Advanced Music Theory | intervals, chords, inversions, tonic-dominant, secondary dominance.

Intervals

Before we define a chord, we must revisit intervals because chords are made of intervals. As stated in the basic theory article, an interval is the distance between two notes and is measured in its amount of half steps/semitones. Intervals can be unisons, 2nds, 3rds, 4ths, 5ths, 6ths, 7ths, or octaves. Also, intervals can be qualified as Major (M), minor (m), diminished (d), Augmented (A) or Perfect (P). Here’s a list of all the intervals:

  • P8 (octave): 12 half steps/semitones
  • M7: 11 h/s
  • m7: 10 h/s
  • M6: 9 h/s
  • m6: 8 h/s
  • P5: 7 h/s
  • d5 or A4 (tritone): 6 h/s
  • P4: 5 h/s
  • M3: 4 h/s
  • m3: 3 h/s
  • M2: 2 h/s
  • m2: 1 h/s
  • P1 (unison): 0 h/s

These are called simple intervals because they fall within an octave. Compound intervals exist (9ths, 10ths, 11ths, etc.), but they’re essentially simple intervals with an octave between (for example, a m9 is a m2 with an octave between the two notes).

Chords

Much of this article will be dedicated to chords because they form the basis through which we can understand harmony. Without chords, it’s hard to take our music beyond either a melody or a bassline paired with drums. Half the battle of writing music is knowing chords because they’re so fundamental to music in general.

Chords are essentially three or more notes played as a unit. Simple chords (called triads because they have three notes) are made by stacking two 3rds on top of each other and the bottom note, or root, of the chord gives it its name. To illustrate this, let’s take C as our bottom note. First, we go up a 3rd to make C – E and then we go up another 3rd from there to make C – E – G, a C chord.  The three notes of a triad have names; the bottom note (C in this case) is called the root (R), the middle note (E) is called the 3rd and the top note (G) is called the 5th.

R – 3rd – 5th

This particular chord is called a C Major chord (CM). All the chords we’ll look at are either Major, minor, diminished, or Augmented. Major chords are made by making a M3 between the root and the 3rd and then making a m3 between the 3rd and the 5th. A minor is made by making a m3 between the root and the 3rd and then making a M3 between the 3rd and the 5th.

So if a C Major (CM) chord is this:

C – M3 (4 h/s) – E – m3 (3 h/s) – G

Then a C minor (Cm) chord looks like this:

C – m3 (3 h/s) – Eb – M3 (4 h/s) – G

“Half the battle of writing music is knowing chords because they’re so fundamental to music in general.”

Here’s a shortcut: the difference between a Major and a minor chord is the 3rd. Taking a Major chord and lowering the 3rd a half step/semitone gives you a minor chord, and taking a minor chord and raising the 3rd a half step/semitone gives you a Major chord. Either way, you’ll still have 7 half steps/semitones between the root and the 5th, which means that Major and minor chords always have a P5.

Chords can also be diminished or Augmented. These kinds of chords are somewhat bizarre and are used must less frequently in music. Diminished chords are made by stacking two m3’s on top of each other, which creates a d5 (6 h/s) between the root and the 5th, and Augmented chords are made by stacking two M3’s, which creates an A5 (8 h/s) between the root and the 5th.

A C diminished (C dim) chord looks like this:

C – m3 (3 h/s) – Eb – m3 (3 h/s) – Gb

And a C Augmented (C Aug) chord looks like this:

C – M3 (4 h/s) – E – M3 (4 h/s) – G#

In general, a diminished chord has a very dissonant sound because it contains a d5, which is one of the most unstable intervals in (Western) music. Every Major or minor scale has one naturally occurring diminished chord in it, unlike the Augmented chord. The Augmented chord is all but completely theoretical and has a strange, almost inquisitive sound to it.

Here’s the formulas for all the types of chords we looked at:

  • Maj: R + M3 + m3 (P5 between R and 5th)
  • min: R + m3 + M3 (P5 between R and 5th)
  • dim: R + m3 + m3 (d5 between R and 5th)
  • Aug: R + M3 + M3 (A5 between R and 5th)

The scales we’re in dictate the chords that we use because the notes (also called scale degrees) from the scales make up the chords. Let’s look at Major scales first using Bb Major as an example. First we have to list all the notes, or scale degrees, in Bb Major (which has two flats):

Bb  C    D   Eb  F    G    A  (notes)

1     2    3    4    5    6    7  (scale degree number)

Then we build triads from all the notes of the scale:

  • Bb – D – F (BbM)
  • C – Eb – G (Cm)
  • D – F – A (Dm)
  • Eb – G – Bb (EbM)
  • F – A – C (FM)
  • G – Bb – D (Gm)
  • A – C – Eb (Adim)

Now let’s look at minor scales using F# minor as an example (which has three sharps):

F#  G# A    B   C#  D   E

1    2    3    4    5    6    7 

We do the same thing as before and build triads using the notes of the scale:

  • F# – A – C# (F#m)
  • G# – B – D (G#dim)
  • A – C# – E (AM)
  • B – D – F# (Bm)
  • C# – E – G# (C#m)
  • D – F# – A (DM)
  • E – G# – B (EM)

Every Major and minor scale has 3 Major chords, 3 minor chords and 1 diminished chord. The way that we notate these chords follows certain conventions: we use Roman numerals to denote with note we start on, and capitalization with or without other markers to denote the quality (M, m, dim) of the chord.

