847/845: Difference between revisions
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'''847/845''', the '''cuthbert comma''', is a [[13-limit]] (also 5.7.11.13 | '''847/845''', the '''cuthbert comma''', is a [[13-limit]] (also [[5.7.11.13 subgroup]]) [[comma]] measuring about 4.09{{cent}}. It is the difference between [[7/5]] and a stack of two [[13/11]]'s. It is also the difference between [[125/121]] and [[175/169]]. | ||
In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, [[637/625]] is about as simple but much larger, and [[2941225/2924207]] is significantly more complex yet still twice as large. | |||
In terms of full 13-limit commas, it is the difference between the following superparticular pairs: | In terms of full 13-limit commas, it is the difference between the following superparticular pairs: | ||
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== Temperaments == | == Temperaments == | ||
[[Tempering out]] this comma in the 13-limit results in the rank-5 cuthbert temperament and enables the [[cuthbert chords]]. | [[Tempering out]] this comma in the 13-limit results in the rank-5 cuthbert temperament and enables the [[cuthbert chords]]. | ||
Tempering it out in the 5.7.11.13 subgroup leads to an extremely efficient rank-3 temperament. | |||
== See also == | == See also == | ||
Revision as of 21:57, 8 April 2026
| Interval information |
847/845, the cuthbert comma, is a 13-limit (also 5.7.11.13 subgroup) comma measuring about 4.09 ¢. It is the difference between 7/5 and a stack of two 13/11's. It is also the difference between 125/121 and 175/169.
In the 5.7.11.13 subgroup, it is the simplest comma of its size (and the smallest of its complexity) by an extremely large margin. For comparison, 637/625 is about as simple but much larger, and 2941225/2924207 is significantly more complex yet still twice as large.
In terms of full 13-limit commas, it is the difference between the following superparticular pairs:
Meanwhile, it can be factorized as 1001/1000 × 2200/2197 or 441/440 × 10648/10647.
Temperaments
Tempering out this comma in the 13-limit results in the rank-5 cuthbert temperament and enables the cuthbert chords.
Tempering it out in the 5.7.11.13 subgroup leads to an extremely efficient rank-3 temperament.