A couple of weeks ago, I posted my survey results about key signatures, sharps and flats. The data drew me to the conclusion that there’s something special about both D major and minor that draws us to them. Armed with this theory, I sought to investigate.
For a long time, I’ve suspected that the reason we’re drawn to certain keys over others has to do with the differences between just intonation and equal temperament. For those who don’t know, just intonation is a system of musical tuning where the frequencies of the notes of the scale are related to one another by ratios. The intervals are all constructed using three basic intervals:
The semitone – 16:15
Minor tone – 10:9
Major tone – 9:8
By combining these, we can come up with the ratios of larger intervals: for example, the ratio of a perfect octave is 2:1, perfect fifth 3:2, perfect fourth 4:3, etc.
There are numerous problems with just intonation. The scale can be tuned in many ways, which make certain chords very consonant, while other chords become less so. If one player tunes in one way and another in a different, there will be beating between them.
Equal temperament was created as a more ‘universal’ form of Western tuning. In order to create a stable set of 12 semitones, each pair is tuned using an identical frequency ratio, usually tuned relative to A= 440Hz. This, however, means that the the more pure chords found in just intonation become lost. It also means that scales sound different – there are slightly different frequencies between key intervals, such as the perfect fifth – which some of us can pick up on. My theory is that these differences in frequencies – and differences in purity of chords – are what draw us to develop ‘favourite’ keys, beyond how easy they are to play on our particular instrument, or our familiarity with them.
To test this theory, I compared the just intonation and equal temperament ratios of six major keys (D, E, C, A and Bb) and six minor keys (Dm, Em, Gm, Am and Fm). These were the keys that came out as the most ‘popular’ as a result of the survey.
Just a small snippet of the calculations!
Interestingly, out of the keys analysed, D/Dm and C had the least consonant octaves – although D was more consonant than that of C. Unlike A, where the A below middle C is at 220Hz and the A above is precisely double at 440Hz, the same couldn’t be said of these two keys – including the most popular! Puzzling. D did, however, have the most consonant perfect fifth (along with Bb, interestingly!). Strangely D had the least consonant perfect fourths, with the most consonant belonging to A. As A is the dominant of D, and its perfect fourth is a D, I wonder if this also plays a part.
However, the most interesting part came when I analysed minor and major thirds. I decided to analyse the tonic triads of each key, as these normally form the basis of any pieces we are playing in a specific key to some extent or other. A major triad is made up of a major third (D to F#) and a minor third (F# to A). When these were analysed, the most consonant major thirds and minor thirds for tonic triads were both for D major. For minor triads (minor and major third), D and A possessed the most consonant minor third, although not major third.
I think, therefore, that one of the reasons we all naturally enjoy D major and minor is because the make up of their tonic triads creates the most consonant chord out of all our of our favourite keys. The fifths are the most consonant, both intervals contained within the triad in D major, and the first minor third interval in the minor (arguably the more important of the two). I feel I should point out that the differences in these ratios are very small – but the human ear is finely tuned! It’s also worth noting that none of the intervals (asides from the octaves, mentioned above) fall precisely on the just intonation ratios.
I intend to further this by repeating the analysis on the other keys that have been missed out – the remaining 6 major and minor keys. The relation of D major/minor’s tuning to just intonation may be one factor in its popularity, but obviously its lack of sharps and flats, and its playability on various instruments definitely factored into it. Are there other keys that are actually closer to just intonation that we are less favourable to because of their ridiculous key signatures..?
To be continued..!
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