Half step motions in the bass are rare in traditional major/minor scale. They play an important rule in establishing the key center and therefore merit separate discussion.
Much of the tonal structure of cadences relies on this motion from 7-8 and 4-3. The tritone between 7 and 4 resolves to the tonic third.
Remember that the original solfège system included them only between ‘mi’ and ‘fa’ – making it easier to identify these by ear. So, this cadential pull involved ‘fa-mi’ and ‘mi-fa’ at the same time!
So ‘mi’ and ‘fa’ in adjacent hexachords are a tritone apart. That’s important to remember! (In modern terminology, ‘ti’ and ‘fa’ are a tritone apart.)
Thinking about the rule of the octave, can we make any generalizations about the relationship between ‘mi’ and the figure and ‘fa’ and the figure?
In Major, we have half steps between 4-3 and 7-8
- Scale degree 3, ‘mi’, generally takes a 6/3 figure
- Scale degree 4, ‘fa’, is takes 5/3, 6/3, or 6/5/3 in ascending Rule of the Octave
- Scale degree 4, ‘fa’, takes 6/4/2 in descending Rule of the Octave. The 4 is an augmented fourth (tritone)
- Scale degree 7, ‘mi’, takes 6/3 or 6/5/3 in ascending Rule of the Octave. The 5 is a diminished fifth (tritone)
- Scale degree 8, ‘fa’, takes 5/3
In minor, there is one added half step, between 6-5, descending only
- In minor descending, scale degree 6, ‘fa’, takes a +6/4/3 (or a 6/4/3)
- In minor descending, scale degree 5, ‘mi’, takes a 5/3 figure with a raised.
So we can generalize:
- ‘mi’ always takes a 6/3 and can take a 6/5/3 ascending (the 5 has to be a diminished fifth)
- ‘fa’ can take 5/3, 6/5/3 or 6/3 ascending, and a 6/4/2 when descending.
We can use our understanding of the ‘mi-fa’ ‘fa-mi’ relationships to emphasize certain harmonies. For example, by converting the 6/3 chord on 3 to a 6/5/3 with a diminished fifth:
In modern analytical terms, this becomes an applied dominant:
We’ve actually used this idea already in descending rule of the octave (this is the concept behind Bach’s prelude in C Major in the first Well Tempered Clavier):
Again in modern analytical terms:
We can also add half steps and consider them ‘mi’ when ascending and ‘fa’ when descending. For example, the 5-6 ascending sequence can become chromatic:
And here’s a similar progression descending.
We can also expand our view of the Rule of the Octave to see it as a series of clausulae – this allows for some interesting expansions and for the creation of sequences from these clausulae