Thirds & Sixths (from Harmonising the Major Scale)

close2u

An explanation in several parts.

Step 1 – Harmonising the major scale

Step 2 – Intervals of thirds

Step 3 – Intervals of Sixths

Step 4 – Sixths are inverted thirds

Step 5 – If an interval has the same two notes, is it a 3rd or a 6th?

Step 6 – Double stop 3rds mapped out in G

Step 7 – Double stop 6ths mapped out in G

Step 8 - thirds and sixths mapped in chord shapes from a harmonised major scale in A

close2u

Step 1 – Harmonising the major scale

Consider G major - using the major scale formula:
whole, whole, half, whole, whole, whole, half  - we get these seven notes:
G  A  B  C  D  E  F#  
String several octaves together and you have:
G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  …

OK?

Now the chords in the scale (called the diatonic chords) are found using a process called ‘harmonising the major scale’.
These chords will all be triads (three note chords).
Take each of the seven notes in turn.
Each note is the root of a chord.
Each chord contains three notes.
One is the root note.
The other two notes are found at intervals of a third from the root.
This means count 1, miss 1, count 1, miss 1, count 1.
Giving the famous 1, 3, 5 chord formula.
To count this, the root note counts as 1.

So:
Chord I
Root note = G
G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  …
Counting:
1, 3, 5 = G, B, D
Chord = G Major

Chord II
Root note = A
A major scale = A, B, C#, D, E, F#, G#
Counting:
1, 3, 5 = A, C#, E
BUT
C# is not a note in the G major scale (the scale we are harmonising).
You see the only note with a ‘C’ in its name in the G major scale is C natural.
And we need to use only the notes in the G major scale when harmonising the G major scale.
Therefore, this third note from our counting must be ‘flattened’ to match the notes found in the G major scale.
So, instead of
1, 3, 5 = A, C#, E
G major scale has
1, b3, 5 = A, C, E
Chord = A minor (a flat 3rd note makes a minor chord)

Chord III
Root note = B
B major scale = B, C#, D#, E, F#, G#, A#
Counting:
1, 3, 5 = B, D#, F#
BUT
D# is not a note in the G major scale (the scale we are harmonising).
You see the only note with a ‘D’ in its name in the G major scale is D natural.
And we need to use only the notes in the G major scale when harmonising the G major scale.
Therefore, this third note from our counting must be ‘flattened’ to match the notes found in the G major scale.
So, instead of
1, 3, 5 = B, D#, F#
G major scale has
1, b3, 5 = B, D, F#
Chord = B minor (a flat 3rd note makes a minor chord)


Chord IV
Root note = C
C major scale = C, D, E, F, G, A, B
Counting:
1, 3, 5 = C, E, G
Chord = C Major

Chord V
Root note = D
D major scale = D, E, F#, G, A, B, C#
Counting:
1, 3, 4, = D, F#, A
Chord = D Major

Chord VI
Root note = E
E major scale = E, F#, G#, A, B, C#, D#
Counting:
1, 3, 5 = E, G#, B
BUT
G is not a note in the G major scale (the scale we are harmonising).
You see the only note with a ‘G’ in its name in the G major scale is G natural.
And we need to use only the notes in the G major scale when harmonising the G major scale.
Therefore, this third note from our counting must be ‘flattened’ to match the notes found in the G major scale.
So, instead of
1, 3, 5 = E, G#, B
G major scale has
1, b3, 5 = E, G, B
Chord = E minor (a flat 3rd note makes a minor chord)

Chord VII
Root note = F#
F# major scale = F#, G#, A#, B, C#, D#, E#
Counting:
1, 3, 5 = F#, A#, C#
BUT
Neither A# nor C# are notes in the G major scale (the scale we are harmonising).
You see the only notes with ‘A’ or ‘C’ in their names in the G major scale are A
natural and C natural.
And we need to use only the notes in the G major scale when harmonising the G major scale.
Therefore, this third note and the fifth note from our counting must be ‘flattened’ to match the notes found in the G major scale.
So, instead of
1, 3, 5 = F#, A#, C#
G major scale has
1, b3, b5 = F#, A, C
Chord = F# diminished (a flat 3rd note and a flat 5th note makes a diminished chord)

close2u

Step 2 –Intervals of thirds

Here is the G major scale again:
G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  …

Intervals of thirds are used to give the first two notes contained in the triads above.

