User:Ganaram inukshuk/4L 3s: Difference between revisions

Line 81:

==Theory==

~~== Theory ==~~

===Low harmonic entropy scales===

=== Low harmonic entropy scales ===

There are two notable harmonic entropy minima:

* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1 ~~((6/5)^6, tempered, equals 3/1)~~.

*[[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1.

* [[myna|Myna temperament]], in which the generator is also 6/5 but 10 of them make a 6/1 ~~((6/5)^10, tempered, equals 6/1)~~, resulting in the intervals 4/3 and 3/2 being absent.

*[[myna|Myna temperament]], in which the generator is also 6/5 but 10 of them make a 6/1, resulting in the intervals 4/3 and 3/2 being absent.

=== Temperament interpretations ===

===Temperament interpretations===

: ''Main article: [[4L 3s/Temperaments]]''

:''Main article: [[4L 3s/Temperaments]]''

4L 3s has the following temperament interpretations:

* ~~With generator size between 5\18 (333.3c) and 11\39 (338.5c):~~ [[Sixix]], ~~corresponding to a step ratio between 3/2 and 6/5~~.

*[[Sixix]], with generators around 338.6¢.

* ~~With generator size between 4\15 (320.0c) and 3\11 (327.3c):~~ [[Orgone]], ~~corresponding to a step ratio between 3/1 and 2/1~~.

*[[Orgone]], with generators around 323.4¢.

* ~~With generator size between 5\19 (315.8c) and 4\15 (320.0c):~~ [[Kleismic]], ~~corresponding to a step ratio between 4/1 and 3/1~~.

*[[Kleismic]], with generators around 317¢.

~~There are other~~ temperaments, ~~particularly~~ [[amity]] and [[myna]], ~~but~~ 7 pitches ~~is not enough~~ to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a [[MODMOS]] or a larger MOS gamut is necessary to access these pitches.

Other temperaments, such as [[amity]] and [[myna]], require more than 7 pitches to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a [[MODMOS]] or a larger MOS gamut is necessary to access these pitches.

== Step ratios ==

==Step ratios==

=== Simple tunings ===

===Simple tunings===

The basic tuning for 4L 3s has a large and small step size of 2 and 1 respectively, which is supported by 11edo. Other small edos include 15edo and 18edo.{{MOS degrees|Scale Signature=4L 3s|Step Ratio=2/1; 3/1; 3/2}}

===Parasoft tunings===

=== Parasoft tunings ===

Parasoft smitonic tunings (4:3 to 3:2) can be considered "meantone smitonic" since it has the following features of [[meantone]] diatonic tunings:

* The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".

*The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".

* The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.

*The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.

These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702¢), and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like chromas. For this reason, parasoft might be the most accessible smitonic tuning range.

Line 116:

Line 114:

Edos include [[18edo]], [[25edo]], and [[43edo]]. Some key considerations include:

* 18edo can be used to make the large and small steps more distinct.

*18edo can be used to make the large and small steps more distinct.

** 18edo smitonic can also be considered a distorted 19edo diatonic.

**18edo smitonic can also be considered a distorted 19edo diatonic.

** 18edo has a major 1-mosstep that is close to 9/8 (203¢).

**18edo has a major 1-mosstep that is close to 9/8 (203¢).

** 18edo's major and minor 4-mossteps are both equally off from 3/2 (702¢).

**18edo's major and minor 4-mossteps are both equally off from 3/2 (702¢).

** 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.

**18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.

* The augmented 2-mosstep of 25edo is very close to 5/4 (386¢).

*The augmented 2-mosstep of 25edo is very close to 5/4 (386¢).

{{MOS degrees|Scale Signature=4L 3s|Step Ratio=3/2; 4/3; 7/5|Genchain Extend=9}}

=== Hyposoft tunings ===

===Hyposoft tunings===

Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327¢ and 333¢. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] smitonic".

Edos include [[11edo]], [[18edo]], and [[29edo]]. (11edo and 18edo not shown in table below.)