Let’s look at the first three chords of F# minor. We denote the F#m chord as “i”; it’s minor so it must be lowercase, and it starts on scale degree 1 so we use the Roman numeral “i” (as opposed to “I” which would be Major)). The G# dim chord would be “iio”; diminished chords are lowercase with the little circle and it starts on scale degree 2. Lastly, the AM chord would be “III”; it’s uppercase because it’s Major and it starts on scale degree 3.

Now that we know how to notate our chords, we can use our Bb Major and F# minor chords as blueprints for all the chords in any Major or minor scale.

List of Major chords:

  • I
  • ii
  • iii
  • IV
  • V
  • vi
  • viio

List of Minor chords:

  • i
  • iio
  • III
  • iv
  • v
  • VI
  • VII

So far we’ve just been looking at triads, chords which only have three notes in them.  However, if we stack another third on top of a triad, we create a 7th chord because we’ve made a 7th between the root and the top note. Now we have three different 3rds and four different notes in a chord.

R – 3rd – 5th – 7th

There are 5 different types of 7th chords that we’ll look at:

  • Major-Major 7th: R + M3 + m3 + M3 (Major chord with a M7 between R and 7th) –> Notation: I7
  • Major-minor 7th: R + M3 + m3 + m3 (Major chord with a m7 between R and 7th, aka Dominant 7th) –> Notation: I7 (usually found on scale degree 5; V7)
  • minor-minor 7th: R + m3 + M3 + m3 (minor chord with a m7 between R and 7th) –> Notation: i7
  • half-diminished 7th: R + m3 + m3 + M3 (diminished chord with a m7 between R and 7th) –> Notation: i∅7
  • fully-diminished 7th: R + m3 + m3 + m3 (diminished chord with a d7 between R and 7th) –> Notation: io7

When applying 7ths to our list of Major and minor chords, there are two things to note. First, both the V in Major and VII in minor are Major-minor 7th chords (dominant 7th chords). Second, both the viio in Major and iio in minor are half-diminished 7th chords; vii∅7 and ii∅7 respectively.

Inversions

Inversion applies to intervals and chords. When we invert an interval or a chord, we either take the bottom note and move it up an octave or we take the top note and move it down an octave. We keep all the same notes but we stack them differently.

Let’s look at intervals first using a M3 as an example. Let’s have our bottom be C which means our top note is E, which gives us C – E. If we were to invert this interval, we would put C up an octave or E down an octave; either way we get E – – – – C. This new interval has 8 half steps/semitones, which makes it a m6. Using this as an example, we can create rules for knowing what an interval inverts to.  First we flip the quality (M, m, P, dim or Aug) of the interval; Major intervals invert to minor and vice versa, diminished invert to Augmented and vice versa, while perfect intervals stay perfect. Then take the interval’s number and subtract it from 9. (i.e., start with M3, M turns into m, 9 – 3 = 6, ergo m6)

List of intervals and their inversions:

  • P8       →    P1
  • M7      →   m2
  • m7      →    M2
  • M6      →    m3
  • m6      →    M3
  • P5        →    P4
  • d5/A4 → A4/d5
  • P4        →     P5
  • M3      →     m6
  • m3      →     M6
  • M2      →     m7
  • m2      →     M7
  • P1       →      P8

An interval and its inversion should always combine to make 12 half steps/semitones.

Just as we can invert intervals, we can also invert chords. Let’s look at triads first. Using AM as an example, we spell out the triad:

A   C#    E 
R – 3rd – 5th

Spelling a chord this way is called root position because the root is in the bass (the root is the lowest note in the chord). Root position is not an inversion. But if we take the root and bump it up an octave, we get an inverted chord.

C#   E         A
3rd – 5th – – R

Spelling a chord this way is called first inversion. It’s defined by the 3rd being in the bass

Starting from first inversion, we can make another inversion by bumping the 3rd up an octave.

E        A   C#
5th – – R – 3rd

Spelling a chord this way is called second inversion. It’s defined by the 5th being in the bass.

