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{{User:IlL/Template:RTT restriction}}
{{Infobox MOS
{{Infobox MOS
| Name = mosh
| Name = mosh
Line 6: Line 5:
| nSmallSteps = 4
| nSmallSteps = 4
| Equalized = 2
| Equalized = 2
| Paucitonic = 1
| Collapsed = 1
| Pattern = sLsLsLs
| Pattern = LsLsLss
}}
}}
{{MOS intro}}


'''3L 4s''' or '''mosh''' is the [[MOS scale]] built from a generator that falls between 1\3 (one degree of [[3edo]] – 400 cents) and 2\7 (two degrees of [[7edo]] – 343 cents).
== Name ==
[[TAMNAMS]] suggests the temperament-agnostic name '''mosh''' for this scale, adopted from an older [[Graham Breed's MOS naming scheme|mos naming scheme]] by [[Graham Breed]]. The name is a contraction of "mohajira-ish".


== Notation ==
== Scale properties ==
{{TAMNAMS use}}


The notation used in this article is sLsLsLs = JKLMNOPJ unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)
=== Intervals ===
{{MOS intervals}}


Thus the 10edo gamut is as follows:
=== Generator chain ===
{{MOS genchain}}


'''J/K@/P&''' '''K/J&''' K&/L@ '''L/M@''' '''M/L&''' M&/N@ '''N/O@''' '''O/N&''' O&/P@ '''P/J@''' '''J'''
=== Modes ===
{{MOS mode degrees}}
 
=== Proposed names ===
The first 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 Heathwaite's names. The third shows which modes are a mixture of which diatonic modes, as discussed in [[#Theory]].
{{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 ==
Mosh can be thought of as a midpoint between two diatonic scales which are two cyclic orders away from each other. For example, sLsLsLs is the midpoint between the Ionian (major, LLsLLLs) and Phrygian (sLLLsLL) modes. You can prove this by simple addition:
 
<pre>
  2 2 1 2 2 2 1 (LLsLLLs)
+ 1 2 2 2 1 2 2 (sLLLsLL)
= 3 4 3 4 3 4 3 (sLsLsLs)
</pre>
The rest of the equivalencies are listed in [[#Proposed names]].
 
=== 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 third (around 350{{c}}) and two of them make a 3/2 (702{{c}}).
* [[Magic]], in which the generator is 5/4 (386{{c}}) and five of them make a 3/1 (1902{{c}}), though the step ratios in this range are very hard to the point of being lopsided.


== 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&nbsp;3s]] (dicoid); the other scales make mos [[3L&nbsp;7s]] (sephiroid).


<!-- todo -->
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".


The two notable harmonic entropy minima with this pattern are neutral third scales ("dicot" / "hemififth" / "mohajira") where two generators make a 3/2, and [[Magic_family|magic]], where the generator is a 5/4 but five of them make a 3/1.
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.


3\10 on this chart represents a dividing line between "neutral third scales" on the bottom (eg. [[17edo_neutral_scale|17edo neutral scale]]), and something else I don't have a name for yet on the top, with [[10edo|10edo]] standing in between. (What do you call this region, dear reader?) Of course, magic is in the top half, but it's a pretty specific scale and doesn't describe the whole range. MOS-wise, the neutral third scales, after three more generations, make MOS [[7L_3s|7L 3s]] ("unfair mosh"); the other scales make MOS [[3L_7s|3L 7s]] ("fair mosh").
=== Ultrasoft ===
[[Ultrasoft]] mosh tunings have step ratios that are less than 4:3, which implies a generator flatter than {{nowrap| 7\24 {{=}} 350{{c}} }}.


In "neutral third scale territory," 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".
Ultrasoft mosh can be considered "meantone mosh". This is because the large step is a "meantone" in these tunings, somewhere between near-10/9 (as in [[38edo]]) and near-9/8 (as in [[24edo]]).


