680edo: Difference between revisions

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=== Odd harmonics ===

== Odd harmonics ==

680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). ~~It also has a very accurate 23rd harmonic, inherited from [[170edo]].~~

680edo retains a reasonable 3rd and 5th [[harmonic]], though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on [[prime]] 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).

~~{{Harmonics in equal|680|columns=21}}~~

=== Subsets and supersets ===

Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high [[prime limit]]s, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as [[37/29]], of which 680edo is a weak [[circle]].

{{Harmonics in equal|680|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 680edo (continued)}}

== Subsets and supersets ==

Since 680 factors into {{factorization|680}}, 680edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|680}}
+{{ED intro}}
-=== Odd harmonics ===
+== Odd harmonics ==
-edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].
+edo retains a reasonable 3rd and 5th [[harmonic]], though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on [[prime]] 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).
-{{Harmonics in equal|680|columns=21}}
-=== Subsets and supersets ===
+Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high [[prime limit]]s, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as [[37/29]], of which 680edo is a weak [[circle]].
+{{Harmonics in equal|680|columns=11}}
+{{Harmonics in equal|680|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 680edo (continued)}}
+== Subsets and supersets ==
 Since 680 factors into {{factorization|680}}, 680edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.
-{{Stub}}
+{{todo|expand}}