1. Virtual Pitch and the Classification of Chords in Minor and Major Keys
Ludger Hofmann-Engl
Development – Croydon Family Groups
ICMPC 10, Sapporo, 28th August 2008

2. Towards the Concept of Virtual Pitch
Rameau (1722) – Root – e.g. C-major has the root c
Ohm (1843) – Overtone series – e.g. Spectrum c – c – g – c – e – g - bb produces the sensation of a tone on c
Schouten (1940) – Residual Pitch – e.g. Spectrum c – g – c – e – g produces the sensation of a tone an octave lower
Terhardt (1982) – Virtual Pitch – e.g. C-major implies the virtual pitch c (3rd degree) as well as f (2nd degree)

3. Virtual Pitch Models Terhardt (1970) Parncutt (1988)
Hofmann-Engl (1990)
Meddis & Hewitt (1991)
Leman (1995)
Balsach (1997)

4. Hofmann-Engl Pitch 𝑉 𝑡 = 𝑖=1 𝑛 𝑤 𝑠 𝑠 𝑖 𝑤 𝑝 𝑠 𝑖 𝑛 𝑤 𝑠 = 𝑐 2 − 𝑏 2 𝑐
𝑉 𝑡 = 𝑖=1 𝑛 𝑤 𝑠 𝑠 𝑖 𝑤 𝑝 𝑠 𝑖 𝑛
𝑤 𝑠 = 𝑐 2 − 𝑏 2 𝑐
𝑤 𝑝 = 1 𝑖
𝑉 𝑐 = 6𝐻ℎ∗ 𝐻ℎ∗ 𝐻ℎ∗ =4.37𝐻ℎ

5. Hofmann-Engl Pitch Distribution to the C-major Chord

6. Chord Classification Systems

7. Correlation Coefficient: r = 0.00174850462373621
Computing Chord Similarities (I) using the Correlation Coefficient of the Hofmann-Engl Pitch Sets
Correlation Coefficient:
r =

8. Correlation Coefficient: r = 0.650462934354965
Computing Chord Similarities using the Correlation Coefficient of the Hofmann-Engl Pitch Sets
Correlation Coefficient:
r =

9. Correlation (similarity) table of chords
VI fetches highest similarity with I, hence: Parallel Tonic and Tonic
IV fetches highest similarity with II, hence: Parallel Subdominant and Subdominant
V fetches highest similarity with III, hence: Parallel Dominant and Dominant

10. The Riemann system appears to be superior to the Roman numeral system
Conclusions
Chord relationships in major/minor keys was established
The Riemann system appears to be superior to the Roman numeral system
The Hofmann-Engl pitch could prove to be a powerful tool determining chord similarities
Chord similarities could be useful in analysis, audio compression and music information retrieval

11. Virtual Pitch and the Classification of Chords in Minor and Major Keys
Ludger Hofmann-Engl
Development – Croydon Family Groups
ICMPC 10, Sapporo, 28th August 2008