The equal temperament has prevailed in our occidental music culture – despite the obvious shortcoming that its intervals are not pure any longer. This was only possible because some substantial advantages offset the flaw of impurity:
1. One single tuning serves all keys: the fundamental tone is freely selectable.
In pure intonation, instruments basically have to be retuned for each key and each fundamental tone. With a harpsichord, this concerns a few strings, but with an organ, this is really a great undertaking in view of the vast number of registers and pipes. The further the keys are apart from each other – i.e. the more crosses and Bs they have – the worse the detuning is. The equal temperament changes all this. Although it is never quite pure, it is never so detuned as it would be in a distant key, either. With the equal temperament, instruments can be used to play in a different key immediately without retuning. Organs, string, wind and any other instruments can now play together in all keys.
2. Free modulation
This advantage is particularly obvious and effective. In the same piece of music, we are now able to change from one key into any other (modulate) – without any break and retuning. With pure intonation, this is only possible in a limited way and for closely related keys. The equal temperament, however, removes all the limits to modulation.
3. “Bonus keys”
The seven tones of the major key do not merely serve to play major scales. On the white keys of the piano, you can play all the seven ecclesiastical modes, depending on which of the seven tones you choose as the fundamental tone:
C: Ionian = major
D: Dorian
E: Phrygian
F: Lydian
G: Mixolydian
A: Aeolian
B: Locrian
The same applies to the pentatonic scales. If you only press the black keys on a piano, for example, you will automatically play pentatonically (penta = five → five black keys). Depending on which tone you choose as the fundamental tone, you will play a different pentatonic scale, for instance the major pentatonic scale (with F sharp as the fundamental tone) or the minor pentatonic scale (with E flat as the fundamental tone).
This principle of generating new scales with the tones of an existing scale also applies to the ‘melodic minor’ scale in jazz. As the major scale, the melodic minor scale is a selection of seven tones, but in intervals that cannot be played with the white keys of the piano alone. With these seven tones of the melodic minor scale, we are again able to construct seven different musical scales which differ greatly in terms of character, merely depending on which tones we choose as the fundamental tone.
4. Polytonality
This stylistic device emerged in the 20th century with Stravinsky and other composers, and it is also used in modern jazz. To achieve this, several keys are mixed; in practice, it is usually two (= bitonality). This appears to be slightly risky at first sight, but if we take into account the resonances (!) correspondingly, it actually sounds quite catchy.
Conclusion from the perspective of the theory of the three worlds
The mathematical purity (Platonic world) of the interval resonances is violated by the temperament, but in such a minimal way that the resulting vibration phenomena (physical world) are produced all the same and the listening experience (mental world) is hardly diminished at all.
On the other hand, free modulation extends the number of possible combinations of the still only twelve tones immensely. This mathematical extension (Platonic world) is audible and exciting (mental world). Through minimal impurity, music gains in variants and richness.
This is a post about the theory of the three worlds.