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 ~~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, such as [[grosstone]] and [[huygens]]. It [[tempering out|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 (a.k.a. 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's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

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

=== Odd harmonics ===

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

=== Subsets and supersets ===

105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.

As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 21edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime.

== Intervals ==

=== 15-odd-limit interval mappings ===

== Instruments ==

=== Lumatone ===

The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]].

~~[[Category:Theory]]~~

~~[[Category:Meantone]]~~

~~[[Category:Equal divisions of the octave]]~~

[[Category:105edo| ]]

[[Category:Equal divisions of the octave|###]]

[[Category:Huygens]]

[[Category:Meantone]]

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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.
+{{Infobox ET}}
+{{ED intro}}
-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, such as [[grosstone]] and [[huygens]]. It [[tempering out|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 (a.k.a. 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's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.
-== 105edo close-up ==
+=== Odd harmonics ===
+{{Harmonics in equal|105}}
-<pre>C . . Dbb B## . . C# . . Db . . . C## . . D</pre>
+=== Subsets and supersets ===
+is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.
+As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 21edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime.
+== Intervals ==
+{{Main|Table of 105edo intervals}}
+=== 15-odd-limit interval mappings ===
+{{Q-odd-limit intervals|105}}
+== Instruments ==
+=== Lumatone ===
+The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]].
-[[Category:Theory]]
-[[Category:Meantone]]
-[[Category:Equal divisions of the octave]]
 [[Category:105edo| ]] <!-- main article -->
+[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Huygens]]
+[[Category:Meantone]]