Fifth-chroma temperaments: Difference between revisions

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[[Category:Temperament collections]]

[[Category:Pages with mostly numerical content]]

== Fifth-chroma temperaments ==

=== Overview ===

==== Primary ====

* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]]

* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]

: * The "5-limit seed" of novamajor is by splitting 25/24 into five parts of ~sqrt(128/125) = (128/125)1/2.

* 77 & 84: [[Absurdity]], the 1\7-period (at least) 29-limit temperament; note that 70edo also supports it in a reduced/slightly modified form

: * The "5-limit seed" of absurdity is by tempering out (81/80)5 / (25/24) (the absurdity comma) so that 25/24 is split into five flattened (underexaggerated) ~81/80's, which is equivalent to interpreting 1\7 as (10/9)/(81/80) because 2 / ((10/9)/(81/80) = 800/729)7 is equal to the square of the absurdity comma.

* 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)

==== Secondary ====

* 77 & 87: [[Restles]]

: * The "2.3.13 seed" of restles is [[lesser tendoneutralic]], which observes (16/13)/(39/32) = 512/507 so that this interval is what divides 25/24 into five in 2.3.5.13 restles.

* 84 & 87: [[Mutt]]

: * The "5-limit seed" of mutt is by splitting 25/24 into 5 parts of ~cbrt(128/125) = (128/125)1/3.

* 77 & 94: [[Tsaharuk]]

* 80 & 87: "alt artoneutral": a version of [[Hemifamity temperaments#Artoneutral|artoneutral]] that uses a different mapping for prime 7

: * This has the same "5-limit seed" as artoneutral, but is actually more optimal for the 5-limit.

* 80 & 94: unnamed? proposed name: "pseudohemipyth"

* 87 & 94: [[Artoneutral]]

: * The "5-limit seed" of artoneutral is by splitting 25/24 into 5 parts of ~sqrt(81/80) = (81/80)1/2.

=== New temperament definitions ===

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Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal, but both edo tunings (77edo and 84edo) have resources to make various elements of 29-limit harmony possible.

==== Novamajor ====

{{ See also | User:Godtone/Augmented-chromatic equivalence continuum }}

This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125|(128/125)5 / (25/24)2]]), is a full 29-limit temperament that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):

This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125|(128/125)5 / (25/24)2]]), is a full no-31's 37-limit temperament (and possibly/likely higher) that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):

* inframinor third: ~15/13

* subminor third: ~7/6

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* subneutral third: ~39/32~50/41~11/9

* superneutral third: ~27/22~16/13

* submajor third: ~36/29~56/45 (note the absence of ~26/21, which here is ambiguous in mapping between submajor third (77edo) and superneutral third (80edo))

* submajor third: ~36/29~46/37~56/45 (note the absence of ~26/21, which here is ambiguous in mapping between submajor third (77edo) and superneutral third (80edo))

* major third: ~5/4

* novamajor third: ~24/19~43/34~19/15

@@ Line 51: / Line 51: @@
 [[Category:Temperament collections]]
+[[Category:Pages with mostly numerical content]]
 == Fifth-chroma temperaments ==
 === Overview ===
 ==== Primary ====
-* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]]
+* 77 & 80: [[Novamajor]], a temperament with a generator of 5/4 * S12, named because the interval colour of ~405{{cent}} is named a "novamajor third" by [[User:Godtone]]
+: * The "5-limit seed" of novamajor is by splitting 25/24 into five parts of ~sqrt(128/125) = (128/125)<sup>1/2</sup>.
 * 77 & 84: [[Absurdity]], the 1\7-period (at least) 29-limit temperament; note that 70edo also supports it in a reduced/slightly modified form
+: * The "5-limit seed" of absurdity is by tempering out (81/80)<sup>5</sup> / (25/24) (the absurdity comma) so that 25/24 is split into five flattened (underexaggerated) ~81/80's, which is equivalent to interpreting 1\7 as (10/9)/(81/80) because 2 / ((10/9)/(81/80) = 800/729)<sup>7</sup> is equal to the square of the absurdity comma.
 * 80 & 84: unnamed? proposed name: "quarterchromatic" from quartering the octave and having a comma-sized generator, four of which functions melodically as a type of chroma (though not as 25/24)
 ==== Secondary ====
 * 77 & 87: [[Restles]]
+: * The "2.3.13 seed" of restles is [[lesser tendoneutralic]], which observes (16/13)/(39/32) = 512/507 so that this interval is what divides 25/24 into five in 2.3.5.13 restles.
+* 84 & 87: [[Mutt]]
+: * The "5-limit seed" of mutt is by splitting 25/24 into 5 parts of ~cbrt(128/125) = (128/125)<sup>1/3</sup>.
 * 77 & 94: [[Tsaharuk]]
 * 80 & 87: "alt artoneutral": a version of [[Hemifamity temperaments#Artoneutral|artoneutral]] that uses a different mapping for prime 7
+: * This has the same "5-limit seed" as artoneutral, but is actually more optimal for the 5-limit.
 * 80 & 94: unnamed? proposed name: "pseudohemipyth"
 * 87 & 94: [[Artoneutral]]
+: * The "5-limit seed" of artoneutral is by splitting 25/24 into 5 parts of ~sqrt(81/80) = (81/80)<sup>1/2</sup>.
 === New temperament definitions ===
@@ Line 70: / Line 78: @@
 Absurdity was extended to the 29-limit; the mapping for prime 29 is obvious via 1\7 = 32/29 being accurate. The mapping for prime 23 is via 23/20 being equated with 8/7, which isn't ideal, but both edo tunings (77edo and 84edo) have resources to make various elements of 29-limit harmony possible.
 ==== Novamajor ====
-{{ See also | Augmented–chromatic equivalence continuum }}
+{{ See also | User:Godtone/Augmented-chromatic equivalence continuum }}
-This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125|(128/125)<sup>5</sup> / (25/24)<sup>2</sup>]]), is a full 29-limit temperament that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):
+This temperament, originally found independently in the form of a peculiar high-badness 5-limit temperament called "Isnes" ("Sensi" backwards, and can be understood as tempering [[19791209299968/19073486328125|(128/125)<sup>5</sup> / (25/24)<sup>2</sup>]]), is a full no-31's 37-limit temperament (and possibly/likely higher) that is better understood not in terms of generators (where it is very complex and doesn't make much musical sense) but instead as making the following intervals equidistant (which can be seen as a guarantee of a certain kind of metaphorical colour palette with certain properties):
 * inframinor third: ~15/13
 * subminor third: ~7/6
@@ Line 80: / Line 88: @@
 * subneutral third: ~39/32~50/41~11/9
 * superneutral third: ~27/22~16/13
-* submajor third: ~36/29~56/45 (note the absence of ~26/21, which here is ambiguous in mapping between submajor third (77edo) and superneutral third (80edo))
+* submajor third: ~36/29~46/37~56/45 (note the absence of ~26/21, which here is ambiguous in mapping between submajor third (77edo) and superneutral third (80edo))
 * major third: ~5/4
 * novamajor third: ~24/19~43/34~19/15