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#REDIRECT [[5L 2s (8/3-equivalent)]]
One way of distinguishing the '''17/12 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 [[17edXI|17EDXI]]<nowiki/>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''==
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==
The 5L 2s MOS scale has this generalized form.
 
*L L L s L L s
 
Insert 2 for L and 1 for s and you'll get the [[12edXI|12EDXI]] diatonic.
 
*2 2 2 1 2 2 1
 
When L=3, s=1, you have 17EDXI of standard practice: 3 3 3 1 3 3 1
 
When L=3, s=2, you have [[19edXI|19EDXI]]: 3 3 3 2 3 3 2
 
When L=4, s=1, you have [[22edXI|22EDXI]]: 4 4 4 1 4 4 1
 
When L=4, s=3, you have [[26edXI|26EDXI]]: 4 4 4 3 4 4 3
 
When L=5, s=1, you have [[27edXI|27EDXI]]: 5 5 5 1 5 5 1
 
When L=5, s=2, you have [[29edXI|29EDXI]]: 5 5 5 2 5 5 2
 
When L=5, s=3, you have [[31EDXI]]: 5 5 5 3 5 5 3
 
When L=5, s=4, you have [[33edXI|33EDXI]]: 5 5 5 4 5 5 4
 
So you have scales where L and s are nearly equal, which approach [[7edXI|7EDXI]]:
 
*1 1 1 1 1 1 1
 
And you have scales where s becomes so small it approaches zero, which would give us [[5edXI|5EDXI]]:
 
*1 1 1 0 1 1 0 = 1 1 1 1 1
 
==Tuning ranges==
===Parasoft to ultrasoft===
"17/12 Flattone" systems, such as [[26edo|26EDXI]].
 
===Hyposoft===
"17/12 Meantone" (more properly "septimal meantone") systems, such as [[31edo|31EDXI]].
 
===Hypohard===
The near-just part of the region is of interest mainly for those interested in 17/12 [[Pythagorean tuning]] and large, accurate EDO systems based on close-to-Pythagorean fifths, such as [[41edXI|41EDXI]] and [[53edXI|53EDXI]]. This class of tunings is called 17/12 [[schisma|schismic]] temperament; these tunings can approximate 5<sup>17/12</sup>-limit harmonies very accurately by [[tempering out]] 17/12 of a small comma called the [[schisma]]. (Technically, 12EDXI tempers out the 17/12 schisma and thus is a 17/12 schismic tuning, but it is nowhere near as accurate as 17/12 schismic tunings can be.)
 
The sharp-of-just part of this range includes so-called "17/12 [[neogothic]]" or "17/12 parapyth" systems, which tune the 17/12 diatonic major third slightly sharply of [[11/7]] (around [[128/81]]) and the diatonic minor third slightly flatly of [[13/11]] (around [[32/27]]). Good 17/12 neogothic EDXIs include [[29edo|29EDXI]] and [[46edXI|46EDXI]]. [[17edXI|17EDXI]] is often considered the sharper end of the neogothic spectrum; its major third at 800 cents is considerably more discordant than in flatter 17/12 neogothic tunings.
 
===Parahard to ultrahard===
"17/12 Archy" systems such as [[17edo|17EDXI]], [[22edo|22EDXI]], and [[27edo|27EDXI]].
 
==Modes==
17/12 Diatonic modes have standard names from classical music theory:
 
{| class="wikitable center-all"
|-
!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==
*[[17_12Meantone7]]
*[[17_12Pythagorean7]]
*[[17_12Supra7]]
*[[17_12Archy7]]
*[[17_12Garibaldi7]]
*[[17_12Cotoneum7]]
 
==Scale tree==
If 4\7 (four degrees of 7EDXI) is at one extreme and 3\5 (three degrees of 5EDXI) 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 12EDXI.
 
If we carry this freshman-summing out a little further, new, larger [[EDXI]]<nowiki/>s pop up in our continuum.
 
Generator ranges:
*Chroma-positive generator: 960 cents (4\7, normalized) to 1028.5714 cents (3\5, normalized)
*Chroma-negative generator: [[Tel:685.7143|685.7143]] cents (2\5, normalized) to 720 cents (3\7, normalized)
 
