100/99

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¢.

Tempering

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 6/5s 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.

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.

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