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, [[tempering_out|tempering 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|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, [[tempering_out|tempering 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 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 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all. | 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 close-up == | == 105edo close-up == | ||
Revision as of 01:55, 13 February 2019
105edo is the equal division of the octave into 105 equal parts of 11.429 cents each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, tempering 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 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
C . . Dbb B## . . C# . . Db . . . C## . . D