vi in bar 2, there are no common tones so we begin by moving the bass by step. To fill in the vi chord, we could try moving the G and D up a step each. However, that gives us a parallel fifth and no third, so we sacrifice contrary motion and move these notes down to the nearest chord tone instead.

To get the iii^6 we put the G in the bass, keep the common E and move the C by step to a B. To go from iii^6 to IV in bar 3, again there are no common tones so we move the bass by step and the other 2 notes to the nearest chord tones. Then on to the I^6, again retaining the common tone.

Up to this stage, the chord progression mirrors Pachelbel's Canon, so we can carry on to complete it. Returning to the IV in bar 4 is just the reverse of the IV to I^6, but then we want to go from IV to V. We must avoid parallel fifths here, so we can't just raise the top two notes of the IV to get the V, since these notes are a fifth apart. So we use contrary motion as shown. This gives a doubled root plus a third, which is acceptable here. With 3 voices we might not always be able to get all 3 components of a triad.

Finally we return to I. Again, we have a doubled root plus a third. We could instead use E and G in the treble to get all three triad notes, but it's a matter of taste as to which sounds better. (I prefer the C and E.)

Parallel sixth progression

The parallel sixth progression isn't really a progression in its own right; it is more of a way to move from one chord to another chord that may not be directly related to the first one. Suppose we want to get from a vi^6 to a cadence to end a phrase. We can use a descending parallel sixth progression as follows:

The series of descending chords are all in first inversion (hence the 'sixth' progression), and can carry on for as long as needed to reach the desired final chord. Here we want to get from the vi^6 to the ii^6 which leads into the V-I cadence.

This example is slightly flawed in that there is a parallel octave in bar 3 between the bass and tenor. We can fix this by shifting the soprano D down an octave (remember that parallel octaves are a problem only if they occur between the same two voices; here the upper F and upper G are in different voices after the switch).

Analysis: Mozart, Piano Sonatas K. 545 & K. 332

K. 545

We'll now look at some actual music to illustrate some of the progressions we've been studying. We'll start with an excerpt from the first movement of Mozart's piano sonata K. 545 in C major:

The bottom two staves show the textural reduction. Bar 2 is actually the same harmony as bar 1, except moved down a step, except the last treble note in bar 2 is raised an octave. This shift by an octave allows Mozart to raise the voicing in general, which shows up in bars 3 and 4.

The same pattern shows up in bars 3 and 4. The first chord in bar 3 is one step down from the first chord in bar 2, then raised an octave, and the second chord in bar 3 is the second chord in bar 2 lowered by one step.

The harmonic analysis shows alternating first inversions and root-position chords. As a reminder, here's the circle of fifths chart again:

We see that the progression of chords in the sonata follows the circle of fifths exactly, with the addition of the alternation between first inversions and roots.

K. 332

Now another Mozart sonata, this time in F minor, using seventh chords:

We start in i, then go to iv^7, omitting the fifth in this chord (recall that if you omit a note from a seventh chord, it should be the fifth). Then we move to a VII^7, again omitting the fifth. This is a major seventh as opposed to a diminished (still awaiting clarification on why this is needed to ensure a transition to the following III^7). In bar 2, the top note is the third and next note down is the seventh. These two notes switch roles in the VII^7, then switch back again for the III^7. This pattern continues all the way through the progression.

From the VII^7, the circle of fifths continues using seventh chords until we reach the V^7. At this point, he uses a deceptive progression, which we'll get to later.

Analysis: Beethoven Op. 109

An example of a 5-6 progression is found in Beethoven's piano sonata Op. 109, third movement. The opening bars look like this, with a textural reduction underneath:

The harmonic rhythm is one beat per chord, and we can see the 5-6 progression by looking at the interval between the lowest and highest note in each chord, which alternates between a fifth and a sixth. We can also see that the chord progression is exactly the 5-6 progression (by doing the roman numeral analysis), up to the first chord in bar 4. At this point, Beethoven stops the progression and switches to a V^7 chord to give a cadence that ends the phrase.

The actual piece is built up by breaking up each chord according to a constant rhythmic pattern.

Analysis: Mozart, K. 279

As an example of parallel sixths, here is an excerpt from Mozart's piano sonata K. 279, third movement:

The harmonic rhythm here is one chord per half beat. The textural reduction shows that all the chords in the first 3 bars are in first inversion and serve to step down from the starting to the ending chord.

In bar 4, there are only a C and an E, but these suggest a IV chord with the fifth (G) missing. The next chord contains 2 Ds and a B. This is an unconventional form of a I^6_4, missing the tonic (G). Better voice leading from this to the following V would be to have the treble with G and B, but because the preceding melody has C and E, it sounds better if there is no jump between that and the I^6_4, although this is a matter of taste; the jump could come here rather than between the I^6_4 and the V, as Mozart has written it.

Analysis: Mozart, K. 533

We now look at another circle of fifths example, but the main objective is to introduce non-chord tones (NCTs) and polyphony.

The example is another Mozart sonata:

If we compare this to the previous Mozart example, we see that the voicing is very similar:

This illustrates that by using different rhythmic patterns, we can achieve very different textures from the same harmonic progression.

Non-chord tones (NCTs)

To get the textural reduction in K. 533, the first bar contains only chord tones. The bass contains only chord tones throughout so we can fill that in first.

Looking now at bar 2, we see the bass suggests an Eb dominant seventh, but if we look in the treble, the four eighth notes in the top voice belong to this chord, except for the F, which is a non-chord tone, so we omit it from the textural reduction.

Looking at the remaining harmonies, we see that the pattern repeats itself in each case, with the second in the group of four eighth notes being a non-chord tone that we omit from the textural reduction. These NCTs are used to decorate the chord tones.

Polyphony

There are 3 voices in this example: the bass and two treble voices. The bass has a rhythm independent of the top 2 voices. Similarly, each of the top two voices has a rhythm that is independent of the other, yet each can be clearly heard. This independence of individual voices is the essence of polyphonic writing.

Non-chord tones

Within a single harmonic beat, a non-chord tone or NCT is any note which does not belong to the chord of that beat. Although in principle, any note outside the chord could be present as a NCT, in classical music there are only a few standard types of NCT that are used. You should always check your compositions to ensure that you use only these types of NCT.

The use of a non-chord tone is determined by how you get into it and how you get out of it. We can step into a NCT or skip (jump by a third or more) into it. Getting out may be done by step or skip, and can be in the same direction or the opposite direction.

Passing and neighbo(u)r tones

A passing tone (labelled 'p' in the diagram) goes in and out by step, and proceeds in the same direction. A neighbour (I'll use the Canadian/British spelling!) tone (labelled 'n') also steps in and out, but the direction reverses.

Escapes and appoggiaturas

In bar 2, we step into a NCT and then skip or leap out of it, since the interval from Bb to G is a third. This type of NCT is an escape NCT. Although the leap out can in principle be any size, typically we step up into the tone and then skip down a third to get out of it.

The appoggiatura is in a sense the inverse of an escape, in that one skips into it and steps out of it.

