Andrew Heathwaite's MOS Investigations - Revision history

BudjarnLambeth: Add links

2024-12-24T07:05:59Z

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BudjarnLambeth: Moved from MOS scales to MOS scale

2023-04-28T22:13:04Z

Moved from MOS scales to MOS scale

← Older revision		Revision as of 22:13, 28 April 2023
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	<span style="">So I posted to XA: "Ok, thinking it over, A and d are interesting choices to describe the additional steps, since they're so general. "d" is the small step minus one chroma, or s-c. But since the chroma itself is L-s, we can define d directly in terms of L a</span><span style="">nd s as 2s-L. Meanwhile, "A" is the large step plus one chroma, or L+c. Described in terms of L and s, "A" is 2L-s. This allows us to quickly compute c, A, and d quickly, given L and s. As one example, Orwell[9] in 53edo has L=7 and s=5. So c=L-s=7-5=2; d=2s-L=2(5)-7=3; and A=2L-s=2(7)-5=9. Sure enough, a MODMOS of Orwell[9] with all four of these steps can be easily generated. Start with 7 5 7 5 7 5 7 5 5 and shift the second tone up by one chroma (which we computed to be 2\53), producing 9 3 7 5 7 5 7 5 5, which generalizes to AdLsLsLss! (Note that in 53edo Orwell[9] 2s=10 and A=9, not equal; while in 22edo's version of Orwell[9] 2s=4 and A=4, a potential 22edo Orwell pun.)</span><span style=""> ... ‎(Oh, and another pun is possible here, since c and d are both 1 degree in 22edo but 2 and 3 degrees, respectively, in 53edo.)"</span>		<span style="">So I posted to XA: "Ok, thinking it over, A and d are interesting choices to describe the additional steps, since they're so general. "d" is the small step minus one chroma, or s-c. But since the chroma itself is L-s, we can define d directly in terms of L a</span><span style="">nd s as 2s-L. Meanwhile, "A" is the large step plus one chroma, or L+c. Described in terms of L and s, "A" is 2L-s. This allows us to quickly compute c, A, and d quickly, given L and s. As one example, Orwell[9] in 53edo has L=7 and s=5. So c=L-s=7-5=2; d=2s-L=2(5)-7=3; and A=2L-s=2(7)-5=9. Sure enough, a MODMOS of Orwell[9] with all four of these steps can be easily generated. Start with 7 5 7 5 7 5 7 5 5 and shift the second tone up by one chroma (which we computed to be 2\53), producing 9 3 7 5 7 5 7 5 5, which generalizes to AdLsLsLss! (Note that in 53edo Orwell[9] 2s=10 and A=9, not equal; while in 22edo's version of Orwell[9] 2s=4 and A=4, a potential 22edo Orwell pun.)</span><span style=""> ... ‎(Oh, and another pun is possible here, since c and d are both 1 degree in 22edo but 2 and 3 degrees, respectively, in 53edo.)"</span>

	[[Category:MOS ~~scales~~]]		[[Category:MOS scale]]

Fredg999 category edits: Categories

2022-08-15T04:49:14Z

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

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Wikispaces>igliashon: Imported revision 390971520 - Original comment:

2012-12-10T14:33:53Z

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Wikispaces>Andrew_Heathwaite: Imported revision 288263754 - Original comment:

2011-12-23T03:47:15Z

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Wikispaces>Andrew_Heathwaite: Imported revision 287859632 - Original comment:

2011-12-21T04:24:28Z

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Wikispaces>Andrew_Heathwaite: Imported revision 287855032 - Original comment:

2011-12-21T03:40:23Z

**Imported revision 287855032 - Original comment: **

← Older revision		Revision as of 03:40, 21 December 2011
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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>2011-12-21 01:43:32 UTC</tt>.<br>		: This revision was by author [[User:Andrew_Heathwaite\|Andrew_Heathwaite]] and made on <tt>2011-12-21 03:40:23 UTC</tt>.<br>
	: The original revision id was <tt>~~287845824~~</tt>.<br>		: The original revision id was <tt>287855032</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>
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	This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).		This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).

