As I was studying music theory (harmonizing a melody in particular) I got confused with the statement:
The interval of a 4th above the bass is considered dissonant and must resolve down by step.
Could you explain what is meant by that?
A dissonance is an unstable sound - two or more tones sounding together that demand a resolution towards a consonance, which is a stable sound. "Resolving" a dissonant interval means that it is followed up by a consonant interval.
Consonances are divided into perfect and imperfect ones. Perfect consonant intervals are most stable; they are the prime, octave and perfect fifth. Imperfect consonant intervals are third and sixth. The second, seventh, as well as the diminished fifth and augmented fourth, are dissonant intervals. If you haven't tried what they sound like, try it.
"Resolving" dissonance is not as simple as putting just a consonance immediately after the dissonance - there are special rules for how the move is made. It would go too far to mention all rules that could apply, but in general, the rule of the thumb is that dissonant intervals are resolved by moving the minimal number of voices by the smallest interval to achieve the nearest imperfect consonant interval. The concrete case you describe fits this pattern: if you have a dissonant perfect fourth, and let the upper voice descend by a step, the next interval will be a (minor or major) third, which is an imperfect consonant interval.
The harmonic interval of a perfect fourth is sometimes treated as a consonance, sometimes as a dissonance.
It is almost always considered a dissonance when there are only two voices producing this interval. Example: bass plays E, and the melody voice above it plays an A.
(The implied chord here is a chord of A, with A being the root and E it's fifth. But since the E is now below the A, it is not a fifth but a fourth, implying a second inversion of the A chord. This interval sounds quite harsh. Typically it is resolved by leaving the bass, and lowering the A by 1 step so that bass and melody now form a third, implying a chord of E (with omitted fifth). There are other ways of resolving it, but this is the example meant by your theory book)
If there are more voices, then if the lowest voice (bass) is a fourth below the voice immediately above it, then the interval of the fourth sounds so prominently, that it is considered a dissonant and needs to be resolved in the same way as in the 2 voice situation.
But a fourth between any two voices above the bass is typically considered a consonance, that is, it needn't be resolved in a special way (provided that all other intervals are consonant).
Let's say you have 3 voices, with in the bass, an A, and the voice above it an E, and then in the upper voice also an A, then there will be a fourth between the upper two voices. However, it doesn't sound as harsh as when it is between the lowest two voices. It's as if the bass stabilizes the sound.
Even if the bass would now have a C, the fourth between the upper two voices would still sound consonant (in fact this would be a Am chord in first inversion).
It is simple, but first you have to understand what an inversion is and how the bass defines harmony. The bass note (i.e. the lowest note) defines the harmony because it is the lowest (fundamental) frequency. Because the bass has such a strong effect on the harmony, whatever note of a chord the bass is playing defines the inversion of the chord being played. If the bass is playing the root then the chord is in root position. If the bass is playing the 3rd of a chord the chord is in first inversion. If the bass is playing the 5th of a chord the chord is in second inversion.
The statement you pointed out pretty much means that a second inversion chord is considered dissonant and must be approached and resolved with care in order to sound right. This music theory lesson gives you a more in depth idea on how to use them. Hope this helps.
It's somewhat subjective. In the 'bass', it depends, to the listener, how low the interval gets played on the instrument. Lower, a third interval, and fourth interval, will sound quite 'muddy'. The harmonics each of the two notes produce are not sounding happy in each other's company.The fifth, however, blends well.
But - if the notes are still in the 'bass', but not too low, they will sound fine together. Particularly if the missing note from the triad is played in the 'treble'.This has the effect of completing the chord.It is odd, because a fourth is effectively an upside-down fifth.
I say subjective, as it also depends which instruments are playing the interval. Some will have harmonics which will not affect the consonance. A distorted guitar, for example, will make the dissonance sound, well, dissonant.
There could also be a different reason the theory writer has said that: the fourth interval may be a suspended one, which wants to come back to a third, or go up to a fifth.
I'm playing Devil's Advocate here a little, as I find there are sweeping statements in some theory that need some explaining, but it's not forthcoming.