In the as-yet unnamed northern territory, 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 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.
* [[38edo]] can be used to tune the diminished and perfect mosthirds near [[6/5]] and [[11/9]], respectively.


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


The various modes of 3L 4s (with [[Modal UDP Notation]] and nicknames coined by [[Andrew Heathwaite]]) are:
The sizes of the generator, large step and small step of mosh are as follows in various ultrasoft mosh tunings.
 
{| class="wikitable right-2 right-3 right-4 right-5 right-6 right-7"
{| class="wikitable"
|-
|-
| style="text-align:center;" | '''Mode'''
!
| style="text-align:center;" | '''UDP'''
! [[24edo]] (supersoft)
| style="text-align:center;" | '''Nickname'''
! [[31edo]]
! [[38edo]]
! [[55edo]]
! JI intervals represented
|-
|-
| | s L s L s L s
| generator (g)
| style="text-align:center;" | 3|3
| 7\24, 350.00
| | bish
| 9\31, 348.39
| 11\38, 347.37
| 16\55, 349.09
| [[11/9]]
|-
|-
| | L s L s L s s
| L ({{nowrap| 4g − octave }})
| style="text-align:center;" | 6|0
| 4\24, 200.00
| | dril
| 5\31, 193.55
| 6\38, 189.47
| 9\55, 196.36
| [[9/8]], [[10/9]]
|-
|-
| | s L s L s s L
| s ({{nowrap| octave − 3g }})
| style="text-align:center;" | 2|4
| 3\24, 150.00
| | fish
| 4\31, 154.84
| 5\38, 157.89
| 7\55, 152.72
| [[11/10]], [[12/11]]
|}
 
=== 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}} }}.
 
The large step is a sharper major second in these tunings than in ultrasoft tunings. These tunings could be considered "parapyth mosh" or "archy mosh", in analogy to ultrasoft mosh being meantone mosh.
 
These identifications are associated with [[beatles]] and [[suhajira]] temperaments.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
| | L s L s s L s
!
| style="text-align:center;" | 5|1
! [[17edo]] (soft)
| | gil
! [[27edo]] (semisoft)
! [[44edo]]
! JI intervals represented
|-
|-
| | s L s s L s L
| generator (g)
| style="text-align:center;" | 1|5
| 5\17, 352.94
| | jwl
| 8\27, 355.56
| 13\44, 354.55
| 16/13, 11/9
|-
|-
| | L s s L s L s
| L ({{nowrap| 4g − octave }})
| style="text-align:center;" | 4|2
| 3\17, 211.76
| | kleeth
| 5\27, 222.22
| 8\44, 218.18
| 9/8, 8/7
|-
|-
| | s s L s L s L
| s ({{nowrap| octave − 3g }})
| style="text-align:center;" | 0|6
| 2\17, 141.18
| | led
| 3\27, 133.33
| 5\44, 137.37
| 12/11, 13/12, 14/13
|}
|}


== Scale tree ==
=== 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}} }}.
 
The large step ranges from a semifourth to a subminor third in these tunings. The small step is now clearly a semitone, ranging from 1\10 (120{{c}}) to 1\13 (92.31{{c}}).


The spectrum looks like this:
The symmetric mode sLsLsLs becomes a distorted double harmonic major in these tunings.


{| class="wikitable"
This range is associated with [[sephiroth]] temperament.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
! |
!
! |
! [[10edo]] (basic)
! |
! [[13edo]] (hard)
! | g
! [[23edo]] (semihard)
! | 2g
! | 3g
! | 4g (-1200)
! | comments
|-
|-
| | 1\3
| generator (g)
| |
| 3\10, 360.00
| |
| 4\13, 369.23
| | 400.000
| 7\23, 365.22
| | 800.000
| | 1200.000
| | 400.000
| style="text-align:center;" |
|-
|-
| |  
| L ({{nowrap| 4g − octave }})
| | 15\46
| 2\10, 240.00
| |
| 3\13, 276.92
| | 391.304
| 5\23, 260.87
| | 782.609
| | 1173.913
| | 365.217
| style="text-align:center;" |
|-
|-
| |  
| s ({{nowrap| octave − 3g }})
| | 14\43
| 1\10, 120.00
| |
| 1\13, 92.31
| | 390.698
| 2\23, 104.35
| | 781.395
|}
| | 1172.093
 
