Aberrismic theory: Difference between revisions

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The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.

* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists a tuning with no deviation from JI, or that the temperament before applying the tuning map has the same rank as the JI subgroup. This also implies that there is only one JI ratio for each interval under such interpretations, unlike in temperaments that temper out commas.

* [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one.

* [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. groundfault terms the tuning of blackdye that makes aberrisma-altered thirds 13/11 and 14/11 ''Flutterpyth blackdye'', since Flutterpyth temperament is restricted to (maximally) 2.3.7.11.13.19 and does not include 5-limit thirds.

{| class="wikitable right-2 right-3 right-4 right-5"

|+ Blackdye intervals in two temperaments

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| 143/80 21/11 280/143

|}

Mathematically, this difference corresponds to choices of <math>\mathbb{Z}</math>-linear maps

<math>\alpha : \mathbb{Z}^3\langle\mathbf{L}, \mathbf{m}, \mathbf{s}\rangle \to A, \ A = \mathsf{JI}\langle 2, p_1, ..., p_s \rangle/\mathrm{ker}(T),</math> (where <math>T</math> is a temperament map in the 2.p1....ps subgroup) determined by differing choices of <math>\alpha(\mathbf{L}), \alpha(\mathbf{m}), \alpha(\mathbf{s}).</math>

== Musical open problems ==

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 The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
 * ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists a tuning with no deviation from JI, or that the temperament before applying the tuning map has the same rank as the JI subgroup. This also implies that there is only one JI ratio for each interval under such interpretations, unlike in temperaments that temper out commas.
-* [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one.
+* [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. groundfault terms the tuning of blackdye that makes aberrisma-altered thirds 13/11 and 14/11 ''Flutterpyth blackdye'', since Flutterpyth temperament is restricted to (maximally) 2.3.7.11.13.19 and does not include 5-limit thirds.
 {| class="wikitable right-2 right-3 right-4 right-5"
 |+ Blackdye intervals in two temperaments
@@ Line 70: / Line 70: @@
 | 143/80<br/>21/11<br/>280/143
 |}
+Mathematically, this difference corresponds to choices of <math>\mathbb{Z}</math>-linear maps
+<math>\alpha : \mathbb{Z}^3\langle\mathbf{L}, \mathbf{m}, \mathbf{s}\rangle \to A, \ A = \mathsf{JI}\langle 2, p_1, ..., p_s \rangle/\mathrm{ker}(T),</math> (where <math>T</math> is a temperament map in the 2.p<sub>1</sub>....p<sub>s</sub> subgroup) determined by differing choices of <math>\alpha(\mathbf{L}), \alpha(\mathbf{m}), \alpha(\mathbf{s}).</math>
 == Musical open problems ==