Chords across keys

[sclay]

As a keen student of theory, I’m always fascinated by new pieces of logic, and in particular how these can possibly help me to both better understand the guitar, and put into practice any possible uses they may have.
Some of these fascinating logical patterns seem obvious when they are revealed to you, but there they often are - hidden away in plain sight.

Here’s a cool one I came across. I’m currently trying to reverse engineer it in my head to understand why it is so. Circle of 5ths will likely help here. Anyway, enough rambling. Naturally, cue @Richard_close2u

Any particular major or minor triad will appear in 3, and only 3, major keys, and in 3, and only 3, minor keys.
So 6 keys in total.

Take C major. It only appears in the keys of C,F,G, and keys Am,Dm,Em

Take Am. It only appears in the keys Am,Dm,Em, and the keys C,F,G

Take D major. It only appears in the keys D, G, A, and the keys Bm,Em,F#m

Take Bm. It only appears in the keys Bm, Em, F#m, and the keys D,G,A

See the pattern? (I’ve used relative keys to make the pattern more obvious).

The 6 keys that a particular chord is a member of, are always the 1,4,5 of the parent key, and, logically the 1,4,5 of the relative key.

So,

C Major chord- C,F,G - Am,Dm,Em

Am chord - Am,Dm,Em - C,F,G

D Major chord- D,G,A - Bm,Em,F#m

Bm chord - Bm,Em,F#m - D,G,A

A cool little relationship that seemingly has its roots in the interplay between intervals in the key, and stacked thirds. Perhaps it will be of some use. Perhaps it may lead to some other, more useful logic. Not sure yet.
I am expecting that Richard will formalise it shortly, and elegantly.

Cheers, Shane

[Lisa_S]

Thanks for sharing this one, Shane!

I was having a kind of a light bulb moment reading through your post and it’s making a lot of sense. I wonder how I could not have seen this 20 years back when I was studying loads of theory during my piano times. Great food for thought.

Cheers - Lisa

[Richard_close2u]

Something to get my teeth into perhaps when I have a bit of time.

[Richard_close2u]

@sclay

As a keen student of theory, I’m always fascinated by new pieces of logic, and in particular how these can possibly help me to both better understand the guitar, and put into practice any possible uses they may have.

A man after my own heart!

@sclay

Circle of 5ths will likely help here.

It helps in many instances … let’s see.

[Richard_close2u]

@sclay

Any particular major or minor triad will appear in 3, and only 3, major keys …

If we take a major scale chart (similar to one you will find in Justin’s Practical Music Theory course) and colour code all triads with a root note of C natural we find, as you describe, three major triads. We also find three minor triads and one diminished triad. Seven triads in total. Necessarily so, as there are exactly seven positions the note C natural could fit within sets of major scales, 1st then 2nd, 3rd, 4th, 5th, 6th and 7th scale degrees.

I have created this graphic to illustrate.

The major triads occur when our root note (C natural) is 1st, 4th and 5th. Of course. The chords built upon those scale degrees of a major scale are always major.

The minor triads occur when our root note (C natural) is 2nd, 3rd and 6th. Of course. The chords built upon those scale degrees are always minor.

The diminished triad occurs when our root note (C natural) is 7th. Of course. The chord built upon that scale degrees is always diminished.

[Richard_close2u]

If we take a major scale chart and modify it so that all scales now start on their 6th scale degrees, we will have a chart showing all matching relative minor scales.
The colour coding system remains as before.
We again find three major triads, three minor triads and one diminished triad. Seven triads in total. Necessarily so, as there are exactly seven positions the note C natural could fit within sets of major scales, 1st then 2nd, 3rd, 4th, 5th, 6th and 7th scale degrees.

I have created this graphic to illustrate.

Now it is the minor triads that occur when our root note (C natural) is 1st, 4th and 5th. Of course. The chords built upon those scale degrees of a minor scale are always minor.

The major triads also switch and occur when our root note (C natural) is 2nd, 3rd and 6th.

The diminished triad occurs when our root note (C natural) is 2nd. As it should.

All scale degrees have in fact moved the same number of places from their positions in the relative major scales.

6th → 1st … (6 → 5 → 4 → 3 → 2 → 1)
7th → 2nd … (7 → 6 → 5 → 4 → 3 → 2)
1st → 3rd … (1 → 7 → 6 → 5 → 4 → 3)
2nd → 4th … (2 → 1 → 7 → 6 → 5 → 4)
3rd → 5th … (3 → 2 → 1 → 7 → 6 → 5)
4th → 6th … ( 4 → 3 → 2 → 1 → 7 → 6)

[Richard_close2u]

“What, no Circle of Fifths?”, I hear you ask.