7th chords are a little different. Because they have 4 notes instead of 3, they have a third inversion. Let’s use A7 as an example (A7 is a Dominant 7th chord; AM7 would be a Major-Major 7th):

Root position:

A   C#    E      G
R – 3rd – 5th – 7th

First Inversion:

C#    E      G   A
3rd – 5th – 7th R

Second Inversion:

E      G   A   C#
5th – 7th R – 3rd

Third Inversion (7th in the bass):

G   A   C#    E
7th R – 3rd – 5th

The way we notate inversions is based on implications. Technically, every root position triad could be notated like this: I53.  This basically denotes that the chord has a 3rd and a 5th using the numbers beside the roman numeral. However, because every triad has a 3rd and a 5th, those numbers are omitted because it’s implied that the chord has a 3rd and a 5th. So it’s written like this: I. We omit things when we notate for the sake of shorthand.

Let’s look a first inversion triad as an example of notation. A first inversion triad still has a 3rd between its bottom note and middle note, but it doesn’t have the interval of a 5th between its bottom and top note; instead it has a 6th. Therefore, we would initially write it like this: I63.  However, since the 3rd is (usually) implied, we can simply notate it like this: I6.

  • With this in mind, here’s the list of notated triad inversions:
  • Root position: I (I53 but 3rd and 5th are implied)
  • First inversion: I6 (I63 but 3rd is implied)
  • Second inversion: I64
  • And here’s the list of notated 7th inversions:
  • Root position: I7
  • First Inversion: I65 (3rd is implied, 5th is included to distinguish from I6)
  • Second Inversion: I43 (6th is implied, 3rd is included to distinguish from third inversion)
  • Third Inversion: I42 (6th is implied)

Inversions are always defined by what’s in the bass, so you can rearrange the notes above the bass note and keep the same inversion. For instance, spelling a chord like this, 3rd – – – – R – – – 5th, is still first inversion because the 3rd is in the bass.

Inversions can be used creatively to spice up your basslines. One way you can do this is by using inversions to create a sequence of adjacent notes in the bass. For instance, if we are in G Major and we have this as our chord progression, I – V6 – vi – I64 – IV – I6 – V64 – I, then we’ve made a bassline that goes down stepwise starting from scale degree 1: G – F# – E – D – C – B – A – G.

Another way we can use inversions creatively is to surround notes in the bass with their adjacent notes. Using G Major again, if we have this partial chord progression, I – V6 – V – vi – I64 – I6 – IV, then we’ve made this in our bassline: G – F# – D – E – D – B – C. The E in the vi chord gets surrounded by its neighbors, F# and D, and the C in the IV chord gets surrounded by its neighbors, D and B. In this way, the chord that’s surrounded becomes a strongly emphasized destination in the music.

Try to think of your own inventive ways to use inversions!

The Tonic-Dominant Relationship

A precursor to this topic is knowing the technical names of the scale degrees in a scale.

Scale degree 1: Tonic
Scale degree 2: Supertonic
Scale degree 3: Mediant
Scale degree 4: Subdominant
Scale degree 5: Dominant
Scale degree 6: Submediant
Scale degree 7: Leading Tone (Major)/Subtonic (minor)

Scale degree 7 is called the leading tone when there’s a half step/semitone between it and scale degree 1, which we have in Major keys.  In minor, we call scale degree 7 the subtonic because there’s a whole step/tone between it and scale degree 1.

Half steps/semitones create strong pulls between two notes. The notes that tend to pull up or down to resolve the tension created by the half steps/semitones are called tendency tones. In Major, the half steps/semitones are between scale degrees 7 & 1 and 3 & 4, so 7 and 4 are tendency tones that move to 1 to 3 respectively.  Likewise in minor, scale degree 2 tends to move to 3 and 6 tends to move to 5 because of where the half steps/semitones lie in a minor scale. This is why the tritone (d5/A4) is such a dissonant interval. Since it’s found on scale degrees 7 and 4 (2 and 6 in minor), it’s basically made of harmonic tension that wants to resolve to 1 and 3 (3 and 5 in minor).

All of this plays into the strongest harmonic relationship in music theory: the tonic-dominant relationship.  The tonic chord (I) is our home plate and acts a baseline for our scale; there’s no sense of tension or movement when we first play a tonic chord. The dominant chord (V) is the opposite; it creates a lot of harmonic tension that seeks to resolve back to the tonic chord. The duality of the tonic and dominant chords is the basis for how we understand tonal harmony.

In Major, the dominant chord (V) seeks to resolve to the tonic chord (I) because it has a leading tone as its 3rd (a dominant triad is spelled 5 – 7 – 2). That means that there is a half-step/semitone between scale degree 7 and scale degree 1, so 7 wants to resolve up to 1, which is the root of the tonic chord.  In a natural minor scale, however, there’s a whole step/tone between scale degree 7 and 1, so our minor dominant chord (v) doesn’t have a strong pull to the tonic (i). The minor dominant chord (v) is some serious weak sauce and you hardly ever see it in music. To beef it up, we have a harmonic minor scale in which we raise scale degree 7 a half step/semitone. Doing this makes a Major dominant chord (V), which now has a strong pull back to the minor tonic chord (i) because there’s a half step/semitone between scale degree 7 and 1. We turned weak sauce (v) into harmonically strong beef sauce (V) by raising its third.

