100/99: Difference between revisions

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[[de:100/99]]

{{Infobox Interval

| Name = Ptolemy's comma~~, ptolemisma~~

| Name = ptolemisma, Ptolemy's comma

| Color name = 1uyy1, luyoyo comma

| Color name = 1uyy1, luyoyo 1sn,<br>Luyoyo comma

| Comma = yes

}}

In [[just intonation]], '''100/99''', the '''~~Ptolemy's comma~~''' or '''~~ptolemisma~~''', is a [[superparticular]] ~~interval~~ 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¢. 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]]. 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.

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 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 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 ==

* [[Ptolemismic triad]]

* [[Rank-4 temperament #Ptolemismic (100/99)]]

* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out

* [[Small comma]]

* [[List of superparticular intervals]]

* [[Gallery of just intervals]]

* [[:File: Demo 100 99 tempered.pdf]]

[[Category:Ptolemismic]]

[[Category:Commas named after polymaths]]

@@ Line 1: / Line 1: @@
 [[de:100/99]]
 {{Infobox Interval
-| Name = Ptolemy's comma, ptolemisma
+| Name = ptolemisma, Ptolemy's comma
-| Color name = 1uyy1, luyoyo comma
+| Color name = 1uyy1, luyoyo 1sn,<br>Luyoyo comma
 | Comma = yes
 }}
-In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval 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¢. 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]]. 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.
+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 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 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 ==
-* [[Ptolemismic triad]]
-* [[Rank-4 temperament #Ptolemismic (100/99)]]
-* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
-* [[Small comma]]
 * [[List of superparticular intervals]]
 * [[Gallery of just intervals]]
+* [[:File: Demo 100 99 tempered.pdf]]
 [[Category:Ptolemismic]]
+[[Category:Commas named after polymaths]]