65/64: Difference between revisions
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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 harmonic | 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]]. | ||
== 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 == | == See also == | ||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
[[Category:Commas with unknown etymology]] | [[Category:Wilsormic]] | ||
[[Category:Commas with unknown etymology]] <!-- Is it named after Erv Wilson? --> | |||
Latest revision as of 08:15, 31 October 2026
| Interval information |
Thoyo comma
reduced,
reduced harmonic
In 13-limit just intonation, 65/64, the wilsorma, is a superparticular interval of around 26.8 ¢, 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 13th harmonic above a 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.
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 "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 complements 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 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.