The scales and the piano
There are many different scales in music, from pentatonic scales ("oriental" sounding scales) to major scales which are more common in Western music.
Why Learn the Scales?
Learning the scales is like learning the ABC's when learning to read: not so exciting, and it doesn't get a lot of practical use in everyday life, but trying to learn piano without learning the scales is like trying to learn to read without first learning the alphabet. It can be done, but the "shortcut" only winds up handicapping a novice player. The counter-argument has been made that the scales mainly use fingers 123, which are already technically strong compared to fourth and fifth fingers, which are weak because of anatomical limitations. This argument's weakness, however, is in realizing that the purpose of learning and practicing scales has little or nothing to do with strengthening fingers 1,2, and 3 - although this is a pleasant benefit. Practicing scales with rigid tempo and in a legato (connected) style helps lay the foundations of rhythm and fluidity of motion.
The scales are essential, because 99% of music is built on them. One can find the scales, or portions thereof, in songs of every complexity, from the simplest one-note rendition of "Mary Had a Little Lamb" to the complicated classical pieces and even almost all modern songs.
When one plays the melody of "Mary had a little lamb" in C major, you can be pretty sure that it doesn't have a wild bluesy part with a lot of black keys. OK, we exaggerated a bit, but we were still very very close. Try to play "Mary had a little lamb" on black keys only. You can make it, and when you can do it, you probably can do it forever. It's easy to remember to only use black keys after all. Conclusion: Scales are very good help in memorizing. If you remember the key of your piece, memorizing eases.
When you try to play some piece by ear, and you have figured out that it is in major key, the rest will come a lot easier. You will now know that the black keys aren't the first candidates for the right notes, if you want to play it in C major. Or if it sounds bluesy, you will presumably need some "blue notes", which you can always find in blues scales (there are various). Conclusion: Knowledge of scales is good for playing by ear.
When you play in parties, and you've finished your highly virtuosic piece and everybody is fascinated by your playing (the situation is fictional, it doesn't happen for real), somebody brings to you sheet music for some piece. But then you see, it has six(!) flats. This is no problem for you, because you instantly know it's either Gb major or Eb minor and what the black and white keys are. And then everybody at the parties considers you as a miracle. Conclusion: Knowledge of scales is absolutely essential for sight reading.
What scales are
In music, a scale is a set of musical notes in order by pitch, either ascending or descending. Western music is divided into whole and half steps. A half step is the distance from one note to the very next note, including black keys. For example, D to D#, Ab to G, or B to C. A whole step is two half steps. For example C to D, E to F#, or Ab to Bb. ALL SCALES RELY ON WHOLE AND HALF STEPS!
If you play any C and play all the white keys up to the next C, you have played the C major scale ascending. Do it backwards and you have played it descending. The scale you have played has a special pattern, what makes it sound major. It's 1-1-½-1-1-1-½. 1 stands for a whole step and ½ for a half step. Try to use 1-1-½-1-1-1-½ for every possible key; it always works; it creates a major sounding scale. Resorting the order of whole and half steps creates different sounding scales -- different modes -- as can be seen below.
Playing C major's relative minor A minor, just start from A and play all the keys up to next A. This scale is actually A natural minor scale. The notes and their order are the same, the start and end point is different. Then start from some other key than A or C. This scale isn't major or minor, it's another mode. Major and natural minor are, in fact, just different modes of each other. In the table below are all the modes. Don't memorize, if you don't want to, the only reason they're there is to help you to understand what minor and major really are. Major and minor just "killed" other modes. But, for example, phrygian mode is still used in flamenco, and all of the modes are used in modal jazz.
Why do major and minor sound so different? Why are the major and minor modes used so much more than the others? The answer lies in chords. So called I IV V chords in major are all major chord. In C major they are I = C major, IV = F major and V = G major. In A minor they are i = A minor, iv = D minor and v = E minor. Note: the roman numerals for minor chords are lower case. The other modes are more odd. You can harmonise many pieces using only I, IV, V or i, iv, v. Or improvise on them: just play something only using the notes of the scale and its I, IV, or V. If anything feels confusing, visit Ricci Adam's Musictheory.net. It's a free site with easy to follow animations.
Each scale degree has a name. I is called tonic, II is supertonic, III is mediant, IV is subdominant, V is dominant, VI is submediant, VII is leading tone. For example, in the key of C Major, if you are referring to the G note, which is the fifth scale degree, you would call it the "dominant."
Table of scaledegrees: I - Tonic - First (and most important) note of the scale II - Supertonic - Second note of the scale III - Mediant - Third note of the scale IV - SubDominant - Fourth (and third most important) note of the scale V - Dominant - Fifth (and second most important) note of the scale VI - Sub Mediant - Sixth note of the scale VII - Leading Tone - Seventh note of the scale (VIII)I - Upper Tonic - First (and most important) note of the scale
The different minors
The difficulty with natural minor scales lies in the fact that there is a whole step between the 7th and 8th notes in the scale. The 7th note in the major scale gives a very strong sense of leading to the final pitch, a desire to 'resolve' the sound back at the root (tonic) pitch, since there is only a half-step between them. But the natural minor scale does not contain that. The harmonic minor scale includes a raised, or form of the 7th scale degree. So in our example in A minor, the 7th scale degree would be raised from G natural to G sharp (# is the symbol for the term 'sharp'.
But doing this added a new difficulty--the distance between the 6th and 7th scale degrees was now more than a whole-step (the musical term for this distance or interval is an 'augmented second'). This sound was not pleasing to composers, and it can be difficult to sing as well, so a new minor scale, the melodic minor, was introduced, which involved raising both the 6th and 7th scale degrees as you ascend (move up in pitch from lower to higher tones), which shortened the distance of that unusual interval back to something more common and easier to sing. However, when you are descending (moving from higher pitch to lower pitch), it is not necessary to have the 6th and 7th scale degrees raised, since you are not leading immediately into the tonic pitch. So, using our example, the A melodic minor scale would be as follows: (ascending) A-B-C-D-E-F#-G#-A (descending) G natural-F natural-E-D-C-B-A. You may also notice then that the melodic minor scale (ascending) only differs from the corresponding Major scale on the third scale degree: A Major includes C#, whereas A melodic minor includes C natural.
List of things to practice with scales
All major scales, harmonic minor scales, melodic minor scales and natural minor scales (natural not so important):
Hands together, hands separately, four octaves, two octaves, ascending, descending, ascending and descending, stacatto, legato, tenuto, crescending, decrescending, softly and loudly and everything between, rhythmic variations, contrary and similar motions.
Fingerings for the major and minor scales
Few different schools exists (in no particular order):
1. Always use fourth finger on the black key. (More specifically: fourth finger on Bb with the right hand, and fourth finger on Gb with the left hand.)
2. "Orthodox" Hanon-fingering
3. Same fingering to every scale