81/80: Difference between revisions

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=== 31edo as splitting the fifth into two, three and nine ===

[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]~~, so that it is very strong in the 2.5.7 subgroup (leading to [[birds]] if you want a better fifth)~~. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is ~~uniquely meantone + valentine. Valentine is a natural [[11-limit]] temperament~~ that ~~tempers [[121/120]] so for this reason might be natural to combine with meantone~~. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle, which interestingly, though a rank 2 temperament, only has [[31edo]] as a [[patent val]] tuning.

[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle, which interestingly, though a rank 2 temperament, only has [[31edo]] as a [[patent val]] tuning.

== As an interval ==

If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w|appoggitura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave.

If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out ~~(which in higher-accuracy contexts causes significant damage to the [[5-limit]])~~, one can easily use it in melodies as either an {{w|appoggitura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions. Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave.

== Relations to other superparticular ratios ==

@@ Line 34: / Line 34: @@
 === 31edo as splitting the fifth into two, three and nine ===
-[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]], so that it is very strong in the 2.5.7 subgroup (leading to [[birds]] if you want a better fifth). A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is uniquely meantone + valentine. Valentine is a natural [[11-limit]] temperament that tempers [[121/120]] so for this reason might be natural to combine with meantone. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle, which interestingly, though a rank 2 temperament, only has [[31edo]] as a [[patent val]] tuning.
+[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle, which interestingly, though a rank 2 temperament, only has [[31edo]] as a [[patent val]] tuning.
 == As an interval ==
+If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out, one can easily use it in melodies as either an {{w|appoggitura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions.  Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave.
-If one wants to treat the syntonic comma as a musical interval in its own right as opposed to tempering it out (which in higher-accuracy contexts causes significant damage to the [[5-limit]]), one can easily use it in melodies as either an {{w|appoggitura}}, an {{w|acciaccatura}}, or a quick passing tone. It is also very easy to exploit in [[comma pump]] modulations, as among the [[Meantone comma pump examples|known examples]] of this kind of thing are familiar-sounding chord progressions.  Furthermore, not tempering out 81/80 both allows wolf intervals like [[40/27]] and [[27/20]] to be deliberately exploited as dissonances to be resolved, and it also allows one to contrast intervals like 5/4 and [[81/64]]. The [[barium]] temperament exploits the comma by setting it equal to exactly 1/56th of the octave.
 == Relations to other superparticular ratios ==