2190edo: Difference between revisions

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2190edo is a very strong [[13-limit]] system; no smaller division has a smaller 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and nothing beats it until [[2684edo|2684]]. A basis for the 13-limit ~~commas~~ is {[[9801/9800]], [[10648/10647]], 105644/105625, [[140625/140608]], 196625/196608}; also [[tempering out|tempered out]] are [[123201/123200]] ~~and~~ 151263/151250. It is not as impressive beyond the 13-limit, though it does well in the 2.3.5.7.11.13.19~~.29~~ [[subgroup]], ~~holding~~ the record of relative error until [[14618edo|14618]].

2190edo is a very strong [[13-limit]] system; no smaller division has a smaller 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and nothing beats it until [[2684edo|2684]]. It is closely related to [[730edo]], the Woolhouse unit system, with which it shares the same tuning in the [[5-limit]], but the [[harmonic]]s [[7/1|7]], [[11/1|11]], and [[13/1|13]] are all mapped differently. A basis for the 13-limit [[comma]]s is {[[9801/9800]], [[10648/10647]], 105644/105625, [[140625/140608]], 196625/196608}; also [[tempering out|tempered out]] are [[123201/123200]], [[151263/151250]], and [[250047/250000]].

It is not as impressive beyond the 13-limit, though it does well in the 2.3.5.7.11.13.19 [[subgroup]], where it holds the record of lowest relative error until [[6079edo|6079]], and the 2.3.5.7.11.13.19.29 subgroup, where it holds the record of lowest relative error until [[14618edo|14618]].

=== Prime harmonics ===

{{Harmonics in equal|2190|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 2190edo (continued)}}

=== Subsets and supersets ===

2190 factors into {{~~factorization~~|~~2190~~}}~~; among its divisors~~ is ~~the~~ Woolhouse unit ~~system, [[730edo|730]]~~.

Since 2190 factors into primes as {{nowrap| 2 × 3 × 5 × 73 }}, 2190edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 12, 15, 30, 73, 146, 219, 365, 438, 730, and 1095 }}. A step of 2190edo is exactly {{frac|1|3}} Woolhouse unit.

[[4380edo]], which doubles 2190edo, provides a good correction to the harmonics 17 and 23.

[[4380edo]], which doubles 2190edo, provides a good correction to the harmonics [[17/1|17]] and [[23/1|23]].

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 {{ED intro}}
-edo is a very strong [[13-limit]] system; no smaller division has a smaller 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and nothing beats it until [[2684edo|2684]]. A basis for the 13-limit commas is {[[9801/9800]], [[10648/10647]], 105644/105625, [[140625/140608]], 196625/196608}; also [[tempering out|tempered out]] are [[123201/123200]] and 151263/151250. It is not as impressive beyond the 13-limit, though it does well in the 2.3.5.7.11.13.19.29 [[subgroup]], holding the record of relative error until [[14618edo|14618]].
+edo is a very strong [[13-limit]] system; no smaller division has a smaller 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and nothing beats it until [[2684edo|2684]]. It is closely related to [[730edo]], the Woolhouse unit system, with which it shares the same tuning in the [[5-limit]], but the [[harmonic]]s [[7/1|7]], [[11/1|11]], and [[13/1|13]] are all mapped differently. A basis for the 13-limit [[comma]]s is {[[9801/9800]], [[10648/10647]], 105644/105625, [[140625/140608]], 196625/196608}; also [[tempering out|tempered out]] are [[123201/123200]], [[151263/151250]], and [[250047/250000]].
+It is not as impressive beyond the 13-limit, though it does well in the 2.3.5.7.11.13.19 [[subgroup]], where it holds the record of lowest relative error until [[6079edo|6079]], and the 2.3.5.7.11.13.19.29 subgroup, where it holds the record of lowest relative error until [[14618edo|14618]].
 === Prime harmonics ===
-{{Harmonics in equal|2190|columns=11}}
+{{Harmonics in equal|2190|columns=9}}
+{{Harmonics in equal|2190|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 2190edo (continued)}}
 === Subsets and supersets ===
-factors into {{factorization|2190}}; among its divisors is the Woolhouse unit system, [[730edo|730]].
+Since 2190 factors into primes as {{nowrap| 2 × 3 × 5 × 73 }}, 2190edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 12, 15, 30, 73, 146, 219, 365, 438, 730, and 1095 }}. A step of 2190edo is exactly {{frac|1|3}} Woolhouse unit.
-[[4380edo]], which doubles 2190edo, provides a good correction to the harmonics 17 and 23.
+[[4380edo]], which doubles 2190edo, provides a good correction to the harmonics [[17/1|17]] and [[23/1|23]].