105edo: Difference between revisions

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'''105edo''' is the [[~~Equal_division_of_the_octave|~~equal division of the octave]] into 105 equal parts of 11.429 [[~~cent|~~cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly ~~fulfils~~ both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[~~tempering_out|~~tempers out]] [[~~81/80|~~81/80]] in the [[~~5-limit|~~5-limit]]; 81/80, [[~~126/125|~~126/125]] and hence 225/224 in the [[~~7-limit|~~7-limit]]; 99/98, 176/175 and 441/440 in the [[~~11-limit|~~11-limit]]; and if we want to push that far, 144/143 in the [[~~13-limit|~~13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[~~Optimal_patent_val|~~optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[~~74edo|~~74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "~~[[cent|~~cent]]" is desired to include them all or be a triangular number.

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

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[[Category:Meantone]]

[[Category:Equal divisions of the octave]]

[[Category:105edo]]

[[Category:105edo| ]]

[[Category:Huygens]]

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-'''105edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 105 equal parts of 11.429 [[cent|cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfils both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempering_out|tempers out]] [[81/80|81/80]] in the [[5-limit|5-limit]]; 81/80, [[126/125|126/125]] and hence 225/224 in the [[7-limit|7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit|11-limit]]; and if we want to push that far, 144/143 in the [[13-limit|13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
+'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
-edo gives the [[Optimal_patent_val|optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo|74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all or be a triangular number.
+edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.
 == 105edo close-up ==
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 [[Category:Meantone]]
 [[Category:Equal divisions of the octave]]
-[[Category:105edo]]
+[[Category:105edo| ]] <!-- main article -->
 [[Category:Huygens]]