EPTThis time we will analyze actual music, while we discuss some new chords.
Part 1: Introduction, and the dominant chord
Part 2: The leading tone, more about the dominant and the subdominant chord
Let's recapitulate the last blog: The concept of functional harmony is based on the effect of a chord versus a tonic. That tonic center needs to be established first. Once this has been done, we relate other notes and chords to the tonic.
So far we learned about the dominant, which is the urge to come back to the tonic, and subdominant, which is much more relaxed. We also learned about the concept of the minor relative chord. That is a minor chord which sounds very similar to the major tonic and can be used as tonic substitute within the piece.
Minor Chords For a Major Scale?
To get chords which fit a major scale, those chords should use notes which appear in that major scale. Some degrees will automatically turn out to get us minor chords. While the overall sound of a minor triad is different to a major one, the minor chords still have a close connection to our melody in major – because we use only the notes from that scale.
![[image loading]](http://i.imgur.com/eUqhcMC.png)
Establishing A Tonic
Let us use the first four bars of the piano song Nightswimming by R.E.M. Interestingly, Mike Mills doesn't seem to play actual chords with his left hand. But the combined notes of both hands form a G chord. Then some other notes follow. The next measure has only one chord, a C.
The next bar plays a G chord again and is followed by a measure which consists a D chord. Then the ostinato, a repeating pattern, begins. So the intro gives us G, C, G, D. Can we make any sense of it?
We recognize that G, surrounded by both C and D, acts as point of reference. Because the tuning ratio of C and D compared to G is so simple, we perceive a stable order when we structure the chords in way that G is the scale root for any other note. G is the root degree, therefore the first degree, the tonic.
We have now a pitch in mind which we use to relate other notes to, which are different in pitch. The note name "G" is arbitrary, but the tonic is the result of harmonic relationships and acts as reference for any future note and chord.
The Key Of Luka
Now we know enough to analyze the chords of another song. We use Luka from Suzanne Vega. Luka is written in F♯-major. That key signature consists of six sharps. It key cannot be simplified, because the enharmonic equivalent is G♭, needing six flats.
On a piano, the F♯-major scale uses all five black keys, plus two white keys. But if F♯ uses six sharps, why do we only use five black keys? F♯-major includes the note E♯, which is played with the white F key. We have no more than five black keys per octave anyway.
As a side note, the black piano keys can be used to play a pentatonic, meaning five-note antique scale. Of course, Luka requires a modern seven-note scale.
Guitar Player Tricks
With equal temperament, all intervals of F-sharp-major are as good or poorly approximated as for the C key or any other key. Since is it not simple to play F-sharp on a guitar with standard tuning, Suzanne Vega uses a capo (a clamp) to effectively shorten the open strings by two frets and thus increases their pitch by two semitones. She actually plays the piece as if it is in E, the capo transposes it to F-sharp. This is possible thanks to equal temperament.
Verse Chords For Luka
The intro of the song consists a progression of chords. They establish the pitch of F-sharp as tonic. Instead of showing the actual chords, we will use the functions. Functions can be described by its name or its degree. The dominant is the fifth scale degree and gets the roman numeral for 5, which is V.
[Tonic] My name is [Dominant] Luka
[Subdominant] I live on the [Dominant] second floor
We see how the chords support the lyrics. We begin with the tonal center, called the tonic. The name "Luka" is accompanied by the dominant, creating tension. The beginning of the next line is accompanied by the subdominant which provides a more relaxed sound. But the information where Luka lives is again accompanied by the dominant, which again creates tension. Remember, the tension towards the tonic is just in our head. Every now and then the song must return to the tonic keep it stable.
The same chord pattern repeats for the next two lines, so it begins with the tonic and ends again with the dominant. But we get interrupted in when it comes to
[Relative key] If you hear [dominant] something late at night
The dominant of the previous line is not resolved to the tonic, but the relative minor key instead. The resolution from V, the dominant, to vi, which is the relative minor key of the tonic, is not really satisfying. The sound of the minor chord sound also supports the lyrics as we recognize that Luka's persona tries to elude the conversion as the chord progression eludes the tonic and uses its relative minor instead.
Another interesting effect provides the line
[Subdominant] Just don't [dominant] ask me what it was
This line is repeated three times, so we have the IV-V chord progression three times, and really ask for a resolution to I, the tonic. As long as Vega in her Luka role repeats this line, we don't return to the stable tonic. This supports the uneasy feeling of Luka.
The Instrumental Part
Despite the depressing lyrics, the song has an upbeat melody and is known for its final instrumental part. The radio version is neatly arranged, using several guitars and a keyboard. Let us have a look at the functions.
Tonic - Dominant - Subdominant - Dominant
This is a very common method: We play the tonic, confirm it with the dominant and go into a standard cadence with IV-V. This is resolved by the first chord of next iteration:
Tonic - Dominant - Subdominant - Dominant
The pattern repeats, but is played slightly differently and uses a higher notes. This can be done by playing some of the chord notes in higher octaves. The function of the chords don't change. The higher overall pitch supports the high spirit. As this progression ends, we go to the tonic's relative chord instead of the real tonic.