Thirds are going to be major or minor.
Major thirds span five half tones (= 5 frets).
Minor thirds span four half tones (= 4 frets).

So:

G to B is a 3rd (a Major 3rd ... 5 half tones = 5 frets & remember the I chord is a Major chord)

A to C is a 3rd (a minor 3rd ... 4 half tones = 4 frets & remember the II chord is a minor chord … remember the b3 note?)

B to D is a 3rd (a minor 3rd ... 4 half tones = 4 frets & remember the III chord is a minor chord)

C to E is a 3rd (a Major 3rd ... 5 half tones = 5 frets & remember the IV chord is a Major chord)

D to F# is a 3rd (a Major 3rd ... 5 half tones = 5 frets & remember the V chord is a Major chord)

E to G is a 3rd (a minor 3rd ... 4 half tones = 4 frets & remember the VI chord is a minor chord)

F# to A is a 3rd ( a minor 3rd ... 4 half tones = 4 frets)

Listed in a simple form here are the thirds from the G major scale:

Major - G to B

minor - A to C

minor - B to D

Major - C to E

Major - D to F#

minor - E to G

minor - F# to A

close2u

Step 3 – Intervals of Sixths

Here is the G major scale again:
G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  …

Intervals of sixths are found by counting six along the scale..

Sixths are going to be major or minor.
Major sixths span ten half tones (= 10 frets).
Minor thirds span nine half tones (= 9 frets).

So:

B to G is a 6th (a minor 6th ... 9 half tones = 9 frets)

C to A is a 6th (a Major 6th ... 10 half tones = 10 frets)

D to B is a 6th (a Major 6th ... 10 half tones = 10 frets)

E to C is a 6th (a minor 6th ... 9 half tones = 9 frets)

F# to D is a 6th (a minor 6th ... 9 half tones = 9 frets)

G to E is a 6th (a Major 6th ... 10 half tones = 10 frets)

A to F# is a 6th (a Major 6th ... 10 half tones = 10 frets)

Listed in simple form here are the sixths from the G major scale:

minor - B to G

Major - C to A

Major - D to B

minor - E to C

minor - F# to D

Major - G to E

Major - A to F#

close2u

Step 4 – Sixths are Inverted Thirds

For a given list of thirds, listing the sixths is easy.
Just reverse the order in each pairing.

3rd = G to B .... so 6th = B to G

3rd = A to C ... so 6th = C to A

3rd = B to D ... so 6th = D to B

3rd = C to E ... so 6th = E to C

3rd = D to F# ... sp 6th = F# to D

3rd = E to G ... so 6th = G to E

3rd = F# to A so 6th = A to F#

How is it that 6ths are inverted 3rds?
Again, string several octaves of the G major scale together and you have:
G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  A  B  C  D  E  F#  G  …

Example 1.
Can you see that G to B is a count of 1, 2, 3 = an interval of three notes = a third?
Look.
Can you see that B to G is a count of 1, 2, 3, 4, 5, 6 = an interval of six notes = a sixth?
Example 2.
Can you see that A to C is a count of 1, 2, 3 = an interval of three notes = a third?
Look.
Can you see that C to A is a count of 1, 2, 3, 4, 5, 6 = an interval of six notes = a sixth?

This works for all such intervals of 3rds and 6ths.
So it is that 6ths are inverted 3rds.

Note:
A Major 3rd inverts to a minor 6ths and vice versa.
This can be seen when listed side by side.

Major 3rd G to B inverts to minor 6th B to G

minor 3rd A to C inverts to Major 6th C to A

minor 3rd B to D inverts to Major 6th D to B

Major 3rd C to E inverts to minor 6th E to C

Major 3rd D to F# inverts to minor 6th F# to D

minor 3rd E to G inverts to Major 6th G to E

minor 3rd F# to A inverts to Major 6th A to F#

close2u

Step 5 – If an interval has the same two notes, is it a 3rd or a 6th?

