Andrew Heathwaite's MOS Investigations: Difference between revisions

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: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-10-~~31 19~~:46:20 UTC</tt>.<br>

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-11-01 11:22:56 UTC</tt>.<br>

: The original revision id was <tt>~~270457072~~</tt>.<br>

: The original revision id was <tt>270653882</tt>.<br>

: The revision comment was: <tt></tt><br>

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**3 2 5 3 3 3 3 .. -4 -3 -2 -1 0 1 _ _ _ _ _ _ _ 9**

~~Did~~ I ~~miss any~~???</pre></div>

Update: Mike Battaglia has made a dedicated page for explaining these modes -- yay! -- see [[Porcupine Temperament Modal Harmony]].

=Orwell[9], meet Porcupine[7]=

I've done a little composing in Orwell[9], which, in 22edo, goes 3 2 3 2 3 2 3 2 2 (where L=3\22 and s=2\22), so I want to apply MODMOS to that. To make a MODMOS here, we alter a tone by a single degree of 22edo, same as we do in Porcupine[7]. This is our "chroma," and it's generated by taking L-s: so in 22edo we have 3\22-2\22=1\22. We wind up with either:

# A permutation of the four large and five small steps, eg. 3 3 2 2 3 2 3 2 2

## How many of these are there? Does anyone know a formula for finding the number of possible permutations when some of the items are interchangable? Here the question is, how many permutations can we make of 4 items of Type A and 5 items of Type B?

# A scale with three step sizes: large, small, and smaller, eg. 3 2 3 2 3 2 3 3 1

## In 22edo, our "smaller" step is the same as our "chroma" (which is the interval that we alter a tone by to produce a MODMOS, L-s). However, this is not the case in larger edos! Look at [[31edo]], where our initial scale is 4 3 4 3 4 3 4 3 3. Now our chroma is 4\31-3\31=1\31 and our "smaller" step is 2\31: 4 3 4 3 4 3 4 4 2! We get our "smaller" step by starting with s (3\31) and taking away a chroma (1\31), so we have 3\31-1\31=2\31.

## So what should we call the "smaller" interval in our scale? Maybe some kind of diminished something-or-another?

# A scale with four steps sizes: large, small, larger and smaller, eg. 4 1 3 2 3 2 3 2 2

## This is generated by starting with L and adding a chroma, so in 22edo it's 3\22+1\22=4\22. In 31edo, that would be 4\31+1\31=5\31, and the scale in question would be 5 2 4 3 4 3 4 3 3.

## So what should we call the "larger" step? Some kind of augmented something-or-another?

## Note that in 22edo, our "larger" step, 4\22, is the same as two of our small steps (2\22+2\22=4\22), even though we generated our "larger" step by adding a chroma to a large step (3\22+1\22). In 31edo, our "larger" step is NOT the same as two of our small steps (4\31+1\31=5\31 does not equal 3\31+3\31=6\31)!

So we can take advantage of the fact that two small steps in 22edo's Orwell[9] (2\22) make one "larger" step (4\22). If 9 tones is a few too many, we can turn some 2+2's into 4's. So for instance, the first example above goes:

3 3 2 2 3 2 3 2 2

3 3 4 3 2 3 4.

But check it out! 3 3 4 3 2 3 4 is a MODMOS of Porcupine[7]! Here's how we can get it by chromatically-altering Porcupine[7] one tone at a time:

3 3 3 3 3 3 4

3 3 4 2 3 3 4

3 3 4 3 2 3 4

And we see, not surprisingly, that this doesn't work the same way in 31edo.

Start with a MODMOS of Orwell[9]: 4 4 3 3 4 3 4 3 3

Combine small steps: 4 4 6 4 3 4 6

4 4 4 4 4 4 7 is as close as we can get to Porcupine[7], and it sure ain't the same. Our chroma (L-s) is 3\31, really different!

