4edo: Difference between revisions

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4edo can be viewed as a [[dual-fifth]] system (the smallest in fact, besides the trivial [[1edo]]), with the tritone and major sixth as the flat and sharp "fifths". The tritone represents 3/2 in the [[patent val]], while the major sixth represents 3/2 in the 4b val (using [[wart notation]]). The 4b val has one of the sharpest mappings of 3/2 of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].

4edo can be seen as a trivial tuning of the [[diminished (temperament)|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves; 4edo is the first edo that is [[~~The~~ Riemann zeta function ~~and tuning~~ #Zeta edo lists|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet|gariberttet]] tuning.

4edo can be seen as a trivial tuning of the [[diminished (temperament)|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves; 4edo is the first edo that is [[Riemann zeta function #Zeta edo lists|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet|gariberttet]] tuning.

=== Odd harmonics ===

{{Harmonics in equal|4}}~~Odd harmonics 3-23 sorted by error (greatest to least): 21, 15, 17, 3, 9, 5, 7, 13, 11, 23, 19~~

~~{{Harmonics in equal|steps=4|start=12|columns=9|collapsed=yes}}~~

~~Odd harmonics 25-41 sorted by error: 35, 29, 41, 25, 31, 33, 37, 39, 27~~

~~{{Harmonics in equal|steps=4|start=21|columns=9|collapsed=yes}}~~

~~Odd harmonics 43-59 sorted by error: 59, 49, 57, 51, 43, 47, 55, 53, 45~~

=== Subsets and supersets ===

@@ Line 17: / Line 17: @@
 edo can be viewed as a [[dual-fifth]] system (the smallest in fact, besides the trivial [[1edo]]), with the tritone and major sixth as the flat and sharp "fifths". The tritone represents 3/2 in the [[patent val]], while the major sixth represents 3/2 in the 4b val (using [[wart notation]]). The 4b val has one of the sharpest mappings of 3/2 of any [[octave]]-repeating equal temperaments, only outmatched by that of [[1edo]], and even falling outside of the 600- to 800-cent range of [[2L 1s]].
-edo can be seen as a trivial tuning of the [[diminished (temperament)|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves; 4edo is the first edo that is [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet|gariberttet]] tuning.
+edo can be seen as a trivial tuning of the [[diminished (temperament)|diminished]] temperament, since it tempers out [[648/625]] (the major diesis) by equating four minor thirds ([[6/5]]) to an octave. Alternately, it can be viewed as a critically flat [[hanson]] or [[myna]] tuning, as both 6 and 10 generators reach the best approximation to the 5th. This interpretation works best if you stretch the octaves; 4edo is the first edo that is [[Riemann zeta function #Zeta edo lists|zeta peak but not zeta peak integer]], which means the point of maximum harmonicity is somewhat further away from pure octaves than the previous two edos. If you compress the octaves instead, it can be interpreted as a critically sharp [[subgroup temperaments #Gariberttet|gariberttet]] tuning.
 === Odd harmonics ===
-{{Harmonics in equal|4}}Odd harmonics 3-23 sorted by error (greatest to least): 21, 15, 17, 3, 9, 5, 7, 13, 11, 23, 19
+{{Harmonics in equal|4}}
-{{Harmonics in equal|steps=4|start=12|columns=9|collapsed=yes}}
-Odd harmonics 25-41 sorted by error: 35, 29, 41, 25, 31, 33, 37, 39, 27
-{{Harmonics in equal|steps=4|start=21|columns=9|collapsed=yes}}
-Odd harmonics 43-59 sorted by error: 59, 49, 57, 51, 43, 47, 55, 53, 45
 === Subsets and supersets ===