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4L 3s: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
|en=4L 3s
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-10 23:07:52 UTC</tt>.<br>
|es=
: The original revision id was <tt>209493246</tt>.<br>
|de=
: The revision comment was: <tt></tt><br>
|ja=4L 3s
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox MOS}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">4L 3s refers to the structure of moment of symmetry scales with generators ranging from 1\4edo (one degree of [[4edo]], 300¢) to 2\7edo (two degrees of [[7edo]], or approx. 342.857¢). The spectrum looks like this:


||  ||  ||  ||  || scale || g in cents || 2g || 3g || 4g ||
{{MOS intro}}
|| 1\4 ||  ||  ||  || 1 0 1 0 1 0 1 || 300.000 || 600.000 || 900.000 || 0.000 ||
4L&nbsp;3s can be seen as a [[Warped diatonic|warped diatonic scale]], where one large step of diatonic ([[5L&nbsp;2s]]) is replaced with a small step.
||  ||  ||  || 5\19 || 4 1 4 1 4 1 4 || 315.789 || 631.579 || 947.368 || 63.158 ||
||  ||  || 4\15 ||  || 3 1 3 1 3 1 3 || 320.000 || 640.000 || 960.000 || 80.000 ||
||  ||  ||  || 7\26 || 5 2 5 2 5 2 5 || 323.077 || 646.154 || 969.231 || 92.308 ||
||  || 3\11 ||  ||  || 2 1 2 1 2 1 2 || 327.273 || 654.545 || 981.818 || 109.091 ||
||  ||  ||  || 8\29 || 5 3 5 3 5 3 5 || 331.034 || 662.069 || 993.013 || 124.138 ||
||  ||  || 5\18 ||  || 3 2 3 2 3 2 3 || 333.333 || 666.667 || 1000.000 || 133.333 ||
||  ||  ||  || 7\25 || 4 3 4 3 4 3 4 || 336.000 || 672.000 || 1008.000 || 144.000 ||
|| 2\7 ||  ||  ||  || 1 1 1 1 1 1 1 || 342.857 || 685.714 || 1028.571 || 171.429 ||


The "northernmost" range, near 4\19, could be called the keemun range or region, representing [[Kleismic family|keemun temperament]].</pre></div>
== Name ==
<h4>Original HTML content:</h4>
{{TAMNAMS name}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;4L 3s&lt;/title&gt;&lt;/head&gt;&lt;body&gt;4L 3s refers to the structure of moment of symmetry scales with generators ranging from 1\4edo (one degree of &lt;a class="wiki_link" href="/4edo"&gt;4edo&lt;/a&gt;, 300¢) to 2\7edo (two degrees of &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;, or approx. 342.857¢). The spectrum looks like this:&lt;br /&gt;
&lt;br /&gt;


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


&lt;table class="wiki_table"&gt;
=== Intervals ===
    &lt;tr&gt;
{{MOS intervals}}
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;scale&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;g in cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2g&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3g&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4g&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1\4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1 0 1 0 1 0 1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;300.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;600.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;900.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5\19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4 1 4 1 4 1 4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;315.789&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;631.579&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;947.368&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;63.158&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4\15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3 1 3 1 3 1 3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;320.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;640.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;960.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;80.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7\26&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5 2 5 2 5 2 5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;323.077&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;646.154&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;969.231&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;92.308&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2 1 2 1 2 1 2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;327.273&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;654.545&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;981.818&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;109.091&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8\29&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5 3 5 3 5 3 5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;331.034&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;662.069&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;993.013&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;124.138&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5\18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3 2 3 2 3 2 3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;333.333&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;666.667&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1000.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;133.333&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4 3 4 3 4 3 4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;336.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;672.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1008.000&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;144.000&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2\7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1 1 1 1 1 1 1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;342.857&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;685.714&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1028.571&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;171.429&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
=== Generator chain ===
The &amp;quot;northernmost&amp;quot; range, near 4\19, could be called the keemun range or region, representing &lt;a class="wiki_link" href="/Kleismic%20family"&gt;keemun temperament&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{MOS genchain}}
 
=== Modes ===
{{MOS mode degrees}}
 
==== Proposed names ====
Alexandru Ianu ([[User:Ayceman|Ayceman]])<ref>Description of ''Sylvian Moon Dance'' mentioning the naming proposal https://musescore.com/user/36772625/scores/6700443 – The theme relates to the mystical nature of the Tribunal and TES lore, which fits smitonic.</ref> has proposed the following mode names relating to the Almsivi in Morrowind (TES):
{{MOS modes
| Mode Names=Nerevarine $
Vivecan $
Lorkhanic $
Sothic $
Kagrenacan $
Almalexian $
Dagothic $
}}
 
== Theory ==
=== Low harmonic entropy scales ===
There are two notable harmonic entropy minima:
* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1.
* [[myna|Myna temperament]], in which the generator is also 6/5 but it takes 10 of them to make a 6/1, meaning that a larger MOS than 4L&nbsp;3s is required to reach 3/2 or 4/3.
 
