(see also Keyboard Video)
| G♯ | D♯ | A♯ | |||
| A | E | B | F♯ | C♯ | |
| F | C | G | D | A | |
| D♭ | A♭ | E♭ | B♭ | F | C |
| G♭ | D♭ |
The following version is in a twelve tone unequal temperament, so you can compare.
(Werckmeister IV tuning, without the final A♯, see below)
Trouble grasping the difference with just intonation? The fragments on the home page may help.
In the original score, the final chord is a major triad C-E-G, concluding a cadence of quite dissonant chords. In 12 tone equal temperament, which renders major triads beating considerably, this final chord has an impressive and majestic character:
(12 tone equal temperament C-E-G)
In just intonation, the final C-E-G chord is an anti-climax; its complete consonance renders the chord very dull - after those dissonant other chords:
(just intonation C-E-G)
Check Alma Redemptoris Mater by G. Dufay where it is 12 tone equal temperament causing the anti-climax.
Therefore, I added an augmented sixth (A♯) to the final chord. That represents a harmonic seventh in that final chord, restoring some of the original character mentioned above. Or at least it prevents the anti-climax demonstrated above:
(just intonation C-E-G-A♯)
But now we hear a sudden slow beating in the final chord, due to the A♯, whose pitch ratio to C is almost 7/4 but not exactly. Tempered just intonation spreads out that beating over all chords:
(tempered just intonation C-E-G-A♯)
For comparison: here is the same in 12 tone equal temperament:
(12 tone equal temperament C-E-G-A♯)
A different approach to curing the anti-climax in the final chord is replacing the E (in the chord C-E-G) by F♭. That changes the major third C-E into a diminished fourth C-F♭ with a pitch ratio 32/25, which is close to 9/7. Check the matrix. The minor third E-G is changed into an augmented second F♭-G with a pitch ratio 75/64, which is close to 7/6.
(tempered just intonation C-F♭-G)
Here, the final chord is quite close to being consonant. And way out of tune if you will. Its character is bright, rather than majestic as in the 12 tone equal temperament version. Is it audacious to call for the F♭ tone just in this chord? It is not used elsewhere in the piece.
The following fragment is the final chord only. It starts out as C-E-G and then slowly shifts to C-F♭-G. When the pitch shift starts, the pitch proportions move away from 4:5:6 and you can hear the beats grow faster. Then when the pitch shift arrives at C-F♭-G, the pitch proportions approach 14:18:21 and you can hear the beats slow down. During the shift, the pitches pass through many other integral pitch proportions, the simplest ones being 18:23:27 and 22:28:33.
(tempered just intonation C-E-G to C-F♭-G)
(The first two measures of the Toccata as a ringtone: download here. Try right click.)