User:Ganaram inukshuk/5L 2s: Difference between revisions

{{Infobox MOS|Tuning=5L 2s}}

==Name==

TAMNAMS suggests the 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'''. Other diatonic-based scales, such as [[Zarlino]], [[blackdye]] and [[diasem]], are 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 to diatonic-based scales, depending on what's contextually the most appropriate.

==Notation==

Intervals are identical to that of standard notation. As such, the usual interval qualities of major/minor and augmented/perfect/diminished apply here.

! rowspan="2" |Interval class

! colspan="2" |Large variety

! colspan="2" |Small variety

|'''1st (unison)'''

|Perfect

|0

|Perfect

|2nd

|3rd

|4th

|5th

|3L + 1s

|Perfect

|6th

|4L + 1s

|Major

|7th

|5L + 1s

|Major

|'''8th (octave)'''

===Note names===

Note names are identical to that of standard notation. Thus, the basic gamut for 5L 2s is the following:

 

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

===Generalizing whole and half steps===

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 (small step), producing [[12edo]]. This can be generalized to form the pattern LLsLLLs with whole-number step sizes for L and s, where L is greater than s. The terms "large step" and "small step" are preferred as most step size pairings cannot be interpreted as "whole" and "half" steps.

!Selected multiples

|4 4 3 4 4 4 3

|5:3

|5 5 3 5 5 5 3

|2:1

|5 5 2 5 5 5 2

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]].

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.

: ''Main article: [[5L 2s/Temperaments]]''

* Meantone, with generators around 696.2¢.

** Flattone, with generators around 693.7¢.

*Schismic, with generators around 702¢.

*Parapyth, with generators around 704.7¢.

*Archy, with generators around 709.3¢.

== Step ratio ranges==

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===

:''Main article: [[Meantone]] and [[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.

Both meantone and flattone are characterized with flattening the perfect 5th (3/2) to achieve major 3rds that are close to 5/4 (386¢), where the latter has major 3rds that are flatter than 5/4.

:''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).

==== 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.

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.

===Parahard and ultrahard step ratios===

Parahard and ultrahard step ratios (3:1 or sharper) correspond to "archy" systems, whose perfect 5th is significantly sharper than just.

== Modes==

!6th

|Major (D)

|Perfect (C)

|Perfect (C)

|Perfect (F)

|Minor (Eb)

5L 2s has a parent scale of 2L 3s, meaning 5L 2s contains 2L 3s as a subset. 5L 2s also has two child scales that both contain 5L 2s as a subset: either 7L 5s (if the step ratio is less than 2:1) or 5L 7s (if the step ratio is greater than 2:1). A step ratio exactly 2:1 will produce 12edo, an equalized form of 5L 7s and 7L 5s.

==Tuning spectrum==

{{SB tree|Depth=1}}

{{SB tree|Depth=2}}

{{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}}

 

*[[Diatonic functional harmony]]