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. :)

Semitone relationships and scales

Look at a guitar, or a piano/synthesizer.

Regardless of where one starts on a guitar, an octave of the note played is found 12 frets above the first fret played. Similarly, in a piano, octaves of notes are separated by 12 semitones.

This leads to the deduction that there is nothing inherently 'special' about the black and white keys on a piano. This is more evident on a guitar, where there are no demarcations for the notes, and all frets bear the same relationships to the adjacent ones.

Essentially, the notes could have just as well been named A through L. There is no reason why 'C#' should be named thus, incorrectly implying that it is in some way related to 'C', and that 'C' has some special significance.

Semitones are mathematically related, however. This is evident by the trained human ear's tendency to 'recognize' a major/minor scale, indicating that the degrees (2212221 in the case of major, 2122122 in the case of minor scales) and the relative frequencies between notes are important for a scale to 'sound' right and for the ear to intuitively identify the root.

The seven modes of a scale (more on this later), however, can be used effectively to 'reprogram'' the ear into accepting something initially unfamiliar as 'correct', making the argument about getting accustomed to set scale degrees null and void.

Physics of the acoustic guitar

Ok, let's get started.

Many musicians who play guitar do not have an idea about the workings of their guitar - the physics and the math that makes playing of a guitar possible.

Throw your mind back to your high-school physics (or check here if your mind refuses to get thrown!) and recall string vibration theory, where,

f = 1/(2L) * √(T/u), where,
f = frequency of vibration of string,
L = Vibrating length of string
T = Tension in string
u = mass/unit length of string

In a guitar (see here for the parts of a guitar)
- The frequency (f) is percieved as the pitch of the note played on the guitar.
- The vibrating length (L) of an open string is the length between the bridge and the nut, and this is changed by fretting any of the frets, decreasing the vibrating length.
- The tension (T) in the string is adjusted using the tuning screws
- The mass/unit length (u) is characteristic of a string, with the thicker bass strings having a higher 'u' than the thinner treble ones.

All these combine to produce the working of a guitar. Fretting a fret, decreases L and increases f. Tightening the tuning screws increases T and hence f. The bass strings have a higher u as compared to the treble strings, and that makes their f lower in comparison (for the same fret).

This simple equation is also the reason it is possible to play harmonics on a guitar. Lightly touching your finger on a vibrating string creates a "node" at that point, effectively decreasing the vibrating length. If the ratio of the new vibrating length to the original is a rational number, a harmonic is formed.

Human ears can only detect the harmonics if the fraction is a simple one with small integers, e.g. 2/1, 3/2, 4/3 etc.

Intro

I intend to publish a number of posts explaining music theory and the math behind music - how music works, why certain types of music "sound" the way they do and invoke certain emotions in us etc.

I'm armed with a guitar and a synthesizer, and will be referring to these instruments quite frequently in the posts. Oh, and I have a degree in Engineering too!

One of the reasons for starting this blog is to keep track of the myriad things about music theory that I scrounge off the Net. I then try these techniques on the guitar and the synthesizer, and, hopefully, make sense out of them.

I'm nowhere near perfect (or even very good) as a musician, nor am I an "expert" in music theory, so there's no need for any "that sounds like a fart through a whistle", or "you suck" or any of that stuff!