=== Temperament interpretations ===
{{main|4L&nbsp;3s/Temperaments}}
4L&nbsp;3s has the following temperament interpretations:
* [[Sixix]], with generators around 338.6{{c}}.
* [[Orgone]], with generators around 323.4{{c}}.
* [[Kleismic]], with generators around 317{{c}}.
 
Other temperaments, such as [[amity]] and [[myna]], require more than 7 pitches to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a [[MODMOS]] or a larger MOS gamut is necessary to access these pitches.
 
== Tuning ranges ==
{{Todo|Populate|comment=Populate with JI ratios from prior edits of this page.|inline=1}}
 
=== Simple tunings ===
The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 11edo, 15edo, and 18edo, respectively.
{{MOS tunings}}
 
=== Parasoft tunings ===
Parasoft smitonic tunings can be considered "meantone smitonic" since it has the following features of [[meantone]] diatonic tunings:
 
* The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".
* The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.
 
These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702{{c}}), and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like chromas. For this reason, parasoft might be the most accessible smitonic tuning range.
 
Edos include [[18edo]], [[25edo]], and [[43edo]]. Some key considerations include:
 
* 18edo can be used to make the large and small steps more distinct, or can be considered a distorted 19edo diatonic.
** 18edo has a major 1-mosstep that is close to 9/8 (203{{c}}).
** 18edo's major and minor 4-mossteps are both equally off from 12edo's diatonic perfect 5th (700{{c}}) by 33.3{{c}}.
** 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* The augmented 2-mosstep of 25edo is very close to 5/4 (386{{c}}).
{{MOS tunings|Step Ratios=3/2; 7/5; 4/3}}
 
=== Hyposoft tunings ===
Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327{{c}} and 333{{c}}. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] smitonic".
 
Edos include [[11edo]] (not shown), [[18edo]], and [[29edo]].
 
{{MOS tunings|Step Ratios=3/2; 5/3; 7/4}}
 
=== Hypohard tunings===
Hypohard smitonic tunings (2:1 to 3:1) have generators between 320{{c}} and 327{{c}}. The major 1-mosstep, or large step, tends to approximate [[8/7]] (231{{c}}) and the major 3-mosstep tends to approximate [[11/8]] (551{{c}}). [[26edo]] approximates these two intervals very well. These JI approximations are associated with [[orgone]] temperament.
 
Other hypohard edos include [[11edo]] (not shown), [[15edo]] and [[37edo]].
 
{{MOS tunings|Step Ratios=3/1; 5/2; 7/3}}
 
=== Parahard tunings ===
Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9{{c}} and 320{{c}}, putting it close to a pure 6/5 (316{{c}}). Stacking six generators and octave-reducing approximates 3/2 (702{{c}}), a diatonic perfect 5th, represented by the diminished 5-mosstep.
 
This range contains very accurate edos such as [[53edo]] and [[72edo]], and has very accurate approximations to many [[low-overtone JI]] intervals, namely basic [[5-limit]] ratios and some ratios involving 13. However, 4L 3s only has one interval of 3/2, so it's suggested to use a larger MOS, such as [[4L 7s]], to achieve 5-limit harmony.
 
These JI approximations are associated with [[kleismic]] temperament, through the 2.3.5.13 extension known as [[Kleismic family#Cata|cata]].
 
Parahard edos smaller than 53edo include [[15edo]] (not shown), [[19edo]], and [[34edo]].
 
{{MOS tunings|Step Ratios=4/1; 11/3; 7/2}}
 
== Scales ==
* [[Orgone7]]
* [[Cata7]]
* [[Myna7]]
 
== Scale tree==
{{MOS tuning spectrum
| 6/5 = [[Amity]]/[[hitchcock]]&nbsp;↑
| 5/4 = [[Sixix]]
| 4/3 = [[Supramin]]
| 13/8 = Golden 4L&nbsp;3s (868.3282{{c}})
| 12/5 = [[Hyperkleismic]]
| 5/2 = [[Orgone]]
| 13/5 = Golden superkleismic
| 8/3 = [[Superkleismic]]
| 11/3 = [[Hanson]]/[[keemun]]
| 6/1 = [[Oolong]]/[[myna]]&nbsp;↓
}}
 
== Music ==
* [[City of the Asleep]], [https://cityoftheasleep.bandcamp.com/album/an-amputated-elliptic-knob-of-the-cryptocurve-regenerates "An Amputated Elliptic Knob of the Cryptocurve Regenerates"] (Various orgone edos)
* [[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
* [[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
 
== References ==
<references />
 
[[Category:Smitonic|*]] <!--Main article-->
[[Category:7-tone scales]]
Retrieved from "https://en.xen.wiki/w/4L_3s"