Rodan: Difference between revisions
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| Mapping = 1; 3 17 -1 -13 | | Mapping = 1; 3 17 -1 -13 | ||
| Pergen = (P8, P5/3) | | Pergen = (P8, P5/3) | ||
| Odd limit 1 = 9 | Mistuning 1 = | | Odd limit 1 = 9 | Mistuning 1 = 5.05 | Complexity 1 = 41 | ||
| Odd limit 2 = (2.3.5.7.11) 21 | Mistuning 2 = | | Odd limit 2 = (2.3.5.7.11) 21 | Mistuning 2 = 5.34 | Complexity 2 = 87 | ||
}} | }} | ||
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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]] (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 [[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. | ||
As can be elucidated by [[S-expression]]s, rodan is very much | As can be elucidated by [[S-expression]]s, rodan is very much an "opposed counterpart" to mothra: the basic equivalence of slendric tempers S7 (49/48) = S8 (64/63), and mothra proceeds to equate it to S6 ([[36/35]]) as well; meanwhile, rodan extends the equivalence in the opposite direction to add S9 (81/80) to it, making it one of the five [[rank-2 temperament]]s definable by equating three adjacent square superparticulars. | ||
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. | ||
| Line 135: | Line 135: | ||
| 151.35 | | 151.35 | ||
| 12/11, 35/32 | | 12/11, 35/32 | ||
| | | 56/51 | ||
|- | |- | ||
| 17 | | 17 | ||
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| 1061.49 | | 1061.49 | ||
| 50/27, 90/49 | | 50/27, 90/49 | ||
| 24/13 | | 24/13 | ||
|- | |- | ||
| 26 | | 26 | ||
| Line 237: | Line 237: | ||
== Chords == | == Chords == | ||
11-limit rodan contains [[essentially tempered | 11-limit rodan contains [[essentially tempered chord]]s of the commas 245/243, 385/384, 441/440, and 896/891. A list of 11-odd-limit [[dyadic chord|dyadically consonant chords]], both essentially tempered and essentially just, can be found at [[Chords of rodan]]. | ||
== Scales == | == Scales == | ||
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== Tunings == | == Tunings == | ||
=== Tuning spectrum === | === Tuning spectrum === | ||
{{see also|Slendric #Tuning spectrum}} | |||
{| class="wikitable center-all left-4 left-5" | {| class="wikitable center-all left-4 left-5" | ||
|- | |- | ||
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[[Category:Rodan| ]] <!-- main article --> | [[Category:Rodan| ]] <!-- main article --> | ||
[[Category:Rank-2 temperaments]] | [[Category:Rank-2 temperaments]] | ||
[[Category:Gamelismic clan]] | |||
[[Category:Sensamagic clan]] | [[Category:Sensamagic clan]] | ||
[[Category: | [[Category:Hemifamity temperaments]] | ||