Intervals are always considered from the lowest note.
If you have two notes, say G and B and wanted to describe the interval, you would call it from the lowest.
If G was the lowest of the two, the interval is a 3rd.
If B was the lowest, the interval is a 6th.

close2u

Step 6 – Double stop 3rds mapped out in G

5 sets of 3rds that lie on adjacent strings.
Root notes are marked in red.
Other notes of the G major scale are marked in black along the two strings in each diagram.
Blue lines connect the 3rds.

http://i234.photobucket.com/albums/ee238/chaucer73/misc/Gdoublestop3rds.jpg

close2u

Step 7 – Double stop 6ths mapped out in G

4 sets of 6ths that lie on strings two apart.
Root notes are marked in red.
Other notes of the G major scale are marked in black along the two strings in each diagram.
Blue lines connect the 6ths.

http://i234.photobucket.com/albums/ee238/chaucer73/misc/Gmajor6thsall.jpg

close2u

Step 8 - thirds and sixths mapped in chord shapes from a harmonised major scale in A

Key of A Major.
Harmonised Major scale ...

A Major - B minor - C# minor - D Major - E Major - F# minor - G# diminished

These are tabbed below as a progression of '4-string small barre chords' starting at fret 5.
Each tab diagram shows fret positions for the chords, actual notes of the chords and scale degree of those notes.
The left hand side shows this information with the 3rds highlighted in yellow, the right hand side shows it with the 6ths highlighted in yellow.
The 3rds and 6ths are based only on the 1 to the 3, the 3 to the 1 for Major chords and the 1 to the b3 and the b3 to the 1 for minor / diminished chords.


This first diagram, based on the A Major chord at fret 5, shows two intervals highlighted:
Interval A to C# ... 1 to 3 = 4 semitones = 2 tones = Major 3rd
Interval C# to A ... 3 to 1 = 9 semitones = 4 1/2 tones = minor 6th


e  --||-----5-----A-----1-----||-----5-----A-----1-----||--
B  --||-----5-----E-----5-----||-----5-----E-----5-----||--
G  --||-----6-----C#---3-----||-----6-----C#---3-----||--
D  --||-----7-----A-----1-----||-----7-----A-----1-----||--
A  --||--------------------------||-------------------------||--
E  --||--------------------------||-------------------------||--



This second diagram, based on the B minor chord at fret 7, shows two intervals highlighted:
Interval B to D ... 1 to b3 = 3 semitones = 1 & 1/2 tones = minor 3rd
Interval D to B ... 3 to 1 = 10 semitones = 5 tones = Major 6th
   
           
e  --||-----7-----B-----1------||-----7-----B-----1------||--
B  --||-----7-----F#----5-----||-----7-----F#----5------||--
G  --||-----7-----D----b3-----||-----7-----D----b3-----||--
D  --||-----9-----B-----1------||-----9-----B-----1------||--
A  --||---------------------------||---------------------------||--
E  --||---------------------------||---------------------------||--


This third diagram, based on the C# minor chord at fret 9, shows two intervals highlighted:
Interval C# to E ... 1 to b3 = 3 semitones = 1 & 1/2 tones = minor 3rd
Interval E to C# ... b3 to 1 = 10 semitones = 5 tones = Major 6th
  
           
e  --||-----9-----C#----1------||-----9-----C#-----1-----||--
B  --||-----9-----G#----5------||-----9-----G#-----5-----||--
G  --||-----9-----E--- -b3-----||-----9------E-----b3-----||--
D  --||-----11----C#----1-----||-----11----C#-----1-----||--
A  --||----------------------------||----------------------------||--
E  --||----------------------------||----------------------------||--

close2u

This fourth diagram, based on the D Major chord at fret 10, shows two intervals highlighted:
Interval D to F# ... 1 to 3 = 4 semitones = 2 tones = Major 3rd
Interval F# to D ... 3 to 1 = 9 semitones = 4 1/2 tones = minor 6th
 
           
e  --||-----10-----D-----1-----||-----10-----D------1-----||--
B  --||-----10-----A-----5-----||-----10-----A------5-----||--
G  --||-----11-----F#----3-----||-----11-----F#----3-----||--
D  --||-----12-----D-----1-----||-----12-----D------1-----||--
A  --||----------------------------||-----------------------------||--
E  --||----------------------------||-----------------------------||--


