1600edo: Difference between revisions

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1600edo is a very strong 37-limit system, being [[consistency|distinctly consistent]] in the [[37-odd-limit]] with a smaller [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]].

It is also the first division past [[311edo|311]] with a lower [[43-limit]] relative error, being ''almost'' consistent in the [[45-odd-limit]], missing [[50/39]] and [[39/25]], both of which being off by ''52,6%'' by [[patent val]] mapping, which is still just an error of 0.3945 cents.

It is also the first division past [[311edo|311]] with a lower [[43-limit]] relative error, being ''almost'' consistent in the [[45-odd-limit]], missing only [[50/39]] and [[39/25]], both of which being off by ''52.6%'' by [[patent val]] mapping, which is still just an error of 0.3945 cents.

In the 5-limit, it supports [[~~kwazy~~]]. In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[12376/12375]] in the 17-limit and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].

In the 7-limit, it supports [[crazy]], it supports In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[4096/4095]] in the [[13-limit]] (allowing [[schisminic chords]]), [[12376/12375]] in the [[17-limit]], [[6860/6859]] in the 19-limit, and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].

=== ~~Odd~~ harmonics ===

=== Prime harmonics ===

{{Harmonics in equal|1600|prec=3|columns=12}}{{Harmonics in equal|1600|columns=12|start=13|prec=3|collapsed=true|title=Approximation of prime harmonics in 1600edo (continued)}}

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Since 1600 factors into {{factorization|1600}}, 1600edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800 }}.

One step of it is the [[relative cent]] for [[16edo|16]]. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for [[interval size measure]]. One step of 1600edo is already used as a measure called ''śata'' in the context of 16edo [[Armodue theory]]. Similar to the [[Mina]] in the [[27-odd-limit]], All [[45-odd limit]] intervals can be written using integer values of śata, being more in tune than out of tune.

== Regular temperament properties ==

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| [[Tetraicosic]]

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[normal ~~lists~~|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

@@ Line 5: / Line 5: @@
 edo is a very strong 37-limit system, being [[consistency|distinctly consistent]] in the [[37-odd-limit]] with a smaller [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than anything else with this property until [[4501edo|4501]].
-It is also the first division past [[311edo|311]] with a lower [[43-limit]] relative error, being ''almost'' consistent in the [[45-odd-limit]], missing [[50/39]] and [[39/25]], both of which being off by ''52,6%'' by [[patent val]] mapping, which is still just an error of 0.3945 cents.
+It is also the first division past [[311edo|311]] with a lower [[43-limit]] relative error, being ''almost'' consistent in the [[45-odd-limit]], missing only [[50/39]] and [[39/25]], both of which being off by ''52.6%'' by [[patent val]] mapping, which is still just an error of 0.3945 cents.
-In the 5-limit, it supports [[kwazy]]. In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[12376/12375]] in the 17-limit and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].
+In the 7-limit, it supports [[crazy]], it supports In the 11-limit, it supports the rank-3 temperament [[thor]]. In higher limits, it tempers out [[4096/4095]] in the [[13-limit]] (allowing [[schisminic chords]]), [[12376/12375]] in the [[17-limit]], [[6860/6859]] in the 19-limit, and due to being consistent higher than 33-odd-limit it enables the essentially tempered [[flashmic chords]].
-=== Odd harmonics ===
+=== Prime harmonics ===
 {{Harmonics in equal|1600|prec=3|columns=12}}{{Harmonics in equal|1600|columns=12|start=13|prec=3|collapsed=true|title=Approximation of prime harmonics in 1600edo (continued)}}
@@ Line 15: / Line 15: @@
 Since 1600 factors into {{factorization|1600}}, 1600edo has subset edos {{EDOs| 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800 }}.
 One step of it is the [[relative cent]] for [[16edo|16]]. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for [[interval size measure]]. One step of 1600edo is already used as a measure called ''śata'' in the context of 16edo [[Armodue theory]]. Similar to the [[Mina]] in the [[27-odd-limit]], All [[45-odd limit]] intervals can be written using integer values of śata, being more in tune than out of tune.
 == Regular temperament properties ==
@@ Line 105: / Line 105: @@
 | [[Tetraicosic]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct