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One way of distinguishing the "diatonic" scale is by considering it a [[MOSScales|moment of symmetry]] scale produced by a chain of "fifths" (or "fourths"). This will include [[12edo]]'s diatonic scale along with the Pythagorean diatonic scale and meantone systems, while excluding just intonation scales that use more than one size of "tone".
{{interwiki
| en = 5L 2s
| de = 5L2s
| es =
| ja = 5L 2s
| ko = 5L2s (Korean)
}}
{{Infobox MOS}}
{{Wikipedia|Diatonic scale}}


It may be misleading to call 5L 2s "diatonic," since other scales called diatonic can be arrived at different ways (through just intonation procedures for instance, or with tetrachords). Also, a composer working with a 5L 2s scale may choose to do something very different than typical diatonic music.
{{MOS intro}}


==Substituting step sizes==
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.


The 5L 2s MOS scale has this generalized form.
Among the most well-known forms of this scale are the Pythagorean diatonic scale, and scales produced by meantone systems (including [[12edo]]).


L L s L L L s
== Name ==
{{TAMNAMS name}} "Mosdiatonic" may also be used for the sake of specificity.


Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice.
== Notation ==
: ''This article assumes [[TAMNAMS]] for naming step ratios.''


2 2 1 2 2 2 1
== Scale characteristics ==
{{TAMNAMS use}}


When L=3, s=1, you have [[17edo]]: 3 3 1 3 3 3 1
=== Intervals ===
{{MOS intervals}}


When L=3, s=2, you have [[19edo]]: 3 3 2 3 3 3 2
=== Generator chain ===
{{MOS genchain}}


When L=4, s=1, you have [[22edo]]: 4 4 1 4 4 4 1
=== Modes ===
{{MOS mode degrees}}


When L=4, s=3, you have [[26edo]]: 4 4 3 4 4 4 3
Diatonic modes have standard names from classical music theory.
{{MOS modes}}


When L=5, s=1, you have [[27edo]]: 5 5 1 5 5 5 1
=== Note names ===
Note names are identical to that of standard notation. Thus, the basic gamut for 5L 2s is the following:  
{{MOS gamut}}


When L=5, s=2, you have [[29edo]]: 5 5 2 5 5 5 2
== Theory ==
=== 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}}.


When L=5, s=3, you have [[31edo]]: 5 5 3 5 5 5 3
=== Generator chain ===
{{MOS genchain}}


When L=5, s=4, you have [[33edo]]: 5 5 4 5 5 5 4
=== 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.


So you have scales where L and s are nearly equal, which approach [[7edo]]:
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.


1 1 1 1 1 1 1
=== Interval categories ===
''See [[5L 2s/Interval categories]]''.


And you have scales where s becomes so small it approaches zero, which would give us [[5edo]]:
== Tuning ranges ==
{{Todo|Verify|inline=1|text=Populate/verify tables}}


1 1 0 1 1 1 0 = 1 1 1 1 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}}


==A continuum of tunings==
=== Ultrasoft tunings ===
So 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 "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:
{{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}}


{| class="wikitable"
=== Parasoft tunings ===
|-
{{See also| Flattone }}
| | 4\7
| |
|-
| |
| | 7\12
|-
| | 3\5
| |
|}


If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
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}}).