4 4 7 1 4 4 7

4 4 7 1 4 1 7

Not even close!</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"><html><head><title>Andrew Heathwaite's MOS Investigations</title></head><body>Ok, this is a page for me, Andrew Heathwaite, to organize my thoughts and questions regarding <a class="wiki_link" href="/MOSScales">Moment of Symmetry Scales</a>.<br />

Line 253:

Line 288:

<br />

~~Did~~ I ~~miss any~~???</body></html></pre></div>

Update: Mike Battaglia has made a dedicated page for explaining these modes -- yay! -- see <a class="wiki_link" href="/Porcupine%20Temperament%20Modal%20Harmony">Porcupine Temperament Modal Harmony</a>.<br />

<br />

<h1 id="toc4"><a name="Orwell[9], meet Porcupine[7]"></a>Orwell[9], meet Porcupine[7]</h1>

<br />

I've done a little composing in Orwell[9], which, in 22edo, goes 3 2 3 2 3 2 3 2 2 (where L=3\22 and s=2\22), so I want to apply MODMOS to that. To make a MODMOS here, we alter a tone by a single degree of 22edo, same as we do in Porcupine[7]. This is our &quot;chroma,&quot; and it's generated by taking L-s: so in 22edo we have 3\22-2\22=1\22. We wind up with either:<br />

<ol><li>A permutation of the four large and five small steps, eg. 3 3 2 2 3 2 3 2 2<ol><li>How many of these are there? Does anyone know a formula for finding the number of possible permutations when some of the items are interchangable? Here the question is, how many permutations can we make of 4 items of Type A and 5 items of Type B?</li></ol></li><li>A scale with three step sizes: large, small, and smaller, eg. 3 2 3 2 3 2 3 3 1<ol><li>In 22edo, our &quot;smaller&quot; step is the same as our &quot;chroma&quot; (which is the interval that we alter a tone by to produce a MODMOS, L-s). However, this is not the case in larger edos! Look at <a class="wiki_link" href="/31edo">31edo</a>, where our initial scale is 4 3 4 3 4 3 4 3 3. Now our chroma is 4\31-3\31=1\31 and our &quot;smaller&quot; step is 2\31: 4 3 4 3 4 3 4 4 2! We get our &quot;smaller&quot; step by starting with s (3\31) and taking away a chroma (1\31), so we have 3\31-1\31=2\31.</li><li>So what should we call the &quot;smaller&quot; interval in our scale? Maybe some kind of diminished something-or-another?</li></ol></li><li>A scale with four steps sizes: large, small, larger and smaller, eg. 4 1 3 2 3 2 3 2 2<ol><li>This is generated by starting with L and adding a chroma, so in 22edo it's 3\22+1\22=4\22. In 31edo, that would be 4\31+1\31=5\31, and the scale in question would be 5 2 4 3 4 3 4 3 3.</li><li>So what should we call the &quot;larger&quot; step? Some kind of augmented something-or-another?</li><li>Note that in 22edo, our &quot;larger&quot; step, 4\22, is the same as two of our small steps (2\22+2\22=4\22), even though we generated our &quot;larger&quot; step by adding a chroma to a large step (3\22+1\22). In 31edo, our &quot;larger&quot; step is NOT the same as two of our small steps (4\31+1\31=5\31 does not equal 3\31+3\31=6\31)!</li></ol></li></ol><br />

So we can take advantage of the fact that two small steps in 22edo's Orwell[9] (2\22) make one &quot;larger&quot; step (4\22). If 9 tones is a few too many, we can turn some 2+2's into 4's. So for instance, the first example above goes:<br />

<br />

3 3 2 2 3 2 3 2 2 <br />

3 3 4 3 2 3 4.<br />

<br />

But check it out! 3 3 4 3 2 3 4 is a MODMOS of Porcupine[7]! Here's how we can get it by chromatically-altering Porcupine[7] one tone at a time:<br />

<br />

3 3 3 3 3 3 4<br />

3 3 4 2 3 3 4<br />

3 3 4 3 2 3 4<br />

<br />

And we see, not surprisingly, that this doesn't work the same way in 31edo.<br />

<br />

Start with a MODMOS of Orwell[9]: 4 4 3 3 4 3 4 3 3<br />

Combine small steps: 4 4 6 4 3 4 6<br />

<br />

4 4 4 4 4 4 7 is as close as we can get to Porcupine[7], and it sure ain't the same. Our chroma (L-s) is 3\31, really different!<br />

