1/5-comma meantone: Difference between revisions

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In '''1/5-comma [[meantone]]~~'''~~ temperament, each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes 2 and 16/15 purely.

{{Interwiki

| en = 1/5-comma meantone

| de = Fünftelkomma-mitteltönig

}}

'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely.

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 [[~~cent~~]]~~s sharper~~.)

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. The augmented sixth is exactly [[225/128]], and for septimal meantone, this means the [[7/4]] is sharp by a [[225/224|septimal kleisma (225/224)]].

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

== Tuning profile ==

[[Projection map]]:

{| class="right-all"

|-

| [⟨ || 1 || 4/5 || -4/5 || -5 || ]

|-

| ⟨ || 0 || 1/5 || 4/5 || 2 || ]

|-

| ⟨ || 0 || 1/5 || 4/5 || 2 || ]

|-

| ⟨ || 0 || 0 || 0 || 0 || ]]

|}

[[Tuning map]]: {{val| 1200 1897.6537 2790.6150 3376.5374 }}

[[Error map]]: {{val| 0 -4.3013 +4.3013 +7.7115 }}

== Music ==

; [[septimal lemon]]

* [https://youtu.be/jlbpQqahZh0 ''Bubbles''] (2025)

[[Category:Meantone]]

[[Category:Historical]]

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-In '''1/5-comma [[meantone]]''' temperament, each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes 2 and 16/15 purely.
+{{Interwiki
+| en = 1/5-comma meantone
+| de = Fünftelkomma-mitteltönig
+}}
+'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely.
-/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 [[cent]]s sharper.)
+The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. The augmented sixth is exactly [[225/128]], and for septimal meantone, this means the [[7/4]] is sharp by a [[225/224|septimal kleisma (225/224)]].
+/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)
+== Tuning profile ==
+[[Projection map]]:
+{| class="right-all"
+|-
+| [⟨ || 1 || 4/5 || -4/5 || -5 || ]
+|-
+| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
+|-
+| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
+|-
+| ⟨ || 0 || 0 || 0 || 0 || ]]
+|}
+[[Tuning map]]: {{val| 1200 1897.6537 2790.6150 3376.5374 }}
+[[Error map]]: {{val| 0 -4.3013 +4.3013 +7.7115 }}
+== Music ==
+; [[septimal lemon]]
+* [https://youtu.be/jlbpQqahZh0 ''Bubbles''] (2025)
 [[Category:Meantone]]
+[[Category:Historical]]