34. The 24-cycle of alternating major/minor 3rds as refinement of the cycle of 5ths with its symbolism for chord progressions. The 2-dimensional hexagonal or square, tonality grid for chords geometric representation.

HERE IS THE 24-CHORDS CYCLE IN THE REVERSE ORDER BY 5TH RATHER THAN BY 4TH WHICH IS THE ACTUAL 

THIS WHEEL IF WE WALK LIKE A ZIG-ZAG ALTERNATING MINOR WITH MAJORS IT IS NOT A WHEEL BY 4THS OR 5THS BUT A WHEEL BY THIRDS (ALTERNATING MINOR MAJOR THIRDS)

Also we may define a 24, cycle in alternating major/minor intervals of  3rds ,  and also alternating major/minor chords , so that consecutive chords are relative chords, and as 3s+4s=7s, it is also a refinement of the cycle of 5ths. In the next we state this cycle in the reverse order, which seems as if a refinement of the cycle of 4ths, with alternating steps of 8  and 9 semitones (small and big interval of 6th). As notes of the consecutive chords we have also the pattern (434343434343434343434343) which is (generalized) long scale as in the definitions of the post 42.

G        C         F         Bb          Eb        Ab         Db          Gb          B            E            A              D              G                 
      Em   Am     Dm        Gm       Cm       Fm        Bbm       Ebm     Abm       Dbm       Gbm         Bm             Em


This cycle of 24-chords seems that it is used for example in software that helps composing songs like the Harmony Navigator 2, see next video
https://www.youtube.com/watch?v=xa6eTlS3uDk
 https://www.youtube.com/watch?v=FfZygNUmFas
This pattern of the chords appears naturally on the fretboard of a bass or guitar tuned on all chords by pure 4ths!  In the next image we see only the positions or arpeggio pof the cmaj chord, and we can easily add the lower relative Am.Then put the same for all other postions of the other chords. The vertical direction from lower to higher notes is thesame as the direction of the 24-cycle of chords. Every vertical path inside a fret, spans with 3 of the positions of the major chords a diatonic scale.

As it is usual to apply  numbers for the chords on the steps of a diatonic scale giving some abstractness to chord progressions, we apply apply also the same here in the 24 double cycle o chords.

We may symbolize any (major) chord of this cycle by capital X, and its next chord in the cycle by X+1, e.g. X=G, X+1=C or X=D, X+1=G etc. Now for the relative minor chords we reserve small variables like x, x+1, and we use the same x as the Capital X for the upper minor relative chords e.g. if X=G, then x=Em, if X=D then x=Bm,  And if x=Em then x+1=Am etc. I is important to realize some recursive equations here like (Xm=x+4) and (xmaj=X-4), where by Xm we denote the minor chord with the same root with  the major X and by xmaj the major chord with the same root as the minor x. 
Under this symbolism the 3 minor relatives chords of a major chord X would be the x, x-1, and x+4=Xm

Thus according to this symbolism an Andalusian cadenza progression like (Dm, C, Bb, A)  (see post 17) would be symbolized  by  (x, X-1, X+1, X-4) or (x, X-1, X+1, X+8), while the jazz progression (ii, V7, I)  would be symbolized by (x, (X-2)7, X-1) or (x+1, (X-1)7, X), where by (X-2)7 we symbolize the dominant seventh version of the major chord X-2. 

A double Andalusian Cadenza (see post 17) e.g.  (Am Dm)->(G-C)->(F Bb)->(E A) will become  (x, x+1)->(X-1,X)->(X+1,X+2)->(X-4,X-3) , from which we see directly the many consecutive positions in the cycle of 4ths and 24 chords  .

The advantage of this symbolism of progressions, is that it is scale-free, and the resolutions X7, X+1 (e.g. C7, F) are directly understood as well as the relatives relations like X-x (e.g. C, Am) etc.

The chords of a diatonic scale in this 24-cycle are easily defined as the chords of an arc of 3 consecutive major chords together with their 4 relative minors. The root of the diatonic scale is the middle major chord. In the symbolism above it is the next arc of 7 chords 
 (x-2, X-1,x-1,X,x,X+1,x+1)=(vii,V,iii,I,vi,IV,ii).