| | 362.791
=== Ultrahard ===
| style="text-align:center;" |
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.
|-
| |
| | 13\40
| |
| | 390.000
| | 780.000
| | 1170.000
| | 360.000
| style="text-align:center;" |
|-
| |
| | 12\37
| |
| | 389.189
| | 778.378
| | 1167.568
| | 356.757
| style="text-align:center;" |
|-
| |
| | 11\34
| |
| | 388.235
| | 776.471
| | 1164.706
| | 352.941
| style="text-align:center;" |
|-
| |
| | 10\31
| |
| | 387.097
| | 774.194
| | 1161.290
| | 348.387
| style="text-align:center;" | <span style="display: block; text-align: center;">[[Würschmidt_family|Würschmidt]] is around here</span>
|-
| |
| |
| | 19\59
| | 386.441
| | 772.881
| | 1159.322
| | 345.763
| style="text-align:center;" |
|-
| |
| | 9\28
| |
| | 385.714
| | 771.429
| | 1157.143
| | 342.857
| style="text-align:center;" |
|-
| |
| | 8\25
| |
| | 384.000
| | 768.000
| | 1152.000
| | 336.000
| style="text-align:center;" |
|-
| |
| |
| | 23\72
| | 383.333
| | 766.667
| | 1150.000
| | 333.333
| style="text-align:center;" |
|-
| |
| |
| | 15\47
| | 382.988
| | 765.957
| | 1148.936
| | 331.915
| style="text-align:center;" |
|-
| |
| | 7\22
| |
| | 381.818
| | 763.636
| | 1145.455
| | 327.273
| style="text-align:center;" |
|-
| |
| |
| | 13\41
| | 380.488
| | 760.976
| | 1141.463
| | 321.951
| style="text-align:center;" | <span style="display: block; text-align: center;">Magic is around here</span>
|-
| |
| |
| | 19\60
| | 380.000
| | 760.000
| | 1140.000
| | 320.000
| style="text-align:center;" |
|-
| |
| |
| | 25\79
| | 379.747
| | 759.494
| | 1139.2405
| | 318.987
| style="text-align:center;" |
|-
| |
| | 6\19
| |
| | 378.947
| | 757.895
| | 1136.842
| | 315.789
| style="text-align:center;" |
|-
| |
| |
| | 11\35
| | 377.143
| | 754.286
| | 1131.429
| | 308.571
| style="text-align:center;" |
|-
| |
| |
| | 16\51
| | 376.471
| | 752.941
| | 1129.412
| | 305.882
| style="text-align:center;" |
|-
| |
| | 5\16
| |
| | 375.000
| | 750.000
| | 1125.000
| | 300.000
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = 4</span>
|-
| |
| |
| | 24\77
| | 374.026
| | 748.052
| | 1122.078
| | 296.104
| style="text-align:center;" |
|-
| |
| |
| | 19\61
| | 373.7705
| | 747.541
| | 1121.3115
| | 295.082
| style="text-align:center;" |
|-
| |
| |
| | 14\45
| | 373.333
| | 746.667
| | 1120.000
| | 293.333
| style="text-align:center;" |
|-
| |
| |
| | 9\29
| | 372.414
| | 744.828
| | 1117.241
| | 289.655
| style="text-align:center;" |
|-
| |
| |
| | 13\42
| | 371.429
| | 742.857
| | 1114.286
| | 285.714
| style="text-align:center;" |
|-
| |
| |
| | 17\55
| | 370.909
| | 741.818
| | 1112.727
| | 283.636
| style="text-align:center;" |
|-
| |
| | 4\13
| |
| | 369.231
| | 738.462
| | 1107.692
| | 276.923
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = 3</span>
|-
| |
| |
| | 23\75
| | 368.000
| | 736.000
| | 1104.000
| | 272.000
| style="text-align:center;" | <span style="display: block; text-align: center;">