4 4 7 1 4 4 7<br />

4 4 7 1 4 1 7<br />

<br />

Not even close!</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-10-31 19:46:20 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-11-01 11:22:56 UTC</tt>.<br>
-: The original revision id was <tt>270457072</tt>.<br>
+: The original revision id was <tt>270653882</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>
@@ Line 132: / Line 132: @@
 **3 2 5 3 3 3 3 .. -4 -3 -2 -1 0 1 _ _ _ _ _ _ _ 9**
-Did I miss any???</pre></div>
+Update: Mike Battaglia has made a dedicated page for explaining these modes -- yay! -- see [[Porcupine Temperament Modal Harmony]].
+=Orwell[9], meet Porcupine[7]=
+I've done a little composing in Orwell[9], which, in 22edo, goes 3 2 3 2 3 2 3 2 2 (where L=3\22 and s=2\22), so I want to apply MODMOS to that. To make a MODMOS here, we alter a tone by a single degree of 22edo, same as we do in Porcupine[7]. This is our "chroma," and it's generated by taking L-s: so in 22edo we have 3\22-2\22=1\22. We wind up with either:
+# A permutation of the four large and five small steps, eg. 3 3 2 2 3 2 3 2 2
+## How many of these are there? Does anyone know a formula for finding the number of possible permutations when some of the items are interchangable? Here the question is, how many permutations can we make of 4 items of Type A and 5 items of Type B?
+# A scale with three step sizes: large, small, and smaller, eg. 3 2 3 2 3 2 3 3 1
+## In 22edo, our "smaller" step is the same as our "chroma" (which is the interval that we alter a tone by to produce a MODMOS, L-s). However, this is not the case in larger edos! Look at [[31edo]], where our initial scale is 4 3 4 3 4 3 4 3 3. Now our chroma is 4\31-3\31=1\31 and our "smaller" step is 2\31: 4 3 4 3 4 3 4 4 2! We get our "smaller" step by starting with s (3\31) and taking away a chroma (1\31), so we have 3\31-1\31=2\31.
+## So what should we call the "smaller" interval in our scale? Maybe some kind of diminished something-or-another?
+# A scale with four steps sizes: large, small, larger and smaller, eg. 4 1 3 2 3 2 3 2 2
+## This is generated by starting with L and adding a chroma, so in 22edo it's 3\22+1\22=4\22. In 31edo, that would be 4\31+1\31=5\31, and the scale in question would be 5 2 4 3 4 3 4 3 3.
+## So what should we call the "larger" step? Some kind of augmented something-or-another?
+## Note that in 22edo, our "larger" step, 4\22, is the same as two of our small steps (2\22+2\22=4\22), even though we generated our "larger" step by adding a chroma to a large step (3\22+1\22). In 31edo, our "larger" step is NOT the same as two of our small steps (4\31+1\31=5\31 does not equal 3\31+3\31=6\31)!
+So we can take advantage of the fact that two small steps in 22edo's Orwell[9] (2\22) make one "larger" step (4\22). If 9 tones is a few too many, we can turn some 2+2's into 4's. So for instance, the first example above goes:
+3 2 2 3 2 3 2 2
+3 4 3 2 3 4.
+But check it out! 3 3 4 3 2 3 4 is a MODMOS of Porcupine[7]! Here's how we can get it by chromatically-altering Porcupine[7] one tone at a time:
+3 3 3 3 3 4
+3 4 2 3 3 4
+3 4 3 2 3 4
+And we see, not surprisingly, that this doesn't work the same way in 31edo.
+Start with a MODMOS of Orwell[9]: 4 4 3 3 4 3 4 3 3
+Combine small steps: 4 4 6 4 3 4 6
+4 4 4 4 4 7 is as close as we can get to Porcupine[7], and it sure ain't the same. Our chroma (L-s) is 3\31, really different!
+4 7 1 4 4 7
+4 7 1 4 1 7
+Not even close!</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;Andrew Heathwaite's MOS Investigations&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Ok, this is a page for me, Andrew Heathwaite, to organize my thoughts and questions regarding &lt;a class="wiki_link" href="/MOSScales"&gt;Moment of Symmetry Scales&lt;/a&gt;.&lt;br /&gt;
@@ Line 253: / Line 288: @@
 &lt;strong&gt;3 2 5 3 3 3 3 .. -4 -3 -2 -1 0 1 _ _ _ _ _ _ _ 9&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;
-Did I miss any???&lt;/body&gt;&lt;/html&gt;</pre></div>
