Introduction

The “rule of the octave” (Italian regola dell’ottava; French règle de l’octave) is a fundamental pedagogical idea in tonal music.^[1] It came to prominence as a teaching tool back in the eighteenth century, especially in Italy and France, and was frequently written down in many different versions for students to learn and use.  It remains extremely useful both as a practical device (for getting used to keyboard configurations in various keys) and as a theoretical concept (for harmonising bass lines).

Basically, the rule of the octave is a standard way of harmonising each degree of a scale presented as a bass line. It has an ascending form and a descending form.

This chapter examines one version (out of a great many subtle variations) of this harmonisation in both major and minor. The version presented here is one of the most well-known and frequently used versions of the rule of the octave and was first notated by the composer and teacher Fedele Fenaroli (1730–1818)—it is often called the “Fenaroli scale.”

Because the rule of the octave had a significant role to play in the past training not only of composers but also keyboard players who were expected to be able to harmonise bass lines (figured and unfigured) at sight, it is most often presented in “keyboard style,” with the ascending and descending scale in the left-hand part, and the realisation in the three-note chords in the right.  However, this does not mean that its use is restricted to this style.

Learning to play the rule of the octave in a range of keys is a very powerful way to develop a tactile feeling for playing in various keys and to learn how to play successions of chords in the right hand with efficient and economical voice-leading. (Students in Italian conservatories in the eighteenth-century were expected to be able to play fluently all versions of the rule of the octave in all twenty-four major and minor keys before being allowed to proceed to more advanced studies!)

Fenaroli’s Rule of the Octave

Example 00 presents Fenaroli’s rule of the octave in C major in one of its three possible voicings—we will set the other two shortly.

The first thing to notice about the rule of the octave is that root position triads are only supported by and in the bass.  This means that the harmonisation only supports I and V in root position and all other degrees of the scale are realised as triads or seventh chords in inversion, as shown in the figuring below the left-hand part.  This is often summarised in the dictum that, aside from the tonic and the dominant, all other degrees of the scale “take the sixth.”  While some exceptions will be found, this is still a very useful rule of thumb in deciding on how to harmonise a bass line in common-practice styles.

A second feature to note is that, in the descending realisation, the rule of the octave requires one non-diatonic pitch: the raised subdominant () over the submediant degree () of the scale—turning what would diatonically be a minor sixth into a major sixth.  The effect of this is to briefly “lean” into the dominant triad in root position on . You might also notice that the chord involving the (in C major: D⁷/A) is the dominant seventh in G major (dominant key in relation to C major) and that the F acts as a leading note, rising up by a semitone to G. Given that in Example 00 we are in C major, this means that the chord above is, effectively, the “dominant of the dominant.” This is what we call a “secondary dominant,” a topic to be covered in depth elsewhere. Notably, while this temporarily strengthens the dominant triad, as we continue down the scale, the effect is immediately neutralised by the harmonisation on the diatonic subdominant ().

A further detail that you might notice is that there are apparantly parallel perfect fifths occurring in the right-hand part in the ascending realisation between the submediant () and leading note ().  These seem rather difficult to avoid if smooth and minimal part motion is to be maintained in this voicing.

Individual Parts in the Rule of the Octave

To break the rule of the octave down a little further, we can examine the three lines that form the right-hand realisation independently.  Because the chord that commences and finishes the rule of the octave is a tonic triad, voiced identically at the start and the finish, this means that each of these lines has the same starting and finishing note, these being:

• tonic (root of I)
• mediant (third of I)
• dominant (fifth of I)

Hence, we can refer to these lines as the “tonic,” “mediant” and “dominant” lines.  Each one is shown separately in Example 00 as a first species (1:1) counterpoint. For each line, the pitch content is very limited, comprising notes either a step above or a step below.

We can see from the interval labels between the parts in Example 00 that most intervals above the bass form perfect or imperfect consonances.  The exceptions to this, circled, have clear contextual explanations. These involve the tritone fa and ti (either as diminished fifth or augmented fourth) resolving to mi and do or a passing note pattern in the bass moving from a fifth to a third, or vice-versa, through a perfect fourth, or from an octave to a third via a major second, against a stationary note.  Invariably these passing-note figures connect notes of I or V.

Voicing  the Rule of the Octave

We can rearrange the voicing of the upper chords, so than any one of the lines above is the upper-most line.  While this changes the “inversion” of the upper chords, it does not change the overall inversion (and the figuring) of the whole texture because the bass line remains unchanged, being the ascending or descending scale.

