Some basic definitions of music

Semitone: the smallest interval in western music


For example B to C, F to F sharp


Tone: 2 semitones


For example C to D, E to F sharp.


An octave is 12 semitones.
If we know the frequency of a note, to find out the frequency for the next semitone we have to multiply by a number slightly larger than 12th root of 2. This is because the human ear brain requires a good sounding octave to be slightly larger than a ratio of 2 to 1 in frequency[euroelectron]. Pitch also depends on intensity [Stevens]


Frequency: number of vibrations per second


Pitch: how high a note sounds to our brain perception.


The first note of a scale is called tonic. The fifth dominant and the 4th subdominant.


Usually the first or even last note of the bass in a musical composition is the tonic.


For example Imagine - John Lennon, begins with bass C and ends with bass C. We can deduce that this song is written in the key of C major. Ie all white or natural notes, C D E F G A B C.


Since the first bass note of Imagine is C by the 135 default rule we can deduce that the first chord is C E G. This is the C major chord.


So the introduction begins with bass C and this chord.


Then the bass goes down to the subdominant which is F. So by default the chord that the right hand plays on the piano is F A C. This is called an F major chord.


But in fact for reasons of smooth change from C E G the right hand plays C F A, which is still an F major (and sounds F major) but with the notes arranged in a different order. See the principle of least change in music [euroelectron].


Then the bass comes back to tonic ie C. So the right hand again plays C E G. But if you listen carefully there is a higher D added. A D is nine natural or white notes above C so the note D with bass C define a interval of 9th which sounds very nice and unexpected. (Later addition, in fact there is a small riff, A, A#, B, D which leads to D somewhat preparing the listener's brain).


So the code for bass or figured bass in this case is not just C but C9.


The D only lasts for a while, the chord is resolved back to simple C, C E G, or C major.


Usually a composer prepares the brain of a listener for a 9th but in this case John Lennon or the arranger or the producer Phil Spector chose an semi unexpected short lasting D that catches the brain's attention in an unexpected way.


The last 3 bass notes of Imagine are F to G to C. Defining a typical subdominant to dominant to tonic progression in this case cadence. And the world will be as one.


References:


Ear brain octave versus engineering octave - euroelectron


Sound and Hearing - S. S. Stevens, Fred Worshofsky - Life science series


Information Theory (something unexpected or less probable to happen contains comtains more information)


Further reading:


Bach Bass Rules:
https://normanschmidt.net/scores/bachjs-general_bass_rules.pdf


The concise Oxford dictionary of music


Enchiridion - Epictetous - PDF