What are the rules of borrowing notes from other scales if any?
Are there any rules of borrowing notes from different scales? If so what are they and what is the scientific explanation of it in terms of sound waves' lengths?
For example I've got a chord progression in a minor scale i IV VI VII. In E minor it would be Em A C D. The chord IV (A major) is wrong, it should be minor (Am). However, many songs use that progression and it sounds ok. How can that be?
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"Should"? Why? Where did you get the idea that a chord sequence should all fit one scale? Mostly it doesn't. Here's a very simple pop-song type sequence. C, C#dim7, Dm7, G7, C, Bb, C. It never strays from the key of C major. But it uses plenty of notes and chords that aren't in the SCALE of C major. And they aren't "borrowed" from anywhere, because it's just fine to use the dim7 on the sharpened tonic and the major chord on the flattened 7th.
We get a lot of people here who confuse a method of improvising with a system of musical theory, and they're all asking variations on the same question - "how can this chord/note be allowed in this key?" They're often guitarists. Someone out there is teaching guitarists a most peculiar musical theory system :-)
There aren't any rules specifically with scales to this matter.
However, it just takes instinct. The people who improvise like this in such a way that they literally create new scales as they play; have been playing for many years and can hear if it will sound right before they even play it!
Using a V chord in this example but can apply to any chord,
It is common that some players will be playing a scale over say.... a V7 chord in some progression and decide to switch from the scale to a V7 arpeggio or V altered arpeggio and add a couple notes into their improvisation. Which combines the scale with the V arpeggio.
There are rules however to adding in notes. To understand them all you have to understand Chord Theory since adding in extra notes into your scale will depend on what type of chord is being played whilst improvising.
Simple example to applying chord theory to improvisation, if the chord in which you are improvising over is a minor (has a flattened third) you're not going to add in a natural third to the scale. Because it would sound horrible!! (unless of course you like the tension)
As for the progression. There aren't any rules to tell you that the AMaj should be a Amin chord. There are recommendations to basic structure, but no rules to say it doesn't sound nice either.
Always remember the most important rule in music:
If it sounds good, it IS good - Duke Ellington
Hope this helps! :)
Generally, the neighboring keys in the circle of fifths are good candidates for any chord borrowing. D-major is only one sharp away from e-minor, i.e. the scales are very similar.
Specifically, VII acts as a dominant in a natural minor scale, and IV is a dominant to that, i.e. a double-dominant. This is arguable in your example though since there's a VI between the IV and the VII.
Why going to neighbours in the circle of fifths works so smoothly can be explained easily enough on grounds of physics: a fifth is a frequency ratio of 3:2, which also occurs very early in the overtone series of most instrument sounds. So, adding a fifth on top of any note that's already in the scale will hardly get you out of the scale at all; in your example, the C? in the A-major IV chord is actually already present as the third harmonic of the scale note F?.
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