28edo: Difference between revisions

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== Octave stretch and compression ==

'''[[ed9|89ed9]]''', a slightly [[octave shrinking|compressed]] tuning of 28edo, makes 28edo potentially useable as a [[dual-fifth]] tuning. It shares the error equally between 28edo's two perfect fifths.

So, where pure-octaves 28edo has a flat fifth with 16.2{{c}} error and a sharp fifth with 28.6{{c}} error, 89ed9 instead has a flat fifth with 21.4{{c}} error and a sharp fifth with 21.4{{c}} error.

21.4{{c}} error is slightly worse than [[5edo]]'s best fifth but slightly better than [[23edo]]'s.

Some might consider this fifth too bad to use, in which case 89ed9 would be a downgrade from pure-octaves 28edo.

Others might consider it just useable enough, in which case 89ed9 would be an upgrade because it has one more useable interval and one less [[wolf interval]].

89ed9 approximates the no-3s 7-, 11-, 13-, 17-, 19- and 23-limits significantly better than pure-octaves 28edo.

; 28edo

* Step size: 42.8571{{c}}, octave size: 1200.00{{c}}

Approximates all no-3s harmonics up to 7 within ''16.9''{{c}}.

Approximates all no-3s harmonics up to 11 & up to 13 within ''16.9''{{c}}.

Approximates all no-3s harmonics up to 17 & up to 19 within ''19.2''{{c}}.

{{Harmonics in equal|28|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 28edo}}

{{Harmonics in equal|28|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 28edo (continued)}}

; [[ed9|89ed9]]

* Step size: 42.7406{{c}}, octave size: 1196.74{{c}}

Approximates all no-3s harmonics up to 7 within ''8.2''{{c}}.

Approximates all no-3s harmonics up to 11 & up to 13 within ''11.4''{{c}}.

Approximates all no-3s harmonics up to 17 & up to 19 within ''11.4''{{c}}.

{{Harmonics in equal|89|9|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 89ed9}}

{{Harmonics in equal|89|9|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 89ed9 (continued)}}

== Scales ==

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 <references/>
+== Octave stretch and compression ==
+'''[[ed9|89ed9]]''', a slightly [[octave shrinking|compressed]] tuning of 28edo, makes 28edo potentially useable as a [[dual-fifth]] tuning. It shares the error equally between 28edo's two perfect fifths.
+So, where pure-octaves 28edo has a flat fifth with 16.2{{c}} error and a sharp fifth with 28.6{{c}} error, 89ed9 instead has a flat fifth with 21.4{{c}} error and a sharp fifth with 21.4{{c}} error.
+.4{{c}} error is slightly worse than [[5edo]]'s best fifth but slightly better than [[23edo]]'s.
+Some might consider this fifth too bad to use, in which case 89ed9 would be a downgrade from pure-octaves 28edo.
+Others might consider it just useable enough, in which case 89ed9 would be an upgrade because it has one more useable interval and one less [[wolf interval]].
+ed9 approximates the no-3s 7-, 11-, 13-, 17-, 19- and 23-limits significantly better than pure-octaves 28edo.
+; 28edo
+* Step size: 42.8571{{c}}, octave size: 1200.00{{c}}
+Approximates all no-3s harmonics up to 7 within ''16.9''{{c}}.
+Approximates all no-3s harmonics up to 11 & up to 13 within ''16.9''{{c}}.
+Approximates all no-3s harmonics up to 17 & up to 19 within ''19.2''{{c}}.
+{{Harmonics in equal|28|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 28edo}}
+{{Harmonics in equal|28|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 28edo (continued)}}
+; [[ed9|89ed9]]
+* Step size: 42.7406{{c}}, octave size: 1196.74{{c}}
+Approximates all no-3s harmonics up to 7 within ''8.2''{{c}}.
+Approximates all no-3s harmonics up to 11 & up to 13 within ''11.4''{{c}}.
+Approximates all no-3s harmonics up to 17 & up to 19 within ''11.4''{{c}}.
+{{Harmonics in equal|89|9|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 89ed9}}
+{{Harmonics in equal|89|9|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 89ed9 (continued)}}
 == Scales ==