Xenharmonic Wiki

5L 2s: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,016 edits
mNo edit summary
Overthink (talk | contribs)
autopatrolled
2,726 edits
 
(95 intermediate revisions by 14 users not shown)
Line 1: Line 1:
{{interwiki
{{interwiki
| en = 5L 2s
| de = 5L2s
| de = 5L2s
| en = 5L 2s
| es =  
| es =  
| ja =  
| ja = 5L 2s
}}
| ko = 5L2s (Korean)
{{Infobox MOS
| Neutral = 3L 4s
}}
}}
{{Infobox MOS}}
{{Wikipedia|Diatonic scale}}
{{MOS intro}}
The familiar pattern of 5 whole steps and 2 half steps, commonly written as WWHWWWH for the major scale, takes on a generalized form of LLsLLLs, where the large and small steps—denoted as ''L''{{'s}} and ''s''{{`s}}—represent whole number step sizes, thus producing different [[edo]]s. These [[step ratio]]s affect the sizes of the diatonic scale's intervals and correspond to different tuning systems.
Among the most well-known forms of this scale are the Pythagorean diatonic scale, and scales produced by meantone systems (including [[12edo]]).
== Name ==
{{TAMNAMS name}} "Mosdiatonic" may also be used for the sake of specificity.
== Notation ==
: ''This article assumes [[TAMNAMS]] for naming step ratios.''
== Scale characteristics ==
{{TAMNAMS use}}
=== Intervals ===
{{MOS intervals}}
=== Generator chain ===
{{MOS genchain}}
=== Modes ===
{{MOS mode degrees}}


One way of distinguishing the '''diatonic''' scale is by considering it a [[MOS scale|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths") with the step combination of '''5L 2s'''. 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.
Diatonic modes have standard names from classical music theory.
{{MOS modes}}


== On the term ''diatonic'' ==
=== Note names ===
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.
Note names are identical to that of standard notation. Thus, the basic gamut for 5L 2s is the following:
{{MOS gamut}}


== Substituting step sizes ==
== Theory ==
The 5L 2s MOS scale has this generalized form.
=== Temperament interpretations ===
{{Main| {{PAGENAME}}/Temperaments }}
5L 2s has several rank-2 temperament interpretations, such as:
* [[Meantone]], with generators around 696.2{{c}}. This includes:
** [[Flattone]], with generators around 693.7{{c}}.
* [[Schismic]], with generators around 702{{c}}.
* [[Leapfrog]], with generators around 704.7{{c}}.
* [[Archy]], with generators around 709.3{{c}}. This includes:
** Supra, with generators around 707.2{{c}}
** [[Superpyth]], with generators around 710.3{{c}}
** [[Ultrapyth]], with generators around 713.7{{c}}.


* L L L s L L s
=== Generator chain ===
{{MOS genchain}}


Insert 2 for L and 1 for s and you'll get the 12EDO diatonic of standard practice.
=== Warped diatonic scales ===
Because of most listeners' familiarity with the 5L 2s diatonic scale, listeners may sometimes experience an effect like pareidolia, hearing 5L 2s even when it isn’t there.  


* 2 2 2 1 2 2 1
A larger scale can be constructed so that it contains chains of 5L 2s, but then breaks the pattern, exploiting that pareidolic effect to surprise and disorient the listener. Scales which have this effect are called [[warped diatonic]] scales.


When L=3, s=1, you have [[17edo|17EDO]]: 3 3 3 1 3 3 1
=== Interval categories ===
''See [[5L 2s/Interval categories]]''.


When L=3, s=2, you have [[19edo|19EDO]]: 3 3 3 2 3 3 2
== Tuning ranges ==
{{Todo|Verify|inline=1|text=Populate/verify tables}}


When L=4, s=1, you have [[22edo|22EDO]]: 4 4 4 1 4 4 1
=== 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.
{{MOS tunings|JI Ratios=Int Limit: 30; Complements Only: 1|Tolerance=20}}


When L=4, s=3, you have [[26edo|26EDO]]: 4 4 4 3 4 4 3
=== Ultrasoft tunings ===
{{See also| Superflat }}
In this range, the major third is so flat that it can best be approximated by [[16/13]], tempering out [[1053/1024]].
{{MOS tunings|Step Ratios=Ultrasoft|JI Ratios=NONE}}


When L=5, s=1, you have [[27edo|27EDO]]: 5 5 5 1 5 5 1
=== Parasoft tunings ===
{{See also| Flattone }}


When L=5, s=2, you have [[29edo|29EDO]]: 5 5 5 2 5 5 2
Parasoft diatonic tunings (4:3 to 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths ([[3/2]], flat of 702{{c}}) to produce major 3rds that are flatter than [[5/4]] (386{{c}}).


