MODMOS scale: Difference between revisions

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-03-14 04:04:37 UTC</tt>.<br>

: The original revision id was <tt>210178680</tt>.<br>

<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">=Near-MOS (NMOS) Scales=  

<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;MODMOS Scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Near-MOS (NMOS) Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Near-MOS (NMOS) Scales&lt;/h1&gt;

&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Basic Approach"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;strong&gt;Basic Approach&lt;/strong&gt;&lt;/h1&gt;

&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Outline"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;strong&gt;Outline&lt;/strong&gt;&lt;/h1&gt;

  &lt;ol&gt;&lt;li&gt;The chromatic vector for an MOS should assume that the MOS is the &lt;a class="wiki_link" href="/Chromatic%20Pairs"&gt;albitonic&lt;/a&gt; scale of a &lt;a class="wiki_link" href="/chromatic%20pairs"&gt;chromatic pair&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;The chromatic vector doesn't have to be defined in terms of ratios, mappings, or &lt;a class="wiki_link" href="/Fokker%20Blocks"&gt;periodicity blocks&lt;/a&gt;. &lt;strong&gt;In general, the chromatic vector c = L-s&lt;/strong&gt;, regardless of what mapping you use and regardless of whether or not the scale is proper or improper.&lt;/li&gt;&lt;li&gt;To apply this systematically to an MOS, we need to define a which mode of the MOS we're making alterations on.&lt;/li&gt;&lt;li&gt;Sharpening one of the notes in an MOS by this vector can be denoted by an accidental. To keep with tradition, we will use the # sign and the scale degree of the base mode that is being altered, where the first note is scale degree 1. Flattening one of the notes can be denoted by another accidental, in this case the b sign.&lt;/li&gt;&lt;li&gt;So, to demonstrate over the LLsLLLs mode of 5L2s (Ionian)&lt;ol&gt;&lt;li&gt;The melodic minor NMOS parent scale is reached by Ionian b3 or Ionian #4.&lt;/li&gt;&lt;li&gt;The harmonic minor NMOS parent scale is reached by Ionian #4 or Ionian b3, b6.&lt;/li&gt;&lt;li&gt;The harmonic major NMOS parent scale is reached by Ionian b6 or Ionian b3&lt;/li&gt;&lt;li&gt;The locrian major NMOS parent scale is reached by Ionian b2,b3 or Ionian #1,#2.&lt;ol&gt;&lt;li&gt;Other NMOS's exist, but they may be wildly improper; above I stick only to the proper NMOS's that exist in 12-tet.&lt;/li&gt;&lt;li&gt;One can also arrive at these same NMOS's by making different alterations.&lt;/li&gt;&lt;li&gt;There are also NMOS's that can be arrived at by three alterations; I chose to not explore that far.&lt;/li&gt;&lt;li&gt;Sometimes a chromatic alteration simply gives you another mode of the same scale.&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;li&gt;The same procedure can be applied to porcupine. In 22-tet, c = L-s = 3\22 - 2\22 = 1\22. So the chromatic vector here is about 55 cents.&lt;/li&gt;&lt;li&gt;Let's say we're performing manipulations on the Lssssss mode (&amp;quot;porcupine major&amp;quot;). In 22-tet, this is 4 3 3 3 3 3 3. Some interesting near-MOS's are&lt;ol&gt;&lt;li&gt;P-major b3 or P-major #7 - 4 2 4 3 3 3 3 - this is a P-major scale where the 5/4 has been replaced by 6/5; for a different mode replace the 11/6 with 15/8.&lt;/li&gt;&lt;li&gt;P-major b4 - 4 3 2 4 3 3 3 - this is a P-major scale where the 11/8 has been replaced by 4/3; this gives it more of a &amp;quot;fractured&amp;quot; and less of a &amp;quot;wind chimes&amp;quot;y sound.&lt;/li&gt;&lt;li&gt;P-major b5 - 4 3 3 2 4 3 3 - this is a P-major scale where the 3/2 has been replaced by an approximate 16/11; this ~650 cent interval can function in certain circumstances as a very flat &amp;quot;false fifth&amp;quot;&lt;/li&gt;&lt;li&gt;P-major b6 - 4 3 3 3 2 4 3 - this is a P-major scale where the 5/3 has been flattened to 8/5. Very gothic sound.&lt;/li&gt;&lt;li&gt;P-major b7 - 4 3 3 3 3 2 4 - this is a P-major scale where the 11/6 has been flattened to an approximate 7/4. Very &amp;quot;otonal&amp;quot; sounding, as an 8:9:10:11:12:14 hexad exists in this scale.&lt;/li&gt;&lt;li&gt;P-major #3 - 4 4 2 3 3 3 3 - this is a P-major scale where the 5/4 has been sharpened to a 9/7. Very &amp;quot;bright and brassy&amp;quot; sounding.&lt;/li&gt;&lt;li&gt;There are many more.&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;li&gt;If the chromatic interval is a generalized version of the &amp;quot;sharp&amp;quot; accidental, then generalized versions of the &amp;quot;half-sharp&amp;quot; accidental also exist.&lt;ol&gt;&lt;li&gt;If you go from the albitonic scale up to the chromatic scale, a chromatic vector c is implied. If you go up one more level to the hyperchromatic MOS, the large step in the chromatic MOS is split into two new intervals. If the albitonic scale was strictly proper, then its s &amp;gt; c, so s is what gets split. Otherwise, c is what gets split. If the scale is proper, no further shades of chromaticism exist.&lt;/li&gt;&lt;li&gt;Regardless of which gets split, the size of the new interval, which we will denote c2, is |c-s|.&lt;/li&gt;&lt;li&gt;Depending on the propriety of the scale you're working with, c2 may or may not be smaller than c, so the &amp;quot;half-sharp&amp;quot; moniker may not always be appropriate.&lt;/li&gt;&lt;li&gt;For meantone, in 31-tet, this interval is the diesis, which I will notate by &amp;quot;^&amp;quot; and &amp;quot;v&amp;quot; for upward and downward alteration, respectively. This leads to such near-near-MOS's as&lt;ol&gt;&lt;li&gt;C D Ev F G A B C - Ionian with a neutral third&lt;/li&gt;&lt;li&gt;C D Ebv F G A B C - In 31-tet, Ebv maps to 7/6, so this may well be thought of as a septimal Dorian scale&lt;/li&gt;&lt;li&gt;C D E F^ G A B C - Ionian with 4/3 replaced with 11/8&lt;/li&gt;&lt;li&gt;C D E F^ G A Bbv C - This is Ionian with 4/3 replaced with 11/8 and 9/5 replaced with 7/4&lt;/li&gt;&lt;li&gt;C D E F^ G Av Bbv C - This is Ionian with 4/3 replaced with 11/8, 9/5 replaced with 7/4, and 5/3 replaced with ~13/8.&lt;/li&gt;&lt;li&gt;As you can see, the more alterations we make, the less this scale starts to resemble the actual meantone MOS that it originated from.&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;li&gt;One can theoretically alter a scale as many times as one wants.&lt;ol&gt;&lt;li&gt;However, it is suggested by Rothenberg that the near-MOS's that will be most useful are those that are proper. The question of how to deal with near-MOS's that are derived from scales which are themselves improper, as in superpyth[7], is left up to future research.&lt;/li&gt;&lt;li&gt;It is also suggested, that, as a problem of managing the complexity of the sheer number of these resulting scales, that if more than two alterations are made, the resultant scale may best be viewed as a new scale in its own right and not a near-MOS of the original scale.&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;/ol&gt;&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Outline for General Algorithm"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Outline for General Algorithm&lt;/h1&gt;