A Compositional Characterization of the Z-relation via Closure and Additivity
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866916057604685824 |
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| author | Joksimović, Aleksa |
| author_facet | Joksimović, Aleksa |
| contents | This paper gives a structural explanation for the Z-relation by modelling pitch-class sets as complete weighted graphs and encoding their interval content in a composition of $n$ via an additivity rule. We introduce the realization number $\mathcal{R}(μ,n)$ -- the count of T/I-inequivalent compositions producing the same interval multiset $μ$ -- and show that Z-relation is precisely the condition $\mathcal{R}(μ,n)\geq 2$. We prove that Z-relation cannot occur at cardinality $k=3$ in any $\mathbb{Z}_n$, construct an explicit primitive Z-related pair at $k=4$ for every $n$ divisible by $4$ with $n\geq 8$, and establish a Scaling Theorem showing that any Z-pair in $\mathbb{Z}_m$ propagates to $\mathbb{Z}_{dm}$ for all $d\geq 1$. All results are verified computationally for $\mathbb{Z}_{12}$ and $\mathbb{Z}_{19}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22948 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Compositional Characterization of the Z-relation via Closure and Additivity Joksimović, Aleksa General Mathematics This paper gives a structural explanation for the Z-relation by modelling pitch-class sets as complete weighted graphs and encoding their interval content in a composition of $n$ via an additivity rule. We introduce the realization number $\mathcal{R}(μ,n)$ -- the count of T/I-inequivalent compositions producing the same interval multiset $μ$ -- and show that Z-relation is precisely the condition $\mathcal{R}(μ,n)\geq 2$. We prove that Z-relation cannot occur at cardinality $k=3$ in any $\mathbb{Z}_n$, construct an explicit primitive Z-related pair at $k=4$ for every $n$ divisible by $4$ with $n\geq 8$, and establish a Scaling Theorem showing that any Z-pair in $\mathbb{Z}_m$ propagates to $\mathbb{Z}_{dm}$ for all $d\geq 1$. All results are verified computationally for $\mathbb{Z}_{12}$ and $\mathbb{Z}_{19}$. |
| title | A Compositional Characterization of the Z-relation via Closure and Additivity |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2605.22948 |