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.

The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''' (thus systematically the '''syntonisma''' and '''ptolemaisma''', not to be confused with the [[ptolemisma]]) 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 {{nowrap|''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.

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 {{Wikipedia|Syntonic comma}}
-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.
+The '''syntonic comma''', also known as the '''Didymus' comma''', the '''meantone comma''' or the '''Ptolemaic comma''' (thus systematically the '''syntonisma''' and '''ptolemaisma''', not to be confused with the [[ptolemisma]]) 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 {{nowrap|''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.