Why we like Diatonic Scales.

I first read about this while reading Rameau's A Treatise on Harmony which is an 18th century text; this isn't new knowledge by any means. In fact, the harmonic series, which I'll get to shortly, was first discovered by Pythagoras using a "monochord"--a one-stringed instrument used for examining properties of waves--in ancient Greece. Although this explanation doesn't sit well with me (I'll get to that later), it is considered by most music theorists to be the correct explanation.

Let me first take a moment to demonstrate how to "read" a keyboard and how to construct a major scale. Below is an illustration of an octave, starting and ending on C, on a keyboard for you to refer to.




Each key represents one half step out of the 12 half steps which make an octave. As you can see, the white keys are given the names C,D,E,F,G,A,B, & C, respectively. These names practically never change. The black keys, however, can be assigned two names... one with a sharp, and one with a flat. Remember that "flat" means one half-step lower, and "sharp" means one half-step higher. Because each key is a half step higher than the one previous, and conversely a half step lower than the one following,  and because sharp denotes being a half-step higher and flat denotes being a half-step lower, it would make sense that the black key between C and D can be called either "C#" or "Db", the one between F and G is "F#" or "Gb". A slight problem arises between E & F and B & C because there is no black key between these notes. In this case, "E#" is actually the same as F, and "Fb" is the same as E; "Cb" is the same as B, and "B#" is the same as C.

By looking at this keyboard, it is easy to see the construction of the major scale as a series of whole- and half-steps. The C major scale is C, D, E, F, G, A, B, C as has been stated before in other posts. You can see by looking at the keys on the keyboard, that to get to D from C you have to move up 2 keys, or 2 half-steps i.e. one whole-step. From D to E is another whole-step; E to F, a half-step; F to G a whole-step; G to A a whole-step; A to B a whole-step; and B to C a half-step. In short, the formula is: W-W-h-W-W-W-h, where "W" denotes a whole-step and "h" a half-step.

Notes like "E#" and "Cb" are used instead of writing "F" or "B", respectively, for this reason: When writing a diatonic scale, each letter of the musical alphabet (A through G) must be used once. So if you start on the note F#, use each letter once, and also follow the W-W-h-W-W-W-h pattern, you get this: F#-G#-A#-B-C#-D#-E#-F#. 

So why do we like this arrangement of notes? The reason lies in what is called the harmonic- or overtone series. The harmonic series is a series of intervals given off by any natural, vibrating object; a pure sine wave does not give off a harmonic series. When a natural tone (called the fundamental tone) is played, it gives off overtones in multiples of its frequency. For instance, if a fundamental tone of 500Hz is played, then overtones of 1000, 1500, 2000, 2500Hz, etc. will also resound.  These overtones diminish in loudness as they get farther and farther away from the fundamental. If  you would notate these out musically, you would get the following progression of notes (using C as the fundamental):

C, C, G, C, E, G, Bb, C, ...

If you spell this out as intervals between the two consecutive notes (not between the fundamental C and the note) it would look like this:

perfect Octave, perfect Fifth, perfect Fourth, major Third, minor Third, major Second. 

Remember that a major 6th is an inversion of a minor third, and a minor 7th is an inversion of a major second. When you include these two intervals, and rearrange the other intervals to suit, you get this:

Unison, major 2nd, major 3rd, perfect 4th, perfect 5th, major 6th, minor 7th, perfect octave. With the exception of the 7th (which would be a Bb), this is a diatonic major scale. C-D-E-F-G-A-B-C. 

If you take the same pattern W-W-h-W-W-W-h, and shift the pattern to start on the sixth note (in our case A) and then circulate through, you get: W-h-W-W-h-W-W. This is the A minor scale. It is also known as "A aeolian" or the "sixth mode of C". A 'mode' is when you take the notes of a major scale, and start on a different scale tone. In the case of the A minor, you use the same notes as a C major scale, but start on A instead of C. 

Now we have to go back and reexamine the overtone series. When you take the intervals and arrange them as was previously done, but this time include the minor 7th, the following scale emerges: C-D-E-F-G-A-Bb-C.  This is actually the mode "C mixolydian" or the "fifth mode of F", because it uses the same notes as an F major scale (F-G-A-Bb-C-D-E-F), but starts on a C, which is the fifth note of the F major scale. 

So in review of the information, here are some things that I find interesting: 

1) If naturally the mixolydian mode is formed instead of the regular diatonic scale, why do we not prefer this mode over any other mode or scale? (the mixolydian mode is the most commonly used mode, though; besides the aeolian, which is the same as the diatonic minor scale.)

2) Why does the major scale on a piano that uses all the white keys (i.e. the "simplest" one) start on C and not A-- after all, A is the beginning of the alphabet and would make the most logical sense? What this suggests to me is that either the piano was intended to 'naturally' play in the key of A minor for whatever reason, or that we as humans naturally prefer the minor scale over the major scale. In other words, was the piano (and musical scales in general) named thusly because it was "intended" to play in A minor rather than C major because we humans liked the minor scale the best, or was it just arbitrary?