896/891

The pentacircle comma or undecimal semicomma, 896/891 (9.68796 cents), is an 11-limit comma with monzo [7 -4 0 1 -1⟩. It is similar to the Didymus or syntonic comma, 81/80, in that it makes a stack of four fifths equal a major third (octave equivalent). In the case of 81/80, however, the major third is 5/4, while with the pentacircle comma, the major third is 14/11. Tempering it out also equates 33/32 and 28/27.

Interval information

Ratio	896/891
Factorization	27 × 3-4 × 7 × 11-1
Monzo	[7 -4 0 1 -1⟩
Size in cents	9.687961¢
Names	pentacircle comma, undecimal semicomma
Color name	s1uz2, Saluzo comma
FJS name	[math]\displaystyle{ \text{m2}^{7}_{11} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	19.6066
Weil norm (log2 max(n, d))	19.6147
Wilson norm (sopfr(nd))	44
Comma size	small
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The pentacircle comma can be factored into two 13-limit superparticular commas, 364/363 (which is [2 -1 0 1 -2 1⟩) and 352/351 (which is [5 -3 0 0 1 -1⟩).

364/363 is the minor minthma or gentle comma, which is the difference between a stack of 14/11 and 13/11 (182/121) and 3/2. If 364/363 is tempered out, a 14/11 major third and a 13/11 minor third become fifth complements; that is, they add up to a perfect fifth. (This isn't necessary for traditional minor and major thirds, because 5/4 × 6/5 = 3/2.)

352/351 is the major minthma. Tempering it out means 13/11 is equated with the pythagorean minor third, 32/27. Tempering out 352/351 also makes 11/9 and 16/13 fifth complements of each other.

Each of these commas has their own essentially tempered chords; see the articles on pentacircle chords, minor minthmic chords, and major minthmic chords.

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented by the sagittal  and is called the 11/7 kleisma, or 11/7k for short, because the simplest interval it notates is 11/7, as for example in A-F⁠ ⁠. The upward version is called 7/11k or 11/7k up and is represented by .

Temperaments

Tempering out 896/891 in the 11-limit leads to the rank-4 pentacircle temperament. This temperament naturally extends to the 13-limit via 352/351 and 364/363. Tridecimal pentacircle is supported by the following edos, using their patent vals: 17, 22, 24, 29, 41, 46, 58, 63, 65, 80, 87, 92, 104, 109, 121, etc. The 2.3.7.11.13-subgroup version of this temperament is known as parapyth, and the 2.3.11/7.13/7-subgroup version of it is known as pepperoni.

See also

• Pentacircle clan, the clan of rank-3 temperaments where it is tempered out
• Small comma
• Cantonpenta, a scale that tempers out the pentacircle comma