Notes On Music (Part 2)

8 min readApr 27, 2019

In the first part of this series, we went from sound waves as a basic building block of music to creating musical tones in a scale. This time, we’ll be picking up where we left off by combining the notes that we made into chords.

Introducing Chords

We concluded our last post by dividing the frequencies between the fundamental and the second harmonic ― an octave ― into twelve equal temperament tones. From that, we built a heptatonic scale out of the seven notes that were closest to being simple ratios. Today we’re going to talk about how we combine those notes into chords.

The simple way to think of chords is just to view them as multiple notes sounding at once. If we play a C an E and a G at same time, we are playing a chord:

When we created our scale, we learned to refer to the notes in it by scale degree, which is really just the counted number of the note in the scale:

1    Tonic
2    Supertonic
3    Mediant
4    Subdominant
5    Dominant
6    Submediant
7    Leading Tone

We form our chords off of these scale degrees. In C major, for example, C is the tonic. In the context of a chord, we call the note the root note — so if we form our chord off of the tonic, we say that the root note is C. The resulting chord in this example is a C major chord.

To distinguish the note C1 and the chord C1, we can use roman numerals:

As you can see, just as I (C) is formed from the root note C, ii (Dm) is formed off of the root note D. Looking at the above, our chords for C major are:

I       C major (C)
ii      D minor (Dm)
iii     E minor (Em)
IV      F major (F)
V       G major (G)
vi      A minor (Am)
vii°    B diminished (B°)

In this chart, we are using lower-case roman numerals to easily recognize minor chords. There are multiple conventions for making this distinction, but I’ll try to sick with upper- and lower-case for consistency.

The most basic type of chord is a triad. In addition to the root note, it contains, as you might guess from the name, two other notes:

Major triads contain a major third and perfect fifth interval (0–4–7 semitones).
Minor triads contain a minor third and a perfect fifth (0–3–7 semitones).
Diminished triads contain a minor third and a flattened fifth (0–3–6 semitones).
Augmented triads contain two major thirds (0–4–8 semitones).

Major, Minor, Diminished and Augmented triads

So, in C major, the major triad I (where our root note is the tonic) is C / E / G:

Let’s look at C major again as triads:

I     C major (C)          C / E / G
ii    D minor (Dm)         D / F / A
iii   E minor (Em)         E / G / B
IV    F major (F)          F / A / C
V     G major (G)          G / B / D
vi    A minor (Am)         A / C / E
vii°  B diminished (B°)    B / D / F

Hopefully you can see how each of these triads was built using the rules that we defined above. Just to be sure, let’s pick ii and walk through the process:

We’re in C major so the second degree of the scale is D — this will be our root note.
We’re building a minor triad, so the next note to include is 3 semitones above D. This is F.
The final note is a perfect fifth above D — 7 semitones. This is A.
So ii is D minor: D / F / A

Extending our Triads

Let’s move beyond triads and look at the seventh chords. The major and minor seventh chords are formed by adding the major or flat 7th to a basic triad. For example:

C Major I (C / E / G) becomes C / E / G / B
C Major vi (A / C / E) becomes A / C / E / G

What is interesting about that is that adding that 7th (putting the B at the end of C / E / G) effectively added an E minor triad to the root (C): When we create the seventh chord C / E / G / B, what we end up with isn’t just C major with an additional B — it is also E minor with an additional C.

If that doesn’t make sense, look at the diagram above. Our 1―3―5 are the notes C / E / G. The seventh is a B. So we have C / E / G / B. The last part of that is the triad E / G / B ― which is E minor. So we hear a C, but also C + E minor.

This minor chord that we created on top of our major chord is known as the upper-structure. Let’s look at the seventh chords in C major. I’ve made the upper-structures bold:

I     C / E / G / B
ii    D / F / A / C
iii   E / G / B / D
IV    F / A / C / E
V     G / B / D / F♯
vi    A / C / E / G
vii°  B / D / F / A♭

Major and minor chords evoke very different feelings: major chords tend to feel open and majestic, while minor chords tend to feel sadder or more intimate. By adding that 7th to our triad, the presence of the minor chord actually introduced some sadness or intimacy to the sound. The lower structure is still there — this is still a major chord — but the upper structure helps shape the sound into something more intimate.

It is easy to imagine two instruments creating a seventh chord: the first instrument playing a major chord (1 ― 3―5) while another hits the 7. In this case, the second instrument makes the chord sound smaller and more intimate: it adds something, but in a sense, it also subtracts from the sound.

Inversions

Octave equivalency — the idea that a note is “the same” even when offset by an octave — applies when building chords. We rely on this principle to create an inversion of a chord. In an inversion, we move some of our notes by an octave. For example, in C major, I could be played as C2 / E2 / G2, but it could also be played as:

E2 / G2 / C3 (first inversion)
G2 / C3 / E3 (second inversion)

Looking at the keys, this gives us:

Looking at C Major I and the first and second inversions in notation, we get:

You might see inversions written as slash chords, where C/E means a C major chord with E as the lowest note (the bass note). In other contexts, inversions can use a notation called figured bass. In figured bass, normal triad would be represented by it’s chord and 5 on top of a 3. This is because the triad is formed by the fifth and third intervals. But, since this is implied, it probably isn’t even written. The first inversion is written with a small 6 on top of a 3, but since the 3 is still implied, it might be written like: IV6. The second inversion is written with a 6 on top of a 4, or when that isn’t possible IV64. Figured bass is mostly the realm of classical musicians. Outside of inversions, you might never encounter it.

So why invert? Inverted chords often have a distinctive color that adds a weakness or even poignancy to the harmony (this is why you’ll seldom hear two second inversions in a row or end a chord progression with one). Perhaps more importantly, inverting chords allows us to begin controlling voicing.

Voicing

As we extended and inverted our chords, we started to see that our initial definition of chords was accurate… but a bit too simplistic.

When we discussed how a second instrument could contribute to creating an upper structure, we caught a glimpse of the notes that we write not as vertical blocks, but as parts that move through a melody and together create harmony ― as individual voices simultaneously sounding notes that combine to be more than just their individual contributions.

Not only can we choose which instruments or voices perform each note, we can arrange the vertical spacing and ordering of the notes in our chords to create harmonic options for how those voices flow together.

Just like octave equivalency makes inversion possible, it allows us to move double notes one octave higher or lower and to spread out the voicing of a chord across multiple octaves.

We can, for example, use inversions to decide which notes are on the top or in the middle of our chords in order to create step-wise smoothing in a bass melody by creating a passing or neighboring bass-motion.

In the next part of this series, this shift of thinking — from vertical blocks of notes into voices that flow throughout a piece of music — will be core to how we approach building chord progressions.

Back to the Beginning

Just for fun, because now we’re starting to get deeper into “music”, let’s go all the way back to our waves and frequency from Part 1, and look at C / E / G:

We know that A4 is 440 Hz and that each semitone increases/decreases by the twelfth root of two logarithmically. That means that C is 130.81 Hz, E is 164.81 Hz, and G is 196 Hz — and we’re playing those all at the same time.

So we can easily graph these notes using any graphing software:

In this example, the root note, C, is darkest. We can add those waves to look at what the chord really looks like:

Adding together the notes

What I hope to demonstrate here is that even when we are thinking in whole numbers instead of a logarithmic scale and the twelfth root of two, we can go back down to that lower level when required.

In the next part of this series, we will build on what we learned about chords to create chord progressions. See you there!