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User:Moremajorthanmajor/5L 2s (major sixth-equivalent): Difference between revisions

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*[[3_4Cotoneum7]]
*[[3_4Cotoneum7]]
==Scale tree==
==Scale tree==
If 4\7 (four degrees of 7EDS) is at one extreme and 3\5 (three degrees of 5EDS) 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:
If 4\7 (four degrees of 7EDS) is at one extreme and 3\5 (three degrees of 5EDS) 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 12EDS:
{| class="wikitable"
{| class="wikitable"
|-
|-
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|}If we carry this freshman-summing out a little further, new, larger EDSs pop up in our continuum.
|}If we carry this freshman-summing out a little further, new, larger EDSs pop up in our continuum.
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="7" |Generator
!Generator
!Cents
!Cents
!Śata
!L
!L
!s
!s
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!Comments
!Comments
|-
|-
|4\7|| || || || || ||
|4\7
|533.333||711.111||1||1||1.000||
|533.333¢ ||1||1||1.000||
|-
|-
| || || || || || ||27\47
| 27\47
|[[Tel:531.1475|531.1475]]||708.196||7||6||1.167||
|531.148¢ ||7||6||1.167||
|-
|-
| || || || || ||23\40||
| 23\40
|530.769||707.692||6||5||1.200||
|530.769¢ ||6||5||1.200||
|-
|-
| || || || || || ||42\73
| 19\33
|530.526||707.368||11||9||1.222||
|530.232¢ ||5||4||1.250||
|-
|-
| || || || ||19\33|| ||
| 15\26
|530.232||706.977||5||4||1.250||
|529.411¢ ||4||3||1.333||
|-
|-
| || || || || || ||53\92
| 11\19
|530.000||706.667||14||11||1.273||
|528.000¢ ||3||2||1.500||L/s = 3/2
|-
|-
| || || || || ||34\59||
| 29\50
|529.870||[[Tel:706.4935|706.4935]]||9||7||1.286||
|527.272¢||8||5||1.600||
|-
|-
| || || || || || ||49\85
| 18\31
|529.730||706.306||13||10||1.300||
|526.829¢||5||3||1.667||3/4 Meantone is in this region
|-
|-
| || || ||15\26|| || ||
| 25\43
|529.411||705.882||4||3||1.333||
|526.316¢||7||4||1.750||
|-
|-
| || || || || || ||56\97
| 32\55
|529.134||705.512||15||11||1.364||
|526.028¢||9||5||1.800||
|-
|-
| || || || || ||41\71||
| 39\67
|529.032||705.376||11||8||1.375||
|525.843¢||11||6||1.833||
|-
|-
| || || || || || ||67\116
| 7\12
|528.947||705.263||18||13||1.385||
|525.000¢||2||1||2.000||Basic 3/4 diatonic
|-
| || || || ||26\45|| ||
|528.814||705.085||7||5||1.400||
|-
| || || || || || ||63\109
|528.617||704.895||17||12||1.417||
|-
| || || || || ||37\64||
|528.571||704.762||10||7||1.429||
|-
| || || || || || ||48\83
|528.440||704.587||13||9||1.444||
|-
| || ||11\19|| || || ||
|528.000||704.000||3||2||1.500||L/s = 3/2
|-
| || || || || || ||51\88
|527.586||703.448||14||9||1.556||
|-
| || || || || ||40\69||
|[[Tel:527.4725|527.4725]]||703.297||11||7||1.571||
|-
| || || || || || ||69\119
|[[Tel:527.3885|527.3885]]||703.185||19||12||1.583||
|-
| || || || ||29\50|| ||
|527.272||703.030||8||5||1.600||
|-
| || || || || || ||76\131
|527.168||702.890||21|| |13||1.615||Golden 3/4 meantone
|-
| || || || || ||47\81||
|527.103||702.804||13||8||1.625||
|-
| || || || || || ||65\112
|527.027||702.703||18||11||1.636||
|-
| || || ||18\31|| || ||
|526.829||702.409||5||3||1.667||3/4 Meantone is in this region
|-
| || || || || || ||61\105
|526.619||702.158||17||10||1.700||
|-
| || || || || ||43\74||
|526.531||702.041||12||7||1.714||
|-
| || || || || || ||68\117
|526.452||[[Tel:701.9355|701.9355]]||19||11||1.727||The generator closest to 3/4 of a just [[3/2]] for EDSs less than 200
|-
| || || || ||25\43|| ||
|526.316||701.754||7||4||1.750||
|-
| || || || || || ||57\98
|526.154||[[Tel:701.5385|701.5385]]||16||9||1.778||
|-
| || || || || ||32\55||
|526.028||701.370||9||5||1.800||
|-
| || || || || || ||39\67
|525.843||701.124||11||6||1.833||
|-
| ||7\12|| || || || ||
|525.000||700.000||2||1||2.000||Basic 3/4 diatonic
(Generators larger than this are proper)
(Generators larger than this are proper)
|-
|-
| || || || || || ||38\65
| 38\65
|524.138||698.851||11||5||2.200||
|524.138¢||11||5||2.200||
|-
| || || || || ||31\53||
|523.944||[[Tel:698.5915|698.5915]]||9||4||2.250||