When L=5, s=3, you have [[31edo|31EDO]]: 5 5 5 3 5 5 3
Edos include [[19edo]], [[26edo]], [[45edo]], and [[64edo]].
{{MOS tunings|Step Ratios=4/3; 7/5; 10/7; 3/2|JI Ratios=Subgroup: 2.3.5.7.13; Int Limit: 27; Complements Only: 1; Tenney Height: 10|Tolerance=20}}


When L=5, s=4, you have [[33edo|33EDO]]: 5 5 5 4 5 5 4
=== Hyposoft tunings ===
{{See also| Meantone }}


So you have scales where L and s are nearly equal, which approach [[7edo|7EDO]]:
Hyposoft diatonic tunings (3:2 to 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702{{c}}) to produce diatonic major 3rds that approximate 5/4 (386{{c}}).


* 1 1 1 1 1 1 1
Edos include [[19edo]], [[31edo]], [[43edo]], and [[50edo]].
{{MOS tunings|Step Ratios=3/2; 5/3; 8/5; 7/4; 2/1|JI Ratios=Subgroup:2.3.5; Int Limit: 40; Tenney Height: 10|Tolerance=15}}


And you have scales where s becomes so small it approaches zero, which would give us [[5edo|5EDO]]:
=== Hypohard tunings ===
: ''See also: [[Pythagorean tuning]] and [[Schismatic family #Schismatic aka helmholtz|schismatic temperament]]''


* 1 1 1 0 1 1 0 = 1 1 1 1 1
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).
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=NONE}}


== Tuning ranges ==
==== Minihard tunings ====
=== Parasoft to ultrasoft ===
Minihard diatonic tunings correspond to Pythagorean tuning and schismatic temperament, characterized by having a perfect 5th that is as close to just (701.96{{c}}) as possible, resulting in a major 3rd of [[81/64]] (407{{c}}).
"Flattone" systems, such as [[26edo|26EDO]].


=== Hyposoft ===
Edos include [[41edo]] and [[53edo]].
"Meantone" (more properly "septimal meantone") systems, such as [[31edo|31EDO]].
{{MOS tunings|Step Ratios=2/1; 7/3; 5/2; 9/4|JI Ratios=Prime Limit:3; Int Limit: 1024|Tolerance=10}}


=== Hypohard ===
==== Quasihard tunings ====
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.)
Quasihard diatonic 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{{c}}).


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.
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.
{{MOS tunings|Step Ratios=Quasihard|JI Ratios=Subgroup: 2.3.7.11.13; Int Limit: 30; Complements Only: 1|Tolerance=15}}


=== Parahard to ultrahard ===
=== Parahard and ultrahard tunings ===
"Archy" systems such as [[17edo|17EDO]], [[22edo|22EDO]], and [[27edo|27EDO]].
{{See also| Archy }}


== Modes ==
Parahard (3:1 to 4:1) and ultrahard (4:1 to 1:0) diatonic tunings correspond to archy systems, with perfect 5ths that are significantly sharper than than 702{{c}}.
Diatonic modes have standard names from classical music theory:  


{| class="wikitable center-all"
Edos include [[17edo]], [[22edo]], [[27edo]], and [[32edo]], among others.
|-
{{MOS tunings|Step Ratios=3/1; 4/1; 5/1; 6/1|JI Ratios=Subgroup: 2.3.7 ; Int Limit: 80; Complements Only: 1|Tolerance=15}}
! Mode
! [[Modal UDP Notation|UDP]]
! Name
|-
| LLLsLLs
| <nowiki>6|0</nowiki>
| Lydian
|-
| LLsLLLs
| <nowiki>5|1</nowiki>
| Ionian
|-
| LLsLLsL
| <nowiki>4|2</nowiki>
| Mixolydian
|-
| LsLLLsL
| <nowiki>3|3</nowiki>
| Dorian
|-
| LsLLsLL
| <nowiki>2|4</nowiki>
| Aeolian
|-
| sLLLsLL
| <nowiki>1|5</nowiki>
| Phrygian
|-
| sLLsLLL
| <nowiki>0|6</nowiki>
| Locrian
|}


