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3L 3s: Difference between revisions

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m Apparently I have to be more specific in the ToDo
 
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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 = triwood
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-11-03 12:51:23 UTC</tt>.<br>
| Periods = 3
: The original revision id was <tt>564967897</tt>.<br>
| nLargeSteps = 3
: The revision comment was: <tt></tt><br>
| nSmallSteps = 3
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>
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<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">Associated with the [[augmented family]], which comprises the only significant minimum of harmonic entropy for these scales.
| Pattern = LsLsLs
||||||||||~ Generator ||~ Cents ||~ Comments ||
}}
|| 0\3 ||  ||  ||  ||  || 0 ||=  ||
||  ||  ||  || 1\15 ||  || 80 ||= L/s = 4 ||
||  ||  ||  ||  || 2\27 || 88.88 ||= Optimal augmented is around here ||
||  ||  ||  ||  ||  || 400/(1+pi) ||  ||
||  ||  || 1\12 ||  ||  || 100 ||= L/s = 3 ||
||  ||  ||  ||  ||  || 400/(1+e) ||  ||
||  ||  ||  ||  || 3\33 || 109.09 ||  ||
||  ||  ||  ||  ||  || 400/(2+phi) ||  ||
||  ||  ||  || 2\21 ||  || 114.29 ||  ||
||  || 1\9 ||  ||  ||  || 133.33 ||= Boundary of propriety for near-MOS
Optimum rank range (L/s=2/1) for MOS ||
||  ||  ||  ||  ||  || 400/(1+sqrt(3)) ||  ||
||  ||  ||  || 3\24 ||  || 150 ||=   ||
||  ||  ||  ||  ||  || 400/(1+phi) ||= Golden triforce ||
||  ||  ||  ||  || 5\39 || 153.85 ||=   ||
||  ||  ||  ||  ||  || 400/(1+pi/2) ||  ||
||  ||  || 2\15 ||  ||  || 160 ||=   ||
|| 1\6 ||  ||  ||  ||  || 200 ||=  ||
The true MOS is LsLsLs but interesting near-MOSs include LLsLss and LLssLs. The MOS is always proper but the other forms are only proper if the generator is larger than 1\9 of an octave (so not augmented).


Out of all **[[Rothenberg propriety|proper]]** six-note MOS scales, this augmented scale probably has the lowest harmonic entropy.</pre></div>
{{MOS intro}}
<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;3L 3s&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Associated with the &lt;a class="wiki_link" href="/augmented%20family"&gt;augmented family&lt;/a&gt;, which comprises the only significant minimum of harmonic entropy for these scales.&lt;br /&gt;


In addition to the true MOS (LsLsLs or sLsLsL), there are also near-MOS patterns of LLsLss and LLssLs, which are only proper if the generator is larger than [[9edo|1\9]].


&lt;table class="wiki_table"&gt;
Out of all ''[[Rothenberg propriety|proper]]'' six-note MOS scales, this augmented scale probably has the lowest harmonic entropy{{Clarify}}.
    &lt;tr&gt;
        &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\3&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;1\15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;80&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;L/s = 4&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;2\27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;88.88&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Optimal augmented is around here&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(1+pi)&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;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;100&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;L/s = 3&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(1+e)&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;109.09&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(2+phi)&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;2\21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;114.29&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;1\9&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;133.33&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Boundary of propriety for near-MOS&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(1+sqrt(3))&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;3\24&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;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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(1+phi)&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Golden triforce&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\39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;153.85&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;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;400/(1+pi/2)&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;2\15&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;160&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\6&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;200&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;


The true MOS is LsLsLs but interesting near-MOSs include LLsLss and LLssLs. The MOS is always proper but the other forms are only proper if the generator is larger than 1\9 of an octave (so not augmented).&lt;br /&gt;
== Intervals ==
&lt;br /&gt;
{{MOS intervals}}
Out of all &lt;strong&gt;&lt;a class="wiki_link" href="/Rothenberg%20propriety"&gt;proper&lt;/a&gt;&lt;/strong&gt; six-note MOS scales, this augmented scale probably has the lowest harmonic entropy.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== Modes==
{{MOS mode degrees}}
 