{| class="wikitable center-all"
!Generator
!Normalized
!L
!s
!L/s
!Comments
|-
|4\7||960.000¢||1||1||1.000||
|-
| 27\47||967.164¢||7||6||1.167||
|-
| 23\40||968.421¢||6||5||1.200||
|-
| 42\73||969.231¢||11||9||1.222||
|-
| 19\33||970.213¢||5||4||1.250||
|-
| 34\59||971.429¢||9||7||1.286||
|-
| 15\26||972.{{Overline|972}}¢||4||3||1.333||
|-
| 56\97||973.913¢||15||11||1.364||
|-
| 41\71||974.{{Overline|2574}}¢||11||8||1.375||
|-
| 67\116||974.{{Overline|54}}¢||18||13||1.385||
|-
| 26\45||975.000¢||7||5||1.400||17/12 [[Flattone]] is in this region
|-
| 63\109||975.484¢||17||12||1.417||
|-
| 37\64||975.824¢||10||7||1.429||
|-
| 48\83||976.271¢||13||9||1.444||
|-
| 11\19||977.{{Overline|7}}¢||3||2||1.500||
|-
| 51\88||979.200¢||14||9||1.556||
|-
| 40\69||979.592¢||11||7||1.571||
|-
| 29\50||980.282¢||8||5||1.600||
|-
| 76\131||980.645¢||21|| |13||1.615||17/12 [[Golden meantone]] (980.7157¢)
|-
| 47\81||980.870¢||13||8||1.625||
|-
| 18\31||981.{{Overline|81}}¢||5||3||1.667||17/12 [[Meantone]] is in this region
|-
| 43\74||982.857¢||12||7||1.714||
|-
| 25\43||983.607¢||7||4||1.750||
|-
| 32\55||984.615¢||9||5||1.800||
|-
| 39\67||985.263¢||11||6||1.833||
|-
|46\79
|985.714¢
|13
|7
|1.857
|
|-
|53\91
|986.047¢
|15
|8
|1.875
|
|-
|60\103
|986.301¢
|17
|9
|1.889
|
|-
|67\115
|986.503¢
|19
|10
|1.900
|
|-
|74\127
|986.{{Overline|6}}¢
|21
|11
|1.909
|
|-
|81\139
|986.802¢
|23
|12
|1.917
|
|-
|88\151
|986.916¢
|25
|13
|1.923
|
|-
|95\163
|987.013¢
|27
|14
|1.929
|
|-
| 7\12||988.235¢||2||1||2.000||Basic 17/12 diatonic <br>(Generators smaller than this are proper)
|-
|45\77
|990.826¢
|13
|6
|2.167
|
|-
| 38\65||991.304¢||11||5||2.200||
|-
| 31\53||992.000¢||9||4||2.250||The generator closest to 17/12 of a just [[3/2]] for EDXIs less than 200
|-
| 55\94||992.481¢||16||7||2.286||17/12 [[Garibaldi]] / [[Cassandra]]
|-
| 24\41||993.103¢||7||3||2.333||
|-
| 41\70||993.{{Overline|93}}¢||12||5||2.400||
|-
| 58\99||994.2857¢||17||7||2.428||
|-
| 17\29||995.122¢||5||2||2.500||
|-
| 44\75||996.226¢||13||5||2.600||
|-
| 71\121||996.491¢||21||8||2.625||17/12 Golden neogothic (996.3946¢)
|-
| 27\46||996.923¢||8||3||2.667||17/12 [[Neogothic]] is in this region
|-
| 37\63||997.753¢||11||4||2.750||
|-
| 47\80||998.230¢||14||5||2.800||
|-
|57\97
|998.540¢
|17
|6
|2.833
|
|-
|67\114
|998.758¢
|20
|7
|2.857
|
|-
|77\131
|998.{{Overline|918}}¢
|23
|8
|2.875
|
|-
|87\148
|999.043¢
|26
|9
|2.889
|
|-
| 10\17||1000.000¢||3||1||3.000||
|-
| 43\73||1001.942¢||13||4||3.250||
|-
| 33\56||1002.532¢||10||3||3.333||
|-
| 23\39||1003.{{Overline|63}}¢||7||2||3.500||
|-
| 36\61||1004.651¢||11||3||3.667||
|-
| 49\83||1005.128¢||15||4||3.750||
|-
| 13\22||1006.452¢||4||1||4.000||17/12 [[Archy]] is in this region
|-
|55\93
|1007.634¢
|17
|4
|4.250
|
|-
| 42\71||1008.000¢||13||3||4.333||
|-
| 29\49||1008.696¢||9||2||4.500||
|-
| 45\76||1009.346¢||14||3||4.667||
|-
| 16\27||1010.526¢||5||1||5.000||
|-
|51\86
|1011.570¢
|16
|3
|5.333
|
|-
| 35\59||1012.048¢||11||2||5.500||
|-
| 19\32||1013.{{Overline|3}}¢||6||1||6.000||
|-
|41\69
|1014.432¢
|13
|2
|6.500
|
|-
| 22\37||1015.385¢||7||1||7.000||
|-
|3\5||1028.571¢||1||0||→ inf||
|}
 
Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include 17/12 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 17/12 Pythagorean tuning itself (well approximated by 31\53) as well as 17/12 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 5-limit than 7-limit intervals: 6:5 and 5:4 as opposed to 7:6 and 9:7.
 
[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]
 
5L 2s contains the pentatonic MOS [[2L 3s (perfect eleventh equivalent)|2L 3s]] and (with the sole exception of the 5L 2s of 12EDXI) is itself contained in a dodecaphonic MOS: either [[7L 5s (perfect eleventh equivalent)|7L 5s]] or [[5L 7s (perfect eleventh equivalent)|5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (988.235¢ normalized).
 
==Related Scales==
{{main| 5L 2s MODMOSes (perfect eleventh equivalent)}} {{main| 5L 2s Muddles (perfect eleventh equivalent)}}
 
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 (perfect eleventh equivalent)/Temperaments}}
 
==Approaches to Functional Harmony==
{{see also| Diatonic functional harmony}}
 
== See also ==
[[5L 2s (8/3-equivalent)]]
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