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, called an epimorphism, v: A → ℤ such that v(S[i]) = i for all i ∈ ℤ.
Epimorphism is strictly stronger than CS. The reader should verify that when one assumes S is CS but not that it is epimorphic, there is a unique mapping v that witnesses that S is CS. Thus S is epimorphic if and only if this mapping v is also linear.
An epimorphic temperament of an epimorphic scale S on a JI subgroup A is a temperament supported by its epimorphism on A. Some temperaments (including vals for small edos) can be used as epimorphic temperaments 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.)