Muddle: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 293747706 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 293766380 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-01-19 | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2012-01-19 23:40:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>293766380</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | 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> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html"> | <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">//Muddling// refers to a process of mapping one periodic scale onto another one; the scale thus produced is a //muddle//. By periodic scale, I mean a scale that repeats "at the something" (octave, tritave, something else). There are two necessary components: a **parent scale** and a **target scale**, or simply "parent" and "target". The parent scale can be any periodic scale at all; the target scale is not exactly a scale -- it's the outline or //shape// of a scale -- and it must be defined in terms of units or degrees which comprise the steps. If the period is an octave, this means the target scale will be a subset of an [[EDO]]. | ||
The simplest sort of muddle is a MOS Muddle, which is a sort of second-order [[MOSScales|MOS Scale]] and is useful for generating usable subsets of larger MOS scales and for navigating [[Regular Temperaments]]. This article will mostly deal with MOS muddles, but it should be remembered that this process can be generalized to other structures. | |||
=MOS Muddles= | |||
In the case of a MOS muddle, the parent scale is any MOS scale large enough that taking subsets of it would be musically useful. The target scale is something like 12122. This could be the form of an actual scale (an MOS subset of [[8edo]], in this example), but as a target scale, we are interested in its general //shape//, which suggests a way of bunching intervals of the parent scale. If we apply the target to an equal-step scale, we arrive at a standard MOS. Eg. If our parent scale is [[8edo]] -- with steps 11111111 -- and our target scale is 12122, then the resulting scale is (1)(11)(1)(11)(1)(11) = 12122 -- the same as our target scale. But if our parent scale is some other MOS scale, say 22222223 (a subset of [[17edo]]), applying the 12122 target scale generates (2)(22)(2)(22)(23) = 24245 -- not an MOS scale. The latter scale, which we can call a //muddle//, has some melodic similarity to the target scale of 12122, but belongs to a different temperament family entirely. Choosing a different mode (rotation) of either the parent scale or the target scale may produce a different muddle. | |||
=Examples= | =Examples= | ||
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===1.=== | ===1.=== | ||
To continue with our example of a parent scale of 22222223 and | To continue with our example of a parent scale of 22222223 and a target shape of 12122, here are all the muddles that can result from different rotations of the parent scale: | ||
* 22222223 parent with 12122 | * 22222223 parent with 12122 target gives (2)(22)(2)(22)(23) = 24245 | ||
* 22222232 parent with 12122 | * 22222232 parent with 12122 target gives (2)(22)(2)(22)(32) = 24245 | ||
* 22222322 parent with 12122 | * 22222322 parent with 12122 target gives (2)(22)(2)(23)(22) = 24254 | ||
* 22223222 parent with 12122 | * 22223222 parent with 12122 target gives (2)(22)(2)(32)(22) = 24254 | ||
* 22232222 parent with 12122 | * 22232222 parent with 12122 target gives (2)(22)(3)(22)(22) = 24344 | ||
* 22322222 parent with 12122 | * 22322222 parent with 12122 target gives (2)(23)(2)(22)(22) = 25244 | ||
* 23222222 parent with 12122 | * 23222222 parent with 12122 target gives (2)(32)(2)(22)(22) = 25244 | ||
* 32222222 parent with 12122 | * 32222222 parent with 12122 target gives (3)(22)(2)(22)(22) = 34244 | ||
Notice that not all of these rotations | Notice that not all of these rotations produce unique muddles. The unique muddles are 24245, 24254, 24344, 25244, and 34244. | ||
===2.=== | ===2.=== | ||
Here is a diagram showing the muddles available with 55755757 parent scale ([[Sensi]][8] | Here is a diagram showing the muddles available with a 55755757 parent scale ([[Sensi]][8] in [[46edo]]) and a 12122 target scale. Note that this combination produces MOS scales as well as muddles. | ||
[[image:sensi_pentatonics.png]] | [[image:sensi_pentatonics.png]] | ||