== Scales ==
== Scales ==
=== Subset and superset scales ===
5L&nbsp;2s has a parent scale of [[2L&nbsp;3s]], a pentatonic scale, meaning 2L&nbsp;3s is a subset. 5L&nbsp;2s also has two child scales, which are supersets of 5L&nbsp;2s:
* [[7L&nbsp;5s]], a chromatic scale produced using soft-of-basic step ratios.
* [[5L&nbsp;7s]], a chromatic scale produced using hard-of-basic step ratios.
12edo, the equalized form of both 7L&nbsp;5s and 5L&nbsp;7s, is also a superset of 5L&nbsp;2s.
=== MODMOS scales and muddles ===
{{Main|5L&nbsp;2s/MODMOSes|5L&nbsp;2s/Muddles}}
=== Scala files ===
* [[Meantone7]] – 19edo and 31edo tunings
* [[Meantone7]] – 19edo and 31edo tunings
* [[Nestoria7]] – 171edo tuning
* [[Nestoria7]] – 171edo tuning
Line 106: Line 134:
* [[Garibaldi7]] – 94edo tuning
* [[Garibaldi7]] – 94edo tuning
* [[Cotoneum7]] – 217edo tuning
* [[Cotoneum7]] – 217edo tuning
* [[Edson7]] – 29edo tuning
* [[Pepperoni7]] – 271edo tuning
* [[Pepperoni7]] – 271edo tuning
* [[Supra7]] – 56edo tuning
* [[Supra7]] – 56edo tuning
* [[Archy7]] – 472edo tuning
* [[Archy7]] – 49edo 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.
{{MOS tuning spectrum
 
| Depth = 6
If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
| 7/5 = [[Flattone]] region
 
| 21/13 = [[Golden meantone]] (696.214{{c}})
Generator ranges:
| 5/3 = [[Meantone]] region
* Bright generator: 685.7143 cents (4\7) to 720 cents (3\5)
| 9/4 = [[Pythagorean tuning]] (701.955{{c}})
* Dark generator: 480 cents (2\5) to 514.2857 cents (3\7)
| 16/7 = [[Garibaldi]] / [[cassandra]]
 
