User:Ganaram inukshuk/5L 2s: Difference between revisions
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:''This is a test page. For the main page, see [[5L 2s]].'' | :''This is a test page. For the main page, see [[5L 2s]].'' | ||
{{MOS intro|Scale Signature=5L 2s}} | {{MOS intro|Scale Signature=5L 2s}} | ||
==Name== | Among the most-well known forms of this scale are the Pythagorean diatonic scale and those 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 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 tetrachords, just intonation, or in general have more than one size of whole tone. Such 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 such scales, depending on what's contextually the most appropriate. | |||
==Notation== | ==Notation== | ||
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|- | |- | ||
!Size | !Size | ||
!Quality | ! Quality | ||
!Size | !Size | ||
!Quality | !Quality | ||
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|Perfect | |Perfect | ||
|0 | |0 | ||
|Perfect | | Perfect | ||
|- | |- | ||
|2nd | |2nd | ||
|L | |L | ||
|Major | | Major | ||
|s | |s | ||
|Minor | |Minor | ||
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|3rd | |3rd | ||
|2L | |2L | ||
|Major | | Major | ||
|L + s | |L + s | ||
|Minor | | Minor | ||
|- | |- | ||
|4th | |4th | ||
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|- | |- | ||
|7th | |7th | ||
|5L + 1s | | 5L + 1s | ||
|Major | |Major | ||
|4L + 2s | |4L + 2s | ||
|Minor | | Minor | ||
|- | |- | ||
|'''8th (octave)''' | |'''8th (octave)''' | ||
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{{MOS gamut|Scale Signature=5L 2s}} | {{MOS gamut|Scale Signature=5L 2s}} | ||
==Theory== | ==Theory == | ||
===Introduction to large and small steps=== | ===Introduction to large and small steps=== | ||
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|- | |- | ||
|5:3 | |5:3 | ||
|5 5 3 5 5 5 3 | | 5 5 3 5 5 5 3 | ||
|[[31edo]] | |[[31edo]] | ||
| | | | ||
|- | |- | ||
|2:1 | |2:1 | ||
|2 2 1 2 2 2 1 | | 2 2 1 2 2 2 1 | ||
|[[12edo]] (standard tuning) | |[[12edo]] (standard tuning) | ||
|[[24edo]], [[36edo]], etc. | |[[24edo]], [[36edo]], etc. | ||
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| | | | ||
|- | |- | ||
|3:1 | | 3:1 | ||
|3 3 1 3 3 3 1 | |3 3 1 3 3 3 1 | ||
|[[17edo]] | |[[17edo]] | ||
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|- | |- | ||
|4:1 | |4:1 | ||
|4 4 1 4 4 4 1 | | 4 4 1 4 4 4 1 | ||
|[[22edo]] | |[[22edo]] | ||
| | | | ||
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===Rank-2 temperament interpretations=== | ===Rank-2 temperament interpretations=== | ||
: ''Main article: [[5L 2s/Temperaments]]'' | :''Main article: [[5L 2s/Temperaments]]'' | ||
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¢. This includes: | *[[Meantone]], with generators around 696.2¢. This includes: | ||
** [[Flattone]], with generators around 693.7¢. | **[[Flattone]], with generators around 693.7¢. | ||
*[[Schismic]], with generators around 702¢. | *[[Schismic]], with generators around 702¢. | ||
*[[Parapyth]], with generators around 704.7¢. | *[[Parapyth]], with generators around 704.7¢. | ||
*[[Archy]], with generators around 709.3¢. This includes: | *[[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¢ | ||
**Ultrapyth, with generators around 713.7¢. | **Ultrapyth, with generators around 713.7¢. | ||
== Step ratio ranges== | ==Step ratio ranges== | ||
=== 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}} | ||
=== Parasoft step ratios === | ===Parasoft step ratios=== | ||
:''Main article: [[Flattone]]'' | :''Main article: [[Flattone]]'' | ||
Parasoft step ratios (between 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}} | Parasoft step ratios (between 4:3 and 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=3/2; 4/3; 7/5; 10/7|Genchain Extend=0, 5}} | ||
===Hyposoft step ratios=== | === Hyposoft step ratios=== | ||
:''Main article: [[Meantone]]'' | :''Main article: [[Meantone]]'' | ||
Hyposoft step ratios (between 3:2 and 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}} | Hyposoft step ratios (between 3:2 and 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 === | ===Hypohard step ratios=== | ||
:''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]'' | :''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 and 5:2) and quasihard range (between 5:2 and 3:1). | The range of hypohard step ratios can be divided into a minihard range (between 2:1 and 5:2) and quasihard range (between 5:2 and 3:1). | ||
==== Minihard step ratios ==== | ====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}} | 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==== | ||
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. | Quasihard step ratios 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¢). | ||
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}} | 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}} | ||
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:''Main article: [[Archy]]'' | :''Main article: [[Archy]]'' | ||
The parahard and ultrahard ranges (between 3:1 and 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}} | The parahard and ultrahard ranges (between 3:1 and 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}} | ||
==Modes== | == Modes == | ||
Diatonic modes have standard names from classical music theory: | Diatonic modes have standard names from classical music theory: | ||
{{MOS modes|Scale Signature=5L 2s}} | {{MOS modes|Scale Signature=5L 2s}} | ||
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|Diminished (Gb) | |Diminished (Gb) | ||
|Minor (Ab) | |Minor (Ab) | ||
|Minor (Bb) | | Minor (Bb) | ||
|Perfect (C) | | Perfect (C) | ||
|} | |} | ||
==Scales== | ==Scales== | ||
===Subset and superset scales=== | === Subset and superset scales=== | ||
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. | 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. | ||
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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}} | 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]] | ||