2. Learn easier and more chord-scales or arpeggios with the three , 2-string, and 3-string chord-triads and their transitions or CAGED and DEA system.

Before this post the reader must study the posts 40 (that classifies intervals), the classification of  2-string triads , and  38 (that classifies 3-string triads) and 35. 

In spite  the fact that I give here an easier metthod to learn the various positions of  the 

arpeggios of the chords  based on the 2-string triads , and CAGED system, I still have doubts for the value of learning the arpeggios, as fas as improvisations is concerned on the highest 3 or 4 strings of the guitar. The arpeggios seems redundant to learn as all notes of an arpeggio of a chord on the highest 3 or 4 strings of the guitar , that do not belong to the chord shape, within the arpeggio, are essential notes of the chord shape , in the DAE system, in one or both of the other two positions of the chord. And learning the 3 positions of each chord is easier than learning all the diffrent arpeggios of each chord.

The same holds if we include all the 6 strings, but this time, we must talk about the shapes of the chords in the CAGED system. In other words:  All notes of an arpeggio of a chord , that do not belong to the chord shape, within the arpeggio, are essentially notes of a chord shape in the CAGED system in one or both of the adjacents  CAGED  positions of the chord.  Actually as patterns of improvisation with notes exclusively of a chord we may very well take the D(C), A, (G)E, shapes, of the CAGED system. The union of all arpgeggios of a chord in the fretboard is also the disjoint union of the 3 shapes-patterns D,A,E or more explicitly the  D(C), A, (G)E, and these cover excatly all notes of the arpeggios of the chord on the fretboard. Why to group the notes in the standard way to present arpeggios, when the knowledge of the chords on the fretboard by the CAGED (or DAE) system also provides another easier to remember grouping of the notes? In the improvisations this alternative way (the shapes D(C), A, (G)E of the CAGED system)  is adequate and easier and is not necessary the move of the notes withn  chord to be with the smalest step, in the 12-notes scale. Each shape-pattern D(C), A, (G)E covers essentially 2 octaves with notes fron the chord. Also this applies to all types of chords like R, Rm, R7, Rmaj7, Rm7 R6, Rm6, Rdim7, Raug etc

Since the chord shapes are essentially one of the three positions-shapes (C)D-(G)A-E, there are simple definitions of the arpeggios based only on the chord shapes, that have significant advantage over other arpeggios shapes.  The arpeggios that have advantages are by combining two consecutive shapes-positions of the chord in the DAE system.
In other words combining the consecutive or adjacent shapes E-D we get an arpeggio and similarly the consecutive or adjacent shapes A-E. The combination D-A is not appropriate as in ascends in the vertical axis of the Fretboard. Now the advantages of the above two types of arpeggios are the next

1) They are direct combinations  of half of each ofthe two consecutive shapes-positions of the chord in the DAE system, so it is direct to remember them when we know the 3 D-A-E positions of the chord (see posts 13,23  ). And they are only of two types E-D and A-E. 
2) They span 2 octaves, by moving also along the length of the fretboard, which enriches the sound of the notes from harder to softer. 
3) As they ascend, the overall move on the fretboard is both from left to right and from lower strings to higher strings, utilizing at most three notes per string. 

Here is a video of the use of pentatonic scale as soloing around any chord!

(actually, the pentatonic scale can be considered the arpeggio of the chord with 2nd and 6th notes too or 6th and added 9th C6add9 )

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

Compared to the standard patterns of arpeggios, they may span one more fret, but the rest of the advantages supersed the advantages of the classical positions of the arpeggios. 

Here is  an example . The E-D arpeggios and of Emajor , E7, Eminor , Eminor7 are  the next correspondingly.
Notice that based on the 2-string major triads as presented later below the next E-D arpeggio can be considered also as composition of 2-string triads, in symbols it is
2SM2+2SM1+2SM1


                                                                 Emajor E-D arpeggio

E7   E-D arpeggio 

Notice that based on the 2-string minor triads as presented later below the next  E-D minor arpeggio can be considered also as composition of 2-string triads, in symbols it is
2Sm1+2Sm1+2Sm2

Em  E-D arpeggio

Em7 E-D arpeggio

Here is  another  example . The A-E arpeggios and of Emajor , E7, Eminor , Eminor7 are  the next correspondingly

Notice that based on the 2-string major triads as presented later below the next A-E arpeggio can be considered also as composition of 2-string triads, sa formula bu thsitime starting at the 5th string and not the 6th string. In symbols it is

2SM2+2SM1+2SM1
                                                    Emajor A-E arpeggio

                                                          E7   A-E arpeggio 

Notice that based on the 2-string minor triads as presented later below the next  A-E minor arpeggio can be considered also as composition of 2-string triads, same fprmula as with the E-D arpeggio but thsi time starting at the 5th string and not the 6th string. In symbols it is
2Sm1+2Sm1+2Sm2

Em  A-E arpeggio

Em7 A-E arpeggio

Nevertheless if one wants to learn the standard shapes of the arpeggios, there is an easier way, by learning first the three 2-string and 3-string triads and their transitions or compositions. We proceed to show how.

