Tones & Harmonics
Tones are made of regular vibrations measured in cycles per second or Hertz. When you play two tones together they will interact. They are in harmony if, on a regular basis, some of their vibrations beat together at the same time. Perhaps we need a picture example:
In the second trace we could add a red-coloured harmonic exactly 1.5 times (or 3/2 as a fraction) higher in tone frequency. You can see that it takes 2 complete cycles of the blue tone to complete until the two tones peak at the same time. During this period the red harmonic has peaked 3 times.
The resultant tone, generated by playing the tonic and the 3/2 harmonic, is shown in green. It too has a regularity or repeatability to its pattern. The wave shape will repeat every two cycles of the tonic tone. The shape of the resultant green tone isn't too important because with real life the tonic and harmonic are unlikely to start off exactly in phase with each other and will therefore result in a more asymmetric shape wave. However, what is important is that the repeatability of the shape will remain at a frequency half that of the tonic. What I have shown you here is the foundation of consonance, perfect fifths and harmony.
Just Intonation versus Equal Temperament
There are two mathematical ways (and practical ways if you know the process) to tune the notes in a scale.
The example in the diagram above shows a harmonic that was not only a perfect harmony with the tonic but also a nice simple ratio made up of small whole numbers. In particular, the denominator of 2 meant that it only took two cycles of the tonic for the waves to peak at the same time. So it was also a very close harmony.
Circle of Fifths
Now we finally know the significance of the fifth, here's a tool that helps you:
- Moving clockwise increases the tone by a fifth and moving counter-clockwise increases the tone by a forth.
- The number of sharps or flats increases as you move away from C / Am
- The addition of the extra sharp or flat in the outer circle echoes the circle of fifths pattern. So you can predict easily which sharp/flat will be added next
- You can translate chords shapes between G-tuning (Baritone/Guitar) and C-tuning (standard uke). E.g. Making a G-Major shape on a Baritone uke is actually a D. You just move 1 place around the circle.
- The SUS2 chord has the same notes as the SUS4 chord of its clockwise neighbour. E.g. FSus2 = CSus4
- The bottom three segments include enharmonic tones
- All five of the adjacent chords are within the key of that root note. E.g. In C Major, you have C, Dm, Am, Em, F &G (Diminished chords not represented on this model)
Circle of Fifths in Music
This model can also be useful in analysing the movement and key changes in a piece of music. Sometimes seemingly odd chords may appear like a major chord when you're expecting a minor chord. Sometimes if you're looking at an unrealiable source, it might well be a mistake but if it is correct, you may find that it is a fifth of a fifth. E.g. You might be playing in C Major and find you have an out of key D Major rather than a D Minor. As described above, Dm is the second weakest harmony chord in the key of C. So a key change to G would be a fairly smooth transition.
Very often a song will resolve back to the tonic chord of that key. The collection of chords that performs this is sometimes called a 'turnaround' or a 'cadence'. A common turnaround (particularly in Jazz) is ii-V-I. E.g. Dm-G-C (or sometimes ii-V7-I giving Dm-G7-C in C Major). The roots of these chords are descending by fifths back to the tonic.
A very common one in Blues is the V-IV-I and again you can see the circle of fifths in action here. In fact in a typical 12-bar Blues pattern you may only see these chords (or very similar).
Another example is sometimes called the "Amen Cadence" which is the IV-I. An example is from the chords F to C in the key of C Major (Try alternative chord versions tab 2,0,1,x to 0,0,0,x and also 2,0,1,3 to 0,0,0,3. I find you get a similar feel when you finish with Csus4 to C because of the F in the Csus4 chord and also with Fm to C.
Note: in Tab notation, 2,0,1,x means fingers on the 2nd fret of the G string, open C string, 1st fret of the E string and don't play the A string. In order not to accidentally play the A string you could use a spare fret finger to touch the A string to dampen it.