Würschmidt comma: Difference between revisions
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The '''Würschmidt comma''' | The '''Würschmidt comma''' ({{monzo| 17 1 -8 }} = '''393216/390625''') is a [[5-limit]] [[comma]] of 11.4 cents. | ||
It is the amount by which eight major thirds fall short of a perfect fifth, octave-reduced: ((5/4)<sup>8</sup> × 393216/390625) / 4 = 3/2. | It is the amount by which eight major thirds fall short of a perfect fifth, octave-reduced: ((5/4)<sup>8</sup> × 393216/390625) / 4 = 3/2. | ||
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Therefore, it is also the amount by which seven major thirds fall short of 24/5 (i.e. 6/5 plus two octaves). In other words, ((5/4)<sup>7</sup> × 393216/390625) / 4 = 6/5. | Therefore, it is also the amount by which seven major thirds fall short of 24/5 (i.e. 6/5 plus two octaves). In other words, ((5/4)<sup>7</sup> × 393216/390625) / 4 = 6/5. | ||
Tempering it out leads to [[ | Tempering it out leads to the [[würschmidt family]] of temperaments. As in [[meantone]], it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. | ||
== See also == | == See also == | ||
Revision as of 12:29, 20 December 2020
| Interval information |
The Würschmidt comma ([17 1 -8⟩ = 393216/390625) is a 5-limit comma of 11.4 cents.
It is the amount by which eight major thirds fall short of a perfect fifth, octave-reduced: ((5/4)8 × 393216/390625) / 4 = 3/2.
Therefore, it is also the amount by which seven major thirds fall short of 24/5 (i.e. 6/5 plus two octaves). In other words, ((5/4)7 × 393216/390625) / 4 = 6/5.
Tempering it out leads to the würschmidt family of temperaments. As in meantone, it implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat.