half note and eighth note stems for voice parts
With voice parts for piano the stems usually go on opposite directions, when they're on the same staff. But I have a book where a half note and an eighth note are one semitone apart but together like a chord. Does this mean that they share the stem? Shouldn't their stems go in opposite directions?
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I believe the situation you're referring to should look something like this:
Note that their positions should be reversed if the 8th is the lower note, and also probably the 8th would be joined via a beam to other 8ths (as in the example at the bottom of this answer).
This is a good explanation from retmusic.com, emphasis mine:
When writing the interval of a second, the stem should be placed between the note heads. The higher pitch is always placed to the right, regardless of stem direction (this results in one of the note heads being “displaced”). [...] When the interval of a second is written with opposite stems, as with multiple voices sharing the same staff, the stems are aligned vertically — in the case of a second, this means that the higher note is placed on the left.
The second bar in the referenced image corresponds to the bolded text. If your example looks like the first bar, then the two notes do not represent separate voices! (Or the editor made a mistake.)
You even draw both stems when both voices share the same pitch:
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