</span>
This range is associated with [[magic]] temperament.
|-
{| class="wikitable right-2 right-3 right-4 right-5 right-6"
| |
| |
| | 19\62
| | 367.742
| | 735.484
| | 1103.226
| | 270.968
| |
|-
| |
| |
| | 15\49
| | 367.347
| | 734.694
| | 1102.041
| | 269.388
| style="text-align:center;" |
|-
| |
| |
| | 11\36
| | 366.667
| | 733.333
| | 1100.000
| | 266.667
| style="text-align:center;" |
|-
|-
| |
!
| |
! [[16edo]] (superhard)
| |
! [[19edo]]
| | 366.256
! [[22edo]]
| | 732.513
! [[41edo]]
| | 1198.77
! JI intervals represented
| | 265.026
| |
|-
|-
| |
| generator (g)
| |
| 5\16, 375.00
| | 7\23
| 6\19, 378.95
| | 365.217
| 7\22, 381.82
| | 730.435
| 13\41, 380.49
| | 1095.652
| 5/4
| | 260.870
| style="text-align:center;" | <span style="display: block; text-align: center;">Modi Sephiratorum (Kosmorsky)</span>
|-
|-
| |  
| L ({{nowrap| 4g − octave }})
| |
| 4\16, 300.00
| | 17\56
| 5\19, 315.79
| | 364.286
| 6\22, 327.27
| | 728.571
| 11\41, 321.95
| | 1092.857
| 6/5
| | 257.143
| style="text-align:center;" |
|-
|-
| |  
| s ({{nowrap| octave − 3g }})
| |
| 1\16, 75.00
| | 10\33
| 1\19, 63.16
| | 363.636
| 1\22, 54.54
| | 727.272
| 2\41, 58.54
| | 1090.909
| 25/24
| | 254.545
| style="text-align:center;" |
|-
| |
| |
| | 13\43
| | 362.791
| | 725.581
| | 1088.372
| | 251.163
| style="text-align:center;" |
|-
| |
| |
| | 16\53
| | 362.264
| | 724.528
| | 1086.7925
| | 249.057
| style="text-align:center;" |
|-
| |
| |
| | 19\63
| | 361.905
| | 723.8095
| | 1085.714
| | 247.619
| style="text-align:center;" |
|-
| |
| | 3\10
| |
| | 360.000
| | 720.000
| | 1080.000
| | 240.000
| style="text-align:center;" | <span style="display: block; text-align: center;">Boundary of propriety</span><span style="display: block; text-align: center;">(generators smaller than this are proper)</span>
|-
| |
| |
| | 38\127
| | 359.055
| | 718.110
| | 1077.165
| | 236.2205
| style="text-align:center;" |
|-
| |
| |
| | 35\117
| | 358.974
| | 717.949
| | 1076.923
| | 235.898
| style="text-align:center;" |
|-
| |
| |
| | 32\107
| | 358.8785
| | 717.757
| | 1076.6355
| | 235.514
| style="text-align:center;" |
|-
| |
| |
| | 29\97
| | 358.763
| | 717.526
| | 1076.289
| | 235.0515
| style="text-align:center;" |
|-
| |
| |
| | 26\87
| | 358.621
| | 717.241
| | 1075.862
| | 234.483
| style="text-align:center;" |
|-
| |
| |
| | 23\77
| | 358.442
| | 716.883
| | 1075.325
| | 233.767
| style="text-align:center;" |
|-
| |
| |
| | 20\67
| | 358.209
| | 716.418
| | 1074.627
| | 232.836
| style="text-align:center;" |
|-
| |
| |
| | 17\57
| | 357.895
| | 715.7895
| | 1073.684
| | 231.579
| style="text-align:center;" |
|-
| |
| |
| | 14\47
| | 357.447
| | 714.894
| | 1072.340
| | 229.787
| style="text-align:center;" |
|-
| |
| |
| | 11\37
| | 356.757
| | 713.514
| | 1070.270
| | 227.027
| style="text-align:center;" |
|-
| |
| |
| |
| | 356.5035
| | 713.007
| | 1069.511
| | 226.014
| |
|-
| |
| |
| | 8\27
| | 355.556
| | 711.111
| | 1066.667
| | 222.222
| style="text-align:center;" | <span style="display: block; text-align: center;">Beatles is around here</span>
|-
| |
| |
| |
| | 354.930
| | 709.859
| | 1064.789
| | 219.718
| style="text-align:center;" | <span style="display: block; text-align: center;">Golden neutral thirds scale</span>
|-
| |
| |
| | 21\71
| | 354.783
| | 709.565
| | 1064.348
| | 219.13
| style="text-align:center;" |
|-
| |
| |
| | 13\44
| | 354.5455
| | 709.091
| | 1063.636
| | 218.182
| style="text-align:center;" |
|-
| |
| |
| |
| | 354.088
| | 708.177
| | 1062.266
| | 216.354
| |
|-
| |
| | 5\17
| |
| | 352.941
| | 705.882
| | 1058.824
| | 211.765
| style="text-align:center;" | <span style="display: block; text-align: center;">Optimum rank range (L/s=3/2)</span>
|-
| |
| |
| | 12\41
| | 351.220
| | 702.439
| | 1053.659
| | 204.878
| style="text-align:center;" | <span style="display: block; text-align: center;">2.3.11 neutral thirds scale is around here</span>
|-
| |
| | 7\24
| |
| | 350.000
| | 700.000
| | 1050.000
| | 200.000
| style="text-align:center;" |
|-
| |
| |
| | 16\55
| | 349.091
| | 698.182
| | 1047.273
| | 196.364
| |
|-
| |
| | 9\31
| |
| | 348.387
| | 696.774
| | 1045.161
| | 193.548
| style="text-align:center;" | <span style="display: block; text-align: center;">Mohajira/dicot is around here</span>
|-
| |
| | 11\38
| |
| | 347.368
| | 694.737
| | 1042.105
| | 189.474
| |
|-
| | 2\7
| |
| |
| | 342.857
| | 685.714
| | 1028.571
| | 171.429
| style="text-align:center;" |
|}
|}


