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The step ratios shown above form a continuum of step ratios. The section Tuning spectrum shows how this is made, as well as a larger spectrum.
The step ratios shown above form a continuum of step ratios. The section Tuning spectrum shows how this is made, as well as a larger spectrum.


==Temperament interpretations==
===Rank-2 temperament interpretations===


: ''Main article: [[5L 2s/Temperaments]]''
: ''Main article: [[5L 2s/Temperaments]]''
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5L 2s has several rank-2 temperament interpretations, such as:
5L 2s has several rank-2 temperament interpretations, such as:


* [[Meantone]], with generators around 696.2¢. These temperaments flatten the perfect 5th (702¢) to produce 5/4 (386¢) for a major 3rd.
* [[Meantone]], with generators around 696.2¢. This includes:
** [[Flattone]], with generators around 693.7¢. These temperaments have major 3rds that are flatter than 5/4.
** [[Flattone]], with generators around 693.7¢.
*[[Schismic]], with generators around 702¢. These temperaments have perfect 5ths that are close to just, producing 81/64 (407¢) for a major 3rd.
*[[Schismic]], with generators around 702¢.
*[[Parapyth]], with generators around 704.7¢. These temperaments have major 3rds that are sharper than 81/64.
*[[Parapyth]], with generators around 704.7¢.
*[[Archy]], with generators around 709.3¢. These temperaments have perfect 5ths that are significantly sharp.
*[[Archy]], with generators around 709.3¢. This includes:
**Supra, with generators around 707.2¢
**Supra, with generators around 707.2¢
**Superpyth, with generators around 710.3¢
**Superpyth, with generators around 710.3¢
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== Step ratio ranges==
== Step ratio ranges==
This section describes step ratio ranges for 5L 2s and what rank-2 temperaments they correspond to. Although the two may have closely correspond with one another, the purpose of step ratios is to describe 5L 2s without necessarily using [[regular temperament theory]].
=== Simple step ratios ===
=== Simple step ratios ===
17edo and 19edo, produced using step ratios of 3:1 and 3:2 respectively, are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=2/1; 3/1; 3/2|Genchain Extend=7}}
17edo and 19edo, produced using step ratios of 3:1 and 3:2 respectively, are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=2/1; 3/1; 3/2|Genchain Extend=7}}
===Soft step ratios===
=== Parasoft step ratios ===
:''Main article: [[Meantone]] and [[Flattone]]''
:''Main article: [[Flattone]]''
Most of the soft step ratio range (1:1 to 2:1) correspond to meantone temperaments. More specifically, the hyposoft range (3:2 to 2:1) corresponds to meantone and the parasoft range (4:3 and 3:2) corresponds to flattone.
Parasoft step ratios (4:3 and 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce major 3rds that are flatter than 5/4 (386¢).{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 4/3; 7/5; 10/7|Genchain Extend=0, 5}}
===Hyposoft step ratios===
:''Main article: [[Meantone]]''
Hyposoft step ratios (3:2 to 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702¢) to produce diatonic major 3rds that approximate 5/4 (386¢).{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 5/3; 7/4; 8/5|Genchain Extend=0, 5}}
===Hypohard step ratios ===
:''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]''
The range of hypohard step ratios can be divided into a minihard range (between 2:1 to 5:2) and quasihard range (between 5:2 to 3:1).
==== Minihard step ratios ====
Minihard step ratios correspond to Pythagorean tuning and schismatic temperament, characterized by having a perfect 5th that is as close to just (701.96¢) as possible, resulting in a major 3rd of 81/64 (407¢).{{MOS degrees|Scale Signature=5L 2s|Step Ratio=7/3; 9/4|Genchain Extend=0, 5}}
 
==== Quasihard step ratios ====
Quasihard step ratios correspond to "neogothic" or "parapyth" systems whose perfect 5th is sharper than just, resulting in major 3rds that are sharper than 81/64.


==== Parasoft step ratios ====
17edo is considered to be on the sharper end of the neogothic spectrum, with a major 3rd that is more discordant than flatter neogothic tunings.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 5/2; 8/3|Genchain Extend=0, 5}}
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 4/3; 7/5; 10/7|Genchain Extend=0, 5}}
===Parahard and ultrahard step ratios===
====Hyposoft step ratios====
:''Main article: [[Archy]]''
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 5/3; 7/4; 8/5|Genchain Extend=0, 5}}
The parahard and ultrahard ranges (3:1 to 1:1) correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 4/1; 5/1; 6/1|Genchain Extend=0, 5}}
===Hard step ratios ===
:''Main article: [[Pythagorean tuning]], [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]], and [[Archy]]''
The range of hard step ratios can be divided into a hypohard range (2:1 to 3:1), parahard range (3:1 to 4:1), and ultrahard range (4:1 to 1:0).
==== Hypohard step ratios ====
The flatter end of the hypohard region corresponds to Pythagorean tuning and schismic temperament, with perfect 5ths that are close to 702¢.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=7/3; 9/4|Genchain Extend=0, 5}}
The sharper end of the hypohard region corresponds to "neogothic" or "parapyth" systems, with perfect 5ths that are sharper than 702¢.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 5/2; 8/3|Genchain Extend=0, 5}}
====Parahard and ultrahard step ratios====
The parahard and ultrahard ranges correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 4/1; 5/1; 6/1|Genchain Extend=0, 5}}


== Modes==
== Modes==
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{{SB tree|Depth=3}}
{{SB tree|Depth=3}}


This process can be repeated to produce a finer continuum of step ratios as shown below, with each ratio producing a different edo.{{Scale tree|5L 2s|depth=6|Comments=7/5:[[Flattone]] is in this region;21/13:[[Golden meantone]] (696.2145¢);5/3:[[Meantone]] is in this region;2/1:(Generators smaller than this are proper);9/4:The generator closest to a just [[3/2]] for EDOs less than 200;16/7:[[Garibaldi]] / [[Cassandra]];21/8:Golden neogothic (704.0956¢);8/3:[[Neogothic]] is in this region;4/1:[[Archy]] is in this region|tuning=5L 2s}}
This process can be repeated to produce a finer, larger continuum of step ratios as shown below, with each ratio producing a different edo.{{Scale tree|5L 2s|depth=6|Comments=7/5:[[Flattone]] is in this region;21/13:[[Golden meantone]] (696.2145¢);5/3:[[Meantone]] is in this region;2/1:(Generators smaller than this are proper);9/4:The generator closest to a just [[3/2]] for EDOs less than 200;16/7:[[Garibaldi]] / [[Cassandra]];21/8:Golden neogothic (704.0956¢);8/3:[[Neogothic]] is in this region;4/1:[[Archy]] is in this region|tuning=5L 2s}}


==See also==
==See also==


*[[Diatonic functional harmony]]
*[[Diatonic functional harmony]]
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