	[[image:map_of_sensi[8].png]]		[[image:map_of_sensi[8].png]][[image:map_of_sensi[11]_correction.png]]

	==An Approach to Chromatic/Enharmonic MODMOS Scales==		==An Approach to Chromatic/Enharmonic MODMOS Scales==
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	This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).<br />		This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).<br />
	<br />		<br />
	<!-- ws:start:WikiTextLocalImageRule:878:&lt;img src=&quot;/file/view/map_of_sensi%5B8%5D.png/287845258/map_of_sensi%5B8%5D.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/map_of_sensi%5B8%5D.png/287845258/map_of_sensi%5B8%5D.png" alt="map_of_sensi[8].png" title="map_of_sensi[8].png" /><!-- ws:end:WikiTextLocalImageRule:878 --><br />		<!-- ws:start:WikiTextLocalImageRule:878:&lt;img src=&quot;/file/view/map_of_sensi%5B8%5D.png/287845258/map_of_sensi%5B8%5D.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/map_of_sensi%5B8%5D.png/287845258/map_of_sensi%5B8%5D.png" alt="map_of_sensi[8].png" title="map_of_sensi[8].png" /><!-- ws:end:WikiTextLocalImageRule:878 --><!-- ws:start:WikiTextLocalImageRule:879:&lt;img src=&quot;/file/view/map_of_sensi%5B11%5D_correction.png/287855018/map_of_sensi%5B11%5D_correction.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/map_of_sensi%5B11%5D_correction.png/287855018/map_of_sensi%5B11%5D_correction.png" alt="map_of_sensi[11]_correction.png" title="map_of_sensi[11]_correction.png" /><!-- ws:end:WikiTextLocalImageRule:879 --><br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-An Approach to Chromatic/Enharmonic MODMOS Scales"></a><!-- ws:end:WikiTextHeadingRule:2 -->An Approach to Chromatic/Enharmonic MODMOS Scales</h2>		<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-An Approach to Chromatic/Enharmonic MODMOS Scales"></a><!-- ws:end:WikiTextHeadingRule:2 -->An Approach to Chromatic/Enharmonic MODMOS Scales</h2>
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	I'm wanting to do a study on the MOS generator spectrum with diagrams. I made two sample diagrams using 31\137edo and 32\137edo. Here they are right next to each other so I can compare and contrast.<br />		I'm wanting to do a study on the MOS generator spectrum with diagrams. I made two sample diagrams using 31\137edo and 32\137edo. Here they are right next to each other so I can compare and contrast.<br />
	<br />		<br />
	<!-- ws:start:WikiTextLocalImageRule:~~879~~:&lt;img src=&quot;/file/view/137edo_MOS_031_demo_correction.png/285785730/137edo_MOS_031_demo_correction.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/137edo_MOS_031_demo_correction.png/285785730/137edo_MOS_031_demo_correction.png" alt="137edo_MOS_031_demo_correction.png" title="137edo_MOS_031_demo_correction.png" /><!-- ws:end:WikiTextLocalImageRule:~~879~~ --><br />		<!-- ws:start:WikiTextLocalImageRule:880:&lt;img src=&quot;/file/view/137edo_MOS_031_demo_correction.png/285785730/137edo_MOS_031_demo_correction.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/137edo_MOS_031_demo_correction.png/285785730/137edo_MOS_031_demo_correction.png" alt="137edo_MOS_031_demo_correction.png" title="137edo_MOS_031_demo_correction.png" /><!-- ws:end:WikiTextLocalImageRule:880 --><br />