In these cases, the directions of stepping and skipping doesn't matter.

A leap on both sides of a NCT is possible, but is very rare, so we won't consider it further.

Anticipations

If we step or skip into a NCT then stay the same to get out, the NCT is known as an anticipation.

Suspensions

In the second half of bar 3, we start with an F major chord. The two eighth notes labelled 'p' are NCTs and satisfy the rules for passing tones. However, we're moving to a C chord in bar 4, so the first F there is a NCT, which resolves down by step to the E. This is a 4-3 suspension, since it starts with a fourth and resolves down to a third. There are a few other types of suspension such as 9-8 and 7-6, but it is important that the resolution is always down by a step. Sometimes other notes appear between the two notes of a suspension; this is known as a delayed resolution, but we won't study these here.

The three notes making up a suspension all have names: the note leading into it is called the preparation, the NCT itself is the suspension and the note following is called the resolution.

Pedals

Finally, in bar 5 we see a pedal NCT. The A in the bass is part of the initial F major chord, but the E diminished chord that follows does not have F as a chord tone so the sustained A in the bass becomes a NCT for this second chord. Such a sustained or held note, usually in the bass, is called a pedal NCT.

Summary

When writing classical music in the style of the late 18th-early 19th century, it is important that you use only the NCTs described here. Other types of music do use different types of NCT, but they will not fit into the classical style.

The main types of NCTs are shown in the table:

Type

Enter

Exit

Direction

passing

step

step

same

neighbour

step

step

opposite

escape

step

skip

either

appoggiatura

skip

step

either

anticipation

step or skip

hold

either on entry

suspension

hold

step down

always down on exit

pedal

hold

hold

n/a

Although there are a few other NCTs that are used occasionally in classical music, for now you should ensure that every NCT used in your compositions is one of these types.

Using NCTs with the circle of fifths

We'll now look at transforming the circle of fifths progression into a 3-voice phrase using polyphony and non-chord tones (NCTs). The circle of fifths is:

Using passing tones

The first job is to transform this four-voice progression into 3 voices. In the treble, the pairs of thirds alternate between the top two and bottom two voices. So we can try projecting the two elements in the third into a single voice, tied together with a NCT.

Each NCT here is a passing tone, since the 3 notes move by step and in the same direction.

Using suspensions and pedals

With the same bass line, we can try a different pattern in the treble, this time using suspensions.

The I chord is broken up into only chord tones, but the IV chord contains a NCT G in the upper voice, which, since the preceding note is the same and the following note is a step down, is a suspension.

Recall that a suspension starts with the preparation note, leading into the suspension NCT itself, followed by the resolution. The suspension here is a 9-8 suspension, as the interval between the suspension note and the bass is a 9th resolving to an 8th. It's recommended that only certain types of suspension (9-8, 7-6 and 4-3) are used, and these only in the upper voice.

The middle voice uses 2 passing tones in the descending eighth note passages, and the first four bars are copies of the initial phrase, transposed down a step on each repetition.

In the bar 4, the pattern changes on the third beat. Here we want a V so the A is a passing tone, with all other notes being chord tones. Bar 5 should resolve to a I, so the first four notes follow this pattern with the D-F being passing tones in the treble. However, we've added a couple of extra chords to give some variation to the finish. The second chord in bar 5 is a viio which makes the third note a passing tone. The bass C is not part of the viio, and it is held from a preceding C and resolves to another C, so this is a pedal NCT. The final progression in bars 5 and 6 is thus I-viio-I.

Using NCTs with the descending 5-6 progression

We can apply similar techniques to the descending 5-6 progression. Here's the harmonic progression:

Starting with the bass, we can try a simple rhythm as shown:

To work out the treble, we can again use a four-note pattern for each chord. To figure out what these patterns could look like, it is often helpful to look first at the interval you want to span, and then work out a note sequence that fits into this interval. For example, the middle voice needs to get from G in the V^6 to E in the vi. so one four-note pattern that achieves this is as shown.

This pattern contains two consecutive passing tones. Although this doesn't fit the strict rule for a passing tone with a chord note on either side, this sort of pattern is quite common especially in scale sequences.

If the pattern had approached the E in bar 2 by using an F at the end of bar 1, we'd get parallel fifths at the end of each bar when this pattern is repeated.

At the end of bar 3, we enter a IV-V-I cadence, so we can't continue the same pattern. We need to get to a IV at the start of bar 4, so if we have an A in the top voice and an F in the bass, we'd like to have a fifth (C) in the middle voice. We alter the four notes in the middle voice at the end of bar 3 to arrive at a C. This uses an appoggiatura (skip in to the F, step out to the E) and a passing tone (D).

In bar 4, we start with the IV and use one passing tone (B), then proceed to the V using a neighbour tone (step up to the A, then step down), then a passing tone to arrive at the final I.

Using NCTs with parallel sixths

The parallel sixths progression is:

The Mozart example earlier is a simple projection of the parallel sixths progression:

Starting with the basic progression, we can use a simple bass line and keep the top voice the same. We can then add decorations from the thirds in the middle voice.

The pattern consists of eighth-16th-16th-eighth, with a passing tone at the third note. In bar 3, we need to make the transition from ii^6 to V, but if we kept the same pattern at the start of bar 3, the middle voice would move from F to G in parallel with the bass, so we'd get parallel octaves. To fix this, we alter the pattern a bit, then add a run of 16th notes to make the final transition from V to I. This scale passage has a neighbour and a couple of consecutive passing tones.

Diatonic substitutions

At the start of the course, we introduced the chart of harmonic progressions:

The two boxes near the end show that we can substitute a ii for a IV or a viio for a V. These are examples of diatonic substitutions. A diatonic substitution is a substitution of one chord with another chord from the same scale.

Substituting viio for V

Let's start with a basic I-V-I progression and see what happens if we substitute a viio for the V. Here's an example in D minor:

In bar 3, we compare the V and viio. They have two common tones (C# and E). If we bring the top G down an octave (bar 4), the A in the V chord is only a step away from the G. If we add in an A to the viio6_4, we get a dominant seventh chord in third inversion (with the seventh at the bottom), giving a V^42. , which has three notes in common with the V.

In bar 5, we use the V^7 in the progression, but we can substitute a viio for this chord by changing the bass so that it's no longer the root of the V^7. Bar 7 shows a viio in root position, which is not often found because the double leading tone (C#) forces the voice leading to resolve to the tonic (D) in the i chord, giving rise to parallel octaves. Usually this chord is found in first inversion instead (bar 9). Although the progression to the root-position i is found, it is more common to resolve the viio6 to a i^6 as in bars 11 and 12.

Substituting viio7 for V^7

The substitution above also works for seventh chords. Again in D minor, we have

Comparing the two chords in bar 1, we find three notes in common and, if we shift the top Bb down an octave, it is a semitone away from the bottom A in the V^7. Thus the only difference between a V^7 and a viio7 is this one semitone shift in a single note.