Notice that strictly speaking of for example X =C, then the x-2=Bm is not the exactly  a correct chord type of the scale as the Bdim (=xdim) is, but in the 24-cycle there are no diminished chords thus the 7th chord is represented there as minor chord. 


The best way to learn the fretboard is by chords and the best way to learn the fretboard by chords is to map the 24-cycle of chords on the fretboard!

A simple way to map the 24-cycles on the on the fretboard is to map the 12-cycle of chords by 4ths, in vertical lines relative to the strings , where three chords of shapes E, A, D are in the vertical line, and the continue the vertical line higher of lower in the fretboard. 


THERE ARE TWO WAYS THE THE WHEEL OF 4THS (OR 5THS) IS REPRESENTED WITH e, a, d SHAPE CHORDS IN THE FRETBOARD
1) THE FORWARD OR ASCENDING METHOD (NO REPETITION OF CHORDS)
2) THE BACKWARD DESCENDING METHOD WITH REPETITION ( A D-SHAPE CHORD IS NOMINALLY  IDENTICAL WITH THE NEXT E-SHAPE CHORD

STILL THE LARGEST ARC OF CHORDS IN THE WHEEL OF 4THS WITHOUT FLATS OR SHARPS THAT IS REPRESENTED IN THE FRETBOARD IS WITH THE BACKWARD METHOD, AND IS ANALYZED IN POST 13 AS THE MAIN 3-NEIGHBORHOODS OF THE FRETBOARD, THE G (1ST), A (2ND) AND b (3RD). 

THE STANDARD GUITAR TUNING IS ALMOST SUB-OPTIMAL IN REPRESENTING IN A QUITE SYMMETRIC WAY THE WHEEL OF 4THS IN THE CHORDS. PROBABLY THE BEST IN IN REPRESENTING IN A QUITE SYMMETRIC WAY THE WHEEL OF 4THS BY CHORDS IN NORMAL POSITION IS THE REGULAR TUNING BY 4THS. BUT IT HAS DIFFICULT SHAPES FOR THE INVERSIONS, WHILE THE STANDARD NOT SO MUCH.





Then the relative chords are discovered as relations of a chord at the vertical line with a chords at the neighboring vertical lines. The rules to do so are the next

In relation with the 24-chords cycle of chords by intervals of 4ths  the DAE system has the next keys and correspondences (with the symbolism of chords on the fretboard as in post 23 ).

 The sequence  X=(nE)Y1, X+1=((n)A)Y2, X+2=((n)D)Y3 is of course a vertical sequence of chords in the fretbaird and a sequence of 3 successive chords in the cycle of 4ths and symbols of the current post. Now after the X+2=((n)D)Y3, the cycle of 4ths continues either lower in the fretboard or higher in the fretboard

1) Lower in the fretboard is X+3=((n-2)A)Y4

2) Higher in the fretboard is X+3=((n+3)E)Y4

From this point of view, the area of the open chords of the guitar, is simply two such vertical 3-sequences of chords on the cycle of 4ths, as the C-shape is essentially a ((n-3)D)Y chord and D-shape and the G-shape is a ((n-2)A)Y chord and A-shape. 

For the relative chords of major chords the rules are :

The (nE)X chord (which means the chord of E shape at the nth fret, and that sounds like X) has as minor lower  relative  chord the ((n-1)Dm)Ym (e.g. F with Dm)  (or ((n+4)Am)Ym) and as upper relative chord the  ((n-3)Am)Zm (or  ((n+4)Em)Ym) 


The (nA)X chord (which means the chord of A shape at the nth fret, and that sounds like X) has as minor lower  relative  chord the ((n+2)Em)Ym (e.g. A with F#m) and as upper relative chord the
 ((n-1)ADm)Zm

The (nD)X chord (which means the chord of D shape at the nth fret, and that sounds like X) has as minor lower  relative  chord the ((n+2)Am)Ym (e.g. D with Bm) and as upper relative chord the ((n+2)Em)Zm

It is easy to see that the shape of lower relative minor chord of  major chord, compared to the shape of the major chord is simply the cycle of letters of the DAE system (D->A->E->D), which is the reverse order of the successive irresolution relations which is  E->A->D->E . That is the lower relative minor of major D shape chord is a minor A shape chord, the lower relative minor of major A shape chord is a minor E shape chord, the lower relative minor of major E shape chord is a minor D shape chord!!! Notice also that to find the lower relatives of the vertical sequence of successive resolutional chords E,A,D on the fretboard, we only need to go either 1 step lower in the fretboard (n-1) or 2 steps higher in the fretboard (n+2). 
One step lower for E and 2 steps higher for A and D, while always the shape of the relative is in the reverse order of the resolutional order E->A->D->E. 

