: Using intervals to build chords Chords in western music are said to be built on intervals of 3rds, such as the root, third and the fifth creating a major chord. The reason this chord sounds harmonically pleasant is due to

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This duplicates other questions, but there are many re-wordings in the various posts so they may be hard to find. Ex.
How does the harmonic series affect consonance?

I think what you want to look at is:

From a historic perspective the chord of nature.

From an acoustic perspective the harmonic series.

And to some extent the psychological perspective of octave equivalence.

The basic idea is that the beginning overtones of the harmonic series form a major chord so a major chord gets imprinted on our ears as resonant, stable, and consonant.

But the harmonic series does not produce all the diatonic tones. So you can't use it to produce the full tertian stack you mentioned, root to thirteenth.

At best the chord of nature/harmonic series is a rationale for why a root position major chord creates a sense of stability. By extension we could say any non-root position chord, non-major triad is unstable and provides a dynamic force to move to a stable chord. But the details of how those unstable harmonies are handled is hardly a 'scientific process.'

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Chords in western music are said to be built on intervals of 3rds....
Well, this ignores developments in harmony dating back at least 100 years and with roots all the way back in the middle ages when perfect fourths were considered consonances. Quartal and quintal harmony are quite common today. You can think of triads as being built in thirds, and you can think of seventh chords and extended seventh chords as being build in thirds too. But this approach can be misleading since in practice chord voicings are what matters. In practice, in some circumstances, an extended seventh chord may be voiced without one or several of the notes derived from the stack-of-thirds construction, or the chord may be voiced in one of many inversions, carrying the interval relationships far from the original stack-of-thirds.
This chord can be said to be consonant from a basis of being harmonically pleasant.
That sounds fine; it is one way to define consonance. Others try to define consonance in terms of harmonic ratios. The first approach ("harmonically pleasant") would seem to have the advantage of being ambiguous; intervals that were once considered dissonant have come to be considered consonant, and the perfect fourth was considered consonant, then dissonant, then consonant again!
These two ways of defining dissonance seem to be talking about different things. Maybe we need better terminology here. When I was young I liked sweet foods and didn't like bitter foods; I thought that only sweet foods were delicious. When I was young I liked consonant harmonies and didn't like dissonant harmonies; I thought that only consonant harmonies were pleasing. Now that I am older, I think that bitter foods can also be delicious, and dissonant harmonies can also be pleasing. Maybe the harmonic ratio people have a point, if we could only stop using consonant to mean "pleasing". I guess I'll have to remain uncommitted on this issue.
What is the scientific process behind the selection of intervals in creating a chord?
There isn't one. You can discover one, if you like. Any such process will be much more complicated than coming to grips with thirds, though. Actual chord voicings are complex, may omit notes, may omit the root note, may emphasize dissonant intervals that can be had through inversion or added tones.
As someone who has played and continues to play a lot of chords, I can tell you that I never think anything like "this chord needs to be more dissonant, I'll change it like so to effect that." Different intervals have distinct sounds; you get to know those sounds and combine them as sounds, not as principles. Sometimes the intervals arise as if by magic, because there is a melody line or other linear structure that suggests them. You get to know the sounds of larger chord structures, i.e., specific chord voicings, and alter them to suit an occasion. You learn what things sound like, and you learn to hear better. This is what I meant in an earlier comment when I said "it usually seems better to approach it from a more phenomenological perspective." If you approach music as an exercise in scientific principles (which may or may not exist), you will miss nearly everything important about music. You are creating sounds to be experienced; you had better experience them before anyone else.
Ultimately, how are chords built through the use of intervals, what are the mechanics?
I'm not quite sure what the question is here. There is no mechanics of chord construction other than that you combine some notes. You can build a chord any way you like; you can combine any notes you like. The question is, does a combination of notes sound "good?" That is a question only you can answer by developing your own harmonic conception. Such things don't lend themselves well to words; you have to get up to your elbows in the music.
Postscript
Of course, there are systematic ways to construct chords (although I'd rather think about developing harmonic knowledge; the term chord tends to imply mashing a bunch of notes together to a lot of people, and harmony is more subtle than this).
You can construct close-voiced chords in thirds, fourths, or fifths, and find all of the inversions. There are systematic ways to approach inversions too: start with drop-2 and drop-3 voicings, then learn about other drop voicing patterns. This will give you a large vocabulary of chord sounds, and you will discover that different voicings of, say a Maj7 chord, have distinctly different qualities.
You can construct chords by taking a voicing that you already know, and adding or altering one or more notes. This is very convenient in practice, and can dramatically expand your harmonic vocabulary.
You can try to enumerate every possiblity on a particular instrument and analyze those chords. Obviously, this won't be too useful if you are composing in a DAW. George van Eps did something like this with his famous (and enormous) three-volume series Harmonic Mechanisms for Guitar. The point of that series is to inquire into the harmonic devices can be played on a guitar, the sounds of those devices, and methods for getting those devices under a player's hands. Ted Greene also did something like this with Chord Chemistry, which had the goal of enumerating nearly every chord form that is playable on the guitar.
The closest I ever come to "using intervals to build chords" is when I play, say a Maj7 chord and think to myself, "there is a minor 2nd available here, and I'd like the sound of that right now." That kind of thinking might happen more or less explicitly for me when I am arranging something, probably more subconciously when I am performing something. When it does happen, I'll add a minor 2nd by adding to or altering a voicing I already know if I don't already have a pat voicing at hand. I can do this because I know lots of voicings, and I know lots of voicings because I started with a few voicings and built up a large vocabulary by doing exactly what I have been describing (starting with a vocabulary of simple voicings, altering them as needed, and adding any interesting discoveries found along the way to your vocabulary). This is pretty much how it seems to work for everyone.
I just glanced at my copy of Chord Chemistry and saw this on the very first page:

