5th-octave temperaments: Difference between revisions

== Thunder ==

Thunder is a weak extension of slendrismic (above), [[rainy]] and [[cata]], with a generator of a slightly sharp ~6/5 (befitting of any [[kleismic]] temperament), three of which making [[26/15]]~[[19/11]]. More interesting though is that the period is [[5edo|1\5]]; slendrismic gives this a very accurate interpretation of [[147/128]] = ([[3/2]])/([[8/7]])² = [[8/7]] * [[1029/1024|S7/S8]] which is a significant interval as it is the "harmonic 5edostep" (in that it's a [[rooted]] (/2^n) interval in the 2.3.7 subgroup that approximates 1\5 very well). Thunder gives a wealth of interpretations to [[5edo]] intervals, which are available everywhere due to 1\5 = 240{{cent}} being the period of Thunder. In fact, Thunder combines many convergents and semiconvergents to intervals of [[5edo]] into a single, high-limit temperament:

1\5 = [[23/20]] = [[31/27]] = [[85/74]] = [[54/47]] (which Thunder also equates with [[63/50]]), and 2\5 = [[33/25]] = [[95/72]] = [[29/22]] = [[62/47]] = [[128/97]] (which Thunder also equates with [[37/28]] and [[120/91]]).

Thunder can be thought of as the [[125edo|125f]] & [[140edo|140]] temperament in the [[37-limit]] add-47 add-97, with both tunings notable in all corresponding limits.  

7-limit Thunder also tempers out the [[4096000/4084101]] (the [[hemfiness comma]]).

Thunder extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that Thunder observes the comma [[441/440]] = ([[21/20]])/([[22/21]]) = S21, as if it didn't, we would have 63/55 also equated with [[8/7]], leading to the [[15edo]] tuning which tempers the [[cloudy comma]]. In the 11-limit, the 5 EDO fourth is interpreted as [[33/25]].

As Thunder is a weak extension of [[cata]], it is naturally at least 13-limit. Cata admits a fairly simple mapping of prime 13 via ([[6/5]])² = [[13/9]] so that a gen above that is [[26/15]] as half of [[3/1]]. As cata tempers [[625/624|625/624 = S25]] and [[676/675|676/675 = S26 = S13/S15]] and as the [[kleisma]] is S25² * S26, this replaces the kleisma in the comma list so that we now move it to the end (as both are 13-limit). For simplicity, we show [[325/324|325/324 = S25 * S26]] and the more structurally important aforementioned comma 676/675, omitting 625/624. It also tempers [[1001/1000]] and [[1716/1715]] in the 13-limit.

In the 17-limit, the 5 EDO fifth is interpreted as [[85/56]] = [[561/560]] * [[50/33]], so that [[17/16]] is reached at 11 periods minus 8 gens at approx. 103{{cent}}; equating it with [[16/15]] might seem natural but is not the route taken due to the precision affording observing their difference, [[256/255]]. (If you do want to equate 17/16 with 16/15, you get the 15 & 155 temperament instead, for which the main tuning is [[155edo]], but you very much do pay for it; notice the errors!) Interestingly, the CTE tuning of the 17-limit of Thunder is practically the same as that of the 29-limit (up to a thousandth of a cent), which is also notable as being where this temperament exhibits the lowest Dirichlet badness. It's also the smallest prime limit where the vals larger than [[140edo]] haven't disappeared from the [[optimal ET sequence]], as from the 19-limit and onwards the optimal ET sequence is always [[15edo|15(ko)]], [[125edo|125f]], [[140edo|140]].

As [[33/25]] and [[95/72]] are both close to the 5 EDO fourth, Thunder extends naturally to the 19-limit by tempering [[2376/2375]] = ([[33/25]])/([[95/72]]) = ([[6/5]])³ / ([[19/11]]) and thus equivalently by tempering ([[26/15]])/([[19/11]]) = [[286/285]]. This is equivalent to tempering [[400/399|400/399 = S20]] = ([[20/19]])/([[21/20]]), which is natural to temper given that we observe [[441/440|441/440 = S21]] as aforementioned.