Example 00, above, has already shown one such voicing. The remaining two are shown in example 00, below.

The Rule of the Octave in the Minor

The rule of the octave can also be used to harmonise a minor scale.  In doing so, the left-hand part uses the ascending and descending forms of the melodic minor, as shown in example 00, which is set in A minor. The figuring is essentially the same as for the major version, aside from requiring indications for raising the leading note.

The use of the melodic minor has two distinctive effects on the qualities of certain chords.  In the ascending version at x, the raised submediant in the bass () results in a major subdominant triad (IV⁶) instead of the usual minor triad (iv⁶). Likewise, in the descending version the use of the subtonic () in the bass line results in a minor dominant triad (v⁶), at y. As it is shown in other parts of this book, this is one of the very few occasions where a minor dominant triad is used in common practice—to harmonise a descending melodic minor bass line in the upper tetrachord of the scale.  Noticeably, where we arrive on in the bass, we return to the expected major dominant triad in root position (V).

A final feature resulting from the use of the melodic minor is found at z in Example 00.  Instead of a dominant seventh chord in second inversion, we have a very distinct and beautiful chord, known as a “French augmented sixth chord” (chord symbol: Fr.), so called because the submediant () and the raised subdominant () create the interval of an augmented sixth—in this case F–D. (It is also possible to have a variant which does not use the , resulting in a diatonic chord, ii^ø.)

Harmonic Analysis of the Rule of the Octave

We have already noted that only two chords stand in root position in the rule of the octave: I, or i, and V.  The remaining chords are (mostly) inversions of diatonic triads or seventh chords.  Complete harmonic analyses of both the major and minor rule of the octave are shown below, using B major and B minor, with different voicings.

Again, referring to the harmonisation of in the descending major scale, the raised subdominant pitch (here E) results in a chord which is the “dominant of the dominant”.  In the tonal context of B major, this is a C⁷/G; this is the dominant seventh to F major (which is V in B major).

To indicate that this is the dominant of the dominant, we use a slash (/) as follows V/V (often pronounced “five four three of five”). Note that this slash has an entirely different meaning to the slash used in (non-Roman-numeral) chord symbols:

• When used with Roman numerals, the slash means “of,” as in V7/V = the dominant seventh chord of the dominant triad (in speech we often contract this to “five seven of five”)
• When used with chord symbols, the slash means “over,” as in D7/F = a D7 chord in first inversion (in speech we usually contract this to “D seven over F sharp”)

For the corresponding chord in the descending minor scale, a French augmented sixth, we designate it with a non-Roman-numeral symbol: Fr. (in some texts, Fr.⁺⁶); augmented sixths are explained in another chapter.

Playing the rule of the octave

Practicing the rule of the octave regularly at a keyboard is highly recommended for anyone learning tonal music theory. As mentioned above, learning to play the rule of the octave in a range of keys is a useful skill.  It gives you an invaluable tactile feeling for the voicing of chords in different keys and it helps develop a feeling for efficient voicing in the right hand as you go from one chord to the next.

Every persons hand is slightly different and there is not fixed rule for exactly what fingers to use. Additionally, the figuring must change in the context of different keys.  However, the following principles are universally applicable:

• The left-hand part always follows the standard fingering for the scale in use
• In the right hand, where a note remains unchanged from one chord to the next, always use the same finger

In example 00 below, the three voicings of the scale are shown in different keys with fingerings.^[2] These are not universal fingerings for all keys. Part of the value of playing the rule of the octave in different keys is that it requires small adjustments to accomodate natural hand positions in different keys which helps enforce a tactile sense of key.  The red ties show notes that remain unchanged from one chord to the next and the red lines in the figuring indicate the corresponding finger that should remain in place (whether you restrict the note or hold it).

Some useful ways top practice the rule of the octave in different keys and voicings is to break it into segments and to hold (rather than replay) keyboard keys that repeat, keeping your finger in place.

Because the first three chords of the ascending rule of the octave are reversed (or retrograded) with the last three notes of the descending rule of the octave, it makes sense to isolate these three chords and play them in both directions. Other segments can be practices unidirectionally.

The following video shows some ways of practicing the rule of the octave as suggested above.

Dissecting the Rule of the Octave

Another way to think about the rule of the octave

We can make this a little more universal by constructing a kind of key-independent schema.  In Example 00 below, we show the major version of the rule of the octave only the scale degree numbers and the corresponding harmonic analysis symbols.