4L 4s: Difference between revisions

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**Imported revision 385876408 - Original comment: **
→Scale tree: replace with template (- golden diminished)
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox MOS
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Name = tetrawood; diminished
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2012-11-25 20:09:11 UTC</tt>.<br>
| Periods = 4
: The original revision id was <tt>385876408</tt>.<br>
| nLargeSteps = 4
: The revision comment was: <tt></tt><br>
| nSmallSteps = 4
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Equalized = 1
<h4>Original Wikitext content:</h4>
| Collapsed = 0
<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">The minimum harmonic entropy scale with this [[MOSScales|MOS]] pattern is [[Jubilismic clan|diminished]][8], basically the familiar [[Octatonic scale|octatonic scale]] of 12edo.
| Pattern = LsLsLsLs
}}
{{MOS intro}}


There are 4 different near-MOSes such that multiples of the period are the only generic intervals with more than two specific representatives: LLsLsLss, LLsLssLs, LLssLLss, LLssLsLs. However, none are strictly proper in 12edo. (They are only proper if the generator is larger than 100 cents, which unfortunately is the wrong direction for good diminished tunings.)
The minimum harmonic entropy scale with this mos pattern is [[Diminished]][8], the familiar [[octatonic scale]] of 12edo.
||||||||||~ Generator ||~ Cents ||~ Comments ||
|| 0\4 ||  ||  ||  ||  || 0 ||=  ||
||  ||  ||  ||  || 1\24 || 50 ||  ||
||  ||  ||  || 1\20 ||  || 60 ||= L/s=4:
generators smaller than this are impractical" ||
||  ||  || 1\16 ||  ||  || 75 ||= L/s=3:
generators smaller than this may be considered "impractical" ||
||  || 1\12 ||  ||  ||  || 100 ||= Diminished is around here
Optimum rank range (L/s=2/1) for MOS ||
||  ||  ||  || 3\32 ||  || 112.5 ||=  ||
||  ||  ||  ||  || 5\52 || 115.38 ||= Golden diminished ||
||  ||  || 2\20 ||  ||  || 120 ||=  ||
|| 1\8 ||  ||  ||  ||  || 150 ||=  ||</pre></div>
<h4>Original HTML content:</h4>
<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 4s&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The minimum harmonic entropy scale with this &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; pattern is &lt;a class="wiki_link" href="/Jubilismic%20clan"&gt;diminished&lt;/a&gt;[8], basically the familiar &lt;a class="wiki_link" href="/Octatonic%20scale"&gt;octatonic scale&lt;/a&gt; of 12edo.&lt;br /&gt;
&lt;br /&gt;
There are 4 different near-MOSes such that multiples of the period are the only generic intervals with more than two specific representatives: LLsLsLss, LLsLssLs, LLssLLss, LLssLsLs. However, none are strictly proper in 12edo. (They are only proper if the generator is larger than 100 cents, which unfortunately is the wrong direction for good diminished tunings.)&lt;br /&gt;


In addition to the true mos form, LsLsLsLs, there are four near-mos forms – LLsLsLss, LLsLssLs, LLssLLss, and LLssLsLs – in which the period and its multiples are the only intervals with more than two varieties. The near-mos forms are only proper if the dark generator is larger than 1\12 (100{{c}}).


&lt;table class="wiki_table"&gt;
== Intervals ==
    &lt;tr&gt;
{{MOS intervals}}
        &lt;th colspan="5"&gt;Generator&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Comments&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0\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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\24&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;50&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&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;1\20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;60&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;L/s=4:&lt;br /&gt;
generators smaller than this are impractical&amp;quot;&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;1\16&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;75&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;L/s=3:&lt;br /&gt;
generators smaller than this may be considered &amp;quot;impractical&amp;quot;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\12&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;100&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Diminished is around here&lt;br /&gt;
Optimum rank range (L/s=2/1) for MOS&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;3\32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;112.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5\52&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;115.38&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Golden diminished&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;2\20&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;120&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1\8&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;150&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
== Modes ==
{{MOS mode degrees}}
 
== Scale tree ==
{{MOS tuning spectrum
| 6/5 = [[Fourfives]]↑
| 13/8 = Unnamed golden tuning
| 2/1 = [[Diminished]]
| 13/5 = Unnamed golden tuning
| 6/1 = [[Quadritikleismic]]↓
}}
 
[[Category:Tetrawood| ]] <!-- main article -->
[[Category:8-tone scales]]

Latest revision as of 12:09, 21 August 2025

↖ 3L 3s ↑ 4L 3s 5L 3s ↗
← 3L 4s 4L 4s 5L 4s →
↙ 3L 5s ↓ 4L 5s 5L 5s ↘ 
┌╥┬╥┬╥┬╥┬┐
│║│║│║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLsLs
sLsLsLsL
Equave 2/1 (1200.0 ¢)
Period 1\4 (300.0 ¢)
Generator size
Bright 1\8 to 1\4 (150.0 ¢ to 300.0 ¢)
Dark 0\4 to 1\8 (0.0 ¢ to 150.0 ¢)
TAMNAMS information
Name tetrawood
Prefix tetrawd-
Abbrev. ttw
Related MOS scales
Parent none
Sister 4L 4s (self)
Daughters 8L 4s, 4L 8s
Neutralized 8edo
2-Flought 12L 4s, 4L 12s
Equal tunings
Equalized (L:s = 1:1) 1\8 (150.0 ¢)
Supersoft (L:s = 4:3) 4\28 (171.4 ¢)
Soft (L:s = 3:2) 3\20 (180.0 ¢)
Semisoft (L:s = 5:3) 5\32 (187.5 ¢)
Basic (L:s = 2:1) 2\12 (200.0 ¢)
Semihard (L:s = 5:2) 5\28 (214.3 ¢)
Hard (L:s = 3:1) 3\16 (225.0 ¢)
Superhard (L:s = 4:1) 4\20 (240.0 ¢)
Collapsed (L:s = 1:0) 1\4 (300.0 ¢)