Notice also that the upper relative of a major chord X , is the lower relative chord of its previous chord in the cycles of 4ths, that is the lower relative chord of X-1. Summarizing upper relative of X=lower relative of X-1.  Therefore we need only memorize the lower relatives if we are familiar with the successive resolutional relations of chords.

While the shape of upper relative minor chord of  major chord, compared to the shape of the major chord is simply the reverse order of cycle of letters of the DAE system (D->E->A->D) exactly as in the relation of successive resolutions . That is the upper relative minor of major D shape chord is a minor E shape chord, the upper relative minor of major A shape chord is a minor D shape chord, and the upper relative minor of major E shape chord is a minor A shape chord!!!

For the relative chords of minor chords:

The (nEm)Xm chord (which means the chord of E shape at the nth fret, and that sounds like X) has as minor lower  relative  chord the ((n-2)D)Y and as upper relative chord the ((n-2)A)Z


The (nAm)Xm chord (which means the chord of A shape at the nth fret, and that sounds like X) has as minor lower  relative  chord the ((n+1)E)Y and as upper relative chord the ((n-2)D)Z

The (nDm)Xm chord (which means the chord of D shape at the nth fret, and that sounds like X) has as minor lower  relative  chord the ((n+1)A)Y and as upper relative chord the ((n+1)E)Z

Of course the in-place change of a chord from minor or major or vice-versa is also a relation of middle  relative chords. 

The 3 ways to play all the chords of a major scale on the fretboard within 4 or 5 frets,  with root-chords as D, A, or E shape and are the next. They are mostly convenient for 3 or 4 string instruments where even the D shape is played easily with all 4-strings (see post 67) 

1) With D-shape as root,   In the symbolism of post 23 the  (nD)X means at n-th fret play the shape D and it sounds as chord X. Here instead of X we will utilize the Latin symbols of the steps in a major scale, as it is standard in Jazz with small if the chord is minor and capital if the chord is major

So the chords I, ii, iii, IV, V, vi, vii, are played on the fretboard  as follows

I=(nD)I, ii=((n+2)Dm)ii , iii=((n)Em)iii, IV=((n+1)E)IV, V=(nA)V, vi=((n+2)Am)vi ,

  vii=((n+1)dim7)vii.

In short the three main major chords I, IV, V are the 

I=(nD)I, IV=((n+1)E)IV, V=(nA)V.

The geometry of the shapes E,A,D vertically and horizontally on the fretboard are as in the following table. We place vertically the E,A,D shapes of the major chords V,I,IV and 3 more minor chords that are complementary by one tone higher to V, I, and one semitone lower to IV, on the fretboard. Notice also that the minor chords are positioned vertically on the fretboard as  the also have mutual successive resolutional relation

(n+3)E=IV	(n+2)Em=iii
	(n+2)Am=iv		nA=V
	(n+2)Dm=ii		nD=I

2) With A-shape as root,  where the I, and V are on the same fret. In the symbolism of post 23 the  (nA)X means at n-th fret play the shape A and it sounds as chord X. Here instead of X we will utilize the Latin symbols of the steps in a major scale, as it is standard in Jazz with small if the chord is minor and capital if the chord is major

So the chords I, ii, iii, IV, V, vi, vii, are played on the fretboard only as shapes A and E as follows

I=(nA)I, ii=((n+2)Am)ii , iii= ((n-1)D)iii,  (nD)IV,  V=(nE)V,  vi=((n+2)Em)vi  

vii=((n)dim7)vii.

In short the three main major chords I, IV, V are the 

I=(nA)I, IV=((n)D)IV, V=(nE)V. 

