Pythagorean tuning: Difference between revisions

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{{Wikipedia|Pythagorean tuning}}
{{Wikipedia|Pythagorean tuning}}
The '''Pythagorean tuning''' is the 3-limit version of [[just intonation]]. It is the rank-2 temperament in the 2.3 subgroup that tempers out no commas. In other words, it is a [[trivial temperament]]. All rank-2 temperaments generated by 3/2 and 2/1 in the 5-limit or higher are extensions of pythagorean.
The '''Pythagorean tuning''' is the 3-limit version of [[just intonation]]. It can be considered a [[trivial temperament|trivial]] rank-2 temperament in the 2.3 subgroup, as it tempers out no commas (providing no additional mappings for intervals other than the pure just structure). As such, all rank-2 temperaments generated by 3/2 and 2/1 in the 5-limit or higher (e.g. meantone) are extensions of pythagorean.


The Pythagorean temperament consists of all intervals generated by a just 3/2 and 2/1. Musically, the 2/1 is most often interpreted as an equave, and as such Pythagorean tuning mirrors the structure of the [[chain of fifths]].
The Pythagorean temperament consists of all intervals generated by a just 3/2 and 2/1. Musically, the 2/1 is most often interpreted as an equave, and as such Pythagorean tuning mirrors the structure of the [[chain of fifths]].