In bar 2, we have a i-V-i progression, but this time the V is a seventh chord in first inversion (root note A, so first inversion gives a note order of C#-E-G-A. The interval C# to A is a sixth, and C# to G is a fifth so it's a V^6_5). We can now substitute a viio7 for the V^6_5 as seen in bar 4. The only change is the shift of the A in the V^6_5 to the Bb in the viio7.

Diatonic substitutions: ii for iv

We refer back to the chart of harmonic progressions:

Just as we can substitute the viio for the V, we can also substitute the ii for the IV (in major) or the iio for the iv (in minor).

Starting with the i-iv-V-i in D minor, we see how the substitution works:

Comparing the iv with the iio6 triad, we see there are two tones in common and the other note is a single step from one triad to the other. The only change between bar 1 and bar 3 is the D in the iv changes to an E in the iio6.

The iio6 is also commonly found followed by a V^7 as shown in bars 5 and 6. The only change from bar 4 to bar 6 is the A in the V is moved to the seventh (G) in the V^7.

We can also make the ii chord a seventh, as shown in bar 7, where it appears in first inversion, making it a iio6_5. The change from the iio6 to the iio6_5 involves changing 2 notes: the E down a step to the D (to give the seventh) and G to the E (to give the root).

Another progression allowed by the chart at the top is I-vi-ii-V, so we can insert a VI (the VI is major in a minor key) between the i and the iio6_5, as shown in bars 9 to 11. Only one note changes in each of the steps i-VI-iio6_5. All these patterns are found in classical music, so it's a matter of artistic taste which one is used.

Diatonic substitutions: V-vi progressions

The two diatonic substitutions we've seen so far have both been between chords that are a third apart (viio for V and ii for IV), meaning that the corresponding triads always have two notes in common. It is not acceptable to substitute any chord with another chord that is a third apart, however. But there is one final substitution that is used.

Bars 1 and 2 shows the standard V-i progression. A third down from i is the VI chord, so we'd like to try a i-VI substitution.

Bar 3 compares the i and VI chords. As with the other substitutions, there are two notes in common. This time, the note (A) in the i that has to change is the root of the V. A first attempt is shown in bars 4 and 5, where the VI chord is in root position. We've written in the bass Bb first, then by applying the rules for voice leading, we should move the A in the iV chord to the nearest chord tone in the VI, which gives us the upper Bb. However, this now produces parallel octaves between bass and tenor, and parallel fifths between bass and soprano (and tenor and soprano, for that matter).

One possibility is to treat the V-VI like a IV-V transition (see 01.05.04). That is, we raise the bass and lower the other 3 notes to the nearest VI chord tone (bars 6 and 7). This works in a major key, but in minor we get an augmented second (Bb-D; see 02.02), which is avoided in harmonic minor keys.

A common solution is to double the third (bars 8 and 9), which is actually acceptable in minor keys.

If we use a V^7 and wish to progress to a VI (bars 10 and 11), we have two tendency tones (01.05.03) ; the C# resolves to a D and the G resolves down to the F. Thus a doubled third is the best solution here as well.

The ii and IV chords both play the role of subdominant, and the viio and V chords are both dominant, so these substitutions are both fairly subtle changes. The VI for i is fundamentally different in that a VI chord can never act as a tonic chord. The V-VI progression is known as a deceptive progression. It doesn't have the sense of closure that a V-I does.

Cadences

A cadence is a harmonic progression that concludes a phrase.

Half-cadences

A half-cadence is a progression that ends a phrase but does not sound like it can end a piece.

Typically, the half-cadence is a progression that ends on a V.

Phrygian half-cadences

A variant half-cadence that is found only in minor is the Phrygian half-cadence, shown in bars 3 and

4 above. It is also a i-iv-V progression, but the iv is in first inversion, so the interval between the iv

and V bass notes is a semitone. It is the lowered sixth scale degree (the Ab) dropping by a semitone

to the root of the V that is characteristic of the Phrygian half-cadence. This is why it can occur only

in minor, since in major, the interval between the sixth scale degree (A for C major) to the root of

the V is a full tone.

Perfect authentic cadences

A phrase ending with a V-i progression can end a piece, as shown in the example below (bars 1 and

2):

This is called a perfect authentic cadence, and must satisfy two properties:

1. Both the V and i chords must be in root position.

2. The melody note in the i must be the first scale degree (that is, the tonic).

Variations in which one of these conditions is violated are called imperfect authentic cadences and are shown in bars 3 to 6. In bar 3, the V is in first inversion, while in bar 6, the melody ends on the

third scale degree. In classical era music, virtually all pieces end with a melody note in the first

scale degree.

It is also acceptable to substitute a viio for the V as shown in bar 7, although this gives an imperfect

authentic cadence. To get a perfect authentic cadence, the chord preceding the i must be a V and the

cadence must obey the two rules above.

Deceptive cadences

By substituting VI for i we can produce a deceptive cadence (bars 9 and 10). The final VI has a

doubled third, but although this isn't ideal, it's the best voice-leading solution in this situation.

Analysis: Beethoven Op. 2 #3, ii

As an example of chord progressions and cadences, we'll have a look at the second movement of Beethoven's piano sonata, Op. 2 #3 (the opus 1 shown in the video is wrong) in E major.

It's a good idea to begin the analysis by actually listening to the music, but unless your ear is well-trained, it can be difficult to pick out the different types of chord.

Beginning with the roman numeral analysis, we get the chords as shown. Note in particular that the V^4_2 in bars 1 and 3 resolves to a I^6 (see 02.01.02). The leading tone (D#) in the V^4_2 resolves to the E in the I^6 and the fourth scale degree (the tendency tone A) resolves down to the G#.

In bar 3, we have a I-ii^6-V progression, with the IV substituted by the ii^6. The ii^6 is a subdominant and the V is a dominant.

The two 32nd notes in bar 4 are both members of the same chord.

The phrase ends with a ii^6 -V, which isn't a conclusive finish, so this is a half-cadence.

The only NCTs occur in bars 1 and 2, and consist of a neighbour (n) and escape (esc) in each case.

Period form

A period form is a collection of phrases that is closed by a perfect authentic cadence (PAC), possibly with other types of cadence in the middle. It is the shortest building block of a piece.

Parallel period form

The parallel period form contains two phrases which are almost identical; the main difference being that the first phrase ends with an inconclusive cadence while the second ends with a PAC.

A good illustration is Schumann's The Wild Horseman from Album for the Young:

The first phrase ends in bar 5 with a transition from i to V, so this is a half-cadence.

There are no NCTs in this piece, and the harmonic analysis is fairly simple. We are in i up to bar 3, where there is a iv^6_4. This is a pedal 6-4 chord (02.01.02). Originally, we said that the pedal 6-4 held the tenor and the bass, but here only the bass is held.

In bar 4 we shift to V with a i^6_4 in the middle. Although we saw the i^6_4 originally used as a cadential 6-4 (02.01.02 again), here it is not part of a cadence and, since the bass note is held, it is another pedal 6-4.

Bar 6 to the first 2 beats of bar 9 is just a repeat of bars 1 to 5. The last 2 notes change to a V-i, which is a PAC.

Using a deceptive cadence

We can add a bit to the second phrase by putting in a deceptive cadence before the PAC. This is done by duplicating bars 7 and 8, and changing the final i in bar 8 to a VI:

This gives a deceptive cadence in bar 8, which when performed would probably be followed by a brief pause before resuming with the final two bars, which now end with a PAC.

The two bars following the deceptive cadence is not a new phrase; rather it is just an extension of the current phrase. The bars following the DC are usually a repeat of what comes before the DC, except they end with a PAC.

This doesn't destroy the parallel period form; it just modifies it.

Melodic writing

Melodic writing is similar to what we've done so far in creating a polyphonic piece from a basic harmonic progression. We'll start with a basic two-voice counterpoint:

This suggests a harmonic progression as shown. The first 4 bars are a common way of opening a piece in classical music. The ii^6 could also be a IV, but the ii^6 is more common. This relatively simple progression was used frequently by classical composers.

Step 1: Elaborate the melody

We can start by applying a rhythm to the treble. The melody and rhythm should be designed to link neighbouring notes.

We use a neighbour note in bar 1 and a passing note in bar 3 to provide the links between the main tones. A good rule (although it's not standard) is the rule of two: in a melody, do it twice and move on. So after the the first four bars we should try to find something different to continue.

For the next bit, we'd like to spring into the opening G in bar 5, and then into the A in bar 6, starting from the C in both cases. The eighth-note passages serve as links between the two main notes again.

The next idea is to reach a climax with a high E in bar 7. To get there, we start with a leap from the closing D in bar 6 to a G in bar 7, and then use an arpeggio to reach the high E. Then a descending scale is used to reach the D on the third beat in bar 8, where we use the eighth-note motif from bars 5 and 6 to lead into the closing I chord.

We can now split the staves into a melody for an instrument such as an oboe with a piano accompaniment.

We can start the accompaniment by filling in the chords for the piano by referring to the harmonic analysis. We keep the piano chords lower than the oboe's melody in order to retain a separation between the two.

The voice-leading between bars 5 and 6 violates a rule from our original vocal writing since the low G moves up to a D (with the C kept as a common tone and the upper G moving by step to the A), since the bass and tenor voices cross (or, if you're viewing it as a shift from G to A, C to D, and G to C, the low G doesn't move to the nearest chord tone). However, for keyboard writing this is fine.

We can impose a rhythm on the accompaniment. Here we do something fairly simple, although a more elaborate accompaniment (with NCTs and something of a melody) is also OK.

A few alterations have been made:

- Octaves in the bass, to add some depth when played on the piano.

- Lowered the treble chords in bars 1 and 4.

- Lowered the oboe's eighth notes in bar 3. This changes the passing tone to an anticipation (03.08 step into the NCT and stay the same getting out).

- Used four-note chords in bars 7 and 8.

All these changes are a matter of personal taste; they're not required by any of the rules we've seen so far.

We now want to add an extension to the end by using the last 4 eighth notes in bar 8.

In bar 9, we change the piano's role from that of an accompaniment to a partner in a duet. We use the eighth-note motif separated by a chord tone between the oboe and piano.

Retardation suspensions

To finish in bar 13 we use a retardation suspension. Between the bass and the oboe there is a 9-8 suspension, which resolves down to the tonic as usual. It's still valid to call this a suspension even though the note immediately preceding the D isn't a D itself. In effect, we're holding over the V^7 chord from the previous bar, and that had a D in the oboe part to open bar 12. Thus the D in bar 13 effectively holds the D from the beginning of bar 12.

The piano's treble part, however, resolves upwards, which violates the rule for suspensions. This is a special kind of suspension known as a retardation, which is common in late 18th-century music.

Chromatic substitutions: the Neapolitan 6 chord

All substitutions we've seen so far have been diatonic substitutions, meaning that one chord belonging to the key in which we're writing is substituted by another chord from the same key. A chromatic substitution is a substitution with a chord not in the same key.

A common example of a chromatic substitution is the Neapolitan 6 chord. To see how this arises, start with a i-ii-V-i progression in F minor:

The ii chord is in first inversion and is a diminished chord. If we lower the root of this chord by a semitone, we convert it into a major chord but, since this new root (Gb) is not part of the scale of F minor, the resulting chord is not a chord from the F minor scale; it is a chromatic substitution. This chord is called a Neapolitan 6 chord and is denoted by N^6. This notation emphasizes the fact that the chord is not from the key being used.

The diminished third

The top note in the N^6 in bar 3 above leads to an E natural in the V chord; this is a diminished third which, although it is technically the same interval as an augmented second (which we're supposed to avoid in minor keys) is actually acceptable when dealing with a N^6.

Using a i^6_4 to resolve a N^6

One way to avoid the augmented second/diminished third is to insert a i^6_4 between the N^6 and the V, as shown in bars 5 and 6. (Although it's not stated in the video, this looks like a cadential 6-4 chord.)

Analysis: Schubert's Der Müller und der Bach

An example in which a Neapolitan 6 chord is found is a song by Schubert:

The piece begins in G minor with a i chord. In bars 4 and 5, the notes suggest a V chord. The low G is being held over from the preceding i and continues to bar 6, when it is again part of a i. This G is therefore a pedal NCT, and the harmony in bars 4 and 5 is V.

Bar 8 contains the N^6, since the Ab is the root of the iio lowered by a semitone, and the C in the bass is the third of the iio, so this is a iio6 with the root lowered, which is the definition of the Neapolitan 6. The diminished third is used from the top Ab in bar 8 to the F# in bar 9, but that's acceptable here.

Ordinarily the iio6 would lead to V for the final cadence, and that's what is happening here with the iio6 substituted by a N^6. The phrase ends with a PAC.

The E in bar 5 is a neighbour NCT since we step in going up and step out going down. There are several passing tones as marked on the score.

Chromatic substitutions: augmented sixth chords

Back in 04.04 we looked at the phrygian half-cadence, which consisted of i-iv^6-V, and is found only in minor keys (bars 1 and 2):

In the Neapolitan 6, we lowered the root by a semitone. If we now raise the root of the iv^6 in the phrygian half-cadence by a semitone, we get an augmented sixth, as seen in bar 3.

The augmented sixth is always seen with the sixth scale degree (Db here) in the bass. The name 'augmented sixth' comes from the interval between the sixth scale degree (the Db) and the augmented root (B natural).

Types of augmented sixth

There are 4 types of augmented sixth; we'll look at 3 of them here. The one shown in bar 3 is the Italian augmented sixth.

The German augmented sixth is shown in bar 5, where we move the F from the Italian to an Ab. This chord contains four notes: bass (Db), third above the bass (F), fifth (Ab) and the augmented sixth (B). In terms of actual notes, the German augmented sixth is the same as a dominant seventh, although not from the same key. The notes Db-F-Ab-B (or C#-F-G#-B) form the dominant seventh in the key of Gb major.

The French augmented sixth is obtained from the Italian by raising the F to a G, so we now have the bass (Db), third (F), fourth (G) and augmented sixth (B).

Voice leading

In bars 5 and 6 above, we have parallel fifths in the bass. We can avoid this by inserting a cadential 6-4 (bars 1 and 2 below).

In general, the lowest note (root) in an augmented sixth (the Db here) always resolves downwards by a semitone; the top note (the augmented sixth; B here) always resolves upwards by a semitone. That is, the top and bottom notes always resolve outwards into an octave.

This explains why the augmented sixth functions differently from a dominant seventh. If we treated the augmented sixth as a V^7, we would need to be in the key of Gb (bar 3, with 6 flats). The V^7 chord here is enharmonically equivalent (has the same notes, but with different names) as the Ger+6 in bar 1. We can't resolve the lower Db or the upper Cb to a C (which would be required if it were being treated as an augmented sixth), since C isn't a note in the Gb major scale. Instead, we'd need to resolve the leading tone (F) to the tonic (Gb). The fourth scale degree (top Cb) resolves to the nearest chord tone (Bb) and the two notes in the bass can be resolved as shown.

With the chord respelled to be a Ger+6 (bar 4), we are back in F minor and can resolve it appropriately.

How a chord is spelled influences how they are voice-led and how they function.

Analysis: Beethoven's bagatelle, op. 119 #1

In this example, we'll study the harmonic analysis of a Beethoven bagatelle:

We start by looking for the phrases. There are half-cadences in bars 5, 9 and 13, and a PAC at the end in bars 16 and 17. Because there are two phrases up to bar 9 and these two phrases are essentially repeated in the second half, this is a double parallel period.

Harmonic analysis of the first and third phrases

Analyzing the harmony in bar 2 is quite tricky. The initial chord (Bb and 2 Ds) looks like it might be a Bb chord, which would be a III. However, composers of Beethoven's period would not normally start a piece with a III chord, so it makes better sense to regard this as a i^6, even though there is no root tone (G).

The second beat in bar 2 is also difficult to analyze, so we'll come back to it later.

The third beat in bar 2 is a i, and in bar 3 we open with a V^6.

In bar 4, the first chord is a i. The second chord is a v^6, that is a minor five chord in first inversion, but instead of the usual F# as the third, we have an F natural, which makes the chord minor. The third beat has the sixth in the bass, and a G for the third and a C# for an an augmented sixth. This is an augmented sixth chord with 3 distinct notes, which is an Italian augmented sixth.

Returning to the second chord in bar 2, we could regard it as a passing chord. (The only reference to passing chords that we've had so far is the passing 6-4 chord in 02.01.02, but this chord isn't a 6-4 chord. Wikipedia defines a passing chord as a nondiatonic chord, but this doesn't seem to satisfy that either, as the three notes it contains belong to a V^4_3. I'm guessing that he's calling it a passing chord since it connects two more mainstream chords.) The V^4_3 chord here isn't complete as it's missing the third, but the other 3 notes fit.

The progression in bar 4 is a common harmonic progression though often in the form i-V^6-iv^6-V. Here the iv^6 is substituted by the It+6.

Bars 10 through 13 duplicate this phrase so the analysis is the same.

The opening notes in bar 1 can be considered a harmonic extension of the i^6 at the start of bar 2, so that makes the Eb a neighbour tone.

The two notes at the end of bar 3 belong more to the harmony in bar 4 than as an extension of the V^6 in bar 3, so it makes most sense to label the C as an appoggiatura.

Harmonic analysis of the second phrase

Bars 2 and 3 use homorhythmic homophony, but the second phrase, starting in bar 5, uses two to one polyphony. To analyze this phrase we need to consider each pair of eighth-notes together with the bass note accompanying it.

The last two notes in bar 5 together with the first two eighth-notes in bar 6 and the bass note all form part of the V. The second beat in bar 6 contains notes from a V^4_2 but missing the fifth.

In bar 7, we start with A-C-D which is a V^4_3 missing the third (F#). Then we have F#-A-C which is a V^6_5, this time missing the root. Then we take the low G and the Bb as the chord tones to give a i, and regard the C as an appoggiatura.

In bar 8, we use the low C and the A as the chord tones giving a iio6. The Bb can be regarded as a NCT carried over from bar 7, so it becomes a suspension. The bass note is a C and the suspension note is Bb, resolving down to A, so this is a 7-6 suspension.

The third beat of bar 8 is a secondary dominant, which we'll get to next week.

Harmonic analysis of the fourth phrase

The analysis of the fourth phrase parallels that of the second up to the beginning of bar 16, where the bass breaks up into 2 eighth notes. The Eb in the bass can be regarded as just a decoration rather than a chord tone, so we still take the C and the A as defining the chord.

The second beat of bar 16 may look like a V if we use the D and A as chord tones and the G as a NCT. However, looking at the pattern of the melody we can think of the A as being carried over from the preceding A so that we can use the D and G as the chord tones, giving a i^6_4. Continuing this pattern, the G in the third beat becomes a suspension so the chord tones are the low D and the F#, giving a V.

It's interesting to note that there are probably differing opinions on how to analyze this bar.

Chromatic substitutions: borrowed chords

A couple of examples of borrowed chords, this time in F major.

We start with a pedal 6-4 chord (02.01.02) in bar 1, with the IV being a major chord since we're in a major key. We can change it to a minor iv in bar 3, by changing the third from D to Db. This is known as borrowing from the parallel mode. There are only two modes (major and minor) used in classical music of this period.

In bar 5 we have a deceptive progression (04.03) of V^7 to vi. A deceptive progression has a different character in minor. In bar 8, we put in the VI chord from the F minor scale (paradoxically, the vi chord in F major is minor and the VI chord in F minor is major). This is still a deceptive progression but delivers a bigger surprise in minor.

Picardy third

Another example of mode mixture is as follows:

Bars 1 and 2 are in D minor, but in bar 4 we change the final i chord to a major I by raising the third (F) a semitone to F#. This third is known as a Picardy third, and is more common in baroque music than classical.

There are many other chords from the other mode that can be borrowed, but these ones are the most common.

Simple 2-voice counterpoint

Counterpoint is a large subject on its own, so we'll look only at a basic introduction in this course.

We begin by writing a bass line. We have two goals:

1. Write a treble line in one-to-one counterpoint, that is, a treble line consisting of quarter notes in the same rhythm as the bass. Here, we use chord tones only.

2. Then we'll elaborate this to provide two notes in the treble for each quarter note in the bass. Here we use both chord tones and NCTs, with a good mix of both.

We need to ensure that, even though we have only two voices, we can still hear the harmonic progression.

One-to-one counterpoint

We'll start in B minor with the bass line shown:

A first inspection shows that the bass line rises by step up to bar 3, then jumps down by an octave. Then there is a deceptive progression (the V-VI). There are i^6_4-V progressions in bars 3 and 5, and a PAC in bars 5 and 6.

Now we can add in the soprano part. Normally, we'd start the piece on the tonic, but here we'll use another common progression. Starting with the third (D), the first chord is still i, since we have the root in the bass and the third in the treble. By descending by steps we get to the i^6 in bar 2, where we again have the root and third. The intervening V^6_4 is a passing 6-4, and in chords like this, it is ok to just hint at the nature of the chord. Here we have only two C#s so we have only the fifth of the chord. We can also keep in mind that when we elaborate this to two-to-one counterpoint, we can use the second note to introduce another note from the chord.

The iio6 in bar 2 adds in the root (C#) and so moves in parallel sixths from the i^6. In bar 3, we move downward again by putting in the root (B). This B is a fourth above the F# and is a tendency tone, so we resolve it down to the A#.

In bar 4, we have little choice but to move up to the B, since going down to G would create an augmented second.

We've been moving down and up, and proper counterpoint needs more of a sense of direction, so we need to establish more of a line to the melody. However, we should never repeated consecutive notes, since this has no direction in the melody.

When writing one-to-one counterpoint, remember the rules from vocal writing back in week 1; avoiding parallel fifths and octaves, etc. Also avoid repeated notes and ensure you can still hear the harmonic progression.

Two-to-one counterpoint

We can now elaborate the one-to-one counterpoint to produce two-to-one counterpoint. At each stage we need to consider whether to use a chord-tone or a NCT. To some extent this is determined by what chord tones are already present in the one-to-one. In bar 1, the initial i already has a root and third, so we can put in a NCT (E) as an escape tone, since it steps in and leaps out.

For the V^6_4, we need to add in another chord tone since we started with only two C#s, so we use an A# as the extra note.

Bar 9 opens with a 5-4 suspension. Here it's ok to repeat the C# from bar 8, since in this case the first note is the anticipation.

Having decided to change the closing tonic to an octave lower in bar 12, we need to work out how to get there. It's a bad idea to have a long scale passage without any interruption, so we break up the downward run in bar 10 with an escape NCT before resuming the downward scale in bar 11.

The passing tone in bar 10 is an A natural, rather than A#. This arises from the melodic minor scale, in which the sixth and seventh degrees are raised on ascending and lowered on descending.

Review

When writing the one-to-one counterpoint, we should check:

1. All notes are chord tones.

2. No voice-leading problems.

3. No repeated notes.

4. Have a good shape.

Two-to-one counterpoint can help fix some of these problems.

1. Determine whether to use chord tones or NCTs.

2. Leaping down by a third and continuing in the same direction is ok. Maybe ok with a fourth but definitely not with a fifth or more (unless you're using an arpeggio).

3. Keep checking the audibility of the harmonic progression.

Guided practice in one-to-one counterpoint

Here we look at a few pseudo-student exercises and point out some flaws in each.

Example 1

The first 5 notes suffer from 'turbulence' in that there is no clear direction, and the melody consists of just two notes in a monotonous pattern. There are also repeated notes in bars 2 and 3-4 which should be avoided.

In bar 6, we have direct fifths (02.02). The two Ds in the i^6_4 leap into a D-A in the V, and voice-leading rules say we should avoid these leaps in the outer voices.

Example 2

The leap up by a fourth in bar 2 suggests going down afterwards, but here the continued upward motion sounds ok too. The F# in bar 3 is a leading tone and should resolve to the tonic, rather than step down to the D. The repeated D-F# in bars 3 and 4 is turbulent.

Although the notes in bars 5 through 7 are quite high, there is a clear climax with the Bb in bar 6.

Example 3 – good

This example is fairly good. There's a clear line from bar 1 up to the G in bar 4, although it's a bit boring, being a straightforward scale. A bit of turbulence with the F#-G repetition in bars 3 through 5. Bars 5 to 7 are fairly high.

Example 4 – better

This example is better than the last one in that it gets rid of the monotonous scale passage. The leap up by a fourth then down by a third in bars 2 and 3 is a common progression. The 2 leaps here make a nice progression, although we shouldn't have too many such repeated leaps. The second beat in bar 3 consists of the two tendency tones (fourth and seventh scale degrees) and these resolve outwards.

The viio6 in bar 4 contains only the third, so needs to include some other notes in the chord when we elaborate into two-to-one counterpoint. Otherwise we could relabel the chord as V^6_4.

There is a single highest note (C in bar 2) which is often desirable.

Guided practice in counterpoint: two-to-one

We start by elaborating the 'better' one-to-one counterpoint we analyzed at the end of the last video:

Example 1

There is an overuse of anticipations here, which tends to slow down the motion or flow of the melody. The introduction of the NCT Bb at the end of bar 2 causes parallel fifths with the D and A at the start of bar 3. This illustrates that good voice leading in one-to-one counterpoint can be disrupted when we introduce two-to-one counterpoint.

There is an augmented second at the end of bar 5 (Eb-F#) which should be avoided. Finally, there is some turbulence in bars 5 through 7. The main problem overall is a lack of direction arising from the anticipations and turbulence towards the end.

Example 2

There are parallel octaves and fifths as shown; these problems with voice-leading arise when going from one-to-one to two-to-one and you need to keep a careful watch for them.

The high D in bar 3 sticks out and doesn't fit in with the flow of the melody. Finally, successive thirds are used too often.

Example 3 – better

The pattern with successive thirds occurs in bar 6.

The V in bar 6 is now really a V^7 since the C is the seventh of this chord. It's possible to resolve the C and A to a Bb in bar 7 instead of the G. The resolution to the G is a common way to end, and resolving to the tonic is important. Basically a matter of taste.

By about halfway through, the original one-to-one counterpoint is essentially forgotten. The leap up to the G in bar 5 provides a clear climax.

The successive leaps of a fourth and a third in bars 3 and 4 is ok, as long as this isn't overused. Here, the D-G-Bb outlines the i triad.

Example 4 – better with N^6 substitution

The ii chord in example 3 is a sub-dominant so it can be substituted with a Neapolitan sixth. This required lowering the G in bar 5 by an octave, which introduces a bit of turbulence. Making the G-Ab an octave higher destroys the climax on the G, since the Ab is higher. Because it ruins the line in bar 5, example 4 is probably inferior to example 3.

Example 5 – mixed note values

This is an example of how the right hand can use different rhythmic patterns. It's not a fully developed example since the left hand still uses constant note values. In more advanced counterpoint, the left hand would also have an independent rhythmic pattern and use some NCTs.

Progressions within progressions: introduction

Secondary dominants

Last week when analyzing Beethoven's bagatelle (05.04) we came across a chord that wasn't part of the key, and also wasn't any of the chromatic substitutions we had encountered so far. That was a secondary dominant which, besides its use here, is also used in modulation (changing key).

In bar 1 above, we start with a I-ii^6_5-V progression. The ii is a fifth above the V. To turn this into a V^6_5 of the V chord, we need to make the chord a major chord (since the V chord is always major). To do this, all we need to change is the F in bar 1 to F# in bar 3. This new chord is not one found in the key (C major), but is found in the key of the V (G major).

(The term 'progressions within progressions' is not commonly used.) We can view the V^6_5/V-V as a V-I progression in the key of G, and since it occurs within a phrase written in C, it is a progression within a progression. The secondary dominant is a chromatic harmony since it is a chord not found in the key of the composition.

