The title really sums it all up but I'll try expand it slightly. Lets say I'm learning guitar and would like to improve my guitar work. Which of the scales should I learn first and why?
It really depends what type of music you want to do; and how deeply you want to understand the mechanics of music itself.
If you just want to get straight into jamming a tune; learn the pentatonic scale, its about the easiest scale to learn and very versatile; something like 70% of the licks in all popular since the mid 60's is pentatonic based, and even before that most blues was pentatonic based (with an added note between the third and the forth intervals)
Here is Pentatonic scale in A
Its a good idea to learn this up and down; learn it well slowly before you start to speed up. A wise man once told me that if you never practice to play things slowly first you never learn how to play slowly. Practice it in different keys; its a good idea to find backing tracks to jam over, be musical.
Here is Blues scale in A (with the added note)
Note the added chromatic interval this where that classic blues sound comes from; its a good idea to try sliding into this note, or bending into it rather than playing it straight all the time; be expressive with it its the blues.
If you want to take this further you could bleed over into the natural minor as the mood takes you, since it differs from the pentatonic by only two notes.
A Minor scale
These scales are all pretty interchangeable in the key of A minor; however learning the minor scale particularly give you a whole new range of options.
A minor contains all of the notes of the Key of C Major, and so is rooted in C major, starting from its 6 interval (Aeolian Mode). This being the case, if you play A minor from its 3rd interval you are playing C Major. Because of this you have access to all of the other modes of the C Major Scale.
B Locrian - C Ionian(Major) - D Dorian - E Phrygian - F Lydian and G Mixolydian
If you start the A minor scale from each of the notes above, and carry on through an octave, you are playing the corresponding mode above. Each of these modes sounds good at certain times (over certain chords (their own)); each mode contains 'deviant'(from the major scale) intervals, which when embellished over the right chords really bring out the sound of the mode.
So technically by 'really' leaning A minor you have learned a whole branch of music theory, actually using this stuff really opens up your playing.
The stuff that I have just mentioned is the essential 'in the box' stuff. Things can get a lot more interesting if you start swapping modes in and out of your root key, from A minor to A Dorian for instance. A Dorian is rooted in G Major/E Minor, so you have the whole set of modes again from a different key signature, but you can remain rooted in the key of A.
Im not going any further since i'm already Waaay past the scope of the answer, so in summary:
Pentatonic/blues is a good first scale to learn if you just want to get playing.
Then if you want to take your knowledge further, the minor scale is only two extra notes, and it gives you access to all of the above and a whole world more.
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I would suggest a major scale. Although, pentanotic and blues scales are easier, because they have less notes and easier shapes, the major scale will give you a great starting point, because all other scales are, in one way or another, modifications of a major scale.
Here's the formula:
Here is C major scale on 5th string:
And the same scale in one position:
Good luck in your playing!
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Silver Light and DRL both give excellent answers outlining scales to start out with, but I think they've missed the important aspect to choosing between the two, what type of music do you play, how do you learn, what do you want to learn?
If you a rock / blues player that wants to get jamming with some friends as quickly as possible, then start with your minor pentatonic scale (often referred to as just pentatonic). It's very accessible and is used by a lot of the guitar greats of the 60s-70s. You can play it over an awful lot of music.
If you are a jazz player, start with your major scale. It's as useful for jazz as the pentatonic is for blues. And more importantly leads us to my last point:
If you want to learn music theory, start with your major scale. All music theory is based off your major scale. Absolutely all of it. If that's what you're interested in, learn it as soon as possible. Certain types of playing will absolutely need this knowledge (see Jazz and to a lesser extent Classical).
Here is your C Major Scale
Here is your C Minor Pentatonic Scale
Note that your first note in each scale is the root note of the scale. Low E, 8th fret is a C, so the scale is a C scale. If you were to move everything down one fret, then you would have a B scale. One more fret, and it's Bb / A#, etc. This is one of the big advantages of learning scales and chords on the guitar, once you know one, you know them all.
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The seventh mode of melodic minor. Also there are plenty of symmetric scales out there, meaning the number of modes (starting points) doesn't necessarily go up to seven.
The C major scale in the first position would be an easy scale to learn first.
I remember when I was 10 or 11 and I'd already learned to play all the first position chords. Then "La Bamba" came out (redone by Los Lobos) and of course I had to figure out that solo... which is the C Major scale played all in the first position.
The good thing about figuring this out is that the scale already has a context - C/F/G, and it has a melody you can hum/sing and a distinct rhythm! Perfect for a beginner!
Don't worry about knowing the scale and the notes - go by the sound of the notes and being able to play with the record/cd/mp3. Use the open strings! Once you can manoeuvre your fingers around the open position - you can then begin to study what you've just learned ;-)