{| class="wikitable"
Edos include [[19edo]], [[26edo]], [[45edo]], and [[64edo]].
! colspan="16" | generator
{{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}}
! | cents
! | L
! | s
! | L/s
! | comments
|-
| 4\7||||||||||||||
|
|
|
|
|
|
|
|||685.714||1||1||1.000||
|-
| ||||||||||||||
|
|
|
|
|
|
|
|''63\110''||687.273||16||15||1.067||
|-
| ||||||||||||||
|
|
|
|
|
|
|''59\103''
|||687.379||15||14||1.071||
|-
| ||||||||||||||
|
|
|
|
|
|''55\96''
|
|||687.500||14||13||1.077||
|-
| ||||||||||||||
|
|
|
|
|''51\89''
|
|
|||687.640||13||12||1.083||
|-
| ||||||||||||||
|
|
|
|''47\82''
|
|
|
|||687.805||12||11||1.091||
|-
| ||||||||||||||
|
|
|''43\75''
|
|
|
|
|||688.000||11||10||1.100||
|-
| ||||||||||||||
|
|''39\68''
|
|
|
|
|
|||688.235||10||9||1.111||
|-
| ||||||||||||||
|''35\61''
|
|
|
|
|
|
|||688.525||9||8||1.125||
|-
| ||||||||||||||31\54
|
|
|
|
|
|
|
|||688.889||8||7||1.143||
|-
| ||||||||||||27\47||
|
|
|
|
|
|
|
|||689.362||7||6||1.167||
|-
| ||||||||||23\40||||
|
|
|
|
|
|
|
|||690.000||6||5||1.200||
|-
| ||||||||19\33||||||
|
|
|
|
|
|
|
|||690.909||5||4||1.250||
|-
| ||||||||||34\59||||
|
|
|
|
|
|
|
|||691.525||9||7||1.286||
|-
| ||||||15\26||||||||
|
|
|
|
|
|
|
|||692.308||4||3||1.333||
|-
| ||||||||||41\71||||
|
|
|
|
|
|
|
|||692.958||11||8||1.375||
|-
| ||||||||26\45||||||
|
|
|
|
|
|
|
|||693.333||7||5||1.400||
|-
| ||||||||||37\64||||
|
|
|
|
|
|
|
|||693.750||10||7||1.429||
|-
| ||||11\19||||||||||
|
|
|
|
|
|
|
|||694.737||3||2||1.500||Optimum rank range (L/s=3/2) diatonic
|-
| ||||||||||||51\88||
|
|
|
|
|
|
|
|||695.455||14||9||1.556||
|-
| ||||||||||||||
|
|
|
|
|
|
|
|||695.644||π||2||1.571||LucyTuning
|-
| ||||||||||40\69||||
|
|
|
|
|
|
|
|||695.652||11||7||1.571||
|-
| ||||||||29\50||||||
|
|
|
|
|
|
|
|||696.000||8||5||1.600||
|-
| ||||||||||||76\131||
|
|
|
|
|
|
|
|||696.183||21||13||1.615||
|-
| ||||||||||||||
|''199\343''
|
|
|
|
|
|
|||696.210||55||34||1.618||
|-
| ||||||||||||||
|
|
|
|
|
|
|
|||696.215||φ||1||1.618||Golden meantone
|-
| ||||||||||||||
|
|''322\555''
|
|
|
|
|
|||696.216||89||55||1.618||
|-
| ||||||||||||||123\212
|
|
|
|
|
|
|
|||696.226||34||21||1.619||
|-
| ||||||||||47\81||||
|
|
|
|
|
|
|
|||696.296||13||8||1.625||
|-
| ||||||18\31||||||||
|
|
|
|
|
|
|
|||696.774||5||3||1.667||Meantone is in this region
|-
| ||||||||||43\74||||
|
|
|
|
|
|
|
|||697.297||12||7||1.714||
|-
| ||||||||||||||
|
|
|
|
|
|
|
|||697.487||√3||1||1.732||
|-
| ||||||||25\43||||||
|
|
|
|
|
|
|
|||697.674||7||4||1.750||
|-
| ||||||||||32\55||||
|
|
|
|
|
|
|
|||698.182||9||5||1.800||
|-
| ||||||||||||39\67||
|
|
|
|
|
|
|
|||698.507||11||6||1.833||
|-
| ||||||||||||||46\79
|
|
|
|
|
|
|
|||698.734||13||7||1.857||
|-
| ||||||||||||||
|''53\91''
|
|
|
|
|
|
|||698.901||15||8||1.875||
|-