[[Category:Scales]]
== Scales ==
[[Category:MOS scales]]
* [[Mohaha7]] – 38\131 tuning
[[Category:Abstract MOS patterns]]
* [[Neutral7]] – 111\380 tuning
* [[Namo7]] – 128\437 tuning
* [[Rastgross1]] – POTE tuning of [[namo]]
* [[Hemif7]] – 17\58 tuning
* [[Suhajira7]] – POTE tuning of [[suhajira]]
* [[Sephiroth7]] – 9\29 tuning
* [[Magic7]] – 46\145 tuning
 
== Scale tree ==
Generator ranges:
* 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)
{{MOS tuning spectrum
| 6/5 = [[Mohaha]] / ptolemy&nbsp;↑
| 5/4 = Mohaha / migration / [[mohajira]]
| 11/8 = Mohaha / mohamaq
| 7/5 = Mohaha / [[neutrominant]]
| 10/7 = [[Hemif]] / [[hemififths]]
| 11/7 = [[Suhajira]]
| 13/8 = Golden suhajira (354.8232{{c}})
| 5/3 = Suhajira / [[ringo]]
| 12/7 = [[Beatles]]
| 13/5 = Unnamed golden tuning (366.2564{{c}})
| 7/2 = [[Sephiroth]]
| 9/2 = [[Muggles]]
| 5/1 = [[Magic]]
| 6/1 = [[Würschmidt]]&nbsp;↓
}}
 
[[Category:Mosh]]
[[Category:7-tone scales]]
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