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"In the four-part writing of diatonic seventh chords the fifth may be omitted, in which case the doubled note in a four-part texture should not be a tendency tone. The most common doubling is the root." (in Harmony text) What is a useful definition for a tendency tone in Harmony theory ?

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In diatonic harmony, tendency tones tend to be one of two things:

  1. The leading tone, which has a tendency to resolve up to tonic (especially when in an outer voice), and
  2. Chordal sevenths, which have a tendency to resolve down by step (no matter what voice they're in).

The logic is as follows: since these tendency tones have a tendency to resolve in a particular way, doubling a tendency tone will result in parallel octaves, which is of course one of the most strident errors in four-part voice leading.

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    And the reason for avoiding parallel octaves is that harmonics moving together as a group are (kind of biologically) perceived as belonging to the same voice, sort of like tracked as one single dot on the radar, so if you have moving parallel octaves, you don't have four voices anymore. :) Or like, if everyone sings in unison, there's in a sense only one voice, even if they're in different octaves. (I know you know this, but just to add to the answer, it's good to know that the "rules" are for a reason, they're not magical) – piiperi Reinstate Monica Jun 6 '20 at 8:56
  • interesting rationale , merging the voices , as the 2 become one ... – Thomas Jun 18 '20 at 17:32
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In a dominant 7th - tonic progression, the notes that form the tritone in the dom7, the 3rd and 7th of the chord, have a strong tendency to resolve to the root and 3rd of the subsequent tonic. (This desire of a dominant to reach a tonic is the harmonic basis of the whole Common Practice thing.)

We avoid doubling these notes because their active nature would predominate too much. And, as @Richard said, because if both doubled instances resolved 'correctly' it would create parallel octaves.

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I looked up tendency tones in several of my textbooks and online and was surprised to read so many different definitions. But, somehow, I've developed the idea the tones of the tonic triad are stable and the other tones have various tendencies to move to those tonic tones. I guess I accumulated this understanding from many sources over time.

Most of it is covered by the resolution of a dominant seventh chord to the tonic, as described in Lessons in Harmony: Complete. Parts I and II, By Arthur Edward Heacox. Using solfege the dominant's tendency tones resolved to tonic are:

  • TI to DO
  • FA to MI
  • RE to DO

...SOL then is either static, being held in the upper voices in typical voice leading, or it leaps to the tonic, but I've always thought of tendency tones and their resolution in terms of step-wise motion.

That leaves out the tone LA. If we think of tendency tone resolution in terms of movement to tonic chord tones, then it follows that LA steps down to SOL. The only reference I could find to that is in Teaching Music Theory, By Jennifer Snodgrass, but the wording is fairly indirect. It's about an ear training warm up where presumably DO MI SOL is first sung to outline the tonic to establish the stable tones and then FA MI, LA SOL, TI DO, and RE DO are sung in turn to apparently demonstrate the various tendency tones resolving to tonic tones. So, we can add this to the list...

  • LA to SOL

Another way to think of this is the tendency tones are contain in two simple, basic voice leading paradigms: V6/5 to I, and IV6/4 to I...

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