When I play octaves, or any chords that require an octave stretch or more, I find that my hand and forearm become very tense to play them accurately. If I try to play relaxed, the octaves are sloppy and messy. This especially becomes bothersome when playing octaves for long periods of time (throughout various songs). How do I fix the tension in my arm/hand when I play octaves without losing the accuracy of the notes I'm playing?
Thank you Bob, very insightful answer.
My addition would be that sometimes you must take two steps back to take three steps forward. Slow the octaves down as much as necessary to play them tension-free. As Bob mentioned, practice you scales in octaves (very slowly), and find the sweet spot where you can reach the octave accurately without tension.
It could take weeks or months to develop octave scales that don't cause tension, so don't feel down about it - it's a natural part of breaking through the skill plateau. You'll be capable of so much more once you've taken the time to work through this.
If you're becoming tense, it's most likely that you are playing them too quickly. If you can play octaves at all, you can play octaves at some slow tempo where you are both relaxed and accurate. Start there. Be patient, and keep at it.
There is a process of building up strength in weaker muscles, but a much bigger part of playing "difficult" passages is economy of effort. People get tense when they try to force a tempo without knowing exactly where to place their hands, arms and fingers.
I have often asked my students to remember how it feels to stumble while running, and manage to recover without falling down. If you remember a time that happened to you, think about how much more effort you had to make to recover than to actually run. It is much the same with your fingers and hands; when they "stumble" (i.e. go slightly to the wrong place), they need a lot more effort to get back on track than they need to actually play what you intend. That's where the tension comes in. So, it's a cliché, I know, but learn to walk before you try to run.
Exactly where to put your hands, arms and fingers is something you have to teach yourself to a very large degree. A teacher can give you guidelines about how to sit (don't use more than the front half of the bench, for example), arm movements, hand placement, finger movement and so on, but they're your hands, arms and fingers, and nobody else's are quite the same as yours.
You have to start very slowly (or so it will probably seem to you), without wrong notes. As you continue, you'll find that speed comes naturally, just like any other repetitive thing that you might do with your hands. At some point, it just becomes automatic and you don't think about it any more. Typing, for example.
I would suggest that you practice your scales in octaves. Start with using 5 on white keys and 4 on black keys. Don't go in for a legato or a staccato, just something in the middle. And make sure that you play only at a tempo where it feels easy. When you get these under your hands, start working on arpeggios in octaves, so you can get more comfortable with leaps.
Do not expect results. Just put in the work. And one day you'll find that some of those "various songs" are not difficult to play.