3L 4s: Difference between revisions

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== Scale properties ==

Mosh can be thought of as a midpoint between two diatonic scales which are two cyclic orders away from each other. For example, LsLsLss is the midpoint between LsLLLsL and LLLsLLs. You can prove this by simple addition.

~~2 1 2 2 2 1 2 (LsLLLsL)~~

=== Intervals ===

~~+2 2 2 1 2 2 1 (LLLsLLs)~~

~~4 3 4 3 4 3 3 (LsLsLss)~~

~~The rest of the equivalencies are listed in [[3L 4s#Proposed names|Proposed names.]]~~

===Intervals===

===Generator chain===

=== Generator chain ===

===Modes ===

=== Modes ===

===Proposed names===

=== Proposed names ===

One set of mode nicknames was coined by [[Andrew Heathwaite]]. The other set was coined by [[User:CellularAutomaton|CellularAutomaton]] and follows the diatonic modes' naming convention by using ancient Greek toponyms that sound similar to the Heathwaite names~~. The third shows which modes are a mixture of which diatonic modes, as discussed earlier~~.

One set of mode nicknames was coined by [[Andrew Heathwaite]]. The other set was coined by [[User:CellularAutomaton|CellularAutomaton]] and follows the diatonic modes' naming convention by using ancient Greek toponyms that sound similar to the Heathwaite names.

{{MOS modes

| Table Headers=

Mode names (Heathwaite) $

Mode names (CA) $

~~Mixed diatonic modes $~~

| Table Entries=

Dril $

Dalmatian $

~~Dorian + Lydian $~~

Gil $

Galatian $

~~Aeolian + Lydian $~~

Kleeth $

Cilician $

~~Aeolian + Ionian $~~

Bish $

Bithynian $

~~Phrygian + Ionian $~~

Fish $

Pisidian $

~~Phrygian + Mixolydian $~~

Jwl $

Illyrian $

~~Locrian + Mixolydian $~~

Led $

Lycian $

~~Locrian + Dorian $~~

}}

==Theory==

== Theory ==

=== Low harmonic entropy scales===

=== Low harmonic entropy scales ===

There are two notable harmonic entropy minima:

*[[Neutral third scales]], such as dicot, hemififth, and mohajira, in which the generator is a neutral 3rd (around 350{{c}}) and two of them make a 3/2 (702{{c}}).

* [[Neutral third scales]], such as dicot, hemififth, and mohajira, in which the generator is a neutral 3rd (around 350{{c}}) and two of them make a 3/2 (702{{c}}).

*[[Magic]], in which the generator is 5/4 (386{{c}}) and 5 of them make a 3/1 (1902{{c}}).

* [[Magic]], in which the generator is 5/4 (386{{c}}) and 5 of them make a 3/1 (1902{{c}}).

==Tuning ranges==

== Tuning ranges ==

3\10 represents a dividing line between "neutral third scales" on the bottom (eg. 17edo neutral scale), and scales generated by submajor and major thirds at the top, with 10edo standing in between. The neutral third scales, after three more generators, make mos [[~~7L 3s~~]] (dicoid); the other scales make mos [[~~3L 7s~~]] (sephiroid).

3\10 represents a dividing line between "neutral third scales" on the bottom (eg. 17edo neutral scale), and scales generated by submajor and major thirds at the top, with 10edo standing in between. The neutral third scales, after three more generators, make mos [[7L 3s]] (dicoid); the other scales make mos [[3L 7s]] (sephiroid).

In dicoid, the generators are all "neutral thirds," and two of them make an approximation of the "perfect fifth." Additionally, the L of the scale is somewhere around a "whole tone" and the s of the scale is somewhere around a "neutral tone".

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In sephiroid, the generators are major thirds (including some very flat ones), and two generators are definitely sharp of a perfect fifth. L ranges from a "supermajor second" to a "major third" and s is a "semitone" or smaller.

=== Ultrasoft===

=== Ultrasoft ===

[[Ultrasoft]] mosh tunings have step ratios that are less than 4:3, which implies a generator flatter than {{nowrap| 7\24 {{=}} 350{{c}} }}.