4L 4s, named tetrawood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 4 small steps, with a period of 1 large step and 1 small step that repeats every 300.0 ¢, or 4 times every octave. Generators that produce this scale range from 150 ¢ to 300 ¢, or from 0 ¢ to 150 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

The minimum harmonic entropy scale with this mos pattern is Diminished[8], the familiar octatonic scale of 12edo.

In addition to the true mos form, LsLsLsLs, there are four near-mos forms – LLsLsLss, LLsLssLs, LLssLLss, and LLssLsLs – in which the period and its multiples are the only intervals with more than two varieties. The near-mos forms are only proper if the dark generator is larger than 1\12 (100 ¢).

Intervals

Intervals of 4L 4s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-tetrawdstep Perfect 0-tetrawdstep P0ttws 0 0.0 ¢
1-tetrawdstep Minor 1-tetrawdstep m1ttws s 0.0 ¢ to 150.0 ¢
Major 1-tetrawdstep M1ttws L 150.0 ¢ to 300.0 ¢
2-tetrawdstep Perfect 2-tetrawdstep P2ttws L + s 300.0 ¢
3-tetrawdstep Minor 3-tetrawdstep m3ttws L + 2s 300.0 ¢ to 450.0 ¢
Major 3-tetrawdstep M3ttws 2L + s 450.0 ¢ to 600.0 ¢
4-tetrawdstep Perfect 4-tetrawdstep P4ttws 2L + 2s 600.0 ¢
5-tetrawdstep Minor 5-tetrawdstep m5ttws 2L + 3s 600.0 ¢ to 750.0 ¢
Major 5-tetrawdstep M5ttws 3L + 2s 750.0 ¢ to 900.0 ¢
6-tetrawdstep Perfect 6-tetrawdstep P6ttws 3L + 3s 900.0 ¢
7-tetrawdstep Minor 7-tetrawdstep m7ttws 3L + 4s 900.0 ¢ to 1050.0 ¢
Major 7-tetrawdstep M7ttws 4L + 3s 1050.0 ¢ to 1200.0 ¢
8-tetrawdstep Perfect 8-tetrawdstep P8ttws 4L + 4s 1200.0 ¢

Modes

Scale degrees of the modes of 4L 4s
UDP Cyclic
order 		Step
pattern 		Scale degree (tetrawddegree)
0 1 2 3 4 5 6 7 8
4|0(4) 1 LsLsLsLs Perf. Maj. Perf. Maj. Perf. Maj. Perf. Maj. Perf.
0|4(4) 2 sLsLsLsL Perf. Min. Perf. Min. Perf. Min. Perf. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 4L 4s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\8 150.000 150.000 1:1 1.000 Equalized 4L 4s
6\44 163.636 136.364 6:5 1.200 Fourfives↑
5\36 166.667 133.333 5:4 1.250
9\64 168.750 131.250 9:7 1.286
4\28 171.429 128.571 4:3 1.333 Supersoft 4L 4s
11\76 173.684 126.316 11:8 1.375
7\48 175.000 125.000 7:5 1.400
10\68 176.471 123.529 10:7 1.429
3\20 180.000 120.000 3:2 1.500 Soft 4L 4s
11\72 183.333 116.667 11:7 1.571
8\52 184.615 115.385 8:5 1.600
13\84 185.714 114.286 13:8 1.625 Unnamed golden tuning
5\32 187.500 112.500 5:3 1.667 Semisoft 4L 4s
12\76 189.474 110.526 12:7 1.714
7\44 190.909 109.091 7:4 1.750
9\56 192.857 107.143 9:5 1.800
2\12 200.000 100.000 2:1 2.000 Basic 4L 4s
Diminished
9\52 207.692 92.308 9:4 2.250
7\40 210.000 90.000 7:3 2.333
12\68 211.765 88.235 12:5 2.400
5\28 214.286 85.714 5:2 2.500 Semihard 4L 4s
13\72 216.667 83.333 13:5 2.600 Unnamed golden tuning
8\44 218.182 81.818 8:3 2.667
11\60 220.000 80.000 11:4 2.750
3\16 225.000 75.000 3:1 3.000 Hard 4L 4s
10\52 230.769 69.231 10:3 3.333
7\36 233.333 66.667 7:2 3.500
11\56 235.714 64.286 11:3 3.667
4\20 240.000 60.000 4:1 4.000 Superhard 4L 4s
9\44 245.455 54.545 9:2 4.500
5\24 250.000 50.000 5:1 5.000
6\28 257.143 42.857 6:1 6.000 Quadritikleismic↓
1\4 300.000 0.000 1:0 → ∞ Collapsed 4L 4s
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