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| nSmallSteps = 4
| nSmallSteps = 4
| Equalized = 2
| Equalized = 2
| Paucitonic = 1
| Collapsed = 1
| Pattern = sLsLsLs
| Pattern = LsLsLss
}}
}}
{{MOS intro}}


'''3L 4s''' 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 ==
== Standing assumptions ==
[[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".
The [[TAMNAMS]] system is used in this article to name 3L 4s intervals and step size ratios and step ratio ranges.


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".)
== Scale properties ==
{{TAMNAMS use}}


Thus the 10edo gamut is as follows:
=== Intervals ===
{{MOS intervals}}


'''J''' '''K''' K&/L@ '''L''' '''M''' M&/N@ '''N''' '''O''' O&/P@ '''P''' '''J'''
=== Generator chain ===
{{MOS genchain}}


== Names ==
=== Modes ===
[[TAMNAMS]] (used by this article) uses the name '''mosh''' for this pattern. This is taken from an older [[MOS naming]] scheme by Graham Breed.
{{MOS mode degrees}}


== Low-harmonic-entropy scales ==
=== 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]].


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, where the generator is a 5/4 but five of them make a 3/1.
=== 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).
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".
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 35: Line 80:


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


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]]).
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]]).


Ultrasoft mosh EDOs include [[24edo]], [[31edo]], [[38edo]], and [[55edo]].
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.
Line 62: Line 107:
| [[11/9]]
| [[11/9]]
|-
|-
| L (4g - octave)
| L ({{nowrap| 4g − octave }})
| 4\24, 200.00
| 4\24, 200.00
| 5\31, 193.55
| 5\31, 193.55
Line 69: Line 114:
| [[9/8]], [[10/9]]
| [[9/8]], [[10/9]]
|-
|-
| s (octave - 3g)
| s ({{nowrap| octave − 3g }})
| 3\24, 150.00
| 3\24, 150.00
| 4\31, 154.84
| 4\31, 154.84
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=== 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}} }}.
Quasisoft tunings of mosh have a step ratio between 3/2 and 5/3, implying a generator sharper than 5\17 = 352.94¢ and flatter than 8\27 = 355.56¢.


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.
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.
Line 98: Line 142:
| 16/13, 11/9
| 16/13, 11/9
|-
|-
| L (4g - octave)
| L ({{nowrap| 4g − octave }})
| 3\17, 211.76
| 3\17, 211.76
| 5\27, 222.22
| 5\27, 222.22
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| 9/8, 8/7
| 9/8, 8/7
|-
|-
| s (octave - 3g)
| s ({{nowrap| octave − 3g }})
| 2\17, 141.18
| 2\17, 141.18
| 3\27, 133.33
| 3\27, 133.33
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=== Hypohard ===
=== Hypohard ===
Hypohard tunings of mosh have a step ratio between 2 and 3, implying a generator sharper than 3\10 = 360¢ and flatter than 4\13 = 369.23¢.
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¢) to 1\13 (92.31¢).
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 symmetric mode sLsLsLs becomes a distorted double harmonic major in these tunings.
The symmetric mode sLsLsLs becomes a distorted double harmonic major in these tunings.
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| 7\23, 365.22
| 7\23, 365.22
|-
|-
| L (4g - octave)
| L ({{nowrap| 4g − octave }})
| 2\10, 240.00
| 2\10, 240.00
| 3\13, 276.92
| 3\13, 276.92
| 5\23, 260.87
| 5\23, 260.87
|-
|-
| s (octave - 3g)
| s ({{nowrap| octave − 3g }})
| 1\10, 120.00
| 1\10, 120.00
| 1\13, 92.31
| 1\13, 92.31
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=== 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&nbsp;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 5\16 = 375¢. 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 MOSes, such as the [[3L 7s]] 10-note MOS, is suggested for getting 5-limit harmony.


This range is associated with [[magic]] temperament.
This range is associated with [[magic]] temperament.
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| 5/4
| 5/4
|-
|-
| L (4g - octave)
| L ({{nowrap| 4g − octave }})
| 4\16, 300.00
| 4\16, 300.00
| 5\19, 315.79
| 5\19, 315.79
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| 6/5
| 6/5
|-
|-
| s (octave - 3g)
| s ({{nowrap| octave − 3g }})
| 1\16, 75.00
| 1\16, 75.00
| 1\19, 63.16
| 1\19, 63.16
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|}
|}


== Modes ==
== Scales ==
 
* [[Mohaha7]] – 38\131 tuning
The various modes of 3L 4s (with [[Modal UDP Notation]] and nicknames coined by [[Andrew Heathwaite]]) are:
* [[Neutral7]] – 111\380 tuning
 
* [[Namo7]] – 128\437 tuning
{| class="wikitable center-all"
* [[Rastgross1]] – POTE tuning of [[namo]]
|-
* [[Hemif7]] – 17\58 tuning
! Mode
* [[Suhajira7]] – POTE tuning of [[suhajira]]
! UDP
* [[Sephiroth7]] – 9\29 tuning
! Nickname
* [[Magic7]] – 46\145 tuning
|-
| LsLsLss
| <nowiki>6|0</nowiki>
| dril
|-
| LsLssLs
| <nowiki>5|1</nowiki>
| gil
|-
| LssLsLs
| <nowiki>4|2</nowiki>
| kleeth
|-
| sLsLsLs
| <nowiki>3|3</nowiki>
| bish
|-
| sLsLssL
| <nowiki>2|4</nowiki>
| fish
|-
| sLssLsL
| <nowiki>1|5</nowiki>
| jwl
|-
| ssLsLsL
| <nowiki>0|6</nowiki>
| led
|}


