Pentacircle comma

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The Pentacircle Comma or undecimal semicomma, 896/891, 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.

Pentacircle is an 11-limit comma with monzo | 7 -4 0 1 -1 >. It is tempered out by the following EDOs, using their patent vals: 5, 12, 17, 19, 22, 24, 27, 29, 36, 41, 44, 46, 51, 58, 60, 63, 65, 68, 80, 82, 85, 87, 90, 92, 104, 109, 114, 121, 123, 126, 128, 131, 133, 136, 138, 145, 150, 155, 167, 169, 172, 174, 177, 184, 189, 191, 196, 201, 208, 213, 218, 230, 232, 237, 242, 254, 259, 271, 278, 283, and 295.

Pentacircle 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 difference between (14/11 * 13/11) and 3/2. If both Pentacircle and 364/363 are tempered out (which implies that 352/351 is also tempered out, of course), a 14/11 major third and a 13/11 minor third together 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 minthma: See the article on minthmic chords.

Finally, see the article on Pentacircle Chords.

Example scales: Cantonpenta is a scale that tempers out the pentacircle comma. Also, the MOSes with an octave period and 17\29 as a generator temper out the Pentacircle comma.