	<!-- ws:start:WikiTextLocalImageRule:~~880~~:&lt;img src=&quot;/file/view/137edo_MOS_032_demo.png/285785372/137edo_MOS_032_demo.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/137edo_MOS_032_demo.png/285785372/137edo_MOS_032_demo.png" alt="137edo_MOS_032_demo.png" title="137edo_MOS_032_demo.png" /><!-- ws:end:WikiTextLocalImageRule:~~880~~ --><br />		<!-- ws:start:WikiTextLocalImageRule:881:&lt;img src=&quot;/file/view/137edo_MOS_032_demo.png/285785372/137edo_MOS_032_demo.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/137edo_MOS_032_demo.png/285785372/137edo_MOS_032_demo.png" alt="137edo_MOS_032_demo.png" title="137edo_MOS_032_demo.png" /><!-- ws:end:WikiTextLocalImageRule:881 --><br />
	<br />		<br />
	Update: I decided to go with <a class="wiki_link" href="/127edo">127edo</a> and have completed the visual study. See <a class="wiki_link" href="/MOS%20Scales%20of%20127edo">MOS Scales of 127edo</a>.<br />		Update: I decided to go with <a class="wiki_link" href="/127edo">127edo</a> and have completed the visual study. See <a class="wiki_link" href="/MOS%20Scales%20of%20127edo">MOS Scales of 127edo</a>.<br />
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	Mike Battaglia has brought up this idea of Porcupine Chromaticism and given <a class="wiki_link" href="/MODMOS%20Scales">MODMOS Scales</a> of Porcupine as specific examples. So to start that exploration, I've made a diagram of all the MOS scales that Porcupine makes possible, starting at Porcupine[7], and terminating at <a class="wiki_link" href="/140edo">140edo</a>, which is arguably an optimal tuning for Porcupine. Take a look:<br />		Mike Battaglia has brought up this idea of Porcupine Chromaticism and given <a class="wiki_link" href="/MODMOS%20Scales">MODMOS Scales</a> of Porcupine as specific examples. So to start that exploration, I've made a diagram of all the MOS scales that Porcupine makes possible, starting at Porcupine[7], and terminating at <a class="wiki_link" href="/140edo">140edo</a>, which is arguably an optimal tuning for Porcupine. Take a look:<br />
	<br />		<br />
	<!-- ws:start:WikiTextLocalImageRule:~~881~~:&lt;img src=&quot;/file/view/porcupine_mos_overview_140edo.jpg/271210382/porcupine_mos_overview_140edo.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/porcupine_mos_overview_140edo.jpg/271210382/porcupine_mos_overview_140edo.jpg" alt="porcupine_mos_overview_140edo.jpg" title="porcupine_mos_overview_140edo.jpg" /><!-- ws:end:WikiTextLocalImageRule:~~881~~ --><br />		<!-- ws:start:WikiTextLocalImageRule:882:&lt;img src=&quot;/file/view/porcupine_mos_overview_140edo.jpg/271210382/porcupine_mos_overview_140edo.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/porcupine_mos_overview_140edo.jpg/271210382/porcupine_mos_overview_140edo.jpg" alt="porcupine_mos_overview_140edo.jpg" title="porcupine_mos_overview_140edo.jpg" /><!-- ws:end:WikiTextLocalImageRule:882 --><br />
	<br />		<br />
	On the XA Facebook page, Paul Erlich showed me some horograms in which the two intervals I call Q and q (for greater and lesser quartertone) switch places, leading me to conclude that <em>there is no standard form for Porcupine[22]</em>. This means that, after a certain point, we have to <em>pick a tuning</em> (pick a side of 22edo for the generator to land on) if we want to explore Porcupine chromaticism that deeply into it, i.e. that far down the generator chain.<br />		On the XA Facebook page, Paul Erlich showed me some horograms in which the two intervals I call Q and q (for greater and lesser quartertone) switch places, leading me to conclude that <em>there is no standard form for Porcupine[22]</em>. This means that, after a certain point, we have to <em>pick a tuning</em> (pick a side of 22edo for the generator to land on) if we want to explore Porcupine chromaticism that deeply into it, i.e. that far down the generator chain.<br />

