Wednesday, January 27, 2016

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

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

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/

For the hexagonal representation of tonality effects see also the 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

AND

Lippens Keyboard:

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

33. The emotions chart of a song.

x-axis=time
z-axies=anxiety-serenity

The reason we choose only these 2 emotional dimensions or 4 emotions is because, the middle not of a chord defnes the sadness-joy dimension, while the upper (dominant) note of the chor defines the anxiety-serenity emotinal dimension

The composed song  must have a definite phrase with start , tension, resolution and end, in the diagram of emotions. The emotions correspond not only to the harmony of the chords but also to the morphology of the melody dynamics and rhythm and of course to the meaning of the lyrics.

Summarizing in  a simplistic way the correspondence of melodic pitch dynamics and the 4-basic emotions in music (joy, sadness, anxiety, serenity) we have
1) Up pitch moves correspond to joy
2) Down pitch moves to sadness
3) Small pitch intervals of 1 or 2 semitones (chromatic or interval of 2nd) correspond to anxiety
4) Large pitch intervals (e.g. 4th, 5th octave etc) correspond to harmony and serenity.

E.g. in order to have mainly joy the next rules apply (see also post  73 )

5) Descend in melody with small steps (1 semitone, and intervals of 2nd or minor 3rds ) and ascend with larger steps (intervals of major 3rds, 5th or octave).Ascending with larger steps that those of descending indicates favor of joy. E.g. Ascend with intervals of 2nd and major 3rd and descend with 2nds and minor 3rds.
6) While ascending, accelerate ascending (meaning use large and larger steps, or larger distances up)Accelerating ascending indicates more joy, while decelerating ascending less joy. The converse with descending. E.g. Ascend by first intervals of 2nd then 3rd and finally 4th or 5th while descend at first with intervals of 5th or 4th, then 3rds and finally 2nds.
7) Melodies that move with steps of minor 2nds and minor 3rds mainly may be called minor feeling melodies, while melodies that move with steps of major 2nds and major 3rds mainly may be called major-feeling melodies.

We must emphasize a basic philosophy  here that when playing the musical instrument, we do not only produce musical sounds but also independently emotions in us. And I do not mean that the musical sounds produce the emotions I mean that our self is producing the emotions parallel and in fitness with the musical sounds

See also post 59 about the 4 basic melodic moves (spike, waving, scaling, isokratic) and their psychological meaning.

1) up/down spike (=large interval of 5th or larger in one or very few steps, extreme emotional effect, excitement , emotional intensity)

2) up/down waving (also called in this blog butterflying, emotional effect of playing either sad of with joy, emotional complication and ambiguity)

3) up/down scaling (direct ascending or descending of notes in a monotone order without waving, emotional effect straight usually with simplicity, emotional clarity)

4) Iso-kratic waving (=horizontal waving with repeating same note, peculiar emotional effect of internal symmetry , and emotional stability )

From these 4 patterns the 1) and 3) are simple and with emotional clarity. The 2) and 4) are with emotional complication and ambiguity.

Another classification would make them 5!

1) Straight scaling up or down (including spikes) in one or more of the melodic speeds (straight sadness or joy). Here the notes of the simplicial submelody are the starting and ending notes.
2) Ascending or descending waving (complex sadness or joy). Here the notes of the simplicial submelody are the starting and ending notes.
3) Flat equilibrium waving (serenity and equilibrium emotion).Here the notes of the simplicial submelody are the upper level and lower level ofthe flat channel.
4) Flat diminishing waving (serenity and diminishing emotions). Here the notes of the simplicial submelody are the starting upper or lower level and h ending note of the diminishing channel
5) Flat expanding waving resolving up or down  (serenity emotions exploding to either sadness or joy). Here the notes of the simplicial submelody are the starting note and the ending note at the upper or lower level of the expanding channel.

MORE PITCH DYNAMICS AND THEIR PSYCHOLOGY

We may create more complex pitch dynamic patterns than these basic with recognizable psychological meaning. Eg. a melodic theme that has a spike up but then falls back to the same pitch level corresponds ton an emotion of "complaining" or "crying" or angry protest that turns to  sadness"
In general when the melody is ascending through repetitive descending melodic themes, or is descending through ascending repetitive melodic themes, the emotion (either joyful or sad) is more dark emotion compared to than when the  the melody is ascending through repetitive ascending melodic themes, or is descending through descending repetitive melodic themes, the emotion (either joyful or sad) in which latter case it is an emotion more straight and transparent.