Okay.
If you insist.

Take a look at this beauty of man and nature.

What were those major keys where we found the C major triad, remind me? C, F and G wasn’t it?

And what minor keys did we find the C major triad? That’s it, Am, Dm and Em.

Holy macaroni. Would you just take a wee look up top, in the 11, 12 and 1 o’ clock positions.
On the outer wheel we have F, C, G. On the inner wheel we have Dm, Am, Em.
The Circle of Fifths places the relative major and minors directly adjacent to one another, major on the outer and minor on the inner.

Which major chord can be found in all of these keys?
B minor, C# minor, F# minor, A major, D major and E major?

[Richard_close2u]

Would it be remiss of me to point out that we have only considered major keys and their relative minors?

Would it be confusing to state that these two types of key are also two parts of the modal system?

Major = Ionian and Minor = Aeolian

[Richard_close2u]

First, looking at the major type modes.

Ionian is just one of three modes whose character is ‘major’. Ionian, Lydian and Mixolydian are the full set.

• Is the C major triad found in C Ionian, C Lydian and C Mixolydian?

• Is the C major triad found in F Ionian, F Lydian and F Mixolydian?

• Is the C major triad found in G Ionian, G Lydian and Mixolydian?

• Do we have 3 x 3 = 9 major modes where we can find the C major triad?

Ionian is identical to the major scale of each so of course the C major triad exists in all three of those Ionian modes.

If C is the root note, the C major triad must necessarily exist as the tonic chord of C Lydian and C Mixolydian because they are major type modes.

Consider F Lydian and F Mixolydian.

The Parent Major Scale of F Lydian is C major. Bingo. Of course that contains the C major triad.
The Parent Major Scale of F Mixolydian is B flat major. As we can see in the major scale chart above, that contains a C minor triad.

Next to consider G Lydian and G Mixolydian.

The Parent Major Scale of G Lydian is D major. As we can see in the major scale chart above, that contains a C# diminished triad.
The Parent Major Scale of G Mixolydian is C major. Aha. As with F Lydian, that contains the C major triad.

Of the nine modes we investigated, two (F Mixolydian and G Lydian) did not contain the C major triad.

Therefore seven major type modes do contain the C major triad.

[Richard_close2u]

Now, looking at the minor type modes.

Aeolian is just one of three modes whose character is ‘minor’. Aeolian, Dorian and Phrygian make up the full set.

• Is the C major triad found in A Aeolian, A Dorian and A Phrygian?

• Is the C major triad found in D Aeolian, D Dorian and D Phrygian?

• Is the C major triad found in E Aeolian, E Dorian and E Phrygian?

• Do we have 3 x 3 = 9 minor modes where we can find the C major triad?

Aeolian is identical to the minor scale of each so of course the C major triad exists in all three of those Aeolian modes. See the Minor Scales chart above.

Consider D Dorian and D Phrygian.

The Parent Major Scale of D Dorian is C major. Bingo. Of course that contains the C major triad.
The Parent Major Scale of D Phrygian is B flat major. As we can see in the major scale chart above, that contains a C minor triad.

Next to consider E Dorian and E Phrygian.

The Parent Major Scale of E Dorian is D major. As we can see in the major scale chart above, that contains a C# diminished triad.
The Parent Major Scale of E Phrygian is C major. Aha. As with D Dorian, that contains the C major triad.

Of the nine modes we investigated, two (D Phrygian and E Dorian) do not contain the C major triad.

Therefore seven minor type modes do contain the C major triad.

[Richard_close2u]

But hey, let’s keep it simple.
All that modal stuff is headache time.

Where’s the C major triad to be found?
In the major keys of its 1st, 4th and 5th scale degrees plus in the minor keys of its 2nd, 3rd and 6th scale degrees.
Or, if you prefer, in the cluster of six, grouped around C on the Circle of Fifths.

[Jamolay]

[sclay]

Eloquent and detailed Richard. A valuable resource.
Yes, once you look at the Circle Of Fifths, it hits you in the face. Might be a handy tidbit if your writing a progression etc, and looking for ideas.

Re, the modes. Very interesting. Can follow your logic here, but am a relative rookie on that subject. One to bookmark.

