Dissonance is an interesting concept in music. As I have learned the idea that dissonance is inherently bad, is a completely outdated concept. For the longest time, classical music has operated under the header of consonance.
That is to say, that is what is easy on the ear was held as good. The name of our intervals reflects that fact. The octave, the unison, the fifth and the fourth are given the title of "perfect" for their inherently consonant sounds.
The second and seventh are relegated to merely being Major or minor, there dissonant quality betraying them. Only trough the work of the modern genre has there been a paradigm shift in regards to dissonance and the means in which music applies it.
So back to the question at hand. The second has a truly uneasy quality about it. It simply is too close to the root. The minor second, in particular, does not sound all that good. It sounds very much like a piano player playing two notes next to each other.
The third on the other hand starts to get better. The Major third is a particularly strong, consonant interval. The minor third also has a different quality but can also work very well in certain context. The quality of the third gives a chord its main character and is the most important part of the chord in a harmonic context
The fourth is a bit awkward. Although it does bear the title of perfect it to me sometimes sits awkwardly between the third and the fifth.
The fifth along with the thirds are the two intervals that form the consonant basis on which chords can be build. The diminished fifth or the devil's interval has a hauntingly beautiful character.
For a fair number of years, this interval was considered to have an evil character and was only used in composition when the devil or things that were considered evil, were composed about.
The Major sixth is another important interval as it has a good sound on stringed instruments. The Seventh is important in the way in which it with the tonic provides the finality on which music can end.
The Major seventh that forms part of the dominant chord of keys in both the harmonic minor and Major keys was the most important development the Baroque era developed and has been the basis of most music the west has produced for nearly five centuries.
The octave is good to provide notes sounding different, do not take away or give anything to the character of the chords. C Major chord still sounds Major regardless how many of the octaves, of the chord notes, are played.