tag:blogger.com,1999:blog-7003066908957086041.comments2023-03-15T06:46:09.220+00:00Clamor Vincit OmniaLemoUtanhttp://www.blogger.com/profile/11110031436968511583noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-7003066908957086041.post-88220345864582333902018-09-10T12:06:15.206+01:002018-09-10T12:06:15.206+01:00One may additionally describe a PC set with a corr...One may additionally describe a PC set with a corresponding circulant matrix. Note that the 'fifthiest' hexatonic scale (221223) has - like most hexatonic scales - a non-invertable circulant matrix (with determinant of zero). In fact the only symmetric 6-sets with non-zero determinants thereof are the pair 122133, 121323 (interval vector <2,2,4,3,2,2>).<br /><br />Of the 12 pairs of dual=inverse asymmetrics, 4 have non-zero circulant determinants - 131223/221313, 122223/222213, 112134/312114 and 211224/221124.<br /><br />Of the 8 quads of dual≠inverse asymmetrics, 3 have non-zero circulant determinants - 121224/221214:221133/112233, 113124/213114:132114/112314 and 121125/211215:211134 311124.<br /><br />The six note oriental scale (131142) is one of those final 3 in that 4-homometry group, but the six note blues scale (321132) - also in that group - is not and has an uninvertible circulant.<br /><br />In total, then, only 8 out of the 35 hexatonic co-homometries have invertible circulants.LemoUtanhttps://www.blogger.com/profile/11110031436968511583noreply@blogger.com