About the 2-string major triads:

https://www.youtube.com/watch?v=4zS_-gz37JA&list=RD4zS_-gz37JA#t=3

We may observe that our major and minor arpeggios are compositions of the two different 2-string triads. A good exercise is to play the arpeggio with waving or butterflying way. That is e.g. E ,G# ,B, G# ,B,  E , B, E, G# etc. Or with 2-steps waving E ,B, G# ,E, B, G# etc.

Here we list the 2-string major triads, in root position symbolized by 2SMi , i=1,2 .
The particular notes do not matter what matters is their shape and mutual intervals.

Ist 2-string major triad 2SM1

2nd 2-string major triad 2SM2

We do not list here their obvious transformations when they start at the 2nd string!

Here we list the 2-string minor triads in root position , symbolized by 2Smi , i=1,2 .

The particular notes do not matter what matters is their shape and mutual intervals.

1st 2-string minor triad 2Sm1

2nd 2-string minor triad 2Sm2

We do not list here their obvious transformations when they start at the 2nd string!

THERE IS ONLY ONE WAY THAT THE ABOVE TRIADS CAN BE COMBINED TO GIVE THE SAME CHORD ARPEGGIO: BY STARTING THEM AGAIN ONE PERFECT FORTH HIGHER. 

When ending at the strings 5th,4th,  this means staring them at the same fret and next string. At the string 3rd this means starting at the next fret on the 2nd string. 

By producing in this way sequences of notes that are all belong to a chord, and by starting one step or 2 steps later, we also produce all the 1st and 2nd inversions 2-strings triads, that we may  also list.

The 1st , 2-string  triad of a diminished chord is  the next , symbolized by 2Sdim1 is the next
 (the example is with a chord Edim)

While the 2nd , 2-string  triad of a diminished chord is  the next symbolized by 2Sdim2 is the next, 

 (the example is with a chord Edim)

By combining  major triads with  dim triads we get a 2 or 3-string quadruplet of a major dominant 7th chord as below.

In the same way by combining a minor triad with a major triad we obtain a 2 or  3-string quadruplet of a minor 7th chord as below.

Similarly by combining major and minor triads we may obtain the  2 or  3-string quadruplet of Major7th chords.

Finally, by repeating such  2 or  3-string quadruplets we obtain larger R7, Rm7, Rmaj7 arpeggios 

Correspondence of chord transitions of chord progressions to the 3-melodic densities or speeds  of the melodies that fit to such chord progressions
(See also posts 30,68)

1) The complementary chords in a 2-chords transition corresponds to the chromatic/diatonic melodic speed or density. 

2) The relative chords in a 2-chords transition corresponds to the middle harmonic melodic speed or density. 

3) The successive resolutional  chords in a 2-chords transition corresponds to the high harmonic melodic speed or density. 

ANGLES IN FRETBOARD AND MELODIC SPEEDS

1) When playing the melodies on the fretboard in the guitar, the chromatic/diatonic speed is played mainly along the length of a string, so it is the zero angle.

2)  When playing the melodies on the fretboard in the guitar, the middle harmonic  speed is played mainly at an angle which relative to the horizontal is about 45 degrees and moves from the keys of the guitar to the sounding body as the melody descends in pitches! This is is because it consists of intervals of 3 or 4 semitones that in two successive strings is such an angle.

3)  When playing the melodies on the fretboard in the guitar, the high  harmonic  speed is played mainly at an vertical  angle  relative to the horizontal because the strings are tuned at intervals of 5 semitones (and one string in 4 semitones). Also the interval of 7 semitones (5th) when played in descending the pitches makes an angle  larger than vertical or 90 degrees (e.g. 135 degrees) and moves from  the the sounding body of the guitar to the keys of the guitar  as the melody descends in pitches!


Open chord shapes instead of common barre shapes
https://www.youtube.com/watch?v=tSiHcOutDc4


ARPEGGIOS AND DEA SYSTEM OF 4-STRING INSTRUMENTS (SEE POST 67)

For the 4-string (double or simple strings) instruments of post 67, that are most of the ethnic music instruments , the chord shapes theory simplifies to the DEA instead of the CAGE of the 6-string guitar. Similarly the arpeggios of the chords, although are not identical with the shapes of the chords in a 6-string guitar, for the above 4-string instruments , they are identical with the chord shape! Thus knowing the chords means knowing their arpeggios of them too, which gives immediately a way of easy improvisation along a chord progression!