36/35: Difference between revisions

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{{Interwiki

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| de = 36/35

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{{Infobox Interval

| Name = septimal quartertone, mint comma

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'''36/35''', the '''septimal quartertone''' (~48.8{{cent~~}}) is~~ the difference between [[10/9]] and [[8/7]], [[7/6]] and [[6/5]], [[5/4]] and [[9/7]], [[14/9]] and [[8/5]], [[5/3]] and [[12/7]]~~, and~~ [[7/4]] and [[9/5]]. It has a numerator which is both the sixth square number and the eighth [[triangular number]], leading to it being the product of two [[superparticular]] commas both as [[64/63]] × [[81/80]] and as [[66/65]] × [[78/77]]; it is also [[45/44]] × [[176/175]], [[51/50]] × [[120/119]], [[128/125]] × [[225/224]], [[50/49]] × [[126/125]] and [[56/55]] × [[99/98]].

'''36/35''', the '''septimal quartertone''' is a [[7-limit]] interval with a size of about 48.8 [[cent]]s. It arises as the difference between various 7-limit [[interval quality|qualities]] of seconds, thirds, sixths, and sevenths: between supermajor and classical major, and between subminor and classical minor. These are the pairs of intervals separated by 36/35:

* [[28/27]] and [[16/15]]

* [[10/9]] and [[8/7]]

* [[7/6]] and [[6/5]]

* [[5/4]] and [[9/7]]

* [[14/9]] and [[8/5]]

* [[5/3]] and [[12/7]]

* [[7/4]] and [[9/5]]

* [[15/8]] and [[27/14]]

It has a numerator which is both the sixth square number and the eighth [[triangular number]], leading to it being the product of two [[superparticular]] commas both as ([[64/63]])⋅([[81/80]]) and as ([[66/65]])⋅([[78/77]]); it is also ([[45/44]])⋅([[176/175]]), ([[51/50]])⋅([[120/119]]), ([[128/125]])⋅([[225/224]]), ([[50/49]])⋅([[126/125]]) and ([[56/55]])⋅([[99/98]]).

== Temperaments ==

When treated as a generator, it is almost a fourth of the septimal whole tone [[28/25]], differing by a [[wizma]]. This allows it to be used as a generator for ~~an extension~~ of [[~~Hemimean family|~~hemimean]] called [[sengagen]], where 5/4, 7/5 and 7/4 map to 8, 12 and 20 quartertones respectively.

When treated as a generator, it is almost a fourth of the septimal whole tone [[28/25]], differing by a [[wizma]]. This allows it to be used as a generator for a temperament of [[hemimean]] called [[sengagen]], where 5/4, 7/5 and 7/4 map to 8, 12 and 20 quartertones respectively.

When treated as a comma to be tempered out, it is known as the '''mint comma''', and tempering it out leads to the [[mint]] temperament. See [[~~mint~~ family]], the family of rank-3 temperaments where it is tempered out, and [[~~mint~~ temperaments]], the collection of rank-2 temperaments where it is tempered out.

When treated as a comma to be tempered out, it is known as the '''mint comma''', and tempering it out leads to the [[mint]] temperament. See [[Mint family]] for the family of rank-3 temperaments where it is tempered out, and [[Mint temperaments]] for the collection of rank-2 temperaments where it is tempered out.

== Etymology ==

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* [[List of superparticular intervals]]

== ~~Notes~~ ==

== References ==

[[Category:Quartertone]]

[[Category:Mint]]

[[Category:Commas named for the intervals they stack]]

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~~[[de:36/35]]~~

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+{{Interwiki
+| en = 36/35
+| de = 36/35
+}}
 {{Infobox Interval
 | Name = septimal quartertone, mint comma
@@ Line 7: / Line 11: @@
 {{Wikipedia|Septimal quarter tone}}
-'''36/35''', the '''septimal quartertone''' (~48.8{{cent}}) is the difference between [[10/9]] and [[8/7]], [[7/6]] and [[6/5]], [[5/4]] and [[9/7]], [[14/9]] and [[8/5]], [[5/3]] and [[12/7]], and [[7/4]] and [[9/5]]. It has a numerator which is both the sixth square number and the eighth [[triangular number]], leading to it being the product of two [[superparticular]] commas both as [[64/63]] × [[81/80]] and as [[66/65]] × [[78/77]]; it is also [[45/44]] × [[176/175]], [[51/50]] × [[120/119]], [[128/125]] × [[225/224]], [[50/49]] × [[126/125]] and [[56/55]] × [[99/98]].
+'''36/35''', the '''septimal quartertone''' is a [[7-limit]] interval with a size of about 48.8 [[cent]]s. It arises as the difference between various 7-limit [[interval quality|qualities]] of seconds, thirds, sixths, and sevenths: between supermajor and classical major, and between subminor and classical minor. These are the pairs of intervals separated by 36/35:
+* [[28/27]] and [[16/15]]
+* [[10/9]] and [[8/7]]
+* [[7/6]] and [[6/5]]
+* [[5/4]] and [[9/7]]
+* [[14/9]] and [[8/5]]
+* [[5/3]] and [[12/7]]
+* [[7/4]] and [[9/5]]
+* [[15/8]] and [[27/14]]
+It has a numerator which is both the sixth square number and the eighth [[triangular number]], leading to it being the product of two [[superparticular]] commas both as ([[64/63]])⋅([[81/80]]) and as ([[66/65]])⋅([[78/77]]); it is also ([[45/44]])⋅([[176/175]]), ([[51/50]])⋅([[120/119]]), ([[128/125]])⋅([[225/224]]), ([[50/49]])⋅([[126/125]]) and ([[56/55]])⋅([[99/98]]).
 == Temperaments ==
-When treated as a generator, it is almost a fourth of the septimal whole tone [[28/25]], differing by a [[wizma]]. This allows it to be used as a generator for an extension of [[Hemimean family|hemimean]] called [[sengagen]], where 5/4, 7/5 and 7/4 map to 8, 12 and 20 quartertones respectively.
+When treated as a generator, it is almost a fourth of the septimal whole tone [[28/25]], differing by a [[wizma]]. This allows it to be used as a generator for a temperament of [[hemimean]] called [[sengagen]], where 5/4, 7/5 and 7/4 map to 8, 12 and 20 quartertones respectively.
-When treated as a comma to be tempered out, it is known as the '''mint comma''', and tempering it out leads to the [[mint]] temperament. See [[mint family]], the family of rank-3 temperaments where it is tempered out, and [[mint temperaments]], the collection of rank-2 temperaments where it is tempered out.
+When treated as a comma to be tempered out, it is known as the '''mint comma''', and tempering it out leads to the [[mint]] temperament. See [[Mint family]] for the family of rank-3 temperaments where it is tempered out, and [[Mint temperaments]] for the collection of rank-2 temperaments where it is tempered out.
 == Etymology ==
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 * [[List of superparticular intervals]]
-== Notes ==
+== References ==
 [[Category:Quartertone]]
 [[Category:Mint]]
 [[Category:Commas named for the intervals they stack]]
-<!-- interwiki -->
-[[de:36/35]]