1/2-comma meantone: Difference between revisions

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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.

For a tuning with so flat a fifth, the 7-limit [[extension]] that preserves [[7-odd-limit]] [[diamond monotone]] is [[flattertone]]. Otherwise the septimal meantone mapping has the sizes of [[7/6]] and [[8/7]] swapped, and the flattone mapping has the sizes of [[7/5]] and [[10/7]] swapped.

== Tuning profile ==

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{| class="right-all"

|-

| [⟨ || 1 || 2 || 4 || -1 || ]

| [⟨ || 1 || 2 || 4 || 10 || ]

|-

| ⟨ || 0 || -1 || -4 || 9 || ]

| ⟨ || 0 || -1 || -4 || -17 || ]

|-

| ⟨ || 0 || 1/2 || 2 || -9/2 || ]

| ⟨ || 0 || 1/2 || 2 || 17/2 || ]

|-

| ⟨ || 0 || 0 || 0 || 0 || ]]

|}

[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 ~~3379~~.~~1833~~ }}

[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3350.4316 }}

[[~~Mistuning~~ map]]: {{val| 0 -10.7531 -21.5063 ~~+10~~.~~3574~~ }}

[[Error map]]: {{val| 0 -10.7531 -21.5063 -18.3944 }}

[[Category:Meantone]]

[[Category:Historical]]

@@ Line 3: / Line 3: @@
 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.
+For a tuning with so flat a fifth, the 7-limit [[extension]] that preserves [[7-odd-limit]] [[diamond monotone]] is [[flattertone]]. Otherwise the septimal meantone mapping has the sizes of [[7/6]] and [[8/7]] swapped, and the flattone mapping has the sizes of [[7/5]] and [[10/7]] swapped.
 == Tuning profile ==
@@ Line 10: / Line 10: @@
 {| class="right-all"
 |-
-| [⟨ || 1 || 2 || 4 || -1 || ]
+| [⟨ || 1 || 2 || 4 || 10 || ]
 |-
-| ⟨ || 0 || -1 || -4 || 9 || ]
+| ⟨ || 0 || -1 || -4 || -17 || ]
 |-
-| ⟨ || 0 || 1/2 || 2 || -9/2 || ]
+| ⟨ || 0 || 1/2 || 2 || 17/2 || ]
 |-
 | ⟨ || 0 || 0 || 0 || 0 || ]]
 |}
-[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3379.1833 }}
+[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3350.4316 }}
-[[Mistuning map]]: {{val| 0 -10.7531 -21.5063 +10.3574 }}
+[[Error map]]: {{val| 0 -10.7531 -21.5063 -18.3944 }}
 [[Category:Meantone]]
 [[Category:Historical]]