@@ Line 1: / Line 1: @@
-{{Infobox MOS|Tuning=5L 2s}}
+{{Infobox MOS|Tuning=5L 2s|debug=1}}
 :''This is a test page. For the main page, see [[5L 2s]].''
 {{MOS intro|Scale Signature=5L 2s}}
-==Name==
+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 name '''diatonic''' for this scale, which commonly refers to a scale with 5 whole steps and 2 small steps.
+== Name==
+[[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'''.
-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.
+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.
-==Notation==
+==Intervals==
+:''This article assumes [[TAMNAMS]] for naming step ratios, mossteps, and mosdegrees.''
+Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (the 0-mosstep and 0-mosdegree) for the unison.
-===Intervals===
+Except for the unison and octave, all [[Interval class|interval classes]] have two [[Interval variety|varieties]] or sizes, denoted using the terms ''major'' and ''minor'' for the large and small sizes, respectively. The exception to this rule are the generators, which use the terms ''augmented'', ''perfect'', and ''diminished'' instead.
-Intervals are identical to that of standard notation. As such, the usual interval qualities of major/minor and augmented/perfect/diminished apply here.
 {| class="wikitable"
-! rowspan="2" |Interval class
+|+5L 2s interval varieties
-! colspan="2" |Large variety
+!Interval class
-! colspan="2" |Small variety
+!Specific intervals
+!Size (in ascending order)
 |-
-!Size
+|'''0-diastep'''
-!Quality
+|'''Perfect 0-diastep (unison)'''
-!Size
+|0
-!Quality
 |-
-|'''1st (unison)'''
+| rowspan="2" |1-diastep
-|0
+|Minor 1-diastep
-|Perfect
+|s
-|0
-|Perfect
 |-
-|2nd
+|Major 1-diastep
 |L
-|Major
-|s
-|Minor
 |-
-|3rd
+| rowspan="2" |2-diastep
+|Minor 2-diastep
+|L + s
+|-
+|Major 2-diastep
 |2L
-|Major
-|L + s
-|Minor
 |-
-|4th
+| rowspan="2" |'''3-diastep'''
+|'''Perfect 3-diastep'''
+|2L + s
+|-
+|Augmented 3-diastep
 |3L
-|Augmented
-|2L + 1s
-|Perfect
 |-
-|5th
+| rowspan="2" |'''4-diastep'''
-|3L + 1s
+|Diminished 4-diastep
-|Perfect
 |2L + 2s
-|Diminished
 |-
-|6th
+|'''Perfect 4-diastep'''
-|4L + 1s
+|3L + s
-|Major
+|-
+| rowspan="2" |5-diastep
+|Minor 5-diastep
 |3L + 2s
-|Minor
 |-
-|7th
+|Major 5-diastep
-|5L + 1s
+|4L + s
-|Major
+|-
+| rowspan="2" |6-diastep
+|Minor 6-diastep
 |4L + 2s
-|Minor
 |-
-|'''8th (octave)'''
+|Major 6-diastep
+|5L + s
+|-
+|'''7-diastep (octave)'''
+|'''Perfect 7-diastep (octave)'''
 |5L + 2s
-|Perfect
-|5L + 2s
-|Perfect
 |}
+A 7-note scale using these intervals will typically use scale degrees that represents one size from each interval class, with the true MOS upholding the step pattern of LLLsLLs, or some rotation thereof. MODMOS scales may be formed this way without upholding the step pattern, thereby creating a non-MOS pattern such as LLLLsLs, or may include alterations that exceed the two varieties typical of a MOS scale.
-===Note names===
+==Notation==
-Note names are identical to that of standard notation. Thus, the basic gamut for 5L 2s is the following:
+:''See [[5L 2s/Notation]]''
-{{MOS gamut|Scale Signature=5L 2s}}
 ==Theory==
-===Generalizing whole and half steps===
+===Introduction to step sizes===
-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.
+:''Main article: [[Scale tree]] and [[TAMNAMS#Step ratio spectrum]]''
+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.
 Different edos are produced by using different ratios of step sizes. A few examples are shown below.
@@ Line 85: / Line 85: @@
 !Step pattern
 !EDO
+! Selected multiples
+|-
+|1:1
+|1 1 1 1 1 1 1
+|[[7edo]]
+|[[14edo]], [[21edo]], etc.
 |-
 |4:3
-|4 4 3 4 4 4 3
+| 4 4 3 4 4 4 3
 |[[26edo]]
+|
 |-
 |3:2
 |3 3 2 3 3 3 2
 |[[19edo]]
+|[[38edo]]
 |-
-|5:3
+| 5:3
-|5 5 3 5 5 5 3
+| 5 5 3 5 5 5 3
 |[[31edo]]
+|
 |-
-|2:1
+| 2:1
 |2 2 1 2 2 2 1
 |[[12edo]] (standard tuning)
+|[[24edo]], [[36edo]], etc.