| 5/2 = [[Dominant (temperament)|Dominant]] region
{| class="wikitable center-all"
| 21/8 = Golden neogothic (704.096{{c}})
! colspan="7" | Bright generator
| 8/3 = [[Neogothic]] region
! Cents
| 7/2 = [[Quasisuper]] region
! L
| 9/2 = [[Superpyth]] region
! s
| 11/2 = [[Quasiultra]] region
! L/s
| 7/1 = [[Ultrapyth]] region
! Comments
}}
|-
| 4\7 || || || || || || || 685.714 || 1 || 1 || 1.000 ||
|-
| || || || || || || 27\47 || 689.362 || 7 || 6 || 1.167 ||
|-
| || || || || || 23\40 || || 690.000 || 6 || 5 || 1.200 ||
|-
| || || || || || || 42\73 || 690.411 || 11 || 9 || 1.222 ||
|-
| || || || || 19\33 || || || 690.909 || 5 || 4 || 1.250 ||
|-
| || || || || || || 53\92 || 691.304 || 14 || 11 || 1.273 ||
|-
| || || || || || 34\59 || || 691.525 || 9 || 7 || 1.286 ||
|-
| || || || || || || 49\85 || 691.765 || 13 || 10 || 1.300 ||
|-
| || || || 15\26 || || || || 692.308 || 4 || 3 || 1.333 ||
|-
| || || || || || || 56\97 || 692.784 || 15 || 11 || 1.364 ||
|-
| || || || || || 41\71 || || 692.958 || 11 || 8 || 1.375 ||
|-
| || || || || || || 67\116 || 693.103 || 18 || 13 || 1.385 ||
|-
| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 || [[Flattone]] is in this region
|-
| || || || || || || 63\109 || 693.578 || 17 || 12 || 1.417 ||
|-
| || || || || || 37\64 || || 693.750 || 10 || 7 || 1.429 ||
|-
| || || || || || || 48\83 || 693.976 || 13 || 9 || 1.444 ||
|-
| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 ||
|-
| || || || || || || 51\88 || 695.455 || 14 || 9 || 1.556 ||
|-
| || || || || || 40\69 || || 695.652 || 11 || 7 || 1.571 ||
|-
| || || || || || || 69\119 || 695.798 || 19 || 12 || 1.583 ||
|-
| || || || || 29\50 || || || 696.000 || 8 || 5 || 1.600 ||
|-
| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || [[Golden meantone]] (696.2145¢)
|-
| || || || || || 47\81 || || 696.296 || 13 || 8 || 1.625 ||
|-
| || || || || || || 65\112 || 696.429 || 18 || 11 || 1.636 ||
|-
| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || [[Meantone]] is in this region
|-
| || || || || || || 61\105 || 697.143 || 17 || 10 || 1.700 ||
|-
| || || || || || 43\74 || || 697.297 || 12 || 7 || 1.714 ||
|-
| || || || || || || 68\117 || 697.436 || 19 || 11 || 1.727 ||
|-
| || || || || 25\43 || || || 697.674 || 7 || 4 || 1.750 ||
|-
| || || || || || || 57\98 || 697.959 || 16 || 9 || 1.778 ||
|-
| || || || || || 32\55 || || 698.182 || 9 || 5 || 1.800 ||
|-
| || || || || || || 39\67 || 698.507 || 11 || 6 || 1.833 ||
|-
| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic <br>(Generators smaller than this are proper)
|-
| || || || || || || 38\65 || 701.539 || 11 || 5 || 2.200 ||
|-
| || || || || || 31\53 || || 701.887 || 9 || 4 || 2.250 || The generator closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 || [[Garibaldi]] / [[Cassandra]]
|-
| || || || || 24\41 || || || 702.409 || 7 || 3 || 2.333 ||
|-
| || || || || || || 65\111 || 702.703 || 19 || 8 || 2.375 ||
|-
| || || || || || 41\70 || || 702.857 || 12 || 5 || 2.400 ||
|-
| || || || || || || 58\99 || 703.030 || 17 || 7 || 2.428 ||
|-
| || || || 17\29 || || || || 703.448 || 5 || 2 || 2.500 ||
|-
| || || || || || || 61\104 || 703.846 || 18 || 7 || 2.571 ||
|-
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
|-
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic (704.0956¢)
|-
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 || [[Neogothic]] is in this region
|-
| || || || || || || 64\109 || 704.587 || 19 || 7 || 2.714 ||
|-
| || || || || || 37\63 || || 704.762 || 11 || 4 || 2.750 ||
|-
| || || || || || || 47\80 || 705.000 || 14 || 5 || 2.800 ||
|-
| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 ||
|-
| || || || || || || 43\73 || 706.849 || 13 || 4 || 3.250 ||
|-
| || || || || || 33\56 || || 707.143 || 10 || 3 || 3.333 ||
|-
| || || || || || || 56\95 || 707.368 || 17 || 5 || 3.400 ||
|-
| || || || || 23\39 || || || 707.692 || 7 || 2 || 3.500 ||
|-
| || || || || || || 59\100 || 708.000 || 18 || 5 || 3.600 ||
|-
| || || || || || 36\61 || || 708.197 || 11 || 3 || 3.667 ||
|-
| || || || || || || 49\83 || 708.434 || 15 || 4 || 3.750 ||
|-
| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || [[Archy]] is in this region
|-
| || || || || || || 42\71 || 709.859 || 13 || 3 || 4.333 ||
|-
| || || || || || 29\49 || || 710.204 || 9 || 2 || 4.500 ||
|-
| || || || || || || 45\76 || 710.526 || 14 || 3 || 4.667 ||
|-
| || || || || 16\27 || || || 711.111 || 5 || 1 || 5.000 ||
|-
| || || || || || || 35\59 || 711.864 || 11 || 2 || 5.500 ||
|-
| || || || || || 19\32 || || 712.500 || 6 || 1 || 6.000 ||
|-
| || || || || || || 22\37 || 713.514 || 7 || 1 || 7.000 ||
|-
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
|}
 
Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.
 
Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.


=== Step ratio diagram ===
[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]
[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]


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¢).
== See also ==
 
* [[Diatonic functional harmony]]
== Related Scales ==
* [[Diatonic]] (disambiguation page)
{{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.
 
== Rank-2 temperaments ==
{{main| 5L 2s/Temperaments }}
 
== Approaches to Functional Harmony ==
{{see also| Diatonic functional harmony}}


[[Category:Diatonic| ]] <!-- main article -->
[[Category:Diatonic| ]] <!-- Main article -->
[[Category:7-tone scales]]
[[Category:7-tone scales]]
Retrieved from "https://en.xen.wiki/w/5L_2s"