Wikispaces>Andrew_Heathwaite: Imported revision 287845824 - Original comment:

2011-12-21T01:43:32Z

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Wikispaces>Andrew_Heathwaite: Imported revision 287584396 - Original comment:

2011-12-20T04:30:35Z

**Imported revision 287584396 - Original comment: **

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← Older revision		Revision as of 04:49, 15 August 2022
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	<span style="">So I posted to XA: "Ok, thinking it over, A and d are interesting choices to describe the additional steps, since they're so general. "d" is the small step minus one chroma, or s-c. But since the chroma itself is L-s, we can define d directly in terms of L a</span><span style="">nd s as 2s-L. Meanwhile, "A" is the large step plus one chroma, or L+c. Described in terms of L and s, "A" is 2L-s. This allows us to quickly compute c, A, and d quickly, given L and s. As one example, Orwell[9] in 53edo has L=7 and s=5. So c=L-s=7-5=2; d=2s-L=2(5)-7=3; and A=2L-s=2(7)-5=9. Sure enough, a MODMOS of Orwell[9] with all four of these steps can be easily generated. Start with 7 5 7 5 7 5 7 5 5 and shift the second tone up by one chroma (which we computed to be 2\53), producing 9 3 7 5 7 5 7 5 5, which generalizes to AdLsLsLss! (Note that in 53edo Orwell[9] 2s=10 and A=9, not equal; while in 22edo's version of Orwell[9] 2s=4 and A=4, a potential 22edo Orwell pun.)</span><span style=""> ... ‎(Oh, and another pun is possible here, since c and d are both 1 degree in 22edo but 2 and 3 degrees, respectively, in 53edo.)"</span>		<span style="">So I posted to XA: "Ok, thinking it over, A and d are interesting choices to describe the additional steps, since they're so general. "d" is the small step minus one chroma, or s-c. But since the chroma itself is L-s, we can define d directly in terms of L a</span><span style="">nd s as 2s-L. Meanwhile, "A" is the large step plus one chroma, or L+c. Described in terms of L and s, "A" is 2L-s. This allows us to quickly compute c, A, and d quickly, given L and s. As one example, Orwell[9] in 53edo has L=7 and s=5. So c=L-s=7-5=2; d=2s-L=2(5)-7=3; and A=2L-s=2(7)-5=9. Sure enough, a MODMOS of Orwell[9] with all four of these steps can be easily generated. Start with 7 5 7 5 7 5 7 5 5 and shift the second tone up by one chroma (which we computed to be 2\53), producing 9 3 7 5 7 5 7 5 5, which generalizes to AdLsLsLss! (Note that in 53edo Orwell[9] 2s=10 and A=9, not equal; while in 22edo's version of Orwell[9] 2s=4 and A=4, a potential 22edo Orwell pun.)</span><span style=""> ... ‎(Oh, and another pun is possible here, since c and d are both 1 degree in 22edo but 2 and 3 degrees, respectively, in 53edo.)"</span>

			[[Category:MOS scales]]