== Scale tree ==
== Scale tree ==
The spectrum looks like this:
Generator ranges:  
 
* Chroma-positive generator: 342.8571{{c}} (2\7) to 400.0000{{c}} (1\3)
{| class="wikitable center-all"
* Chroma-negative generator: 800.0000{{c}} (2\3) to 857.1429{{c}} (5\7)
! colspan="6" | Generator
{{MOS tuning spectrum
! Cents
| 6/5 = [[Mohaha]] / ptolemy&nbsp;↑
! L
| 5/4 = Mohaha / migration / [[mohajira]]
! s
| 11/8 = Mohaha / mohamaq  
! L/s
| 7/5 = Mohaha / [[neutrominant]]
! Comments
| 10/7 = [[Hemif]] / [[hemififths]]
|-
| 11/7 = [[Suhajira]]
| 2\7 || || || || || || 342.857 || 1 || 1 || 1.000 ||
| 13/8 = Golden suhajira (354.8232{{c}})
|-
| 5/3 = Suhajira / [[ringo]]
| || || || || || 11\38 || 347.368 || 6 || 5 || 1.200 || Mohaha/ptolemy
| 12/7 = [[Beatles]]
|-
| 13/5 = Unnamed golden tuning (366.2564{{c}})
| || || || || 9\31 || || 348.387 || 5 || 4 || 1.250 || Mohaha/mohajira
| 7/2 = [[Sephiroth]]
|-
| 9/2 = [[Muggles]]
| || || || || || 16\55 || 349.091 || 9 || 7 || 1.286 ||
| 5/1 = [[Magic]]
|-
| 6/1 = [[Würschmidt]]&nbsp;↓
| || || || 7\24 || || || 350.000 || 4 || 3 || 1.333 ||
}}
|-
| || || || || || 19\65 || 350.769 || 11 || 8 || 1.375 || Mohaha/mohamaq
|-
| || || || || 12\41 || || 351.220 || 7 || 5 || 1.400 || Mohaha/maqamic
|-
| || || || || || 17\58 || 351.724 || 10 || 7 || 1.428 || Hemif
|-
| || || 5\17 || || || || 352.941 || 3 || 2 || 1.500 || L/s = 3/2
|-
| || || || || || 18\61 || 354.098 || 11 || 7 || 1.571 || Suhajira
|-
| || || || || 13\44 || || 354.545 || 8 || 5 || 1.600 ||
|-
| || || || || || 21\71 || 354.783 || 13 || 8 || 1.625 || Golden suhajira
|-
| || || || 8\27 || || || 355.556 || 5 || 3 || 1.667 || Beatles/ringo
|-
| || || || || || 19\64 || 356.250 || 12 || 7 || 1.714 || Beatles
|-
| || || || || 11\37 || || 356.757 || 7 || 4 || 1.750 ||
|-
| || || || || || 14\47 || 357.447 || 9 || 5 || 1.800 ||
|-
| || 3\10 || || || || || 360.000 || 2 || 1 || 2.000 || Basic mosh<br>(Generators smaller than this are proper)
|-
| || || || || || 13\43 || 362.264 || 9 || 4 || 2.250 ||
|-
| || || || || 10\33 || || 363.636 || 7 || 3 || 2.333 ||
|-
| || || || || || 17\56 || 364.286 || 12 || 5 || 2.400 ||
|-
| || || || 7\23 || || || 365.217 || 5 || 2 || 2.500 ||
|-
| || || || || || 18\59 || 366.102 || 13 || 5 || 2.600 || Unnamed golden tuning
|-
| || || || || 11\36 || || 366.667 || 8 || 3 || 2.667 ||
|-
| || || || || || 15\49 || 367.347 || 11 || 4 || 2.750 ||
|-
| || || 4\13 || || || || 369.231 || 3 || 1 || 3.000 || L/s = 3/1
|-
| || || || || || 13\42 || 371.429 || 10 || 3 || 3.333 ||
|-
| || || || || 9\29 || || 372.414 || 7 || 2 || 3.500 || Sephiroth
|-
| || || || || || 14\45 || 373.333 || 11 || 3 || 3.667 ||
|-
| || || || 5\16 || || || 375.000 || 4 || 1 || 4.000 ||
|-
| || || || || || 11\35 || 377.143 || 9 || 2 || 4.500 || Muggles
|-
| || || || || 6\19 || || 378.947 || 5 || 1 || 5.000 || Magic
|-
| || || || || || 7\22 || 381.818 || 6 || 1 || 6.000 ||
|-
| 1\3 || || || || || || 400.000 || 1 || 0 || → inf ||
|}


[[Category:Scales]]
[[Category:Mosh]]
[[Category:Abstract MOS patterns]]
[[Category:7-tone scales]]
Retrieved from "https://en.xen.wiki/w/3L_4s"