Rhodium: Difference between revisions

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Rhodium is a rank-2 temperament which splits the octave into 45 parts and has a generator that is mapped to [[55/32]]. It is named after the 45th element.

'''Rhodium''' is a [[rank-2]] [[regular temperament|temperament]] which splits the [[octave]] into 45 parts and has a generator that is mapped to [[55/32]]. It is named after the 45th element by [[Eliora]] in 2023.

~~For technical data see:~~ [[~~Ragismic microtemperaments#Rhodium~~]]

As a [[weak extension]] of the [[ennealimmal]] temperament, rhodium has the same mapping as ennealimmal in the 7-limit – [[27/25]] for 1\9, [[7/6]] for 2\9, [[63/50]] for third-octave and [[49/36]] for 4\9. However, it also provides a mapping of the Alpharabian quarter-tone [[33/32]] to 2\45 and thereby also tempers out the [[quartisma]]. Since rhodium tempers out [[4225/4224]], a single period is mapped to an interval that stands both for ~[[66/65]] and ~[[65/64]] and two of them make 33/32, and ten thus make 7/6.

~~==Theory==~~

~~From a classical regular temperament theory perspective, rhodium is just an extension~~ of the [[ennealimmal ~~temperament~~]], ~~and respectively~~ has the same ~~mappings for~~ the ~~9edo subset~~ - [[27/25]] for 1\9, [[7/6]] for 2\9, [[63/50]] for third-octave and [[49/36]] for 4\9.

~~However, it also provides a mapping of the Alpharabian quarter-tone~~ [[~~33/32~~]] ~~to 2\45~~ and ~~thereby also tempers out~~ the [[~~quartisma~~]]~~. Since rhodium tempers out~~ [[~~4225/4224~~]], ~~a single period is mapped to an interval that stands both for ~~~[[~~66/65~~]] and ~[[~~65/64~~]] ~~and two~~ of ~~them make 33/32, and ten thus make 7/6~~.

[[1665edo]] is the optimal patent val for rhodium both in the 11-limit and in the 13-limit. Other notable equal temperaments which support rhodium include [[270edo]] and [[1395edo]], which are [[The Riemann zeta function and tuning|zeta edos]], and [[1125edo]], which while is not zeta, is still a multiple of 45 consistent in the 15-odd-limit.

[[~~1665edo~~]] is the optimal patent val for rhodium both in the 11-limit and in the 13-limit. Other notable equal temperaments which support rhodium include [[270edo]] and [[1395edo]], which are [[The Riemann zeta function and tuning|zeta edos]], and [[1125edo]], which while is not zeta, is still a multiple of 45 consistent in the 15-odd-limit.

For technical data see [[Ragismic microtemperaments #Rhodium]].

~~=== Relationship to Ben Johnston's notation ===~~

[[Category:Rhodium| ]]

Rhodium tempers 65/64 to 1\45 and 33/32 to 2\45. These intervals are used in Ben Johnston notation to translate a nearby simple interval into a usually more complex tridecimal (65/64) or undecimal (33/32) version. As such, using symbols 11 and 13 while playing in rhodium enables modulation by various steps of 45edo.

[[Category:Rank-2 temperaments]]

[[Category:Rank 2]]

[[Category:Ennealimmal]]

[[Category:~~Quartismic~~]]

@@ Line 1: / Line 1: @@
-Rhodium is a rank-2 temperament which splits the octave into 45 parts and has a generator that is mapped to [[55/32]]. It is named after the 45th element.
+'''Rhodium''' is a [[rank-2]] [[regular temperament|temperament]] which splits the [[octave]] into 45 parts and has a generator that is mapped to [[55/32]]. It is named after the 45th element by [[Eliora]] in 2023.
-For technical data see: [[Ragismic microtemperaments#Rhodium]]
+As a [[weak extension]] of the [[ennealimmal]] temperament, rhodium has the same mapping as ennealimmal in the 7-limit – [[27/25]] for 1\9, [[7/6]] for 2\9, [[63/50]] for third-octave and [[49/36]] for 4\9. However, it also provides a mapping of the Alpharabian quarter-tone [[33/32]] to 2\45 and thereby also tempers out the [[quartisma]]. Since rhodium tempers out [[4225/4224]], a single period is mapped to an interval that stands both for ~[[66/65]] and ~[[65/64]] and two of them make 33/32, and ten thus make 7/6.
-==Theory==
-From a classical regular temperament theory perspective, rhodium is just an extension of the [[ennealimmal temperament]], and respectively has the same mappings for the 9edo subset - [[27/25]] for 1\9, [[7/6]] for 2\9, [[63/50]] for third-octave and [[49/36]] for 4\9.
-However, it also provides a mapping of the Alpharabian quarter-tone [[33/32]] to 2\45 and thereby also tempers out the [[quartisma]]. Since rhodium tempers out [[4225/4224]], a single period is mapped to an interval that stands both for ~[[66/65]] and ~[[65/64]] and two of them make 33/32, and ten thus make 7/6.
+[[1665edo]] is the optimal patent val for rhodium both in the 11-limit and in the 13-limit. Other notable equal temperaments which support rhodium include [[270edo]] and [[1395edo]], which are [[The Riemann zeta function and tuning|zeta edos]], and [[1125edo]], which while is not zeta, is still a multiple of 45 consistent in the 15-odd-limit.
-[[1665edo]] is the optimal patent val for rhodium both in the 11-limit and in the 13-limit. Other notable equal temperaments which support rhodium include [[270edo]] and [[1395edo]], which are [[The Riemann zeta function and tuning|zeta edos]], and [[1125edo]], which while is not zeta, is still a multiple of 45 consistent in the 15-odd-limit.
+For technical data see [[Ragismic microtemperaments #Rhodium]].
-=== Relationship to Ben Johnston's notation ===
+[[Category:Rhodium| ]] <!-- main article -->
-Rhodium tempers 65/64 to 1\45 and 33/32 to 2\45. These intervals are used in Ben Johnston notation to translate a nearby simple interval into a usually more complex tridecimal (65/64) or undecimal (33/32) version. As such, using symbols 11 and 13 while playing in rhodium enables modulation by various steps of 45edo.
+[[Category:Rank-2 temperaments]]
-[[Category:Rank 2]]
+[[Category:Ennealimmal]]
-[[Category:Quartismic]]