+Update: Mike Battaglia has made a dedicated page for explaining these modes -- yay! -- see &lt;a class="wiki_link" href="/Porcupine%20Temperament%20Modal%20Harmony"&gt;Porcupine Temperament Modal Harmony&lt;/a&gt;.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Orwell[9], meet Porcupine[7]"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Orwell[9], meet Porcupine[7]&lt;/h1&gt;
+ &lt;br /&gt;
+I've done a little composing in Orwell[9], which, in 22edo, goes 3 2 3 2 3 2 3 2 2 (where L=3\22 and s=2\22), so I want to apply MODMOS to that. To make a MODMOS here, we alter a tone by a single degree of 22edo, same as we do in Porcupine[7]. This is our &amp;quot;chroma,&amp;quot; and it's generated by taking L-s: so in 22edo we have 3\22-2\22=1\22. We wind up with either:&lt;br /&gt;
+&lt;ol&gt;&lt;li&gt;A permutation of the four large and five small steps, eg. 3 3 2 2 3 2 3 2 2&lt;ol&gt;&lt;li&gt;How many of these are there? Does anyone know a formula for finding the number of possible permutations when some of the items are interchangable? Here the question is, how many permutations can we make of 4 items of Type A and 5 items of Type B?&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;li&gt;A scale with three step sizes: large, small, and smaller, eg. 3 2 3 2 3 2 3 3 1&lt;ol&gt;&lt;li&gt;In 22edo, our &amp;quot;smaller&amp;quot; step is the same as our &amp;quot;chroma&amp;quot; (which is the interval that we alter a tone by to produce a MODMOS, L-s). However, this is not the case in larger edos! Look at &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, where our initial scale is 4 3 4 3 4 3 4 3 3. Now our chroma is 4\31-3\31=1\31 and our &amp;quot;smaller&amp;quot; step is 2\31: 4 3 4 3 4 3 4 4 2! We get our &amp;quot;smaller&amp;quot; step by starting with s (3\31) and taking away a chroma (1\31), so we have 3\31-1\31=2\31.&lt;/li&gt;&lt;li&gt;So what should we call the &amp;quot;smaller&amp;quot; interval in our scale? Maybe some kind of diminished something-or-another?&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;li&gt;A scale with four steps sizes: large, small, larger and smaller, eg. 4 1 3 2 3 2 3 2 2&lt;ol&gt;&lt;li&gt;This is generated by starting with L and adding a chroma, so in 22edo it's 3\22+1\22=4\22. In 31edo, that would be 4\31+1\31=5\31, and the scale in question would be 5 2 4 3 4 3 4 3 3.&lt;/li&gt;&lt;li&gt;So what should we call the &amp;quot;larger&amp;quot; step? Some kind of augmented something-or-another?&lt;/li&gt;&lt;li&gt;Note that in 22edo, our &amp;quot;larger&amp;quot; step, 4\22, is the same as two of our small steps (2\22+2\22=4\22), even though we generated our &amp;quot;larger&amp;quot; step by adding a chroma to a large step (3\22+1\22). In 31edo, our &amp;quot;larger&amp;quot; step is NOT the same as two of our small steps (4\31+1\31=5\31 does not equal 3\31+3\31=6\31)!&lt;/li&gt;&lt;/ol&gt;&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
+So we can take advantage of the fact that two small steps in 22edo's Orwell[9] (2\22) make one &amp;quot;larger&amp;quot; step (4\22). If 9 tones is a few too many, we can turn some 2+2's into 4's. So for instance, the first example above goes:&lt;br /&gt;
+&lt;br /&gt;
+3 2 2 3 2 3 2 2 &lt;br /&gt;
+3 4 3 2 3 4.&lt;br /&gt;
+&lt;br /&gt;
+But check it out! 3 3 4 3 2 3 4 is a MODMOS of Porcupine[7]! Here's how we can get it by chromatically-altering Porcupine[7] one tone at a time:&lt;br /&gt;
+&lt;br /&gt;
+3 3 3 3 3 4&lt;br /&gt;
+3 4 2 3 3 4&lt;br /&gt;
+3 4 3 2 3 4&lt;br /&gt;
+&lt;br /&gt;
+And we see, not surprisingly, that this doesn't work the same way in 31edo.&lt;br /&gt;
+&lt;br /&gt;
+Start with a MODMOS of Orwell[9]: 4 4 3 3 4 3 4 3 3&lt;br /&gt;
+Combine small steps: 4 4 6 4 3 4 6&lt;br /&gt;
+&lt;br /&gt;
+4 4 4 4 4 7 is as close as we can get to Porcupine[7], and it sure ain't the same. Our chroma (L-s) is 3\31, really different!&lt;br /&gt;
+4 7 1 4 4 7&lt;br /&gt;
+4 7 1 4 1 7&lt;br /&gt;
+&lt;br /&gt;
+Not even close!&lt;/body&gt;&lt;/html&gt;</pre></div>