3L 4s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Restore wiki markup
Tag: Undo
Line 15: Line 15:
== Scale properties ==
== Scale properties ==
{{TAMNAMS use}}
{{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.


2 1 2 2 2 1 2 (LsLLLsL)
=== Intervals ===
 
+<u>2 2 2 1 2 2 1 (LLLsLLs)</u>
 
4 3 4 3 4 3 3 (LsLsLss)
 
The rest of the equivalencies are listed in [[3L 4s#Proposed names|Proposed names.]]
===Intervals===
{{MOS intervals}}
{{MOS intervals}}


===Generator chain===
=== Generator chain ===
{{MOS genchain}}
{{MOS genchain}}


===Modes ===
=== Modes ===
{{MOS mode degrees}}
{{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
{{MOS modes
| Table Headers=
| Table Headers=
Mode names<br />(Heathwaite) $
Mode names<br />(Heathwaite) $
Mode names<br />(CA) $
Mode names<br />(CA) $
Mixed diatonic<br />modes $
| Table Entries=
| Table Entries=
Dril $
Dril $
Dalmatian $
Dalmatian $
Dorian + Lydian $
Gil $
Gil $
Galatian $
Galatian $
Aeolian + Lydian $
Kleeth $
Kleeth $
Cilician $
Cilician $
Aeolian + Ionian $
Bish $
Bish $
Bithynian $
Bithynian $
Phrygian + Ionian $
Fish $
Fish $
Pisidian $
Pisidian $
Phrygian + Mixolydian $
Jwl $
Jwl $
Illyrian $
Illyrian $
Locrian + Mixolydian $
Led $
Led $
Lycian $
Lycian $
Locrian + Dorian $
}}
}}


==Theory==
== Theory ==
=== Low harmonic entropy scales===
=== Low harmonic entropy scales ===
There are two notable harmonic entropy minima:
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&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".
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.
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}} }}.
[[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]].
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.
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 }})
| L ({{nowrap| 4g − octave }})
Line 111: Line 95:
| 6\38, 189.47
| 6\38, 189.47
| 9\55, 196.36
| 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}} }}.
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
| 13\44, 354.55
|16/13, 11/9
| 16/13, 11/9
|-
|-
| L ({{nowrap| 4g − octave }})
| 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
| 2\17, 141.18
|3\27, 133.33
| 3\27, 133.33
| 5\44, 137.37
| 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}} }}.
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
| 4\13, 369.23
|7\23, 365.22
| 7\23, 365.22
|-
|-
|L ({{nowrap| 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 ({{nowrap| octave − 3g }})
| s ({{nowrap| octave − 3g }})
|1\10, 120.00
| 1\10, 120.00
| 1\13, 92.31
| 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.
This range is associated with [[magic]] temperament.
Line 192: Line 176:
|-
|-
!
!
![[16edo]] (superhard)
! [[16edo]] (superhard)
![[19edo]]
! [[19edo]]
![[22edo]]
! [[22edo]]
![[41edo]]
! [[41edo]]
! JI intervals represented
! 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 }})
| L ({{nowrap| 4g − octave }})
|4\16, 300.00
| 4\16, 300.00
| 5\19, 315.79
| 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\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:  
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
{{MOS tuning spectrum
| 6/5 = [[Mohaha]] / ptolemy&nbsp;↑  
| 6/5 = [[Mohaha]] / ptolemy&nbsp;↑  

Revision as of 09:56, 23 March 2025

↖ 2L 3s ↑ 3L 3s 4L 3s ↗
← 2L 4s 3L 4s 4L 4s →
↙ 2L 5s ↓ 3L 5s 4L 5s ↘ 
┌╥┬╥┬╥┬┬┐
│║│║│║│││
│││││││││
└┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLss
ssLsLsL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 2\7 to 1\3 (342.9 ¢ to 400.0 ¢)
Dark 2\3 to 5\7 (800.0 ¢ to 857.1 ¢)
TAMNAMS information
Name mosh
Prefix mosh-
Abbrev. mosh
Related MOS scales
Parent 3L 1s
Sister 4L 3s
Daughters 7L 3s, 3L 7s
Neutralized 6L 1s
2-Flought 10L 4s, 3L 11s
Equal tunings
Equalized (L:s = 1:1) 2\7 (342.9 ¢)
Supersoft (L:s = 4:3) 7\24 (350.0 ¢)
Soft (L:s = 3:2) 5\17 (352.9 ¢)
Semisoft (L:s = 5:3) 8\27 (355.6 ¢)
Basic (L:s = 2:1) 3\10 (360.0 ¢)
Semihard (L:s = 5:2) 7\23 (365.2 ¢)
Hard (L:s = 3:1) 4\13 (369.2 ¢)
Superhard (L:s = 4:1) 5\16 (375.0 ¢)
Collapsed (L:s = 1:0) 1\3 (400.0 ¢)

3L 4s, named mosh in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 4 small steps, repeating every octave. Generators that produce this scale range from 342.9 ¢ to 400 ¢, or from 800 ¢ to 857.1 ¢.