← Older revision		Revision as of 14:33, 10 December 2012
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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>~~2011~~-12-~~23 03~~:47:15 UTC</tt>.<br>		: This revision was by author [[User:igliashon\|igliashon]] and made on <tt>2012-12-10 14:33:53 UTC</tt>.<br>
	: The original revision id was <tt>~~288263754~~</tt>.<br>		: The original revision id was <tt>390971520</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>
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	* MB writes about Porcupine's [[MODMOS Scales\|MODMOS]] scales (which I will deal with more below), summarizing, "<span class="commentBody">In short, when you're playing in porcupine, you should never feel like you're limited to just the 7 or 8-note MOS. Just freeform modify notes by L-s as much as you want, deliberately, in a willful attempt to explore porcupine chromaticism. It's even easier than meantone chromaticism.</span>"		* MB writes about Porcupine's [[MODMOS Scales\|MODMOS]] scales (which I will deal with more below), summarizing, "<span class="commentBody">In short, when you're playing in porcupine, you should never feel like you're limited to just the 7 or 8-note MOS. Just freeform modify notes by L-s as much as you want, deliberately, in a willful attempt to explore porcupine chromaticism. It's even easier than meantone chromaticism.</span>"
	* MB: "I<span class="commentBody">n porcupine, bIII/bIII/bIII = IV/IV. This is the same thing as saying that 6/5 * 6/5 * 6/5 = 4/3 * 4/3</span>."		* MB: "I<span class="commentBody">n porcupine, bIII/bIII/bIII = IV/IV. This is the same thing as saying that 6/5 * 6/5 * 6/5 = 4/3 * 4/3</span>."
	* ~~Igliashon Jones~~ argues that Porcupine doesn't do that great in the 5-limit after all, saying, "<span class="commentBody">Its only real selling-point over optimal meantone is simpler 7-limit and 11-limit approximations, but that assumes that these are a good in their own right and thus worth sacrificing some 5-limit efficiency; for anyone other than a dyed-in-the-wool xenharmonist, that's a questionable assumption to make.</span>" (As for me, I want those 7- and 11-limit approximations, and I could care less about a 5-limit temperament to rival meantone. I don't compose in 5-limit temperaments, period.)		* Someone argues that Porcupine doesn't do that great in the 5-limit after all, saying, "<span class="commentBody">Its only real selling-point over optimal meantone is simpler 7-limit and 11-limit approximations, but that assumes that these are a good in their own right and thus worth sacrificing some 5-limit efficiency; for anyone other than a dyed-in-the-wool xenharmonist, that's a questionable assumption to make.</span>" (As for me, I want those 7- and 11-limit approximations, and I could care less about a 5-limit temperament to rival meantone. I don't compose in 5-limit temperaments, period.)
	* In response to the above, Keenan Pepper says, "<span class="commentBody">You mentioned that almost every interval in the diatonic scale is a 9-limit consonance? Well, every interval in porcupine[7] is an 11-limit consonance! 1/1 10/9 9/8 6/5 5/4 4/3 11/8 16/11 3/2 8/5 5/3 16/9 9/5 2/1. Bam!</span>" (This is relevant to my work, which assumes composers want 11-limit approximations.)		* In response to the above, Keenan Pepper says, "<span class="commentBody">You mentioned that almost every interval in the diatonic scale is a 9-limit consonance? Well, every interval in porcupine[7] is an 11-limit consonance! 1/1 10/9 9/8 6/5 5/4 4/3 11/8 16/11 3/2 8/5 5/3 16/9 9/5 2/1. Bam!</span>" (This is relevant to my work, which assumes composers want 11-limit approximations.)
	* I (Andrew Heathwaite) added, "<span class="commentBody">...maybe another description for what Porcupine is good for is a gateway from 5 and 7 to 11, for those comfortable with the former and curious about the latter. As a full 11-limit temperament, it is efficient and easy.</span>"		* I (Andrew Heathwaite) added, "<span class="commentBody">...maybe another description for what Porcupine is good for is a gateway from 5 and 7 to 11, for those comfortable with the former and curious about the latter. As a full 11-limit temperament, it is efficient and easy.</span>"
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	<br />		<br />
	I'm going to zoom in on <a class="wiki_link" href="/Porcupine">Porcupine Temperament</a>, which has been mentioned on the Facebook Xenharmonic Alliance page recently as a xenharmonic alternative to Meantone. Here's a little list of some of the things that were mentioned, so they can be collected in one place and not lost forever in the impenetrable Facebook Caverns:<br />		I'm going to zoom in on <a class="wiki_link" href="/Porcupine">Porcupine Temperament</a>, which has been mentioned on the Facebook Xenharmonic Alliance page recently as a xenharmonic alternative to Meantone. Here's a little list of some of the things that were mentioned, so they can be collected in one place and not lost forever in the impenetrable Facebook Caverns:<br />
