5L 2s: Difference between revisions
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One way of distinguishing the | {{MOS intro}} | ||
One way of distinguishing the diatonic scale is by considering it a [[MOS scale|moment of symmetry]] scale. Among the most well-known variants of this MOS proper are [[12edo|12EDO]]'s diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS. | |||
== On the term ''diatonic'' == | ==On the term ''diatonic''== | ||
In [[TAMNAMS]] (which is the convention on all pages on scale patterns on the wiki), [[diatonic]] exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate. | In [[TAMNAMS]] (which is the convention on all pages on scale patterns on the wiki), [[diatonic]] exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as [[Zarlino]], [[blackdye]] and [[diasem]], are called ''[[Detempering|detempered]]'' (if the philosophy is [[RTT]]-based) or ''deregularized'' (RTT-agnostic) ''diatonic scales''. The adjectives ''diatonic-like'' or ''diatonic-based'' may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate. | ||
== Substituting step sizes == | ==Substituting step sizes== | ||
The 5L 2s MOS scale has this generalized form. | The 5L 2s MOS scale has this generalized form. | ||
* L L L s L L s | *L L L s L L s | ||
Insert 2 for L and 1 for s and you'll get the 12EDO diatonic of standard practice. | Insert 2 for L and 1 for s and you'll get the 12EDO diatonic of standard practice. | ||
* 2 2 2 1 2 2 1 | *2 2 2 1 2 2 1 | ||
When L=3, s=1, you have [[17edo|17EDO]]: 3 3 3 1 3 3 1 | When L=3, s=1, you have [[17edo|17EDO]]: 3 3 3 1 3 3 1 | ||
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So you have scales where L and s are nearly equal, which approach [[7edo|7EDO]]: | So you have scales where L and s are nearly equal, which approach [[7edo|7EDO]]: | ||
* 1 1 1 1 1 1 1 | *1 1 1 1 1 1 1 | ||
And you have scales where s becomes so small it approaches zero, which would give us [[5edo|5EDO]]: | And you have scales where s becomes so small it approaches zero, which would give us [[5edo|5EDO]]: | ||
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* 1 1 1 0 1 1 0 = 1 1 1 1 1 | * 1 1 1 0 1 1 0 = 1 1 1 1 1 | ||
== Tuning ranges == | == Tuning ranges== | ||
=== Parasoft to ultrasoft === | ===Parasoft to ultrasoft=== | ||
"Flattone" systems, such as [[26edo|26EDO]]. | "Flattone" systems, such as [[26edo|26EDO]]. | ||
=== Hyposoft === | ===Hyposoft=== | ||
"Meantone" (more properly "septimal meantone") systems, such as [[31edo|31EDO]]. | "Meantone" (more properly "septimal meantone") systems, such as [[31edo|31EDO]]. | ||
=== Hypohard === | ===Hypohard=== | ||
The near-just part of the region is of interest mainly for those interested in [[Pythagorean tuning]] and large, accurate EDO systems based on close-to-Pythagorean fifths, such as [[41edo|41EDO]] and [[53edo|53EDO]]. This class of tunings is called [[schisma|schismic]] temperament; these tunings can approximate 5-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12EDO tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.) | The near-just part of the region is of interest mainly for those interested in [[Pythagorean tuning]] and large, accurate EDO systems based on close-to-Pythagorean fifths, such as [[41edo|41EDO]] and [[53edo|53EDO]]. This class of tunings is called [[schisma|schismic]] temperament; these tunings can approximate 5-limit harmonies very accurately by [[tempering out]] a small comma called the [[schisma]]. (Technically, 12EDO tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.) | ||