| ||||||||||||||
|
|''60\103''
|
|
|
|
|
|||699.029||17||9||1.889||
|-
| ||7\12||||||||||||
|
|
|
|
|
|
|
|||700.000||2||1||2.000||Boundary of propriety (generators smaller than this are proper)
|-
| ||||||||||||||
|
|''59\101''
|
|
|
|
|
|||700.990||17||8||2.125||
|-
| ||||||||||||||
|''52\89''
|
|
|
|
|
|
|||701.124||15||7||2.143||
|-
| ||||||||||||||45\77
|
|
|
|
|
|
|
|||701.299||13||6||2.167||
|-
| ||||||||||||38\65||
|
|
|
|
|
|
|
|||701.539||11||5||2.200||
|-
| ||||||||||31\53||||
|
|
|
|
|
|
|
|||701.887||9||4||2.250||
|-
| ||||||||||||||
|
|
|
|
|
|
|
|||701.955||||||2.260||Pythagorean (g = 3/2 ; L=9/8 ; s=256/243)
|-
| ||||||||24\41||||||
|
|
|
|
|
|
|
|||702.409||7||3||2.333||
|-
| ||||||||||41\70||||
|
|
|
|
|
|
|
|||702.857||12||5||2.400||
|-
| ||||||17\29||||||||
|
|
|
|
|
|
|
|||703.448||5||2||2.500||
|-
| ||||||||||44\75||||
|
|
|
|
|
|
|
|||704.000||13||5||2.600||
|-
| ||||||||||||||115\196
|
|
|
|
|
|
|
|||704.082||34||13||2.615||
|-
| ||||||||||||||
|''186\317''
|
|
|
|
|
|
|||704.101||55||21||2.619||
|-
| ||||||||||||71\121||
|
|
|
|
|
|
|
|||704.132||21||8||2.625||
|-
| ||||||||27\46||||||
|
|
|
|
|
|
|
|||704.348||8||3||2.667||
|-
| ||||||||||||||
|
|
|
|
|
|
|
|||704.607||e||1||2.718||
|-
| ||||||||||37\63||||
|
|
|
|
|
|
|
|||704.762||11||4||2.750||
|-
| ||||||||||||47\80||
|
|
|
|
|
|
|
|||705.000||14||5||2.800||
|-
| ||||10\17||||||||||
|
|
|
|
|
|
|
|||705.882||3||1||3.000||
|-
| ||||||||||||||
|
|
|
|
|
|
|
|||706.447||π||1||3.142||
|-
| ||||||||||33\56||||
|
|
|
|
|
|
|
|||707.143||10||3||3.333||
|-
| ||||||||23\39||||||
|
|
|
|
|
|
|
|||707.692||7||2||3.500||
|-
| ||||||||||36\61||||
|
|
|
|
|
|
|
|||708.197||11||3||3.667||
|-
| ||||||13\22||||||||
|
|
|
|
|
|
|
|||709.091||4||1||4.000||(No-5's) superpyth is in this region
|-
| ||||||||||29\49||||
|
|
|
|
|
|
|
|||710.204||9||2||4.500||
|-
| ||||||||16\27||||||
|
|
|
|
|
|
|
|||711.111||5||1||5.000||
|-
| ||||||||||19\32||||
|
|
|
|
|
|
|
|||712.500||6||1||6.000||
|-
| ||||||||||||22\37||
|
|
|
|
|
|
|
|||713.514||7||1||7.000||
|-
| ||||||||||||||25\42
|
|
|
|
|
|
|
|||714.286||8||1||8.000||
|-
| ||||||||||||||
|''28\47''
|
|
|
|
|
|
|||714.894||9||1||9.000||
|-
| ||||||||||||||
|
|''31\52''
|
|
|
|
|
|||715.385||10||1||10.000||
|-
| ||||||||||||||
|
|
|''34\57''
|
|
|
|
|||715.790||11||1||11.000||
|-
| ||||||||||||||
|
|
|
|''37\62''
|
|
|
|||716.129||12||1||12.000||
|-
| ||||||||||||||
|
|
|
|
|''40\67''
|
|
|||716.418||13||1||13.000||
|-
| ||||||||||||||
|
|
|
|
|
|''43\72''
|
|||716.667||14||1||14.000||
|-
| ||||||||||||||
|
|
|
|
|
|
|''46\77''
|||716.883||15||1||15.000||
|-
| ||||||||||||||
|
|
|
|
|
|
|
|''49\82''||717.073||16||1||16.000||
|-
| 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.
=== Hyposoft tunings ===
{{See also| Meantone }}


