Detempering
In regular temperament theory, detempering is the process of taking a tempered tuning system and replacing each of its pitches with one or more pitches from its preimage, that is, the just or tempered pitches that the pitch represents. It is the opposite of tempering. Specifically, a detempered system (aka detemperament or detempering) has each pitch of a tempered system (according to a fixed regular temperament) replaced with some set of interpretations of the pitch under the temperament mapping. If exactly one interpretation is used for each degree of a scale, then the detempered scale is called a one-to-one detempering. Ideally the resultant detempered scale will have a compact lattice. A higher rank temperament is also called a detempering of a lower-rank temperament if the lower-rank temperament results from tempering out one or more commas in the higher-rank temperament. For example, meantone is a detempering of 12edo.
Detempering is one way among many to create a neji, or a JI scale approximating a given scale.
One-to-one detemperings of equal temperaments
The following are two equivalent definitions for one-to-one detemperings of an equal temperament:
- A JI scale is a one-to-one detempering of an ET if each note of the equal temperament is matched to exactly one JI note which tempers to the note.
- A JI scale S is epimorphic if on the JI subgroup [math]A \leq \mathbb{Q}_{\gt 0}[/math] generated by the intervals of S, there exists a val v: A → ℤ (which can be called an epimorphism) such that v(S[i]) = i for all i ∈ ℤ.
The two terms are equivalent because if a detempering of an n-note equal temperament v is one-to-one, then second definition follows by the additivity of v, and given the second definition, injectivity is immediate.
The property is strictly stronger than constant structure (CS). When one assumes S is a CS but not that it is a one-to-one detempering, there is a unique set map [math]v : \{\text{intervals of $S$}\} \to \mathbb{Z}[/math] that witnesses that S is a CS and satisfies v(S[i]) = i for all i. Thus a CS scale S is a one-to-one detempering if and only if this mapping v extends to a linear map on the entirety of A.
The second definition extends naturally to asking whether a higher-dimensional mapping [math]S:\mathbb{Z}^n \to P[/math] for an arbitrary codomain [math]P[/math] of relative pitches is epimorphic, in the same sense of there existing an abelian group [math]A[/math] and a linear map [math]v : A \to \mathbb{Z}^n[/math] such that [math]v(S(x)) = x.[/math] This can be of practical interest: one might ask whether an isomorphic keyboard mapping [math]S : \mathbb{Z}^2 \to P[/math] is epimorphic.
Temperaments supported by vals for one-to-one detemperings have occasionally been considered. Some temperaments (including vals for small edos) can be viewed this way for small one-to-one detemperings despite their relatively low accuracy:
- The 2.3.5 temperament dicot supports nicetone (3L2M2s), blackdye (5L2M3s) and superzarlino (a 17-note epimorphic scale) scale structures.
- The 2.3.7 temperament semaphore supports archylino (2L3M2s), diasem (5L2M2s), and other scales in the Tas series.
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] \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 a val associated with the the Ptolemaic diatonic scale. Indeed, 7edo supports dicot temperament.
Facts
Definition: constant structure (CS)
Given a periodic scale [math]S : \mathbb{Z} \to (0,\infty)[/math] (with codomain written as ratios from S(0) = 1 in the linear frequency domain), let [math]C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}[/math] be the set of k-steps of S. Then S is a constant structure (CS) if for any [math]i, j \in \mathbb{Z}, i \neq j,[/math] we have [math]C_i \cap C_j = \varnothing.[/math]
One-to-one detemperings of ETs are CSes
If the steps of a CS scale are linearly independent, then the scale is a one-to-one detempering of an ET
Theorem: Suppose S is a 2/1-equivalent increasing constant structure JI scale of length n. Let [math]C_1[/math] be the set of 1-steps of S, and suppose that [math]C_1[/math] is a basis for the JI subgroup A generated by it. Then there exists an val [math] 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]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 a CS but not a one-to-one detempering. 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.)
Terminology
As it is a common concept, one-to-one detemperings of ETs have also been called by a number of other names in xen theory, including transversal, epimorphic scale, and strong CS.
Deregularization
The term deregularization can be used as a JI-agnostic alternative to detempering.
In this sense, diasem (LMLSLMLSL) is a deregularization of semiquartal (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S.