Epimorphic scale
A JI scale S is epimorphic if on the JI subgroup [math]\displaystyle{ A \leq \mathbb{Q}_{\gt 0} }[/math] generated by the intervals of S, there exists a linear map v: A → ℤ, called an epimorphism, such that v(S[i]) = i for all i ∈ ℤ.
Epimorphism is strictly stronger than constant structure (CS). When one assumes S is CS but not that it is epimorphic, there is a unique set map [math]\displaystyle{ v : \{\text{intervals of $S$}\} \to \mathbb{Z} }[/math] that witnesses that S is CS and satisfies v(S[i]) = i for all i. Thus a CS scale S is epimorphic if and only if this mapping v extends to a linear map on the entirety of A.
This definition extends naturally to asking whether a higher-dimensional mapping [math]\displaystyle{ S:\mathbb{Z}^n \to P }[/math] for an arbitrary codomain [math]\displaystyle{ P }[/math] of relative pitches is epimorphic, in the same sense of there existing an abelian group [math]\displaystyle{ A }[/math] and a linear map [math]\displaystyle{ v : A \to \mathbb{Z}^n }[/math] such that [math]\displaystyle{ v(S(x)) = x. }[/math] This can be of practical interest: one might ask whether an isomorphic keyboard mapping [math]\displaystyle{ S : \mathbb{Z}^2 \to P }[/math] (for a theoretical infinite 2D isomorphic keyboard) is epimorphic.
Temperament supported by epimorphisms for epimorphic scales have occasionally been considered. Some temperaments (including vals for small edos) can be viewed this way for small epimorphic scales despite their relatively low accuracy:
Example
Consider the Ptolemaic diatonic scale, {9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}, which is nicetone with L = 9/8, M = 10/9, and s = 16/15. This scale is epimorphic because we can apply ⟨7 11 16], the 7edo patent val, to map the intervals into the number of scale steps:
[math]\displaystyle{ \left(\begin{array} {rrr} 7 & 11 & 16 \end{array} \right) \left(\begin{array}{rrrrrrr} -3 & -2 & 2 & -1 & 0 & -3 & 1 \\ 2 & 0 & -1 & 1 & -1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \end{array}\right) = \left(\begin{array}{rrrrrrr} 1 & 2 & 3 & 4 & 5 & 6 & 7 \end{array}\right) }[/math]
where the columns of the 3×7 matrix are the scale intervals written in monzo form. Hence, 7edo (equipped with its patent val) is an epimorphic temperament of the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.
Facts
Definition: constant structure (CS)
Given a periodic scale [math]\displaystyle{ S : \mathbb{Z} \to (0,\infty) }[/math] (with codomain written as ratios from S(0) = 1 in the linear frequency domain), let [math]\displaystyle{ C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\} }[/math] be the set of k-steps of S. Then S is constant structure (CS) if for any [math]\displaystyle{ i, j \in \mathbb{Z}, i \neq j, }[/math] we have [math]\displaystyle{ C_i \cap C_j = \varnothing. }[/math]
Epimorphic scales are CS
If the steps of a CS scale are linearly independent, then the scale is epimorphic
Theorem: Suppose S is a 2/1-equivalent increasing constant structure JI scale of length n. Let [math]\displaystyle{ C_1 }[/math] be the set of 1-steps of S, and suppose that [math]\displaystyle{ C_1 }[/math] is a basis for the JI subgroup A generated by it. Then there exists an epimorphism [math]\displaystyle{ v: A \to \mathbb{Z} }[/math] which is a val of n-edo (and a similar statement holds for other equaves).
(The condition of [math]\displaystyle{ C_1 }[/math] being a basis rather than merely a generating set cannot be omitted, since the scale {5/4, 32/25, 2/1} is CS but not epimorphic. The converse of this conditional also fails, as {9/8, 5/4, 3/2, 25/16, 2/1} is epimorphic under 5edo's patent val.)