12276edo: Difference between revisions

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~~{{novelty}}{{stub}}~~{{Infobox ET}}

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps.

12276edo's step size is sometimes called a '''prima''', a term proposed by [[Lillian Hearne]], when used as an interval size unit.

== Theory ==

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps. It is the smallest [[atomic]] EDO inside its [[5-odd-limit]] [[diamond tradeoff]] tuning range.

=== As an interval size measure ===

The prima is useful for measurement of 11-limit intervals and commas. Given that 12276edo factors as 22 × 32 × 11 × 31, a prima is a whole number division of one degree of [[12edo]], [[22edo]], [[31edo]], [[99edo]] and [[198edo]]. The [[Pythagorean comma]] is represented by 240 prima, and the [[syntonic comma]] by 220 (and the [[schisma]] is therefore represented as 20 prima). A prima is almost exactly three [[tuning unit]]s. As one degree of [[12edo]] is 1023 prima, one [[cent]] is exactly 10.23 prima.

=== Prime harmonics ===

=== ~~Interval size measure~~ ===

=== Subsets and supersets ===

12276edo factors as 22 × 32 × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s.

12276edo factors into primes as {{nowrap| 22 × 32 × 11 × 31 }}, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s.

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

[[Category:3-limit record edos|#####]]

[[Category:12276edo| ]]

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-{{novelty}}{{stub}}{{Infobox ET}}
+{{Infobox ET}}
-{{EDO intro|12276}}
+{{ED intro}}
-is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps.
+edo's step size is sometimes called a '''prima''', a term proposed by [[Lillian Hearne]], when used as an interval size unit.
+== Theory ==
+is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps. It is the smallest [[atomic]] EDO inside its [[5-odd-limit]] [[diamond tradeoff]] tuning range.
+=== As an interval size measure ===
+The prima is useful for measurement of 11-limit intervals and commas. Given that 12276edo factors as 2<sup>2</sup> × 3<sup>2</sup> × 11 × 31, a prima is a whole number division of one degree of [[12edo]], [[22edo]], [[31edo]], [[99edo]] and [[198edo]]. The [[Pythagorean comma]] is represented by 240 prima, and the [[syntonic comma]] by 220 (and the [[schisma]] is therefore represented as 20 prima). A prima is almost exactly three [[tuning unit]]s. As one degree of [[12edo]] is 1023 prima, one [[cent]] is exactly 10.23 prima.
 === Prime harmonics ===
 {{Harmonics in equal|12276|columns=11}}
-=== Interval size measure ===
+=== Subsets and supersets ===
-edo factors as 2<sup>2</sup> × 3<sup>2</sup> × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s.
+edo factors into primes as {{nowrap| 2<sup>2</sup> × 3<sup>2</sup> × 11 × 31 }}, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s.
-[[Category:Equal divisions of the octave|#####]] <!-- 5-digit number -->
+[[Category:3-limit record edos|#####]] <!-- 5-digit number -->
-[[Category:12276edo| ]] <!-- main article -->