The art of counterpoint: a Mazzola-type three-voice first-species counterpoint model
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| Format: | Preprint |
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2026
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| _version_ | 1866917551240380416 |
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| author | Okumura, Ren Shibayama, Takuro |
| author_facet | Okumura, Ren Shibayama, Takuro |
| contents | We propose a three-voice extension of Mazzola's algebraic model of first-species counterpoint. A three-voice sonority is represented as \(a+\varepsilon_1 .b+\varepsilon_2 .c\), where \(a\) is the lower voice and \(b,c\) are the lower--middle and lower--upper interval classes. Starting from a strong dichotomy \((X/Y)\) of \(\mathbb Z_{2k}\), we introduce a harmonic mask \(H\subseteq X\times X\) and define the admissible three-voice consonances \(X_H\). We then construct a Hichert-type successor relation by applying the ordinary two-voice admitted-successor mechanism to the active pairwise projections. In the twelve-tone Fuxian case, the proposed mask has 26 admissible bass-rooted interval pairs, including complete and incomplete sonorities, and yields a finite directed successor structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01102 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The art of counterpoint: a Mazzola-type three-voice first-species counterpoint model Okumura, Ren Shibayama, Takuro Rings and Algebras Combinatorics 00A65, 05C20, 20K01, 20B25 We propose a three-voice extension of Mazzola's algebraic model of first-species counterpoint. A three-voice sonority is represented as \(a+\varepsilon_1 .b+\varepsilon_2 .c\), where \(a\) is the lower voice and \(b,c\) are the lower--middle and lower--upper interval classes. Starting from a strong dichotomy \((X/Y)\) of \(\mathbb Z_{2k}\), we introduce a harmonic mask \(H\subseteq X\times X\) and define the admissible three-voice consonances \(X_H\). We then construct a Hichert-type successor relation by applying the ordinary two-voice admitted-successor mechanism to the active pairwise projections. In the twelve-tone Fuxian case, the proposed mask has 26 admissible bass-rooted interval pairs, including complete and incomplete sonorities, and yields a finite directed successor structure. |
| title | The art of counterpoint: a Mazzola-type three-voice first-species counterpoint model |
| topic | Rings and Algebras Combinatorics 00A65, 05C20, 20K01, 20B25 |
| url | https://arxiv.org/abs/2606.01102 |