If we examine the diatonic seventh chords, we find there is only one that consists of a major triad with a minor seventh, and that is the V^7 or dominant seventh. (For reference, the other seventh chords contain either a minor triad with a minor seventh (ii^7, iii^7, vi^7), a major triad with a major seventh (I^7, IV^7) or a diminished triad with a minor seventh (viio7). Thus any time we get a major triad with a minor seventh, it acts as a dominant seventh. Calling the secondary dominant a II^6_5 doesn't explain the dominant relationship between the chord and the following V.

Example in G minor

In this example we start with a i-iv^6_4-i progression with all chords being diatonic. In bar 3 we change the iv to a major chord, giving an example of a chord borrowed from the major key. In bars 5 and 6 we have same chord, but this time its function is different, as it leads into a VII. In this case, the E acts as a leading tone resolving to the F in bar 6, so the VII acts as a tonic within an internal progression. Since the IV chord is a fifth above the VII and is major in the key of the VII (which is F), this chord acts as a dominant chord relative to the VII.

The secondary dominant always changes the function of a chord, so in order to classify a chord we need to look at its context. The augmented sixth chords we looked at last week, although chromatic substitutions, still played the role of the sub-dominant, since they resolved to the dominant.

Progressions within progressions – examples

Chords a 2nd apart (upward)

Bar 1 shows a ii^7 modified by raising the third a semitone to F#, giving a V^7/V. All the chromatic chords we've seen so far involve raising one of the tones in a diatonic chord by a semitone. The chromaticism in bar 1 uses an F#, but it's hidden in a middle voice. We can expose the chromaticism a bit more by putting the F# in the bass as in bar 3, giving a V^6_5/V.

Chords a 3rd apart (downward)

Now suppose we want to go from a I chord to the chord a third below (which is the vi chord if we're in a major scale). We can insert a V/vi chord in the progression to do this. To get this chord, start with the fifth above the vi which is D. This is the third in the key of Bb and is a minor chord in that key. To get the V/vi, we raise the third by a semitone (F#) to get a major chord.

Chords a 4th apart (upward)

To go from I up a fourth to the IV, we can progress via a V^7/IV. This chord is the I^7 with the seventh lowered by a semitone (to give a major triad with a minor seventh).

Another option is in bars 3 and 4, where we put the seventh in the bass, giving a V^4_2/IV. This gives a smooth stepwise progression in the bass, and generates a more subtle sound.

Chords a 5th apart (upward)

Now suppose we want to get from i in G minor to V upwards via a secondary dominant. The V relative to the V is the ii, but in minor the ii is actually a iio (that is, diminished) chord. To conver this to a major dominant we therefore need to increase both the third and fifth by a semitone, so we change the C to C# and the Eb to E as seen in bar 1.

We can also use a V^7/V (missing the fifth (E) here) as in bar 3.

Progressions within progressions: secondary leading tone chords

Chords a 2nd apart (upward)

Recall from 01.03 in our discussion of basic progressions that viio (leading tone) diminished chords can substitute for V (dominant) chords. We can therefore substitute a leading tone diminished chord for the secondary dominant in bar 1.

We looked at substituting a diminished seventh chord for a V chord in the video on diatonic substitutions (04.01), where we were in a minor key. A diminished seventh chord consists of 3 consecutive minor thirds and as such, its minor key form consists entirely of chord tones. In a major key, however, the seventh must be lowered a semitone. In the example above, if we want a diminished seventh relative to the V (that is, relative to G major), the root tone is the seventh in the G major scale (F#) and we add three successive minor thirds on top of that, giving A, C and Eb. Here, we can accomplish this by moving the D in the V^6_5/V up a semitone to Eb as seen in bar 3.

Chords a 3rd apart (downward)

Here we start with a V/vi in Bb major, leading to a vi. The dominant relative to vi is the dominant in G major, which is D major, so the V/vi is a D major chord in root position. If we want to substitute a diminished seventh for this, we first work out the root tone, which is the seventh scale degree in G, or F#. Thus, we get the same chord as we had in bar 3 of the first example, except this time we don't need to lower the seventh since it is already an Eb due to the key signature of Bb major. In this case, the seventh (F#) isn't in the root position; rather the chord is in first inversion with the third (A) in the bass, so we get a viio6_5/vi chord.

The Alberti bass

The Alberti bass is a simple, repeated pattern often found in the bass in classical-era music. It takes the triad on which the harmony is based and breaks it up into a low-high-middle-high pattern:

Some variations are possible, but there are some things that shouldn't be changed:

1. Always start with the lowest note.

2. Always use 3 notes.

3. Always use chord tones.

Following these rules doesn't leave a lot of scope for variations, but here are a couple:

Bars 1-2 use low-high-middle-high followed by middle-high-middle-high, while bars 3 and 4 use low-middle-high-middle followed by high-middle-high-middle.

Elaborating progressions

Starting with the basic I-IV-V-I progression with all chords in root position (bars 1-4), how can we elaborate this?

The first thing we could try is an inversion, say of the IV chord, in bar 6.

In bars 11-12 we insert a couple of extra bars, using a passing 6-4 (02.01.02) between the IV^6 and the IV.

In the phrase from bars 15 to 21, we've added another I^6_4 between the IV and V, and since the V is part of a PAC, this is a cadential 6-4.

Now we can try introducing some progressions with progressions. Bar 15 above drops a fifth to bar 16 so that is one place where we could add this. Also, because a cadential 6-4 is a 'V in disguise', we should be able to add a secondary dominant between bars 18 and 19.

Here's the modified phrase:

First, consider the secondary dominant in bar 2. This is the I chord in the original key, but in order for it to fulfil its role as a dominant, it needs to be a seventh chord. The destination is the IV^6 so we should voice lead the secondary dominant so that it leads smoothly into the IV^6. The logic here is that in this interior progression, the IV^6 is acting like a tonic (I) chord, and when we introduced the V^4_2 chord (02.01.02) we see that voice-leading considerations cause it to resolve to a i^6. Therefore, since the IV^6 is acting like a I^6, we could try using a V^4_2/IV as the secondary dominant.

For bar 6, taking the I^6_4 in its role as a V, the fifth above the V is the ii. To make this a secondary dominant, we need to make it a major chord which requires moving the F up to F#. Since the F# is midway between the low F in bar 5 and the low G in bar 7, we can put this F# in the bass, making the chord in first inversion. To voice-lead the treble, we keep the two common tones. Moving the F to the nearest chord tone (F#) would give us parallel octaves and also double the leading tone, so we move it to the D instead.

Mozart's K. 331 sonata – rounded binary form

We'll analyze the theme from the first movement of Mozart's K. 331 piano sonata in A.

Analysis of first half

First, we'll do some harmonic analysis. In bar 1, we start with I and can take the entire bar to be in the same chord.

Bar 2 starts with a V in first inversion, but the D in the treble makes this a seventh chord. We treat this D as a chord tone rather than a NCT because it lasts a long time and is a significant part of the melody. Also, we leap into and out of the D, which isn't done with a NCT.