=Comments= | =Comments= | ||
Muddles always have more than two sizes of step -- either three or four sizes. Whereas MOS scales have two varieties of interval for each interval class (eg. a "large step" and a "small step"), muddles have potentially two varieties within each variety (eg. two sizes of "small step" and two sizes of "large step"). Parent | MOS Muddles always have more than two sizes of step -- either three or four sizes. Whereas MOS scales have two varieties of interval for each interval class (eg. a "large step" and a "small step"), muddles have potentially two varieties within each variety (eg. two sizes of "small step" and two sizes of "large step"). Parent scales that are close to equal (eg. [[Maximal evenness|maximally even]] scales) will produce muddles that are closer in sound to the target scale. Larger parent scales contain more potential muddles than smaller ones, just as larger [[EDO]]s contain more potential MOS scales than smaller ones. | ||
=Non-MOS Muddles= | |||
As mentioned at the beginning, in a muddle, the parent scale can be any kind of periodic scale at all, including but not limited to [[MODMOS scales]], [[MOS Cradle|MOS Cradle scales]], other muddles, scales of [[Regular Temperaments|temperaments]] with rank higher than 2, [[Just Intonation]] scales, etc. The target scale is a little less flexible, but it could be at least a MODMOS scale, a MOS Cradle scale or any other MOS subset or subset of a periodic equal-step scale. It's just important that the total number of "units" in the target scale (eg. the MOS Cradle 23132 has 2+3+1+3+2=11 units) be the same as the number of tones in the parent scale (thus the 23132 target scale must be applied to a scale of 11 tones). | |||
<span class="commentBody">As one example of a muddle with a non-MOS parent</span><span class="text_exposed_show">, if you take overtones 16-32 as the parent scale, you can apply a 2322232 target scale and get 1/1, 9/8, 21/16, 23/16, 25/16, 27/16, 15/8, 2/1.</span></pre></div> | |||
<h4>Original HTML content:</h4> | <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"><html><head><title>Muddle</title></head><body> | <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"><html><head><title>Muddle</title></head><body><em>Muddling</em> refers to a process of mapping one periodic scale onto another one; the scale thus produced is a <em>muddle</em>. By periodic scale, I mean a scale that repeats &quot;at the something&quot; (octave, tritave, something else). There are two necessary components: a <strong>parent scale</strong> and a <strong>target scale</strong>, or simply &quot;parent&quot; and &quot;target&quot;. The parent scale can be any periodic scale at all; the target scale is not exactly a scale -- it's the outline or <em>shape</em> of a scale -- and it must be defined in terms of units or degrees which comprise the steps. If the period is an octave, this means the target scale will be a subset of an <a class="wiki_link" href="/EDO">EDO</a>.<br /> | ||
<br /> | |||
The simplest sort of muddle is a MOS Muddle, which is a sort of second-order <a class="wiki_link" href="/MOSScales">MOS Scale</a> and is useful for generating usable subsets of larger MOS scales and for navigating <a class="wiki_link" href="/Regular%20Temperaments">Regular Temperaments</a>. This article will mostly deal with MOS muddles, but it should be remembered that this process can be generalized to other structures.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="MOS Muddles"></a><!-- ws:end:WikiTextHeadingRule:0 -->MOS Muddles</h1> | |||
<br /> | |||
In the case of a MOS muddle, the parent scale is any MOS scale large enough that taking subsets of it would be musically useful. The target scale is something like 12122. This could be the form of an actual scale (an MOS subset of <a class="wiki_link" href="/8edo">8edo</a>, in this example), but as a target scale, we are interested in its general <em>shape</em>, which suggests a way of bunching intervals of the parent scale. If we apply the target to an equal-step scale, we arrive at a standard MOS. Eg. If our parent scale is <a class="wiki_link" href="/8edo">8edo</a> -- with steps 11111111 -- and our target scale is 12122, then the resulting scale is (1)(11)(1)(11)(1)(11) = 12122 -- the same as our target scale. But if our parent scale is some other MOS scale, say 22222223 (a subset of <a class="wiki_link" href="/17edo">17edo</a>), applying the 12122 target scale generates (2)(22)(2)(22)(23) = 24245 -- not an MOS scale. The latter scale, which we can call a <em>muddle</em>, has some melodic similarity to the target scale of 12122, but belongs to a different temperament family entirely. Choosing a different mode (rotation) of either the parent scale or the target scale may produce a different muddle.