{{MOS degrees|Scale Signature=4L 3s|Step Ratio=5/3|Genchain Extend=3}}

=== Hypohard tunings ===

===Hypohard tunings===

Hypohard smitonic tunings (2:1 to 3:1) have generators between 320¢ and 327¢. The major 1-mosstep, or large step, tends to approximate [[8/7]] (231¢) and the major 3-mosstep tends to approximate [[11/8]] (551¢). [[26edo]] approximates these two intervals very well. These JI approximations are associated with [[orgone]] temperament.

Other hypohard edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].

{{MOS degrees|Scale Signature=4L 3s|Step Ratio=2/1; 3/1; 5/2; 7/3|Genchain Extend=3}}

===Parahard tunings===

=== Parahard tunings ===

Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9¢ and 320¢, putting it close to a pure 6/5 (316¢), six of which and octave-reducing approximates 3/2 (702¢), a diatonic perfect 5th.

Line 143:

Line 140:

Other edos smaller than 53edo include [[15edo]], [[19edo]], and [[34edo]].

{{MOS degrees|Scale Signature=4L 3s|Step Ratio=3/1; 4/1; 7/2; 11/3|Genchain Extend=3}}

===Tuning examples===

=== Tuning examples ===

''(add tuning examples)''

== Modes ==

==Modes==

{{MOS modes|Scale Signature=4L 3s|Mode Names=Nerevarine; Vivecan; Lorkhanic; Sothic; Kagrenacan; Almalexian; Dagothic}}

==Approaches==

~~== Approaches ==~~

*[[4L 3s/Inthar's approach]]

* [[4L 3s/Inthar's approach]]

== Scales ==

==Scales==

=== Scala files ===

===Scala files===

* [[Orgone7]]

*[[Orgone7]]

* [[Cata7]]

*[[Cata7]]

* [[Myna7]]

*[[Myna7]]

== Music ==

==Music==

''(add music)''

== Tuning spectrum ==

==Tuning spectrum==

{{Scale tree|4L 3s|Comments=6/5: [[Amity]]/[[hitchcock]]↑;

5/4: [[Sixix]];