The geometry of the shapes E,A,D vertically and horizontally on the fretboard are as in the following table.We place vertically the E,A,D shapes of the major chords V,I,IV and 3 more minor chords that are complementary by one tone higher to V, I, and one semitone lower to IV, on the fretboard. Notice also that the minor chords are positioned vertically on the fretboard as  the also have mutual successive resolutional relation

(n+2)Em=iv		nE=V
(n+2)Am=ii		nA=I
		nD=IV	(n-1)Dm=iii

3) With E-shape as root,   In the symbolism of post 23 the  (nE)X means at n-th fret play the shape E and it sounds as chord X. Here instead of X we will utilize the Latin symbols of the steps in a major scale, as it is standard in Jazz with small if the chord is minor and capital if the chord is major

So the chords I, ii, iii, IV, V, vi, vii, are played on the fretboard  as follows

I=(nE)I, ii=((n+1)Em)ii , iii=((n-1)Am)iii, IV=((n)A)IV, V=((n-2)D)V, vi=((n-1)Dm)vi ,

  vii=((n-1)dim7)vii.

In short the three main major chords I, IV, V are the 

I=(nE)I, IV=((n)A)IV, V=((n-2)D)V.

(n+2)Em=ii		nE=I
		nA=IV	(n-1)Am=iii
		nDm=iv		(n-2)D=V

2-dimensional hexagonal  tonality grid for chords geometric representation

This hexagonal grid,is defined by the following rules

1) Horizontal sequences of notes differ by the interval of perfect 5th (7 semitones) or its inverse perfect 4th (5 semitones). In the image below from left to right the interval is perfect 5th.

2) Diagonal sequences of notes differ by an interval of  major 3rd  (4 semitones) o minor 3rd (3 semitones). The diagonal left to right and up to down is a minor 3rd. While the diagonal left to right and below to up is  amjor 3rd.

Chords, are represented as triangles or rombuses in this hexagonal grid.

(See e.g. the visualization software for music MAM, http://www.musanim.com/

http://www.musanim.com/player/MAMPlayerUserGuide.pdf )

For the hexagonal representation of tonality effects see also the schismatic temperament 

https://en.wikipedia.org/wiki/Schismatic_temperament

As an alternative we may define a square grid. The rules are:

1) Horizontlly from left to right the interval  is a perfect 4th (5 semitones)

2) Vertically from down to up it is alternating 3rd major and 3rd minor intervals.


More about hexagonal ingenious keyboards here

Terpstra Keyboard :

https://www.youtube.com/watch?v=QUJ2oND3cdg
 AND

Lippens Keyboard:

https://www.youtube.com/watch?v=MHfG20HvPHA

In the latter square grid, every 3x3 square are the notes of a diatonic scale. The relation of scales and chords with common notes are directly asilly visible. Roots of alternating major moinor relative chords are on vertical lines.

We may also define a Z(12)^3   ring, of the chords , where each component Z(12) is the cycle of 4ths, and neigborhood chords among the three componenets are always relative chords.

The 5 -triads in successive resolution harmonic relation on the fretboard.

The best way to learn the fretboard is without mental images but only the feeling of the notes at each fret.But this takes too much practice and familiarization with the fretboard.
On the other hand the best way to learn all the fretboard through mental images,rather than feeling,is not by patterns of scales, neither by the names of all the notes of the frets, but rather with sufficient many chord-shapes that almost cover all the fretboard. And even better  if these chords are organized in to easy repeating patterns. Here we describe a method, based on the triads of chords in shapes of E, A, D, so that each is relative to its previous, at the harmonic relation of successive resolution in the cycle of 4ths (see also post 30, 23).

Here we list the chords of shapes E, A, D,on the notes of the e4-string

e4, g4, a4, b4, d5, 

For the symbolism of chords placed on the fretboard, see post 23

chords (E- shape) V	chords (A- shape) I	chords (D- shape) IV
e4  (0E)E	(0A)A	(0D)D
g4 (3E)A	(3A)C	(3D)F
a4 (5E)A	(5A)D	(5D)G
b4 (7E)B	(7A)E	(7D)A
d5 (10E)D	(10A)G	(10D)C

When adding the minor chords of the diatonic scale, if the roots is an A-shape we have the following positions