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.
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}}).


[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]
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}}


5L 2s contains the pentatonic MOS [[2L_3s|2L 3s]] and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either [[7L_5s|7L 5s]] or [[5L_7s|5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700c).
=== Hypohard tunings ===
: ''See also: [[Pythagorean tuning]] and [[Schismatic family #Schismatic aka helmholtz|schismatic temperament]]''


== Rank-2 temperaments ==
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).
Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the 5L 2s range (the [[period]] is always 1\1). The temperaments are listed roughly in order of increasing generator size and child temperaments are extensions of low-[[complexity]] parent temperaments.
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=NONE}}
=== Meantone (12&19, 2.3.5) ===
Period: 1\1


Optimal ([[POTE]]) generator: ~3/2 = 696.239
==== Minihard tunings ====
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}}).


EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
Edos include [[41edo]] and [[53edo]].
{{MOS tunings|Step Ratios=2/1; 7/3; 5/2; 9/4|JI Ratios=Prime Limit:3; Int Limit: 1024|Tolerance=10}}


Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
==== Quasihard tunings ====
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}}).


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
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.
<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>
{{MOS tunings|Step Ratios=Quasihard|JI Ratios=Subgroup: 2.3.7.11.13; Int Limit: 30; Complements Only: 1|Tolerance=15}}
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #Gens up
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 696.2
| 3/2
|-
| 2
| 192.5
| 9/8, 10/9
|-
| 3
| 888.7
| 5/3
|-
| 4
| 385.0
| 5/4
|-
| 5
| 1081.2
| 15/8
|-
| 6
| 577.434
| 25/18
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [{{val|1 0 -4}}, {{val|0 1 4}}]


Comma: 81/80
=== Parahard and ultrahard tunings ===
{{See also| Archy }}


Mapping generator: ~3
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}}.


[[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5)
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}}


[[Tuning Ranges of Regular Temperaments|nice range]]: [694.786, 701.955] (1/3 comma to Pythagorean)
== 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.


[[Tuning Ranges of Regular Temperaments|strict range]]: [694.786, 701.955]
=== MODMOS scales and muddles ===
{{Main|5L&nbsp;2s/MODMOSes|5L&nbsp;2s/Muddles}}


{{EDOs|legend=1| 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb }}
=== Scala files ===
* [[Meantone7]] – 19edo and 31edo tunings
* [[Nestoria7]] – 171edo tuning
* [[Pythagorean7]] – Pythagorean tuning
* [[Garibaldi7]] – 94edo tuning
* [[Cotoneum7]] – 217edo tuning
* [[Edson7]] – 29edo tuning
* [[Pepperoni7]] – 271edo tuning
* [[Supra7]] – 56edo tuning
* [[Archy7]] – 49edo tuning


[[Wedgie]]: {{wedgie|1 4 4}}
== Scale tree ==
 
{{MOS tuning spectrum
[[Badness]]: 0.00736
| Depth = 6
</div></div>
| 7/5 = [[Flattone]] region
==== Flattone (19&26, 2.3.5.7.13) ====
| 21/13 = [[Golden meantone]] (696.214{{c}})
Period: 1\1
| 5/3 = [[Meantone]] region
 
| 9/4 = [[Pythagorean tuning]] (701.955{{c}})
Optimal ([[POTE]]) generator: ~3/2 = 693.7498
| 16/7 = [[Garibaldi]] / [[cassandra]]
 
| 5/2 = [[Dominant (temperament)|Dominant]] region
EDO generators: [[19edo|11\19]], [[26edo|15\26]], [[45edo|26\45]], [[64edo|37\64]]
| 21/8 = Golden neogothic (704.096{{c}})
 
