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Generally, theory books mention that one should double the root of the chord. This is followed most of the time, except by some composers who choose to double the 3rd or the 5th of the chord.

When attempting theory exams, can you double the 3rd or the 5th of the chord as well? When does one double the 3rd or the 5th of a chord? Or is it that composers can choose which note of the chord to double? Is there a rule regarding doubling notes of the chord other than the root?

There is a difference in what composers do and what is acceptable in a theory exam.

In an exam, usually you don't double the third of a major chord, but you can double it if the chord is minor. Doubling the root or the fifth is the safe choice. To see what note of the chord you'd double, you have to see the preceding as well as the following chord. You'll see that sometimes you cannot double the root, so you'll have to double the fifth/third or vice versa. There isn't a single rule to follow for this, you have to practice harmony.

Exams aside, when you double the root, you make the chord sound somewhat stronger, as far as tonality is concerned. It's not uncommon in piano pieces, for the left hand to play the root in octaves (not only in classical harmony, but in modern music as well)

When attempting theory exams...

Ask the person who will grade the exam.

Whatever rule they want you to follow for the exam will be contradicted by actual practice.

You can easily find examples of a root position tonic chord with the third doubled and the fifth omitted.

Walter Piston's Harmony has a simple rule: in root position double the root, for inversions double one of the tonal degrees of the key. The logic is clear enough. Doubling tonal degrees with reinforce the tonality. In terms of actual historic practice, I defer to Piston to have know better than me. I like the rule, because it is logical and I can remember it.

I once took an exam where in order to get a correct answer I had to contradict what I knew to be true from actual composer's scores and Piston's rule.

The rule comes down to two things: voice leading and avoiding parallel fifths/octaves.

In an SATB arrangement you're going to double a pitch, or leave one voice with nothing to do. Each voice should be moving smoothly, and generally following the guidelines of counterpoint. You'll need to look at what each voice is doing, and look at what chord comes next to make the decision on what gets doubled.

Beyond that, the hard and fast rule is that the last chord is voice with the root doubled, and it should be in the bass and soprano.

As I understand it, this question has to do with chord voicing. In practice the note most often doubled is the tonic. This reinforces the overall sound of the chord and stresses the key-center. Doubling the fifth strengthens the "stability" of the chord, and doubling the third emphasizes the major or minor aspect of the chord. When we choose to double notes in a chord, we are choosing which chord voicing to use in an arrangement and how one chord relates to another in a chord progression. When one chord changes to another, the highest note in the chord establishes a melodic relationship to the highest note of the next chord and the lowest note establishes the inversion of the chord. As I play around with different voicings, I can hear for myself what works and what doesn't.

The sense of an exam is that you will have the chance to show that you know more than the experts.

That you are able to discuss a question and not searching the one only correct answer, to demonstrate that you have the competence to break rules and argue why your point is also possible and defend it. Books are often written by a follower or disciples of a certain theory in the style of a catechism.

I agree with Michael Curtis supporting especially the last sentence.

So you can surprise your experts when you ask: do you mean the rules of the Vienna epoque or Baroque, and if Haydn ... the early Haydn or the later? It‘s good to know the principles of different authors as you can better understand, learn and memorize the rules if you can compare them with each other’s. It‘s interesting to see how different authors came to different rules in different times. (Only with this position you can show that you know something and don‘t only repeat what you have read in books!)

The 2 pages are copied of the „Harmonielehre“ (de la Motte). He shows that the doubling of the third (basstone, first inversion) respectively the root tone of the triads of major chords in root position has changed and varied in the two Epoques and that applying the rules of the Classic-period to the music of Bach and Händel is absolutely wrong (German ed.) inaccurate (English ed.)