My last post was all about D major and D minor – and why, out of the most popular keys, it appeared to be the most popular of all (see my survey results post for the results of my key signature survey).
D had EQ frequency ratios closest to those of JI for the perfect fifth, intervals in the major triad, and first interval (minor third) in the minor triad out of the favourite keys, when compared to the just intonation frequencies. This, I theorised, was one of the reasons that people naturally like D major and minor, along with the more usual reasons such as playability, and a non-outrageous number of accidentals.
However, the last post concentrated just on the six major and minor keys that came out as the favourite. What about those that I had neglected – couldn’t they also have frequencies very close (or equal!) to those found in equal temperament, yet are neglected due to personal preference?
Italics symbolises keys with the same ‘difference’ (e.g. Keys E, F, G, G#, A, Bb and B all share the same octave frequency difference (in this case, none) ).
Closer to ET frequency ratio < > Further from ET frequency ratio
| Octave | E | F | G | G# | A | Bb | B | Db | D | C# | C | F# | |
| Perf 5th | F# | G# | D | Bb | A | C# | C | F | E | Eb | G | B | |
| Perf 4th | C# | Eb | A | F | C | E | G# | G | B | Bb | D | F# | |
| Major triad: | |||||||||||||
| Maj3rd (1-3) | D | E | G | C | A | Bb | F | Eb | C# | B | G# | F# | |
| Min3rd (3-5) | D | Bb | A | F# | E | C | F | G# | G | C# | Eb | B | |
| Min triad: | |||||||||||||
| Min3rd (1-3) | F# | D | A | C# | G# | E | Bb | C | B | F | G | Eb | |
| Maj3rd (3-5) | B | C# | E | A | G | D | C | F# | F | Bb | G# | Eb |
I haven’t yet had much time to analyse this – I’m sure you can look at it and draw your own conclusions. However, some initial observations:
– The tonic triad of D major is the most true to just intonation
– Other major tonic triads that are relatively true (i.e. some of the ‘most’ true!) include E, A and C.
– The most true of the full minor triads are C# and A minors, followed by D.
– The least true minor triads are Eb, F and Bb.
However, I believe that it may be the quality of the initial minor third interval in a minor triad is particularly important, rather than the quality of both the constituent intervals.
Of course, all these figures only apply to anything tuned precisely to equal temperament. Pianos are the obvious instrument – however, they’re not always tuned perfectly either! I suppose, therefore, that these figures are only completely true for computer-produced sound, where it is mathematically calculating the frequency at which something should occur (MIDI, I presume, works like this?). While other instruments work generally in equal temperament, it is easy to alter the frequency of a pitch subtly in order to make it more appealing, which may sway towards just intonation.
This all interests me from a research point of view, or else I wouldn’t be investing time in wondering and calculating everything. However, I do wonder if there is any way this can be used in a compositional way? If a certain key’s initial interval of a minor third is ‘less’ true, does it make the key sound ‘less’ minor? It seems that there’s yet more research to be done… as always!
Later this week I’ll be posting about Summer Rain, my work in progress for solo marimba, and about how I’m ‘structuring’ this non-structured work!
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