| 8/3 = [[Neogothic]] region
Scales (Scala files): [[Flattone12]]
| 7/2 = [[Quasisuper]] region
 
| 9/2 = [[Superpyth]] region
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
| 11/2 = [[Quasiultra]] region
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)</div>
| 7/1 = [[Ultrapyth]] region
<div class="mw-collapsible-content">
}}
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #Gens up
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7.13, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 693.7
| [[3/2]]
|-
| 2
| 187.5
| [[9/8]], [[10/9]]
|-
| 3
| 881.2
| [[5/3]]
|-
| 4
| 375.0
| [[5/4]], [[16/13]]
|-
| 5
| 1068.7
| 15/8, [[24/13]]
|-
| 6
| 562.5
| [[18/13]]
|-
| 7
| 56.2
|
|-
| 8
| 750.0
| [[20/13]]
|-
| 9
| 243.7
| [[8/7]]
|-
| 10
| 937.5
| [[12/7]]
|-
| 11
| 431.2
| [[9/7]]
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 17 10}}, {{val|0 1 4 -9 -4}}]
 
[[7-odd-limit|7-limit]] minimax
 
[{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }},
{{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}<nowiki>]</nowiki>
 
[[Eigenmonzo]]s: 2, 7/5
 
[[9-odd-limit|9-limit]] minimax
 
[{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }},
{{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}<nowiki>]</nowiki>
 
Eigenmonzos: 2, 9/7
 
valid range: [692.308, 694.737] (26 to 19)
 
nice range: [692.353, 701.955]
 
strict range: [692.353, 694.737]
 
Mapping generator: ~3
 
Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
 
[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}
 
[[Generator]]s: 2, 3
 
{{EDOs|legend=1| 7, 19, 26, 45 }}
 
[[Badness]]: 0.0386
</div></div>
 
==== Septimal meantone (19&12, 2.3.5.7) ====
 
Period: 1\1
 
Optimal ([[POTE]]) generator: 696.495
 
EDO generators: [[12edo|7\12]], [[19edo|11\19]], [[31edo|18\31]], [[43edo|25\43]], [[50edo|29\50]]
 
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|+
|-
! #Gens up
! Cents <ref>octave-reduced</ref>
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
|-
| 0
| 0.00
| 1/1
|-
| 1
| 696.5
| 3/2
|-
| 2
| 193.0
| 9/8, 10/9
|-
| 3
| 889.5
| 5/3
|-
| 4
| 386.0
| 5/4
|-
| 5
| 1082.5
| 15/8, 28/15
|-
| 6
| 579.0
| 7/5
|-
| 7
| 75.5
| 21/20, 25/24, 28/27
|-
| 8
| 772.0
| 14/9, 25/16
|-
| 9
| 268.5
| 7/6
|-
| 10
| 965.0
| 7/4
|-
| 11
| 461.4
| 21/16
|}
<references/></div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
 
[[Comma]]s: 81/80, 126/125
 
7 and [[9-odd-limit|9-limit]] minimax
 
[{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
 
[[Eigenmonzo]]s: 2, 5
 
[[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 12)
 
nice range: [694.786, 701.955]
 
strict range: [694.786, 700.000]
 
Mapping generator: ~3
 
Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
 
[[Wedgie]]: {{wedgie|1 4 10 4 13 12}}
 
[[Vals]]: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb
 
[[Badness]]: 0.0137
</div></div>
===== Meanpop (31&50, 2.3.5.7.11) =====
*-13 gens = 11/8
*Good EDO gens: 29\50
===== Huygens (31&43, 2.3.5.7.11) =====
[[Period]]: 1\1
 
[[POTE generator]]: ~3/2 = 696.967
 
EDO generators: [[43edo|25\43]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mappings|Period-generator mapping]]: [&lt;1 0 -4 -13 -25|, &lt;0 1 4 10 18|]
 
