9801/9800
Ratio | 9801/9800 |
Factorization | 2^{-3} × 3^{4} × 5^{-2} × 7^{-2} × 11^{2} |
Monzo | [-3 4 -2 -2 2⟩ |
Size in cents | 0.17664752¢ |
Name | kalisma |
Color name | 1oorrgg-2, Bilorugu comma |
FJS name | [math]\text{M}{-2}^{11,11}_{5,5,7,7}[/math] |
Special properties | square superparticular, reduced |
Tenney height (log_{2} nd) | 26.5173 |
Weil height (log_{2} max(n, d)) | 26.5174 |
Wilson height (sopfr (nd)) | 64 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~1.19851 bits |
Comma size | unnoticeable |
S-expressions | S99, S33 / S35 |
open this interval in xen-calc |
9801/9800, the kalisma, sometimes described as Gauss' comma, is an unnoticeable 11-limit comma measuring about 0.18 ¢. It is the smallest 11-limit superparticular interval.
It can be described as the difference between 99/98 and 100/99, and between 99/70 and its octave complement, 140/99. It is also the difference between 245/243 and 121/120, and a stack of two 11/7's and 81/80 against 5/2.
It factors into the two smallest 13-limit superparticular commas: 9801/9800 = (10648/10647)(123201/123200).
Temperaments
Tempering out this comma leads to the kalismic temperament, which splits the octave into two equal parts, each representing 99/70~140/99. Tempering it out also means that 10/9 and 11/7 are 1/2-octave apart, as well as are 11/10 and 14/9. Odd-numbered edos cannot temper it out. See Rank-4 temperament #Kalismic (9801/9800) for some technical details. See Kalismic temperaments for a collection of rank-3 temperaments where it is tempered out.
Etymology
This comma was named kalisma by Margo Schulter in 2000 from the Greek root kal- ("beautiful")^{[1]}. Gene Ward Smith, not aware of Margo's work, proposed gaussisma in 2004, reasoning that D. H. Lehmer claimed Carl Friedrich Gauss had mentioned the ratio^{[2]}.