Quarter-comma meantone: Difference between revisions

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<span style="display: block; text-align: right;">Other languages: [[:de:viertelkomma-mitteltönig Deutsch]]</span>

<span style="display: block; text-align: right;">Other languages: [[:de:viertelkomma-mitteltönig|Deutsch]]</span>

Quarter comma meantone is the tuning of [[Meantone_family|meantone temperament]] which makes the fifth the fourth root of 5, or in other words 696.578 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is also the tuning where the major tone is the exact logarithmic average (e.g. cents average) between the greater tone of 9/8 and the lesser tone of 10/9, and hence it can be argued it is the only tuning which is strictly "mean tone". It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit [[Diamonds|diamond]] is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.

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	<span style="display: block; text-align: right;">Other languages: [[:de:viertelkomma-mitteltönig Deutsch]]</span>		<span style="display: block; text-align: right;">Other languages: [[:de:viertelkomma-mitteltönig\|Deutsch]]</span>

	Quarter comma meantone is the tuning of [[Meantone_family\|meantone temperament]] which makes the fifth the fourth root of 5, or in other words 696.578 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is also the tuning where the major tone is the exact logarithmic average (e.g. cents average) between the greater tone of 9/8 and the lesser tone of 10/9, and hence it can be argued it is the only tuning which is strictly "mean tone". It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit [[Diamonds\|diamond]] is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.		Quarter comma meantone is the tuning of [[Meantone_family\|meantone temperament]] which makes the fifth the fourth root of 5, or in other words 696.578 cents. This means the fifth is flattened by 1/4 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 5.377 cents, hence the name quarter-comma or 1/4-comma meantone. It is also the tuning where the major tone is the exact logarithmic average (e.g. cents average) between the greater tone of 9/8 and the lesser tone of 10/9, and hence it can be argued it is the only tuning which is strictly "mean tone". It is the minimax tuning for 5-limit meantone, meaning the maximum error on the 5-limit [[Diamonds\|diamond]] is minimized. It is also the minimax tuning for septimal meantone in the 7- and 9-limit odd limits, and for meanpop, the version of 11-limit meantone which tunes 11/8 to the doubly diminished fifth, C-Gbb. Moreover, historically it was the predominant tuning of Western common-practice music in the latter part of the Renaissance and the early modern (17th century) era.