	<ul><li>Keenan Pepper writes about how Porcupine tempers 27/20, 15/11 and 25/18 all to the 11/8 approximation, which, he claims, is a stronger consonance than any of the intervals mentioned.</li><li>Mike Battaglia writes about how 81/80 is &quot;tempered in&quot; to 25/24, making it melodically useful instead of an &quot;irritating mystery interval&quot; which &quot;introduces pitch drift&quot;.</li><li>MB writes about Porcupine's <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scales (which I will deal with more below), summarizing, &quot;<span class="commentBody">In short, when you're playing in porcupine, you should never feel like you're limited to just the 7 or 8-note MOS. Just freeform modify notes by L-s as much as you want, deliberately, in a willful attempt to explore porcupine chromaticism. It's even easier than meantone chromaticism.</span>&quot;</li><li>MB: &quot;I<span class="commentBody">n porcupine, bIII/bIII/bIII = IV/IV. This is the same thing as saying that 6/5 * 6/5 * 6/5 = 4/3 * 4/3</span>.&quot;</li><li>~~Igliashon Jones~~ argues that Porcupine doesn't do that great in the 5-limit after all, saying, &quot;<span class="commentBody">Its only real selling-point over optimal meantone is simpler 7-limit and 11-limit approximations, but that assumes that these are a good in their own right and thus worth sacrificing some 5-limit efficiency; for anyone other than a dyed-in-the-wool xenharmonist, that's a questionable assumption to make.</span>&quot; (As for me, I want those 7- and 11-limit approximations, and I could care less about a 5-limit temperament to rival meantone. I don't compose in 5-limit temperaments, period.)</li><li>In response to the above, Keenan Pepper says, &quot;<span class="commentBody">You mentioned that almost every interval in the diatonic scale is a 9-limit consonance? Well, every interval in porcupine[7] is an 11-limit consonance! 1/1 10/9 9/8 6/5 5/4 4/3 11/8 16/11 3/2 8/5 5/3 16/9 9/5 2/1. Bam!</span>&quot; (This is relevant to my work, which assumes composers want 11-limit approximations.)</li><li>I (Andrew Heathwaite) added, &quot;<span class="commentBody">...maybe another description for what Porcupine is good for is a gateway from 5 and 7 to 11, for those comfortable with the former and curious about the latter. As a full 11-limit temperament, it is efficient and easy.</span>&quot;</li></ul><br />		<ul><li>Keenan Pepper writes about how Porcupine tempers 27/20, 15/11 and 25/18 all to the 11/8 approximation, which, he claims, is a stronger consonance than any of the intervals mentioned.</li><li>Mike Battaglia writes about how 81/80 is &quot;tempered in&quot; to 25/24, making it melodically useful instead of an &quot;irritating mystery interval&quot; which &quot;introduces pitch drift&quot;.</li><li>MB writes about Porcupine's <a class="wiki_link" href="/MODMOS%20Scales">MODMOS</a> scales (which I will deal with more below), summarizing, &quot;<span class="commentBody">In short, when you're playing in porcupine, you should never feel like you're limited to just the 7 or 8-note MOS. Just freeform modify notes by L-s as much as you want, deliberately, in a willful attempt to explore porcupine chromaticism. It's even easier than meantone chromaticism.</span>&quot;</li><li>MB: &quot;I<span class="commentBody">n porcupine, bIII/bIII/bIII = IV/IV. This is the same thing as saying that 6/5 * 6/5 * 6/5 = 4/3 * 4/3</span>.&quot;</li><li>Someone argues that Porcupine doesn't do that great in the 5-limit after all, saying, &quot;<span class="commentBody">Its only real selling-point over optimal meantone is simpler 7-limit and 11-limit approximations, but that assumes that these are a good in their own right and thus worth sacrificing some 5-limit efficiency; for anyone other than a dyed-in-the-wool xenharmonist, that's a questionable assumption to make.</span>&quot; (As for me, I want those 7- and 11-limit approximations, and I could care less about a 5-limit temperament to rival meantone. I don't compose in 5-limit temperaments, period.)</li><li>In response to the above, Keenan Pepper says, &quot;<span class="commentBody">You mentioned that almost every interval in the diatonic scale is a 9-limit consonance? Well, every interval in porcupine[7] is an 11-limit consonance! 1/1 10/9 9/8 6/5 5/4 4/3 11/8 16/11 3/2 8/5 5/3 16/9 9/5 2/1. Bam!</span>&quot; (This is relevant to my work, which assumes composers want 11-limit approximations.)</li><li>I (Andrew Heathwaite) added, &quot;<span class="commentBody">...maybe another description for what Porcupine is good for is a gateway from 5 and 7 to 11, for those comfortable with the former and curious about the latter. As a full 11-limit temperament, it is efficient and easy.</span>&quot;</li></ul><br />
	<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Porcupine Chromaticism"></a><!-- ws:end:WikiTextHeadingRule:14 -->Porcupine Chromaticism</h1>		<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Porcupine Chromaticism"></a><!-- ws:end:WikiTextHeadingRule:14 -->Porcupine Chromaticism</h1>
	<br />		<br />

