Invertible counterpoint at the tenth and twelfth

Why other inversion intervals exist

In invertible counterpoint at the octave, one voice is moved by 8 scale degrees. But you can also move it by 10 or 12 degrees (or other intervals): the inversion no longer produces the simple inversion of each interval but different intervals, and every inversion interval imposes its own rules. The advantage is huge: you get harmonically new versions of the same material, and in some cases you can superimpose the original and the inversion to obtain three voices from two. Bach devotes two numbers of The Art of Fugue BWV 1080 to these procedures: Contrapunctus 9 "alla Duodecima" and Contrapunctus 10 "alla Decima".

The sum trick

To find what an interval turns into: new interval = (inversion interval + 1) − original interval.

• At the octave: 9 − x.
• At the tenth: 11 − x.
• At the twelfth: 13 − x.

At the tenth

• Thirds become octaves and sixths become fifths: so parallel thirds and sixths, the bread and butter of two-voice writing, produce parallel octaves and fifths after the inversion. Rule: no consecutive thirds or sixths; prefer contrary and oblique motion.
• Fifth ↔ sixth and octave ↔ third: all consonant, so consonant material stays consonant.
• Suspensions: 7–6 becomes 4–5 (unusable), 4–3 becomes 7–8 (weak), 2–3 becomes 9–8 (good). Use with caution.
• The bonus: if the line is well written, you can play the original voice and its version inverted at the tenth at the same time: you get the same line doubled in thirds/tenths above another voice, that is, three real parts. This is the principle that makes Contrapunctus 10 so rich.

At the twelfth

• Thirds ↔ tenths: thirds remain compound thirds, so parallel thirds are allowed.
• The sixth becomes a seventh: this is the critical point. Every sixth must be treated as a dissonance (passing or neighbor tone); no parallel sixths.
• Fifth ↔ octave: both perfect, watch out for parallels.
• Suspensions: 4–3 becomes 9–10 (good), 7–6 becomes 6–7 (unusable), 2–3 becomes 11–10 (that is, 4–3, good).
• Musically, inversion at the twelfth is equivalent to transposing one voice by a fifth: it is perfect for fugues, where the material must fit under the subject (tonic) and under the answer (dominant).

Example in A minor (at the twelfth)

Upper line: A–B–C–D–E (quarter notes)

Lower line: A–G#–A (half notes with a passing tone), built so as to form 8, 10 and 3 on strong beats, never sixths on strong beats.

Moving the lower line a twelfth up (that is, an octave plus a fifth), the thirds become tenths and the octave becomes a fifth: the result sounds like the same idea "seen from the dominant".

How to practice

1. On paper or in the piano roll, write a table of the transformations above the staff.
2. Compose two 4-bar voices in C major, already thinking of the desired inversion (tenth: contrary motion only; twelfth: no sixths on strong beats).
3. Transpose the lower voice: at the tenth, move it by a diatonic tenth (15 or 16 semitones depending on the scale degree: recalculate every note within the scale, do not use a fixed chromatic transposition). At the twelfth, move it by an octave plus a diatonic fifth.
4. Listen to both versions; for the tenth, also try playing the original + inversion together.

Where to find it

• Bach, The Art of Fugue, Contrapunctus 9 and 10.
• Bach, Goldberg Variations BWV 988: the canons at different intervals (at the unison, at the second… at the ninth) are a close relative of this technique.
• Film and video game writing: composers who need to restate a theme with a countermelody in different keys or registers often exploit combinations at the tenth to get doublings in thirds "for free".

Related topics

Invertible counterpoint at the octave, Canon and imitation, Fugue: subject and answer, Three-part counterpoint.