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Examples--1."></a><!-- ws:end:WikiTextHeadingRule:4 -->1.</h3> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | To continue with our example of a parent scale of 22222223 and a target shape of 12122, here are all the muddles that can result from different rotations of the parent scale:<br /> | ||
<ul><li>22222223 parent with 12122 target gives (2)(22)(2)(22)(23) = 24245</li><li>22222232 parent with 12122 target gives (2)(22)(2)(22)(32) = 24245</li><li>22222322 parent with 12122 target gives (2)(22)(2)(23)(22) = 24254</li><li>22223222 parent with 12122 target gives (2)(22)(2)(32)(22) = 24254</li><li>22232222 parent with 12122 target gives (2)(22)(3)(22)(22) = 24344</li><li>22322222 parent with 12122 target gives (2)(23)(2)(22)(22) = 25244</li><li>23222222 parent with 12122 target gives (2)(32)(2)(22)(22) = 25244</li><li>32222222 parent with 12122 target gives (3)(22)(2)(22)(22) = 34244</li></ul>Notice that not all of these rotations produce unique muddles. The unique muddles are 24245, 24254, 24344, 25244, and 34244.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Examples--2."></a><!-- ws:end:WikiTextHeadingRule:6 -->2.</h3> | |||
<br /> | <br /> | ||
Here is a diagram showing the muddles available with a 55755757 parent scale (<a class="wiki_link" href="/Sensi">Sensi</a>[8] in <a class="wiki_link" href="/46edo">46edo</a>) and a 12122 target scale. Note that this combination produces MOS scales as well as muddles.<br /> | |||
< | <!-- ws:start:WikiTextLocalImageRule:30:&lt;img src=&quot;/file/view/sensi_pentatonics.png/293730366/sensi_pentatonics.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/sensi_pentatonics.png/293730366/sensi_pentatonics.png" alt="sensi_pentatonics.png" title="sensi_pentatonics.png" /><!-- ws:end:WikiTextLocalImageRule:30 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Comments"></a><!-- ws:end:WikiTextHeadingRule:8 -->Comments</h1> | ||
<br /> | <br /> | ||
MOS Muddles always have more than two sizes of step -- either three or four sizes. Whereas MOS scales have two varieties of interval for each interval class (eg. a &quot;large step&quot; and a &quot;small step&quot;), muddles have potentially two varieties within each variety (eg. two sizes of &quot;small step&quot; and two sizes of &quot;large step&quot;). Parent scales that are close to equal (eg. <a class="wiki_link" href="/Maximal%20evenness">maximally even</a> scales) will produce muddles that are closer in sound to the target scale. Larger parent scales contain more potential muddles than smaller ones, just as larger <a class="wiki_link" href="/EDO">EDO</a>s contain more potential MOS scales than smaller ones.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Non-MOS Muddles"></a><!-- ws:end:WikiTextHeadingRule:10 -->Non-MOS Muddles</h1> | ||
<br /> | <br /> | ||
As mentioned at the beginning, in a muddle, the parent scale can be any kind of periodic scale at all, including but not limited to <a class="wiki_link" href="/MODMOS%20scales">MODMOS scales</a>, <a class="wiki_link" href="/MOS%20Cradle">MOS Cradle scales</a>, other muddles, scales of <a class="wiki_link" href="/Regular%20Temperaments">temperaments</a> with rank higher than 2, <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a> scales, etc. The target scale is a little less flexible, but it could be at least a MODMOS scale, a MOS Cradle scale or any other MOS subset or subset of a periodic equal-step scale. It's just important that the total number of &quot;units&quot; in the target scale (eg. the MOS Cradle 23132 has 2+3+1+3+2=11 units) be the same as the number of tones in the parent scale (thus the 23132 target scale must be applied to a scale of 11 tones).<br /> | |||
<br /> | <br /> | ||
< | <span class="commentBody">As one example of a muddle with a non-MOS parent</span><span class="text_exposed_show">, if you take overtones 16-32 as the parent scale, you can apply a 2322232 target scale and get 1/1, 9/8, 21/16, 23/16, 25/16, 27/16, 15/8, 2/1.</span></body></html></pre></div> | ||