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7L 3s (4/1-equivalent): Difference between revisions

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{{Infobox MOS}}
{{Infobox MOS}}
'''7L 3s<4/1>''' (sometimes called '''diaquadic''') refers to a non-octave [[MOS scale]] family with a period of a [[4/1]] and which has 7 large and 3 small steps. These scales are in some ways a double octave clone of [[5L 2s|diatonic scale]]s because they are generated from perfect fifths and have the [[7edo]]/[[7ed4]] 685-cent flat fifth and [[5edo]]/10ed4's 720-cent sharp fifth at the ends of the tuning spectrum. However, ~3/2 is now the chroma-negative rather than chroma-positive generator, equalized and collapsed tunings are now swapped (equalized is 720-cent fifth and collapsed is 685-cent fifth), and the basic tuning is now the 705-cent gentle fifth from [[17edo]]/[[17ed4]] instead of 12edo's fifth. Diaquadic scales are also more widely spaced than diatonic scales, closer melodically to an octave-period pentatonic scale.
{{MOS intro}}


== Intervals ==
These scales are in some ways a double octave clone of [[5L 2s|diatonic scale]]s because they are generated from perfect fifths and have the [[7edo]]/[[7ed4]] 685{{c}} flat fifth and [[5edo]]/10ed4's 720{{c}} sharp fifth at the ends of the tuning spectrum. However, ~3/2 is now the chroma-negative rather than chroma-positive generator, equalized and collapsed tunings are now swapped (equalized is 720{{c}} fifth and collapsed is 685{{c}} fifth), and the basic tuning is now the 705-cent gentle fifth from [[17edo]]/[[17ed4]] instead of 12edo's fifth. Diaquadic scales are also more widely spaced than diatonic scales, closer melodically to an octave-period pentatonic scale.
{|class="wikitable"
|-
!Dark generators
!colspan="2"|Notation (J = 1/1)
[[17ed4]] cents
!
!Bright generators
!colspan="2"|Notation (J = 1/1)
![[17ed4]] cents
|-
|0
|J
|perfect 0-step
|1694.16
|
|0
|J
|perfect 10-step
|282.36
|-
|1
|M
|perfect 3-step
|705.90
|
|1
|Q
|perfect 7-step
|1694.16
|-
|2
|Pb
|minor 6-step
|1411.80
|
|2
|N
|major 4-step
|988.26
|-
|3
|Sb
|minor 9-step
|2117.70
|
|3
|K
|major 1-step
|282.36
|-
|4
|Lb
|minor 2-step
|423.54
|
|4
|R
|major 8-step
|1976.52
|-
|5
|Ob
|minor 5-step
|1129.44
|
|5
|O
|major 5-step
|1270.62
|-
|6
|Rb
|minor 8-step
|1835.34
|
|6
|L
|major 2-step
|564.72
|-
|7
|Kb
|minor 1-step
|141.18
|
|7
|S
|major 9-step
|2258.88
|-
|8
|Nb
|minor 4-step
|847.08
|
|8
|P
|major 6-step
|1552.98
|}


== Modes ==
== Scale properties ==
* LLLsLLsLLs 9|0
{{TAMNAMS use}}
* LLsLLLsLLs 8|1
* LLsLLsLLLs 7|2
* LLsLLsLLsL 6|3
* LsLLLsLLsL 5|4
* LsLLsLLLsL 4|5
* LsLLsLLsLL 3|6
* sLLLsLLsLL 2|7
* sLLsLLLsLL 1|8
* sLLsLLsLLL 0|9


=== Intervals ===
{{MOS intervals}}
=== Generator chain ===
{{MOS genchain}}
=== Modes ===
{{MOS mode degrees}}


== Scale tree ==
== Scale tree ==
{|class="wikitable"
{{MOS tuning spectrum
|-
| 5/3 = Quarchy is around here
!Dark generator
| 3/1 = Tetrominant is around here
!Cents
| 4/1 = Meanquad is around here
!L
}}
!s
 
