Didacus: Difference between revisions

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=== Isomorphism with Sirius ===

One of the more peculiar properties of the Didacus temperament is its relationship with the [[3.5.7 subgroup]] temperament [[Sirius]]. Sirius tempers out [[3125/3087]] which is the ~~Don Page comma~~ between ~~7/5 and~~ [[5/3]]~~: it is the difference between 5/3~~ stacked twice and 7/5 stacked thrice, so that 5/3 spans three generators (representing (5/3)/(7/5) = [[25/21]]) and 7/5 spans two. Therefore [[7/3]], the [[tritave]]-reduced harmonic 7, is split into 5 equal parts, 3 of which represent 5/3, the tritave-reduced harmonic 5, and we can see that the 3:5:7 chord in Sirius and 4:5:7 chord in Didacus are isomorphic to each other's complement.

One of the more peculiar properties of the Didacus temperament is its relationship with the [[3.5.7 subgroup]] temperament [[Sirius]]. Sirius tempers out [[3125/3087]] which is the difference between [[5/3]] stacked twice and 7/5 stacked thrice, so that 5/3 spans three generators (representing (5/3)/(7/5) = [[25/21]]) and 7/5 spans two. Therefore [[7/3]], the [[tritave]]-reduced harmonic 7, is split into 5 equal parts, 3 of which represent 5/3, the tritave-reduced harmonic 5, and we can see that the 3:5:7 chord in Sirius and 4:5:7 chord in Didacus are isomorphic to each other's complement.

Even more interestingly, Sirius also has [[6L 1s (3/1-equivalent)|6L 1s]] and [[6L 7s (3/1-equivalent)|6L 7s]] MOS scales, which have the same shape as the 7- and 13-note MOSes of Didacus, and a descendant 19-note MOS scale. While the cardinalities of scales diverge between the temperaments after 19 notes, a quite analogous hexatonic picture to the previous section can be constructed for Sirius, and in sharing this structure despite the massive stretch between octaves and tritaves, Didacus and Sirius provide a unique avenue for transferring consonant octave-repeating no-threes harmony into consonant tritave-repeating no-twos harmony.

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 === Isomorphism with Sirius ===
-One of the more peculiar properties of the Didacus temperament is its relationship with the [[3.5.7 subgroup]] temperament [[Sirius]]. Sirius tempers out [[3125/3087]] which is the Don Page comma between 7/5 and [[5/3]]: it is the difference between 5/3 stacked twice and 7/5 stacked thrice, so that 5/3 spans three generators (representing (5/3)/(7/5) = [[25/21]]) and 7/5 spans two. Therefore [[7/3]], the [[tritave]]-reduced harmonic 7, is split into 5 equal parts, 3 of which represent 5/3, the tritave-reduced harmonic 5, and we can see that the 3:5:7 chord in Sirius and 4:5:7 chord in Didacus are isomorphic to each other's complement.
+One of the more peculiar properties of the Didacus temperament is its relationship with the [[3.5.7 subgroup]] temperament [[Sirius]]. Sirius tempers out [[3125/3087]] which is the difference between [[5/3]] stacked twice and 7/5 stacked thrice, so that 5/3 spans three generators (representing (5/3)/(7/5) = [[25/21]]) and 7/5 spans two. Therefore [[7/3]], the [[tritave]]-reduced harmonic 7, is split into 5 equal parts, 3 of which represent 5/3, the tritave-reduced harmonic 5, and we can see that the 3:5:7 chord in Sirius and 4:5:7 chord in Didacus are isomorphic to each other's complement.
 Even more interestingly, Sirius also has [[6L 1s (3/1-equivalent)|6L 1s]] and [[6L 7s (3/1-equivalent)|6L 7s]] MOS scales, which have the same shape as the 7- and 13-note MOSes of Didacus, and a descendant 19-note MOS scale. While the cardinalities of scales diverge between the temperaments after 19 notes, a quite analogous hexatonic picture to the previous section can be constructed for Sirius, and in sharing this structure despite the massive stretch between octaves and tritaves, Didacus and Sirius provide a unique avenue for transferring consonant octave-repeating no-threes harmony into consonant tritave-repeating no-twos harmony.