1/2-comma meantone: Difference between revisions

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In '''1/2-comma [[meantone]]~~'''~~ temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

'''1/2-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb val flat 5th is audibly indistinguishable from a closed system of half comma meantone, with an error measured in thousandths of a cent.

In this system, the major thirds are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb [[val]] flat 5th is audibly indistinguishable from a closed system of half-comma meantone, with an error measured in thousandths of a cent.

For a tuning with so flat a fifth, the 7-limit [[extension]] is [[flattone]] instead of septimal meantone, or [[7/6]] and [[8/7]] would swap their sizes.

== Tuning profile ==

[[Projection map]]:

{| class="right-all"

|-

| [⟨ || 1 || 2 || 4 || -1 || ]

|-

| ⟨ || 0 || -1 || -4 || 9 || ]

|-

| ⟨ || 0 || 1/2 || 2 || -9/2 || ]

|-

| ⟨ || 0 || 0 || 0 || 0 || ]]

|}

[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3379.1833 }}

[[Mistuning map]]: {{val| 0 -10.7531 -21.5063 +10.3574 }}

[[Category:Meantone]]

[[Category:Historical]]

@@ Line 1: / Line 1: @@
-In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).
+'''1/2-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).
-In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb val flat 5th is audibly indistinguishable from a closed system of half comma meantone, with an error measured in thousandths of a cent.
+In this system, the major thirds are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb [[val]] flat 5th is audibly indistinguishable from a closed system of half-comma meantone, with an error measured in thousandths of a cent.
+For a tuning with so flat a fifth, the 7-limit [[extension]] is [[flattone]] instead of septimal meantone, or [[7/6]] and [[8/7]] would swap their sizes.
+== Tuning profile ==
+[[Projection map]]:
+{| class="right-all"
+|-
+| [⟨ || 1 || 2 || 4 || -1 || ]
+|-
+| ⟨ || 0 || -1 || -4 || 9 || ]
+|-
+| ⟨ || 0 || 1/2 || 2 || -9/2 || ]
+|-
+| ⟨ || 0 || 0 || 0 || 0 || ]]
+|}
+[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3379.1833 }}
+[[Mistuning map]]: {{val| 0 -10.7531 -21.5063 +10.3574 }}
 [[Category:Meantone]]
 [[Category:Historical]]