13/11: Difference between revisions

| Name = tridecimal minor third, <br> Neo-Gothic minor third

In [[13-limit]] [[just intonation]], '''13/11''' is '''the tridecimal minor third''' (or '''[[Neo-Gothic]] minor third'''), measuring about 289.2¢. It is the difference between the 11th and 13th [[harmonic]]s. The (octave-reduced) 11th harmonic ([[11/8]], about 551.3¢) and 13th harmonic ([[13/8|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 can even function as such in a 13-limit 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|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|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.)

 

* [[22/13]] - its [[octave complement]]

* [[33/26]] - its [[fifth complement]]

* [[gentle chords]]

* [[:File:Ji-13-11-csound-foscil-220hz.mp3]] - another sound example

* [http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant] (earliest description of 13:11 as the "Neo-Gothic" minor third)