Semitone mathematics and perfect fourths/fifths

Last post was about the independence of semitones and their relationships to scales and modes.

Now let's look at the mathematical relation between semitones, and how it ties up with the perfect fourths and fifths of a major scale.

Take a look at the guitar again. Every interval of 12 frets leads to an octave of the initial note, whether from the 1st to the 13th or the 2nd to the 14th etc. Since we've already discussed that none of the frets are inherently special or different from another, these frets must be related in some manner.

Measure the widths of the 1st and the 13th frets, or the 2nd and the 14th etc. It is always found that the fret 12 frets above is half the width of the original. Further, the point on the string 12 frets above a note precisely divides the vibrating length in two, hence doubling the frequency.

Thus, a frequency scaling of 2 is found for a difference of 12 frets, leading to the conclusion that each fret leads to a frequency scaling of 2^(1/12) over the previous,
i.e. f(2nd fret) = 2^(1/12) * f(1st fret).

The whole guitar is tuned by setting a standard as A = 440 Hz. This is the frequency that you hear when you lift the receiver on the phone, enabling musicians to easily tune their instruments.

Now, consider a fifth of a note. It is 7 semitone intervals away from the root. Pull out your calculator and calculate 2^(7/12). It's almost 1.5.

Similarly, the fourth. It is 5 semitones away from the root. Calculate again to get 2^(5/12), almost 1.33 = 4/3.

Also, 2/(4/3) = 3/2 i.e the octave is a perfect fifth away from the perfect fourth. This would probably be known to musicians from the Circle of Fifths anyway (more on that later).

It's because these two notes have such simple relationships to the root that they have similar key signatures (more on this too, later!). Very interesting to note is the fact that in a given major scale, only the perfect fourth and the perfect fifth too have major chords that fit in the scale.

For e.g. in the C major scale (all the natural notes, white keys on the piano: C, D, E, F, G, A ,B, C), the only major chords other than C major (the root chord) are
the fourth : F major (F + A + C),
and the fifth: G major (G + B + D).

That actually reminded me of cadences, and jazz chord progressions, but, since there are so many "more on that later"s and this topic is logically complete, I'll sign off. I need to figure out a way to write these articles by applying "non-spaghetti code" techniques that I've learned in my computer programming career. :)