13/11: Difference between revisions

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In [[13-limit]] [[just intonation]], '''13/11''' is the '''tridecimal minor third''', '''neo-Gothic minor third''' or '''major minthmic minor third''', measuring about 289.2¢. It is the difference between the [[11/1|11th]] and [[13/1|13th]] [[harmonic]]s. The (octave-reduced) 11th harmonic ([[11/8]], about 551.3¢) and 13th harmonic ([[13/8]], about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third – it is a [[352/351|major minthma (352/351)]] narrower than the [[32/27|Pythagorean minor third (32/27)]]. It can even function as such in a 13-limit [[neogothic|neo-Gothic]] minor triad of 22:26:33, with a [[3/2]] perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant [[16/11]] as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.

13/11 is the classic [[mediant]] between the simpler and more familiar ratios [[6/5]] and [[7/6]], as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = [[20/17]], the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = [[19/16]], the overtone minor third of [[19-limit]] JI, about 297.5¢). (See the diagram below.)

13/11 is the classic [[mediant]] between the simpler and more familiar ratios [[6/5]] and [[7/6]], as it can be given as (6+7)/(5+6). This puts it in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = [[20/17]], the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = [[19/16]], the overtone minor third of [[19-limit]] JI, about 297.5¢). (See the diagram below.)

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 In [[13-limit]] [[just intonation]], '''13/11''' is the '''tridecimal minor third''', '''neo-Gothic minor third''' or '''major minthmic minor third''', measuring about 289.2¢. It is the difference between the [[11/1|11th]] and [[13/1|13th]] [[harmonic]]s. The (octave-reduced) 11th harmonic ([[11/8]], about 551.3¢) and 13th harmonic ([[13/8]], about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third – it is a [[352/351|major minthma (352/351)]] narrower than the [[32/27|Pythagorean minor third (32/27)]]. It can even function as such in a 13-limit [[neogothic|neo-Gothic]] minor triad of 22:26:33, with a [[3/2]] perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant [[16/11]] as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.
-/11 is the classic [[mediant]] between the simpler and more familiar ratios [[6/5]] and [[7/6]], as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = [[20/17]], the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = [[19/16]], the overtone minor third of [[19-limit]] JI, about 297.5¢). (See the diagram below.)
+/11 is the classic [[mediant]] between the simpler and more familiar ratios [[6/5]] and [[7/6]], as it can be given as (6+7)/(5+6). This puts it in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = [[20/17]], the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = [[19/16]], the overtone minor third of [[19-limit]] JI, about 297.5¢). (See the diagram below.)
 {| class="wikitable center-all"