Rodan: Difference between revisions

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'''Rodan''' is a [[regular temperament|temperament]] of the [[~~gamelismic clan~~]] ~~and one~~ of ~~the notable~~ [[~~extension~~]]s to [[~~slendric~~]]. ~~Like slendric~~, it is ~~generated by an~~ [[8/7]]~~, three of which gives~~ [[3/2]]~~. In addition~~ it ~~finds~~ [[~~5/1~~|~~harmonic 5~~]] ~~at +17 generator steps~~, ~~and the second defining comma for it is~~ [[~~245~~/~~243~~]].

'''Rodan''' is an [[extension]] of the [[slendric]] [[regular temperament|temperament]], which divides [[3/2]] into three equal intervals representing [[8/7]] (tempering out the gamelisma, [[1029/1024]]), to the full [[7-limit]] such that 17 of these [[generators]] [[stacking|stack]] to reach the interval class of the [[5/4|5th harmonic (5/4)]]. It tempers out [[245/243]], making it a [[sensamagic clan|sensamagic temperament]], so that [[5/3]] is divided into two intervals of [[9/7]]; and it tempers out [[5120/5103]], making it also a [[hemifamity temperaments|hemifamity temperament]], so that [[9/8]] stacks thrice into [[10/7]].

~~Reasonable~~ tunings ~~would~~ be ~~between~~ [[~~41edo~~]] and [[~~46edo~~]]. [[~~87edo~~]] ~~makes~~ for a ~~very recommendable option~~.

Unlike [[mothra]], which flattens the fifth to a [[meantone]] fifth, the fifth of rodan is slightly sharp of just, ranging from that of [[41edo]] to that of [[46edo]]. As a result, the [[256/243|diatonic minor second]] is compressed, and the interval known as the [[quark]], which represents [[49/48]], [[64/63]], and in rodan also [[81/80]], is even smaller than it is in tunings of slendric with a nearly just fifth. This entails that the [[MOS scale]]s of rodan [[cluster MOS|cluster]] even more strongly around [[5edo]], although this can be thought of as an advantage in that it simplifies the conceptualization of rodan's inventory of intervals; rather than directly using MOS scales, which are either extremely imbalanced or overly large, an approach to rodan may involve picking and choosing which intervals from each [[pentatonic]] category to keep in the scale.

As for further extensions, slendric temperaments often find [[55/32]] at 4 generator steps (tempering out [[385/384]] and [[441/440]]), giving new interpretations to the quark as [[55/54]] and [[56/55]]; 55/32 is particularly accurate in the tuning subrange appropriate for rodan, and so harmonic 11 can easily be found at -13 generator steps. The diatonic minor third ([[32/27]]) in hemifamity temperaments represents the square root of [[7/5]], for which [[13/11]] is a good interpretation (tempering out [[352/351]] and [[847/845]]), which turns out to place harmonic 13 at -22 generator steps. Finally, [[17/13]] is a good interpretation of the slendric subfourth comprising two generators, otherwise equated to [[21/16]] (tempering out [[273/272]] and [833/832]]), and this places harmonic 17 at -20 generator steps. Thus proceeds the canonical extension of rodan out to the [[17-limit]].

See [[Gamelismic clan #Rodan]] for more information.

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-'''Rodan''' is a [[regular temperament|temperament]] of the [[gamelismic clan]] and one of the notable [[extension]]s to [[slendric]]. Like slendric, it is generated by an [[8/7]], three of which gives [[3/2]]. In addition it finds [[5/1|harmonic 5]] at +17 generator steps, and the second defining comma for it is [[245/243]].
+'''Rodan''' is an [[extension]] of the [[slendric]] [[regular temperament|temperament]], which divides [[3/2]] into three equal intervals representing [[8/7]] (tempering out the gamelisma, [[1029/1024]]), to the full [[7-limit]] such that 17 of these [[generators]] [[stacking|stack]] to reach the interval class of the [[5/4|5th harmonic (5/4)]]. It tempers out [[245/243]], making it a [[sensamagic clan|sensamagic temperament]], so that [[5/3]] is divided into two intervals of [[9/7]]; and it tempers out [[5120/5103]], making it also a [[hemifamity temperaments|hemifamity temperament]], so that [[9/8]] stacks thrice into [[10/7]].
-Reasonable tunings would be between [[41edo]] and [[46edo]]. [[87edo]] makes for a very recommendable option.
+Unlike [[mothra]], which flattens the fifth to a [[meantone]] fifth, the fifth of rodan is slightly sharp of just, ranging from that of [[41edo]] to that of [[46edo]]. As a result, the [[256/243|diatonic minor second]] is compressed, and the interval known as the [[quark]], which represents [[49/48]], [[64/63]], and in rodan also [[81/80]], is even smaller than it is in tunings of slendric with a nearly just fifth. This entails that the [[MOS scale]]s of rodan [[cluster MOS|cluster]] even more strongly around [[5edo]], although this can be thought of as an advantage in that it simplifies the conceptualization of rodan's inventory of intervals; rather than directly using MOS scales, which are either extremely imbalanced or overly large, an approach to rodan may involve picking and choosing which intervals from each [[pentatonic]] category to keep in the scale.
+As for further extensions, slendric temperaments often find [[55/32]] at 4 generator steps (tempering out [[385/384]] and [[441/440]]), giving new interpretations to the quark as [[55/54]] and [[56/55]]; 55/32 is particularly accurate in the tuning subrange appropriate for rodan, and so harmonic 11 can easily be found at -13 generator steps. The diatonic minor third ([[32/27]]) in hemifamity temperaments represents the square root of [[7/5]], for which [[13/11]] is a good interpretation (tempering out [[352/351]] and [[847/845]]), which turns out to place harmonic 13 at -22 generator steps. Finally, [[17/13]] is a good interpretation of the slendric subfourth comprising two generators, otherwise equated to [[21/16]] (tempering out [[273/272]] and [833/832]]), and this places harmonic 17 at -20 generator steps. Thus proceeds the canonical extension of rodan out to the [[17-limit]].
 See [[Gamelismic clan #Rodan]] for more information.