Rodan: Difference between revisions

{{Infobox regtemp

| Mapping = 1; 3 17 -1 -13

| Generators = 8/7 | Generators tuning = 234.4 | Optimization method = CWE

| Odd limit 1 = 9 | Mistuning 1 = 5.05 | Complexity 1 = 21

| Odd limit 2 = 11-limit 21 | Mistuning 2 = 5.34 | Complexity 2 = 36

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.

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.

When speaking of interval categories, as an extension of slendric it is possible to use a system [[Slendric#Interval categories|designed for slendric]], where notes are inflected from the diatonic [[chain of fifths]] by means of the prefixes "super" and "sub", such that three of these inflections stack to a diatonic minor second. Such an inflection is equivalent to a quark, and due to the quark's versatile nature in rodan as a generalized comma, simple pental and septimal intervals tend to be represented by single-quark offsets from the diatonic spine. These "super" and "sub" prefixes are equivalent to "up" and "down" in [[#Notation|the notation]] system, where lifts and drops would represent a single edostep in [[ups and downs notation]] in the context of larger rodan edos that map this comma to multiple edosteps.  

{| class="wikitable sortable center-1 center-2 center-3 right-4"

! rowspan="2" | # !! rowspan="2" | Extended <br> diatonic <br> interval !! rowspan="2" | Ups and downs <br> notation !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios

| 56/51

| 24/13

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]].

! Edo<br>generator

[[Category:Hemifamity temperaments]]