We talked briefly about the Diaschismic comma in Chapter 11. Let’s discuss it further here. As I said in Chapter 11, the 2048/2025 ratio of four perfect fifths plus two major thirds up is 19.5 cents flatter than the nearest octave. That’s covering a big swath of harmonic territory, and it is such a jolt away from the tonic that it tends to threaten or even dissolve the feeling of a tonal center—which is exactly why the 19th century Romantics and even some 18th century composers like Rameau and Bach loved it so much. Sometimes it is employed within a single key, but it is often used as a means of catapulting the music into a distantly related modulation. In 12-eq, a single note on the keyboard can represent the Diaschismic pair. In 53-eq, the Diaschismic comma of 19.5 cents is very closely approximated by a single step in 53-eq (22.64 cents). Here again what a Diaschisma looks like within C:
Figure 14-8 Diaschisma
.C – G – D — A — E
The 19.5 cent gap between the A1b and the G1# can be used as one of the biggest harmonic leaps you can take in 12-eq. The Diaschisma can be used to blur the distinction between the dominant seventh and the augmented sixth (bVI#6) chords, the best-known of which are Italian, German and French Augmented Sixth chords. In 53-eq, the German bVI chord in C would be A1b, C, E1b, F1#. In 12-eq, that same chord would by necessity sound like an Ab dominant seventh chord: Ab-C-Eb-F# (same as Gb). I chose to talk about the diashisma alongside the seventh chords because they sound identical or similar in 12-eq but function very differently in 53-equal. Once again, the comma that is hidden in 12-eq is now exposed as two separate pitches in 53-eq. And the augmented sixth chords sound much less dissonant than do dominant seventh chords in 53-eq. Allaudin Mathieu has an excellent analysis of the Diaschisma in Chapter 32 of his book, Harmonic Experience. In 53-eq, you can hear that the augmented sixth chords have a beautiful, lush richness to them. They also function differently. The dominant seventh is pulled towards the tonic. The German Sixth pulls you in a completely different direction!
Figure 14-9: VI chords; Dominant 7th Pythagorean vs German bVI (see notes above). The first musical example before, played on a piano sampler, starts with the harmonic path of the Diaschisma. The second example plays only the chords notated below, with wind instrument samples.
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