Performance Record of J.S. Bach – Invention No. 2 in C Minor, BWV 773

Overview: A performance record of J.S. Bach’s Invention No. 2 in C Minor (BWV 773), recorded in October 2025 by Reiji at age 9. This piece exhibits unique characteristics where the 11th harmonic produces minimal interference under subharmonic consideration. Even after modulation, the rational multiples of the 11th harmonic align closely with the F undecimal (~1/4-tone), resulting in a natural structure free from harmonic conflict. The performance emphasizes the clarity of these harmonic and subharmonic relationships throughout the tonal progression Cm → Gm → B♭ → Cm.

Note: All content on this page is originally explained by Reiji in Japanese. The English version is translated by AI and structured by a parent, with Reiji's final approval.

Reiji's Words and Ideas

• This piece has a characteristic in which the 11th harmonic shows minimal interference when considering subharmonics.
• Even after modulation, the harmonics that are rational multiples of 11 approximate the F undecimal (~1/4-tone), forming a structure that avoids interference throughout the piece.
• The tonal progression is Cm → Gm → B♭ → Cm.
• 1. C Minor (Cm)
  Chord tones: C, E♭, G
  The 11th harmonic of C corresponds to F♯–49 cent (F+51 cent).
• 2. G Minor (Gm)
  Chord tones: G, B♭, D
  The 11th harmonic of G corresponds to C♯–49 cent (C+51 cent).
  Since this does not approximate F♯–49 cent, dividing the frequency by 3 (a rational proportion) yields F♯–47 cent (F+49 cent).
• Note: For consistency, the fundamental frequencies are unified according to 12-tone equal temperament (12-TET). In pure just intonation, numerous ratios (11/8, 7/5, 45/32, 10/7) exist depending on the reference tone. Therefore, this analysis employs 12-TET approximations for clarity.
• 3. B♭ Major
  Chord tones: B♭, D, F
  The 11th harmonic of F is B–49 cent (B♭+51 cent). Raising it by a perfect fifth (×3 frequency ratio) yields F♯–47 cent (F+49 cent).
• 4. C Minor (Recapitulation)
  Same as section 1.
• Considering these factors, I performed this piece while consciously emphasizing the 11th harmonic components around the 50-cent region. Although higher harmonics are difficult for the ear to perceive, I focused on bringing out the clarity of the 11th harmonic without interference.
• Subharmonic Analysis (Minor Sections)
  Tonic chord (Cm): C, E♭, G
  Listening to the subharmonics of the 5th (G) yields: G, G, C–2 cent, G, E♭+15 cent, C–2 cent.
  When rounded to 12-TET, they align as C, E♭, G—the tonic chord. Therefore, I emphasized resonance with subharmonics that align with the tonic in minor sections.
• Harmonic Analysis (Major Sections)
  After modulation to B♭ major, the tonic chord (B♭, D, F) produces harmonics: B♭, B♭, F+2 cent, B♭, D–14 cent, F+2 cent.
  Rounded to 12-TET, these form B♭, D, F, perfectly matching the tonic chord. Hence, in the major sections, I emphasized upper harmonics to enrich the sound.
• This represents my personal interpretation of the piece.

* These are personal reflections by the performer.