Mint temperaments: Difference between revisions

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This is a collection of low complexity, high error [[regular temperament|temperaments]] [[~~tempering~~ out]] the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[triangular number]], and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate [[6/5]] with [[7/6]], [[5/4]] with [[9/7]], and [[7/4]] with [[9/5]], so minor and major thirds and sixths are intervals of 5 and 7 at the same time.

This is a collection of low [[complexity]], high [[error]], [[regular temperament|temperaments]] which [[temper out]] the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[triangular number]], and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate [[6/5]] with [[7/6]], [[5/4]] with [[9/7]], and [[7/4]] with [[9/5]], so minor and major thirds and sixths are intervals of 5 and 7 at the same time.

Temperaments discussed elsewhere include

* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]

* [[Dominant (temperament)|Dominant]] (+64/63) → [[Meantone family #Dominant|Meantone family]]

* ''[[~~Hexadecimal~~]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]

* ''[[Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]

* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]

* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]

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* ''[[Gorgo]]'' (+1029/1024) → [[Gamelismic clan #Gorgo|Gamelismic clan]]

* ''[[Hystrix]]'' (+160/147) → [[Porcupine family #Hystrix|Porcupine family]]

* [[Diminished (temperament)|Diminished]] (+50/49) → [[~~Dimipent~~ family #Diminished~~|Dimipent~~ family]]

* [[Diminished (temperament)|Diminished]] (+50/49) → [[Diminished family #Septimal diminished|Diminished family]]

* ''[[Smate]]'' (+2048/1875) → [[Smate family #Septimal smate|Smate family]]

* ''[[Darkstone]]'' (+1875/1792) → [[Magic family #Darkstone|Magic family]]

* ''[[Ripple]]'' (+2560/2401) → [[Ripple family #Septimal ripple|Ripple family]]

* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]

== Progression ==

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''

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: mapping generators: ~2, ~15/14

~~{{Multival|legend=1| 5 3 7 -7 -3 8 }}~~

[[Optimal tuning]]s:

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: mapping generators: ~2, ~7/6

~~{{Multival|legend=1| 6 7 5 -3 -9 -8 }}~~

[[Optimal tuning]]s:

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: mapping generators: ~2, ~21/20

~~{{Multival|legend=1| 7 11 3 1 -15 -24 }}~~

[[Optimal tuning]]s:

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: mapping generators: ~2, ~7/6

~~{{Multival|legend=1| 3 2 4 -4 -2 4 }}~~

[[Optimal tuning]]s:

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[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

[[Category:Mint temperaments| ]]

[[Category:Mint| ]]

[[Category:Rank 2]]

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-This is a collection of low complexity, high error [[regular temperament|temperaments]] [[tempering out]] the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[triangular number]], and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate [[6/5]] with [[7/6]], [[5/4]] with [[9/7]], and [[7/4]] with [[9/5]], so minor and major thirds and sixths are intervals of 5 and 7 at the same time.
+{{Technical data page}}
+This is a collection of low [[complexity]], high [[error]], [[regular temperament|temperaments]] which [[temper out]] the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[triangular number]], and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate [[6/5]] with [[7/6]], [[5/4]] with [[9/7]], and [[7/4]] with [[9/5]], so minor and major thirds and sixths are intervals of 5 and 7 at the same time.
 Temperaments discussed elsewhere include
 * [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
 * [[Dominant (temperament)|Dominant]] (+64/63) → [[Meantone family #Dominant|Meantone family]]
-* ''[[Hexadecimal]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
+* ''[[Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
 * ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
 * [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
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 * ''[[Gorgo]]'' (+1029/1024) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
 * ''[[Hystrix]]'' (+160/147) → [[Porcupine family #Hystrix|Porcupine family]]
-* [[Diminished (temperament)|Diminished]] (+50/49) → [[Dimipent family #Diminished|Dimipent family]]
+* [[Diminished (temperament)|Diminished]] (+50/49) → [[Diminished family #Septimal diminished|Diminished family]]
 * ''[[Smate]]'' (+2048/1875) → [[Smate family #Septimal smate|Smate family]]
 * ''[[Darkstone]]'' (+1875/1792) → [[Magic family #Darkstone|Magic family]]
 * ''[[Ripple]]'' (+2560/2401) → [[Ripple family #Septimal ripple|Ripple family]]
+* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]
 == Progression ==
+{{Distinguish| Progress }}
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''
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 : mapping generators: ~2, ~15/14
-{{Multival|legend=1| 5 3 7 -7 -3 8 }}
 [[Optimal tuning]]s:
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 : mapping generators: ~2, ~7/6
-{{Multival|legend=1| 6 7 5 -3 -9 -8 }}
 [[Optimal tuning]]s:
@@ Line 193: / Line 192: @@
 : mapping generators: ~2, ~21/20
-{{Multival|legend=1| 7 11 3 1 -15 -24 }}
 [[Optimal tuning]]s:
@@ Line 246: / Line 243: @@
 : mapping generators: ~2, ~7/6
-{{Multival|legend=1| 3 2 4 -4 -2 4 }}
 [[Optimal tuning]]s:
@@ Line 313: / Line 308: @@
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Mint temperaments| ]] <!-- main article -->
 [[Category:Mint| ]] <!-- key article -->
 [[Category:Rank 2]]