Tonic relative - Dominant
The relative chord is played in a quite high pitch, adding to the emotion. While the relative chord sounds quite similar to the tonic, we expect the song to not end here because we want to go back to the real tonic.
The song ends with a standard cadence:
Subdominant - Dominant - Tonic
The subdominant in this part is played in a very high pitch, the instrumental reaches its emotional climax. Then it ends on a satisfying V-I cadence.
Modifying Our Chords
Many popular songs, Luka included, use only four chords: The tonic, the subdominant, the dominant, and the relative key of the tonic. To get a bit more variety, chords can be played as suspended chord where the third is replaced by another interval, often the fourth.
Another way is to add notes to the triad. The addition of the sixth and ninth degree are especially common. The addition of the seventh degree is also used, most notably for the dominant.
While changes in chords allow to arrange an accompaniment with more diversity, we are still using just four degrees. But we have a diatonic scale with seven degrees. Let us have a look at the other possible diatonic functions.
Contrasting The Tonic
The third degree of a major scale results in a minor chord and is therefore the iii degree with lower-case letters. It can function as mediant, because this chord is in the middle between the tonic and its dominant. The mediant shares two notes with the tonic triad.
Another function of iii is being the relative minor of the dominant. If we are in C, the dominant is G. Its relative minor is Em, where 'm' marks the minor property. But Em is also the iii for C.
There are even more possibilities to interpret this degree. The iii is in opposite direction to the vi degree. Of course, vi is the relative minor, as we remember. Being in this sense the opposite of the relative minor, iii can be considered the gegenklang or contrast chord.
Exceeding The Tonic
The second degree is very interesting. Because it is a whole tone above the tonic, it can be called the supertonic. It is a minor triad and therefore the ii. This degree also be viewed as the relative minor of the subdominant.
There even are more ways of thinking about this chord degree. Two dominants on top on each other would result in a ninth degree, but shifted down by an octave – which doesn't change the actual note – the ninth is the same as the second degree. Therefore, ii could also be considered the dominant of the dominant and therefore a secondary dominant.
Before we will discuss the final chord degree, the seventh, let us analyze a well-known piece of music.
![[image loading]](http://i.imgur.com/BwYV9bg.png)
Canon in D
Johann Pachelbel, a German composer, lived about 300 years ago. His work is mostly forgotten. Except for one piece: Kanon in D. It has a distinctive chord progression which you hear in too many songs of today, including bad hip-hop songs. That is why Pachelbel has a place in a blog about modern music.
Why did he write his canon in D? At Pachelbel's time, equal temperament was not yet fully developed, so each key sounded more or less pure. In this context, D is one of the rather pure keys. But we shouldn't overstate key color here. Many string instruments sound very good with the notes D, A or G anyway, because the standard tuning allows the open strings to resonate in harmonic relationships.
Johann Pachelbel wrote his canon using basso continuo. This part of the baroque era played an important role in developing the harmony rules we still use today.
German Notation
Since Pachelbel was German, let me show the canon chord progress in German notation. Small-letter chords are minor, the other major.
D - A - h - fis - G - D - G - A
Fis means F-sharp. In German music notation, "is" is note suffix is for sharp. But what about that H note? In English notation, that would be B. But Germans know a B note, too. It is B-flat.
If that isn't weird enough, what is B-doubleflat in German? Now just B-flat, since the German B includes a flat? No, now it is derived again from H and called Heses. The "es" suffix is for flat. Two "es" make a doubeflat. There are exceptions for the suffix uses as well, "es" for some notes it is replaced with just "s".
If you are German and used to it, it all makes sense. Otherwise, well … let us have a look at the Pachelbel chord progression in English notation:
D - A - Bm - F♯m - G - D - G - A
Of course, we are not so much interested in the chord names, but rather how they function.
Analysis Of The Chords Of Canon In D
The first measures of the canon use only bass notes, no chords. But the root note of a triad carries the meaning of the degree function anyway. Basso continuo means that the bass notes are complemented by other instruments to full chords. The actual canon piece is quite nice, we only analyze the bass part as if they were chords.
We have the first chord of the canon. As the dominant (the V) follows, we now are sure that the first chord was the tonic, indeed. Those two chords make most sense if we view them as I-V, since they are a fifth away, which is a very simple and stable interval. The V also creates an expectation to get back to I as the V includes notes with high tension to I.
Instead we hear vi now, which is the relative minor of I. This kind of interrupted cadence is common in music, so we are not too surprised. The following chord is a iii. In our context we hear it as relative minor of the dominant. Let's have an overview of the first four chords:
Tonic - Dominant - Tonic Relative - Dominant Relative
The next chord is a IV. This is the subdominant of the tonic. So the somewhat difficult iii degree is followed by the relaxed sound of IV. We are also back at major chord degrees. The next chord is the tonic itself. It is one octave lower than the chord we started with, adding to the relaxation.
