4edo: Difference between revisions

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

When viewed from a [[regular temperament]] perspective, 4edo can be seen as a tuning of the [[Dimipent family #Dimipent|dimipent temperament]], since it tempers [[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]] scale, 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 critically sharp [[~~Chromatic_pairs~~#Gariberttet|Gariberttet]].

When viewed from a [[regular temperament]] perspective, 4edo can be seen as a tuning of the [[Dimipent family #Dimipent|dimipent temperament]], since it tempers [[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]] scale, 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 critically sharp [[Subgroup temperaments#Gariberttet|Gariberttet]].

=== Harmonics ===

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 edo can ne viewed as a [[dual-fifth]] system (the smallest in fact), with the tritone and major sixth as the flat and sharp "fifths". The tritone is the the [[patent val]] while the major sixth is in 4b-edo (using [[wart notation]]). 4b-edo has one of the sharpest mappings of [[3/2]] of any [[octave]]-repeating ET, only outmatched by that of [[1edo]], and even falling outside of the 600-800 cent range of [[2L 1s]].
-When viewed from a [[regular temperament]] perspective, 4edo can be seen as a tuning of the [[Dimipent family #Dimipent|dimipent temperament]], since it tempers [[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]] scale, 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 critically sharp [[Chromatic_pairs#Gariberttet|Gariberttet]].
+When viewed from a [[regular temperament]] perspective, 4edo can be seen as a tuning of the [[Dimipent family #Dimipent|dimipent temperament]], since it tempers [[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]] scale, 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 critically sharp [[Subgroup temperaments#Gariberttet|Gariberttet]].
 === Harmonics ===