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The names of the intervals is derived from their position within a traditional major/minor scale, where if starting on A, E is the 5th note you would play as you ascended the scale, and D the 4th. (Since this gap exists in both Major and minor scales, these intervals are known as "perfect". Intervals such as the 3rd, which is in a different place in a Major Scale or Minor Scale, are usually defined as major or minor 3rd.
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Tell me why we have perfect intervals. Why can't a 5th, for example, be major or minor?
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A major scale for example ascends by a tone, tone, semitone, tone, tone, tone, semitone. A minor scale ascends in a pattern of tone, semitone, tone, tone, semitone, tone, tone.
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Scales are not laid out this way. It's customary, and far more useful, to use numbers to represent the scale degrees or positions:
1, 2, 3, 4, 5, 6, 7, 8
In a major scale, the two half-steps occur between 3 and 4 and also 7 and 8.
In a minor scale, the two half-steps occur between 2 and 3 and also 5 and 6.
For every major, there is a corresponding or relative minor. Both scales use the identical notes.
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The actual notes within that sequence are defined by the root note, that which is defined as first in the scale.
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The term root note is used for the lowest note in a chord. It's customary to refer to the notes in a scale as:
1. Tonic
2. Super tonic
3. Mediant
4. Subdominant
5. Dominant
6. Submediant
7. Subtonic/Leading tone
We usually call a minor 7th the subtonic and the major 7th the leading tone. Each one of these positions has certain task to perform. The tonic is the foundation note of the scale. The subdominant and dominant are called that precisely because they are the dominant notes in the scale. They are also inverses of each other. They are the twin pillars holding the scale up. The submediant and mediant add color to the scale via major and minor intervals. The leading tone adds restlessness to the scale and wants to propel it back to the tonic again so that the scale can rest. We say that it wants to
resolve. Likewise, the supertonic wants to lead way from the tonic.
Now you may wonder why the mediant at position 3 comes before the submediant at 6. It's because the straight-line scale we are accustomed to is misleading as hell. The tonic is really the note around which the others revolve so it is in the center. The dominant is a 5th above and the subdominant is a 5th below it. The closest notes to the tonic are the supertonic a step above and the leading tone a half-step below it. Both are restless and want to lead away from or back to the tonic. The mediant is a major 3rd above the tonic and the submediant is a minor 3rd below but since they are squeezed between the supertonic and leading tone and the dominant and subdominant, they are in the middle or median positions.
So a scale is like a solar system rather than a ladder (scale is derived from the Italian scala or ladder). The tonic is like the sun in the center, the supertonic and leading tone flank the tonic. Then the mediant and submediant flank the supertonic and leading otne and then the dominant and subdominant flank the mediant and submediant.
Hope that wasn't too confusing.
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You can also get musical systems, such as indian music, which deal with microtones, and split the octave into more than the traditional 12 semitones, such as systems with as 19 or more possible notes, though I don't know how many notes these musical systems tend to use to define the scale.
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Scales are assembled using the Fibonacci Sequence. This sequence starts off with two numbers and the third is the sum of the first two. For example:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. Notice how each number is the sum of the two before it? Scales follow this same sequence--5 (pentatonic), 7 (diatonic), 12 (chromatic). The next scale would be 19 notes and then 31 after that. They have built 19-tone keyboards but they are unruly. The cool thing about the Fibonacci Sequence is that the closer to infinity one gets, the more closer any two adjacent numbers in the sequence divide out to Phi or phi. Phi is 1.618 (it's actually irrational) and phi is 0.618 and is the reciprocal or Phi. The same is true of the scales, the closer to infinity they get, the closer any two adjacent scales will divided out to Phi or phi. It's a characteristic of the Fibonacci Sequence. You can look around the web for all the cool stuff there is about Phi and phi and the Golden Section and the Golden Mean and the Golden Spiral. It's really neat.