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Given a numerical semigroup ring $R=k[\![S]\!]$, an ideal $E$ of $S$ and anodd element $b \in S$, the numerical duplication $S \! \Join^b \! E$ is anumerical semigroup, whose associated ring $k[\![S \! \Join^b \! E]\!]$ sharesmany properties with the Nagata's idealization and the amalgamated duplicationof $R$ along the monomial ideal $I=(t^e \mid e\in E)$. In this paper we studythe associated graded ring of the numerical duplication characterizing when itis Cohen-Macaulay, Gorenstein or complete intersection. We also study when itis a homogeneous numerical semigroup, a property that is related to the factthat a ring has the same Betti numbers of its associated graded ring. On theway we also characterize when ${\rm gr}_{\mathfrak m}(I)$ is Cohen-Macaulay andwhen ${\rm gr}_{\mathfrak m}(\omega_R)$ is a canonical module of ${\rmgr}_{\mathfrak m}(R)$ in terms of numerical semigroup's properties, where$\omega_R$ is a canonical module of $R$.

Tangent cones of monomial curves obtained by numerical duplication