==Scale tree==
{{todo|inline=1|complete table|Convert manually-generated scale tree table to MOS tuning spectrum without losing information from the existing table}}
{| class="wikitable center-all"
! colspan="6" rowspan="2" |Generator
! colspan="2" |Cents
! rowspan="2" |L
! rowspan="2" |s
! rowspan="2" |L/s
! rowspan="2" | Comments
|-
! Chroma-positive
! Chroma-negative
|-
|1\6|| || ||  || || ||200.000||200.000 || 1||1||1.000||
|-
| || ||  ||  || ||6\33||218.182|| 181.818||6||5||1.200||
|-
| || ||  ||  ||5\27|| ||222.222|| 177.778||5||4||1.250||
|-
| || ||  ||  || ||9\48||225.000|| 175.000||9||7||1.286||Oodako
|-
|  ||  || ||4\21|| || ||228.571|| 171.429||4||3||1.333||
|-
| || ||  ||  || ||11\57||231.579||168.421 || 11 ||8||1.375||
|-
| || || || ||7\36|| ||233.333|| 166.667||7||5||1.400||
|-
| || ||  ||  || ||10\51||235.294||164.706 || 10 ||7||1.428||
|-
| || ||3\15 ||  || || ||240.000|| 160.000|| 3||2||1.500||
|-
| || ||  ||  || ||11\54||244.444||155.556 || 11 ||7||1.571||
|-
| || || || ||8\39|| ||246.154|| 153.846||8||5||1.600||Triforce
|-
| ||  ||  || || ||13\63||247.619||152.381 || 13 ||8||1.625|| Unnamed golden tuning
|-
| || || ||5\24|| || ||250.000|| 150.000||5||3||1.667||
|-
| || ||  ||  || ||12\57||252.632||147.368 || 12 ||7||1.714||
|-
| || || || ||7\33|| ||254.545|| 145.455||7||4||1.750||
|-
| || ||  ||  || ||9\42||257.143|| 142.857||9||5||1.800||
|-
| ||2\9 ||  || || || ||266.667||133.333 || 2||1||2.000||Basic triwood
|-
| ||  ||  || ||  || 9\39||276.923|| 123.077||9||4||2.250||
|-
| || ||  ||  ||7\30|| ||280.000|| 120.000||7||3||2.333||Deflated (optimal around here)
|-
| || || || || ||12\51||282.353||117.647 || 12 ||5||2.400||
|-
| || ||  || 5\21||  ||  ||285.714|| 114.286||5||2||2.500||
|-
| || ||  ||  || ||13\54||288.889||111.111 || 13 ||5||2.600|| Unnamed golden tuning
|-
| || || || ||8\33|| ||290.909|| 109.091||8||3||2.667||
|-
| || ||  ||  || ||11\45||293.333||106.667 || 11 ||4||2.750|| August
|-
| || ||3\12|| || || ||300.000|| 100.000|| 3||1||3.000||Trug (optimal around here)
|-
| || || || || ||10\39||307.692||92.308||10||3||3.333||Augene
|-
| || || || ||7\27|| ||311.111|| 88.889||7 ||2 || 3.500||
|-
| || ||  ||  || ||11\42||314.286||85.714||11||3||3.667||
|-
| || || ||4\15 || || ||320.000|| 80.000||4||1 || 4.000||Inflated
|-
| ||  ||  || || ||9\33 ||327.273|| 72.727||9 || 2|| 4.500||
|-
| || ||  ||  ||5\18|| ||333.333|| 66.667||5 ||1 || 5.000||
|-
| || ||  ||  || ||6\21||342.857|| 57.143||6 || 1|| 6.000||Hemiug↓, hemiaug↓
|-
|1\3 ||  || || || || ||400.000||0.000||1 || 0||→ inf||
|}
 
[[Category:Triwood| ]]
[[Category:6-tone scales]]
<!-- main article -->
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