5th-octave temperaments: Difference between revisions
m →Thunderclysmic: missing optimal patent vals |
m →Thunderclysmic: remove wrong duplicate misplaced Optimal ET sequences |
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=== 11-limit === | === 11-limit === | ||
Thunderclysmic extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that Thunderclysmic 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]]. | Thunderclysmic extends naturally to the 11-limit by tempering [[385/384]] = ([[147/128]])/([[63/55]]) (or equivalently [[6250/6237]]). Note that Thunderclysmic 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]]. | ||
[[Subgroup]]: [[11-limit|2.3.5.7.11]] | [[Subgroup]]: [[11-limit|2.3.5.7.11]] | ||
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=== 13-limit === | === 13-limit === | ||
As Thunderclysmic 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]])<sup>2</sup> = [[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 S25<sup>2</sup> * 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. | As Thunderclysmic 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]])<sup>2</sup> = [[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 S25<sup>2</sup> * 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. | ||
[[Subgroup]]: [[13-limit|2.3.5.7.11.13]] | [[Subgroup]]: [[13-limit|2.3.5.7.11.13]] | ||
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=== 17-limit === | === 17-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 Thunderclysmic 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]]. | 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 Thunderclysmic 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]]. | ||
[[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]] | [[Subgroup]]: [[17-limit|2.3.5.7.11.13.17]] | ||