The IV is then played again and followed by a V. That means, we play a IV-V which is should lead us to the tonic. And since we are at the end of our progression and repeat it, we play the I degree – the tonic – indeed. The second part therefore is:
Subdominant - Tonic - Subdominant - Dominant
I dare to day that only because of Pachelbel we today have popsongs using more than four chords. Because his famous canon uses five.
The Seventh Degree
Let us go back to C-major, as we don't have any sharps or flats in here. The seventh degree if C-major is B. In German, it is H … okay, let us stop this confusion and continue with English notation.
How does the seventh degree triad look? It is B-D-F. The interval B-D is a minor third, so we could consider this chord a minor chord. That means we need lower-case letters for this VII degree and write it as vii.
But the overarching interval, B-F, is no perfect fifth. It is a diminished fifth. This triad is therefore a diminished triad, which is marked with a °. That means, the seventh degree in a major scale is vii°.
Using equal temperament, the diminished fifth is equivalent to a tritone.
![[image loading]](http://i.imgur.com/AbG6wDI.png)
The Tritone Is Its Own Inversion
The tuning ratio of B and F is extremely difficult. It is the most difficult tuning ratio possible in a scale. That means, the interval B-F is very instable because we cannot discern any useful relationship of the pitches.
B-F spans three whole tone steps, giving the tritone interval its name. An inversed tritone, or the octave complement of a tritone, is again a tritone. But as always, we can – and should – explain an interesting interval with fifths.
A fifth is a pitch increase by 50%, or a pitch difference by factor 1.5. For example, a single whole tone represents two fifths. If we go up in fifths from C, we get via G to D. C-D is a whole tone, which also stands for two fifths as we just demonstrated.
A tritone spans six fifths. That is the widest harmonic distance possible.
No More Than Six Fifths
Twelve perfect fifths add up to almost exactly seven octaves. Octaves do not create new notes, so we almost exactly get to the point where we started from. If only the "almost exactly" would be "exactly".
What equal temperament does, is to slightly temper the fifth so that twelve of them do exactly produce the same note, only some octaves higher. Octaves doesn't matter here.
That means, the thirteenth step now results in exactly the first note, leaving us only twelve different notes. We essentially have complete circle.
Think of an analogue clock. It has twelve hours, but that is an accident. The notes are not derived from the clock, but from the aforementioned completion of twelve fifths to an integer number of octaves.
Let us rather focus on how we can get from any point – any fifth step, or hour of a clock – to any other. If we can go in one direction only, let's say clockwise, we need up to eleven steps. But if we can go in either direction, no more than six steps are needed.
The B-F interval spans six steps of fifths. That prevents us to recognize a direction: Those six fifths could be meant to be either upwards or downwards, or in clock terms either clockwise or counterclockwise. This interesting property adds to the instability of the tritone interval.
Creating Leading Tones With The Tritone
Because the tritone produced by the B-F interval is so instable, we perceive both B and F as leading tones. That means, we ask for a resolution to a stable interval with using the smallest change in pitch possible. The smallest possible pitch change is of course a semitone.
B-D-F can be very nicely resolved to C-E-G, a C-major triad chord. For one, we resolve F by going a semitone down to E, the major third of C. But more important is that we resolve B by going up a semitone to C, the tonic of C-major.
This makes the vii° a dominant degree. Because of its quite harsh sound caused by the overarching interval of a diminished fifth, which is equivalent to a tritone, it is not as often used as dominant as the V.
The Dominant Seventh – The Most Used Tetrad Today
The seventh of the tonic is the leading tone and its chord, as just showed, has a strong dominant effect. But what happens when we think of the regular dominant triad, the V degree, and add its seventh? We expand the triad to a tetrad. Let's look at in in C-major. The dominant is G, and the dominant triad is G-B-D. The seventh of G-major is F♯, but since our tonic is C-major, we are allowed to use only notes from the C-major scale, so that we get F. That means, the seventh of the dominant G is F, so the dominant seventh chord is G-B-D-F.
This is no cheap trick, we restricted ourselves to scale-notes with triads already. We just expand this concept to tetrads. Because it is a chord on the fifth degree, but with an added seven to the triad, its chord degree is V7. V is the dominant degree, the fifth. The 7 marks the addition of the seventh degree counted from that fifth.
G-B-D-F includes both the V and the vii° chord. Even if we have no further context and hear just this chord, the tritone interval of B-F makes B a leading tone, which asks for a resolution to C. We also hear G, the dominant of C. That are two arrows pointing to C. That means, we hear this chords as the dominant of C no matter what. If we confirm it with a C chord, we created a tonic with only two chords.
The strong dominant effect of the dominant seventh chord makes this the most used tetrad today. Resolving this chord with the tonic really resolves the extremely high tension and instability of the dominant seventh chord, getting us from tenseness to ease.
