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 {{nowrap|''n''2/(''n''2 − 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.

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''2/(''n''2 − 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.

== Comma pumps ==

The familiar vi-ii-V-I progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.

== Use in recorded music ==

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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 {{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.
+/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.
+== Comma pumps ==
+The familiar vi-ii-V-I progression requires that 81/80 be tempered out in order for the root in the vi chord to be the same as the root in the final I chord. If 81/80 is not tempered out, the new root will be 81/80 lower than the original root.
+{{Todo|inline=1|add sound example}}
 == Use in recorded music ==