 |-
 |5:2
-|5 5 2 5 5 5 2
+| 5 5 2 5 5 5 2
 |[[29edo]]
+|
 |-
 |3:1
 |3 3 1 3 3 3 1
 |[[17edo]]
+|[[34edo]]
 |-
 |4:1
 |4 4 1 4 4 4 1
 |[[22edo]]
+|
+|-
+|1:0
+|1 1 0 1 1 1 0
+|[[5edo]]
+|[[10edo]], [[15edo]], etc.
 |}
-Edos that are multiples of the examples above can be reached by entering non-simplified step ratios. For example, edos that are multiples of 12 are reached by using larger values whose ratio simplifies to 2:1, such as 4:2 for [[24edo]] and 12:6 for [[72edo]]. The step sizes may be called whole and half in this case.
+Edos that are multiples of the examples above can be reached by entering non-simplified step ratios. For example, edos that are multiples of 12 are reached by using larger values whose ratio simplifies to 2:1, such as 4:2 for [[24edo]].
-A spectrum of step ratios can be produced by starting with the ratios 1:1 and 1:0 and repeatedly finding the [[Mediant|mediants]] between adjacent ratios. The first three iterations are shown below, yielding the step ratios previously mentioned.{{SB tree|Depth=1}}
+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]].
-{{SB tree|Depth=2}}
-{{SB tree|Depth=3}}
-Larger edos, such as [[53edo]] (step ratio 9:4), can be reached by repeatedly expanding the tuning spectrum. A larger tuning spectrum can be found in this page's tuning spectrum section.
-The step ratios 1:1 and 1:0 represent the extremes of the tuning spectrum. 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 size of the small step approaches 0 relative to the size of the large step, 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).
 ===Temperament interpretations===
-: ''Main article: [[5L 2s/Temperaments]]''
+:''Main article: [[5L 2s/Temperaments]]''
-L 2s has several temperament interpretations, such as:
+L 2s has several rank-2 temperament interpretations, such as:
-* Flattone, with a generator size around 692.3¢ to 694.7¢, corresponding to a step ratio within the parasoft range (4:3 to 3:2).
+*[[Meantone]], with generators around 696.2¢. This includes:
-*Meantone, with a generator size around 696¢, corresponding to a step ratio within the hyposoft range (3:2 to 2:1).
+**[[Flattone]], with generators around 693.7¢.
-*Schismic, with a generator size around 702¢ (just perfect 5th, or 3/2), corresponding to a step ratio within the hypohard range (2:1 to 3:2).
+*[[Schismic]], with generators around 702¢.
-*Parapyth, with a generator size ranging between 702¢ and 705¢, corresponding to a step ratio within the quasihard to range (5:2 to 3:1).
+*[[Parapyth]], with generators around 704.7¢.
-*Archy, with a generator size greater than 705¢, corresponding to a step ratio within the parahard to ultrahard range (greater than 3:1).
+*[[Archy]], with generators around 709.3¢. This includes:
+**Supra, with generators around 707.2¢
+**Superpyth, with generators around 710.3¢
+**Ultrapyth, with generators around 713.7¢.
-== Step ratios==
+==Tuning ranges==
-===Parasoft===
+===Simple tunings===
+[[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.
+===Parasoft tunings ===
 :''Main article: [[Flattone]]''
-Parasoft tunings (step ratios between 4:3 to 3:2) correspond to flattone temperaments, characterized by flattening the perfect 5th (3/2) so it makes the diatonic major 3rd flatter than 5/4 (386¢). Compatible edos include 26edo, 45edo, and 64edo.
+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¢).
-{{MOS degrees|Scale Signature=5L 2s|Step Ratio=4/3; 7/5; 10/7}}
-===Hyposoft===
+Edos include [[19edo]], [[26edo]], [[45edo]], and [[64edo]].
+===Hyposoft tunings===
 :''Main article: [[Meantone]]''
-Hyposoft tunings (step ratio between 3:2 to 2:1) correspond to meantone temperaments, characterized by flattening the perfect 5th (3/2) to achieve a diatonic major 3rd that approximates 5/4 (386¢). Compatible edos include 19edo, 31edo, and 50edo.
+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¢).
-{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 5/3; 8/5}}
-===Hypohard ===
+Edos include [[19edo]], [[31edo]], [[43edo]], and [[50edo]].
-:''Main articles: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]''
+===Hypohard tunings===
-The range of hypohard tunings can be divided into a minihard range (step ratios 2:1 to 5:2) and quasihard range (step ratios 5:2 to 3:1).
+:''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]''
-==== Minihard ====
+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).
-Minihard tunings 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 81/64 (407.8¢) for its major 3rd. Compatible edos include 41edo, and 53edo.
+====Minihard tunings====
-{{MOS degrees|Scale Signature=5L 2s|Step Ratio=7/3; 9/4}}
+Minihard tunings 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¢).