Although the D is the fourth scale degree which normally would be resolved down by a step, the leap down to the tonic is ok here as this is piano writing where some of the vocal rules can be relaxed.

Bar 3 uses a new progression, with a vi^7 going to V^6. The main component of this first phrase is the set of parallel thirds between the bass and treble. Another structure central to the phrase is the common E that runs throughout the phrase.

In bar 4, there is a ii^6 with the top E sixteenth note being an appoggiatura (leap into, step out) and the second sixteenth note (D) being a chord tone. This leads to a cadential I^6_4-V half-cadence. The harmony in bars 7 and 8 is the same as in bars 3 and 4, but at a faster tempo, and instead of ending on a V, we end on I for a PAC.

In both cases, the cadential I^6_4 lasts longer than the V which follows it.

The form of this first section is A-A', that is, it splits into two sections that are the same except for the cadences at the end. This is another example of a parallel period form. The texture of this first section is homorhythmic homophony.

Analysis of second half

The second half (B section) should have some contrast with the first (A) half. We start with some new material and then return to material from the A half.

In bar 9, we start with I and then get a IV^6_4 which is a pedal 6-4 since the bass note is repeated from the preceding I. In bar 11, we can include the D in the treble as a chord tone, making the chord a V^6_5.

The second chord in bar 12 can be interpreted as a triad if we take the D# as the root, the F# as the third and the A as the fifth. As the triad is made up of two minor thirds, it is a diminished triad. In major, the only diminished triad is the leading tone (viio) triad. D# is the leading tone in E major, and E is the fifth up from A, so that makes this chord a viio6/V.

The harmony from bar 13 to the start of 16 is the same as before. The third chord in bar 16 is a retardation suspension (04.07), Since the G# and B are held over from the preceding V and resolve to the A-C#, making the harmony here a I.

The second chord in bar 17 is a passing chord (02.01.01). We met this type of chord originally as a passing 6-4 going to a i^6, but here the chord is a seventh chord in second inversion, giving a V^4_3; it still resolves to a I^6.

Rounded binary form

The overall structure of the piece is A-A'-B-A'. That is the first half consists of a parallel period, but the second half consists of a new theme (B) followed by a return to material from the first half. This form defines a rounded binary form.

The contrast between the first and second halves is that in part A, homorhythmic homophony is used, while in the second half, the left and right hands could be played by different instruments. To develop contrasts, look in the 'toolbox' of techniques to see what effects can be added to the music.

Guided practice – secondary dominants and Alberti bass

Starting with the harmonic progression given, we write in the bass notes in bars 1 to 7 with the view to modifying these using inversions when we complete the harmony.

In bars 8 to 10, we insert a secondary dominant in front of each chord in the following bar, again putting in the bass note in root position. A cadential 6-4 is inserted in bar 13.

In bars 15 to 21 we insert 3-note chords in the bass (3 notes since we're going to convert each chord into an Alberti bass), with a view to keeping the bass line as smooth as possible. This requires using inversions in most places. Since all the secondary dominants are seventh chords, we look at the melody note for each of these chords and use the remaining 3 notes to make up the bass chord.

Finally, we convert the bass into Alberti bass:

Content

1. Write like Mozart – lecture notes

3. Major Diatonic Chords

1. Triads

2. Seventh chords

4. Minor Diatonic Chords

1. Minor Triads

2. Minor Sevenths

5. Other chord types

6. Voicing

1. Vocal Ranges

2. Open and Close Structure

3. Two Things to Avoid

4. Doubling

7. Basic Progressions

8. Voice Leading

9. Voice Leading with the Common Tone Approach

10. Voice leading without common tones (I-IV-I and I-V-I)

11. Resolution of the V7 chord

12. Voice leading from IV-V

13. Voice leading with chords that have a mediant relationship

14. Texture types

15. Voice leading example – 1

16. Voice leading example – 2

17. Inversion notation

18. Progressions with inversions – 1

19. Progressions with inversions – 2

20. More voice leading

21. Using inversions (A Demonstration)

22. Keyboard voicing

23. Creating accompaniment

24. Textural reduction

25. Reversing Schubert

26. Circle of fifths progressions

27. Descending 5-6 progressions

28. Parallel sixth progression

29. Analysis: Mozart, Piano Sonatas K. 545 & K. 332

30. Analysis: Beethoven Op. 109

31. Analysis: Mozart, K. 279

32. Analysis: Mozart, K. 533

33. Non-chord tones

34. Using NCTs with the circle of fifths

35. Using NCTs with the descending 5-6 progression

36. Using NCTs with parallel sixths

37. Diatonic substitutions

38. Diatonic substitutions: ii for iv

39. Diatonic substitutions: V-vi progressions

40. Cadences

41. Analysis: Beethoven Op. 2 #3, ii

42. Period form

43. Melodic writing

44. Chromatic substitutions: the Neapolitan 6 chord

45. Analysis: Schubert's Der Müller und der Bach

46. Chromatic substitutions: augmented sixth chords

47. Analysis: Beethoven's bagatelle, op. 119 #1

48. Chromatic substitutions: borrowed chords

1. Picardy third

49. Simple 2-voice counterpoint

50. Guided practice in one-to-one counterpoint

51. Guided practice in counterpoint: two-to-one

52. Progressions within progressions: introduction

53. Progressions within progressions – examples

54. Progressions within progressions: secondary leading tone chords

55. The Alberti bass

56. Elaborating progressions

57. Mozart's K. 331 sonata – rounded binary form

58. Guided practice – secondary dominants and Alberti bass

Table of Contents

Write like Mozart – lecture notes

Major Diatonic Chords

Triads

Seventh chords

Minor Diatonic Chords

Minor Triads

Minor Sevenths

Other chord types

Voicing

Vocal Ranges

Open and Close Structure

Two Things to Avoid

Doubling

Basic Progressions

Voice Leading

Voice Leading with the Common Tone Approach

Voice leading without common tones (I-IV-I and I-V-I)

Resolution of the V7 chord

Voice leading from IV-V

Voice leading with chords that have a mediant relationship

Texture types

Voice leading example – 1

Voice leading example – 2

Inversion notation

Progressions with inversions – 1

Progressions with inversions – 2

More voice leading

Using inversions (A Demonstration)

Keyboard voicing

Creating accompaniment

Textural reduction

Reversing Schubert

Circle of fifths progressions

Descending 5-6 progressions

Parallel sixth progression

Analysis: Mozart, Piano Sonatas K. 545 & K. 332

Analysis: Beethoven Op. 109

Analysis: Mozart, K. 279

Analysis: Mozart, K. 533

Non-chord tones

Using NCTs with the circle of fifths

Using NCTs with the descending 5-6 progression

Using NCTs with parallel sixths

Diatonic substitutions

Diatonic substitutions: ii for iv

Diatonic substitutions: V-vi progressions

Cadences

Analysis: Beethoven Op. 2 #3, ii

Period form

Melodic writing