11/9: Difference between revisions

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In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]. As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. As it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]]. It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]. As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. Since it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]]. It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including [[7edo]], [[10edo]], [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== Approximation ==

== See also ==

* [[7edo]]

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-In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]].  As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. As it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.
+In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]].  As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. Since it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.
 In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including [[7edo]], [[10edo]], [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].
+== Approximation ==
+{{Interval edo approximation|11/9}}
 == See also ==
 * [[7edo]]