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Ultrasoft mosh edos include [[24edo]], [[31edo]], [[38edo]], and [[55edo]].

*[[24edo]] can be used to make large and small steps more distinct (the step ratio is 4/3), or for its nearly pure 3/2.

* [[24edo]] can be used to make large and small steps more distinct (the step ratio is 4/3), or for its nearly pure 3/2.

*[[38edo]] can be used to tune the diminished and perfect mosthirds near [[6/5]] and [[11/9]], respectively.

* [[38edo]] can be used to tune the diminished and perfect mosthirds near [[6/5]] and [[11/9]], respectively.

These identifications are associated with [[mohajira]] temperament.

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|-

!

![[24edo]] (supersoft)

! [[24edo]] (supersoft)

![[31edo]]

! [[31edo]]

![[38edo]]

! [[38edo]]

![[55edo]]

! [[55edo]]

!JI intervals represented

! JI intervals represented

|-

|generator (g)

| generator (g)

|7\24, 350.00

| 7\24, 350.00

|9\31, 348.39

| 9\31, 348.39

|11\38, 347.37

| 11\38, 347.37

|16\55, 349.09

| 16\55, 349.09

|[[11/9]]

| [[11/9]]

|-

| L ({{nowrap| 4g − octave }})

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| 6\38, 189.47

| 9\55, 196.36

|[[9/8]], [[10/9]]

| [[9/8]], [[10/9]]

|-

|s ({{nowrap| octave − 3g }})

| s ({{nowrap| octave − 3g }})

|3\24, 150.00

| 3\24, 150.00

|4\31, 154.84

| 4\31, 154.84

|5\38, 157.89

| 5\38, 157.89

|7\55, 152.72

| 7\55, 152.72

|[[11/10]], [[12/11]]

| [[11/10]], [[12/11]]

|}

===Quasisoft===

=== Quasisoft ===

Quasisoft tunings of mosh have a step ratio between 3/2 and 5/3, implying a generator sharper than {{nowrap| 5\17 {{=}} 352.94{{c}} }} and flatter than {{nowrap| 8\27 {{=}} 355.56{{c}} }}.

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|-

!

![[17edo]] (soft)

! [[17edo]] (soft)

![[27edo]] (semisoft)

! [[27edo]] (semisoft)

![[44edo]]

! [[44edo]]

!JI intervals represented

! JI intervals represented

|-

|generator (g)

| generator (g)

|5\17, 352.94

| 5\17, 352.94

|8\27, 355.56

| 8\27, 355.56

| 13\44, 354.55

|16/13, 11/9

| 16/13, 11/9

|-

| L ({{nowrap| 4g − octave }})

|3\17, 211.76

| 3\17, 211.76

|5\27, 222.22

| 5\27, 222.22

|8\44, 218.18

| 8\44, 218.18

|9/8, 8/7

| 9/8, 8/7

|-

|s ({{nowrap| octave − 3g }})

| s ({{nowrap| octave − 3g }})

| 2\17, 141.18

|3\27, 133.33

| 3\27, 133.33

| 5\44, 137.37

|12/11, 13/12, 14/13

| 12/11, 13/12, 14/13

|}

===Hypohard===

=== Hypohard ===

Hypohard tunings of mosh have a step ratio between 2 and 3, implying a generator sharper than {{nowrap| 3\10 {{=}} 360{{c}} }} and flatter than {{nowrap| 4\13 {{=}} 369.23{{c}} }}.

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|-

!

![[10edo]] (basic)

! [[10edo]] (basic)

![[13edo]] (hard)

! [[13edo]] (hard)

![[23edo]] (semihard)

! [[23edo]] (semihard)

|-

|generator (g)

| generator (g)

|3\10, 360.00

| 3\10, 360.00

| 4\13, 369.23

|7\23, 365.22

| 7\23, 365.22

|-

|L ({{nowrap| 4g − octave }})

| L ({{nowrap| 4g − octave }})

| 2\10, 240.00

| 3\13, 276.92

|5\23, 260.87

| 5\23, 260.87

|-

|s ({{nowrap| octave − 3g }})

| s ({{nowrap| octave − 3g }})

|1\10, 120.00

| 1\10, 120.00

| 1\13, 92.31

|2\23, 104.35

| 2\23, 104.35

|}

=== Ultrahard===

=== Ultrahard ===

Ultra tunings of mosh have a step ratio greater than 4/1, implying a generator sharper than {{nowrap| 5\16 {{=}} 375{{c}} }}. The generator is thus near a [[5/4]] major third, five of which add up to an approximate [[3/1]]. The 7-note mos only has two perfect fifths, so extending the chain to bigger mosses, such as the [[~~3L 7s~~]] 10-note mos, is suggested for getting 5-limit harmony.

