65/64: Difference between revisions

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In [[~~Just_intonation|Just Intonation~~]], 65/64, the wilsorma, is a [[~~superparticular|~~superparticular]] interval of around 26.8¢, nearly a quarter of a semitone or eighth of a tone~~. It belongs to the 13-prime limit, which means that the highest prime in the ratio is 13~~. 65 is 5 times 13, which means that 65/64 can be treated as a harmonic ~~13th~~ above a harmonic ~~5th~~ or vice versa. It is the difference between [[~~5/4|~~5/4]] and [[~~16/13|~~16/13]]; [[~~8/5|~~8/5]] and [[~~13/8|~~13/8]]; [[~~13/12|~~13/12]] and [[~~16/15|~~16/15]]; [[~~15/8|~~15/8]] and [[~~24/13|~~24/13]], [[~~13/10|~~13/10]] and [[~~32/25|~~32/25]]; [[~~20/13|~~20/13]] and [[~~25/16|~~25/16]], and of course, infinitely many other pairs of just intervals.

{{Infobox Interval

| Name = wilsorma

| Color name = 3oy1, thoyo 1sn,<br>Thoyo comma

| Comma = yes

}}

In [[13-limit]] [[just intonation]], '''65/64''', the '''wilsorma''', is a [[superparticular]] interval of around 26.8{{cent}}, nearly a quarter of a semitone or eighth of a tone. 65 is 5 times 13, which means that 65/64 can be treated as a [[13/1|13th harmonic]] above a [[5/1|5th harmonic]] or vice versa. It is the difference between [[5/4]] and [[16/13]]; [[8/5]] and [[13/8]]; [[13/12]] and [[16/15]]; [[15/8]] and [[24/13]], [[13/10]] and [[32/25]]; [[20/13]] and [[25/16]], and of course, infinitely many other pairs of just intervals. It differs from the septimal comma [[64/63]] by [[4096/4095]] and from the syntonic comma [[81/80]] by [[325/324]].

See: [[~~Gallery_of_Just_Intervals|~~Gallery of ~~Just Intervals~~]]

== Notation ==

This interval is the 13th-partial chroma (13-limit formal comma) in [[Ben Johnston's notation]], denoted simply with the number "13", while its reciprocal is denoted as "{{invert|13}}" (a turned "13"). If the base note is C, then [[13/8]] is represented by C–A♭13.

== Temperaments ==

[[Tempering out]] this comma in the 2.5.13 subgroup leads to the rank-2 '''wilsormatic''' temperament, which has a generator tuned to about 370–375 cents that represents both 5/4 and 16/13, or in the 13-limit, the rank-5 '''wilsormic''' temperament. In either case, it turns 5/4 and 13/8 into [[octave complement]]s of one another. This is particularly useful in many [[13-limit]] [[magic family]] extensions, as it means they are very simply mapped to plus and minus one generator.

However, it is natural to equate [[14/13]] and [[15/14]] with 13/12[[~]]16/15, thus leading to the [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament [[negri]], which splits the perfect fourth [[4/3]] into four equal parts, each being one step of the 12::16 segment of the [[harmonic series]].

== See also ==

* [[Gallery of just intervals]]

[[Category:Wilsormic]]

[[Category:Commas with unknown etymology]]

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-In [[Just_intonation|Just Intonation]], 65/64, the wilsorma, is a [[superparticular|superparticular]] interval of around 26.8¢, nearly a quarter of a semitone or eighth of a tone. It belongs to the 13-prime limit, which means that the highest prime in the ratio is 13. 65 is 5 times 13, which means that 65/64 can be treated as a harmonic 13th above a harmonic 5th or vice versa. It is the difference between [[5/4|5/4]] and [[16/13|16/13]]; [[8/5|8/5]] and [[13/8|13/8]]; [[13/12|13/12]] and [[16/15|16/15]]; [[15/8|15/8]] and [[24/13|24/13]], [[13/10|13/10]] and [[32/25|32/25]]; [[20/13|20/13]] and [[25/16|25/16]], and of course, infinitely many other pairs of just intervals.
+{{Infobox Interval
+| Name = wilsorma
+| Color name = 3oy1, thoyo 1sn,<br>Thoyo comma
+| Comma = yes
+}}
+In [[13-limit]] [[just intonation]], '''65/64''', the '''wilsorma''', is a [[superparticular]] interval of around 26.8{{cent}}, nearly a quarter of a semitone or eighth of a tone. 65 is 5 times 13, which means that 65/64 can be treated as a [[13/1|13th harmonic]] above a [[5/1|5th harmonic]] or vice versa. It is the difference between [[5/4]] and [[16/13]]; [[8/5]] and [[13/8]]; [[13/12]] and [[16/15]]; [[15/8]] and [[24/13]], [[13/10]] and [[32/25]]; [[20/13]] and [[25/16]], and of course, infinitely many other pairs of just intervals. It differs from the septimal comma [[64/63]] by [[4096/4095]] and from the syntonic comma [[81/80]] by [[325/324]].
-See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]
+== Notation ==
+This interval is the 13th-partial chroma (13-limit formal comma) in [[Ben Johnston's notation]], denoted simply with the number "13", while its reciprocal is denoted as "{{invert|13}}" (a turned "13"). If the base note is C, then [[13/8]] is represented by C–A♭13.
+== Temperaments ==
+[[Tempering out]] this comma in the 2.5.13 subgroup leads to the rank-2 '''wilsormatic''' temperament, which has a generator tuned to about 370–375 cents that represents both 5/4 and 16/13, or in the 13-limit, the rank-5 '''wilsormic''' temperament. In either case, it turns 5/4 and 13/8 into [[octave complement]]s of one another. This is particularly useful in many [[13-limit]] [[magic family]] extensions, as it means they are very simply mapped to plus and minus one generator.
+However, it is natural to equate [[14/13]] and [[15/14]] with 13/12[[~]]16/15, thus leading to the [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament [[negri]], which splits the perfect fourth [[4/3]] into four equal parts, each being one step of the 12::16 segment of the [[harmonic series]].
+== See also ==
+* [[Gallery of just intervals]]
+[[Category:Wilsormic]]
+[[Category:Commas with unknown etymology]] <!-- Is it named after Erv Wilson? -->

65/64: Difference between revisions

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