...although learning many nice inversions or different ways to play
the same chord is essential to a good guitarist, by itself it is
nothing. It is far better to know only a few nice chords and know how
to use them than to know thousands of chords without knowing where to
put them or how they relate to each other.... Some chords that are not too pleasing by themselves are excellent when used with the right combination of other chords. [Emphasis in the original]

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A chord is formed from two or more of these pitches which depending on the pitches utilized will dictate the consonance or dissonance of a chord. Lets say that is the mechanics. Through what approach are intervals for a chord selected from that understanding?

The most common approach is that:

A scale is developed that presents a set of intervallic possibilities. The intervals in the scale will be chosen such that there are a number of ways to play frequencies with simple ratios between them; there will often be the potential for some more dissonant intervals too.
Alongside the development of such a scale, a framework for harmonic practice using the scale evolves with some awareness of the levels of consonance between the intervals, but also according to subjective cultural preferences.
Individual composers create chords according to some balance between these cultural practices and their own tastes.

You might wonder, if it's somewhat agreed-on how the level of consonance of a chord can be calculated, why there's so much subjectivity. The answer is that there are many other considerations to what makes a chord 'good' than some nominal static level of consonance and dissonance, such as

maintaining good voice leading, and perhaps also the specific motion of the bass voice and melody line
adhering to the kinds of chords expected in a particular style, or a harmonic pallete set up within a song
choosing notes that a given instrument can play easily
respecting emotional associations (a simple example is that people tend to think of a minor chord as 'sad'; it's a question of debate whether this is purely subjective or not, but this consideration certainly goes beyond that of consonance/dissonance)
the way that tuning and timbral variances will change the nature of the chord
achieving the desired phrasing and cadences
the value of surprise

The vast majority of composers aren't consciously considering specific, measured levels of consonance and dissonance at all when they pick out the notes in their chords; they're building on higher-level constructs, subjective preferences, and using their mind's ear.

Perhaps a shorter answer would be: If you're using the 12-tone (chromatic) scale to make music, or any subset such as a major or minor scale, then a great deal of the consideration of levels of consonance and dissonance has already been done for you.