[[Comma]]s: 81/80, 126/125, 99/98
 
[[11-odd-limit|11-limit]] minimax
 
[{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }},
{{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}<nowiki>]</nowiki>
 
[[Eigenmonzo]]s: 2, 11/9
 
valid range: [696.774, 700.000] (31 to 12)
 
nice range: [691.202, 701.955]
 
strict range: [696.774, 700.000]
 
Mapping generator: ~3
 
[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
 
[[Generator]]s: 2, 3
 
{{EDOs|legend=1| 12, 31, 43, 74, 105, 198be }}
 
[[Badness]]: 0.0170
</div></div>
 
=== Schismic/Garibaldi (12&29, 2.3.5.7) ===
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 702.085
 
EDO generators: [[41edo|24\41]], [[53edo|31\53]], [[94edo|55\94]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Commas: 225/224, 3125/3087
 
7-limit minimax tuning:
 
7-limit: [|1 0 0 0&gt;, |5/3 1/15 0 -1/15&gt;,
|5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;<nowiki>]</nowiki>
 
[[Eigenmonzo|Eigenmonzo]]s: 2, 7/6
 
9-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;,
|5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;<nowiki>]</nowiki>
 
Eigenmonzos: 2, 9/7
 
Mapping generator: ~3
 
Map: [&lt;1 0 15 25|, &lt;0 1 -8 -14|]
 
Wedgie: &lt;&lt;1 -8 -14 -15 -25 -10||
 
{{EDOs|legend=1| 12, 29, 41, 53, 94, 241c, 335cd, 576cd }}
 
Badness: 0.0216
</div></div>
==== 41&53, 2.3.5.7.11.13.19 ====
*-14 gens = 7/4
*23 gens = 11/8
*20 gens = 13/8
 
=== Parapyth (29&17, 2.3.7.11.13) ===
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 704.745
 
EDO generators: [[17edo|10\17]], [[29edo|17\29]], [[46edo|27\46]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mapping|Period-generator mapping]]: [&lt;1 0 -21 -14 -9|, &lt;0 1 15 11 8|]
 
Commas: 169/168, 352/351, 364/363
 
Gencom: [2 3/2; 169/169 352/351 364/363]
 
Gencom mapping: [&lt;1 1 0 -6 -3 -1|, &lt;0 1 0 15 11 8|]
 
EDOs: 17, 46, 63
 
[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
</div></div>
 
=== Archy (17&5, 2.3.7) ===
Comma: 64/63
 
[[Lp_tuning|POL2 generator]]: ~3/2 = 709.321
 
Gencom: [2 3/2; 64/63]
 
Gencom mapping: [&lt;1 1 0 4|, &lt;0 1 0 -2|]
 
Map: [&lt;1 2 2|, &lt;0 -1 2|]
 
EDOs: 5, 12, 17, 22, 27, 137bc
 
[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
 
==== Supra (17&22, 2.3.7.11) ====
Commas: 64/63, 99/98
 
[[POTE_tuning|POTE generator]]: ~3/2 = 707.192
 
Gencom: [2 3/2; 64/63 99/98]
 
Gencom mapping: [&lt;1 1 0 4 7|, &lt;0 1 0 -2 -6|]
 
Sval map: [&lt;1 0 6 13|, &lt;0 1 -2 -6|]
 
EDOs: 5, 12, 17, 39c, 56c
 
[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
 
==== Superpyth (22&27, 2.3.5.7) ====
Commas: 64/63, 245/243
 
POTE generator: ~3/2 = 710.291
 
Map: [&lt;1 0 -12 6|, &lt;0 1 9 -2|]
 
Wedgie: &lt;&lt;1 9 -2 12 -6 -30||
 
EDOs: 5, 17, 22, 27, 49
 
Badness: 0.0323
 
==== Ultrapyth (27&37, 2.3.7.13/10) ====
*4 gens = 13/10


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


[[Category:Scales]]
== See also ==
[[Category:MOS Scales]]
* [[Diatonic functional harmony]]
[[Category:Diatonic]]
* [[Diatonic]] (disambiguation page)


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