5L 2s: Difference between revisions

Among the most well-known forms of this scale are the diatonic scale of [[12edo]], the Pythagorean diatonic scale, and scales produced by meantone systems.

[[TAMNAMS]] suggests the temperament-agnostic name '''diatonic''' for this scale, which commonly refers to a scale with 5 whole steps and 2 small steps. Under TAMNAMS and for all scale pattern pages on the wiki, '''the term ''diatonic'' exclusively refers to 5L 2s'''.

The term ''diatonic'' may also refer to scales produced using [[Tetrachord|tetrachords]], [[just intonation]], or in general have more than one size of whole tone. Such scales, such as [[Zarlino]], [[blackdye]] and [[diasem]], are specifically called ''[[Detempering|detempered]] diatonic scales'' (for an RTT-based philosophy) or ''deregularized diatonic scales'' (for an RTT-agnostic philosophy). The terms ''diatonic-like'' or ''diatonic-based'' may also be used to refer such scales, depending on what's contextually the most appropriate.

{{MOS gamut|Scale Signature=5L 2s}}

==Theory ==

===Introduction to step sizes===<!-- The 5L 2s page already had an introduction to step sizes, but this may be worth moving to its own page. -->

The familiar pattern of 5 whole steps and 2 half steps, commonly written as WWHWWWH for the major scale, has step sizes of 2 (whole step) and 1 (half step), producing [[12edo]]. This can be generalized into the form LLsLLLs, with whole-number sizes for the large steps and small steps, denoted as "L" and "s" respectively.

!Step ratio (L:s)

All step ratios lie on a spectrum from 1:1 to 1:0, referred to on the wiki as a scale tree. The step ratios 1:1 and 1:0 represent the limits for valid step ratios. A step ratio that approaches 1:1, where the large and small step are equal to one another, approaches [[7edo]], and a step ratio that approaches 1:0, where the small step "collapses" to zero, approaches [[5edo]].

TAMNAMS has names for regions of this spectrum based on whether they are "soft" (between 1:1 and 2:1) or "hard" (between 2:1 and 1:0).

5L 2s has several rank-2 temperament interpretations, such as:

==Tuning ranges==

[[17edo]] and [[19edo]] 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}}

:''Main article: [[Flattone]]''

Parasoft tunings (4:3 to 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths ([[3/2]], flat of 702¢) to produce major 3rds that are flatter than [[5/4]] (386¢).

Edos include [[19edo]], [[26edo]], [[45edo]], and [[64edo]].{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 4/3; 7/5; 10/7|Genchain Extend=0, 5}}

===Hyposoft tunings ===

:''Main article: [[Meantone]]''

Hyposoft tunings (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¢).

Edos include [[19edo]], [[31edo]], [[43edo]], and [[50edo]].{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 5/3; 7/4; 8/5|Genchain Extend=0, 5}}

===Hypohard tunings===

:''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]''

The range of hypohard tunings can be divided into a minihard range (2:1 to 5:2) and quasihard range (5:2 to 3:1).

Edos include [[41edo]] and [[53edo]].{{MOS degrees|Scale Signature=5L 2s|Step Ratio=7/3; 9/4|Genchain Extend=0, 5}}

==== Quasihard tunings ====

Quasihard tunings correspond to "neogothic" or "parapyth" systems whose perfect 5th is slightly sharper than just, resulting in major 3rds that are sharper than 81/64 and minor 3rds that are slightly flat of [[32/27]] (294¢).

Edos include [[17edo]], [[29edo]], and [[46edo]]. 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}}

===Parahard and ultrahard tunings===

:''Main article: [[Archy]]''

Parahard (3:1 to 4:1) and ultrahard tunings (4:1 to 1:0) correspond to archy systems, with perfect 5ths that are significantly sharper than than 702¢.

===Scala files===

*[[Meantone7]] – 19edo and 31edo tunings

*[[Nestoria7]] – 171edo tuning

*[[Cotoneum7]] – 217edo tuning

{{Scale tree|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}}

==See also==

* [[Diatonic functional harmony]]