We can make our beef sauce (V) even beefier though. To pack our dominant chord with even more nutritious tension, we can add a 7th to it. When we do this, we add scale degree 4 as our 7th (5 – 7 – 2 – 4), which wants to resolve down to scale degree 3, the 3rd of the tonic chord.  This dominant 7th chord (V7) we’ve created has an internal tritone (d5) between scale degrees 7 and 4, which makes this chord the one of the most harmonically tense chords in music theory.

The vii* chord can also be used like a dominant chord, which makes sense because it’s basically V7 without scale degree 5.  The vii* chord, also called the leading tone chord, is spelled 7 – 2 – 4, which means it’s a diminished chord because of the d5 between scale degrees 7 and 4. vii∅7 is spelled 7 – 2 – 4 – 6, with scale degree 6 usually resolving down to scale degree 5.

I’ll stress it again: the tonic-dominant relationship is central to understanding harmony. It’s so fundamental that the functions of all the chords in our Major and minor scales refer to either the tonic or the dominant.

There are three different harmonic functions that our chords can fall into:

  • Tonic: chords that act as our tonal “home”
  • Pre-dominant: chords of mild harmonic tension
  • Dominant: chords of strong harmonic tension

Here’s the list of which categories our chords fall into:

Tonic Pre-dominant  Dominant
Major: I  vi   iii ii   IV   vi  iii  V  viio
minor:   i   III iio  iv   VI V  VII

 

There are a few things to note here. V in minor is Major, so we’re using the harmonic minor. The v chord in natural minor is hardly used, so I’m not even going to categorize it. Its counterpoint in Major is iii and that one is seen a fair amount. Formal music education says that it’s strictly a tonic-functioning chord used as a tonic extension, but I’ll argue that it largely acts as a dominant chord to vi, the chord of the relative minor.

Generally, the progression of the harmonic functions goes tonic to pre-dominant to dominant (a common chord progression is I – IV – V). This should be treated more as a guideline than as a rule because pre-dominants can go back to tonic (I – IV – I – V – I is common) and sometimes dominants can go back to pre-dominants (I – IV – I – V – IV – I is a common blues progression).

Secondary Dominance

As stated above, the tonic-dominant relationship is powerful. It’s so strong that we can even apply it to any Major or minor chord in our two scales. When we apply it in this way, we are employing secondary dominance. Essentially, we can treat a chord that’s not the tonic chord (I/i) as if it were a tonic chord by preceding it with a chord that acts as its dominant-functioning chord. In order to do this, we have to alter some of the scale degrees in the key that we’re in.

Let’s use an example to illustrate secondary dominance.  Let’s say that we’re in C Major and we have this chord progression: CM – Dm – GM – CM (I – ii – V – I).  If we want to use secondary dominance on Dm (ii), we precede it with a chord that acts as its dominant-functioning chord.  The distance between a regular tonic and dominant chord is 5 scale degrees (up a P5 from the tonic), so we precede Dm (ii) with the chord 5 scale degrees up from it, Am (vi). However, Am (vi) is minor and would be a weak sauce dominant chord, so we make it AM to make a leading tone up to D to beef it up. This AM chord would we notated as V/ii, “the five of two,” and now acts as a suitable secondary-dominant chord for Dm (ii). Our chord progression now looks like this: CM – AM – Dm – GM – CM (I – V/ii – ii – V – I). We can also secondary dominant seventh chords, which would be A7 (V7/ii) in this example.

We can also use leading tone chords in secondary dominance. Secondary leading tone chords are made by going a half step/semitone down from the root of a chord and building a diminished chord tthat precedes the target chord. Using our previous example, we would go a half step/semitone down from Dm to C# and use it as the root of a diminished chord, C#dim. This chord is notated as viio/ii, “the seven of two,” in this case. Applying this to our previous chord progression, we would have CM – C#dim – Dm – GM – CM (I – viio/ii – V – I).

One last thing to say about secondary dominance is that diminished and Augmented chords can never be the target of a secondary dominant chord. Only Major and minor chords can have secondary dominants, which makes sense because they have a P5 in them.  Going down a P5 is a powerful movement and is the heart of the tonic-dominant relationship.

Alex Gagliano

Alex Gagliano

Contributor

Isaac Schiller is a seasoned musician and amateur producer. He’s been playing piano since he was 6, writing music since he was 13 and producing EDM since he was 19. Although he is currently writing a pop song to advertise a friend’s business product, he experiments with trance and house music under his moniker, ORAK.
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