| E | B | F♯ | C♯ | G♯ | ||||||||
| C | G | D | A | E | B | F♯ | C♯ | G♯ | ||||
| E♭ | B♭ | F | C | G | D | A | E | B | F♯ | C♯ | ||
| E♭ | B♭ | F | C | G | D | A | E | |||||
| B♭ | F | C |
Within the research for this website, this piece was among the first attempted in just intonation; chosen because of the term cromatica. Would chromatic music sound good in just intonation?
The piece has 12 different note names in the score. Ordered by fifths: E♭, B♭, F, C, G, D, A, E, B, F♯, C♯, G♯. So it contains E♭ and not D♯ etc. These 12 are exactly the tones in the 12 tone mean tone tuning:
(12 tone meantone tuning)
As you may know, this tuning has a wolf fifth: the interval that connects the final G♯ back to the initial E♭. When we change E♭ enharmonically to D♯, the interval becomes a pure fifth in name: G♯-D♯. But the pitch ratio is quite different from the ideal value of 3/2 (= 1.5), namely 1.5359965928, which is 743.01 cents, or almost half a semitone too wide.
So Frescobaldi only writes the notes that are part of the 12 tone meantone tuning. He also avoids writing the wolf fifth. But he does write harmonic diminished fourths (C♯-F, B-E♭ and G♯-C). On an ordinary keyboard, they can easily be mistaken for major thirds (D♭-F, B-D♯ and A♭-C). But their pitch ratios are quite different: 32/25. Because that is so near the simple ratio 9/7, diminished fourths are almost consonant in their own right. And tempered just intonation makes that even a tiny bit better. Here is a major third B-D♯ followed by a diminished fourth B-E♭:
(tempered just intonation)
Anyway, have you ever wondered how a wolf fifth would actually sound? They pop up when we transpose the score to a key for which 12 tone meantone tuning is not suitable, for instance a semitone up:
(12 tone meantone tuning a semitone up)
In actual music practice such a transposition would never be made in this tuning on a 12 tone keyboard, because so many intervals then sound bad.