Ultra tunings of mosh have a step ratio greater than 4/1, implying a generator sharper than {{nowrap| 5\16 {{=}} 375{{c}} }}. The generator is thus near a [[5/4]] major third, five of which add up to an approximate [[3/1]]. The 7-note mos only has two perfect fifths, so extending the chain to bigger mosses, such as the [[3L 7s]] 10-note mos, is suggested for getting 5-limit harmony.

This range is associated with [[magic]] temperament.

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|-

!

![[16edo]] (superhard)

! [[16edo]] (superhard)

![[19edo]]

! [[19edo]]

![[22edo]]

! [[22edo]]

![[41edo]]

! [[41edo]]

! JI intervals represented

|-

|generator (g)

| generator (g)

|5\16, 375.00

| 5\16, 375.00

|6\19, 378.95

| 6\19, 378.95

|7\22, 381.82

| 7\22, 381.82

|13\41, 380.49

| 13\41, 380.49

|5/4

| 5/4

|-

| L ({{nowrap| 4g − octave }})

|4\16, 300.00

| 4\16, 300.00

| 5\19, 315.79

|6\22, 327.27

| 6\22, 327.27

|11\41, 321.95

| 11\41, 321.95

|6/5

| 6/5

|-

|s ({{nowrap| octave − 3g }})

| s ({{nowrap| octave − 3g }})

|1\16, 75.00

| 1\16, 75.00

| 1\19, 63.16

|1\22, 54.54

| 1\22, 54.54

|2\41, 58.54

| 2\41, 58.54

|25/24

| 25/24

|}

==Scales==

== Scales ==

*[[Mohaha7]] – 38\131 tuning

* [[Mohaha7]] – 38\131 tuning

*[[Neutral7]] – 111\380 tuning

* [[Neutral7]] – 111\380 tuning

*[[Namo7]] – 128\437 tuning

* [[Namo7]] – 128\437 tuning

*[[Rastgross1]] – POTE tuning of [[namo]]

* [[Rastgross1]] – POTE tuning of [[namo]]

*[[Hemif7]] – 17\58 tuning

* [[Hemif7]] – 17\58 tuning

*[[Suhajira7]] – POTE tuning of [[suhajira]]

* [[Suhajira7]] – POTE tuning of [[suhajira]]

*[[Sephiroth7]] – 9\29 tuning

* [[Sephiroth7]] – 9\29 tuning

*[[Magic7]] – 46\145 tuning

* [[Magic7]] – 46\145 tuning

==Scale tree==

== Scale tree ==

Generator ranges:

*Chroma-positive generator: 342.8571{{c}} (2\7) to 400.0000{{c}} (1\3)

* Chroma-positive generator: 342.8571{{c}} (2\7) to 400.0000{{c}} (1\3)

*Chroma-negative generator: 800.0000{{c}} (2\3) to 857.1429{{c}} (5\7)

* Chroma-negative generator: 800.0000{{c}} (2\3) to 857.1429{{c}} (5\7)

