I’m talking about keys of course – not the notes themselves!
We all have favourite scales and keys. My current ones seem to have tonic notes between C and E flat. Why is that? Why, when we tend to use an equally tempered scale, does one scale sound differently to us than another? There are a number of potential reasons. Some scales may fall more naturally under your fingers than another, and these may become favoured – even in circumstances when you’re not playing your choice instrument. I think there’s more to it than that, however. I’ve spoken to many composers with favoured keys for different emotional scenarios (like myself), and refuse to believe that there’s no scientific reasoning behind this.
I posted about this originally almost two years ago in 2011, but I’ve recently had a few debates on Twitter about whether all keys are equal. For the sake of argument I’m talking about an equally tempered piano, where it has been tempered to ensure all keys sound nice and are playable.
I’ve discussed equal temperament and just intonation before. In just intonation, the key factor is that everything is based on ratios, whereas in equal temperament frequencies have been adjusted so that all the keys sound nice, and so that instruments can play in any key and modulate between them easily. The scales, therefore, aren’t the same as they were in just intonation – there are slight alterations, frequency wise, to ensure they all sound ok. These alterations are, in my opinion, what leads to the differences in key.
Universals – and the importance of the harmonic series
I’m very interested in universal principles in music – trends that seem to be constant across music across the world. An example is the recent BBC magazine article on the universal nature of lullabies. The pentatonic scale is another I’ve discussed on this blog before. The octave, being the first interval in the harmonic series, is one of these universal principles, and one present throughout music – as is the fifth (the third partial/second overtone). For example, fifths occur in Indian scales and gamelan music, as well as in the Western classical tradition we’re all familiar with. I’m going to focus in on this particular interval.
The perfect fifth
The ratio of a perfect fifth, as we call it, is also defined in science by the harmonic series as 3:2 – or, in my spread sheet, as 66.66667% (2 divided by 3). This is also how it’s calculated in just intonation keys. So, when a fifth is played in a just intonation scale, it is the pure perfect fifth, using its proper ratios. However, when we come to the equally tempered scale, the frequency of each note has been adjusted, meaning that this pure perfect fifth no longer exists – it has been adjusted slightly to create the equally tempered scales.
The interesting thing is how it has been tempered. I decided to investigate whether the ET had resulted in a change to the ratios – e.g. if the 3:2 ratio still existed in the fifths, making them pure. I took the ET frequencies for fifths, and calculated the ratios (column on the right, below).
I then compared them to the ratio of the pure perfect fifth in the JI scale (left). As you can see, none are pure perfect fifths. Interestingly, however, they’re not all identical. Many are very similar, but there are a couple of odd ones. The 0.00058 is F# major, and the 0.00092 is B.
I think it’s these differences that make each key sound different – the variations in the ratios of the frequencies. There is actually little variation in the fifths, but when you come to other intervals – such as major and minor thirds – the variations in ratios are more apparent. I go into a little more detail in my earlier blog entries – one about D major/minor (which I think is are popular keys due to their close relationship to the JI ratios), and some analysis of other keys .
I intend to carry out some further research on this topic. Do we all have the same instinctive feelings about different keys – do they evoke similar moods in different people? I know pieces in a certain key help evoke certain moods for me, but is that a tool that a composer can use to subconsciously influence the listener? Maybe I should be doing some analysis on works from music history to see if certain composers preferred certain keys, or if they used keys as one of the tools in their toolbox.
Whatever your level of engagement with musical keys is, maybe the thought that there is something unique about each one will impact how you use or interpret them in the future!