100/99: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = Ptolemy's comma | | Name = ptolemisma, Ptolemy's comma | ||
| Color name = 1uyy1, luyoyo 1sn,<br>Luyoyo comma | | Color name = 1uyy1, luyoyo 1sn,<br>Luyoyo comma | ||
| Comma = yes | | Comma = yes | ||
}} | }} | ||
In [[just intonation]], '''100/99''', the ''' | 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 == | == 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/ | 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 == | == See also == | ||
* [[List of superparticular intervals]] | * [[List of superparticular intervals]] | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[:File: Demo 100 99 tempered.pdf]] | |||
[[Category:Ptolemismic]] | [[Category:Ptolemismic]] | ||
[[Category:Commas named after polymaths]] | [[Category:Commas named after polymaths]] | ||
Latest revision as of 08:39, 5 November 2025
| Interval information |
Ptolemy's comma
Luyoyo comma
reduced
In just intonation, 100/99, the ptolemisma or Ptolemy's comma, is a small 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.
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.