User:Ganaram inukshuk/5L 2s: Difference between revisions
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!Step pattern | !Step pattern | ||
!EDO | !EDO | ||
!Selected multiples | |||
|- | |||
|1:1 | |||
|1 1 1 1 1 1 1 | |||
|[[7edo]] | |||
|[[14edo]], [[21edo]], etc. | |||
|- | |- | ||
|4:3 | |4:3 | ||
|4 4 3 4 4 4 3 | |4 4 3 4 4 4 3 | ||
|[[26edo]] | |[[26edo]] | ||
| | |||
|- | |- | ||
|3:2 | |3:2 | ||
|3 3 2 3 3 3 2 | |3 3 2 3 3 3 2 | ||
|[[19edo]] | |[[19edo]] | ||
|[[38edo]] | |||
|- | |- | ||
|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. | |||
|- | |- | ||
|5:2 | |5:2 | ||
|5 5 2 5 5 5 2 | |5 5 2 5 5 5 2 | ||
|[[29edo]] | |[[29edo]] | ||
| | |||
|- | |- | ||
|3:1 | |3:1 | ||
|3 3 1 3 3 3 1 | |3 3 1 3 3 3 1 | ||
|[[17edo]] | |[[17edo]] | ||
|[[34edo]] | |||
|- | |- | ||
|4:1 | |4:1 | ||
|4 4 1 4 4 4 1 | |4 4 1 4 4 4 1 | ||
|[[22edo]] | |[[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]] | 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]]. | ||
The step ratios 1:1 and 1:0 represent extreme tunings that technically do not form a valid 5L . 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 | 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=== | ===Temperament interpretations=== | ||
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: ''Main article: [[5L 2s/Temperaments]]'' | : ''Main article: [[5L 2s/Temperaments]]'' | ||
5L 2s has several temperament interpretations, such as: | 5L 2s has several rank-2 temperament interpretations, such as: | ||
* Flattone, with a generator around 693.7¢. | * Flattone, with a generator around 693.7¢. | ||
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=== Simple tunings === | === Simple tunings === | ||
17edo and 19edo are the smallest edos that offer | 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}} | ||
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=2/1; 3/1; 3/2|Genchain Extend=7}} | |||
===Parasoft=== | ===Parasoft=== | ||
:''Main article: [[Flattone]]'' | :''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¢) | 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¢). | ||
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=4/3; 7/5; 10/7}} | {{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 4/3; 7/5; 10/7}} | ||
===Hyposoft=== | ===Hyposoft=== | ||
:''Main article: [[Meantone]]'' | :''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¢) | 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 closely approximates 5/4 (386¢). | ||
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 5/3; 7/4; 8/5}} | {{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/2; 5/3; 7/4; 8/5}} | ||
===Hypohard === | ===Hypohard === | ||
:''Main | :''Main article: [[Pythagorean tuning]] and [[Schismatic family#Schismatic aka Helmholtz|schismatic temperament]]'' | ||
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). | 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). | ||
==== Minihard ==== | ==== Minihard ==== | ||
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 | 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. | ||
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=7/3; 9/4}} | {{MOS degrees|Scale Signature=5L 2s|Step Ratio=7/3; 9/4}} | ||
==== Quasihard ==== | ==== 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 | 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. | ||
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. | 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}} | {{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 5/2; 8/3}} | ||
===Parahard and ultrahard=== | ===Parahard and ultrahard=== | ||
Parahard and ultrahard tunings (step ratio 3:1 or sharper) correspond to "archy" systems | Parahard and ultrahard tunings (step ratio 3:1 or sharper) correspond to "archy" systems, whose perfect 5th is significantly sharper than just. | ||
{{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 4/1; 5/1}} | {{MOS degrees|Scale Signature=5L 2s|Step Ratio=3/1; 4/1; 5/1}} | ||
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==Tuning spectrum== | ==Tuning spectrum== | ||
{{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}} | 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. | ||
{{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}} | |||
==See also== | ==See also== | ||
*[[Diatonic functional harmony]] | *[[Diatonic functional harmony]] | ||