Chords

Major chords

A chord is a set of 3 or more notes played simulteanously. A triad is a chord composed of 3 notes. A tetrad is composed of 4 notes.
Major triad chords are composed of a root note, a note 4 semitones above the root (the major third), and a note 7 semitones above the root (the fifth). We notate the intervals composing the chord using curly brackets:
Major chord = { 0 4 7 }
We also notate the sequence of intervals between notes of the chord in square brackets []. We obtain it by subtracting each number with the previous one:
Major chord = { 0 4 7 } = [ 4-0, 7-4, 12-7] = [ 4 3 5 ]

Using our clock notation, a 0 (C) Major chord looks like this:

C Major =  0 + { 0 4 7 } = (0 4 7) = (C E G)

If we want to find the notes composing a 2 (D) Major chord, we just have to add  { 0, 4, 7 } to the root note 2.

D Major = 2 + { 0 4 7 } = ( 2 6 9 ) = (D F A)

Interestingly, the shape of the triangle is exactly the same. When we transpose the root note, we are actually doing a rotation of the triangle.

Minor chords

Using the previous notation,

Minor chord = { 0, 3, 7 } = [ 3, 4, 5 ]

C Minor = 0 + { 0 3 7 } = ( 0 3 7 )

Seventh chords

• A Major seventh chord is constructed by adding a fourth note to a major triad, 4 semitones above the last note (a major seventh). A C major seventh is noted C7M
• In a Minor seventh chord, the fourth note is 3 semitones above the third note (minor seventh).  A C minor seventh is noted C7m.
• A Dominant seventh chord is a major triad with a fourth note 3 semitones above the third note. (minor seventh). A C dominant seventh is noted C7.

M7 chord = { 0 4 7 11 } = [ 4 3 4 1 ]

m7 chord = { 0 3 7 10 } = [ 3 4 3 2 ]

7 chord = { 0 4 7 10} = [ 4 3 3 2 ]

C7M = 0 + { 0 4 7 11 }

 

C7m = 0 + { 0 3 7 10 }

This is quite interesting: both 7M and 7m chords have a trapezoid shape when drawn in the note clock. This shape has symmetry axe, but I am not sure whether this has any relationship with the harmony of the chord.
On the other hand, the dominant seventh is asymmetric:

C7 = 0 + { 0 3 7 10}

Application to the guitar fretboard

 

Major triad, E shape

Here is a method to to find any major triad (3 notes) on the fretboard. Bear in mind that in any operation, we may have to add or substract 12 to always stay in the [0, 11] range.

1. Find the root note. We know from the previous post that the standard tuning is ( 4 9 2 7 11 4 ), so if I want to play a 4 Maj (E Maj) chord, the first note of the chord can be on the open 6th string, or on the 7th fret of the 5th string (9+7 = 16, 16-12 = 4), or .... For this example let's use the open 6th string.
2. Find a note on the 5th string that is part of the chord and that can be played without stretching fingers too much. We know that our chord structure is { 0, 4, 7 }, and the that the 5th string is 5 semitones above the 6th string.
• Can I play the root note, +0 semitones above the root ? +0 (interval to play) - 5 (number of semitones between 6th and 5th string) = -5, -5 + 12 = 7th fret. I have not used any of my fingers yet, so I could play it. But let's carry on and see if we could play other notes of the chord.
• Can I play the major third, +4 semitones above the root ? +4 -5  = -1, -1+12 = 11th fret. It's even further away, let's continue
• Can I play the fifth, +7 semitones above the root ? +7 -5 = 2nd fret. This fret is much closer to the open string on the 6th string, so it looks like it is the best fit, let's use this.
3. Find a note on the 4th string. The 4th string is 5 semitones above the 5th string, hence it is 10 semitones above the 6th string. Let's repeat the same process.
• Play a root +0 ? +0 -10 (number of semitones between 6th and 4th string) = -10, -10 + 12 = 2nd fret.
• Play a major third +4 ? +4-10 = -6, -6+12 = 6th fret.
• Play a fifth +7 ? +7 -10 = -3, -3+12 = 9th fret.
4. Find a note on the 3rd string, which is 15 semitones above the 6th string. 
• +0 ? +0 -15 = -15, -15+12 = 3rd fret. 
• +4 ? +4 -15 = -11, -11+12 = 1st fret.
• +7 ? +7 -15 = -8, -8+12 = 4th fret
5. Find a note on the 2nd string, which is 19 semitones above the 6th string. 
• -19 is equivalent to -19 + 12 = -7
• +0 ? +0 -7 = -7, -7+12 = 5th fret
• +4 ? +4-7 = -3, -3+12 = 9th fret
• +7 ? +7-7 = 0th fret (open string)
6. Find a note on the 1st string, which is 24 semitones above the 6th string.
• -24 is equivalent to -24 + 12 + 12 = -0
• +0 ? +0 -0 = 0th fret
• +4 ? +4-0 = 4th fret
• +7 ? +7-0 = 7th fret
7. Let's play it !

4 Maj (E Maj)

 

This is the E-type shape of chord in the CAGED system. Notice that the +0 and +7 intervals are played many times on several strings. This explains why this shape of chord sounds more full than other types.
As you probably already know, you can use the same chord shape to play other chords, by just moving your hand down the neck.
For instance if you want to play a 7 Maj (G Maj) chord, just move your hand by 3 frets (7 Maj - 4 Maj = 3)

7 Maj (G Maj)

 

 Major triad, C shape

Using the same method, we can work out other chord shapes.

Here are the details for finding the C shape:

1. We want to play a 0 Maj (C Maj) chord, starting from the 5th string. The 5th string played open is 9, so if I want to play a 0, I need to press a finger on the 0 - 9 = -9, -9+12 = 3rd fret
2. 4th string: 
• Play a root +0 ? +0 -5 (number of semitones between 5th and 4th string) +3 (fret played on the 5th string) = -2, -2+12 = 10th fret. Impossible to play unless you have huge hands !
• Play a major third +4 ? +4 -5 +3 = 2nd fret. This is the closest to the 3rd fret
• Play a fifth +7 ? +7 -5 +3 = 5th fret
3. 3rd string:
• +0 ? +0 -10 +3 = -7, -7+12 = 5th fret
• +4 ? +4 -10 +3 = -3, -3+12 = 9th fret
• +7 ? +7 -10 +3 = 0th fret (open string)
4. The 2nd string is 14 semitones above the 5th string. We can simplify the calculations by always adding -14 + 3 = -11, -11+12 = +1
• +0 ? +0 +1 = 1st fret
• +4 ? +4 +1 = 5th fret
• +7 ? +7 +1 = 8th fret
5. The 1st string is 19 semitones above the 5th string. With the simplification we need to add -19 + 3 = -16, -16+12 = -4 to each chord interval.
• +0 ? +0 -4 = -4, -4 + 12 = 8th fret
• +4 ? +4 -4 = 0th fret
• +7 ? +7 -4 = 3rd fret
6. The resulting chord looks like this on the fretboard:

0 Maj (C Maj)

 

Other chords

As an excercise, you can work out the other CAGED shapes by solving the same puzzle and this information:

• The A shape starts on the 5th string
• The G shape starts on the 6th string
• The D shape starts on the 4th string

Minor chords are obtained by using { 0 3 7 } intervals, M7 by using {0 4 7 11}, ... (see above)