Andrew Heathwaite's MOS Investigations: Difference between revisions

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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">Ok, this is a page for me, Andrew Heathwaite, to organize my thoughts and questions regarding [[MOSScales|Moment of Symmetry Scales]].

Others are more than welcome to correct obvious errors, add clearly-demarked related material, and comment through the comments tab. My approach may be a little different than yours, but hopefully our approaches are compatible and you can tell me what you think. I certainly don't know everything, which is why this is an investigation!

=Porcupine Temperament=  

Others are more than welcome to correct obvious errors, add clearly-demarked related material, and comment through the comments tab. My approach may be a little different than yours, but hopefully our approaches are compatible and you can tell me what you think. I certainly don't know everything, which is why this is an investigation!&lt;br /&gt;

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I'm going to zoom in on &lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine Temperament&lt;/a&gt;, 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:&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;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.&lt;/li&gt;&lt;li&gt;Mike Battaglia writes about how 81/80 is &amp;quot;tempered in&amp;quot; to 25/24, making it melodically useful instead of an &amp;quot;irritating mystery interval&amp;quot; which &amp;quot;introduces pitch drift&amp;quot;.&lt;/li&gt;&lt;li&gt;MB writes about Porcupine's &lt;a class="wiki_link" href="/MODMOS%20Scales"&gt;MODMOS&lt;/a&gt; scales (which I will deal with more below), summarizing, &amp;quot;&lt;span class="commentBody"&gt;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.&lt;/span&gt;&amp;quot;&lt;/li&gt;&lt;li&gt;MB: &amp;quot;I&lt;span class="commentBody"&gt;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&lt;/span&gt;.&amp;quot;&lt;/li&gt;&lt;li&gt;Igliashon Jones argues that Porcupine doesn't do that great in the 5-limit after all, saying, &amp;quot;&lt;span class="commentBody"&gt;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.&lt;/span&gt;&amp;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.)&lt;/li&gt;&lt;li&gt;In response to the above, Keenan Pepper says, &amp;quot;&lt;span class="commentBody"&gt;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!&lt;/span&gt;&amp;quot; (This is relevant to my work, which assumes composers want 11-limit approximations.)&lt;/li&gt;&lt;li&gt;I (Andrew Heathwaite) added, &amp;quot;&lt;span class="commentBody"&gt;...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.&lt;/span&gt;&amp;quot;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

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Mike Battaglia has brought up this idea of Porcupine Chromaticism and given MODMOS Scales 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 &lt;a class="wiki_link" href="/140edo"&gt;140edo&lt;/a&gt;, which is arguably an optimal tuning for Porcupine. Take a look:&lt;br /&gt;

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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 &lt;em&gt;there is no standard form for Porcupine[22]&lt;/em&gt;. This means that, after a certain point, we have to &lt;em&gt;pick a tuning&lt;/em&gt; (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.&lt;br /&gt;

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The following modes are given in steps of 22edo. They are rotations of one moment of symmetry scale with two step sizes: a neutral tone (3\22) and a large whole tone (4\22). On the right is a contiguous chain of 7 tones separated by 6 iterations of the Porcupine generator. Modes in bold have a 3/2 approximation above the bass -- this can be verified easily by looking at the chain. The perfect fifth approximation is -3g, so every mode with a &amp;quot;-3&amp;quot; in the chain has a perfect fifth over the bass.&lt;br /&gt;

&lt;strong&gt;4 3 3 3 3 3 3 .. -6 -5 -4 -3 -2 -1 0&lt;/strong&gt;&lt;br /&gt;

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The following list includes all the modes (hopefully) that can be generated by shifting one tone of Porcupine[7] by one quartertone interval (chroma), which is one degree in 22edo. This produces scales with three step sizes -- in addition to a neutral tone (3\22) and large whole tone (4\22) there is now a semitone as well (2\22). In addition, two scales (and their rotations of course) have a 5-step subminor third. Underscores represent gaps in the chain of Porcupine generators. Note that lowering a tone by one quartertone interval (chroma) means sending it forward 7 spaces in the chain of generators, while raising a tone by one chroma means sending it backward 7 spaces in the chain of generators. This is how we wind up with such large gaps in the chain. Again, modes with perfect fifths from the bass are bolded.&lt;br /&gt;

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;

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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;

Not even close!&lt;br /&gt;

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This is getting silly! We need better names.....&lt;br /&gt;