5-limit: Difference between revisions

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The octave equivalence classes of 5-limit or ''quinquimal'' intervals can usefully be depicted on a lattice diagram, either as a [[wikipedia: Hexagonal lattice |hexagonal lattice]] or as a [[wikipedia: Square lattice|square lattice]]; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[wikipedia:Hexagonal tiling|hexagonal tiling]].

[[~~EDO~~]]~~s which do relatively well in approximating the~~ 5-limit ~~(harmonics 3 and 5) are~~ {{EDOs| 2, 3, 7, 9, 10, 12, 19~~, 22~~, 31, 34, 53, 118, 289, 323, 441~~, 494~~, 559, 612, 1171, 1783, 2513, 3684, 4296, … ~~}} ([[OEIS: A060525]])~~

== Edo approximation ==

A list of edos with progressively better tunings ([[TE error]]) for 5-limit intervals: {{EDOs| 2, 3, 4, 5, 7, 12, 19, 31, 34, 46, 53, 118, 171, 289, 323, 388, 441, 559, 612, 1171, 1783, 2513, 3684, 4296 }}, …

Another ~~approach~~ is to ~~find EDOs which have better approximations for~~ [[5-odd-limit]] intervals ~~than all smaller EDOs. This results in~~ {{EDOs| 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, … }} ([[OEIS: ~~A054540]]~~)

Another list of edos that do relatively well in approximating harmonics 3 and 5: {{EDOs| 2, 3, 7, 9, 10, 12, 19, 22, 31, 34, 53, 118, 289, 323, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, … }} ({{OEIS|A060525}}<ref>The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to four of the simple ratios of musical harmony: 5/4, 4/3, 3/2 and 8/5. It is yet to be found out what criterion is used. </ref>)

Another list of edos that do relatively well in approximating [[5-odd-limit]] intervals: {{EDOs| 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, … }} ({{OEIS|A054540}}<ref>The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3. It is yet to be found out what criterion is used. </ref>)

== Syntonic comma pairs ==

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* [[Harmonic limit]]

* [[5-limit commas]]

== Notes ==

[[Category:5-limit| ]]

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 The octave equivalence classes of 5-limit or ''quinquimal'' intervals can usefully be depicted on a lattice diagram, either as a [[wikipedia: Hexagonal lattice |hexagonal lattice]] or as a [[wikipedia: Square lattice|square lattice]]; this can be done automatically by [[Scala]]. If the intervals are depicted with maximum symmetry as a hexagonal lattice, then the corresponding 5-limit triads define a [[wikipedia:Hexagonal tiling|hexagonal tiling]].
-[[EDO]]s which do relatively well in approximating the 5-limit (harmonics 3 and 5) are {{EDOs| 2, 3, 7, 9, 10, 12, 19, 22, 31, 34, 53, 118, 289, 323, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, … }} ([[OEIS: A060525]])
+== Edo approximation ==
+A list of edos with progressively better tunings ([[TE error]]) for 5-limit intervals: {{EDOs| 2, 3, 4, 5, 7, 12, 19, 31, 34, 46, 53, 118, 171, 289, 323, 388, 441, 559, 612, 1171, 1783, 2513, 3684, 4296 }}, …
-Another approach is to find EDOs which have better approximations for [[5-odd-limit]] intervals than all smaller EDOs. This results in {{EDOs| 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, … }} ([[OEIS: A054540]])
+Another list of edos that do relatively well in approximating harmonics 3 and 5: {{EDOs| 2, 3, 7, 9, 10, 12, 19, 22, 31, 34, 53, 118, 289, 323, 441, 494, 559, 612, 1171, 1783, 2513, 3684, 4296, … }} ({{OEIS|A060525}}<ref>The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to four of the simple ratios of musical harmony: 5/4, 4/3, 3/2 and 8/5. It is yet to be found out what criterion is used. </ref>)
+Another list of edos that do relatively well in approximating [[5-odd-limit]] intervals: {{EDOs| 1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, … }} ({{OEIS|A054540}}<ref>The description reads: a list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3. It is yet to be found out what criterion is used. </ref>)
 == Syntonic comma pairs ==
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 * [[Harmonic limit]]
 * [[5-limit commas]]
+== Notes ==
+<references/>
 [[Category:5-limit| ]] <!-- main article -->