The Circle

 Circle of 5ths / 4ths showing all the notes of each key as radial spokes:

Notes of each spoke are arranged in descending 3rds starting from the major tonic inward toward the center.

The circle organizes all the diatonic keys based on perfect 5th (clockwise) and perfect 4th (counter-clockwise) intervals. The tonics of the major keys (Ionian Mode) form the outer ring, accompanied by the relative minor tonics (Aeolian Mode) placed one level in toward the center.

We can extend the logic further by adding all the notes of each key in descending 3rds, just as the minor tonic is a 3rd below the major. So, starting from C at the top of the circle, we read the notes C, A, F , D, B, G, E ... essentially a descending arpeggio.

The upper and lower case letters in the diagram represent the major and minor qualities of each 3rd interval for which that note is the (root) ...  and can also be treated as the finals of their respective modes: Ionian, Lydian and Mixolydian have major 3rds, so they are upper case. Dorian, Phrygian, Aeolian and Locrian have minor 3rds, so they are lower case.

From the outside in, the modes are: Ionian, Aeolian, Lydian, Dorian, Locrian, Mixolydian and Phrygian. 

Numerically, they are I, vi, IV, ii, viiº, V, and iii.

The "Circle of Fifths" is something to ponder. A schematic for the diatonic system which reveals many interval relationships in a way that comports well with the nature of music. 

Paradoxically, if we approach musical concepts as circular, as a continuum with not fixed beginning or end, many things become clearer. Starting points and ending points in music are essentially contextual.

The Speed of Pitch

What we call "octaves' are the phenomenon of a 2:1 frequency ratio. Our perception of octaves as "the same note" seems to be a deeply embedded sense. No one questions the practice of giving two different notes the same name when they are one octave, two octaves, three octaves or more apart from each other.

Why is this?

What is it about this 2:1 ratio that we sense as "the same"?

Consider the effect of clapping at a steady speed, then doubling the speed of clapping. Or have another person clap along with you but at double the speed and stay in sync.

You'll notice that the claps will perfectly align every other beat, so that the lower speed clap is effectively embedded in the faster one. The ratio of the frequency of your clapping is the same as an octave. This provides a clue as to the reason octaves are perceived as the same.

In the ear, sound is processed by setting in motion some 15,000 tiny specialized hair cells, each of which has between 50 and 300 filaments projecting from it, called stereocilia. These hairs are contained in a fluid filled spiral shaped structure called the cochlea. The hairs and the stereocilia in each ear resonate with specific frequencies and transmit that information to the auditory cortex of the brain which has specialized neurons for each frequency and range of frequencies. So your brain is always sensing the frequencies of every sound and is able to recognize the ratios between those freq
uencies.

Exactly how the brain can calculate the precise frequency ratios between all the pitches we hear is quite complex and still somewhat mysterious, but it is probably connected to the brain's ability to sense time as sequential events based on it's own brainwave frequencies and the embedding of resonations in its neural network. Regardless of the precise mechanism, it's clear that we are very good at perceiving the ratios between notes regardless of their absolute pitch.

In other words; if you hear a melody in one key, and then the same melody in another key, you recognize it as "the same" because you recognize the ratios between the frequencies of every note in the melody, rather than relying only on the absolute pitches of the individual notes. The melody of the song Happy Birthday in the key of C would be:

and In the key of F:

Now, clearly the notes are not the same in both keys, but that doesn't prevent us from hearing both of them as the same song, even if one is higher in pitch and one is lower overall.

Our awareness of pitch as proportional relates to our sense of time and, therefore, to our sense of motion ... which is in turn connected to our experience with gravity and other physical forces. All this is at the heart of why we naturally regard higher (faster in time) frequencies as physically "up" and lower (slower in time) frequencies as physically "down". 

Because differences in note frequencies are not really tangible on a conscious level (we can't discern the individual oscillations of hundredths of a second) music affects us subconsciously, subliminally. We feel it without knowing exactly why.

We can consciously identify a 4/4 beat ... we can clap and we can count the beats. So our relationship to rhythm is not so mysterious. We can be satisfied that dancing to a beat is a relatively simple matter. The mysterious nature of our perception of melody and harmony is does not mean that melody and harmony are any less real than the beating of a drum or tapping your foot. Sustained tones are just faster oscillations. The same essential physical phenomenon as individual drum beats. 

The video below explores this subject from a practical and philosophical perspective: