81/80: Difference between revisions

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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. 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 should be so bold as 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 [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|acciaccatura]], or a quick passing tone. In addition, 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 ~~the~~ wolf intervals~~, particularly~~ [[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]].

If one should be so bold as 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 [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|acciaccatura]], or a quick passing tone. In addition, 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]].

[[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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 /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. 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 should be so bold as 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 [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|acciaccatura]], or a quick passing tone.  In addition, 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 the wolf intervals, particularly [[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]].
+If one should be so bold as 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 [[Wikipedia:Appoggiatura|appoggitura]], an [[Wikipedia:Acciaccatura|acciaccatura]], or a quick passing tone.  In addition, 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]].
 [[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>.