100/99: Difference between revisions

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In [[just intonation]], '''100/99''', the '''ptolemisma''' or '''Ptolemy's comma''', is a [[small comma|small]] [[superparticular ratio|superparticular]] [[comma]] of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢.

In [[just intonation]], '''100/99''', the '''ptolemisma''' or '''Ptolemy's comma''', is a [[small comma|small]] [[superparticular ratio|superparticular]] [[comma]] of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].

== Temperaments ==

Tempering out this interval in the 11-limit results in the rank-4 ptolemismic temperament (→ [[Rank-4 temperament #Ptolemismic (100/99)]]). Equal ~~temperaments~~ in which 100/99 is tempered out include [[22edo]], [[41edo]] and others. In such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]'s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5~~. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]]~~.

[[Tempering out]] this interval in the 11-limit results in the rank-4 ptolemismic temperament (→ [[Rank-4 temperament #Ptolemismic (100/99)]]). [[Equal temperament]]s in which 100/99 is tempered out include [[22edo]], [[41edo]] and others. In such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]'s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

See [[Ptolemismic clan]] for the clan of rank-3 temperaments where it is tempered out.

== Approximation ==

If the interval is not tempered and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/~~99s~~ and the octave, is only 0.56 cents wide. Nice.

If the interval is not tempered out and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99's and the octave, is only 0.56 cents wide. Nice.

== See also ==

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-In [[just intonation]], '''100/99''', the '''ptolemisma''' or '''Ptolemy's comma''', is a [[small comma|small]] [[superparticular ratio|superparticular]] [[comma]] of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢.
+In [[just intonation]], '''100/99''', the '''ptolemisma''' or '''Ptolemy's comma''', is a [[small comma|small]] [[superparticular ratio|superparticular]] [[comma]] of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].
 == Temperaments ==
-Tempering out this interval in the 11-limit results in the rank-4 ptolemismic temperament (→ [[Rank-4 temperament #Ptolemismic (100/99)]]). Equal temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others. In such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]'s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].
+[[Tempering out]] this interval in the 11-limit results in the rank-4 ptolemismic temperament (→ [[Rank-4 temperament #Ptolemismic (100/99)]]). [[Equal temperament]]s in which 100/99 is tempered out include [[22edo]], [[41edo]] and others. In such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]'s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.
 See [[Ptolemismic clan]] for the clan of rank-3 temperaments where it is tempered out.
 == Approximation ==
-If the interval is not tempered and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99s and the octave, is only 0.56 cents wide. Nice.
+If the interval is not tempered out and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99's and the octave, is only 0.56 cents wide. Nice.
 == See also ==