36/35: Difference between revisions

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{{Interwiki

| en = 36/35

| 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]]

[[~~Ben Johnston's notation~~]] ~~denotes this interval with "7<~~/~~span>"~~ (~~a turned 7~~), and ~~the reciprocal 35~~/~~36 with an ordinary 7~~.

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 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 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]] 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 ==

The name ''mint comma'' was given by [[Mike Battaglia]] in 2012, for minor third because "it mixes 7/6 and 6/5 together into one minty interval"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103732.html Yahoo! Tuning Group | ''Quartonic'']</ref>. Before that, it had been known as the ''quartonic comma'', which refers to another comma today.

== Sagittal notation ==

== Notation ==

In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \!) }} and is called the '''35 medium diesis''', or '''35M''' for short, because the simplest ~~ratio~~ it notates is 16~~:35~~, as for example in C:D{{sagittal | \!) }}.

=== Ben Johnston's notation ===

In [[Ben Johnston's notation]], this interval is denoted with "{{invert|7}}" (a turned "7"), and the reciprocal 35/36 with an ordinary "7". If the base note is C, then [[7/4]] is reprented by C–Bb7.

=== Sagittal notation ===

In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \!) }} and is called the '''35 medium diesis''', or '''35M''' for short, because the simplest interval it notates is 35/1 = 5×7 (equiv. 35/16), as for example in C-D{{nbhsp}}{{sagittal | \!) }}. The upward version is called '''1/35M''' or '''35M up''' and is represented by {{sagittal| /|) }}.

== See also ==

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* [[List of superparticular intervals]]

== ~~Notes~~ ==

== References ==

[[Category:Quartertone]]

[[Category:Mint]]

~~~~

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

[[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]]
-[[Ben Johnston's notation]] denotes this interval with "<span style="display: inline-block; transform: rotate(180deg)">7</span>" (a turned 7), and the reciprocal 35/36 with an ordinary 7.
+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 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 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]] 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 ==
 The name ''mint comma'' was given by [[Mike Battaglia]] in 2012, for <u>min</u>or <u>t</u>hird because "it mixes 7/6 and 6/5 together into one minty interval"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_103732.html Yahoo! Tuning Group | ''Quartonic'']</ref>. Before that, it had been known as the ''quartonic comma'', which refers to another comma today.
-== Sagittal notation ==
+== Notation ==
-In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \!) }} and is called the '''35 medium diesis''', or '''35M''' for short, because the simplest ratio it notates is 16:35, as for example in C:D{{sagittal | \!) }}.
+=== Ben Johnston's notation ===
+In [[Ben Johnston's notation]], this interval is denoted with "{{invert|7}}" (a turned "7"), and the reciprocal 35/36 with an ordinary "7". If the base note is C, then [[7/4]] is reprented by C–Bb7.
+=== Sagittal notation ===
+In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | \!) }} and is called the '''35 medium diesis''', or '''35M''' for short, because the simplest interval it notates is 35/1 = 5×7 (equiv. 35/16), as for example in C-D{{nbhsp}}{{sagittal | \!) }}. The upward version is called '''1/35M''' or '''35M up''' and is represented by {{sagittal| /|) }}.
 == See also ==
@@ Line 26: / Line 44: @@
 * [[List of superparticular intervals]]
-== Notes ==
+== References ==
 [[Category:Quartertone]]
 [[Category:Mint]]
-<!-- interwiki -->
+[[Category:Commas named for the intervals they stack]]
-[[de:36/35]]