!L/s
!Comments
|-
|3\[[10ed4]]
|720.000
|1
|1
|1.000
|
|-
|14\[[47ed4]]
|714.894
|5
|4
|1.250
|-
|11\[[37ed4]]
|713.514
|4
|3
|1.333
|
|-
|8\[[27ed4]]
|711.111
|3
|2
|1.500
|
|-
|13\[[44ed4]]
|709.091
|5
|3
|1.667
|Quarchy is around here
|-
|5\[[17ed4]]
|705.882
|2
|1
|2.000
|Basic diaquadic
|-
|12\[[41ed4]]
|702.439
|5
|2
|2.500
|
|-
|7\[[24ed4]]
|700.000
|3
|1
|3.000
|Tetrominant is around here
|-
|9\[[31ed4]]
|696.774
|4
|1
|4.000
|Meanquad is around here
|-
|11\[[38ed4]]
|694.737
|5
|1
|5.000
|
|-
|2\[[7ed4]]
|685.714
|1
|0
| -> ∞
|
|}
== Temperaments ==
== Temperaments ==
There are two major rank-2 temperament interpretations of diaquadic, corresponding to double octave clones of [[meantone]] and [[archy]].
There are two major rank-2 temperament interpretations of diaquadic, corresponding to double octave clones of [[meantone]] and [[archy]].
=== Meanquad ===
=== Meanquad ===
''Main article: [[Meanquad]]''
''Main article: [[Meanquad]]''
* [[Subgroup]]: 4.5.6
* [[Subgroup]]: 4.5.6
* [[Comma list]]: [[81/80]]
* [[Comma list]]: [[81/80]]
* {{Mapping|legend=1|1 0 1|0 4 1}}
* [[Support]]ing ETs: 7, 31, 38, 24, 17, 10, 69, 45, 55, 11[-5], 52, 41, 13[+5], 27[+5]  
* [[Support]]ing ETs: 7, 31, 38, 24, 17, 10, 69, 45, 55, 11[-5], 52, 41, 13[+5], 27[+5]  
* [[CTE tuning]]: ~[[3/2]] = 697.214
* [[CTE tuning]]: ~[[3/2]] = 697.214
==== Tetrominant ====
==== Tetrominant ====
* [[Subgroup]]: 4.5.6.7
* [[Subgroup]]: 4.5.6.7
* [[Comma list]]: [[36/35]], [[64/63]]
* [[Comma list]]: [[36/35]], [[64/63]]
* {{Mapping|legend=1|1 0 1 2|0 4 1 -2}}
* [[Support]]ing ETs: 7, 17, 24, 10, 31, 41, 27[+5], 38[+7], 55[+7], 58[+5, +7], 65[+5, +7], 44[+5], 75[+5, +7]  
* [[Support]]ing ETs: 7, 17, 24, 10, 31, 41, 27[+5], 38[+7], 55[+7], 58[+5, +7], 65[+5, +7], 44[+5], 75[+5, +7]  
* [[CTE tuning]]: ~[[3/2]] = 699.622
* [[CTE tuning]]: ~[[3/2]] = 699.622
=== Quarchy ===
=== Quarchy ===
* [[Subgroup]]: 4.6.7
* [[Subgroup]]: 4.6.7
* [[Comma list]]: [[64/63]]
* [[Comma list]]: [[64/63]]
* {{Mapping|legend=1|1 2 0|0 1 -2}}
* [[Support]]ing ETs: 10, 17, 7, 27, 37, 44, 13, 24, 47, 11[+7], 23, 57, 16, 41  
* [[Support]]ing ETs: 10, 17, 7, 27, 37, 44, 13, 24, 47, 11[+7], 23, 57, 16, 41  
* [[CTE tuning]]: ~[[3/2]] = 709.595
* [[CTE tuning]]: ~[[3/2]] = 709.595
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