We will integrate a few pieces of information to play/understand melodic arpeggios [build chords] up a single string. Our example is for E, so this can be done on either E string, and can be applied to any single string.
First, let's play the scale again. Ascend & descend. Improvise with it for one minute.
We can see the Major scale pattern [221-2221] quite easily using basic math with fret numbers: 2 - 0 = 2, 4 - 2 = 2, 5 - 4 = 1, etc.
To build chords [triads & 7ths] from a Major scale, we select each tone as a root, then select every other note [EON]. For triads, we go EON until we have 3 tones. For 7th chords, we go until we have 4 tones.
We are creating the chords, but can only play them as melodic arpeggios, since we are using only a single string. Here we go...
The 7 for each chord is in gray, triads in black.
Play just the triads, then play the 7ths. Your ear can figure this out.
One reason it is helpful to do this exercise is that by using simple math, we can clearly see the tone spacing for the chord qualities.
Example: a Major 7 type chord [I & IV chord for a Major key] is a 4-3-4 [half steps between adjacent components].
For EMajor7, the basic math works out like this → 4 - 0 = 4, 7 - 4 = 3, 11 - 4 = 4. This yields 4-3-4.
Another way to view it: the Major 7 chord is a Major triad [4-3] + 11 [the type of 7th]. The 11 is 4 + 3 + 4 [the half step spacing between each].
We don't typically play melodic arpeggios like this, in a line - at least not for very long, within a solo - [better options exist - position arpeggios], yet, it is an effective exercise to deepen our linear understanding of the board. Integrate it into the mix.