Rodan: Difference between revisions
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{{Infobox | {{Infobox regtemp | ||
| Title = Rodan | | Title = Rodan | ||
| Subgroups = 2.3.5.7, 2.3.5.7.11 | | Subgroups = 2.3.5.7, 2.3.5.7.11 | ||
| Comma basis = [[245/243]], [[1029/1024]] (7-limit); <br> [[245/243]], [[385/384]], [[441/440]] (11-limit) | | Comma basis = [[245/243]], [[1029/1024]] (7-limit); <br> [[245/243]], [[385/384]], [[441/440]] (11-limit) | ||
| Edo join 1 = 41 | Edo join 2 = 46 | | Edo join 1 = 41 | Edo join 2 = 46 | ||
| | | Mapping = 1; 3 17 -1 -13 | ||
| Generators = 8/7 | Generators tuning = 234.4 | Optimization method = CWE | |||
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], …, [[5L 36s]], [[41L 5s]] | | MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], …, [[5L 36s]], [[41L 5s]] | ||
| Pergen = (P8, P5/3) | | Pergen = (P8, P5/3) | ||
| Odd limit 1 = 9 | Mistuning 1 = 5.05 | Complexity 1 = | | Odd limit 1 = 9 | Mistuning 1 = 5.05 | Complexity 1 = 21 | ||
| Odd limit 2 = | | Odd limit 2 = 11-limit 21 | Mistuning 2 = 5.34 | Complexity 2 = 36 | ||
}} | }} | ||
'''Rodan''' is one of the notable [[extension]]s of the [[slendric]] [[regular temperament|temperament]], which divides the perfect fifth, [[3/2]], into three equal intervals representing [[8/7]] ([[tempering out]] the gamelisma, [[1029/1024]]), reaching the full [[7-limit]] such that 17 of these [[generators]] [[stacking|stack]] to reach the interval class of the [[5/1|5th harmonic]]. 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 [[ | '''Rodan''' is one of the notable [[extension]]s of the [[slendric]] [[regular temperament|temperament]], which divides the perfect fifth, [[3/2]], into three equal intervals representing [[8/7]] ([[tempering out]] the gamelisma, [[1029/1024]]), reaching the full [[7-limit]] such that 17 of these [[generators]] [[stacking|stack]] to reach the interval class of the [[5/1|5th harmonic]]. 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 [[Aberschismic temperaments|aberschismic (formerly hemifamity) temperament]], so that [[9/8]] stacks thrice into [[10/7]]. | ||
Unlike in [[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]] (with [[87edo]] being an essentially optimal tuning). 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 (see [[#As a detemperament of 5et]]). 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. | Unlike in [[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]] (with [[87edo]] being an essentially optimal tuning). 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 (see [[#As a detemperament of 5et]]). 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. | ||
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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 [[11/1|harmonic 11]] can easily be found at -13 generator steps. It is also worth mentioning that this equates the diatonic major third to [[14/11]], tempering out [[896/891]]. A [[2.3.7.11 subgroup]] version of rodan, known as '''radon''', uses exclusively this mapping and forgoes interpreting the 5th harmonic. | 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 [[11/1|harmonic 11]] can easily be found at -13 generator steps. It is also worth mentioning that this equates the diatonic major third to [[14/11]], tempering out [[896/891]]. A [[2.3.7.11 subgroup]] version of rodan, known as '''radon''', uses exclusively this mapping and forgoes interpreting the 5th harmonic. | ||
Toward the [[13-limit]], the diatonic minor third ([[32/27]]) in | Toward the [[13-limit]], the diatonic minor third ([[32/27]]) in aberschismic 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 [[13/1|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 [[17/1|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. | See [[Gamelismic clan #Rodan]] for more information. | ||
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[[Category:Gamelismic clan]] | [[Category:Gamelismic clan]] | ||
[[Category:Sensamagic clan]] | [[Category:Sensamagic clan]] | ||
[[Category: | [[Category:Aberschismic temperaments]] | ||