Some instructive remarks in the composition of the melody based on the chord progression

1) In the part of the chord progression with minor chords, utilize descending melodic moves so that sadness from melody and sadness from harmony fit.Similarly ascending melodic moves for  major chords.
2) In the sad melody parts of the melody (and minor chords) utilize rhythmic patterns that start with faster notes and end with slower notes, and the reverse for the happy part (and major chords).
3) In a triad or 7 nth 4-notes chord the most characteristic notes are the middle 2nd note (in 1-3-5 interval notation  is the 3) and the 7 nth (if it exists). So for the anxiety part of the melodic moves we may utilize 1-semitone trills around these two notes, or waving with 1 or 2 semitones steps and notes outside the chord in the interval of minor 3rd (3 semitones) of the chord. Alternatively instead of trill or small amplitude waves we may utilize chromatic monotone scaling by steps of 1 semitone , or scaling with steps by intervals of 2nd of the scale,  that go from these previous notes of the chord to the same such notes in the next octave. But always make sure that the notes of the chord sound in the average longer, than the notes of these anxiety transition moves with notes outside the chord.
4) Alternate up (happy) and down (sad) pitch moves , or chromatic moves (anxiety), with harmonic (on chord notes) moves (serenity-harmony).
5) Utilize at least 2 octaves, or even 3 for the melodic moves repeating the notes of the underlying chord on the next octaves , so there is sufficient space for melodic moves, to express with sufficiency the emotions.
6) For the duality of emotions anxiety-serenity, it may be utilized also harmonic waves or monotone scaling over 2 octaves at least,  on the notes of the chord, but also chromatic trill wave over the notes of this wave or scaling (modulated wave on wave or move) and then return to the pure harmonic wave or scaling on the notes of the chord.
7) A chromatic wave by 1-semitones steps or all notes of the scale (steps by intervals of 2nd) that goes up and down at least 2 octaves, corresponds to a chord sub-progression of the song , of our choice that utilizes almost all the chords of the scale!

After the chord progression and simplicial submelody we chose,
THE DEFINITION OF MELODIC BRIDGES THAN LINK TWO SUCCESSIVE CHORDS BETWEEN THEM AND START AND END AT THE NOTES OF  THE SIMPLICIAL SUBMELODY.

1) WHICH CHORD-TRANSITIONS (PAIRS OF CHORDS) WILL HAVE A MELODIC BRIDGE! (Usually the chord-trasnitions that are in resolutional relation, or resolutional-like relation)

2) THEN WHICH BRIDGES WILL BE ISOMORPHIC IN PITCH AND RHYTHMIC DYNAMIC SHAPE AND WHICH DIFFERENT, DEFINING THEREFORE A PARTITIONING IN THE BRIDGES.

3) THEN IF IN EACH EQUIVALENCE CLASS OF  ISOMORPHIC MELODIC BRIDGES IN THIS PARTITIONING, THE BRIDGES ARE  EVENTUALLY ASCENDING OR DESCENDING (This besides the emotional significance, determines also where to play the chord in one of the 3 neighborhoods of the fretboard)

4) FINALLY  HOW IN EACH EQUIVALENCE CLASS OF  ISOMORPHIC MELODIC BRIDGES IN THE PARTITIONING, THE COMPLICATED PITCH DYNAMIC SHAPE  OR WAVING AND RHYTHM WILL BE AS A REPETITION  OF SUCH PATTERNS OF PREVIOUS ISOMORPHIC MELODIC BRIDGES, OR VARIATION OF  SUCH PATTERNAS S SO NOT TO BE TOO BORING. (This pitch dynamic shape has again a significant emotional meaning)

(The post has not been written yet completely)

Tuesday, January 26, 2016

32. Cycles of chords of a diatonic scale: Root based cycles. The minor (sad) and major (joyful) cycles. Closed cycles of relative chords. The open cycle of chords by intervals of 4ths.

The diatonic progressions is the sequence (iii->vi->ii->V->I->IV->VII->iii), This progression leads from sadness to joy, from the triad of minor chords to the triad of major chordsIn the symbolism of the 24-cycle of  chords the diatonic scale is the arc of the next chords   (x-2, X-1,x-1,X,x,X+1,x+1)=(vii,V,iii,I,vi,IV,ii). (see post 34).

