## Abstract

A “tertial element” is a primary interval (y, z) such that z is a generic “klang generator” e of a non-empty harmonic tone system (Sect. 8.1). A “proto-tonal system” is a harmonic tone system such that the quintic element (c, d) satisfies c is as close as possible to alog2(3/2); moreover, assuming a non-empty system, there exists a tertial element (y, z) such that y is as close as possible to alog2(5/4). It is proven that a proto-tonal system satisfies a = 12, b = 7 = c, d = 4, and (if the system is not empty) e = 2. Section 8.2 revisits the theory of categorical equal temperament from a historical and historical-theoretical perspective, now that a = 12 has been established. Finally, Sect. 8.3 considers an alternative to the theory of proto-tonality, based on Clough and Douthett’s (1991) theory of “maximally even sets.” It is shown that the alternative theory is weaker than the proposed theory in terms of explanatory power.

Original language | English |
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Title of host publication | Computational Music Science |

Publisher | Springer Nature |

Pages | 127-143 |

Number of pages | 17 |

DOIs | |

State | Published - 2013 |

### Publication series

Name | Computational Music Science |
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ISSN (Print) | 1868-0305 |

ISSN (Electronic) | 1868-0313 |

### Bibliographical note

Publisher Copyright:© 2013, Springer-Verlag Berlin Heidelberg.

## Keywords

- Categorical Principle
- Complex Tone
- Harmonic System
- Pitch Interval
- Tone System