Musical intervals have mathematical constraints on how they work. The basic principles are simple, but the consequences involve lots of fractions, irrational numbers, and exponents. If you are maths-phobic, you have been warned.

When more than one note is playing, the mathematical relationship between the two notes is important. Polyphonic music normally relies on harmonics: for example, doubling a frequency gives you a note an octave higher, so a tone at 440 Hz and another tone at 880 Hz combine to give you a chord which is a consonant octave. Also important are intervals with the relative frequencies 3:2, 4:3, 5:4, and 6:5. (Western music doesn't normally go beyond that, at least not strictly.) In musical terms these intervals correspond to:

• 2:1 - octave, e.g. C to C
• 3:2 - perfect fifth, e.g. C to G
• 4:3 - perfect fourth, e.g. C to F
• 5:4 - true major third, e.g. C to E
• 6:5 - true minor third, e.g. C to Eb

Already here we've got the basis for some intervals in the scale being differently spaced: just looking at the intervals 3:2 and 4:3 gives us a tone difference between the upper notes in each chord, and 4:3 and 5:4 produce a semitone difference. So here's the core of the answer to your question about E to F, and F to G.

Of course there are complications. The difference between 4:3 and 5:4 is supposed to be a semitone, and so is the difference between 5:4 and 6:5, but they're not the same difference -- the interval in the first one is 16:15, or 1.066667, but the second one is 25:24, or 1.0416667.

In addition, you can't build an octave out of either of these different "semitones": that is, neither of them is a twelfth of an octave. (16:15)^12 = 2.169, and (25:24)^12 = 1.632. Pretty big difference when your target is 2. And you can't build an octave out of six tones formed from the difference between a fifth and a fourth: the difference there is 9:8, but (9:8)^6 = 2.027.

So while the main harmonic intervals tabulated above are non-arbitrary, tones and semitones are, not quite invented, but conventional. There has to be a compromise that people can agree on. There are many different ways of tuning their frequencies to get to a compromise solution. For example, in antiquity, Ptolemy preferred a tuning system that divided a perfect fourth into intervals that we would call (approximately) 1.5 - 1.65 - 1.82 semitones. Other cultures prefer different systems. Many tuning systems feature intervals smaller than a semitone. Still, something approximating a modern western tone does often appear: in contrast to Ptolemy, Eratosthenes preferred a tuning system that was close enough to modern semitones and tones that you wouldn't spot the difference.

Even if you opt for a 12-tone semitone scale, like in modern western music, there are many possibilities for how you tune the intervals. For the last 200 years the preferred tuning in western music has been equal temperament, where every tone and every semitone are identical to every other tone and semitone. This is incompatible with using true, perfectly tuned, harmonic intervals, because each semitone has a relative frequency of ^(12)√2, or 2^(1/12). And that's an irrational number, so you can't combine them to make exactly 3:2 or 4:3. In equal temperament, a fifth comes out as 1.49831, not 1.5, and a fourth comes out as 1.33484, not 1.3333. We've basically decided that it comes 'close enough'.

Prior to 1800 or so there were other options, called meantone temperaments, which featured true harmonic intervals to varying extents. But the mathematics don't admit any tuning system where all the intervals are true harmonic intervals. Harmonic intervals require rational numbers; identical intervals mean irrational numbers.

For further reading there are many options. Murray Barbour's Tuning and Temperament: A Historical Survey (1951) is on the Internet Archive, or for something more recent (and more basic) you could try Erich Neuwirth's Musical Temperaments (1997). There's plenty of more specialist stuff out there too. For the ancient examples I mentioned, the source is Ptolemy's Harmonics (2.73-4, on "diatonic genera"), which is available in a 1999 translation by Jon Solomon.