{{MOS tuning spectrum

| 6/5 = [[Mohaha]] / ptolemy ↑

@@ Line 15: / Line 15: @@
 == Scale properties ==
 {{TAMNAMS use}}
-Mosh can be thought of as a midpoint between two diatonic scales which are two cyclic orders away from each other. For example, LsLsLss is the midpoint between LsLLLsL and LLLsLLs. You can prove this by simple addition.
-1 2 2 2 1 2 (LsLLLsL)
+=== Intervals ===
-+<u>2 2 2 1 2 2 1 (LLLsLLs)</u>
-3 4 3 4 3 3 (LsLsLss)
-The rest of the equivalencies are listed in [[3L 4s#Proposed names|Proposed names.]]
-===Intervals===
 {{MOS intervals}}
-===Generator chain===
+=== Generator chain ===
 {{MOS genchain}}
-===Modes ===
+=== Modes ===
 {{MOS mode degrees}}
-===Proposed names===
+=== Proposed names ===
-One set of mode nicknames was coined by [[Andrew Heathwaite]]. The other set was coined by [[User:CellularAutomaton|CellularAutomaton]] and follows the diatonic modes' naming convention by using ancient Greek toponyms that sound similar to the Heathwaite names. The third shows which modes are a mixture of which diatonic modes, as discussed earlier.
+One set of mode nicknames was coined by [[Andrew Heathwaite]]. The other set was coined by [[User:CellularAutomaton|CellularAutomaton]] and follows the diatonic modes' naming convention by using ancient Greek toponyms that sound similar to the Heathwaite names.
 {{MOS modes
 | Table Headers=
 Mode names<br />(Heathwaite) $
 Mode names<br />(CA) $
-Mixed diatonic<br />modes $
 | Table Entries=
 Dril $
 Dalmatian $
-Dorian + Lydian $
 Gil $
 Galatian $
-Aeolian + Lydian $
 Kleeth $
 Cilician $
-Aeolian + Ionian $
 Bish $
 Bithynian $
-Phrygian + Ionian $
 Fish $
 Pisidian $
-Phrygian + Mixolydian $
 Jwl $
 Illyrian $
-Locrian + Mixolydian $
 Led $
 Lycian $
-Locrian + Dorian $
 }}
-==Theory==
+== Theory ==
-=== Low harmonic entropy scales===
+=== Low harmonic entropy scales ===
 There are two notable harmonic entropy minima:
-*[[Neutral third scales]], such as dicot, hemififth, and mohajira, in which the generator is a neutral 3rd (around 350{{c}}) and two of them make a 3/2 (702{{c}}).
+* [[Neutral third scales]], such as dicot, hemififth, and mohajira, in which the generator is a neutral 3rd (around 350{{c}}) and two of them make a 3/2 (702{{c}}).
-*[[Magic]], in which the generator is 5/4 (386{{c}}) and 5 of them make a 3/1 (1902{{c}}).
+* [[Magic]], in which the generator is 5/4 (386{{c}}) and 5 of them make a 3/1 (1902{{c}}).
-==Tuning ranges==
+== Tuning ranges ==
-\10 represents a dividing line between "neutral third scales" on the bottom (eg. 17edo neutral scale), and scales generated by submajor and major thirds at the top, with 10edo standing in between. The neutral third scales, after three more generators, make mos [[7L 3s]] (dicoid); the other scales make mos [[3L 7s]] (sephiroid).
+\10 represents a dividing line between "neutral third scales" on the bottom (eg. 17edo neutral scale), and scales generated by submajor and major thirds at the top, with 10edo standing in between. The neutral third scales, after three more generators, make mos [[7L&nbsp;3s]] (dicoid); the other scales make mos [[3L&nbsp;7s]] (sephiroid).
