All our senses operate on a logarithmic principles, meaning that changes are not linear but follow multiplication or division.

For example, going up an octave means multiplying a frequency by 2, not adding something to it.

These all are octaves:

• 1 Hz – 2 Hz
• 10 Hz – 20 Hz
• 100 Hz – 200 Hz
• 1000 Hz – 2000 Hz
• 10,000 Hz – 20,000 Hz

Hearing is a distance sense as opposed to a contact sense (we can even hear around corners), meaning that in evolutionary terms it was vital for sensing danger. Vision is secondary to hearing in the sense that it confirms what we perceive by hearing. Hearing gives us advance warning, vision confirms the observation.

When a sound wave arrives at our ear, it first passes through the tapering ear canal to the ear drum (tympanum), where the small bones in the middle ear convey it to the oval window of the cochlea. Inside the cochlea, the vibrations of the sound wave are picked up by hair cells and converted to nerve impulses fed into the brain. The process is a complicated one, and we cannot go into too much detail here. What we do need to know is that our hearing begins to develop before birth and grows substantially very early in childhood. Hearing also involves learned and cultural components.

Music is the most abstract of all art forms, and accordingly every person interprets music in their own way, and even in different ways on different occasions. One person’s comfort music is another person’s boredom.

Music perception and speech perception occupy slightly different regions of the brain. Music is more comprehensive and serves as a foundation for other experiences. This is why music is widely used for therapy purposes. People who have suffered a stroke and can no longer form words to speak may nevertheless be able to sing songs, complete with lyrics.

Let’s say we raise the pitch of a tone repeatedly by a fifth, which means multiplying the frequency by a factor of 3/2 each time. If we do this 12 times and then come back down 7 octaves (dividing the frequency by 2 each time), we arrive at a pitch slightly higher than the one we started from. This phenomenon was first noted and puzzled over by the Greek philosopher Pythagoras, and it is after him that this discrepancy is known as the Pythagorean comma or diatonic comma. It is about 23.46 cents (where a cent is 1/100 of an equally tempered semitone), or nearly one fourth of a semitone. It follows from this that if we use natural tuning based on simple ratios, the intervals will vary in size except for the ‘perfect’ ones.

In the Baroque era, harpsichords had to be retuned for each key, but organs could not practically be retuned (admittedly it would have been laborious to tinker with all the pipes between each piece). As instruments with fixed tuning became more common, it became necessary to develop tuning systems where the principal intervals would be as consonant as possible in as many keys as possible. The ultimate result of this development was equal temperament (which, interestingly, was also invented about 3,000 years ago in China but abandoned because it sounded bad).

In equal temperament, the octave is the only perfect interval (with a simple ratio of 1:2), and all semitones are equal, the ratio being 1 to the 12th root of 2. The unfortunate result of this is that the purity of the perfect fifth and major third is lost. Fortunately, natural intonation can still be achieved by choirs, string ensembles and wind ensembles.

The word ‘chromatic’ goes back to the Greek word khroma, meaning colour. Pitches not belonging to a particular key were originally referred to as ‘chromatic’ because they coloured the music. Much of the original impact of chromatic music has been lost with equal temperament. For example, there is a keyboard work from the Renaissance era that depicts two fleets in a naval battle. The point in this work is that the harpsichord would be tuned to the basic key in just intonation, meaning that everything sounded natural in that key. The theme for the battle was the same throughout, but for the winning side the theme was played in the basic key and for the losing side the theme was played in a key a tritone apart, which means that all chords and intervals would be distorted. If we play this work on an equally tempered piano, this clever idea is lost. It simply sounds as if the same music is being played in two different keys.

One of the most popular tuning systems in the Renaissance era was meantone temperament. This involved selecting one or more intervals (usually thirds and fifths) and tweaking their frequency ratios to make all chords sound reasonably in tune.

Here is a computer-generated performance of the Prelude in C sharp major from Bach’s Das Wohltemperierte Klavier in a meantone temperament.
You will notice that as the music progresses away from the main key, the intervals become less pure.

Here is a melody where we find first F# as a leading note from below to G and then Gb as a leading note from above to F.

Play the opening measures a few times, emphasising the F# as a leading note. Then play the remaining measures with a crescendo. Pay particular attention to F# and Gb. Do you hear any difference between them? Ask your friends to listen too!

Sometimes F# and Gb can sound different to us (in pitch or timbre), because our brain perceives them relative to the harmonic environment and interprets them differently (leading tone from below or above). Our brain creates an expectation of where the pitch should resolve.

Facts:

• Our senses work on a logarithmic basis, meaning that differences can be expressed with ratios, not linear progression
• Natural tuning or just intonation is based on ratios derived from the overtone series, containing several intervals that have the same name but are subtly different (e.g. several minor thirds, major seconds, etc.)
• Equal temperament divides an octave into 12 equal semitones. In this system, none of the intervals except the octave coincides with the simple ratios of just intonation