Music Math (pt 2, now with less math)

Hopefully now we have a basic understanding of how harmonics work related to the mechanical properties of sound. Last time the major chord was dissected, this time we’ll find out what happens when the lowest common factor is not 2, but a different prime number.

The perfect 5^th is the note exactly in the middle of an octave related to the root of the chord, but as I pointed out last time the 3^rd interval isn’t exactly in the middle of the 5^th and the root as you might expect.

We will start in the key of A minor this time, since its key signature is the same as that of C major. Even though it is a minor key, the perfect 5^th of A0 (27.5), E1 (41.25), is still equidistant from and A1 (55). But now, when we add the 3^rd , C1 (32.7), it doesn’t come out anywhere near as close to being right in between the root and the 5^th as before It is 2.2Hz off, as opposed to only .3Hz before.

Remember that the notes are not evenly spaced, aside from the perfect 5^th , and since decimals are involved there are infinite number of factorizations. The major chords come out very nicely, because by design they are supposed to sound pleasing, minor chords have a little bit of dissonance to them so the frequency gaps are shifted slightly.

So let’s do some rounding for both our sakes.

These all have a lowest common factor of about 3, which is the next prime number after 2. The mathematicians following along probably foresaw this outcome, based on the fact that major chords are based on 2’s. Just like last time you can add as many notes above or below with a lowest common factor of about 3 and it will have the same characteristic sound of a minor chord.

There are two more types of 3 note chords (triads), the augmented chord and the diminished chord.

http://upload.wikimedia.org/wikipedia/commons/a/a9/Pitch_constellation_triads.svg

These two chords do not necessarily share a common factor, and sound very dissonant to the ear because they have very few recurring nodes in common.

The augmented chord is simply raising the top note (5^th) by one half step, which removes the perfect 5^th from the chord and gives it quite a different characteristic sound from a major chord. Back in the key of C major, to make an augmented chord starting on C0, we need C0 (16.35), E0 (20.6) and G#0 (25.96). The difference is only 1.46 Hz, but now they no longer share the same multiples as before. Because of this, these chords are not very common because it is hard to find a good place for them in melodies.

The diminished chord is opposite, the 5^th is lowered by a half step, as well as the 3^rd. This gives us Gflat0 (23.12) instead of G#0 and Eflat0 instead of E.. Again the difference is only 1.38Hz, but the interval between the 3^rd and the 5^th becomes much closer together, which means very few recurring common nodes. The augmented chord has a distinctly strange sound, but the diminished chord sometimes can be mistaken for a minor chord. Want to know what a diminished chord sounds like? listen to any 1-note breakdown by any metal band post-2007. In fact the typical format for a breakdown is to play a minor chord on the 7th of the key (which is naturally half-diminished) in whatever heavy, repetitive rhythm you come up with, then a fully diminished version of the same exact chord and rhythm. Observe:

This is where music theory gets a little less simple to understand, as chords become 4 notes tall. This is technically as many notes as you can have in a chord for it to be classified traditionally, otherwise you get into chromatic or poly-tonal chords which are really only used in key-less music and as sound effects for scary moments in movies. These won’t be discussed because they are so rare. 4 note chords, or seventh chords, are the root, 3rd, 5th, and now the 7th also.

What makes seventh chords more complicated, is there can be major intervals between the root and the 3rd, and minor intervals between the  root and the 7^th, or two minor intervals or two major intervals. Then, mix in the option for fully or half diminished intervals and augmented intervals and the number of  seventh chords that are possible is 8 as opposed to 4 triad types.

To imagine how the frequencies of a seventh chord will interact with each other, think of how each interval will react separately. a perfect 5th (existing in all major and minor chords) will always have every other node in common with the root, a major third then also adding another doubling of frequency inside of that. The 3rd will react with the 7th the same way the root react with the 5th when it is a major seventh, but this is the key to understanding layered harmonics. The interaction between any of the two notes will effect the overall sound of the chord. if you change the interval between 3rd and 7th to be diminished or augmented,  the frequencies amount of nodes in common that the previous major seventh had completely changes, and the location of the nodes on the waves change too. Lowering one note in a seventh chord by half a step changes the chord to a dominant seventh which has a completely different feeling. The best example is raising the 5th of a major seventh chord will turn it into a augmented chord which doesn’t sound natural at all, even though 3 of the intervals remain the same.

Now that we understand how chords are just the interaction of different frequencies of sounds combined, you might have wished I never told you any of this because its boring and you will never use this information ever again. Recently I wrote another two-part post about the over-simplification of music in our industry and why the easiest music to make is what sells the best. Now think about everything I just told you,  all the different types of chords that can exist, of the infinite different combination of notes.

The methods mainstream music use for creating music are the ones that I learned in music theory 100, one semester 3 credit hour class. THAT is my point. they all get away with music theory 100 style because nobody ever takes music theory 100. Its like the old joke about the piano teacher who has to stay just one lesson ahead of the student, but the student never shows up to the first lesson so the teacher quits while they’re ahead. Even though the theories and ideas exist for much more complex and interesting ideas, nobody ever even thinks about it because they never learned anything about it. My goal is that I can educate the small number of you who have stuck with this blog enough so that you understand the laziness and exploitation of the mainstream music industry.