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)

Clock notation

I am a guitar player, who has learned playing exclusively with tablatures. Now I am a bit tired of playing like a robot without understanding why the songs I like are written as they are. I can feel that there is an underlying structure and logic common to many songs. I have tried to read and understand some material on music theory and harmony, but I generally find it non-logical, confusing, and difficult to apply to guitar playing.
In this blog I will try to explain harmony by using numbers and geometric figures, which in my opinion is more natural, logical, and easy to remember. I hope this will be helpful for guitarists or keyboard players without a background in music theory.

Rock around the clock

You probably already know the major scale, C D E F G A B C, or Do Re Mi Fa Sol La Si Do. The distance or interval between two notes is expressed in tones and semitones.  2 semitones = 1 tone. You may also know that there are two semitones (one tone) between most notes, such as C (Do) and D (Re), but just one semitone between E and F and between B and C:

• C 2 D 2 E 1 F 2 G 2 A 2 B 1 C

This notation maps nicely to the white keys of the piano keyboard:

• When there is a black key between two white keys, the keys are separated by 2 semitones. 
• When there is no black key (B - C and E - F), the keys are separated by 1 semitone. 
• A black key is 1 semitone above the preceding white key. The note's name is suffixed with a sharp #.

Therefore, unlike the guitar, the piano keyboard is not uniform: two adjacent white keys can be separated by 1 or 2 semitones, depending on their position. This is why some people build alternative keyboards: http://www.altkeyboards.com

Now, why on earth do we use consecutive letters, which look like a regular interval, when the intervals between letters are not all the same ? This makes calculating intervals between two notes quite cumbersome. For example the interval between C and F (3 letters "above" C) is 2.5 tones, but the interval between F and B (3 letters "above" F) is 3 tones...
Playing music with traditional notation looks like doing arithmetic with roman numerals !

Fortunately, there is a simpler notation, called clock or integer modulo 12 notation:

Each note is named with a number, from 0 to 11. With this notation, calculating intervals between two notes is as simple as doing additions and subtractions:

• The interval between 0 (C) and 5 (F) is 5-0 = 5 semitones (2.5 tones). 
• The interval between 5 (F) and 11 (B) is 11-5 = 6 semitones (3 tones).
• The interval between 11 (B) and 2 (D) is 2 + 12 - 11 = 3 semitones (1.5 tones). 

Whenever an operation's result is above 12 or below 0, just subtract / add 12 to stay in the 0 to 11 range.
The 0 (C) major scale becomes:
0 2 2 2 4 1 5 2 7 2 9 2 11 1 0

Now, try to memorize 2 2 1 2 2 2 1, the sequence of intervals of the major scale. Start with any key note, add these numbers, and you obtain the notes of the major scale in any key.
For example, the 5 (F) major scale is:
5 2 7 2 9 1 10 2 0 2 2 2 4 1 5
F   G  A   A#  C   D   E   F

Interval names

Now we know how to calculate the interval between two notes, using the number of semitones. However most musicians prefer to use these awkward names:

Semitones	Name	Position of the note in the C major scale
0	Unison	1st - C
1	Minor second
2	Second	2nd - D
3	Minor third
4	Major third	3rd - E
5	Perfect fourth	4th - F
6	Augmented fourth or Diminished fifth
7	Perfect fifth	5th - G
8	Minor sixth
9	Major sixth	6th - A
10	Minor seventh
11	Major seventh	7th - B
12	Octave	8th - C

Which one do you prefer ? Names or numbers ? I personally find the names of the intervals quite confusing. When someone talks about a "fifth", I hear 5, so I tend to think that there are 5 semitones or 5 tones, when there are in fact 7 semitones in a fifth... Admittedly, there is a logic behind these names: they are named in relation to the position of the note in the C major scale.

 Now, I am not going to change the way millions of musicians call intervals, but as a coping strategy I memorized the table above and mentally translate an interval name into its number of semitones.

Application to the guitar fretboard

The guitar fretboard is divided in semitones, which means that we can again do simple arithmetic to work out how to play the major scale in any key.
The standard guitar tuning is E A D G B E, from the 6th (biggest) string to the 1st (thinnest) string. Using the clock diagram above, this tuning translates into 4 9 2 7 11 4 in the clock notation.
By subtracting these numbers in pairs, we can work out the intervals between the strings:

• Between 6th and 5th: 9 - 4 = 5 semitones
• Between 5th and 4th: 2 - 9 = -7, -7 + 12 = 5 (when we obtain a negative number, add 12)
•  ...
• Between 3rd and 2nd: 11 - 7 = 4 
• Between 2nd and 1st: 4 - 11 = -7, -7 + 12 = 5
• 4 5 9 5 2 5 7 4 11 5 4

We can see that the interval between two strings is 5 semitones, except between the 3rd (G) and 2nd (B) strings, where we have 4 semitones. This is why some people prefer to tune their guitar in "All fourths" (5 semitones between every string), or in "Major thirds" (4 semitones between strings).

Using this knowledge, let's play the 5 (F) major scale on the guitar. As mentioned above, the intervals of the major scale are 2212221.

• We need to find the root note of the scale, 5 or F or Fa. The 6th string of the guitar is the note 4 (E). So if I press the 1st fret, I am going to play a 4 + 1 = 5 (F)
•  The second note of the scale is 2 semitones above the root, so I play the 1st fret + 2 semitones = 3rd fret
• The third note is again 2 semitones above the previous note. I can play the 3rd fret + 2 semitones = 5th fret on the 6th string. But since the 6th and 5th strings are separated by 5 semitones, I can also play the 5th string open, because 5th fret - 5 semitones = 0
• The fourth note is 1 semitone above the previous note. We play it on the 0th fret + 1 semitone = 1st fret of the 5th string.
• ... and so on. When your fingers are stretching, you can play a note on the following string if you subtract 5 to the number of frets, or 4 if you are going from the 3rd to 2nd string. 

The result is the following:

This means that you no longer have to rote learn the major scale in all keys and in all positions. You just have to remember three things and you can work it out !

• 2212221, the intervals of the major scale
• 4 9 2 7 11 4, the standard tuning of the guitar
• When you speak with other musicians or read partitions, you need to know the correspondence between note letters and note integers: C D E F   G A B C => 0 2 4 5   7 9 11 0

As an exercise, try to use this trick to play a 8 (G#) major scale, starting on the 4th string.

In my next post I will explain how to find and play chords using the clock notation.