81/80: Difference between revisions

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The '''syntonic comma'''- also known as the '''Didymus comma''', the '''meantone comma''' or the '''Ptolemaic comma'''- with a frequency ratio '''81/80''', is helpful for comparing [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus, for those who have no interest or desire to utilize such wolves in composition, tempering out 81/80 is desirable. This gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone|meantone temperament]], hence the name meantone comma.

81/80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]]. Like [[16/15]], [[625/624]], [[2401/2400]] and [[4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because n<sup>2</sup>/(n<sup>2</sup>-1) = n/(n-1) ÷ (n+1)/n (which is to say 81/80 is a [[square superparticular]]). 81/80 is in fact the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. That the numerator is a fourth power entails that the wider of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.

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 [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|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, 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.

[[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.

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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.

== 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 (which in higher-accuracy contexts causes significant damage to the [[5-limit]]), one can easily use it in melodies as either an [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|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 ==

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 The '''syntonic comma'''- also known as the '''Didymus comma''', the '''meantone comma''' or the '''Ptolemaic comma'''- with a frequency ratio '''81/80''', is helpful for comparing [[3-limit]] and [[5-limit]] [[just intonation]]. Adding or subtracting this comma to/from any 3-limit [[ratio]] with an [[odd limit]] of 27 or higher creates a 5-limit ratio with a much lower odd-limit. Thus potentially dissonant 3-limit harmonies can often be sweetened via a commatic adjustment. However, adding/subtracting this comma to/from any 3-limit ratio of odd limit 3 or less (the 4th, 5th or 8ve), creates a wolf interval of odd limit 27 or higher. Any attempt to tune a fixed-pitch instrument (e.g. guitar or piano) to 5-limit just intonation will create such wolves, thus, for those who have no interest or desire to utilize such wolves in composition, tempering out 81/80 is desirable. This gives a tuning for the [[Tone|whole tone]] which is intermediate between 10/9 and 9/8, and leads to [[Meantone|meantone temperament]], hence the name meantone comma.
 /80 is the smallest [[superparticular]] interval which belongs to the [[5-limit]]. Like [[16/15]], [[625/624]], [[2401/2400]] and [[4096/4095]] it has a fourth power as a numerator. Fourth powers are squares, and any superparticular comma with a square numerator is the ratio between two wider successive superparticular intervals, because n<sup>2</sup>/(n<sup>2</sup>-1) = n/(n-1) ÷ (n+1)/n (which is to say 81/80 is a [[square superparticular]]). 81/80 is in fact the difference between [[10/9]] and [[9/8]], the product of which is the just major third, [[5/4]]. That the numerator is a fourth power entails that the wider of these two intervals itself has a square numerator; 9/8 is the interval between the successive superparticulars 4/3 and 3/2.
-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 [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|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, 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.
 [[Monroe Golden]]'s ''Incongruity'' uses just-intonation chord progressions that exploit this comma<ref>[http://untwelve.org/interviews/golden UnTwelve's interview to Monroe Golden]</ref>.
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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.
+== 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 (which in higher-accuracy contexts causes significant damage to the [[5-limit]]), one can easily use it in melodies as either an [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|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 ==