140edo: Difference between revisions

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== Theory ==

140edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]] and [[23-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list.

140edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]], [[23-limit|23-]], and [[29-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list.

In the 5-limit, 140et [[tempering out|tempers out]] [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the {{nowrap| 53 & 87 }} temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].

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 == Theory ==
-edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]] and [[23-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list.
+edo is related to [[70edo]], from which it inherits the slightly sharp tuning of the [[3/1|3rd]] [[harmonic]] and the slightly flat tunings of the [[11/1|11th]], [[13/1|13th]] and [[17/1|17th]] harmonics, but the [[5/1|5th]] and [[7/1|7th]] harmonics are much improved, enabling it to approximate [[JI]] of various [[prime limit|limits]]. Its peak form is in the [[17-limit|17-]], [[19-limit|19-]], [[23-limit|23-]], and [[29-limit]], despite the obvious lack of [[consistency]] in the corresponding [[odd limit]]s. In fact, the only inconsistently mapped intervals in the [[17-odd-limit]] are [[11/9]] and its [[octave complement]], though with the [[23-odd-limit]], [[19/11]], [[19/17]], [[23/18]], [[23/19]] and their octave complements are also added to that list.
 In the 5-limit, 140et [[tempering out|tempers out]] [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the {{nowrap| 53 & 87 }} temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].