Name

TAMNAMS suggests the temperament-agnostic name mosh for this scale, adopted from an older mos naming scheme by Graham Breed. The name is a contraction of "mohajira-ish".

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 3L 4s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-moshstep Perfect 0-moshstep P0moshs 0 0.0 ¢
1-moshstep Minor 1-moshstep m1moshs s 0.0 ¢ to 171.4 ¢
Major 1-moshstep M1moshs L 171.4 ¢ to 400.0 ¢
2-moshstep Diminished 2-moshstep d2moshs 2s 0.0 ¢ to 342.9 ¢
Perfect 2-moshstep P2moshs L + s 342.9 ¢ to 400.0 ¢
3-moshstep Minor 3-moshstep m3moshs L + 2s 400.0 ¢ to 514.3 ¢
Major 3-moshstep M3moshs 2L + s 514.3 ¢ to 800.0 ¢
4-moshstep Minor 4-moshstep m4moshs L + 3s 400.0 ¢ to 685.7 ¢
Major 4-moshstep M4moshs 2L + 2s 685.7 ¢ to 800.0 ¢
5-moshstep Perfect 5-moshstep P5moshs 2L + 3s 800.0 ¢ to 857.1 ¢
Augmented 5-moshstep A5moshs 3L + 2s 857.1 ¢ to 1200.0 ¢
6-moshstep Minor 6-moshstep m6moshs 2L + 4s 800.0 ¢ to 1028.6 ¢
Major 6-moshstep M6moshs 3L + 3s 1028.6 ¢ to 1200.0 ¢
7-moshstep Perfect 7-moshstep P7moshs 3L + 4s 1200.0 ¢

Generator chain

Generator chain of 3L 4s
Bright gens Scale degree Abbrev.
9 Augmented 4-moshdegree A4moshd
8 Augmented 2-moshdegree A2moshd
7 Augmented 0-moshdegree A0moshd
6 Augmented 5-moshdegree A5moshd
5 Major 3-moshdegree M3moshd
4 Major 1-moshdegree M1moshd
3 Major 6-moshdegree M6moshd
2 Major 4-moshdegree M4moshd
1 Perfect 2-moshdegree P2moshd
0 Perfect 0-moshdegree
Perfect 7-moshdegree		 P0moshd
P7moshd
−1 Perfect 5-moshdegree P5moshd
−2 Minor 3-moshdegree m3moshd
−3 Minor 1-moshdegree m1moshd
−4 Minor 6-moshdegree m6moshd
−5 Minor 4-moshdegree m4moshd
−6 Diminished 2-moshdegree d2moshd
−7 Diminished 7-moshdegree d7moshd
−8 Diminished 5-moshdegree d5moshd
−9 Diminished 3-moshdegree d3moshd

Modes

Scale degrees of the modes of 3L 4s
UDP Cyclic
order 		Step
pattern 		Scale degree (moshdegree)
0 1 2 3 4 5 6 7
6|0 1 LsLsLss Perf. Maj. Perf. Maj. Maj. Aug. Maj. Perf.
5|1 3 LsLssLs Perf. Maj. Perf. Maj. Maj. Perf. Maj. Perf.
4|2 5 LssLsLs Perf. Maj. Perf. Min. Maj. Perf. Maj. Perf.
3|3 7 sLsLsLs Perf. Min. Perf. Min. Maj. Perf. Maj. Perf.
2|4 2 sLsLssL Perf. Min. Perf. Min. Maj. Perf. Min. Perf.
1|5 4 sLssLsL Perf. Min. Perf. Min. Min. Perf. Min. Perf.
0|6 6 ssLsLsL Perf. Min. Dim. Min. Min. Perf. Min. Perf.

Proposed names

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

Modes of 3L 4s
UDP Cyclic
order		 Step
pattern		 Mode names
(Heathwaite)		 Mode names
(CA)
6|0 1 LsLsLss Dril Dalmatian
5|1 3 LsLssLs Gil Galatian
4|2 5 LssLsLs Kleeth Cilician
3|3 7 sLsLsLs Bish Bithynian
2|4 2 sLsLssL Fish Pisidian
1|5 4 sLssLsL Jwl Illyrian
0|6 6 ssLsLsL Led Lycian

Theory

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 ¢) and two of them make a 3/2 (702 ¢).
  • Magic, in which the generator is 5/4 (386 ¢) and 5 of them make a 3/1 (1902 ¢).

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).

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 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 mosh tunings have step ratios that are less than 4:3, which implies a generator flatter than 7\24 = 350 ¢.

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.

