THE RULE OF THE OCTAVE The Rule of the Octave is actually a small compound of rules. The first rule is that only the tonic and the dominant (scale degrees 1 and 5) are truly stable, and therefore only those scale degrees should carry a root-position triad.
In modern analytical symbols and in figured-bass tradition, the 5/3 figure (note that a slash between the figures, instead of stacking them, is common in prose because it is easier to typeset) is generally assumed whenever a figure is not given, and therefore it is not usually provided. The second rule is the corollary of the first: that all other tones are effectively unstable, and therefore should take the “chord of the sixth” (the old term for a first-inversion triad, though it still has some value in its own right). This chord is created by adding both a third and a sixth (usually diatonic) above the bass tone. This is most completely figured as 6/3, but the third is usually implied, so it is more commonly figured with just a 6.
As stated, these behave as a first-inversion triad, so the root of the chord is actually a third below the base note; so a step up in the bass notes constitutes a step down in chord roots. All the remaining rules in the Rule of the Octave are optional, but they have the distinct advantage that, when completed, each scale degree carries a chord that is unique to that scale degree, so that not only does the scale degree imply the chord, but the chord also implies the scale degree. The third rule is that any chord of the sixth that progresses up or down a step to a stable chord (i.e., progresses to 1 or 5) can take a dissonance: inverted seventh chords resolving according to the cycle of fifths. In the ascending form, this means that the 4 will take a first-inversion diatonic seventh chord rooted on the second scale degree, while the 7 will take a first-inversion dominant seventh chord (obviously rooted on the fifth scale degree). Both of these chords are figured as 6/5/3, but once again, the 3 is implied, so the more usually figure is just 6/5. It is important to remember that the chordal 7th must resolve down. Resolving the leading tone is not a concern, because it is in the bass, and the bass line resolves it by definition.
In the descending form, is it the 6 and the 2 that precede the stable “perfect” chords. Both of these take the dissonance equivalent to the modern second-inversion seventh chord, figured 6/4/3 – in this case, the clash between the 4 and the 3 prevents us from simply assuming the intervallic 3, but we do typically assume the intervallic 6 (which is chord tone 3), so the more typical figure is 4/3 (this chord was once called the “Chord of the Small Sixth”). Scale degree 2 supports the inversion of the dominant seventh chord. Scale degree 6 supports a seventh chord rooted on the second scale degree (just as 4 did, since they both resolve to 5); however, it is typical to raise scale degree 4 in an upper voice, to make the chord into a secondary dominant (still rooted on the same tone), figured #6/4/3; this was not possible on scale degree 4 because the tone that would have to be altered is the bass note itself, and we do not normally change the given bass line. As always, the seventh chord tone (the intervallic 3) should resolve downwards; in addition, the leading tones (including the secondary leading tone over 6) need to resolve upwards (at least if they are in the soprano, but it is also desirable, though not mandatory, in inner voices).
Note that the superfluous 8 figure over the 5/3 chords is given here to make a point about leading-tone resolution. The final rule in the Rule of the Octave complex is that 3, while not strictly stable, is at least semi-stable, and while it takes chord of the sixth (i.e., first-inversion triad), it can also be approach by a dissonant (sc. seventh) chord. Since this chord is a tonic chord, approaching it from either above or below is accomplished by a dominant seventh chord; what varies is the inversion. From 2, the second inversion (4/3) is used (just as when 2 descends to 1, since both are over 2 and resolve to a tonic chord). From 4, third inversion is used (6/4/2, usually just given as 4/2, or sometimes even just 2), placing the chordal 7 in the bass (which resolves downward by definition). This last chord is the only seventh chord in the more complex Rule of the Octave that does relate to the triad that was used without the seventh.
Putting this all together, we now have the final version of the Rule of the Octave as it is traditionally given.
The Rule of the Octave in the minor key is not substantially different than in the major key – certainly not in the simplified (triads only) form. There are a few observations worth making, however. First, all the dominant chords are major, using the raised 7 leading tone, except for when descending from 7 in the bass, which cannot be raised in the upper voices because it isn’t raised in the bass. Second, it should be noted that although the melodic minor is used for the bass, the ascending scale does not necessarily use chords from the melodic minor scale. The chord supported by 6 obviously comes from the melodic minor scale, but the chord supported by 4 does not; it retains the lower 6, producing a half-diminished chord that is characteristic of the minor mode. On the way back down, the 6 is lowered again, and of course carries a chord of the sixth. If it is an unaltered triad (in the most basic form of the Rule of the Octave), its progression to 5 creates the socalled Phrygian Half-Cadence. However, if the chord instead contains #4 as its intervallic sixth, as in the major mode, this creates the so-called Italian Sixth chord, one of varieties of the augmented sixth chord, in which the sixth is one semitone wider than even a major sixth. It is important that the 6 resolve down and the #4 resolve upward. If an additional dissonance is added to this tone, a raised fourth above the bass (the diatonic 2), which produces the French Sixth chord, another familiar variety of the augmented sixth chord. Here is the complete Rule of the Octave for the minor scale:
