99edo: Difference between revisions

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Extending it to the [[11-limit]] requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the {{val| 99 157 230 278 '''343''' }} (99e) val, it tempers out [[243/242]], [[441/440]], [[540/539]] and [[896/891]], and is an excellent tuning for the 11-limit version of hemififths temperament. Using the [[patent val]], 99edo is the [[optimal patent val]] for the rank-4 temperament tempering out [[121/120]]; zeus, the rank-3 temperament tempering out 121/120 and [[176/175]]; [[hemiwür]], one of the rank-2 11-limit extensions of hemiwürschmidt; and [[hitchcock]] (an 11-limit amity extension), the rank-2 temperament which also tempers out [[2200/2187]]. The same can be said of the mapping for [[13/1|13]], with the 99ef val tempering out [[144/143]], [[196/195]], 352/351 and [[364/363]], and its patent val tempering out [[169/168]], [[351/350]] and [[352/351]]. Hence 99edo, in spite of the fact that it tunes 11 and 13 relatively badly, is an important 13-limit tuning in more than one way.

Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve [[diamond monotone]] in the [[31-odd-limit]].

Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve [[diamond monotone]] in the [[31-odd-limit]], though it fails in the [[33-odd-limit]] due to mapping [[33/32]] to 6 steps, while [[32/31]] is mapped to 5 steps.

=== Prime harmonics ===

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 Extending it to the [[11-limit]] requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the {{val| 99 157 230 278 '''343''' }} (99e) val, it tempers out [[243/242]], [[441/440]], [[540/539]] and [[896/891]], and is an excellent tuning for the 11-limit version of hemififths temperament. Using the [[patent val]], 99edo is the [[optimal patent val]] for the rank-4 temperament tempering out [[121/120]]; zeus, the rank-3 temperament tempering out 121/120 and [[176/175]]; [[hemiwür]], one of the rank-2 11-limit extensions of hemiwürschmidt; and [[hitchcock]] (an 11-limit amity extension), the rank-2 temperament which also tempers out [[2200/2187]]. The same can be said of the mapping for [[13/1|13]], with the 99ef val tempering out [[144/143]], [[196/195]], 352/351 and [[364/363]], and its patent val tempering out [[169/168]], [[351/350]] and [[352/351]]. Hence 99edo, in spite of the fact that it tunes 11 and 13 relatively badly, is an important 13-limit tuning in more than one way.
-Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve [[diamond monotone]] in the [[31-odd-limit]].
+Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve [[diamond monotone]] in the [[31-odd-limit]], though it fails in the [[33-odd-limit]] due to mapping [[33/32]] to 6 steps, while [[32/31]] is mapped to 5 steps.
 === Prime harmonics ===