View source for 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 out|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}_{>0}</math> generated by the intervals of ''S'', there exists a [[val]] {{nowrap|''v'': ''A'' → ℤ}} (which can be called an '''epimorphism''') such that {{nowrap|''v''(''S''[''i'']) {{=}} ''i''}} for all {{nowrap|''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 {{nowrap|''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 [[support]]ed by vals for one-to-one detemperings have occasionally been considered. Some [[temperament]]s (including [[val]]s 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]] (3L&nbsp;2m&nbsp;2s), [[blackdye]] (5L&nbsp;2m&nbsp;3s) and superzarlino (a 17-note epimorphic scale) scale structures.
* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L&nbsp;3m&nbsp;2s), [[diasem]] (5L&nbsp;2m&nbsp;2s), and other scales in the [[Generator sequence|Tas series]].

=== Example ===
Consider the Ptolemaic diatonic scale, {{nowrap|{9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}<nowiki/>}}, which is nicetone with {{nowrap|L {{=}} 9/8|M {{=}} 10/9}}, and {{nowrap|s {{=}} 16/15}}. This scale is epimorphic because we can apply {{val| 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 {{nowrap|''S''(0) {{=}} 1}} in the linear frequency domain), let <math>C_k = \{ S[i+k]/S[i] : i \in \mathbb{Z}\}</math> be the [[interval class|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 ====
{{proof|contents=
Let {{nowrap|''v'': ''A'' → ℤ}} be the val associated with ''s''. Let <math>x \in C_j.</math> Then there exists <math>i > 0</math> such that <math>S[i+j]/S[i] = x.</math> Suppose by way of contradiction there exist <math>k \neq j</math> and <math>i > 0</math> such that <math>S[i+k]/S[i] = x.</math>

Then <math>v(x) = v(S[i+j]/S[i]) = v(S[i+j]) - v(S[i]) = i + j - i = j,</math> but also <math>v(x) = v(S[i^\prime+k]/S[i^\prime]) = v(S[i^\prime+k]) - v(S[i^\prime]) = k,</math> a contradiction.
}}

==== 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 {{nowrap|{5/4, 32/25, 2/1}<nowiki/>}} is a CS but not a one-to-one detempering. The converse of this conditional also fails, as {{nowrap|{9/8, 5/4, 3/2, 25/16, 2/1}<nowiki/>}} is epimorphic under [[5edo]]'s [[patent val]].)

{{proof|contents=
Define the linear map <math>v:A \to \mathbb{Z}</math> by defining <math>v(\mathbf{s}) = 1</math> for any step <math>\mathbf{s} \in C_1</math> and extending uniquely by linearity. Then for any <math>i \in \mathbb{Z}</math> we have <math>v(S[i]) = v(S[i]/S[i-1]\cdots S[1]) = v(S[i]/S[i-1]) + \cdots + v(S[1]) = i,</math> whence ''v'' is a one-to-one detempering. That <math>v(2) = n</math> is also automatic.
}}

=== 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''.
[[Category:Scale]]

== Lifting ==
The term ''lifting'' can be used as a [[JI-agnostic]] alternative to ''detempering''.

In this sense, [[diasem]] (LMLSLMLSL) is a lifting of [[semiquartal]] (LSLSLSLSL) which "detempers" the S step of semiquartal into two steps sizes M and S.

== Examples ==
* [[Ringer scale]]s
* [[Fantasy detempers]]
* [[87edo/13-limit detempering]]
* [[Diasem]]
* [[Beautiful 27]]

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Method]]
[[Category:Detempering| ]] <!-- main article -->
[[Category:Well temperament]]