@@ Line 81: / Line 81: @@
 {{MOS gamut|Scale Signature=4L 3s}}
+==Theory==
-== Theory ==
+===Low harmonic entropy scales===
-=== Low harmonic entropy scales ===
 There are two notable harmonic entropy minima:
-* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1 ((6/5)^6, tempered, equals 3/1).
+*[[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1.
-* [[myna|Myna temperament]], in which the generator is also 6/5 but 10 of them make a 6/1 ((6/5)^10, tempered, equals 6/1), resulting in the intervals 4/3 and 3/2 being absent.
+*[[myna|Myna temperament]], in which the generator is also 6/5 but 10 of them make a 6/1, resulting in the intervals 4/3 and 3/2 being absent.
-=== Temperament interpretations ===
+===Temperament interpretations===
-: ''Main article: [[4L 3s/Temperaments]]''
+:''Main article: [[4L 3s/Temperaments]]''
 L 3s has the following temperament interpretations:
-* With generator size between 5\18 (333.3c) and 11\39 (338.5c): [[Sixix]], corresponding to a step ratio between 3/2 and 6/5.
+*[[Sixix]], with generators around 338.6¢.
-* With generator size between 4\15 (320.0c) and 3\11 (327.3c): [[Orgone]], corresponding to a step ratio between 3/1 and 2/1.
+*[[Orgone]], with generators around 323.4¢.
-* With generator size between 5\19 (315.8c) and 4\15 (320.0c): [[Kleismic]], corresponding to a step ratio between 4/1 and 3/1.
+*[[Kleismic]], with generators around 317¢.
-There are other temperaments, particularly [[amity]] and [[myna]], but 7 pitches is not enough to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a [[MODMOS]] or a larger MOS gamut is necessary to access these pitches.
+Other temperaments, such as [[amity]] and [[myna]], require more than 7 pitches to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a [[MODMOS]] or a larger MOS gamut is necessary to access these pitches.
-== Step ratios ==
+==Step ratios==
-=== Simple tunings ===
+===Simple tunings===
 The basic tuning for 4L 3s has a large and small step size of 2 and 1 respectively, which is supported by 11edo. Other small edos include 15edo and 18edo.{{MOS degrees|Scale Signature=4L 3s|Step Ratio=2/1; 3/1; 3/2}}
+===Parasoft tunings===
-=== Parasoft tunings ===
 Parasoft smitonic tunings (4:3 to 3:2) can be considered "meantone smitonic" since it has the following features of [[meantone]] diatonic tunings:
-* The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".
+*The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".
-* The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.
+*The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.
 These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702¢), and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like chromas. For this reason, parasoft might be the most accessible smitonic tuning range.
@@ Line 116: / Line 114: @@
 Edos include [[18edo]], [[25edo]], and [[43edo]]. Some key considerations include:
-* 18edo can be used to make the large and small steps more distinct.
+*18edo can be used to make the large and small steps more distinct.
-** 18edo smitonic can also be considered a distorted 19edo diatonic.
+**18edo smitonic can also be considered a distorted 19edo diatonic.
-** 18edo has a major 1-mosstep that is close to 9/8 (203¢).
+**18edo has a major 1-mosstep that is close to 9/8 (203¢).
-** 18edo's major and minor 4-mossteps are both equally off from 3/2 (702¢).
+**18edo's major and minor 4-mossteps are both equally off from 3/2 (702¢).
-** 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
+**18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
-* The augmented 2-mosstep of 25edo is very close to 5/4 (386¢).
+*The augmented 2-mosstep of 25edo is very close to 5/4 (386¢).
 {{MOS degrees|Scale Signature=4L 3s|Step Ratio=3/2; 4/3; 7/5|Genchain Extend=9}}
-=== Hyposoft tunings ===
+===Hyposoft tunings===
 Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327¢ and 333¢. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] smitonic".
 Edos include [[11edo]], [[18edo]], and [[29edo]]. (11edo and 18edo not shown in table below.)
 {{MOS degrees|Scale Signature=4L 3s|Step Ratio=5/3|Genchain Extend=3}}
-=== Hypohard tunings ===
+===Hypohard tunings===
 Hypohard smitonic tunings (2:1 to 3:1) have generators between 320¢ and 327¢. The major 1-mosstep, or large step, tends to approximate [[8/7]] (231¢) and the major 3-mosstep tends to approximate [[11/8]] (551¢). [[26edo]] approximates these two intervals very well. These JI approximations are associated with [[orgone]] temperament.
 Other hypohard edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
 {{MOS degrees|Scale Signature=4L 3s|Step Ratio=2/1; 3/1; 5/2; 7/3|Genchain Extend=3}}
+===Parahard tunings===
-=== Parahard tunings ===
 Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9¢ and 320¢, putting it close to a pure 6/5 (316¢), six of which and octave-reducing approximates 3/2 (702¢), a diatonic perfect 5th.
@@ Line 143: / Line 140: @@
 Other edos smaller than 53edo include [[15edo]], [[19edo]], and [[34edo]].
 {{MOS degrees|Scale Signature=4L 3s|Step Ratio=3/1; 4/1; 7/2; 11/3|Genchain Extend=3}}
+===Tuning examples===
-=== Tuning examples ===
 ''(add tuning examples)''
-== Modes ==
+==Modes==
 {{MOS modes|Scale Signature=4L 3s|Mode Names=Nerevarine; Vivecan; Lorkhanic; Sothic; Kagrenacan; Almalexian; Dagothic}}
+==Approaches==
-== Approaches ==
+*[[4L 3s/Inthar's approach]]
-* [[4L 3s/Inthar's approach]]
-== Scales ==
+==Scales==
-=== Scala files ===
+===Scala files===
-* [[Orgone7]]
+*[[Orgone7]]
-* [[Cata7]]
+*[[Cata7]]
-* [[Myna7]]
+*[[Myna7]]
-== Music ==
+==Music==
 ''(add music)''
-== Tuning spectrum ==
+==Tuning spectrum==
 {{Scale tree|4L 3s|Comments=6/5: [[Amity]]/[[hitchcock]]↑;
 /4: [[Sixix]];