4edo: Difference between revisions

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We can also add more kinds of chords, for instance the subminor ([[6:7:9:10|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126|1–9/7–3/2–9/5]]) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.

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 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 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 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.

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 We can also add more kinds of chords, for instance the subminor ([[6:7:9:10|1–7/6–3/2–5/3]]) and supermajor ([[70:90:105:126|1–9/7–3/2–9/5]]) to the mix, and by encoding which kind of tetrad a note reconstitute a version of 9-odd-limit tetradic harmony, again changing the harmonic content of a note without changing its 4edo skeletal position.
-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 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 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 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.

4edo: Difference between revisions

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