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Music Theory

I'm always irked by how music theory is taught incorrectly (or not taught at all). So here is my attempt to make it as simple as possible to the common person.

The widely-used notation involves the keys: C, D, E, F, G, A and B. All these keys except E and B have a sharp key (C#, D#, F#, G#, A#). These 12 keys make up an octave. It has been agreed that the A key of the 4th octave (referred to as A4) should have a frequency of 440 Hz (440 vibrations per second). This is true of all musical instruments (that I know of). The only factor that separates A4 of piano from A4 of any other instrument is the secondary character of these vibrations (also called timbre). A4 is also the 48th key (since the first key is A1).

Frequency is an objective quality whereas pitch is subjective. The relationship between the two is non-linear. A successive doubling of frequency is perceived as a linear increase in pitch. Hence, A4 is 440 Hz, A5 is 880 Hz, A6 is 1760 Hz, and so on. In order to maintain this linear increase in pitch between the keys from A4 to A5, the frequency increases by a factor of \(2^{\frac{1}{12}}\) for each key. Hence, the frequency of the \(n^{th}\) key is given by: \[ f(n) = 440 \times 2^{\frac{n - 48}{12}} \ Hz \] That's it. Everything beyond this is just notation and semantics. Certain notes sound good when played together because the ratio between their frequencies is something like 3:2 or 4:3. On the other hand, some notes sound terrible when played together because the ratio between their frequencies is an irrational number like \(\sqrt{2}:1\). Now lets see how much of music theory I can speedrun:

1. Those 12 keys in an octave are called a chromatic scale. The interval between two consecutive notes is called a semitone. The mode of assigning frequencies to them is called 12-equal temperament. There are other variations too
2. The most popular progression of notes, called the major scale, is composed of 7 notes obtained by starting from a specific key and then followed by 2 full steps (2 notes), 1 half step (1 note), 3 full steps and 1 half step. For example, in C major scale, the first key we start from is C_n and we end up at C_n+1. There are many other scales, and they evoke different kinds of emotions
3. A pair of notes that feel "good" being played together (owing to an integer ratio between their frequencies) are said to be consonant. Otherwise, they're dissonant
4. A melody refers to a bunch of notes that sound "good" when played in succession. A harmony refers to a succession of notes and chords
5. A chord is a collection of three or more notes that are either played together or broken (like in arpeggios). There are strict rules regarding their construction. For example, the C major chord is composed of the root (C), the third note in its scale (E) and the fifth note in its scale (G). Once again, the way how a chord "feels" comes down to the ratio of the frequencies of its notes
6. The rhythm lays down how successions are ordered throughout the composition. The dynamics dictates the variation in relative amplitude of successive notes. Articulation is about how much each note's sound attacks into the next note. Cadence is the extent to which notes are played with the right rhythm, dynamics and articulation.
7. Sheet music/musical notation is a handy way to convey all of this information in a succinct manner, independent of instrument.