This fifth diagram, based on the E Major chord at fret 12, shows two intervals highlighted:
Interval E to G# ... 1 to 3 = 4 semitones = 2 tones = Major 3rd
Interval G# to E ... 3 to 1 = 9 semitones = 4 1/2 tones = minor 6th
 
           
e  --||-----12-----E------1------||-----12-----E------1------||--
B  --||-----12-----B------5------||-----12-----B------5------||--
G  --||-----13-----G#----3------||-----13-----G#----3-----||--
D  --||-----14-----E------1------||-----14-----E------1------||--
A  --||------------------------------||----------------------------||--
E  --||------------------------------||----------------------------||--


This sixth diagram, based on the F# minor chord at fret 14, shows two intervals highlighted:
Interval F# to A ... 1 to b3 = 3 semitones = 1 & 1/2 tones = minor 3rd
Interval A to F# ... 3 to 1 = 10 semitones = 5 tones = Major 6th
 
           
e  --||-----14-----F#----1-----||-----14-----F#----1-----||--
B  --||-----14-----C#----5-----||-----14-----C#----5-----||--
G  --||-----14-----A----b3-----||-----14-----A----b3-----||--
D  --||-----16-----F#----1-----||-----16-----F#----1-----||--
A  --||----------------------------||-----------------------------||--
E  --||----------------------------||-----------------------------||--



This seventh diagram, based on the G# diminished chord at fret 16, shows two intervals highllighted:
Interval G# to B ... 1 to b3 = 3 semitones = 1 & 1/2 tones = minor 3rd
Interval A to F# ... 3 to 1 = 10 semitones = 5 tones = Major 6th

          
e  --||-----16-----G#----1-----||-----16-----G#----1-----||--
B  --||-----15-----D----b5-----||-----15-----D----b5-----||--
G  --||-----16-----B----b3-----||-----16-----A----b3-----||--
D  --||-----18-----G#----1-----||-----18-----G#----1-----||--
A  --||-----------------------------||-----------------------------||--
E  --||-----------------------------||-----------------------------||--

close2u

So, the harmonised major scale formula of chords :

Major – minor – minor – Major – Major – minor – diminished

gives rise to a matching pattern of 3rds (diminished is tricky and gives a minor 3rd)

Major – minor – minor – Major – Major – minor – minor

And because the interval of a 6th is an inversion of the interval of a 3rd, the reverse holds true for the 6ths and the inverse pattern arises:

Minor – Major – Major – minor – minor – Major – Major



In summary, and with reference to the 1, 3 and 1 for Major chords and 1, b3, 1 for minor / diminished chords:

a Major chord contains a Major 3rd and a minor 6th

a minor chord contains a minor 3rd and a Major 6th

a diminished chord contains a minor 3rd and a Major 6th


I hope that is helpful / useful.

jellyroll

Thank you! Should the penultimate sentence say "a minor chord contains a minor third and a major 6th"? Or have I misunderstood? Edit: OP's post now edited.

close2u

@jellyroll
Yes - well spotted - thanks. Edit / correction now done.

close2u

A variety of videos on practicing, using and applying.

close2u

This video appeared in my subscription list today ,,, double stop 6ths & 3rds

Phil_aka_Pip

@close2u

IIRC the spelling of a diminished triad is R b3 b5, and the dim7 chord has bb7 on top. A doubly flattened 7th looks like a major 6th when you finger it, but is it correct to spell it as 6 instead of bb7?

viz

No because it's a minor chord with a minor 6th (which isn't in the dim7 chord, but is part of the scale - it's the note below the one in question), therefore the note in question is a 7th not a 6th, and must be named bb7.

Phil_aka_Pip

@viz I get the last part of what you said (ie i agree about the bb7)  but I didn't get the minor chord with a minor 6th bit.

viz

Thread resurrection time, even though Pip's not here at the mo - I was trying to say that the name "6th" is already used up (it's the minor 6th above the tonic) so you can't use it for that bb7 note.