An octave is 12 semitones. However, half an octave is 7 semitones – and the other half is 5 semitones!
How is this so? Surely half of 12 equals 6?
Frequencies
Each note pitch produces a repeating sound wave. Lower notes produce longer waves which repeat more slowly, whereas higher notes produce shorter waves which repeat more quickly. The speed at which a sound wave repeats is it’s frequency, measured in Hz (Hertz). 1 Hz = 1 wave cycle per second.
Composite Wave
When two (or more) notes are played together, their sound waves combine to form a composite wave. This wave also has a frequency. Playing two notes produces three!!!
The frequency of this combination wave is also a note. For example, below is an interval of a major 3rd.
Consonant intervals (intervals that sound musical) have frequencies which are closely related. The composite wave’s frequency is an octave of one of the two notes that make up the interval.
Dissonant (musically unpleasant) intervals such as a semitone or a tritone (augmented 4th/diminished 5th) have frequencies that are not closely related. As a result it takes many cycles of each note before they meet up to produce one cycle of the composite. The composite wave has a low frequency unrelated to either note which, if below our ability to detect pressure waves as continuous sound, can be felt as a disturbance known as beats or beating.
For more on beats see The Secret To Tuning: How To Tune An Instrument To A Reference Note.
Octaves
When two notes are an octave apart, their sounds match so well together that we think of them more as being in different registers rather than as completely different notes. Notes which are whole octaves apart are considered to be different versions of the same note, to the extent that they share the same name.
When two notes are an octave apart, the upper note is 2x the frequency of the lower note. For example, if A = 440 Hz then the next A an octave higher is 880 Hz.
The composite wave is 440 Hz, the same as the lower note.
Half An Octave
Half an octave is half-way between the frequencies of the two notes. In the above example, half an octave is half-way between 440 Hz and 880 Hz, which is 660 Hz.
660 Hz is E, 7 semitones above A 440 Hz.
Two Halves
- A to E, the lower half of the octave, is 7 semitones
- E to A, the upper half of the octave, is 5 semitones
- A to E, the lower half, is a perfect 5th
- E to A, the upper half, is a perfect 4th
If you’re wondering why a 5th plus a 4th is an 8th, please visit B5. Inversions Of Intervals.
Let’s look at the composite wave’s frequency of each half.
The interval between A 440 Hz and E 660 Hz has a frequency ratio of 3:2. That is, it tales 3 cycles of E and 2 cycles of A to form the composite wave. The composite’s frequency is 220 Hz, the A an octave below the played note A 440. This reinforces the lower note of the interval, making it stronger.
The interval between E 660 Hz and A 880 Hz has a frequency ratio of 4:3. The composite’s frequency is also 220 Hz, which is 2 octaves below the played note A 880. This reinforces the upper note of the interval, making it stronger.
In other words, the upper half of an octave, a perfect 4th, behaves upside down compared to the lower half, a perfect 5th.
- In a perfect 5th, the lower note is stronger
- In a perfect 4th, the upper note is stronger
Perfect 5ths and perfect 4ths are literally inversions of each other!
Half An Octave In Scales And Melodies
In a scale, the 5th note, the note half an octave above the root note, is called the dominant. The dominant has a double function:
- The half-octave point is as far away from the root note as you can get
- It is also a strong supporter of the root note, as seen by the composite wave examples
The dominant provides a polar opposite point allowing melodies to venture away from the root note and to return from.
This is easily demonstrated in the most simple melody of all, the scale. By splitting it in two, we can see that the first half of the scale leads away from the root note and towards the dominant and the second half of the scale leads from the dominant up to (the octave of) the root note.
In the example below I’ll use the major scale but it works equally well for the melodic minor.
Perfect 5ths And Perfect 4ths In Chords
The presence of a a perfect 5th or perfect 4th in a chord helps us to identify the root note. The root note will be the lower note of a perfect 5th/the upper note of a perfect 4th.
If a chord contains more than one perfect 5th (or perfect 4th), the chord has more than one possible root note and its interpretation is determined by the musical context.
For example, the notes A C E G could be seen as either
- Am7
an A minor chord; A C E, plus a minor 7th; G, or - C6
a C major chord; C E G, plus a major 6th; A
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