There is also the inverse or descending  diatonic progression which is the (I->V->ii->vi->iii->VII->IV->I)

In a diatonic scale, the triad of minor chords (sad triad) is the (iii->vi->ii) where the (iii, vi) and (vi,ii) are consecutive in the cycle of pure 4ths, with standard resolutions (iii7-> vi) , (vi7->ii) and the
(ii, iii) are complementary chords, in other words all of their notes give all the notes of the scale except one.

The triad of joy or triad of major chords  is the (V, I, IV) , where the (V, I) and (I,IV) are consecutive in the cycle of pure 4ths, with standard resolutions (V7-> I) , (I7->IV) and the  (IV, V) are complementary chords, in other words all of their notes give all the notes of the scale except one.

The bridge between these two triads is the well known jazz progression (ii7, V7, I) , where again  the (ii, V) and (V,I) are consecutive in the cycle of pure 4ths, with standard resolutions (ii7-> V) , (V7->I), and  the  (ii, I) are complementary chords, in other words all of their notes give all the notes of the scale except one.

Alternative closures
The diatonic progression closes also to a cycle by utilizing the triad progressions
(IV->IV#7->VII7->iii) or (IV7->VIIb->vi).
Or IV->V7->I or IV->ii7->V7->I
Or IV->IVdim7->G7->I   (see e.g, Bach prelude and Fugue C major, BWV 846)
Or IV->IV#dim7->VII7->iii

The 4-chords (3333444)=(C,D#,F# ,A,C,E,G#,C) is  closed cycle of relative chords:   Cdim7, Am, Eaug, G#

The 7-chords (3334434)=(C,D#,F#,A,C#,F,G#,C) is  closed cycle of relative chords: Cdim, D#dim, F#m, Aaug, C#, Fm, G#

The 7-chords diatonic closed cycle of relative chords is the  (4343343)=(CEGBDFAC):
C, Em,G, Bdim, Dm, F,Am

The 24-chords chromatic closed cycle of relative chords (434343434343434343434343):

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

We may add two more series based on that mnot-major chords with the same root are relative chords too, so as to havea 2-dimensional grid based of the relation of relatives.

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

This grid which is also a table as below shows clearly the chords that are  harmonically in series(interval of 4th or 5th) and chords that are harmonically in parallel (intervals of 3rds or 6ths). But it does not show of course the relations of chords that are melodically in series

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

 Gbm Bm Em Am Dm Gb B E A D Bbm Ebm Abm Dbm Gbm Bm Bb Eb Ab Db Gb B

It is often very instructive to chart the chords of  a  song over this 24-cycle of relatives (or double cycle of 4ths) or the above 2-dimensional grid.

Most often a song is a sequence of connected intervals or arcs of chords in the cycle of 12ths shifting by relatives to a corresponding similar connected arc in the parallel cycle of 4th in the overall cycle of fifths. We call this concept a harmonic multi-tonality. Simple  tonality is simply 3 -successive major chords in the 12-cycle of 4ths.

Conversely any connected sequence of arcs of chords of this 24-cycle of chords (defining harmonic multi-tonality), is the chord progression  of a nice song with nice sounding modulations. Normally in harmonic multi-tonality  we are keeping the qualities major-minor as in the 24-cycle but a more free approach allows altering them , from minor to major and vice versa or to more complicated qualities like 7th, 6ths etc.  The same for chord progression for improvisation (see post 11) . To the rule of harmonic multi-tonality in the 24-cycle, we may allow as transition to a next chord, a shift by one semitone or tone of the root of the current chord. (see also post 30)

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

FOR PLACING THE HARMONIC CYCLE OF 24-CHORDS ON THE GUITAR FRETBOARD SEEING THUS THE CHORD-RELATION OF RELATIVES, AND CHORD RELATION OF DOMINANT7-ROOT RESOLUTION AS FRETBOARD-RELATIONS OF THE CHORD SHAPES IN THE DAE SYSTEM SEE POST 44.

But we may make some partial remarks about placement on the fretboard

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

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-1)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 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. 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 fretboard and a sequence of 3 successive chords in the cycle of 4ths and symbols of the post 23, and 34. 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  (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 and as upper relative chord the
((n-1)Dm)Zm (or ((n+2)Am)Ym )

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 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). 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!!!

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.