-==== Quasihard ====
-Quasihard tunings correspond to "neogothic" or "parapyth" systems whose perfect 5th is sharper than just, resulting in major 3rds that are sharper than 81/64. Compatible edos include 29edo and 46edo.
-edo is considered to be on the sharper end of the neogothic spectrum, with a major 3rd that is more discordant than flatter neogothic tunings.
+Edos include [[41edo]] and [[53edo]].
-{{MOS degrees|Scale Signature=5L 2s|Step Ratio=5/2; 8/3; 3/1}}
+==== Quasihard tunings====
-===Parahard and ultrahard===
+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¢).
-Parahard and ultrahard tunings (step ratio 3:1 or sharper) correspond to "archy" systems. Compatible edos include 17edo and 22edo.
-{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 4/1}}
-== Modes==
+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.
+===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¢.
+Edos include [[17edo]], [[22edo]], [[27edo]], and [[32edo]], among others.
+==Modes==
 Diatonic modes have standard names from classical music theory:
+{{MOS modes|Scale Signature=5L 2s}}
+Each mode has the following scale degrees, reached by raising or lowering certain naturals by a chroma.
+{| class="wikitable"
+|+TODO: Make this into a template
+! colspan="2" |Mode
+! colspan="8" |Scale degree (on C)
+|-
+!UDP
+!Step pattern
+!1st
+!2nd
+!3rd
+!4th
+!5th
+! 6th
+!7th
+!8th
+|-
+|<nowiki>6|0</nowiki>
+|LLLsLLs
+|Perfect (C)
+| Major (D)
+|Major (E)
+|Augmented (F#)
+|Perfect (G)
+|Major (A)
+|Major (B)
+|Perfect (C)
+|-
+|<nowiki>5|1</nowiki>
+|LLsLLLs
+|Perfect (C)
+|Major (D)
+|Major (E)
+|Perfect (F)
+|Perfect (G)
+|Major (A)
+|Major (B)
+| Perfect (C)
+|-
+|<nowiki>4|2</nowiki>
+|LLsLLsL
+|Perfect (C)
+|Major (D)
+|Major (E)
+|Perfect (F)
+|Perfect (G)
+|Major (A)
+|Minor (Bb)
+|Perfect (C)
+|-
+|<nowiki>3|3</nowiki>
+|LsLLLsL
+| Perfect (C)
+|Major (D)
+|Minor (Eb)
+|Perfect (F)
+|Perfect (G)
+|Major (A)
+|Minor (Bb)
+|Perfect (C)
+|-
+|<nowiki>2|4</nowiki>
+|LsLLsLL
+|Perfect (C)
+|Major (D)
+|Minor (Eb)
+| Perfect (F)
+|Perfect (G)
+|Minor (Ab)
+|Minor (Bb)
+|Perfect (C)
+|-
+|<nowiki>1|5</nowiki>
+|sLLLsLL
+|Perfect (C)
+|Minor (Db)
+| Minor (Eb)
+|Perfect (F)
+|Perfect (G)
+|Minor (Ab)
+|Minor (Bb)
+|Perfect (C)
+|-
+|<nowiki>0|6</nowiki>
+|sLLsLLL
+|Perfect (C)
+|Minor (Db)
+|Minor (Eb)
+|Perfect (F)
+|Diminished (Gb)
+|Minor (Ab)
+|Minor (Bb)
+|Perfect (C)
+|}
+== Generator chain ==
+''Explain how the scale can also be generated by stacking 6 generating intervals in any combination of up or down from the root. Also explain how this can be extended further to 11 generators to produce chromatic scales.''
-{{MOS modes|Scale Signature=5L 2s}}
 ==Scales==
 ===Subset and superset scales===
-L 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.
+L 2s has a parent scale of [[2L 3s]], a pentatonic scale, meaning 2L 3s is a subset. 5L 2s also has the two child scales, which are supersets of 5L 2s:
+*[[7L 5s]], a chromatic scale produced using soft-of-basic step ratios.
+*[[5L 7s]], a chromatic scale produced using hard-of-basic step ratios.
+edo contains 5L 2s as the equalized form of both 5L 7s and 7L 5s.
 ===MODMOS scales and muddles===
@@ Line 181: / Line 306: @@
 *[[Archy7]] – 472edo tuning
-==Tuning spectrum==
+==Scale tree ==
-{{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}}
+{{MOS tuning spectrum
+| Scale Signature = 5L 2s
+| Depth = 6
+| 7/5 = [[Flattone]] is in this region
+| 21/13 = [[Golden meantone]] (696.2145{{c}})
+| 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{{c}})
+| 8/3 = [[Neogothic]] is in this region
+| 4/1 = [[Archy]] is in this region
+}}
 ==See also==
+* [[Diatonic functional harmony]]
-*[[Diatonic functional harmony]]

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