Virtual Pitch and the Classification of Chords in Minor and Major Keys


L. Hofmann-Engl

London

in Proceedings of ICMPC10, 2008, Sapporo, Japan

Abstract

1. Background

There are three main systems describing chords within major and minor keys; the Roman numeral system, the Anglo-Saxon name system (e.g. tonic and super tonic) and the Riemann functional system (e..g tonic and tonic parallel). The Riemann system is internationally largely unknown but is interesting because it establishes functional relationships between the chords. However, the question is whether this functionality has any psychological basis.

2. Aims

It is the goal of this paper to demonstrate that by computing the similarity between tonal chords, by implementing Hofmann-Engl's virtual pitch model, the functional relationship between tonal chords exists in the fashion as proposed by Riemann.

3. Main Contribution

Hofmann-Engl's virtual pitch model has been tested in a number of cases (Hofmann-Engl, 1991, 1999, 2004 & 2006) and remains at this point unchallenged. This model produces to a given chord a series of 12 virtual pitches (c, c#, .., b) fetching values between 0 Hh and 6 Hh. Hence two chords can be compared via regression analysis of the values fetched for each pair for c, c# .., b. Applying this methodology to c-major, we obtain strong correlations for the c major (tonic) and a minor chord (parallel tonic), for the f major (sub-dominant) and d minor chord (parallel sub-dominant) and for the g major (dominant) and e minor chord (parallel dominant). It also reveals that the dominant is minimal similar to the tonic explaining the tension between the two functional chords. Hence, Riemann's system appears to have psychological validity.

4. Implications

It can be argued that both the Roman numeral system and the Anglo-Saxon name system are inferior as they do not point to any relationships between tonal chords and that perhaps the Riemann system ought to be used as an international standard.

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Erratum:

Table 4 in the paper is incorrect. It should read:



I

II

III

IV

V

VI

I

1

0.00

0.45

0.20

0.20

0.65

II

0.00

1

0.11

0.65

0.38

0.27

III

0.45

0.11

1

0.36

0.65

0.27

IV

0.20

0.65

0.36

1

0.05

0.20

V

0.20

0.38

0.65

0.05

1

0.00

VI

0.65

0.27

0.27

0.20

0.00

1


This renders the last paragrah of section V incorrect. But it does not challenge the validity of the paper nor the usefulness of the Riemann system.

 


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