118th-octave temperaments: Difference between revisions

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[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets.

[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].

=== Parakleischis ===

== Parakleischis ==

118edo and its ~~multiples are members of both~~ [[parakleismic]] and [[Schismatic family|schismic]], and ~~from this it derives its name~~.

118edo's is an excellent 5-limit system and its comma basis constitutes the [[parakleismic]] and [[Schismatic family|schismic]] temperaments together. Parakleischis retains the 5-limit mapping from 118edo and leaves other harmonics as independent generators.

[[Subgroup]]: 2.3.5.7

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[[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]

~~Mapping~~ generators: ~15625/15552, ~7

: mapping generators: ~15625/15552, ~7

[[Optimal tuning]] ([[~~POTE~~]]): ~7/4 = 968.7235

[[Optimal tuning]] ([[CTE]]): ~7/4 = 968.7235

{{Optimal ET sequence|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}

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Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]

Optimal tuning (~~POTE~~): ~7/4 = 968.5117

: mapping generators: ~176/176, ~7

Optimal tuning (CTE): ~7/4 = 968.5117

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Badness: 0.049316

==== Centenniamajor ====

=== Centenniamajor ===

Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.

Centenniamajor is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of centenniamajor temperament is to expand on that. Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.

Subgroup: 2.3.5.7.11

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Badness: 0.122

=== Oganesson ===

== Oganesson ==

Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that [[28561/28560]] is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.

@@ Line 1: / Line 1: @@
-[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets.
+[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].
-=== Parakleischis ===
+== Parakleischis ==
-edo and its multiples are members of both [[parakleismic]] and [[Schismatic family|schismic]], and from this it derives its name.
+edo's is an excellent 5-limit system and its comma basis constitutes the [[parakleismic]] and [[Schismatic family|schismic]] temperaments together. Parakleischis retains the 5-limit mapping from 118edo and leaves other harmonics as independent generators.
 [[Subgroup]]: 2.3.5.7
@@ Line 10: / Line 10: @@
 [[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]
-Mapping generators: ~15625/15552, ~7
+: mapping generators: ~15625/15552, ~7
-[[Optimal tuning]] ([[POTE]]): ~7/4 = 968.7235
+[[Optimal tuning]] ([[CTE]]): ~7/4 = 968.7235
 {{Optimal ET sequence|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}
@@ Line 25: / Line 25: @@
 Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]
-Optimal tuning (POTE): ~7/4 = 968.5117
+: mapping generators: ~176/176, ~7
+Optimal tuning (CTE): ~7/4 = 968.5117
 {{Optimal ET sequence|legend=1| 118, 354, 472 }}
@@ Line 31: / Line 33: @@
 Badness: 0.049316
-==== Centenniamajor ====
+=== Centenniamajor ===
-Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.
+Centenniamajor is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of centenniamajor temperament is to expand on that. Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.
 Subgroup: 2.3.5.7.11
@@ Line 61: / Line 63: @@
 Badness: 0.122
-=== Oganesson ===
+== Oganesson ==
 Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that [[28561/28560]] is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.