  • 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.

These identifications are associated with mohajira temperament.

The sizes of the generator, large step and small step of mosh are as follows in various ultrasoft mosh tunings.

24edo (supersoft) 31edo 38edo 55edo JI intervals represented
generator (g) 7\24, 350.00 9\31, 348.39 11\38, 347.37 16\55, 349.09 11/9
L (4g − octave) 4\24, 200.00 5\31, 193.55 6\38, 189.47 9\55, 196.36 9/8, 10/9
s (octave − 3g) 3\24, 150.00 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 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.

These identifications are associated with beatles and suhajira temperaments.

17edo (soft) 27edo (semisoft) 44edo JI intervals represented
generator (g) 5\17, 352.94 8\27, 355.56 13\44, 354.55 16/13, 11/9
L (4g − octave) 3\17, 211.76 5\27, 222.22 8\44, 218.18 9/8, 8/7
s (octave − 3g) 2\17, 141.18 3\27, 133.33 5\44, 137.37 12/11, 13/12, 14/13

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 ¢.

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 symmetric mode sLsLsLs becomes a distorted double harmonic major in these tunings.

This range is associated with sephiroth temperament.

10edo (basic) 13edo (hard) 23edo (semihard)
generator (g) 3\10, 360.00 4\13, 369.23 7\23, 365.22
L (4g − octave) 2\10, 240.00 3\13, 276.92 5\23, 260.87
s (octave − 3g) 1\10, 120.00 1\13, 92.31 2\23, 104.35

Ultrahard

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 mosses, such as the 3L 7s 10-note mos, is suggested for getting 5-limit harmony.

This range is associated with magic temperament.

16edo (superhard) 19edo 22edo 41edo JI intervals represented
generator (g) 5\16, 375.00 6\19, 378.95 7\22, 381.82 13\41, 380.49 5/4
L (4g − octave) 4\16, 300.00 5\19, 315.79 6\22, 327.27 11\41, 321.95 6/5
s (octave − 3g) 1\16, 75.00 1\19, 63.16 1\22, 54.54 2\41, 58.54 25/24

Scales

Scale tree

Generator ranges:

  • Chroma-positive generator: 342.8571 ¢ (2\7) to 400.0000 ¢ (1\3)
  • Chroma-negative generator: 800.0000 ¢ (2\3) to 857.1429 ¢ (5\7)
Scale tree and tuning spectrum of 3L 4s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
2\7 342.857 857.143 1:1 1.000 Equalized 3L 4s
11\38 347.368 852.632 6:5 1.200 Mohaha / ptolemy ↑
9\31 348.387 851.613 5:4 1.250 Mohaha / migration / mohajira
16\55 349.091 850.909 9:7 1.286
7\24 350.000 850.000 4:3 1.333 Supersoft 3L 4s
19\65 350.769 849.231 11:8 1.375 Mohaha / mohamaq
12\41 351.220 848.780 7:5 1.400 Mohaha / neutrominant
17\58 351.724 848.276 10:7 1.429 Hemif / hemififths
5\17 352.941 847.059 3:2 1.500 Soft 3L 4s
18\61 354.098 845.902 11:7 1.571 Suhajira
13\44 354.545 845.455 8:5 1.600
21\71 354.930 845.070 13:8 1.625 Golden suhajira (354.8232 ¢)
8\27 355.556 844.444 5:3 1.667 Semisoft 3L 4s
Suhajira / ringo
19\64 356.250 843.750 12:7 1.714 Beatles
11\37 356.757 843.243 7:4 1.750
14\47 357.447 842.553 9:5 1.800
3\10 360.000 840.000 2:1 2.000 Basic 3L 4s
Scales with tunings softer than this are proper
13\43 362.791 837.209 9:4 2.250
10\33 363.636 836.364 7:3 2.333
17\56 364.286 835.714 12:5 2.400
7\23 365.217 834.783 5:2 2.500 Semihard 3L 4s
18\59 366.102 833.898 13:5 2.600 Unnamed golden tuning (366.2564 ¢)
11\36 366.667 833.333 8:3 2.667
15\49 367.347 832.653 11:4 2.750
4\13 369.231 830.769 3:1 3.000 Hard 3L 4s
13\42 371.429 828.571 10:3 3.333
9\29 372.414 827.586 7:2 3.500 Sephiroth
14\45 373.333 826.667 11:3 3.667
5\16 375.000 825.000 4:1 4.000 Superhard 3L 4s
11\35 377.143 822.857 9:2 4.500 Muggles
6\19 378.947 821.053 5:1 5.000 Magic
7\22 381.818 818.182 6:1 6.000 Würschmidt ↓
1\3 400.000 800.000 1:0 → ∞ Collapsed 3L 4s
Retrieved from "https://en.xen.wiki/index.php?title=3L_4s&oldid=187822"