The sharp-of-just part of this range includes so-called "[[neogothic]]" or "parapyth" systems, which tune the diatonic major third slightly sharply of [[81/64]] (around 414 to 423 cents) and the diatonic minor third slightly flatly of [[32/27]] (around 282 to 290 cents). Good neogothic EDOs include [[29edo|29EDO]] and [[46edo|46EDO]]. [[17edo|17EDO]] is often considered the sharper end of the neogothic spectrum; its major third at 423 cents is considerably more discordant than in flatter neogothic tunings. | The sharp-of-just part of this range includes so-called "[[neogothic]]" or "parapyth" systems, which tune the diatonic major third slightly sharply of [[81/64]] (around 414 to 423 cents) and the diatonic minor third slightly flatly of [[32/27]] (around 282 to 290 cents). Good neogothic EDOs include [[29edo|29EDO]] and [[46edo|46EDO]]. [[17edo|17EDO]] is often considered the sharper end of the neogothic spectrum; its major third at 423 cents is considerably more discordant than in flatter neogothic tunings. | ||
=== Parahard to ultrahard === | ===Parahard to ultrahard=== | ||
"Archy" systems such as [[17edo|17EDO]], [[22edo|22EDO]], and [[27edo|27EDO]]. | "Archy" systems such as [[17edo|17EDO]], [[22edo|22EDO]], and [[27edo|27EDO]]. | ||
== Modes == | ==Modes== | ||
Diatonic modes have standard names from classical music theory: | Diatonic modes have standard names from classical music theory: | ||
{{MOS modes}} | {{MOS modes}} | ||
== Scales == | ==Scales== | ||
* [[Meantone7]] – 19edo and 31edo tunings | * [[Meantone7]] – 19edo and 31edo tunings | ||
* [[Nestoria7]] – 171edo tuning | * [[Nestoria7]] – 171edo tuning | ||
* [[Pythagorean7]] – Pythagorean tuning | *[[Pythagorean7]] – Pythagorean tuning | ||
* [[Garibaldi7]] – 94edo tuning | *[[Garibaldi7]] – 94edo tuning | ||
* [[Cotoneum7]] – 217edo tuning | * [[Cotoneum7]] – 217edo tuning | ||
* [[Pepperoni7]] – 271edo tuning | *[[Pepperoni7]] – 271edo tuning | ||
* [[Supra7]] – 56edo tuning | *[[Supra7]] – 56edo tuning | ||
* [[Archy7]] – 472edo tuning | *[[Archy7]] – 472edo tuning | ||
== Scale tree == | ==Scale tree== | ||
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO. | If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO. | ||
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Generator ranges: | Generator ranges: | ||
* Bright generator: 685.7143 cents (4\7) to 720 cents (3\5) | *Bright generator: 685.7143 cents (4\7) to 720 cents (3\5) | ||
* Dark generator: 480 cents (2\5) to 514.2857 cents (3\7) | *Dark generator: 480 cents (2\5) to 514.2857 cents (3\7) | ||
{{Scale tree|depth=7|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}} | {{Scale tree|depth=7|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}} | ||
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5L 2s contains the pentatonic MOS [[2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L 5s]] or [[5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700¢). | 5L 2s contains the pentatonic MOS [[2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L 5s]] or [[5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700¢). | ||
== Related Scales == | ==Related Scales == | ||
{{main| 5L 2s MODMOSes }} ''and [[5L 2s Muddles]]'' | {{main| 5L 2s MODMOSes }} ''and [[5L 2s Muddles]]'' | ||
Because the diatonic scale is so widely used, it should be no surprise that there are a number of noteworthy scales of different sorts related to this MOS. | Because the diatonic scale is so widely used, it should be no surprise that there are a number of noteworthy scales of different sorts related to this MOS. | ||
== Rank-2 temperaments == | ==Rank-2 temperaments == | ||
{{main| 5L 2s/Temperaments }} | {{main| 5L 2s/Temperaments }} | ||
== Approaches to Functional Harmony == | == Approaches to Functional Harmony== | ||
{{see also| Diatonic functional harmony}} | {{see also| Diatonic functional harmony}} | ||
[[Category:Diatonic| ]] <!-- main article --> | [[Category:Diatonic| ]] <!-- main article --> | ||
[[Category:7-tone scales]] | [[Category:7-tone scales]] | ||