28edt: Difference between revisions
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From a no-twos perspective, 28edt has a nearly-pure 5th harmonic (which tritave-reduces to [[5/3]]), with 41/28 being a continued fraction [[convergent]] to log<sub>3</sub>(5). It poorly approximates harmonic 7, but has an acceptable approximation of harmonic 11, so it makes sense to consider it as a [[3.5.11 subgroup]] system. In the 3.5.11 subgroup, it tempers out the commas [[6655/6561]] and [[4-cent comma|1953125/1948617]]. The former means it supports [[deneb]] temperament, which equates 3 flat [[11/9]]s to [[5/3]], and the latter means it supports [[delta centauri]] temperament, which equates 9 near-just [[5/3]]s tritave-reduced to [[11/9]] (although delta centauri is a [[microtemperament]] and larger EDTs like [[71edt]] offer much better tunings of it). [[56edt]] rectifies the approximation of the 7th harmonic. | From a no-twos perspective, 28edt has a nearly-pure 5th harmonic (which tritave-reduces to [[5/3]]), with 41/28 being a continued fraction [[convergent]] to log<sub>3</sub>(5). It poorly approximates harmonic 7, but has an acceptable approximation of harmonic 11, so it makes sense to consider it as a [[3.5.11 subgroup]] system. In the 3.5.11 subgroup, it tempers out the commas [[6655/6561]] and [[4-cent comma|1953125/1948617]]. The former means it supports [[deneb]] temperament, which equates 3 flat [[11/9]]s to [[5/3]], and the latter means it supports [[delta centauri]] temperament, which equates 9 near-just [[5/3]]s tritave-reduced to [[11/9]] (although delta centauri is a [[microtemperament]] and larger EDTs like [[71edt]] offer much better tunings of it). [[56edt]] rectifies the approximation of the 7th harmonic. | ||
28edt has a very accurate [[8/1|8th harmonic]], and as such, one step is very close to every third step of [[53edo]], and the tritave-reduced 8th harmonic [[8/3]] and tritave-reduced 8th subharmonic [[9/8]] are equally well-approximated. It tempers out the [[schisma]] in the 8.3.5 subgroup, albeit a really strange tuning due to being a multiple of 4edt and being index-2 in BP. It can also be used for the 3.5 | 28edt has a very accurate [[8/1|8th harmonic]], and as such, one step is very close to every third step of [[53edo]], and the tritave-reduced 8th harmonic [[8/3]] and tritave-reduced 8th subharmonic [[9/8]] are equally well-approximated. It tempers out the [[schisma]] in the 8.3.5 subgroup, albeit a really strange tuning due to being a multiple of 4edt and being index-2 in BP. It can also be used for the 3.8.5.11 extension of deneb temperament equating +5 generators to 8/3. | ||
Similar to how 14edt can be seen as [[9edo]] with significantly [[stretched and compressed tuning|stretched]] [[octave]]s (~23{{c}}), 28edt can be seen as [[18edo]] with the same stretched octave. Note that using the stretched octave will map 8/1 to an inconsistent approximation rather than the aforementioned accurate 8/1. Considering that [[29edt]] is a compressed 18edo, neither 28edt or 29edt are truly equivalent to 18edo (these two basically correspond to 18edo's dual fifths). | Similar to how [[14edt]] can be seen as [[9edo]] with significantly [[stretched and compressed tuning|stretched]] [[octave]]s (~23{{c}}), 28edt can be seen as [[18edo]] with the same stretched octave. Note that using the stretched octave will map 8/1 to an inconsistent approximation rather than the aforementioned accurate 8/1. Considering that [[29edt]] is a compressed 18edo, neither 28edt or 29edt are truly equivalent to 18edo (these two basically correspond to 18edo's dual fifths). | ||
== Harmonics == | == Harmonics == | ||
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!3.5.11 | !3.5.11 | ||
!3.5 | !3.8.5.11 | ||
|- | |- | ||
| 1 | | 1 | ||
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| 510.714 | | 510.714 | ||
| [[125/81]] | | [[125/81]] | ||
| [[ | | [[99/64]] | ||
| Fq/Gy | | Fq/Gy | ||
|- | |- | ||