Intervals represented on the keyboard referenced from the note C

An important observation is the numbering is founded on the western major scale having the melodic interval series of root note, tone, tone, semi-tone, tone, tone, tone, semi-tone. The piano keyboard is designed around the western major scale, playing from C to high C, the numbers 1 2 3 4 5 6 7, the distance of each interval from the root 1 is measured in semi-tones, example a -3rd interval is three semi-tones, a 6th interval is nine semi-tones.  

The numbers are reference from the root 1 of the scale , in the image below, this is the key of C 1 2 3 4 5 6 7. When the key changes, the numbers will appear at different locations, but the Western major scale will always have the melodic interval series of root note, tone, tone, semi-tone, tone, tone, tone, semi-tone and the harmonic interval series of 1 2 3 4 5 6 7. The intervals which appear out of key -2 -3 -5 +5 -7 are named from the intervals within the key signature. The root of any major scale will always be the number 1. When you learn all your the major scales, you're able to identify all the intervals.

So these number are not glued to the piano keys, they are relative and appear in different locations in different key signatures.  

Interval distances

Remember the fifth rule

Once you're capable of finding fifths quickly on the piano, all you need to do is remember how to find the other intervals from the table below, and you'll have the foundations to create any chord on the piano. Chords are named after the root note, so if you know all the notes of the piano instantly, then you know all your chords with these simple rules.  If you don't know the name of the chord you're creating, that isn't too important for beginners, many chords are descriptive of the intervals in their character, so just be aware of the intervals you include in the chord construction.

I have all the chord names in my book, "The Secrets To Playing Music by Ear- The Akousunesis Method".

Guitar neck notes

You can consider the guitar neck as six individual pianos playing different ranges. If you know all the notes on the piano, by overlaying the six pianos onto each string of the guitar, then notes are easy to visualise. All the white notes are in the key of C, each fret is a semi-tone apart. Use the notes B-C and E-F as you reference notes. You need to know the note names to find the root of a chord, then use interval knowledge to build chords, rather than just remembering on chord shapes like most the guitarists. Learn how to build chords using interval knowledge, eventually you'll remember chord shapes also, but you shouldn't be your starting point because you'll make it a habit then refuse to expand your knowledge like most of the guitarists I have known, stuck in their musical ruts. 

If you're able to visualise all the white notes, the black notes are easier to find as the white notes become your reference locations. If you manage to remember the BC and EF locations, then the single white note between C and E is D, and the two empty white notes in the middle of F and B are G and A. 

Guitar neck intervals

Once you have located the root note of a chord you wish to build, hand shapes should be memorised in terms of the intervals played. One hand shape can make many chords. The following three diagrams show how the interval numbers are all referenced from the root note 1, the bass string. When you slide the root left or right to a new note location on the same string, all the intervals also slide. So if you know one hand shape defining character, by sliding the shape along the strings you know many chords. By simply know your intervals, you will know many chords.

Again, when you slide the root left or right to a new note location on the same string, all the intervals also slide. So if you know one hand shape defining character, by sliding the shape along the strings you know many chords. 

Observe how when we move from pattern 1 to pattern 3, the last two high note strings, B and E, the interval numbers change, this is because the guitar is tuned with fourth intervals between the string except the higher strings G -> B which is a major third interval. 

Overtones

When vibrating a string, a string produces overtones as shown below. The diagram below shows a three octave range starting on any note X. The black dots show the zero crossing points, the red dots the peaks and the blue dots the troughs.  Notice the amount of points which line up with the fifth and the octaves giving it a smooth sound to the ear. The blue arrows show the melodic interval distance as you move up the diagrams. Observe how the frequency is double at the octave intervals, fundamental X, 2X and 3X.

All twelve intervals represented as sine waves within an octave range

The following diagram below shows the intervals within a single octave range, from the 1st to 2nd octave as shown in the diagram above. The 1st octave is in blue and is our reference which doesn't move. The black sine wave starts with double the frequency shown on the octave diagram and is stretched across each diagram until being the same frequency as the reference frequency show in diagram -2.

Overtone amplitude visually shown

The following diagram shows separate waveform overtones and respective amplitude levels. Any tone (timbre) may be produced by adding all the waves together into one waveform and adjusting the amplitude levels of each overtone. Each overtone is called a harmonic, the main harmonics are the doubling in frequencies and the partial harmonics are the other intervals generated, the fifth, third and minor seventh.