 In dicoid, the generators are all "neutral thirds," and two of them make an approximation of the "perfect fifth." Additionally, the L of the scale is somewhere around a "whole tone" and the s of the scale is somewhere around a "neutral tone".
@@ Line 78: / Line 62: @@
 In sephiroid, the generators are major thirds (including some very flat ones), and two generators are definitely sharp of a perfect fifth. L ranges from a "supermajor second" to a "major third" and s is a "semitone" or smaller.
-=== Ultrasoft===
+=== Ultrasoft ===
 [[Ultrasoft]] mosh tunings have step ratios that are less than 4:3, which implies a generator flatter than {{nowrap| 7\24 {{=}} 350{{c}} }}.
@@ Line 84: / Line 68: @@
 Ultrasoft mosh edos include [[24edo]], [[31edo]], [[38edo]], and [[55edo]].
-*[[24edo]] can be used to make large and small steps more distinct (the step ratio is 4/3), or for its nearly pure 3/2.
+* [[24edo]] can be used to make large and small steps more distinct (the step ratio is 4/3), or for its nearly pure 3/2.
-*[[38edo]] can be used to tune the diminished and perfect mosthirds near [[6/5]] and [[11/9]], respectively.
+* [[38edo]] can be used to tune the diminished and perfect mosthirds near [[6/5]] and [[11/9]], respectively.
 These identifications are associated with [[mohajira]] temperament.
@@ Line 93: / Line 77: @@
 |-
 !
-![[24edo]] (supersoft)
+! [[24edo]] (supersoft)
-![[31edo]]
+! [[31edo]]
-![[38edo]]
+! [[38edo]]
-![[55edo]]
+! [[55edo]]
-!JI intervals represented
+! JI intervals represented
 |-
-|generator (g)
+| generator (g)
-|7\24, 350.00
+| 7\24, 350.00
-|9\31, 348.39
+| 9\31, 348.39
-|11\38, 347.37
+| 11\38, 347.37
-|16\55, 349.09
+| 16\55, 349.09
-|[[11/9]]
+| [[11/9]]
 |-
 | L ({{nowrap| 4g − octave }})
@@ Line 111: / Line 95: @@
 | 6\38, 189.47
 | 9\55, 196.36
-|[[9/8]], [[10/9]]
+| [[9/8]], [[10/9]]
 |-
-|s ({{nowrap| octave − 3g }})
+| s ({{nowrap| octave − 3g }})
-|3\24, 150.00
+| 3\24, 150.00
-|4\31, 154.84
+| 4\31, 154.84
-|5\38, 157.89
+| 5\38, 157.89
-|7\55, 152.72
+| 7\55, 152.72
-|[[11/10]], [[12/11]]
+| [[11/10]], [[12/11]]
 |}
-===Quasisoft===
+=== Quasisoft ===
 Quasisoft tunings of mosh have a step ratio between 3/2 and 5/3, implying a generator sharper than {{nowrap| 5\17 {{=}} 352.94{{c}} }} and flatter than {{nowrap| 8\27 {{=}} 355.56{{c}} }}.
@@ Line 130: / Line 114: @@
 |-
 !
-![[17edo]] (soft)
+! [[17edo]] (soft)
-![[27edo]] (semisoft)
+! [[27edo]] (semisoft)
-![[44edo]]
+! [[44edo]]
-!JI intervals represented
+! JI intervals represented
 |-
-|generator (g)
+| generator (g)
-|5\17, 352.94
+| 5\17, 352.94
-|8\27, 355.56
+| 8\27, 355.56
 | 13\44, 354.55
-|16/13, 11/9
+| 16/13, 11/9
 |-
 | L ({{nowrap| 4g − octave }})
-|3\17, 211.76
+| 3\17, 211.76
-|5\27, 222.22
+| 5\27, 222.22
-|8\44, 218.18
+| 8\44, 218.18
-|9/8, 8/7
+| 9/8, 8/7
 |-
-|s ({{nowrap| octave − 3g }})
+| s ({{nowrap| octave − 3g }})
 | 2\17, 141.18
-|3\27, 133.33
+| 3\27, 133.33
 | 5\44, 137.37
-|12/11, 13/12, 14/13
+| 12/11, 13/12, 14/13
 |}
-===Hypohard===
+=== Hypohard ===
 Hypohard tunings of mosh have a step ratio between 2 and 3, implying a generator sharper than {{nowrap| 3\10 {{=}} 360{{c}} }} and flatter than {{nowrap| 4\13 {{=}} 369.23{{c}} }}.
@@ Line 165: / Line 149: @@
 |-
 !
-![[10edo]] (basic)
+! [[10edo]] (basic)
-![[13edo]] (hard)
+! [[13edo]] (hard)
-![[23edo]] (semihard)
+! [[23edo]] (semihard)
 |-
-|generator (g)
+| generator (g)
-|3\10, 360.00
+| 3\10, 360.00
 | 4\13, 369.23
-|7\23, 365.22
+| 7\23, 365.22
 |-
-|L ({{nowrap| 4g − octave }})
+| L ({{nowrap| 4g − octave }})
 | 2\10, 240.00
 | 3\13, 276.92
-|5\23, 260.87
+| 5\23, 260.87
 |-
-|s ({{nowrap| octave − 3g }})
+| s ({{nowrap| octave − 3g }})
-|1\10, 120.00
+| 1\10, 120.00
 | 1\13, 92.31
-|2\23, 104.35
+| 2\23, 104.35
 |}
-=== Ultrahard===
+=== Ultrahard ===
-Ultra tunings of mosh have a step ratio greater than 4/1, implying a generator sharper than {{nowrap| 5\16 {{=}} 375{{c}} }}. The generator is thus near a [[5/4]] major third, five of which add up to an approximate [[3/1]]. The 7-note mos only has two perfect fifths, so extending the chain to bigger mosses, such as the [[3L 7s]] 10-note mos, is suggested for getting 5-limit harmony.
+Ultra tunings of mosh have a step ratio greater than 4/1, implying a generator sharper than {{nowrap| 5\16 {{=}} 375{{c}} }}. The generator is thus near a [[5/4]] major third, five of which add up to an approximate [[3/1]]. The 7-note mos only has two perfect fifths, so extending the chain to bigger mosses, such as the [[3L&nbsp;7s]] 10-note mos, is suggested for getting 5-limit harmony.
 This range is associated with [[magic]] temperament.
@@ Line 192: / Line 176: @@
 |-
 !
-![[16edo]] (superhard)
+! [[16edo]] (superhard)
-![[19edo]]
+! [[19edo]]
-![[22edo]]
+! [[22edo]]
-![[41edo]]
+! [[41edo]]
 ! JI intervals represented
 |-
-|generator (g)
+| generator (g)
-|5\16, 375.00
+| 5\16, 375.00
-|6\19, 378.95
+| 6\19, 378.95
-|7\22, 381.82
+| 7\22, 381.82
-|13\41, 380.49
+| 13\41, 380.49
-|5/4
+| 5/4
 |-
 | L ({{nowrap| 4g − octave }})
-|4\16, 300.00
+| 4\16, 300.00
 | 5\19, 315.79
-|6\22, 327.27
+| 6\22, 327.27
-|11\41, 321.95
+| 11\41, 321.95
-|6/5
+| 6/5
 |-
-|s ({{nowrap| octave − 3g }})
+| s ({{nowrap| octave − 3g }})
-|1\16, 75.00
+| 1\16, 75.00
 | 1\19, 63.16
-|1\22, 54.54
+| 1\22, 54.54
-|2\41, 58.54
+| 2\41, 58.54
-|25/24
+| 25/24
 |}
-==Scales==
+== Scales ==
-*[[Mohaha7]] – 38\131 tuning
+* [[Mohaha7]] – 38\131 tuning
-*[[Neutral7]] – 111\380 tuning
+* [[Neutral7]] – 111\380 tuning
-*[[Namo7]] – 128\437 tuning
+* [[Namo7]] – 128\437 tuning
-*[[Rastgross1]] – POTE tuning of [[namo]]
+* [[Rastgross1]] – POTE tuning of [[namo]]
-*[[Hemif7]] – 17\58 tuning
+* [[Hemif7]] – 17\58 tuning
-*[[Suhajira7]] – POTE tuning of [[suhajira]]
+* [[Suhajira7]] – POTE tuning of [[suhajira]]
-*[[Sephiroth7]] – 9\29 tuning
+* [[Sephiroth7]] – 9\29 tuning
-*[[Magic7]] – 46\145 tuning
+* [[Magic7]] – 46\145 tuning
-==Scale tree==
+== Scale tree ==
 Generator ranges:
-*Chroma-positive generator: 342.8571{{c}} (2\7) to 400.0000{{c}} (1\3)
+* Chroma-positive generator: 342.8571{{c}} (2\7) to 400.0000{{c}} (1\3)
-*Chroma-negative generator: 800.0000{{c}} (2\3) to 857.1429{{c}} (5\7)
+* Chroma-negative generator: 800.0000{{c}} (2\3) to 857.1429{{c}} (5\7)
 {{MOS tuning spectrum
 | 6/5 = [[Mohaha]] / ptolemy&nbsp;↑