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	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>2011-12-21 03:40:23 UTC</tt>.<br>		: This revision was by author [[User:Andrew_Heathwaite\|Andrew_Heathwaite]] and made on <tt>2011-12-21 04:24:28 UTC</tt>.<br>
	: The original revision id was <tt>~~287855032~~</tt>.<br>		: The original revision id was <tt>287859632</tt>.<br>
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	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>
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	This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).		This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).

	[[image:map_of_sensi[8].png]][[image:map_of_sensi[11]~~_correction~~.png]]		[[image:map_of_sensi[8].png]][[image:map_of_sensi[11]_correction2.png]]

	==An Approach to Chromatic/Enharmonic MODMOS Scales==		==An Approach to Chromatic/Enharmonic MODMOS Scales==
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	This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).<br />		This diagram is an experiment in combining the two ways I tend to visualize MOS scales -- as a chain of generators (x-axis) and as particular steps in pitch-space (y-axis).<br />
	<br />		<br />
	<!-- ws:start:WikiTextLocalImageRule:878:&lt;img src=&quot;/file/view/map_of_sensi%5B8%5D.png/287845258/map_of_sensi%5B8%5D.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/map_of_sensi%5B8%5D.png/287845258/map_of_sensi%5B8%5D.png" alt="map_of_sensi[8].png" title="map_of_sensi[8].png" /><!-- ws:end:WikiTextLocalImageRule:878 --><!-- ws:start:WikiTextLocalImageRule:879:&lt;img src=&quot;/file/view/map_of_sensi%5B11%~~5D_correction~~.png/~~287855018~~/map_of_sensi%5B11%~~5D_correction~~.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/map_of_sensi%5B11%~~5D_correction~~.png/~~287855018~~/map_of_sensi%5B11%~~5D_correction~~.png" alt="map_of_sensi[11]~~_correction~~.png" title="map_of_sensi[11]~~_correction~~.png" /><!-- ws:end:WikiTextLocalImageRule:879 --><br />		<!-- ws:start:WikiTextLocalImageRule:878:&lt;img src=&quot;/file/view/map_of_sensi%5B8%5D.png/287845258/map_of_sensi%5B8%5D.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/map_of_sensi%5B8%5D.png/287845258/map_of_sensi%5B8%5D.png" alt="map_of_sensi[8].png" title="map_of_sensi[8].png" /><!-- ws:end:WikiTextLocalImageRule:878 --><!-- ws:start:WikiTextLocalImageRule:879:&lt;img src=&quot;/file/view/map_of_sensi%5B11%5D_correction2.png/287859614/map_of_sensi%5B11%5D_correction2.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/map_of_sensi%5B11%5D_correction2.png/287859614/map_of_sensi%5B11%5D_correction2.png" alt="map_of_sensi[11]_correction2.png" title="map_of_sensi[11]_correction2.png" /><!-- ws:end:WikiTextLocalImageRule:879 --><br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-An Approach to Chromatic/Enharmonic MODMOS Scales"></a><!-- ws:end:WikiTextHeadingRule:2 -->An Approach to Chromatic/Enharmonic MODMOS Scales</h2>		<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-An Approach to Chromatic/Enharmonic MODMOS Scales"></a><!-- ws:end:WikiTextHeadingRule:2 -->An Approach to Chromatic/Enharmonic MODMOS Scales</h2>

Andrew Heathwaite's MOS Investigations - Revision history

BudjarnLambeth: Add links

BudjarnLambeth: Moved from MOS scales to MOS scale

Fredg999 category edits: Categories

Wikispaces>FREEZE at 00:00, 17 July 2018

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