The harmonic series that you describe was discovered to be related to our perception of consonance and dissonance by the German Physicist Herman Helmholtz in the late 1800s. The concept was known to us (the human race) for a lot longer than that. Helmholtz was out to provide an objective description of why some intervals are considered "unpleasant" and other harmonious. The human factor may still be subjective. You cannot force me to say I like the perfect 5th and cringe at the minor 2nd, but the opinion of the masses is at play here. My point is that the objective measure you have read about may not be the last word on the subject and just one of many ideas. It is also describing qualities of WESTERN CLASSICAL music and as such is biased by race and ethnicity. In other cultures sounds that a Westerner might consider "dissonant" are judged as beautiful and harmonious. Also, the actual frequencies of notes have changed over the centuries and the harmonic sequence may not have always been present in the intervals, and some that were present may be lost! So, in reality you can use tones any way you want to create the effect you want with your music.
You mention the 4th as an example that doesn't quite fit the paradigm. In fact many people judge the 4th to be very dissonant. The 4th is an inverted 5th and under the correct circumstances can sound consonant, and in others dissonant. I think that how we perceive things is relative to the tonal landscape.
That being said there are some "classic" uses of these ideas in western classical music that are part of the modern tradition.
Consonance: This is usually harmonious and intervals are not EITHER consonant or dissonant but have a degree of each quality. Generally 3rds and 6ths are judged to be consonant.
Dissonant: This is usually thought of as "clashing" or "tense". Seconds, sevenths and the diminished 5th are judged as dissonant.
The 5th is really quite obnoxious! We generally do NOT want 5th harmony and frequently voice chords so that it is absent or on combination with other intervals to "soften" it. The reason is that there are TOO many matching harmonics and this creates an opportunity for the interval to seem louder than one would predict by adding two incoherent sources. Some texts refer to it as a "resonant" interval.
In any case one classic use of dissonance and consonance in classical and modern western music is the authentic cadence or resolution. We commonly use the chord change V7 --> I at the end of musical phrases. The V7 chord has a diminished 5th from the two notes (Ti, Fa), or (7, 4), of the key and the I has the interval M3 from (1, 3). There is a natural movement (7, 4) --> (1, 3), 7-->8(1) and 4-->3 or dim5 --> M3. The dim5 is sometimes referred to as the most tense interval and M3 the most harmonious. The idea is to create harmonic tension and release it. This "feels good" or is an aural representation of other types of tension --> release. It could represent a fight that resolves, a sickness that is cured, being lost then found (reference to Amazing Grace, and Mahler's 4th symphony) or whatever you think of.
In terms of "approaching" consonance and/or dissonance this is it's classic use. In Jazz we see this idea hyper extended. A lot of classical music just moves diatonically from I-->V and back, or I-->IV and I-->V but ends on V7-->I. Jazz player will often treat each chord as a temporary I and insert its V7 thus create the sound and feel of resolution more often (I'm glossing over this quite a bit). For example the simple set of changes I-->IV-->V for any rock tune (or country, or classical, ...) might by augmented to read, I-->I7-->IV-->#IV-dim-->V9-->V7-->I or some other variant. Each place is "walked into" from an appropriate chord that emphasizes the cadence, or the tension release feel produced by moving dissonant intervals into consonant intervals.
Once you realize this you can really go hog wild with deepening the dissonance in the chord stream and some players (myself included) become fascinated with the tension to the point where you don't want to release it. In Jazz this could be what is described as being "Out" harmonically. One can create this tension to a lesser degree by simply using modified chords that have embellishments. The two features of the V7-->I movement are the D5-->M3 and the chromatic movement of 7->8 and 4->3. So introducing any new chromaticism will generate this tension -> release feel to a lesser degree. Again you hear this played with a lot in Jazz.
Going even deeper it stands to reason that if you used an interval smaller than a half step you could create even MORE dissonance. On instruments like the guitar where we can bend strings it is very common to bend slightly out of tune (1/4 tone or 1/2 of a 1/2 step) and bring the note back in tune. This can sound very haunting and when done right really lave a mark on the listener. One player who comes to mind as a master of this is Jeff Beck. But all blues players make use of this to some degree.
Last point I'd make is that "dissonance" and "consonance" of intervals are somewhat frequency dependent. So a M2 played by a Bass will sound muddy and you may even hear rapid beating of the fundamentals which is a physic based cause of "dissonance" according to Helmholtz. However the same interval played by a violin in the higher register will not be that dissonant.
People love playing with this phenomenon and I'll leave you with an example I use when playing the classical guitar. When I play the A on the second fret of the G string it will excite the n = 2 harmonic of the open A string (this is the same note). The excitation can be so great as to make the harmonic as loud as the plucked note after a short time. I then bend the plucked note slightly until I hear the beat frequency and try to match an 8th note or 16th note tremolo in the tempo of the tune I'm playing. This is not an attempt to replace the tremolo technique but it sound really cool, almost etheric. And even though it's horribly out of tune by matching tempo it sound "correct". This is a way of manipulating what the audience hears and feels by manipulating the physics of the instrument. I will eventually drop the bent note down which creates the classic feel of tension release.