What does “playing in thirds” mean?
If one is playing (double stops) in thirds in C (i.e., playing the scale of C in thirds) one plays the scale of C (C, D, E, F, G, A, B, C) simultaneous with (E, F, G, A, B, C, D, E). In other words: CE, DF, EG, FA, GB, AC, BD, CE.
To figure out what to play, one consults the key of C, notices that E is the third in that scale, and so on...
Why is the third associated with the rest of the notes in that scale not the third relative to each of those notes (i.e., DF#, EG#, FA…)? What is the rationale behind saying: playing 'DF, EG, etc.' is playing in thirds?
(This may appear a silly question for a piano player for whom each finger position is different. But for a guitar player, the question seems more reasonable. Also, it seems absurd when you extend it to all scales and realize all the thirds you'd be playing are the same, regardless of scale. However, it's very relevant if you're trying to figure out how to accompany a melody, note-by-note in thirds [Is it signature-same throughout the piece, regardless of chord change?].)
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You can play a succession of major thirds if you wish, and it's a good and interesting sound. This is just about what 'playing a scale in thirds' means. It normally means just what it says, play THAT scale in thirds, not play e.g. C major with E major on top (which would maintain the major 3rd interval).
Be careful when mechanically 'adding thirds' under a melody. If you don't want to distort the harmony, it's a good idea to make sure that, on the strong beats at least, the harmony note fits the prevailing chord. Sometimes this means adding a third OVER the melody (or a sixth under).
Of course, in a tune in C major, a D chord (or a Bb, Eb, Fm etc. etc.) will often appear. Your harmonisation in thirds will need to take notice.
Thirds come in two varieties - major and minor (for the sake of this question).Let's take your example of key C major. Playing any note from that key, a third above it will be the next letter name but one.So, C+E, E+G, F+A etc. Sometimes it will produce an interval of a maj 3, sometimes a min 3. This keeps all the notes played diatonic, so within the key (C).
You'll find, if harmonising a line in thirds, that the notes will pretty well go with the underlying chord. Thus, in a Dm bar, which you'll find in a lot of pieces in C, If the tune goes D-E-F-D, the thirds harmony will be F-G-A-D, in other words, m3-m3-M3-m3. Your idea of putting F# and/or G# will become noticably wrong if you try it, and listen.
Occasionally, usually during a modulation, the 3rd needs to be as you would like. Example, in C, but modulating into G, play a D, and the 3rd might be F# rather than F, because that gives a leading tone a senitone under G, the new target. Try the two, and you'll hear how the F# third is more convincing.
To try to answer your query to alephzero, playing chromatically, use your ears. Either the maj 3 or the min 3 will sound better. Guess which one to use.
Try playing C-E, D-F#, E-G# while somebody else is playing a C major chord, and see what it sounds like compared with C-E, D-F, E-G.
If you want your "thirds" to fit into the harmony for the piece, they need to stay within the notes of the scale.
Of course if you are playing a Jazz piece based on a whole-tone scale, then C-E, D-F#, E-G# is the "correct" option - but your question doesn't seem to be about that scenario.
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