896/891: Difference between revisions

| Ratio = 896/891

| Name = pentacircle comma, undecimal semicomma

| Color name = s1uz2, Saluzo comma

The '''pentacircle comma''' or '''undecimal semicomma''', '''896/891''' (9.68796 [[cent]]s), is an [[11-limit]] [[comma]] with monzo {{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]].

 

The pentacircle comma can be factored into two [[13-limit]] [[superparticular]] commas, [[364/363]] (which is {{monzo| 2 -1 0 1 -2 1 }}) and [[352/351]] (which is {{monzo| 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 complement]]s; 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]].

 

In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{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{{nbhsp}}{{sagittal | )!( }}. The upward version is called '''7/11k''' or '''11/7k up''' and is represented by {{sagittal| )|( }}.

 

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: {{EDOs|17, 22, 24, 29, 41, 46, 58, 63, 65, 80, 87, 92, 104, 109, 121}}, etc. The [[2.3.7.11.13 subgroup|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]].