What is a simple way to determine the notes in a scale?
For example, let's say, that somehow you determined that a song you want to play, is in the key F# major. How can you quickly recall or figure out the notes that are included in the F# major scale?
Scales are built on interval formulas. Diatonic scales (no added sharps or flats other than what is in the key signature) use only half steps and whole steps. With a diatonic scale you can follow the same interval pattern for major scales and one for minor scales. These patterns (as seen in the links) are Whole Whole Half Whole Whole Whole Half which is often notated as WWHWWWH
for major and Whole Half Whole Whole Half Whole Whole notated as WHWWHWW for minor.
*Note - An interval is the distance between two notes. (ex: C to C# is a half step)
Using these interval patterns you can start at your root note (the first note of the scale) and follow the pattern. For example with a C major scale you start with the root (which is C) and use the major interval pattern to find the rest of the scale. Which would be this:
root(C), whole step(D), whole step(E), half step(F), whole step(G), whole step(A), whole step(B), half step(C)
I would suggest trying a couple major and minor keys using the correct pattern to get the hang of it. If you get stuck look at a key signature chart or a circle of fifths to make sure you have the right notes in the key.
Here is a list of all the scale interval patterns if you are working with more than just diatonic scales:
Major Scale: R, W, W, H, W, W, W, H
Natural Minor Scale: R, W, H, W, W, H, W, W
Harmonic Minor Scale: R, W, H, W, W, H, 1 1/2, H (notice the step and a half)
Melodic Minor Scale: going up is: R, W, H, W, W, W, W, H
going down is: R, W, W, H, W, W, H, W
Dorian Mode is: R, W, H, W, W, W, H, W
Mixolydian Mode is: R, W, W, H, W, W, H, W
Ahava Raba Mode is: R, H, 1 1/2, H, W, H, W, W
A minor pentatonic blues scale (no sharped 5) is: R, 1 1/2, W, W, 1 1/2, W
R - root
W - whole step
H - half step
1 1/2 - a step and a half
I hope that helps. Good luck!
MattCamp's is a thorough answer, but I feel that the question is too vague. If the OP is asking purely about SCALES, then this would probably work, but would depend a lot on the instrument in question. If it's keyboard related, then # and b being quoted may help find a way round (black notes = # & b), but if it was a guitar, for example, it's not much help. Each note looks the same ! For scales on the guitar, patterns work far better. Not particularly technical - but who says music making has to be ?
Why would one want to learn (or know) scales? You don't learn the alphabet to be able to speak. They're there to mix up to make tunes, so either way, learning scales by whatever method should only be a means to an end. By the time the OP has learned all the formulae for the different sorts of scales (why no pentatonics ?) he may as well have learned them note by note by rote.
Now the question is a lot clearer - I think the answers so far all go along lines that are too technical.Your answer as a guitarist is to learn PATTERNS on the fretboard. When you've learned just one shape (2 octaves worth) for each of the main scales - major, natural minor, maj. and min. pentatonic. and the variant called blues, you'll have enough patterns and shapes to cope. To change key, just move up or down. This is not a complete panacea - it doesn't exist - but it'll take you a long way, safely.
Guitars are amongst very few instruments that this trick works for, so feel lucky. Try a phrase you know, in the key you know, then move it along the fretboard. You keep the same relative fingerings, but you'll be playing in a new key. Goodluck !
@Neil Meyer has covered what the cycle of fifths is (well, half of it), but it works the same way with the flats.
Since you just want to play/jam (and I guess at the moment you don't want to write down on a formal piece of paper anything) I suggest you do this:
Start with the C major scale (or A minor) that has only natural notes. That means the notes it contains are C D E F G A B.
Now, look at this picture:
I know it might seem confusing at the beginning, but you'll get used to it. The notes you're going to need in the C major scale are the colored ones (as well as the open strings).
Just play them again and again and again and you'll get used to them eventually (that's what I did). It might be easier to play the notes (of C major scale) on just the first 5-6 frets at first,then add one more fret,then one more and so on..
Then, when you feel comfortable enough, move on to the next scale (on the cycle of fifths), so that there aren't that many differences on the key (only one sharp or flat) and you do the same thing.
This is a good picture to help you with it. You start from the C and then move on on the right or left till you reach the last one.
Each step on the right adds a sharp (#) to your key. That means G major scale has one sharp, D major scale has 2 sharps etc.. The cycle works the same way if you go to the left. The only difference is that you add a flat (b) to your key.
If this picture confuses you, here are some tips: Each pizza piece contains the chord (C), then the notes that the chord contains (C E G) and then the relative minor chord (Am).
Edit: If you don't want to use the cycle for whatever reason of yours, you can just Google 'C major scale' or 'F minor scale' or whatever to find out what notes the scale you are interested in contains and then use the first image to find them on the fretboard.
I'd venture that you're looking for something fairly simple that can help you construct a scale on the spot.
Here is what I use for guitar and piano:
I use tetrachords because they are the building blocks of scales.
Most scales are basically two tetrachords separated by a whole tone (except for lydian, locrian and lydian dominant and locrian#2 but just remember that they all start with an L and they're the ones in which the tetrachords are separated by a half tone;)
Now to the tetrachords themselves: the beauty of the thing is that there is a limited number of tetrachords (major, phrygian, minor, whole, diminished and harmonic) They are easy to learn and you may not even need all of them depending on the kind of music you play.
In this system, the F# major scale is two identical tetrachords (separated by a whole tone as most of the time) called major tetrachords (WWH)
Another example: If I wanted the mixolydian scale of Eb, I'd play a major tetrachord starting on Eb and a minor tetrachord separated again by a whole tone.
It works well for me and I was able to come up with a lydian dominant scale the other during rehearsal in no time when our teacher told us that that scale should be heard over the second chord in Take the A train. They also help figure out scales across the neck if you're a guitarist and distance yourself from Berkley fingerings.
If your mind works like mine, I'd be glad to fill you in on the specifics. Let me know.
For example, let's say, that somehow you determined that a song you want to play, is in the key F# major. How can you quickly recall or figure out the notes that are included in the F# major scale?
OK here is how I teach it.
I made this chart to help learn the notes in any Key. I'm just learning so if you find something wrong with it please comment. 
For Bass the 1st through 7th patterns are 3 notes per sting exercise.
The odd arrangement at the bottom left is a key finder. I made this a straight across tune, no skewing the strings like 1st and 2nd on guitar. Use the sharp at bottom right and align over chart with bottom let hole over root. To me the shapes on the fret board look like a repeating pattern #769L. although the # shape isn't completely filled in. I can see it both ways from the nut to the bridge and the bridge to the nut.
The answers above are very good; you should know how to figure this out on you own. Though, if it helps, there are a few online tools that can tell you this information (and more). Here's a good one: http://www.musictheorysite.com/scale-generator/
What ever happened to finding scale notes from do re mi fa so la te do ? Works for me. Start on any root note for your scale if its sounds wright yippee. Use it all over the neck and note the sharps, when you train your ear to this way of improvising you will never be lost to play a melodic line that will fit with any song. You will soon find and recall all the notes on the fret board, scales are nothing more than a bunch of sharps and flats. You use these odd notes to create musical flavor around the basic major scale that fits what you want to play, By playing this way you will be playing scales, majors, minors and all the odd ones the only difference is you wont even know it, You do not need to know why these notes fit, The only note you need is the one that sounds good. Scale gods are not the greatest players, music is about representing a feeling. learning to play a million notes all over the shop parrot fashion is not going to make you a better player
Think of semitones.
Major:
2 2 1
2 2 2
Minor:
2 1 2
2 1 2