|-
| || || || || || ||55\94
|[[Tel:523.8095|523.8095]]||698.413||16||7||2.286||
|-
| || || || ||24\41|| ||
|523.636||698.182||7||3||2.333||
|-
| || || || || || ||65\111
|523.490||697.987||19||8||2.375||
|-
| || || || || ||41\70||
|523.404||697.872||12||5||2.400||
|-
| || || || || || ||58\99
|523.308||697.744||17||7||2.428||
|-
| || || ||17\29|| || ||
|523.077||697.436||5||2||2.500||
|-
| || || || || || ||61\104
|522.857||697,142||18||7||2.571||
|-
| || || || || ||44\75||
|522.772||697.030||13||5||2.600||
|-
| || || || || || ||71\121
|522.699||[[Tel:696.9325|696.9325]]||21||8||2.625||Golden 3/4 neogothic
|-
| || || || ||27\46|| ||
|522.581||696.774||8||3||2.667||3/4 Neogothic is in this region
|-
| || || || || || ||64\109
|522.449||696.599||19||7||2.714||
|-
| || || || || ||37\63||
|522.352||[[Tel:696.4705|696.4705]]||11||4||2.750||
|-
| || || || || || ||47\80
|522.222||696.296||14||5||2.800||
|-
| || ||10\17|| || || ||
|521.739||695.652||3||1||3.000||L/s = 3/1
|-
| || || || || || ||43\73
|521.212||[[Tel:694.9495|694.9495]]||13||4||3.250||
|-
|-
| || || || || ||33\56||
| 31\53
|521.053||694.737||10||3||3.333||
|523.944¢||9||4||2.250
|-
|-
| || || || || || ||56\95
| 24\41
|520.930||694.574||17||5||3.400||
|523.636¢||7||3||2.333||
|-
|-
| || || || ||23\39|| ||
| 17\29
|520.755||694.340||7||2||3.500||
|523.077¢||5||2||2.500||
|-
|-
| || || || || || ||59\100
| 27\46
|520.588||694.118||18||5||3.600||
|522.581¢||8||3||2.667||3/4 Neogothic is in this region
|-
|-
| || || || || ||36\61||
| 10\17
|520.482||693.976||11||3||3.667||
|521.739¢||3||1||3.000||L/s = 3/1
|-
|-
| || || || || || ||49\83
| 13\22
|520.354||693.805||15||4||3.750||
|520.000¢||4||1||4.000||3/4 Archy is in this region
|-
|-
| || || ||13\22|| || ||
| 29\49
|520.000||693.333||4||1||4.000||3/4 Archy is in this region
|519.403¢||9||2||4.500||
|-
|-
| || || || || || ||42\71
| 16\27
|519.588||[[Tel:692.7835|692.7835]]||13||3||4.333||
|518.919¢||5||1||5.000||
|-
|-
| || || || || ||29\49||
| 19\32
|519.403||692.537||9||2||4.500||
|518.182¢||6||1||6.000||
|-
|-
| || || || || || ||45\76
| 22\37
|519.231||692.308||14||3||4.667||
|517.647¢||7||1||7.000||
|-
|-
| || || || ||16\27|| ||
|3\5
|518.919||691.892||5||1||5.000||
|514.286¢||1||0||→ inf||
|-
| || || || || || ||35\59
|[[Tel:518.5185|518.5185]]||691.358||11||2||5.500||
|-
| || || || || ||19\32||
|518.182||690.909||6||1||6.000||
|-
| || || || || || ||22\37
|517.647||690.196||7||1||7.000||
|-
|3\5|| || || || || ||
|514.286||685.714||1||0||→ inf||
|}Tunings above 7\12 on this chart are called "positive tunings" (as they greaten the size of the fifth) and include 3/4 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 above 7\12 on this chart are called "positive tunings" (as they greaten the size of the fifth) and include 3/4 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.


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5L 2s contains the pentatonic MOS [[2L_3s (major sixth equivalent) |2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L_5s (major sixth equivalent) |7L 5s]] or [[5L_7s (major sixth equivalent) |5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (675¢, 700$).
5L 2s contains the pentatonic MOS [[2L_3s (major sixth equivalent) |2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L_5s (major sixth equivalent) |7L 5s]] or [[5L_7s (major sixth equivalent) |5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (675¢, 700$).
==Related Scales==
==Related Scales==
{{main| 5L 2s (major sixth equivalent) MODMOSes }} ''and [[5L 2s (major sixth equivalent) Muddles]]''
{{main| 5L 2s (major sixth equivalent) MODMOSes }} ''and [[5L 2s (major sixth equivalent) Muddles]]''
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== See also ==
== See also ==
[[5L 2s (5/3-equivalent)]] - classical tuning
[[5L 2s (5/3-equivalent)]] - classical tuning
[[5L 2s (22\13-equivalent)]] - Neogothic tuning
[[5L 2s (12/7-equivalent)]] - Septimal tuning
[[5L 2s (12/7-equivalent)]] - Septimal tuning
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