105edo: Difference between revisions

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'''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''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each.

~~{{Primes in edo|105}}~~

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.

== Theory ==

105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone. 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 of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val.

== ~~105edo close-up~~ ==

=== Odd harmonics ===

~~<pre>C~~ . . ~~Dbb B## . . C# . . Db . . . C##~~ . . ~~D</pre>~~

=== Miscellany ===

105 is fairly 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 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.

== Scales ==

Since 105edo has a step of 11.429 cents, it also allows one to use its [[mos scale]]s as [[circulating temperament]]s, which it is the first triangular edo to do{{clarify}}.

~~Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments, which it is the first triangular edo to do.~~

{| class="wikitable"

|+Circulating temperaments in 105edo

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 {{Infobox ET}}
-'''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''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each.
-{{Primes in edo|105}}
-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.
+== Theory ==
+edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone. 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 of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val.
-== 105edo close-up ==
+=== Odd harmonics ===
-<pre>C . . Dbb B## . . C# . . Db . . . C## . . D</pre>
+{{Harmonics in equal|105}}
+=== Miscellany ===
+is fairly 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 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.
+== Scales ==
+Since 105edo has a step of 11.429 cents, it also allows one to use its [[mos scale]]s as [[circulating temperament]]s, which it is the first triangular edo to do{{clarify}}.
-Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments, which it is the first triangular edo to do.
 {| class="wikitable"
 |+Circulating temperaments in 105edo