8edt: Difference between revisions

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As the double of [[~~4edt|~~4edt]]~~, harmonically~~, it is ~~analogous to~~ 10edo ~~in that~~ the ~~harmonic chain is doubled~~. However, ~~doing so does not make it a schismatic temperament like 10edo, because~~ the full 3:5:7 triad is already present in 4edt. ~~In any case, as a multiple of 4edt~~, ~~I will say it~~ is the ~~widest variety of "Black-Extraterrestrial-Tree" temperament~~.

== Theory ==

As the double of [[4edt]], it is the analog of [[10edo]] being the double of [[5edo]]. However, the full 3:5:7 triad is already present in 4edt which is unlike the situation in 10edo where 4:5:6 gains a new better approximation than the sus4 triad in [[5edo]]. More precisely, 8edt is [[enfactoring|enfactored]] in the 3.5.7 subgroup.

What it does introduce are flat ~~pseudooctaves~~ and sharp 3:2's, making it related to 5edo melodically.

What it does introduce are flat [[2/1]] pseudo-octaves and sharp [[3/2]] perfect fifths, making it related to 5edo melodically. It is equivalent to 5edo with the [[3/1]] made just, by compressing the octave by 11.3 cents, which has the side effect of bringing the step size slightly closer to [[8/7]].

~~0: 1/1 0.000 unison, perfect prime~~

=== Harmonics ===

1~~: 237.744 cents 237.744~~

~~2: 475.489 cents 4/~~3

~~3: 713.233 cents 713.233~~

~~4: 950.978 cents 5/3~~

~~5: 1188.722 cents 1188.722~~

~~6: 1426.466 cents 1426.466~~

== Interval table ==

~~7: 1664.211 cents 1664.211~~

~~8: 3/1 1901.955 perfect 12th~~

== ~~Prime harmonics~~ ==

[[Category:~~edt]]~~

[[Category:Macrotonal]]

~~[[Category:tritave]]~~

~~[[category:macrotonal~~]]

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 {{Infobox ET}}
+{{ED intro}}
-As the double of [[4edt|4edt]], harmonically, it is analogous to 10edo in that the harmonic chain is doubled. However, doing so does not make it a schismatic<!--?!--> temperament like 10edo, because the full 3:5:7 triad is already present in 4edt. In any case, as a multiple of 4edt, I will say it is the widest variety of "Black-Extraterrestrial-Tree" temperament.
+== Theory ==
+As the double of [[4edt]], it is the analog of [[10edo]] being the double of [[5edo]]. However, the full 3:5:7 triad is already present in 4edt which is unlike the situation in 10edo where 4:5:6 gains a new better approximation than the sus4 triad in [[5edo]]. More precisely, 8edt is [[enfactoring|enfactored]] in the 3.5.7 subgroup.
-What it does introduce are flat pseudooctaves and sharp 3:2's, making it related to 5edo melodically.
+What it does introduce are flat [[2/1]] pseudo-octaves and sharp [[3/2]] perfect fifths, making it related to 5edo melodically. It is equivalent to 5edo with the [[3/1]] made just, by compressing the octave by 11.3 cents, which has the side effect of bringing the step size slightly closer to [[8/7]].
-: 1/1 0.000 unison, perfect prime
+=== Harmonics ===
+{{Harmonics in equal|8|3|1|}}
-: 237.744 cents 237.744
+{{Harmonics in equal|8|3|1|intervals=prime}}
-: 475.489 cents 4/3
-: 713.233 cents 713.233
-: 950.978 cents 5/3
-: 1188.722 cents 1188.722
-: 1426.466 cents 1426.466
+== Interval table ==
+{{Interval table}}
-: 1664.211 cents 1664.211
-: 3/1 1901.955 perfect 12th
-== Prime harmonics ==
-{{Harmonics in equal|8|3|1|intervals=prime}}
-[[Category:edt]]
+[[Category:Macrotonal]]
-[[Category:tritave]]
-[[category:macrotonal]]