Cheers, Shane

[Richard_close2u]

@sclay

A cool relationship that seemingly has its roots in the interplay between intervals in the key, and stacked thirds … I am expecting that Richard will formalise it shortly, and elegantly.

All I wrote above shows the what, the where, the when. It does not really explain the how or the why. And you are right, intervals and stacked thirds play the significant part here.

I shall make extensive use of linear diagrams based on taking a note circle, stretched to linear fashion then doubled to show two full repeats. That will lead to multiple viewings of a two octave C major scale with information below each.

[Richard_close2u]

Here is a double span note circle shown in linear, not circular, format.
It stretched from C (as we will be working with the C major scale) to C two octaves higher.
Then we see a two octave C major scale with scale degrees shown below.

It is only the C major scale that will be carried forward.

[Richard_close2u]

To continue, essential prior knowledge and understanding concerns the major scale formula and the process of chord construction by stacking thirds from a scale and the formula for major, minor and diminished triads.

The major scale formula is:

Tone - Tone - Semitone - Tone - Tone - Tone - Semitone

Stacking thirds to construct chords simply means ‘count one, miss one, count one, miss one, count one’. This gives a triad whose notes are all one third from the preceding note.

C major scale
C - D - E - F - G - A - B - C - D - E - F - G - A - B - C

C major scale - examples of stacking thirds to create the first three triads / chords

• C - D - E - F - G - A - B - C - D - E - F - G - A - B - C

• C - D - E - F - G - A - B - C - D - E - F - G - A - B - C

• C - D - E - F - G - A - B - C - D - E - F - G - A - B - C

Major triads = R, 3, 5
Minor triads = R, b3, 5
Diminished triads = R, b3, b5

[Richard_close2u]

C major scale and C major triad

We have a C major scale with intervals shown (following the major scale formula).
Then we have a further C major scale with the 1st, 3rd and 5th scale degrees in bold. This is the method of stacking thirds. It gives the first triad from the scale, the tonic chord of C major

This major triad contains a major third from its root to its 3 then a minor third from its 3 to its 5.
Below, we see a graphic which we will use throughout - one which shows the size and shape for all major triads.

[Richard_close2u]

Here is a Major triad.

The purpose of depicting the major triad in this manner is to use it as a ‘measuring tool’ as we search for other major triads.
We will see it move, step by step, along the C major scale so that the R (root note) of the triad measuring tool sits directly below each of the seven scale degrees in turn.

[Richard_close2u]

Root note D

When the R is placed under the 2nd scale degree, the note D, the major triad tool does not match the stacked thirds exactly. The note F is a semitone lower than the corresponding 3 in the tool. That means we are seeing a flat 3 and the D triad is D minor (R, b3, 5).

Root note E

When R is placed under the 3rd scale degree, the note E, the major triad tool does not match the stacked thirds exactly. The note G is a semitone lower than the corresponding 3 in the tool. That means we are seeing a flat 3 and the E triad is E minor (R, b3, 5).

Root note F

When R is placed under the 4th scale degree, the note F, the major triad tool is an exact match for the stacked thirds. We have a major triad - F major (R, 3, 5).

Root note G

When R is placed under the 5th scale degree, the note G, the major triad tool is an exact match for the stacked thirds. We have a major triad - G major (R, 3, 5).

Root note A

When R is placed under the 6th scale degree, the note A, the major triad tool does not match the stacked thirds exactly. The note C is a semitone lower than the corresponding 3 in the tool. That means we are seeing a flat 3 and the A triad is A minor (R, b3, 5).

Root note B

When R is placed under the 7th scale degree, the note B, the major triad tool does not match the stacked thirds exactly. The notes D and F are both a semitone lower than the corresponding 3 and 5 in the tool. That means we are seeing a flat 3 and a flat 5 so the B triad is B diminished (R, b3, b5).

[Richard_close2u]

Our Major triad measuring tool has shown that the major scale gives us three and only three major triads. Three have their ‘3’ one semitone lower - these are all minor triads. One has its ‘3’ and its ‘5’ one semitone lower - this is a diminished triad.
We used only the C major scale. All major scale follow the same major scale formula so all would give exactly the same result.
The fact of having three and only three major triads follows directly from the logic of the note circle, the major scale formula, the process of stacking thirds to construct chords (triads) and the intervals within the triads. All are inextricably linked and all can be used to explain the others.