A common principle in communication seems to be to enhance the differences between categories. For example, females and males enhance the sex differences by adding some others, e g in clothing, and architects choose different colors for different parts of houses, e.g. windows, roof, basement etc. Likewise musicians tend to enhance differences between scales by playing the major third interval wide and the minor third interval narrow. Given the principles of harmony in traditional tonal music, this implies a need for narrow minor second intervals, too.
If the scale tones are derived from a set of pure fifth intervals, a tuning with narrow minor seconds emerges. This scale is called Pythagorean tuning. If used, the melodic intervals sound just fine but the harmonies generate disturbing beats.
Chords in Pythagorean tuning
| C major chord in equal tuning | |
| C major chord in Pythagorean tuning |
Melodic intonation
In music practise, also quasi-Pythagorean tunings occur, with scale tones tuned somewhere between equally tempered and true Pythagorean. Here you can compare the same excerpt played in equally tempered and quasi-Pythagorean tuning.
H Purcell, Fancy for 3 instruments
| Equal temperament | |
| Melodic intonation, k=2 | |
| Melodic intonation, k=4 |
Harmonic intonation
To avoid the beats in sustained chords, just tuning must be used, but then the major third is narrow and the minor third is wide. Here you can compare the same excerpt as played in equally tempered and just tuning.
| C major chord in equal tuning | |
| C major chord in just tuning |
H Purcell, Fancy for 3 instruments
| Equal temperament | |
| Harmonic intonation (just intonation) |
Mixed intonation
The solution used in the performance grammar is to use a quasi-Pythagorean tuning at the tone onsets and then to slowly approach just tuning, so that beats are gradually reduced. Note how the beats in long chords disappear after a while.
H Purcell, Fancy for 3 instruments
| Equal temperament | |
| Mixed intonation, k=1.2 | |
| Mixed intonation, k=1.6 |