When changing from 3/4 to 6/8 does the eighth note stay the same? Same thing, does the eighth note stay the same length when switching from 4/4 to 12/8? Thank you!
My answer would actually be different for your two examples. That is, given no other indication in the score, when executing a time signature change from 3/4 to 6/8, I would keep the 8th note constant, whereas when changing from 4/4 to 12/8, I would keep the beat constant, making the 4/4 quarter equivalent to the 12/8 dotted quarter. But, if the 4/4 - 12/8 change was happening in a flurry of other changes involving 2/4, 5/8, 7/8, etc., it is far more likely that the 8th note is meant to stay constant.
So I would say there are no hard and fast rules to determine this, just some tendencies and "defaults" that exist in the literature. Some clues that might lend a hint to this would be subdivisions in the former time signature that are re-notated in the latter; for example, eighth note triplets in 4/4 that continue as regular 8th notes in 12/8 would indicate the beat staying constant as I mentioned above. But in most cases, especially those that are ambiguous or would otherwise deviate from the norm, the score editor or composer will (or should) indicate either a raw tempo marking for the new time signature, or (more preferably) a rhythmic equivalence... something like:
e = e 3| 6| 4| 8|
q = q. 4| 12| 4| 8|
In some cases, you'll actually need to do some math to figure out what the new "tempo marking" is given a complicated rhythmic equivalence, but since musicians think in terms of rhythm (and not in terms of BPM) this is usually the better option. If there's no logical or musical rhythmic equivalence, then a straight tempo marking will probably be the way to go.
In the absence of unambiguous markings, you can always consult existing recordings, or (in the case of unrecorded works) the composer themselves.
I think that unless there is a change of tempo indicated on the score, your eight note will have to be faster when changing from */4 to */8 since you will now have to play 3 eight notes per beat instead of 2.
For classical scores, it is mostly determined by the style and usually known intuitively. For example 3/4 can be a waltz and a latter 6/8 can be a 3/8 + 3/8 triplet then q=1.5e but no need for an indication.
But when scoring contemporary pieces, you should always indicate clearly whether q=q or q=e (or whatever) as it is very hard to relate most pieces written today with any style or notation. It is always possible to encounter strange progressions like 5/8 - 4/4 - 17/16...etc. The player will never know what was in your mind at the time you were scoring that part.
You should not approach them the same way even if they have the same amount of notes. 3/4 is Simple Triple time while 6/8 is compound duple time.
Where 3/4 has a strong air of waltzes about it 6/8 time not so much. You have three groups of crotchets versus two groups of dotted crotchets. This may sound like I'm just being pedantic but there must be acceptance of the two strong beats in the bar in 6/8 time versus the three strong beats in 3/4 time.
If you are writing something, and you are maintaining the tempo through the passage, don't change the meter if:
- it is a case of temporarily moving between 6/8 and 3/4 (if the 8th note is constant). This is how Brahms did it (from Op. 117 #1, with an indicated meter of 6/8): It's a standard rhythmic effect called hemiola. Grouping the notes is sufficiently clear.
- it is a temporary use of triplets in the time of a quarter note in 4/4 time - just mark them as triplets.
Do change the meter if:
you are doing something like moving permanently from a 3 beat (3/4) to a 2 beat (6/8), where the beat remains constant but the subdivision of the beat does not. In that case, add a tempo marking that a dotted quarter note equals a quarter note from before the change. If you are alternating 3 beats and 2, it is probably wiser to write 3/4 and 2/4, with triplets written in the time of the quarter notes in the 2/4 bar.
you are making a permanent change from 4/4 to 12/8 when the actual pulse remains the same, but the division of the pulse changes. The advice applies as (1) with regards to adding a tempo marking.
You are making a permanent change from 3/4 to 6/8 with the eighth note constant. Add "♪=♪".
you are making a change from 4/4 to 12/8 and the value of the eighth note remains the same. Add a tempo marking that a eighth note equals a preceding eighth note.
My advice for changes in the opposite direction than specified in these last 4 items is, of course, pretty much the same, although the tempo markings will be reversed in (1) and (2).
Edit: If you are doing a lot of bopping between 3/4 and 6/8 at a constant eighth note, consider adding a time signature of 3/4(6/8) (or vice versa) at the beginning of the passage. The intent of this kind of time signature is pretty clear.