37edo: Difference between revisions

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=== Miscellaneous properties ===

37edo has the sharpest fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. The sharpest mapping of 7 where 9/8 is no greater than 8/7 is 30\37, and the sharpest possible mapping of harmonic 15 where diamond monotone is achieveable is 34\37, where 15/14 is equated with 14/13~13/12. Here 5/4 is mapped to 12\37, and 10/9 is mapped to 5\37. Equating both 11/10 and 12/11 with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to 3/2. If the fifth was any sharper, then harmonics 7 and 15 would have to be flatter. Then, 5 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is [[19edo#Miscellaneous properties|11\19]].

37edo has the sharpest fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. The sharpest mapping of 7 where 9/8 is no greater than 8/7 is 30\37, and the sharpest possible mapping of harmonic 15 where diamond monotone is achieveable is 34\37, where 15/14 is equated with 14/13~13/12. Here 5/4 is mapped to 12\37, and 10/9 is mapped to 5\37. Equating both 11/10 and 12/11 with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to 3/2. If the fifth was any sharper, then harmonics 7 and 15 would have to be flatter. Then, 5 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. 37edo is, in fact, diamond monotone in the 15-odd-limit (see [[Monotonicity limits of small EDOs]]). Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is [[19edo#Miscellaneous properties|11\19]].

== Intervals ==

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 === Miscellaneous properties ===
-edo has the sharpest fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. The sharpest mapping of 7 where 9/8 is no greater than 8/7 is 30\37, and the sharpest possible mapping of harmonic 15 where diamond monotone is achieveable is 34\37, where 15/14 is equated with 14/13~13/12. Here 5/4 is mapped to 12\37, and 10/9 is mapped to 5\37. Equating both 11/10 and 12/11 with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to 3/2. If the fifth was any sharper, then harmonics 7 and 15 would have to be flatter. Then, 5 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is [[19edo#Miscellaneous properties|11\19]].
+edo has the sharpest fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. The sharpest mapping of 7 where 9/8 is no greater than 8/7 is 30\37, and the sharpest possible mapping of harmonic 15 where diamond monotone is achieveable is 34\37, where 15/14 is equated with 14/13~13/12. Here 5/4 is mapped to 12\37, and 10/9 is mapped to 5\37. Equating both 11/10 and 12/11 with 10/9 makes the mappings for 9/8, 10/9, 11/10, and 12/11 add up to 3/2. If the fifth was any sharper, then harmonics 7 and 15 would have to be flatter. Then, 5 would have to be flatter, and therefore 10/9 as well, and at least one of 11/10 and 12/11 would have to be mapped wider than 10/9 for 9/8, 10/9, 11/10, and 12/11 to add up to 3/2. 37edo is, in fact, diamond monotone in the 15-odd-limit (see [[Monotonicity limits of small EDOs]]). Therefore, 22\37 is the sharpest fifth where 15-odd-limit diamond monotone is possible. The flattest fifth where 15-odd-limit diamond monotone is possible is [[19edo#Miscellaneous properties|11\19]].
 == Intervals ==