Orwell: Difference between revisions

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In the 11-limit, two generators are equated to [[15/11]] and [[11/8]] (meaning [[99/98]] and [[121/120]] are tempered out). This means that three stacked generators makes the [[orwell tetrad]] 1–7/6–11/8–8/5, a chord in which every interval is a (tempered) 11-odd-limit consonance. Other such chords in undecimal orwell are the [[keenanismic chords]] and the [[swetismic chords]].

Compatible equal temperaments include [[22edo]], [[31edo]], [[53edo]], and [[84edo]]. Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit. It tempers out the semicomma in the 5-limit, and so belongs to the [[semicomma family]]. In the 7-limit it tempers out [[225/224]], [[1728/1715]], [[2430/2401]] and [[6144/6125]], and in the 11-limit, 99/98, [[121/120]], [[176/175]], [[385/384]] and [[540/539]]. By adding [[275/273]] to the list of commas it can be extended to the 13-limit as [[Semicomma family #Orwell|tridecimal orwell]], and by adding instead [[66/65]], [[Semicomma family #Winston|winston temperament]]. See [[Orwell extensions]] for details about 13-limit extensions.

Compatible [[equal temperaments]] include [[22edo]], [[31edo]], [[53edo]], and [[84edo]]. Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit.

See [[Semicomma family #Orwell]] for technical details.

== Interval chain ==

== Theory ==

=== Tempering information ===

It tempers out the [[semicomma]] (2109375/2097152) in the 5-limit, and so belongs to the [[semicomma family]]. In the 7-limit it tempers out 225/224, 1728/1715, [[2430/2401]] and [[6144/6125]], and in the 11-limit, 99/98, 121/120, [[176/175]], [[385/384]] and [[540/539]]. By adding [[275/273]] to the list of commas it can be extended to the [[13-limit]] as [[Semicomma family #Orwell|tridecimal orwell]], and by adding instead [[66/65]], [[Semicomma family #Winston|winston temperament]]. See [[Orwell extensions]] for details about 13-limit extensions.

=== Interval chain ===

Odd harmonics 1–21 and their inverses are in '''bold'''.

{| class="wikitable center-1 right-2"

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<nowiki/>* In 11-limit CWE tuning, octave reduced

== Chords and harmony ==

=== Chords and harmony ===

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 In the 11-limit, two generators are equated to [[15/11]] and [[11/8]] (meaning [[99/98]] and [[121/120]] are tempered out). This means that three stacked generators makes the [[orwell tetrad]] 1–7/6–11/8–8/5, a chord in which every interval is a (tempered) 11-odd-limit consonance. Other such chords in undecimal orwell are the [[keenanismic chords]] and the [[swetismic chords]].
-Compatible equal temperaments include [[22edo]], [[31edo]], [[53edo]], and [[84edo]]. Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit. It tempers out the semicomma in the 5-limit, and so belongs to the [[semicomma family]]. In the 7-limit it tempers out [[225/224]], [[1728/1715]], [[2430/2401]] and [[6144/6125]], and in the 11-limit, 99/98, [[121/120]], [[176/175]], [[385/384]] and [[540/539]]. By adding [[275/273]] to the list of commas it can be extended to the 13-limit as [[Semicomma family #Orwell|tridecimal orwell]], and by adding instead [[66/65]], [[Semicomma family #Winston|winston temperament]]. See [[Orwell extensions]] for details about 13-limit extensions.
+Compatible [[equal temperaments]] include [[22edo]], [[31edo]], [[53edo]], and [[84edo]]. Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit.
 See [[Semicomma family #Orwell]] for technical details.
-== Interval chain ==
+== Theory ==
+=== Tempering information ===
+It tempers out the [[semicomma]] (2109375/2097152) in the 5-limit, and so belongs to the [[semicomma family]]. In the 7-limit it tempers out 225/224, 1728/1715, [[2430/2401]] and [[6144/6125]], and in the 11-limit, 99/98, 121/120, [[176/175]], [[385/384]] and [[540/539]]. By adding [[275/273]] to the list of commas it can be extended to the [[13-limit]] as [[Semicomma family #Orwell|tridecimal orwell]], and by adding instead [[66/65]], [[Semicomma family #Winston|winston temperament]]. See [[Orwell extensions]] for details about 13-limit extensions.
+=== Interval chain ===
 Odd harmonics 1–21 and their inverses are in '''bold'''.
 {| class="wikitable center-1 right-2"
@@ Line 132: / Line 136: @@
 <nowiki/>* In 11-limit CWE tuning, octave reduced
-== Chords and harmony ==
+=== Chords and harmony ===
 {{Main| Chords of orwell }}
 {{See also| Functional harmony in rank-2 temperaments }}