Ukulele Harmony Theory

It is easy to be confused by so much terminology in music. The vocabulary can often make it difficult to gain true understanding of simple music theory. In this section I wanted to provide an overview of stuff related to harmonics, dominance, consonance and dissonance. This will help create an understanding of the perfect fifth and the circle of fifths. More importantly, it will explain why certain notes and chords tend to work better than others. You'll see that I take scientific approach to this but hopefully explained simply.

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:

Let's say the top blue coloured tone is the "tonic" or root note in a scale or chord.

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.

Just Intonation
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.

Just Intonation tuning works just like this. Notes in the scale are formed by increasing the frequency of the tonic by a nice simple fraction. Here you can see the ratios of each note in a Major scale. The 2nd "C" is double the frequency and there's our friend the 3/2 fraction under the fifth note in the C Major scale. 3/2 is the simplest fraction by which to increase the frequency of the tonic by (without of course doubling it to raise it by an octave). Therefore, the fifth is the most consonant and harmonious note in a scale and why it is called the dominant. The B is the seventh note in the scale and as you can see, it has the ugliest fraction. It is still in harmony but nowhere near as close as the fifth.

With good ears and some experience, we could therefore tune a piano to get G from tuning it against a C.

The next note we could find is the perfect fifth of G. In the key of G Major, this happens to be a D (rather than D# or Db - See my chord theory page to see why). In my table above, G = 3/2. If we multiply that by another 3/2 (IE a perfect fifth) we end up with a fraction 9/4. This is the relationship of the D up in next octave with the C in this octave. So to get the D in the same octave shown in the table, we simply halve 9/4(IE multiply by 1/2) to get a fraction of 9/8 and sure enough, there it is in the table.

In the key of D the perfect fifth of the D tonic is A (rather than A# or Ab). If raise D by a perfect fifth we have 9/8 X 3/2 = 27/16. This is not the same as 5/3.
  • 27/16 = 1.6875
  • 5/3 = 1.6667
These two values are quite close - Maybe to the ear, they are close enough. What has happened is that the choice of the fraction for A in the Just Intonation scale of C Major has been simplified from 27/16 to 5/3 in order for it to have a simpler fraction (with a lower common denominator of 3 instead of 16). This gives the note A a better harmonic relationship to C at the sacrifice of its relationship to D. This makes Just Intonation a little inflexible across different musical keys but very good within the key it is tuned to.

What we have demonstrated now is that we can go round in a circle of fifths to find the other notes (including sharps and flats). If you carry on raising by the perfect fifth, you will hit all 12 semitones in the diatonic scale. We have also demonstrated that mathematically the tuning wouldn't be exact but it would be fairly close.

Equal Temperament
In reality we tune most instruments (including fretted and keyboard instruments) in equal temperament. We use 12 semitones in a diatonic scale tuned to 12-TET (Twelve Tone Equal Temperament). This means the ratio between any two semitones is exactly the same. This allows us to play in different keys and still preserve exactly the same tone relationships when we change the musical key. With Just Intonation, we have seen that the frequency relationships do deviate slightly when we change key.

Equal temperament has the advantage of flexibility across keys but Just Intonation produces exact harmonies. Just Intonation can still be used in 'A Capella' groups and certain instruments but with ukuleles, guitars and pianos, 12-TET is used.

How Accurate is 12-TET?
I did some calculations in a spreadsheet using exact frequencies of notes in C-Major. To get the exact frequencies of the notes you need to start with a standard "A" of either 220Hz or 440Hz (or any multiple depending on your octave preference). You then use the following formula:
Next Semitone = Previous Semitone multiplied by (2^(-1/12))

I then compared the frequencies with those of the 'Just Intonation' scale and produced this table. All the notes of our usual (12-TET) tuning were well within 1% of those produced by the 'Just Intonation' scale I showed above.

What was also interesting was that the 7th note (in the case of C Major, its the B) would be more closely represented in the Just Intonation method by using the fraction 17/9. So I included this addition to the table for the hell of it. However, I can understand why the 15/8 was chosen - It is a simpler fraction (lower denominator value) and therefore a closer harmony than 17/9.

So the notes we play on a piano, guitar or ukulele can't and theoretically should never be an exact harmony because they use 12-TET tuning, they are, however, close enough to do a pretty good job. I have listened to the differences between both tuning types and I really struggle to hear the differences between them. If detectable, you should hear low frequency alias tones produced by the peaks slowly moving in and out of phase with each other. Either way, I would struggle to tune my ukulele accurately enough to replicate this difference - I doubt my frets have an accurate enough intonation too. So I'm just not used to hearing 100% perfect harmonies anyway.

Circle of Fifths
Now we finally know the significance of the fifth, here's a tool that helps you:
  • Find how many sharps or flats in a key
  • Work out what the sharpes or flats actually are
  • See which minor and major scales are enharmonic
  • See which chords go well together
The version I have shown here shows the Majors in blue, corresponding enharmonic minors in purple and the sharps and flats of their keys in outer circle.

Characteristics
  • 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.

Examples
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.